Elements of Fluid Dynamics

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CFD Open Series Revision 1.85.4

Elements of Fluid Dynamics Ideen Sadrehaghighi,

Ph.D.

Flow instability

Votex Shedding of a Cylinder

Wing Tip Vortex

Great Wave by Kanagawa

ANNAPOLIS, MD

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Contents 1

Introduction ................................................................................................................................ 12

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Some Preliminary Concepts in Fluid Mechanics ............................................................ 15

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Brief Review of Thermodynamics ....................................................................................... 27

Linear and Non-Linear Systems .................................................................................................................... 15 Mathematical Definition.................................................................................................... 15 Linear Algebraic Equation ............................................................................................. 15 Nonlinear Algebraic Equations ..................................................................................... 15 Differential Equation ......................................................................................................... 16 Ordinary Differential Equation ..................................................................................... 16 Partial Differential Equation ......................................................................................... 16 Total Differential ................................................................................................................................................. 17 Lagrangian vs Eulerian Description ............................................................................................................ 17 Fluid Properties ................................................................................................................................................... 18 Kinematic Properties ......................................................................................................... 18 Thermodynamic Properties ............................................................................................... 18 Transport Properties.......................................................................................................... 18 Other Misc. Properties ...................................................................................................... 18 Stream Lines .......................................................................................................................................................... 19 Viscosity .................................................................................................................................................................. 19 Vorticity................................................................................................................................................................... 19 Vorticity vs Circulation ....................................................................................................... 20 Conservative and Non-Conservative forms of PDE............................................................................... 21 Physical .............................................................................................................................. 21 Mathematical..................................................................................................................... 21 How to choose which one to use?..................................................................................... 22 Divergence Theorem - Control Volume Formulation........................................................................... 22 General Transport Equation............................................................................................................... 22 Newtonian Fluid ...................................................................................................................................... 23 Some Flow Field Phenomena ............................................................................................................. 23 Viscous Dissipation ............................................................................................................ 23 Diffusion............................................................................................................................. 24 Convection ......................................................................................................................... 24 Dispersion .......................................................................................................................... 24 Advection ........................................................................................................................... 24 Inviscid vs. Viscous................................................................................................................................. 24 Steady-State vs. Transient ................................................................................................................... 25 Flow Field Classification ...................................................................................................................... 25

Pressure .................................................................................................................................................................. 27 Perfect (Ideal) Gas .............................................................................................................................................. 27 Total Energy .......................................................................................................................................................... 27 Thermodynamic Process ................................................................................................................................. 27 First Law of Thermodynamics ....................................................................................................................... 27 Second Law of Thermodynamics .................................................................................................................. 28

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Isentropic Relation ............................................................................................................................................. 28 Static (Local) Condition .................................................................................................................................... 29 Stagnation (Total) Condition .......................................................................................................................... 29 Total Pressure (Incompressible)...................................................................................................... 29 Pressure Coefficient ............................................................................................................................... 30 Application of 1st Law to Turbomachinery................................................................................... 30 Moment of Momentum .................................................................................................... 31 Euler‘s Pump & Turbine Equations .......................................................................... 31 Case Study - Application of 1st and 2nd Laws of Thermodynamics to Single Stage Turbo Machines ........................................................................................................................... 32

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Aerodynamics Distinction...................................................................................................... 34 Aerodynamic Practical Application ............................................................................................................. 34 Aerodynamic Forces and Moments ............................................................................................................. 35 Leading-Edge Flow as a Governing Factor in Leading-Edge Vortex Initiation in Unsteady Airfoil Flows ...................................................................................................................................................................... 36 Identification of LEV initiation from CFD data ................................................................... 36 Measure of Compressibility & Compressible vs. Incompressible Flows ...................................... 37 Speed of Sound ..................................................................................................................................................... 38 Mach Number ....................................................................................................................................................... 38 Sonic Boom ............................................................................................................................................................ 39 Shock Waves.......................................................................................................................................................... 40 Difficulties for Shock Wave Detection ............................................................................... 41 Traditional Shock Detection Methods ............................................................................... 41 Compressible 1D Shock Waves Relations .......................................................................... 42 Oblique Shock Crossing Interaction ................................................................................... 44 Experimental Data ........................................................................................................ 44 Case Study – 2D Oblique Shock Crossing Numerical Simulations ..................................... 44 Discussion ..................................................................................................................... 45 Effect of Heating Loads on Shock-Shock Interaction in Hypersonic Flows........................ 46 Case Study – University of Buffalo Research Center (CUBRC) Test Case ..................... 48 Computational Contributions ....................................................................................... 49 Computational Results ................................................................................................. 50 Quasi -1D Correlation Applied to Variable Area Ducts ................................................. 51 Case Study – Unsteady Phenomena in Supersonic Nozzle Flow Separation ................... 52 Background and Literature Survey ............................................................................... 52 Experimental Setup for Flow Facility ............................................................................ 55 Results of Plume Pitot Pressure ................................................................................... 55 Wall Pressure Statistics................................................................................................. 55 Correlations between Wall Pressure Ports.................................................................. 56 Correlations Between Wall Pressure Ports and Dynamic Pitot Probe ......................... 56 Concluding Remarks .................................................................................................... 58 Study of Aerothermal Loads in the Presence of Edney Type IV Interaction .............................. 58 Case Study 1 - Shock/Shock Interaction in a 2D Flow Around Circular Cylinder ............... 60 Case Study 2 - Flow Over Space Shuttle Orbiter with External Tank................................. 61 Concluding Remarks .......................................................................................................... 62

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Viscous Flow ............................................................................................................................... 64

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Qualitative Aspects of Viscous Flow ............................................................................................................ 64 No-Slip Wall Condition....................................................................................................... 64 Flow Separation ................................................................................................................. 64 Pressure Drag vs Skin Friction Drag ................................................................................... 65 Laminar vs Turbulent Flows ............................................................................................... 66 Skin Friction ....................................................................................................................... 67 Aerodynamic Heating ........................................................................................................ 67 Reynolds Number ............................................................................................................................................... 68 Reynolds Number Effects in Reduced Model .................................................................... 69 Case Study 1 - Scaling and Skin Friction Estimation in Flight using Reynold Number ....... 69 Interaction Between Shock Wave and Boundary Layer ............................................... 70 Reynolds Number Scaling ............................................................................................. 71 Discrepancy in Flight Performance and Wind Tunnel Testing...................................... 71 Flow Separation Type (A - B) ........................................................................................ 72 Over-Sensitive Prediction in Flight Performance ......................................................... 73 Aerodynamic Prediction ............................................................................................... 74 Skin Friction Estimation ................................................................................................ 74 Case Study 2 - Reynolds Number Effects Compared To Semi-Empirical Methods............ 76 Scaling Effects due to Reynolds Number ...................................................................... 77 Direct and Indirect Reynolds Number .......................................................................... 77 CFD Calculations ........................................................................................................... 78 Description of the CFD Code ........................................................................................ 78 Mesh Generation .......................................................................................................... 78 Residual & Mesh Dependence ..................................................................................... 79 Results and Discussion.................................................................................................. 79 Reynolds Number Scaling ............................................................................................. 80 Reynolds Number Scaling using Semi Empirical Skin Friction Methods ....................... 82 Inspection of Local Boundary Layer Properties for Varying Reynolds Number ...... 86 Concluding Remarks ................................................................................................ 87

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Boltzmann Method (LBM) ...................................................................................................... 89

Preliminaries and Background ...................................................................................................................... 89 Approaches ........................................................................................................................ 89 Dilute Gas Regimes ....................................................................................................... 90 Book Keeping ..................................................................................................................... 90 Kinetic Theory .................................................................................................................... 91 Maxwell Distribution Function .......................................................................................... 92 Boltzmann Transport Equation.......................................................................................... 93 The BGKW Approximation ................................................................................................. 94 Lattices & DnQm Classification .......................................................................................... 95 Lattice Arrangements ........................................................................................................ 95 1D Lattice Boltzmann Method (D1O2) .............................................................................. 96 2D Lattice Boltzmann Method (D2Q9) .............................................................................. 97 The Navier-Stokes Equations (NS) .............................................................................................................. 98 NS Equation in Non-Inertial Frame of Reference .............................................................. 99 Some Basic Functional Analysis ......................................................................................... 99 Fourier Series and Hilbert Spaces ............................................................................... 100 Weak vs. Genuine (Strong) Solution........................................................................... 100

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Boundary Layer Theory ................................................................................................................................ 100 Scaling Analysis for Boundary Layer Equation................................................................. 102 3D Boundary Layer ..................................................................................................... 102 Thermal Boundary Layer ............................................................................................ 103 Boundary Layer Separation ............................................................................................. 104 Adverse Pressure Gradient .............................................................................................. 104 Influencing Parameters.................................................................................................... 105 Internal Separation .......................................................................................................... 105 Effects of Boundary Layer Separation ............................................................................. 105

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Conservation Laws and Governing Equations ............................................................. 107

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Linear PDEs and Model Equations ................................................................................... 135

Control Volume Approach ............................................................................................................................ 107 Integral Forms of Conservation Equations............................................................................................ 107 Mathematical Operators................................................................................................................................ 108 Conservation of Mass (Continuity Equation) ....................................................................................... 109 Centrifugal and Coriolis forces ................................................................................................................... 109 Conservation of Momentum (Newton 2nd Law) .................................................................................. 110 Conservation of Energy (1st Law of Thermodynamics) .................................................................. 110 Scalar Transport Equation ........................................................................................................................... 111 Vector Form of N-S Equations..................................................................................................................... 111 Orthogonal Curvilinear Coordinate .............................................................................................. 112 Cylindrical Coordinate for Governing Equation ............................................................... 113 Generalized Transformation to N-S Equation .......................................................................... 114 Coupled and Uncoupled (Segregated) Flows ........................................................................... 117 Simplification to N-S Equations (Parabolized) ........................................................................ 117 Non-dimensional Numbers in Fluid Dynamics........................................................................ 118 Prandtl Number ............................................................................................................... 119 Nusset Number ................................................................................................................ 119 Rayleigh Number ............................................................................................................. 120 Other Dimensionless Number ......................................................................................... 120 Non-Dimensionalizing (Scaling) of Governing Equations ................................................ 120 Incompressible Navier-Stokes Equation .................................................................................... 122 Porous Medium ............................................................................................................... 122 Literature Survey ................................................................................................... 123 Some Insight into Physical Consideration of Porous Medium .............................. 124 Velocity–Pressure Formulation ....................................................................................... 126 Derivation of Volume Average N-S Equations (VANS) .......................................... 126 Discussion .............................................................................................................. 127 Vorticity Consideration in Incompressible Flow .................................................................... 128 Inviscid Momentum Equation (Euler)......................................................................................... 130 Steady-Inviscid–Adiabatic Compressible Equations ........................................................ 130 Velocity Potential Equation ............................................................................................. 131 Hierarchy of Inviscid Fluid Equation ........................................................................................... 132

Mathematical Character of Basic Equations ......................................................................................... 135 Nonsingular Transformation............................................................................................ 136 The 'Par-Elliptic' problem ................................................................................................ 136

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Exact (Closed Form) Solution Methods to Model Equations.......................................................... 137 Linear Wave Equation (1st Order) .................................................................................... 137 Inviscid Burgers Equation ................................................................................................ 137 Diffusion (Heat) Equation ................................................................................................ 138 Viscous Burgers Equation ................................................................................................ 138 Tricomi Equation.............................................................................................................. 139 2D Laplace Equation ........................................................................................................ 139 Boundary Conditions .................................................................................................. 139 Poisson’s Equation ........................................................................................................... 140 The Advection-Diffusion Equation ................................................................................... 140 The Korteweg-De Vries Equation..................................................................................... 140 Helmholtz Equation ......................................................................................................... 140 Exact Solution Methods ................................................................................................... 140 Solution Methods for In-Viscid (Euler) Equations ............................................................................. 141 Method of Characteristics ............................................................................................... 141 Linear Systems ................................................................................................................. 141 Non-Linear Systems ......................................................................................................... 143

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Boundary Conditions ............................................................................................................ 146 Naming Convention for Different Types of Boundaries ................................................................... 146 Dirichlet Boundary Condition .......................................................................................... 146 Von Neumann Boundary Condition................................................................................. 146 Mixed or Combination of Dirichlet and von Neumann Boundary Condition .................. 146 Robin Boundary Condition .............................................................................................. 146 Cauchy Boundary Condition ............................................................................................ 146 Periodic (Cyclic Symmetry) Boundary Condition ............................................................. 147 Generic Boundary Conditions .......................................................................................... 147 Wall Boundary Conditions ........................................................................................................................... 147 Velocity Field ................................................................................................................... 147 Pressure ........................................................................................................................... 148 Scalars/Temperature ....................................................................................................... 148 Common inputs for wall boundary condition ............................................................ 148 Symmetry Planes .............................................................................................................................................. 149 Inflow Boundaries ........................................................................................................................................... 149 Velocity Inlet .................................................................................................................... 149 Pressure Inlet ................................................................................................................... 150 Mass Flow Inlet ................................................................................................................ 150 Inlet Vent ......................................................................................................................... 150 Outflow Boundaries ........................................................................................................................................ 151 Pressure Outlet ................................................................................................................ 151 Pressure Far-Field ............................................................................................................ 151 Outflow ............................................................................................................................ 152 Outlet Vent ...................................................................................................................... 152 Exhaust Fan ...................................................................................................................... 152 Free Surface Boundaries ............................................................................................................................... 152 Velocity Field and Pressure.............................................................................................. 152 Scalars/Temperature ....................................................................................................... 153 Pole (Axis) Boundaries .................................................................................................................................. 153

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Periodic Flow Boundaries ............................................................................................................................ 153 Non-Reflecting Boundary Conditions (NRBCs) ................................................................................... 154 Case Study 1 - Turbomachinery Application of 2D Subsonic Cascade............................. 154 Case Study 2 - CAA Application of Airfoil Turbulence Interaction Noise Simulation ...... 155 Turbulence Intensity Boundaries ................................................................................................. 156 Turbulence Intensity ........................................................................................................ 156 Immersed Boundaries ....................................................................................................................... 156 Free Surface Boundary ...................................................................................................................... 157 The Kinematic Boundary Condition ................................................................................. 157 The Dynamic Boundary Condition ................................................................................... 157 Other Boundary Conditions ............................................................................................................. 158 Further Remarks .................................................................................................................................. 158

List of Tables

Table 4.1 Classification of Mach Number ................................................................................................................ 39 Table 4.2 Theoretical Detachment D and Neumann N Conditions Q the Flow Deflection Angle and a is the Shock-Wave Angle ................................................................................................................................................. 44 Table 4.3 CUBRC Test Conditions (V, H, SGL, R in inches) ............................................................................... 49 Table 4.4 Nomenclature for Unsteady Phenomena in Supersonic Nozzle Flow Separation .............. 52 Table 5.1 Mesh and Residual Dependence on CD in Drag counts relative to the baseline mesh with a Residual of -5.5 - (Courtesy of Pettersson and Rizzi).......................................................................................... 79 Table 5.2 Comparison of the Extrapolated Data and CFD in Drag Counts at Reynolds number 56 M ....................................................................................................................................................................................................... 84 Table 8.1 Classification of the Euler equation on different regimes ......................................................... 141

List of Figures Figure 1.1 Hierarchy of Basic Fluid Flow ................................................................................................................ 13 Figure 2.1 Description of Flow: Lagrangian (left) and Eulerian (right) .................................................... 17 Figure 2.2 Stream Lines around an Airfoil & Cylinder ....................................................................................... 19 Figure 2.3 Viscosity effects in parallel plate ........................................................................................................... 19 Figure 2.4 A sink Vortex flow over a drain and history of a rolle up of a vortex over time................ 20 Figure 2.5 Circulation (Right) vs. Vorticity (Left) ................................................................................................ 20 Figure 2.6 A region V bounded by the surface S = ∂V with the surface normal n .................................. 22 Figure 2.7 Diffusion Process in Physics.................................................................................................................... 24 Figure 2.8 Transient Case of Vortex Shedding over a Cylinder ...................................................................... 25 Figure 2.9 Physical Aspects of a Typical Flow Field............................................................................................ 26 Figure 3.1 Control Volume showing sign convention for heat and work transfer ................................. 30 Figure 3.2 Control Volume for a Generalized Turbomachine ......................................................................... 31 Figure 3.3 Schematic section of Single Stage Turbomachine .......................................................................... 32 Figure 4.1 Schematic of Lift and Drag ....................................................................................................................... 35 Figure 4.2 Vorticity Plots from CFD (first row), Flow Visualization from Experiment (second row) ....................................................................................................................................................................................................... 37 Figure 4.3 An F/A-18 Hornet Creating a Vapor Cone at Transonic Speed ................................................ 38 Figure 4.4 Evolution of Shock Wave .......................................................................................................................... 39 Figure 4.5 Illustration of a sonic boom as received by human ears ............................................................. 40 Figure 4.6 Contour Concentration Examples at M = 5 ; From Left to Right ; (a) Mach number (b) Pressure (c) Density (d) Temperature ........................................................................................................................ 40 Figure 4.7 Solution of Shock Capturing for Euler Equations.......................................................................... 41

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Figure 4.8 Qualitative Depiction of 1D Flow Through Normal and Oblique Shocks ............................. 42 Figure 4.9 Oblique Shock Reflections on a Channel Flow (M=2 AoA=15˚) ............................................. 43 Figure 4.10 Schematic of the Experimental Configuration used by the Ivanov Group and Sample Laser Light-Sheet Visualization from the same M = 4, α =37 degrees, b/w=3.75, g/w=0.3 .................. 45 Figure 4.11 Isobars Demonstrating Hysteresis in 2D Simulations of (Schmisseur and Gaitonde) . 46 Figure 4.12 Schematic of Edney Type IV Shock-Shock Interaction .............................................................. 47 Figure 4.13 Schematic of the CUBRC Edney IV Interaction Generator (Courtesy of Holden) .......... 48 Figure 4.14 Experimental and Numerical Results for the Conditions of CUBRC Run #38 ................. 50 Figure 4.15 Contours of Constant Translational Temperature for; a) Navier-Stokes and b) DSMC Solutions for CUBRC Run #44 .......................................................................................................................................... 51 Figure 4.17 Compressible Flow in Converging-Diverging Ducts (Nozzles and Diffusers) ........ Error! Bookmark not defined. Figure 4.16 Comparison on Computation vs Theory for an Oblique Shock in 2D Channel Flow .......................................................................................................................................... Error! Bookmark not defined. Figure 4.18 Oblique Shock Relationship .................................................................................................................. 52 Figure 4.19 Primary Jet Flow at Mach 0.9 Surrounded by an annular secondary flow at Nozzle Pressure Ratio NPR =1.7 (a) Secondary Nozzle is Convergent; (b) Secondary Nozzle is ConvergentDivergent – (Courtesy of 36) .............................................................................................................................................. 53 Figure 4.20 Schematic of Supersonic Nozzle Flow Separation (Top), vs. schlieren Image (Bottom) , (Courtesy of Papamoschou, D., Zill) – (Courtesy of 36) ........................................................................................... 54 Figure 4.21 RMS Total Pressure Profile of Jet Plume at x/he = 0.5 for Straight Nozzle (Ae/At =1) and Convergent-Divergent Nozzle (Ae/At =1.6) – (Courtesy of 36) .................................................................... 55 Figure 4.22 RMS Wall Static Pressure Fluctuation vs. Nozzle Pressure Ratio Presenting Different Flow Regimes – Courtesy of 36 .......................................................................................................................................... 56 Figure 4.23 Cross Correlations of Upper and Lower Wall Transducers for Various Flow Regimes – Courtesy of 36 ........................................................................................................................................................................... 56 Figure 4.24 Coherence Between Upper and Lower Walls (C12), DPP and Lower Wall (C13), and DPP and Upper Wall (C23) – Courtesy of 36 .................................................................................................................. 57 Figure 4.25 Translation Paths of Dynamic Pitot Probe. Red points Indicate Measurement Locations – Courtesy of 36................................................................................................................................................... 57 Figure 4.26 Six types of shock/shock interaction as classified by Edney .................................................. 59 Figure 4.27 Dependencies of Space Shuttle Flight Altitude on Velocity ..................................................... 59 Figure 4.28 Iso-lines Near Cylinder (a) Pressure (b) Mach Number .......................................................... 60 Figure 4.29 Comparison with Experimental Values on Cylinder Surface .................................................. 61 Figure 4.31 Mesh Generated after Solution-Adaptive Refinement............................................................... 61 Figure 4.33 Surface Pressure and Pressure iso-lines in the Symmetry Plane ......................................... 62 Figure 4.32 Iso-lines of (a)Pressure and (b) Mach number Near the Orbiter Nose .............................. 62 Figure 4.34 Distributions of (a) Pressure and (b) Heat Flux Along the Nose in the Symmetry Plane ....................................................................................................................................................................................................... 63 Figure 5.1 Boundary Layer Flow along a Wall ...................................................................................................... 64 Figure 5.2 Airflow Separating from a Wing at a High Angle of Attack ........................................................ 64 Figure 5.3 Detached Flow induced by adverse pressure gradient................................................................ 65 Figure 5.4 Schematic of Velocity Profiles for Laminar vs Turbulent Flows .............................................. 66 Figure 5.5 Drag on Slender & Blunt Bodies ............................................................................................................ 66 Figure 5.6 Illustrating the calculation of Skin Friction ...................................................................................... 67 Figure 5.7 Quantitate Aspects of Viscous Flow ..................................................................................................... 68 Figure 5.8 Effects of Reynolds Number in Inertia vs Viscosity....................................................................... 68 Figure 5.9 Drag Coefficient versus Reynolds Number for a 1:5 Model and a Car (Courtesy of 35) .. 69 Figure 5.10 Flow features sensitive to Reynolds number for a cruise condition on a wing section ....................................................................................................................................................................................................... 70

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Figure 5.11 Schematic representation of direct and indirect Reynolds number effects ..................... 71 Figure 5.12 Comparison of C-141 Wing Pressure Distributions Between Wind Tunnel and Flight ....................................................................................................................................................................................................... 72 Figure 5.13 Flat plate Skin Friction correlations comparison ........................................................................ 76 Figure 5.14 Cut of the volume mesh along the sweep........................................................................................ 78 Figure 5.15 Convergence history for the adapted mesh at Reynolds number 20 M - (Courtesy of Pettersson and Rizzi) ........................................................................................................................................................... 79 Figure 5.16 Wing Colored by Cp Contours - (Courtesy of Pettersson and Rizzi).................................... 80 Figure 5.17 Simulation Criterion as a Function of Reynolds Number for a Recrit at Reynolds Number 50 M - (Courtesy of Pettersson and Rizzi)................................................................................................. 81 Figure 5.18 Skin Friction Estimated with Karman-Shoenherr and Sommer-Short Methods anchored to CFD data at Reynolds Number 38 M - (Courtesy of Pettersson and Rizzi) .......................... 83 Figure 5.19 Skin Friction Estimated with Karman-Shoenherr and Sommer-Short methods for each part of the aircraft separately, anchored to CFD data at Reynolds Number 38 M - (Courtesy of Pettersson and Rizzi) ........................................................................................................................................................... 83 Figure 5.20 Numerical Fit of Drag Due to Pressure - (Courtesy of Pettersson and Rizzi) .................. 84 Figure 5.21 HTP seen from above, positions where ........................................................................................... 87 Figure 6.1 Flow Regimes for Diluted Gas ................................................................................................................ 90 Figure 6.2 Simulations Spectrum of Rarefied Gas ............................................................................................... 91 Figure 6.3 Position and velocity vector for a particle after and before applying a force, F ................ 93 Figure 6.4 Real Molecules versus LB particles ...................................................................................................... 95 Figure 6.5 Lattice arrangements for velocity vectors for typical 1D, 2D and 3D Discretization ...... 96 Figure 6.6 Schematics of solving 2D Lattice Boltzmann Model...................................................................... 98 Figure 6.7 The development of the boundary layer for flow over a flat plate ...................................... 101 Figure 6.8 Definition of Boundary Layer Thickness: (a) Standard Boundary Layer (u = 99%U), (b) .................................................................................................................................................................................................... 101 Figure 6.9 Example of Subsonic 3D Boundary Layer ...................................................................................... 103 Figure 6.10 Representation Boundary Later Velocity Profile (Courtesy of Sturm et al.) ................ 104 Figure 7.1 Control Volume bondcorresponding control surface S ............................................................ 107 Figure 7.2 Centrifugal and Coriolis force.............................................................................................................. 110 Figure 7.3 Relation between Cartesian and Cylindrical; coordinate......................................................... 113 Figure 7.4 Conditions and Mathematical Character of N-S and its variation ........................................ 118 Figure 7.5 Some Methods for Simplifying Governing Equations ................................................................ 121 Figure 7.6 Earliest Forms of Porous ....................................................................................................................... 123 Figure 7.7 Effect of surface machining on the same numerically generated porous sample: ........ 124 Figure 7.8 Sketch of a Porous Medium, with l*f and l*s the Characteristic Lengths of the................ 125 Figure 7.9 Evolution of a Vortex Tube in pyroclastic flows .......................................................................... 129 Figure 7.10 Condition and Mathematical Character of Inviscid Flow ..................................................... 132 Figure 7.11 Hierarchy of Flow Equations............................................................................................................. 133 Figure 8.1 Two-way interchange of information between Parabolic and Elliptic flows .................. 136 Figure 8.2 Solution of linear Wave equation....................................................................................................... 137 Figure 8.3 Formulation of discontinuities in non-linear Burgers (wave) equation ........................... 138 Figure 8.4 Rate of Decay of solution to diffusion equation ........................................................................... 138 Figure 8.5 Solution to Laplace equation ............................................................................................................... 139 Figure 8.6 Solution to Poisson's equation ............................................................................................................ 140 Figure 8.7 Characteristics of Linear equation .................................................................................................... 142 Figure 8.8 Characteristics of nonlinear solution point ................................................................................... 144 Figure 9.1 Mixed Boundary Conditions................................................................................................................. 146 Figure 9.2 Symmetry Plane to Model one Quarter of a 3D Duct ................................................................ 149 Figure 9.3 Pole (Axis) Boundary .............................................................................................................................. 153

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Figure 9.4 Periodic Boundary ................................................................................................................................... 153 Figure 9.5 Pressure contours plot for 2nd order spatial discretization scheme ................................... 155 Figure 9.6 Aero-Acoustics Application for NRBC’ ............................................................................................. 155 Figure 9.7 Immersed Boundaries ............................................................................................................................ 156 Figure 9.8 Sketch Exemplifying the conditions at a Free Surface Formed by the Interface Between Two Fluids ............................................................................................................................................................................. 157

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Preface This note is intended for all undergraduate, graduate, and scholars of Fluid Mechanics. It is not completed and never claims to be as such. Therefore, all the comments are greatly appreciated. In assembling that, I was influenced with sources from my textbooks, papers, and materials that I deemed to be important. At best, it could be used as a reference. I also would like to express my appreciation to several people who have given thoughts and time to the development of this article. Special thanks should be forwarded to the authors whose papers seemed relevant to topics, and consequently, it appears here©. Finally I would like to thank my wife, Sudabeh for her understanding and the hours she relinquished to me. Their continuous support and encouragement are greatly appreciated. Ideen Sadrehaghighi June 2018

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1 Introduction Fluid mechanics is the study of fluids either in motion (fluid dynamics) or at rest (fluid statics) and the subsequent effects of the fluid upon the boundaries, which may be either solid surfaces or interfaces with other fluids. Both gases and liquids are classified as fluids, and the number of fluids engineering applications is enormous: breathing, blood flow, swimming, pumps, fans, turbines, airplanes, ships, rivers, windmills, pipes, missiles, icebergs, engines, filters, jets, and sprinklers, to name a few. When you think about it, almost everything on this planet either is a fluid or moves within or near a fluid1. There is a clear hierarchy of physical models to choose from. The most general model under routine use is at the level of the fluid molecule where the motion of individual molecules is tracked and intermolecular interactions are simulated. For example, this level of approximation is required for the rarefied gases encountered during the reentry of spacecraft in the upper atmosphere. Although this model can be certainly used at lower speeds and altitudes, it becomes prohibitively expensive to track individual molecules under non-rarefied conditions. Thus, another mathematical model is needed. In moving from the molecular description to the continuum model we basically performed an averaging process over the molecules to obtain bulk quantities such as temperature and pressure. It turns out that averaging is one of the primary means of simplifying our mathematical model. For example, if we average the Navier–Stokes equations in one spatial dimension, then we are left with the twodimensional Navier–Stokes equations. As long as the process that we wish to simulate is approximately two-dimensional then this will be an adequate model. Of course, we can continue by averaging over two spatial dimensions or even over all three directions if we are only interested in the variation of mean quantities. The 1D, 2D analyses are discussed in details later on. Given the hierarchy of mathematical models, and the selection in Figure 1.1, it is possible, under certain circumstances, to make further approximations that take into account special physical characteristics of the flow under consideration. For example, Prandlt’s landmark discovery that viscous effects are primarily limited to a boundary layer near a solid surface has led to the boundary layer equations which are a special form of the Navier–Stokes equations that are considerably easier to solve numerically. Outside of the boundary layer, which means most of the flow in the case of an aircraft, the flow is generally inviscid and the viscous terms in the Navier–Stokes equations can be dropped leading to the Euler equations. If there are no shock waves in the flow, then further simplification can be obtained by using the potential flow equations, the compressible Navier-Stokes equations. Many of the most important aspects of these relations are nonlinear and, as a consequence, often have no analytic solution 2-3. The ultimate goal of fluid dynamics is to understand the physical events that occur in the flow of fluids around and within designated objects. These events are related to the action and interaction of phenomena such as dissipation, diffusion, convection, shock waves, slip surfaces, boundary layers, and turbulence. In the field of aerodynamics, all of these phenomena are governed by the compressible Navier-Stokes equations. Since there is no analytical solutions, therefore, the idea of Computational Fluid Dynamics (CFD) comes to mind where the flow equation being discretized and solved with appropriate simplification. For fluids which are sufficiently dense to be a continuum, do not contain ionized species, and have flow velocities small in relation to the speed of light, the momentum equations for Newtonian fluids are the Navier–Stokes equations, which is a non-linear

Frank M. White, “Fluid Mechanics”, 4th Edition, McGraw Hill Company. Collis,, S,, “An Introduction to Numerical Analysis for Computational Fluid Mechanics”, Sandia National Laboratories Albuquerque, New Mexico 87185 and Livermore, California 94550. 3 Lomax, H., and Pulliam, T.,H., “Fundamentals of Computational Fluid Dynamics”, NASA Ames Research Center, 1999. 1 2

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set of differential equations that describes the flow of a fluid whose stress depends linearly on flow velocity gradients and pressure. The simplified equations do not have a general closed-form solution, so they are primarily of use in Computational Fluid Dynamics. The equations can be simplified in a number of ways, all of which make them easier to solve. In some cases, further simplification is allowed to appropriate fluid dynamics problems to be solved in closed form. Fluid Dynamics

Continuous

Rarefied Gas Dynamics

No

Yes

Boltzmans Linear Theory

Conservation Laws

Gas Dynamics

Aerodynamics

Hydrodynamicse rodynamics

Inviscid

Bernoulli’s Eqs.

Viscous

Potential Eqs.

Euler Eqs.

Figure 1.1

Navier-Stokes Eqs.

Boundary Layer Eqs.

Hierarchy of Basic Fluid Flow

The conservation laws are used to solve fluid dynamics problems, and may be written in integral or differential form. Mathematical formulations of these conservation laws may be interpreted by considering the concept of a control volume. A control volume is a specified volume in space through which air can flow in and out. Integral formulations of the conservation laws consider the change in mass, momentum, or energy within the control volume. Differential formulations of the conservation

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laws applies Stokes' theorem to yield an expression which may be interpreted as the integral form of law applied to an infinitesimal volume at a point within the flow. Mass continuity (conservation of mass) is the rate of change of fluid mass inside a control volume must be equal to the net rate of fluid flow into the volume4. Physically, this statement requires that mass is neither created nor destroyed in the control volume, and can be translated into the integral form of the continuity equation. All fluids are compressible to some extent, that is, changes in pressure or temperature will result in changes in density. However, in many situations the changes in pressure and temperature are sufficiently small that the changes in density are negligible. In this case the flow can be modeled as an incompressible flow. Otherwise the more general compressible flow equations must be used. For conservation Momentum, Newton’s famous 2nd law was applied, and Energy make use of 1st law of Thermodynamics. For flow of gases, to determine whether to use compressible or incompressible fluid dynamics, the Mach number of the flow is to be evaluated. As a rough guide, compressible effects can be ignored at Mach numbers below approximately 0.3. Therefore, it is safe to assume incompressible flow. For liquids, whether the incompressible assumption is valid depends on the fluid properties (specifically the critical pressure and temperature of the fluid) and the flow conditions (how close to the critical pressure the actual flow pressure becomes). Acoustic problems always require allowing compressibility, since sound waves are compression waves involving changes in pressure and density of the medium through which they propagate.

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Wikipedia, “Fluid Dynamics “, the free encyclopedia

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2 Some Preliminary Concepts in Fluid Mechanics Linear and Non-Linear Systems In physical sciences, a nonlinear system is a system in which the change of the output is not proportional to the change of the input5. Nonlinear problems are of interest to engineers, physicists6 and mathematicians and many other scientists because most systems are inherently nonlinear in nature. Nonlinear systems may appear chaotic, unpredictable or counterintuitive, contrasting with the much simpler linear systems. Typically, the behavior of a nonlinear system is described in mathematics by a nonlinear system of equations, which is a set of simultaneous equations in which the unknowns (or the unknown functions in the case of differential equations) appear as variables of a polynomial of degree higher than one or in the argument of a function which is not a polynomial of degree one. In other words, in a nonlinear system of equations, the equation(s) to be solved cannot be written as a linear combination of the unknown variables or functions that appear in them. Systems can be defined as non-linear, regardless of whether or not known linear functions appear in the equations. In particular, a differential equation is linear if it is linear in terms of the unknown function and its derivatives, even if nonlinear in terms of the other variables appearing in it. As nonlinear equations are difficult to solve, nonlinear systems are commonly approximated by linear equations (linearization). This works well up to some accuracy and some range for the input values, but some interesting phenomena such as solitons, chaos and singularities are hidden by linearization. It follows that some aspects of the behavior of a nonlinear system appear commonly to be counterintuitive, unpredictable or even chaotic. Although such chaotic behavior may resemble random behavior, it is absolutely not random. For example, some aspects of the weather are seen to be chaotic, where simple changes in one part of the system produce complex effects throughout. This nonlinearity is one of the reasons why accurate long-term forecasts are impossible with current technology. Mathematical Definition Linear Algebraic Equation In mathematics, a linear function (or map) f(x) is one which satisfies both of the following properties:  

Additivity or Superposition: f(x + y) = f(x) + f(y) Homogeneity: f(αx ) = α f(x)

Additivity implies homogeneity for any rational α, and, for continuous functions, for any real α. For a complex α, homogeneity does not follow from additivity. For example, an ant linear map is additive but not homogeneous. The conditions of additivity and homogeneity are often combined in the superposition principle f (αx + βy) = αf (x) + βf y). An equation written as f (x) = C is called linear if f (x ) is a linear map (as defined above) and nonlinear otherwise. The equation is called homogeneous if C = 0. The definition f(x) = C is very general in that x can be any sensible mathematical object (number, vector, function, etc.), and the function f(x) can literally be any mapping, including integration or differentiation with associated constraints (such as boundary values). Condition f(x) contains differentiation with respect to x , the result will be a differential equation. Nonlinear Algebraic Equations Nonlinear algebraic equations, which are also called polynomial equations, are defined by equating polynomials to zero. For example, x2+x−1=0. For a single polynomial equation, root-finding 5 6

From Wikipedia, the free encyclopedia. Gintautas, V. "Resonant forcing of nonlinear systems of differential equations". Chaos. 18, 2008.

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algorithms can be used to find solutions to the equation (i.e., sets of values for the variables that satisfy the equation). However, systems of algebraic equations are more complicated; their study is one motivation for the field of algebraic geometry, a difficult branch of modern mathematics. It is even difficult to decide whether a given algebraic system has complex solutions. Nevertheless, in the case of the systems with a finite number of complex solutions, these systems of polynomial equations are now well understood and efficient methods exist for solving them7. Differential Equation A system of differential equations is said to be nonlinear if it is not a linear system. Problems involving nonlinear differential equations are extremely diverse, and methods of solution or analysis are problem dependent. Examples of nonlinear differential equations are the Navier–Stokes equations in fluid dynamics and the Lotka–Volterra equations in biology. One of the greatest difficulties of nonlinear problems is that it is not generally possible to combine known solutions into new solutions. In linear problems, for example, a family of linearly independent solutions can be used to construct general solutions through the superposition principle. A good example of this is one-dimensional heat transport with Dirichlet boundary conditions, the solution of which can be written as a timedependent linear combination of sinusoids of differing frequencies; this makes solutions very flexible. It is often possible to find several very specific solutions to nonlinear equations, however the lack of a superposition principle prevents the construction of new solutions. Ordinary Differential Equation First order ordinary differential equations are often exactly solvable by separation of variables, especially for autonomous equations. For example, the nonlinear equation du/d x = −u2 has u=1/x +C as a general solution (and also u = 0 as a particular solution, corresponding to the limit of the general solution when C tends to infinity). The equation is nonlinear because it may be written as d u/d x + u2 = 0 and the left-hand side of the equation is not a linear function of u and its derivatives. Note that if the u2 term were replaced with u, the problem would be linear (the exponential decay problem). Second and higher order ordinary differential equations (more generally, systems of nonlinear equations) rarely yield closed form solutions, though implicit solutions and solutions involving nonelementary integrals are encountered. Common methods for the qualitative analysis of nonlinear ordinary differential equations include:      

Examination of any conserved quantities, especially in Hamiltonian systems. Examination of dissipative quantities analogous to conserved quantities. Linearization via Taylor expansion. Change of variables into something easier to study. Bifurcation theory. Perturbation methods (can be applied to algebraic equations too).

Partial Differential Equation The most common basic approach to studying nonlinear partial differential equations is to change the variables (or otherwise transform the problem) so that the resulting problem is simpler (possibly even linear). Sometimes, the equation may be transformed into one or more ordinary differential equations, as seen in separation of variables, which is always useful whether or not the resulting ordinary differential equation(s) is solvable. Another common (though less mathematic) tactic, often seen in fluid and heat mechanics, is to use scale analysis to simplify a general, natural equation in a certain specific boundary value problem. For example, the (very) nonlinear Navier-Stokes equations can be simplified into one linear partial differential equation in the case of transient, laminar, one Lazard, D. (2009). "Thirty years of Polynomial System Solving, and now?". Journal of Symbolic Computation. 44 (3): 222–231. 7

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dimensional flow in a circular pipe; the scale analysis provides conditions under which the flow is laminar and one dimensional and also yields the simplified equation.

Total Differential Let Q(x, y, z, t) represent any property of fluid. If Dx, Dy, Dz and Dt represent arbitrary changes in four independent variables, then total differential change in

DQ x, y,z, t  

Q Q Q Q Dx  Dy  Dz  Dt x y z t

Eq. 2.1

Lagrangian vs Eulerian Description The Lagrangian specification of flow field is a way of looking a fluid motion where observer follows an individual parcel as it moves through space and time. This could be analog as sitting in a boat and drifting down the river. The equations of motion that arise from this approach are relatively simple because they result from direct application of Newton’s second law. But their solutions consist merely of the fluid particle spatial location at each instant of time, as depicted in Figure 2.1 (left). This figure shows two different fluid particles and their particle paths for a short period of time. Notice that it is the location of the fluid parcel at each time that is given, and this can be obtained directly by solving the corresponding equations8. The notation X1(0) represents particle #1 at time t = 0, with X denoting the position vector (x, y, z) T.

Figure 2.1

Description of Flow: Lagrangian (left) and Eulerian (right)

Alternatively, the Eulerian is a way of looking at fluid motion that focuses at specific locations in space and time. This could be visualized as sitting on the bank of river and watching the parcel pass the fixed locations. As noted above, this corresponds to a coordinate system fixed in space, and within which fluid properties are monitored as functions of time as the flow passes fixed spatial locations. Figure 2.1 (right) is a simple representation of this situation. It is evident that in this case we need not be explicitly concerned with individual fluid parcels or their trajectories. Moreover, the flow velocity will now be measured directly at these locations rather than being deduced from the time

J. M. McDonough, “Lectures In Elementary Fluid Dynamics: Physics, Mathematics and Applications”, Departments of Mechanical Engineering and Mathematics University of Kentucky, © 1987, 1990, 2002, 2004, 2009. 8

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rate-of-change of fluid parcel location in a neighborhood of the desired measurement points9. Within fluid mechanics, one’s first interest is the fluid velocity where Eulerian description would be more suitable. On the other hand, for solid mechanics where particle displacement is the interest, Lagrangian description would be more appropriate. The Eulerian operation of fluid particles could be best depicted by Total/Substantial Derivative and derived easily with aid of Eq. 2.1 as

D ∂ = + 𝐕. ⏟ 𝛁 ⏟ Dt ∂t Conservative Local

Eq. 2.2 Where the terms on the RHS are called the local and conservative derivatives respectfully10. The conservative term has the unfortunate distinction of being the non-linear term and source of great mathematical difficulties. Complete knowledge of Eq. 2.2 is often the solution to problem of fluid mechanics of interest.

Fluid Properties Kinematic Properties These could include (Linear Velocity, Angular Velocity, Vorticity, Acceleration, and Strain Rate). Strictly speaking these are properties of flow field itself rather than fluid, and are related to fluid motion. Thermodynamic Properties Includes (Pressure, Density, Temperature, Enthalpy, Entropy, Specific Heat, Prantle Number Pr, Bulk Modulus, and Coefficient of Thermal Expansion)11. Within thermodynamics, a physical property is any property that is measurable and whose value describes a state of a physical system. Physical properties can often be categorized as being either intensive or extensive quantities, according to how the property changes when the size (or extent) of the system changes. Accordingly, an intensive property is one whose magnitude is independent of the size of the system. An extensive property is one whose magnitude is additive for subsystems12. Transport Properties These includes (Viscosity, Thermal Conductivity, and Mass Diffusivity). They properties that bear to movement or transport of momentum, heat, and mass respectively. Each of three coefficients relates flux or transport to the gradient of property. Viscosity relates momentum flux to velocity gradient, Thermal Conductivity relates heat flux to temperature, gradient, and diffusion coefficients related the mass transport to the concentration gradient. Other Misc. Properties Those could include (surface tension, vapor pressure, eddy diffusion coefficients, surface accommodation coefficient.

See above. White, Frank M. 1974: “Viscous Fluid Flow”, McGraw-Hill Inc. 11 White, Frank M. 1974: “Viscous Fluid Flow”, McGraw-Hill Inc. 12 McNaught, A. D.; Wilkinson, A.; Nic, M.; Jirat, J.; Kosata, B.; Jenkins, A. (2014). IUPAC. Compendium of Chemical Terminology, 2nd ed. (the "Gold Book"). 2.3.3. Oxford: Blackwell Scientific Publications. 9

10

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Stream Lines An important concept in the study of aerodynamics concerns the idea of streamlines. According to NASA, a streamline is a path traced out by a massless particle as it moves with the flow. It is easiest to visualize a streamline if we move along with the body (as opposed to moving with the flow). Figure 2.2 shows the computed streamlines around an airfoil and around a cylinder. In both cases, we move with the object and the flow proceeds from left to right. Since the streamline is traced out by a moving particle, at every point along the path the velocity is tangent to the path. Since there is no normal component of the velocity along the path, mass cannot cross a streamline. The mass contained between any two streamlines remains the same throughout the flow field. We can use Bernoulli's equation to relate the pressure and velocity along the streamline. Since no mass passes through the surface of the airfoil (or cylinder), the surface of the object is a streamline.

Figure 2.2

Stream Lines around an Airfoil & Cylinder

Viscosity A measure of the importance of friction in fluid flow. Viscosity is a fluid property by virtue of which a fluid offers resistance to shear stresses. Consider a fluid in 2D steady shear between two infinite plates h apart, as shown in the Figure 2.3. The bottom plate is fixed, while the upper plate is moving at a steady speed of U. It turns out that the velocity profile, u(y) is linear, i.e. u(y) = U y/h. Also notice that the velocity of the fluid matches that of the wall at both the top and bottom walls. This is known as the no slip condition. The coefficient of Viscosity (μ) is often considered constant, but in reality is a function of both Figure 2.3 Viscosity effects in parallel plate Pressure and Temperature, or μ = μ (T, P). A widely used approximation resulted from kinetic theory by Sutherland (1893) using the formula 3 2

μ  T  T0  S   μ 0  T0  T  S

Eq. 2.3

Where S is an effective temperature, called Sutherland’s Constant and subscripts 0 refer as to reference values.

Vorticity Vorticity ω, being twice the angular velocity, is a measure of local spin of fluid element given by curl

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of velocity as

ω   V

Eq. 2.4 In 3D flow, vorticity (ω) is in plane of flow and perpendicular to stream lines as depicted in Figure 2.4 and Eq. 2.4. By definition, if ω = 0, then the flow labeled irrotational. By Croce’s theorem, the gradient of stagnation pressure is normal to both velocity vector and vorticity vector; thus it lies in the plane of the paper and normal to V. Consequently, the stagnation pressure, P0, is constant along each streamline and varies between streamlines only if vorticity is present13.

Figure 2.4

A sink Vortex flow over a drain and history of a rolle up of a vortex over time

Vorticity vs Circulation The fluid circulation defined as the line integral of the velocity V around any closed curve C. There are distinct differences in circulation and vorticity. Circulation is a macroscopic measure of the rotation of a fluid element is defined as line integral of velocity field along a fluid element, therefore, it is a scalar quantity. Vorticity on the other hand, is microscopic measure of the rotation of a fluid element at any point is defined as the curl of velocity vector. It is a vector quantity. As far as the physical meaning is concerned, circulation can be thought as the amount of 'push' one feels while moving along a closed boundary or path. Vorticity Figure 2.5 Circulation (Right) vs. Vorticity however has nothing to do with a path, it is defined (Left) at a point and would indicate the rotation in the flow field at that point (see Figure 2.5). So, if an infinitesimal paddle wheel is imagined in the flow, it would rotate due to non-zero vorticity. 13

A., S., Shapiro, “Film Notes for Vorticity”, MIT.

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Conservative and Non-Conservative forms of PDE There are two folds to the question of differences between Conservative Vs Non-Conservative forms; namely physical and mathematical14. Physical We drive the governing equations by considering a finite control volume. This control volume may be fixed in space with the fluid moving through it or the control volume may be moving with the fluid in a sense that same fluid particles are always remain inside the control volume. If the first case is taken then the governing equations will be in conservation form else these will be in nonconservation form. The difference between the conservative and non-conservative forms is related to the movement of the control volume in the fluid flow. While deriving the equations of motion if we keep the control volume fixed and write the flow equations, they are called the equations in conservation forms. For example the continuity equation for incompressible flow in rectangular coordinates. On the other hand, if we focus on the same particles in motion and keep the control volume moving with them, the equations are called non-conservation equations, here, the same particles remain in the control volume. Prime examples of conservative and non-conservative forces are Gravity and Friction forces, respectively. Mathematical Splitting the partial derivatives for the purpose of discretization. For example, consider the term ∂ (ρu)/∂x in conservative form

( u) ( u)i  ( u)i 1  x Δx Eq. 2.5 The non-conservative form of the same can be written a

( u) u ρ u  u i 1 ρ  ρi 1 ρ u  ρi i  ui i x x x Δx Δx Eq. 2.6 The difference is obvious. While the original derivative is mathematically the same, the discrete form is not. To demonstrate this, consider a 4 point grid for conservative one (Eq. 2.5)

( u)1  ( u) 0 ( u) 2  ( u)1 ( u)3  ( u) 2   Δx Δx Δx

Eq. 2.7

And corresponding non-conservative (Eq. 2.6):

ρ1

u1  u 0 ρ  ρ0 u  u2 ρ  ρ2 u  u1 ρ  ρ1  u1 1  ρ2 2  u2 2  ρ3 3  u3 3 Δx Δx Δx Δx Δx Δx

Eq. 2.8

Those arguments just show that the non-conservative form is different, and in some ways harder. But 14

Physics Stack Exchange.

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why is it called non-conservative? For a derivative to be conservative, it must form a telescoping series. In other words, when you add up the terms over a grid, only the boundary terms should remain and the artificial interior points should cancel out. Now let's look at the non-conservative form: So now, you end up with no terms canceling! Every time you add a new grid point, you are adding in a new term and the number of terms in the sum grows. In other words, what comes in does not balance what goes out, so it's non-conservative. How to choose which one to use? Now, more to the point, when do you want to use each scheme? If your solution is expected to be smooth, then non-conservative may work. For fluids, this is shock-free flows. If you have shocks, or chemical reactions, or any other sharp interfaces, then you want to use the conservative form. Overall, if there is PDE which represents a physical conservative statement, this means that the divergence of a physical quantity can be identified in the equation, as the case in general conservation equations later.

Divergence Theorem - Control Volume Formulation In vector calculus, the divergence theorem, also known as Green Gauss's theorem, is a result that relates the flow (that is, flux) of a vector field through a surface to the behavior of the vector field inside the surface, (see Figure 2.6). On fluid, using the integral relations to calculate the net fluxes of mass, momentum and energy passing through a finite region of flow. The rate of change of any property Q within control volume could be defined as

dQ dQ ∂ dQ =∬ ρV. dA + ∭ ρVdV dt dm ∂t dm CS

Figure 2.6 A region V bounded by the surface S = ∂V with the surface normal n

or

CV

⃗ ) dV = ∯(F ⃗ .n ∭(∇. F ⃗ )ds V

S

Eq. 2.9 Which could be applied to any property such as mass, momentum and energy and dQ/dm being the amount of Q per unit mass of particle. With the aid of divergence theorem, , the surface integral could be converted to volume integral, and the result could be integrated over a fixed volume15.

General Transport Equation Part of the transport process attributed to the fluid motion alone or simply, the transport of a property by fluid movement. In relation to general transport process of a variable Q, this could be envisioned as Eq. 2.10. Thus, conservation principles can be expressed in terms of differential equations that describe all relevant transport mechanisms, such as convection (also called advection), diffusion, and dispersion. Each terms described below as:

15

White, Frank M. 1974: “Viscous Fluid Flow”, McGraw-Hill Inc.

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∂ ∂ ∂ ∂Q (ρQ) + (ρUj Q) = (ΓQ ) + S⏟ Q ⏟ ∂t ∂x ∂x ∂xj ⏟j ⏟j Source Transient ⏟ Convection Diffusion Transport

Eq. 2.10

1 Ui Q= e φ {ΓQ

Mass Conservation Momentum Conservation Energy Conservation Scalar Diffusion Coefficient

Newtonian Fluid In continuum mechanics, a Newtonian fluid is a fluid in which the viscous stresses arising from its flow, at every point, are linearly proportional to the local strain rate, the rate of change of its deformation over time. That is equivalent to saying those forces are proportional to the rates of change of the fluid's velocity vector as one moves away from the point in question in various directions. More precisely, a fluid is Newtonian only if the tensors that describe the viscous stress and the strain rate are related by a constant viscosity tensor that does not depend on the stress state and velocity of the flow. If the fluid is also isotropic (that is, its mechanical properties are the same along any direction), the viscosity tensor reduces to two real coefficients, describing the fluid's resistance to continuous shear deformation and continuous compression or expansion, respectively. One also defines a total stress tensor σ that combines the shear stress τ, with conventional (thermodynamic) pressure p. The stress-shear equation then becomes

σ= −

∂ui ∂uj + μ( + ) ∂x ∂x i ⏟ j Normal Stress pδ ⏟ij

Shear Stress

Eq. 2.11 On the other hand, a non-Newtonian fluid is a fluid that does not follow Newton's Law of Viscosity. Most commonly, the viscosity (the gradual deformation by shear or tensile stresses) of nonNewtonian fluids is dependent on shear rate or shear rate history. Some non-Newtonian fluids with shear-independent viscosity, however, still exhibit normal stress-differences or other nonNewtonian behavior. Many salt solutions and molten polymers are non-Newtonian fluids, as are many commonly found substances such as ketchup, custard, toothpaste, starch suspension honey, paint, blood, and shampoo16.

Some Flow Field Phenomena Viscous Dissipation Embodies the concept of a dynamical system where important mechanical models such as waves or oscillations, loss energy over time, typically from friction or turbulence. The lost energy converted to heat. For a viscous flow over a body, the kinetic energy decreased under influence of friction. This 16

From Wikipedia.

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lost kinetic energy reappears in the form of internal energy of the fluid, hence causing the temperature to rise. This phenomenon is called viscous dissipation within fluid17. Diffusion This is a physical process that occurs in a flow of gas in which some property is transported by the random motion of the molecules of the gas. Diffusion is related to the stress tensor and to the viscosity of the gas. Heat conduction, turbulence, and the generation of boundary layer are the result of diffusion in the flow. Diffusion is an equal exchange of species where the final state would be a uniform mixture. An ordinary example would be pouring cream into coffee until diffusion produces a uniform mixture. Another example could be release of a gas mixture in room. Standing on one side of room as gas released on the other side, soon we notice the odor diffused to our side replacing some of our air which diffuses to the other side. In other word, diffusion is the transport of Figure 2.7 Diffusion Process in Physics mass, energy and momentum as the result of molecular movement, express in mathematical language by multiplying some constant by the first gradient of quantity of interest. Therefore, a distinguishing feature of diffusion is that it results in mixing or mass transport, without requiring bulk motion or bulk flow (see Figure 2.7). Heat transfer and viscous flow are both diffusive phenomena. Convection Refers to the fluid motion that results from forces acting upon or within it (pressure, viscosity, gravity, etc.). Dispersion Is the combined effects of convention and diffusion? We talked about smoke dispersion from the chimney, which is result of convective (the wind blowing it), diffusive (smoke diffusive in the air), the bouncy forces (hot air rises).18 Advection Refers to the convection of a scalar concentration and very significant. Examples include 1st order linear wave equation. In other word, advection is the transport mechanism of a fluid from one location to another, and is dependent on motion and momentum of that fluid.

Inviscid vs. Viscous A major facet of a gas or liquid is the ability of the molecules to move rather freely. As molecules move, they transport their mass, momentum, and energy from one location to another. This transportation on a molecular scale gives rise to the phenomena of mass diffusion, viscosity (friction), and thermal conduction. All real flows exhibit such phenomena and such flows call viscous flows and to be discussed in detail later. In contrast, a flow which doesn’t experience any of these called Inviscid flow. In-viscid flows do not truly exist in nature, however, many practical aerodynamic flows where the influence of transport phenomena is small, could be modeled as Inviscid.

17 18

Anderson, John D. 1984: “Fundamentals of Aerodynamics”, McGraw Hills Inc. CFD online forum, 2006.

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Steady-State vs. Transient

An important factor in fluid flow analysis is its dependence to time. Simply put, Steady flow is a flow when field variables are independent of time, where for transient, they are. This dependence, or lack of it, could change the mathematical character of governing equation, as to be discussed later, therefore, altering the solution method. The question to be asked is when a flow could be classified as a transient flow? This is not easy as it sounds since most depend on their expertise and problem in hand. Nevertheless, most agree that majority of the flows are transient by nature (turbulent flows) unless proven otherwise. To that end, a useful, but time sensitive method would be to run the flow in steady-state and check the converging residuals. If there are large oscillations in outputs, then there is good chance that flow is Transient and not steady. But if the residuals exhibit a relatively smooth convergence rate, then the flow is steady. Figure 2.8 shows a test case of vortex shedding for flow over a cylinder which is inherently transient19.

Figure 2.8 Figure 1

Transient Case of Vortex Shedding over a Cylinder

Transient test case of vortex shedding over a cylinder

Flow Figure Field 1.1 Classification Transient test case of Vortex shedding over a cylinder In general, the fluid flows equations could be classified in terms of its Physical and Mathematical aspects of it. Mathematically, they can classified as Elliptic, Hyperbolic, or Parabolic, depending on flow as being Subsonic, Transonic or Supersonic, or any combination of two. This will be dealt in details later on. Physically, they can be classified via Figure 2.9.

19

Anderson, John D. 1984: “Fundamentals of Aerodynamics”, McGraw Hills Inc.

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Internal vs. External Laminar vs. Turbulent

Vicous vs. Invscid

Physical Flow Field

Newtonian vs. NonNewtonian

Attached vs. Detached

Figure 2.9

Compressi ble vs. Incompres sible

Steady vs. Unsteady

Physical Aspects of a Typical Flow Field

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3

Brief Review of Thermodynamics Pressure

Pressure is the limiting form of force per unit area20

d𝐅 p = lim ( ) as dA → 0 dA Eq. 3.1

Perfect (Ideal) Gas Cases when gas particles are far enough to be able to neglect the influence of intermolecular forces. For an ideal gas the equation of state is:

PρRT

or

Pv  RT

where

v 

1 specific volume ρ

Eq. 3.2

Total Energy The total energy of a system E consists of internal energy (e), kinetic energy (KE), and potential energy (PE) as

1 E  e  KE  PE , KE  mV 2 , PE  mgz 2

Eq. 3.3

Where internal energy (e), specific enthalpy (h), are related as

h  e  pv

e  e (T) & h  h (T)

Eq. 3.4

Thermodynamic Process The thermodynamic process is divided to three categories of :  Adiabatic - one in which no heat is added to or taken away from the system  Reversible - one in which no dissipative phenomena occur, i.e., where the effects of viscosity, thermal conductivity, and mass diffusion are absent  Isentropic - one which is both adiabatic and reversible

First Law of Thermodynamics Due to molecular motion of a gas, the heat added or work done on the system causes a change in energy

δq + δw = δE

Eq. 3.5 which is also called the steady-state energy equation.

20

Anderson, John D. 1984: “Fundamentals of Aerodynamics”, McGraw Hills Inc.

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Second Law of Thermodynamics In an essence the 2nd law of thermodynamics complements the 1st law by ascertaining the proper direction of a process by defining a new state variable called entropy,

ds 

δq T

Eq. 3.6 For an adiabatic process (δq = 0), Eq. 3.6 becomes,

ds  0 Eq. 3.7 Eq. 3.6 and Eq. 3.7 are forms of the second law of thermodynamics. The second law tells us in what direction a process will take place. A process will proceed in a direction such that the entropy of the system always increases or, at best, stay the same. For a reversible process where δw = -pdv and δq = Tds, the more utilitarian form of 2nd law could be devised as

Tds  de  pdv Tds  dh  vdp

if

dh  de  pdv  vdp

Eq. 3.8

Integrating for a calorically perfect gas, both R and CP constant,

s 2  s1  c p ln

T2 p  R ln 2 T1 p1

Eq. 3.9

Isentropic Relation For an isentropic process which is both adiabatic and reversible, Eq. 3.9, ds = 0, could be manipulated to γ

p 2  ρ 2   T2      p1  ρ1   T1 

γ γ 1

(2.11)

Eq. 3.10 The above mentioned equation (Eq. 3.10) is very important as it relates pressure, density, and temperature for an isentropic process21. It stems from the 1st law of thermodynamics and definition of entropy and basically is an energy relation for an isentropic process. Why so important or why is it frequently used? When it seems so restrictive requiring both adiabatic and reversible conditions? The answer rests in the fact that large number of practical compressible flow problems could be assumed isentropic as dissipative effects are confined to a thin boundary layer, where outside, the flow could be assumed to be isentropic22.

21 22

Anderson, John D. 1984: “Fundamentals of Aerodynamics”, McGraw Hills Inc. Anderson, John D. 1984: “Fundamentals of Aerodynamics”, McGraw Hills Inc.

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Static (Local) Condition These are quantities when riding along with the gas at the local flow velocity.

Stagnation (Total) Condition Represents quantities when fluid elements are brought to rest adiabatically. The values are expected to change. In particular, the value of temperature, denoted by T0 where the corresponding enthalpy h0=CpT0 for a calorically perfect gas. From an steady, in-viscid, adiabatic energy equation23,

h

V2  const  h 0 2

Eq. 3.11 Along a stream line. Analog to this, since h0 = CpT0, thus,

h 0  Cp T0

 T0  const

Eq. 3.12 Using Eq. 3.11 and Eq. 3.12 as an special case of energy equation, and defining Cp in terms of velocity, for a calorically perfect gas the ratio of total temperature to static temperature could be express in terms of Mach number as Eq. 3.13. Similarly, total pressure p0 and total density ρ0 are defined as the properties if the fluid elements brought to rest isentropically by using:

T0 γ 1 2  1 M T 2 γ

P0 γ - 1 2  γ1   1  M  P 2   1

ρ0 γ  1 2  γ1   1  M  ρ 2   Eq. 3.13

Total Pressure (Incompressible) For incompressible flow from Bernoulli’s Equation, without any body force, the pressure is the sum of static and dynamic pressures as,

⏟ P = ⏟ P + total

static

1 2 ρV ⏟ 2 dynamic

Eq. 3.14 Note – Although the stagnation and total terminology are been used indiscriminately for pressure, in general the total pressure also dependent on another identity call gravitational head (ρgz). Therefore, for completeness, the total pressure could be represented is

23

Same as above.

30

Ptotal  Pstatic  Pdynamic  Pgravitational head Eq. 3.15 Where for most cases the Pgravitational head is ignored, therefore, the total and stagnation pressures are assumed to be the same. In essence, total pressure is the constant in Bernoulli’s equations.

Pressure Coefficient The non-dimensional pressure coefficient could be derived with the aid of Eq. 3.14 as

Cp 

p  p q

where q  

1 ρ V 2

Eq. 3.16 For incompressible flow, the, Cp could be expressed in terms of velocities a

 V C p  1    V

  

2

Eq. 3.17

Application of 1st Law to Turbomachinery

Figure 3.1

Control Volume showing sign convention for heat and work transfer

Figure 3.1 shows a control volume representing a turbomachine, through which fluid passes at a steady rate of mass flow ṁ, entering at position 1 and leaving at position 2. Energy is transferred from the fluid to the blades of the turbomachines, positive work being done (via shaft) at the rate Ẇx. In the general case, positive heat transfer takes place at the rate Ǭ from the surrounding to the control volume24. Thus,

1 (h2 − h1 ) + (c22 − c12 ) + ⏟ 𝐐̇ − 𝐖̇x = ṁ [ ⏟ g(z2 − z1 ) ] ⏟ 2 internal energy potential energy kinetic energy ⏟ total energy per unit mass

Eq. 3.18 where h is the specific enthalpy, 1/2c2 the kinetic energy per unit mass, and gz is potential energy per unit mass. Apart from hydraulic machines, the contribution of the last term in Eq. 3.18 is small and usually ignored. Defining the stagnation enthalpy h = h0+1/2c2 and assuming g(z2 - z1) is negligible, it becomes

𝐐̇ − 𝐖̇ 𝑥 = ṁ(h02 − h01 )

Eq. 3.19 Most turbomachinery flow processes are adiabatic (or very nearly so), and it is permissible to write Ǭ=0. Therefore, for work producing machines (turbines) Wx > 0 so that S.L. Dixon and C.A. Hal, “Fluid Mechanics and Thermodynamics of Turbomachinery”, 6th edition, ISBN: 978-185617-793-1. 24

31

𝐖̇ 𝑥 = ṁ(h02 − h01 )

Eq. 3.20 and work for absorbing machines (compressor) is negative of that. Moment of Momentum In dynamics much useful information is obtained by employing Newton’s second law in the form where it applies to the moments of forces. This form is of central importance in the analysis of the energy transfer process in turbomachines. For a system of mass m, the vector sum of the moments of all external forces acting on the system about some arbitrary axis A-A fixed in space is equal to the time rate of change of angular momentum of the system about that axis, i.e.

𝛕A = m

d (r𝑐 ) dt 𝜃

Eq. 3.21 where r is distance of the mass center from the axis of rotation measured along the normal to the axis and cθ the velocity component mutually perpendicular to both the axis and radius vector r. For a control volume the law of moment of momentum can be obtained. Figure 3.2 shows the control volume enclosing the rotor of a generalized turbomachine. Swirling fluid enters the control volume at radius r1 with tangential velocity cθ1 and leaves at radius r2 with tangential velocity cθ2. For onedimensional steady flow Eq. 3.22 which states that, the sum of the moments of the external forces acting on fluid temporarily occupying the control volume is equal to the net time rate of efflux of angular momentum from the control volume. Euler‘s Pump & Turbine Equations For a pump or compressor rotor running at angular velocity 0, the rate at which the rotor does work on the fluid is

𝛕A = ṁ(r2 cθ2 − r1 cθ1 )

Figure 3.2

Control Volume for a Generalized Turbomachine

𝛕𝐀 Ω = m(U2 cθ2 − U1 cθ1 )

Eq. 3.23 where the blade speed U = Ω r. Thus the work done on the fluid per unit mass or specific work, is

∆𝐖𝐜 =

𝐖̇𝐜 𝛕𝐀 = = (U2 cθ2 − U1 cθ1 ) > 0 ṁ ṁ

Eq. 3.24 This equation is referred to as Euler’s pump equation.

32

𝐖̇𝐭 ∆𝐖t = = (U1 cθ1 − U2 cθ2 ) > 0 ṁ

Eq. 3.25 For a turbine the fluid does work on the rotor and the sign for work is then Eq. 3.25 will be referred to as Euler’s turbine equation. Case Study - Application of 1st and 2nd Laws of Thermodynamics to Single Stage Turbo Machines Fluid flow in turbo machines always varies in time, though it is assumed to be steady when a constant rate of power generation occurs on an average. This is due to small load fluctuations, unsteady flow at blade tips, the entry and the exit, separation in some regions of flow etc., which cannot avoided, no matter how good the machine and load stabilization may be. This assumption permits the analysis of energy and mass transfer by using the steady state control volume equations. Assuming further that there is a single inlet (1) and a single outlet (2) for the turbo machine across the sections of which the velocities, pressures, temperatures and other relevant properties are uniform, one writes the steady flow equation of the First Law of Thermodynamics within a control volume for turbo machinery in the form:

Q + m ̇ ( h1 +

V22 V22 ̇ + gz) = Power(out) + m ( h2 + + gz2 ) 2 2

Eq. 3.26 Where Q is the rate of energy transfer as heat cross the CV, Power(out) is the power output, and ṁ is the mass flow rate. In cooperating the total enthalpy relation h0 = h + V2/2 + gz, and assuming an adiabatic process (Q = 0). Eq. 3.26 could be rearranged per unit mass flow as

P p V2 ∇h0 = −W = − = −∇ ( + = gz) ṁ ρ 2 Eq. 3.27 Therefore, the energy transfer as work is numerically equal to the change in stagnation (total) enthalpy of the fluid between the inlet and the outlet of the turbo machine. In a turbo machine, the energy transfer between the fluid and the blades can occur only by dynamic action, i.e., through an exchange of momentum between the rotating blades (Figure 3.3, location 3) and the flowing fluid. It thus follows that all the work is done when the fluid flows over the rotorblades and not when it flows over the stator-blades. As an example, considering a turbo machine with a single stator-rotor combination shown schematically in Figure 3.3. Let points 1 and 2 represent respectively the inlet and the exit of the Figure 3.3 Schematic section of Single Stage stator. Similarly, points 3 and 4 represent Turbomachine the corresponding positions for the rotor blades. Then ideally for flow between points 1and 2, there should be no stagnation enthalpy changes since no energy transfer as heat or

33

work occurs in the stator. Thus, ho1 = ho2. For flow between points 3 and 4 however, the stagnation enthalpy change may be negative or positive, depending upon whether the machine is powergenerating or power-absorbing. Hence, ho3 > ho4 if the machines develops power (compressor), and if ho3 < ho4, the machine needs a driver and absorbs power (turbine). If the system is perfectly reversible and adiabatic with no energy transfer as work, no changes can occur in the stagnation properties (enthalpy, pressure and temperature) between the inlet and the outlet of the machine. But all turbo machines exchange work with the fluid and also suffer from frictional as well as other losses. The effect of the losses in a power-generating machine is to reduce the stagnation pressure and to increase entropy so that the network output is less than that in an ideal process. The corresponding work input is higher in a power-absorbing machine as compared with that in an ideal process. In order to understand how this happens, consider the Second Law equation of state,

T0 ds 0  dh 0  v 0 dp o

(i.e., Tds  dh  vdp)

Eq. 3.28

When applied to stagnation properties. Hence,

 δw  v0dp0  T0ds 0

Eq. 3.29

In a power-generating machine, dpo is negative since the flowing fluid undergoes a pressure drop when mechanical energy output is obtained. However, the 2nd law requires that Todso ≥ δq, but as δq = 0, then To dso ≥ 0. The sign of equality applies only to a reversible process which has a work output w = – vodpo > 0. In a real machine, Todso > 0, and represents the decrease in work output due to the irreversibility in the machine.

34

4 Aerodynamics Distinction A distinction between solids, liquids, and gases can be made in a simplistic sense as follows. Put a solid object inside a larger, closed container. The solid object will not change; its shape and boundaries will remain the same. Now put a liquid inside the container. The liquid will change its shape to conform to that of the container and will take on the same boundaries as the container up to the maximum depth of the liquid. Now put a gas inside the container. The gas will completely fill the container, taking on the same boundaries as the container. The word fluid is used to denote either a liquid or a gas. A more technical distinction between a solid and a fluid can be made as follows. When a force is applied tangentially to the surface of a solid, the solid will experience a finite deformation, and the tangential force per unit area the shear stress will usually be proportional to the amount of deformation. In contrast, when a tangential shear stress is applied to the surface of a fluid, the fluid will experience a continuously increasing deformation, and the shear stress usually will be proportional to the rate of change of the deformation25.The most fundamental distinction between solids, liquids, and gases is at the atomic and molecular level. In a solid, the molecules are packed so closely together that their nuclei and electrons form a rigid geometric structure, “glued” together by powerful intermolecular forces. In a liquid, the spacing between molecules is larger, and although intermolecular forces are still strong they allow enough movement of the molecules to give the liquid its “fluidity.” In a gas, the spacing between molecules is much larger (for air at standard conditions, the spacing between molecules is, on the average, about 10 times the molecular diameter). Hence, the influence of intermolecular forces is much weaker, and the motion of the molecules occurs rather freely throughout the gas. This movement of molecules in both gases and liquids leads to similar physical characteristics, the characteristics of a fluid quite different from those of a solid. Therefore, it makes sense to classify the study of the dynamics of both liquids and gases under the same general heading, called fluid dynamics. On the other hand, certaindifferences exist between the flow of liquids and the flow of gases; also, different species of gases (say, N2, He, etc.) have different properties. Therefore, fluid dynamics is subdivided into three areas as follows:   

Hydrodynamics - flow of liquids Gas dynamics - flow of gases Aerodynamics - flow of air

These areas are by no means mutually exclusive; there are many similarities and identical phenomena between them. Also, the word “aerodynamics” has taken on a popular usage that sometimes covers the other two areas.

Aerodynamic Practical Application Acceding to [Anderson]26, aerodynamics is an applied science with many practical applications in engineering. No matter how elegant an aerodynamic theory may be, or how mathematically complex a numerical solution may be, or how sophisticated an aerodynamic experiment may be, all such efforts are usually aimed at one or more of the following practical objectives: 1. The prediction of forces and moments on, and heat transfer to, bodies moving through a fluid (usually air). For example, we are concerned with the generation of lift, drag, and moments on airfoils, wings, fuselages, engine nacelles, and most importantly, whole airplane configurations. We want to estimate the wind force on buildings, ships, and other surface 25 26

John D. Anderson, Jr., “Fundamentals of Aerodynamics”, 5th Edition, McGraw-Hill Companies, 2011. John D. Anderson, Jr., “Fundamentals of Aerodynamics”, 5th Edition, McGraw-Hill Companies, 2011

35

vehicles. We are concerned with the hydrodynamic forces on surface ships, submarines, and torpedoes. We need to be able to calculate the aerodynamic heating of flight vehicles ranging from the supersonic transport to a planetary probe entering the atmosphere of Jupiter. These are but a few examples. 2. Determination of flows moving internally through ducts. We wish to calculate and measure the flow properties inside rocket and air-breathing jet engines and to calculate the engine thrust. We need to know the flow conditions in the test section of a wind tunnel. We must know how much fluid can flow through pipes under various conditions. A recent, very interesting application of aerodynamics is high-energy chemical and gas-dynamic lasers, which are nothing more than specialized wind tunnels that can produce extremely powerful laser beams. The applications in item 1 come under the heading of external aerodynamics since they deal with external flows over a body. In contrast, the applications in item 2 involve internal aerodynamics because they deal with flows internally within ducts. In external aerodynamics, in addition to forces, moments, and aerodynamic heating associated with a body, we are frequently interested in the details of the flow field away from the body. For example is the flow associated with the strong vortices trailing downstream from the wing tips of large subsonic airplanes such as the Boeing 747. What are the properties of these vortices, and how do they affect smaller aircraft which happen to fly through them? The above is just a sample of the countless applications of aerodynamics.

Aerodynamic Forces and Moments The aerodynamic forces and moments on a body are due to only two basic sources: Pressure and Shear Stress distributions over the body. Both have dimensions of force per unit area where pressure acts normal to surface and shear tangential. The net effect of P and τ distributions integrated over the complete body surface is a resultant aerodynamic force R and moment M on the body. Lift is the perpendicular component of R w.r.t. free stream while Drag represents the parallel27. Therefore, source of aerodynamic Lift, Drag, and Moments on the body are the pressure and shear stress distributions integrated over the body. To better represent these forces, dimensionless coefficients of Lift, CL, drag, CD, and moment, CM, introduced as

CL 

L D M , CD  , CM  q S q S q  SL

Eq. 4.1 Where q∞ is the previously defined dynamic pressure, ½ ρV2 , and reference area S and reference length L are Figure 4.1 Schematic of Lift and Drag Coefficients vs Angle of Attack on a Airfoil chosen to pertain to given geometric shape; for (Courtesy of Anderson) different shapes , S and L may be different things. For example, for an airplane wing S is the plan form area and L the mean chord length. However for a sphere, S would be the cross-section area while L is the diameter. More information such as lifting airfoil and finite wing theory, and other relevant topics, 27

Anderson, John D. 1984: “Fundamentals of Aerodynamics”, McGraw Hills Inc.

36

can be obtained in any aerodynamic specific text books such as [J. D. Anerson]28. Generic variations for CL and CD versus angle of attack (α) are sketched in Figure 4.1. Note that CL increases linearly with α until an angle of attack is reached when the wing stalls, the lift coefficient reaches a peak value, and then drops off as α is further increased. The maximum value of the lift coefficient is denoted by CL,max, as illustrated.

Leading-Edge Flow as a Governing Factor in Leading-Edge Vortex Initiation in Unsteady Airfoil Flows A leading-edge suction parameter (LESP) that is derived from potential flow theory as a measure of suction at the airfoil leading edge is used to study initiation of leading-edge vortex (LEV) formation by [Ramesh et al.]29. The LESP hypothesis is presented, which states that LEV formation in unsteady flows for specified airfoil shape and Reynolds number occurs at a critical constant value of LESP, regardless of motion kinematics. The hypothesis is seen to hold except in cases with slow-rate kinematics which evince significant trailing-edge separation. Low-Reynolds-number flows at low speeds and small scales, despite being incompressible and non-thermodynamic, are rife with complexity owing to the effects of viscosity and flow separation. Much research on this topic in the twenty-first century has been driven by interest in micro-air vehicle (MAV) design, a problem at the interface between low Re fluid mechanics and flight vehicle engineering. The design problem in this regime has been driven by seeking bio-inspiration from insects which employ flapping flight at high dimensionless rates of motion (reduced frequencies) to achieve remarkable flying prowess. It has been shown that the single most important aerodynamic phenomenon largely responsible for the success of flapping flight at low Reynolds numbers is the leading-edge vortex (LEV). The conditions under which such LEVs develop on rounded-leading-edge airfoils form the subject of this study and are investigated with a large set of unsteady test cases using experiments, computations, and theoretical methods. Two-dimensional problems without additional complexity involving span-wise flow and wingtip vortices are considered here and serve as a starting point for more complex investigations. LEV formation is initiated by reversed flow at the airfoil surface in the vicinity of the leading edge, followed by the formation of a free shear layer. The free shear layer then builds up into a vortex, which traverses the airfoil chord and convects into the wake. Research contributions on LEV formation have largely arisen from the rotorcraft community and the more recent low-Re/MAV community. Identification of LEV initiation from CFD data The procedure used in this research for identifying the initiation of LEV formation from CFD skin friction information is illustrated here with the baseline case listed. Figure 4.2 presents results from experiments and CFD for the baseline case at four instants during the motion. The upper surface skin friction (Cf ) distributions from CFD (on the third row of the figure) are examined at various time instants of the motion to identify several key steps that lead to the formation of the LEV. The flow features leading to LEV formation have been discussed by several authors. The four time instants at (a)–(d), are used to highlight the following flow features:

John D. Anderson, Jr., “Fundamentals of Aerodynamics”, 5th Edition, McGraw-Hill Companies, 2011. Kiran Ramesh, Kenneth Granlund, Michael V. Ol, Ashok Gopalarathnam, Jack R. Edwards, “Leading-edge flow criticality as a governing factor in leading-edge vortex initiation in unsteady airfoil flows”, Theoretical Computational Fluid Dynamics, 2017. 28 29

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(a) Attached flow - Before the initiation of the LEV formation, the flow is attached at the leading edge. The attached boundary layer is thin and the Cf is positive. (b) Onset of reversed flow - LEV formation is first preceded by the formation of a small region of reversed flow near the leading edge of the airfoil, signaled by appearance of counterclockwise vorticity near the surface and a small region of negative Cf . (c) Initiation of LEV formation - Next, a small region of clockwise vorticity starts to develop at the surface within the region of counterclockwise vorticity seen in (b). This manifests as a spike in the negative Cf distribution that reaches up to zero and subsequently becomes a region of positive Cf within the region of negative Cf distribution. This flow feature signals the formation of the shear layer in which there is an eruption of surface flow into the mainstream. As in previous work, the instant when the spike in the negative Cf region first reaches the zero value is taken as the time corresponding to initiation of LEV formation. (d) Formation and feeding of the LEV - The eruption of surface flow, initiated in (c), results in a plume of clockwise vorticity flowing into the mainstream. During these time instants, there are several spikes in the Cf distribution corresponding to positive-Cf regions embedded within a larger negativeCf region. (a)

Figure 4.2

(b)

(c )

(d)

Vorticity Plots from CFD (first row), Flow Visualization from Experiment (second row)

Measure of Compressibility & Compressible vs. Incompressible Flows A flow is classified as being compressible or incompressible depending on the level of variation of density during flow. Physically, compressibility is the fractional change in volume of the fluid element per unit change in pressure or isothermal compressibility

1  dv  τ  -   v  dp  T

for v 

1 1 dρ  τ ρ ρ dp

where

ρ  ρ (p,T)

Eq. 4.2

Therefore, for incompressible flow or constant ρ, the compressibility of gas (τ = 0). In contrast, if ρ = ρ (p, T), then flow is considered compressible. There are number of aerodynamic problems that could be considered incompressible without any determinable loss of accuracy. For example flow for liquids could be considered incompressible, and hence most hydrodynamic problem assume ρ = constant. Also the flow of gases at low Mach number (M∞ < 0.3) is essentially incompressible. This is

38

not true for high speed flow when the density fluctuations are apparent and must be treated as compressible30.

Speed of Sound The speed of sound, a molecular phenomenon, in a calorically perfect gas is given by

a  γRT

Eq. 4.3

This is a function of temperature only and related to the average molecular velocity. It is also relates to compressibility of gas, τ by

a

1 ρτ

Eq. 4.4

The lower the compressibility, the higher the speed of sound. For an incompressible flow, τ = 0, then speed of sound is theoretically infinite. The Mach number M = (V/a) is therefore, zero. Hence, the incompressible flow could be theoretically characterized as zero Mach number flow.

Mach Number In fluid dynamics, the Mach number (M or Ma) is a dimensionless quantity representing the ratio of flow velocity past a boundary to the local speed of sound 31-32,

M

V c

Eq. 4.5

Where M is the Mach number, V is the local flow velocity with respect to the boundaries (either internal, such as an object immersed in the flow, or external, like a channel), and c is the speed of sound in the medium. The local speed of sound, and thereby the Mach number, depends on the condition of the surrounding medium, in particular the temperature and pressure. Figure 4.3 shows an F/A-18 creating a vapor cone at transonic speed just before reaching Mach 1 (By Ensign John Gay, U.S. Figure 4.3 An F/A-18 Hornet Creating a Vapor Cone at Transonic Speed Navy). The Mach number is primarily used to determine the approximation with which a flow can be treated as an incompressible flow. The medium can be a gas or a liquid. While the terms "subsonic" and "supersonic," in the purest sense, refer to speeds below and above the local speed of sound respectively, aerodynamicists often use the same terms to talk about particular ranges of Mach Anderson, John D. 1984: “Fundamentals of Aerodynamics”, McGraw Hills Inc. Young, Donald F.; Bruce R. Munson; Theodore H. Okiishi; Wade W. Huebsch (2010). A Brief Introduction to Fluid Mechanics (5 Ed.). John Wiley & Sons. p. 95. 32 Graebel, W.P. (2001). Engineering Fluid Mechanics. Taylor & Francis. p. 16. 30 31

39

values. This occurs because of the presence of a "transonic regime" around M = 1 where approximations of the Navier-Stokes equations used for subsonic design actually no longer apply; the simplest explanation is that the flow locally begins to exceed M = 1 even though the freestream Mach number is below this value. Meanwhile, the "supersonic regime" is usually used to talk about the set of Mach numbers for which linearized theory may be used, where for example the (air) flow is not chemically reacting, and where heat-transfer between air and vehicle may be reasonably neglected in calculations. In the following table (see Table 4.1), the "regimes" or "ranges of Mach values" are referred to, and not the "pure" meanings of the words "subsonic" and "supersonic". Generally, NASA defines "high" hypersonic as any Mach number from 10 to 25, and re-entry speeds as anything greater than Mach 24. Aircraft operating in this regime include the Space Shuttle and various space planes in development. Further details regarding the Mach number regimes can be obtained from [Anderson]33. Regime Mach knots mph Km/h m/s Subsonic < 0.8 < 530 < 609 < 980 < 273 Transonic 0.8-1.2 530-794 609-914 980-1470 273-409 Supersonic 1.2-5.0 794-3308 915-3806 1470-6126 410-1702 Hypersonic 5.0-10.0 3308-6615 3806-7680 6125-12251 1702-3403 High-Hypersonic 10.0-24.0 6615-16537 7680-19031 12251-30626 3403-8508 Re-Entry Speeds > 24.0 > 16537 > 19031 > 30626 > 8508 Table 4.1

Classification of Mach Number

Sonic Boom

A sonic boom is the sound associated with the shock waves created by an object traveling through the air faster than the speed of sound. Sonic booms generate significant amounts of sound energy,

Figure 4.4 33

Evolution of Shock Wave

John D. Anderson, “Fundamentals of Aerodynamics”, McGraw Hill Inc. pp.37-39, 1976.

40

sounding much like an explosion to the human ear34. The crack of a supersonic bullet passing overhead or the crack of a bullwhip are examples of a sonic boom in miniature. Contrary to popular belief, a sonic boom does not occur only at the moment an object crosses the speed of sound; and neither is it heard in all directions emanating from the speeding object. Rather the boom is a continuous effect that occurs while the object is travelling at supersonic speeds. But it only affects observers that are positioned at a point that intersects an imaginary geometrical cone behind the object. As the object moves, this imaginary cone also moves behind it and when the cone passes over the observer, they will briefly experience the boom. When an aircraft passes through the air it creates a series of pressure waves in front of it and behind it, similar to the bow and stern waves created by a boat. These waves travel at the speed of sound and, as the speed of the object increases, the waves are Figure 4.5 Illustration of a sonic boom as received by forced together, or compressed, human ears because they cannot get out of the way of each other. Eventually they merge into a single shock wave, which travels at the speed of sound, a critical speed known as Mach 1, and is approximately 1,235 km/h (767 mph) at sea level and 20 °C (68 °F). In smooth flight, the shock wave starts at the nose of the aircraft and ends at the tail. Because the different radial directions around the aircraft's direction of travel are equivalent (given the "smooth flight" condition), the shock wave forms a Mach cone, similar to a vapor cone, with the aircraft at its tip (see Figure 4.5).

Shock Waves A shock wave is a very thin region in a supersonic compressible flow across which there is a large variation in the flow properties. Because there variation occur in such a short distance, viscosity and heat conductivity play a dominant role in the structure of shocks. These will be revisited later while Figure 4.4 displays shock wave for different flow regions as applicable to a jet fighter. Another examples in detecting of shocks would be the concentration of contour lines. As a discontinuity in the flow field, the contour line of Mach number, pressure, density and temperature all concentrate

Figure 4.6

34

Wikipedia.

Contour Concentration Examples at M = 5 ; From Left to Right ; (a) Mach number (b) Pressure (c) Density (d) Temperature

41

near the shock wave. For slip plane and contact discontinuity, pressure contour lines do not concentrate. Figure 4.6 displays contour lines of all relevant of flow features exemplifying a bow shock in presence of blunt body, in supersonic flow35. Difficulties for Shock Wave Detection There exist several difficulties for shock detection from CFD result:  The numerical dissipation and oscillation in CFD may cause some shock waves to be undetected. The numerical dissipation smears the discontinuity in the flow field, and makes weak shock waves undetectable.  The numerical oscillation produces structures similar to weak shock waves just near real shock wave, and thus may lead to false detection results.  The similarity among shock waves and other discontinuous flow structures like slip lines can lead to incorrect detection results.  The graphical display of shock detection result is also a problem for three-dimensional and multiple shock waves. Traditional Shock Detection Methods According to investigation by [Ziniu, et al. ]36, traditionally, through the contour lines one may detect shock waves since near shock waves the contour lines are concentrated. While pressure, density, temperature and Mach number contour lines all concentrate near shock waves, only pressure contour is recommended for shock detection, because the others cannot distinguish shock wave from slip line, shear layer or contact discontinuity. However, it is still difficult to get a clear view of the shock wave structure from pressure contour. Also contour method cannot give a direct way to display shock surfaces in three-dimensional flow field. These disadvantages highly restrict the use of this contour shock detection method. Another traditional shock detection method is to plot the isosurface of Mach number. It is convenient to approximate the shock surface around the vehicle by displaying a Mach number iso-surface just a little lower than the freestream Mach number. This method can yield a full shock wave surface in three-dimensional flow field. However, it may produce

Computed with firstorder Godunov’s Method

Figure 4.7

Computed with second-order two-step Lax-Wendroff Method

Solution of Shock Capturing for Euler Equations

Wu Ziniu , Xu Yizhe, Wang Wenbin, Hu Ruifeng, “Review of shock wave detection method in CFD postprocessing”, Chinese Journal of Aeronautics, 2013. 36 see Previous. 35

42

too many surfaces other than desired shock wave surfaces, and become helpless to detect shock wave except the leading shock wave. In numerical solutions of fluid flow with discontinuities (shock wave) by the shock-capturing method. The shock wave can be smoothed by low-order scheme or there are spurious oscillations near shock surface by high-order scheme37, as shown in Figure 4.7, where solid lines represent analytical solution, and dotted lines represent computed result. In the classical boundary shock-fitting method, shock wave must be introduced explicitly as outer flow boundary, which depends on experimental, theoretical or numerical-based knowledge on shock shape and location. While in the floating shock-fitting method proposed by Moretti38, shock waves are detected through Rankine–Hugoniot jump condition and the method of characteristics, which may be applicable to shock detection in post-processing. Compressible 1D Shock Waves Relations An undesirable side effect to supersonic, compressible flow is the phenomena called shock wave that almost always associated with aerodynamic losses and should be avoid. A shock wave is a thin region across which flow properties exhibit a large gradient39. On molecular level, the disturbance due to an obstacle is propagated upstream via molecular collisions (momentum) at approximately the local speed of sound. If the upstream flow is subsonic, the disturbances have no problem working their way upstream, thus giving the incoming flow plenty of time to move out of way of obstacle. On the other hand, if upstream flow is supersonic, the disturbances cannot work their way upstream, but rather at some finite distance from the obstacle. This disturbance wave pile up and coalesce, forming a thin standing wave in front of the body. Hence, the physical generation of shocks and expansion wave is due to propagation of information via molecular collisions and due to the fact that in supersonic flow this information cannot work its way into certain region of supersonic flow. The shock wave is usually at an oblique angle to the flow, attached or detached, however, there are many cases that it could be the stronger normal type. In both cases, the pressure increases almost discontinuously across the wave.

Figure 4.8

Qualitative Depiction of 1D Flow Through Normal and Oblique Shocks

Figure 4.8, the qualitative changes across the wave is noted, for region 1 ahead, and region 2 behind, with normal shock (left), and oblique shock (right). The pressure, density, temperature, and entropy increases across the shock, whereas the total pressure, Mach number, velocity decreases. Since the Toro EF. “Riemann solvers and numerical methods for fluid dynamics”. 3rd Edition, Berlin: Springer; 2009. Moretti G. “Experiments in multi-dimensional floating shock fitting”. Polytechnic Institute of Brooklyn, Brooklyn, NY, PIBAL, Report No. 73–18, 1973. 39Anderson, John D. 1984: “Fundamentals of Aerodynamics”, McGraw Hills Inc. 37 38

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flow across the shock is adiabatic (no external heating), the total enthalpy is constant across. Behind the oblique shock the flow remain usually supersonic, but weaker. For normal shock, the downstream flow is always subsonic. U1 and U2 are normal component of velocity. The quantities downstream could be directly evaluated by upstream values40. Another relation vital to oblique shock wave is the relations between deflection angle θ and wave angle β in relation to upstream Mach number (M1) as

tan θ  2 cot β

M12 sin 2β  1 M12 (γ  cos 2 )  2

Eq. 4.6 This is an important relationship between upstream Mach number, M1, deflection angle θ, wave deflection β, and should be analyzed thoroughly. Using known ϴ we could obtain the tangential velocity components (Ut1, Ut2), and use of previously relationship to obtain the downstream values as:

ρ1U1  ρ 2 U 2 ,

p1  ρ1U12  p 2  ρ 2 U 22 , U t1  U t2 , h1 

Eq. 4.7 Other consideration in obtaining ϴ include:

U12 U2  h2  2 2 2

1- For any upstream Mach number, M1, there exists a maximum deflection angle, θmax, where there is no solution exists for straight oblique shock. Instead, nature establishes a curved shock wave, detached from the body. 2- For any values less than θmax, there are two straight oblique shock solutions, denoting to weak and strong shock solutions. 3- If θ=0, then β=90 degrees and therefore normal shock results. 4- For a fixed θ, increasing the upstream Mach number M1, causes the shock becomes stronger and closer to the body (β decreases). This would cause stronger dissipative effects near surface (shear and thermal conductivity), clearly an undesirable effect in thermal management of body. The physical effects of oblique shock discussed above are very important. Yet another feature is the shock interactions and reflections. An impinging oblique shock on a surface would not simply disappear but rather weakens and reflects, provided the flow on the surface preserves the tangential quantities. Figure 4.9 exhibits the reflection of oblique shock wave on an in-viscid channel flow,

Figure 4.9 40

Oblique Shock Reflections on a Channel Flow (M=2 AoA=15˚)

Anderson, John D. 1984: “Fundamentals of Aerodynamics”, McGraw Hills Inc.

44

generated by its edges with free-stream Mach number of 2.0, AOA = 15˚ and slip wall boundary conditions. Oblique Shock Crossing Interaction Oblique crossing shock interactions in steady flows with conditions in the dual-solution domain were examined both numerically and experimentally41. Two and 3D simulations were compared with the measured Mach-stem heights determined from flow visualization. Hysteresis in the shock-reflection configuration was demonstrated for two-dimensional solutions with the same limits of the hysteresis loop observed by both participating research groups. The transition from regular to Mach reflection occurs very near the theoretical detachment condition, while for decreasing flow deflection angles the transition from Mach to regular reflection occurs before the flow deflection angle has decreased to the Neumann condition. Both the two and three-dimensional solutions show reasonable agreement between the numerical and experimental Mach-stem heights on the interaction centerline. Additionally, the 3D simulations closely predict the span-wise variation of the Mach-stem height observed in the experiments. While numerical simulations of the oblique crossing shock interaction appear to predict the shock structure with reasonable accuracy, the question of which reflection configuration (Mach or regular) is correct for flows in the dual-solution domain remains an issue. Computations of these interactions accurately predict the resulting flow as long as the correct initial conditions (either Mach or regular reflection) are employed. Outstanding questions remain regarding which configuration should be expected in real flows within the dual solution domain and what initial conditions should be employed to ensure prediction of the same shock structure observed in real flows. Experimental Data Experimental data for evaluation of the two-dimensional numerical simulations of crossing oblique shock interactions were collected by the Ivanov group in tunnel T313 at ITAM, Novosibirsk. A schematic of the experimental Mach # ɵD αD ɵN αN geometry is shown in . The wedges could be rotated symmetrically about 5 27.7 39.3 20.9 30.9 the trailing edges. The freestream Mach number was 5 and the Reynolds Table 4.2 Theoretical Detachment D and Neumann N number, based on the wedge Conditions Q the Flow Deflection Angle and a is the Shockcompression surface length, w, was 2 Wave Angle million. The wedge aspect ratio was b/w = 3.75, while the distance between the trailing edge of the wedge and the plane of symmetry was g/w = 0.42. The theoretical compression and shock angles for the Neumann and detachment conditions are listed in Table 4.2 where θ is the flow deflection angle and α is the resulting shockwave angle. (See also Figure 4.10). Case Study – 2D Oblique Shock Crossing Numerical Simulations Two-dimensional simulations for the geometry and flow conditions defined above were performed by the Ivanov Group at the Institute for Theoretical and Applied Mechanics (ITAM), Novosibirsk, Russia and [Schmisseur and Gaitonde] at the United Stated Air Force Research Laboratory (AFRL). Both groups used upwind-biased schemes to compute the flows and a common procedure for incorporating changes in the wedge angle. The converged solution at a previous wedge angle was used as the initial condition for the next solution with an incremental change in wedge angle. The computations of the Ivanov Group were carried out with a multi-block shock capturing Euler TVD code using MUSCL reconstruction of the HLLE (Harten-Lax-van Leer-Einfeldt) solver. Time integration was accomplished with a third-order Runge-Kutta scheme. In the computations of 41

S. Walker and J.D. Schmisseur, “CFD Validation of Shock-Shock Interaction Flow Fields”, RTO-TR-AVT-007-V3.

45

[Schmisseur and Gaitonde], two different methods are considered for the discretization of the inviscid components of the governing equations, the Roe scheme and the van Leer scheme. The upwind-biased MUSCL method is used for reconstruction. The viscous fluxes in the governing equations are evaluated with standard second-order central differences. The equations are integrated in time with an implicit approximately-factored scheme. Newton-like sub-iterations are incorporated to accelerate convergence.

Figure 4.10 Schematic of the Experimental Configuration used by the Ivanov Group and Sample Laser Light-Sheet Visualization from the same M = 4, α =37 degrees, b/w=3.75, g/w=0.3

Discussion Hysteresis in the shock-reflection configuration for solutions in the dual-solution domain was verified by computing a sequence of solutions with first gradually increasing and then gradually decreasing compression wedge angles. For a few of the solutions where the wedge angle was gradually increased from an (Regular Reflection) RR solution, uniform flow initial conditions were employed. In terms of the final solution attained, for the demonstration of hysteresis, this approach was similar to that in which the initial condition was a converged RR solution at a lesser wedge angle. (When starting from a uniform flow initial condition the developing shock system propagates away from the fin surface and reflects regularly in the initial transient). See the discussion in [Ivanov, et al.]42. All solutions for decreasing fin angles utilized the previous solution for the higher angle as an initial condition to obtain the new solution at the lower angle. The phenomenon of hysteresis is seen in Figure 4.11 which shows the shock structure computed by [Schmisseur & Gaitonde] for two sets of computations with g/w = 0.42 and g/w = 0.34. In the dual-solution domain, when the initial condition was a regular reflection or uniform flow the RR configuration persisted until the theoretical value of the detachment condition. As may be seen in Figure 4.11, for the cases with g/w = 0.42 the transition from RR to (Mach Reflection) MR occurred for a flow deflection angle between 27 and 28 degrees. For the cases with g/w = 0.34 smaller wedge angle increments were employed and the range for transition from RR to MR was narrowed to wedge angles between 27.5 and 27.85 degrees. This 42 Ivanov M.S., Markelov G.N., Kudryavtsev A.N., and Gimelshein S.F.,

transition in steady flows”, AIAA Journal, Vol. 36, No. 11, 1998.

“Numerical analysis of shock wave reflection

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range of values is in close agreement with the theoretical value for detachment, 27.7 degrees (see Table 4.2). For solutions with MR initial conditions the MR pattern persists through much of the dual solution space, but transitions to an RR configuration before the Neumann condition is reached. For complete analysis and discussion, please consult the paper by [Walker and Schmisseur]43.

Figure 4.11

Isobars Demonstrating Hysteresis in 2D Simulations of (Schmisseur and Gaitonde)

Effect of Heating Loads on Shock-Shock Interaction in Hypersonic Flows The heating rates generated by shock-shock interactions can result in some of the most severe heating loads imposed on the thermal protection systems of hypersonic lifting bodies and airbreathing propulsion systems, as investigated by [Walker and Schmisseur]44. In regions near a leading edge heating levels up to 30 times those encountered in an undisturbed stagnation flow can be generated. In these regions the strong gradients, unsteadiness and transitional nature of the flow combine to make accurate prediction of the flow field a challenging endeavor. The severe heating loads developed in a shock-shock interaction were first studied in detail following the X-15 scramjet program where shock-shock heating resulted in a structural failure in the pylon supporting the scramjet engine45. A series of studies were conducted in the late 1960s where the main focus was

43 S.

Walker* and J.D. Schmisseur, “CFD Validation of Shock-Shock Interaction Flow Fields”, RTO-TR-AVT-007-V3 See Previous. 45 Watts J.D., “Flight experience with shock impingement and interference heating on the X-15-2 research airplane”, NASA TM X-1669, 1968. 44

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the interaction between a shock wave and vertical fin. The studies by [Edney]46 of shock interactions on spherical configurations, coupled with his analysis of various interaction geometries that can be developed over cylinders and struts, provided the basic groundwork for the semi-empirical prediction of these flows. Another similar study conducted by [Chettle et al.]47 where hypersonic research engine model was attached to the underside of the aircraft, and the oblique shock generated by the wing of the X-15 interacted with the bow shock of the engine support pylon48. This resulted in catastrophic damage to the pylon and incineration of part of the protective skin. This damage was due to an increase in the peak heat transfer and pressure at the surface of the pylon, as a simple demonstration of an overall schlieren image in Figure 4.12. Understanding and controlling these interactions is a key part of moving forward in creating and maintaining a viable hypersonic program. A review of these earlier studies together with measurements of heat transfer and pressure distribution in regions of shockshock interaction over cylindrical leading edges in laminar, transitional and turbulent interaction regions were presented by [Holden et al.]49. Historically, the empirical modeling of these flows has been based on defining a small stagnation region downstream of a jet like flow such as that observed in the Type IV interaction. However, experiments have revealed that it is extremely difficult to distinguish between the heating loads generated by the various flow field elements such as the strong viscous effects or transitional Figure 4.12 Schematic of Edney Type IV Shocknature of the shear layer50. It has been Shock Interaction observed that the heating rates derived from an Edney IV shock-shock interaction vary widely depending on whether the flow is laminar, transitional or turbulent as well as whether the gas is considered perfect or real. In laminar flow, both Navie.r-Stokes and DSMC predictions have compared well with experiments if well-defined grid resolution studies are performed. When the shear layers or the boundary layers in the reattachment region become transitional a significant increase in the heating load results. Edney also demonstrated that the jet like model proposed for the Type IV interaction was highly sensitive to the specific heat ratio and the freestream Mach number through the sensitivity of the compression processes to these parameters. Specifically, he concluded that real gas effects could lower specific B. Edney, Anomalous heat transfer and pressure distributions on blunt bodies at hypersonic speeds in the presence of an impinging shock. Rep. 115, Flygtekniska Forsoksanstalten (The Aeronautical Research Institute of Sweden), Stockholm, 1968. 47 A. Chettle1, E. Erdem1, and K. Kontis, “Edney IV Interaction Studies at Mach 5”, Conference Paper · July 2013. 48 J. Watts, “Flight experience with shock impingement and interference heating on the X-15-2 research airplane”, NASA Technical Reports, NASA TM (X-1669), (1968). 49 Holden M.S., Moselle J.R., Lee J., Weiting A.R., and Glass C., “Studies of aerothermal loads generated in regions of shock-shock interaction in hypersonic flow”, NAS1-17721, 1991. 50 S. Walker and J.D. Schmisseur, “CFD Validation of Shock-Shock Interaction Flow Fields”, RTO-TR-AVT-007-V3. 46

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heat ratio, and result in significant increases in heating in these regions. Experimental studies of this have yielded conflicting results and computational techniques, both Navier-Stokes and DSMC, have shown 50% increases in peak heating rates over ideal gas heating predictions51. In summary, the Edney IV shock-shock interaction flow field is a complex shock-shock interaction relevant to high-speed airframe-propulsion system. Given the geometric simplicity of the shock generators and the numerical challenges associated with accurate simulation of the resulting flow field, it is an excellent test case for CFD validation studies for propelled vehicles in hypersonic flight. Case Study – University of Buffalo Research Center (CUBRC) Test Case at the Calspan-University of Buffalo Research Center (CUBRC) developed the second Edney IV shock-shock interaction test case. Flow conditions, experimental setup, and detailed measurements are reported in53. CUBRC conducted an extensive series of studies over a range of Mach numbers from 10 to 16 to define the aerothermal loads generated in regions of shock-shock interaction from the rarefied to the fully continuum turbulent flow regimes. Detailed heat transfer and pressure measurements were made in the 48-inch, 96-inch and LENS shock tunnels. The results of these studies were analyzed to provide guidance to predict the heating enhancement factors in laminar, transitional, and Figure 4.13 Schematic of the CUBRC Edney IV Interaction Generator turbulent flows. The (Courtesy of Holden) experimental data presented in this section are for fully laminar flows. The CUBRC model configuration is shown in Figure 4.13. The regions of shock-shock interaction studied were generated over a series of cylindrical leading edge configurations with nose radii of 0.351, 0.953, and 3.81 cm. Each of these leading edges was densely instrumented with heat transfer instrumentation placed to have a circumferential resolution less than 1 degree. The thin-film instrumentation was deposited on a low conductivity surface to minimize measurement errors associated with lateral conduction in the large heat transfer gradients generated in the region of peak heating. The high-frequency response of the thin-film instrumentation was also a key factor in accurately determining the heating distribution for shock-shock interactions, which exhibited intrinsic flow unsteadiness. The flow conditions for this study produced perfect gas, planar flow field, laminar flow shock-shock interactions. For the exact flow conditions of test runs 38, 43, 44, and 105, (please see Table 4.3). The experimental data includes surface temperature, heat transfer, pressure distributions and Schlieren photographs. [Holden]52

51 Carlsen A.B. and

Wilmoth R.G., “Monte Carlo simulation of a near-continuum shock-shock interaction problem”, AIAA 27th Thermophysics Conference, Nashville, TN, 1992. 52 Holden M.S.,” A review of the aerothermal characteristics of laminar, transitional, and turbulent shock-shock interaction regions in hypersonic flows”, AIAA 98-0899, 1998. 53 Holden M.S., “Database of aerothermal measurements in hypersonic flow for CFD validation”, CUBDAT Version 2.2 CDROM, Calspan-University of Buffalo Research Center, 1999.

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Table 4.3

CUBRC Test Conditions (V, H, SGL, R in inches)

Computational Contributions The three contributors for this test case were Dr Domenic D’Ambrosio, Politecnico di Torino, Torino, Italy, Dr Graham Candler, University of Minnesota, Minneapolis, MN, and Dr. Iain Boyd, University of Michigan, Ann Arbor, MI. D’Ambrosio’s numerical technique was the same as that described above for the ONERA test case. Results were obtained using a coarse (75 X 150) and a fine (250 X 300) grid. Candler’s computations were performed with a CFD code that uses second-order accurate modified Steger-Warming flux vector splitting and an implicit parallel time integration method. The grid consisted of 382 points in the circumferential direction, and 256 points normal to the cylinder surface. The grid was exponentially stretched from the surface, and care was taken to have sufficient near-wall resolution to capture the large flow gradients at the surface. Because of the relatively low enthalpy conditions of the experiments, chemical reactions were not considered, however ibrational relaxation of the gas was allowed. A vibrational equilibrium free-stream was assumed. Standard transport property models were used for the gas as in [Candler and Mac Cormack]. Candler computed post-shock conditions using the experimental free-stream conditions and the 10 degree turning angle of the shock generator. The post-shock conditions were then used as inflow conditions everywhere below a specified distance from the cylinder centerline. This distance was adjusted until the maxima in the heat transfer rate and surface pressure were located at the same point on the cylinder as in the experiments. This approach was required since very slight differences in the location of the shock generator relative to the cylinder result in large differences in the structure of the shock interaction. The DSMC solutions provided by Boyd were performed using the MONACO code: a general, objectoriented, cell-based, parallelized implementation of the DSMC method developed by Dietrich and [Boyd]54. MONACO employs the Variable Soft Sphere (VSS) collision model of [Koura et al]55, a variable rotational energy exchange probability model of [Boyd]56 and the variable vibrational energy exchange probability model of [Vijayakumar et al.]57 . The flow conditions here do not involve chemical reactions. Simulations of particle/wall interaction employ accommodation and momentum reflection coefficients of 0.85. The present simulations employ grids of 512 by 512 cells (Run 105) and 1024 by 1024 cells (Run 43), which give maximum sizes of 2 local mean free paths. The time step Dietrich, S. and Boyd, I. D., “Scalar optimized parallel implementation of the direct simulation Monte Carlo method”, Journal of Computational Physics, Vol. 126, pp. 328-342, 1996. 55 Koura, K. and Matsumoto, H., “Variable soft sphere molecular model for air species”, Physics of Fluids A, Vol. 4, pp. 1083-1085, 1992. 56 Boyd, I. D., “Analysis of rotational non-equilibrium in standing shock waves of Nitrogen”, AIAA Journal, Vol. 28, pp. 1997-1999, 1990. 57 Vijayakumar, P., Sun, Q. and Boyd, I. D., “Detailed models of vibrational-translational energy exchange for the direct simulation Monte Carlo method”, Physics of Fluids, Vol. 11, pp. 2117-2126, 1999. 54

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employed in both simulations is (10)-9 sec and this is less than the local mean time between collisions everywhere. The total numbers of particles employed is 2 million (Run 105) and 8 million (Run 43). This allows the minimum number of particles per cell to be everywhere greater than 4. Computational Results A comparison of the numerical results with the measured CUBRC Run 38 data of Holden is shown in Figure 4.14. Both surface pressure and surface non-dimensional heat transfer are plotted in angular coordinates around the cylinder. Inspection reveals that both the calculated pressure coefficient and heat transfer ratio are severely over-predicted. In a similar manner as described above, D’Ambrosio corrected this over-prediction by averaging his CFD results over the experimental measurement resolution. However, for the CUBRC experiments, this resolution was undetermined at the time of the computations. If similar averaging schemes were utilized, improved agreement between simulation and experiment resulting from the CFD averaging process would suggest that the experiment is highly unsteady and that the data are actually average quantities of an unsteady, oscillating impinging jet.

Figure 4.14

Experimental and Numerical Results for the Conditions of CUBRC Run #38

The sensitivity of the solutions to the shock impingement location may have an important effect on the interpretation of the experimental data. Slight variations in the free-stream conditions result in changes in the shock impingement location, which substantially change the surface quantities. Thus, it is possible that the experimental results represent some averaging of the shock impingement location. This would tend to broaden the peaks and reduce their magnitudes, as seen in the comparison between the computations and experiments. Another reason may be that because the cylinders are small in diameter, and the pressure instrumentation is limited, there are some cases where the actual peak pressure falls between two transducers and is not fully recorded. This may be the case, especially with Run 43, where there is almost no experimental peak pressure coefficient. (In Run 105, the cylinder was too small to incorporate any pressure sensors, so there is no experimental pressure data available for this case). The off-peak surface pressure coefficient beneath the interaction location is not well predicted by either Navier-Stokes or DSMC for Run 43, while the heat transfer ratio comparisons are much better. The good off peak heat transfer comparisons may be a function of the better resolution of heat transfer instrumentation.

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Also, both Navier-Stokes and DSMC methods required several milliseconds to reach steadystate solutions and for some shock impingement locations, the solutions never did stabilize. Run 43 is a particularly strong interaction, and the supersonic jet impinging on the surface is likely to be unstable. Of course, the capacity for the present CFD simulations to accurately capture this unsteadiness is suspect. The translational temperature contours for both Navier-Stokes and DSMC are shown in Figure 4.15 (a , b). Both methods predict similar flow fields. Overall, considering both peak and off-peak regions, Run 44 represents the best that DSMC and Navier-Stokes methods can offer. However, it is still apparent that there are many inconsistencies between the computational methods and the experimental data that need to be resolved.

Figure 4.15 Contours of Constant Translational Temperature for; a) Navier-Stokes and b) DSMC Solutions for CUBRC Run #44

Quasi -1D Correlation Applied to Variable Area Ducts Following the trend developed for 1D shock relations, and expanding on the idea that the area could change A = A(x). But the area variations are moderate and the components in y and z are small relative to x, enabling p = p(x), ρ = ρ(x), u = u(x), etc. Most supersonic wind tunnels could fall within such assumption. Where the momentum equation is seen previously as Euler’s equation holding along a stream line. Now we see that holds for quasi 1-D flow. Manipulating the continuity relation with some help from momentum, yield an important relation between velocity and area called the area-velocity relation as below

dA du  (M 2  1) A u

Eq. 4.8

Depending the character of coefficient (M2 -1), and assuming positive values u and A meant an increase in du or dA, following observations could be made, 1. For subsonic flows (M < 1), the coefficient in parentheses is negative. Hence an increase in velocity (positive du) is associated with a decrease in area (negative dA). Likewise, a decrease in velocity (negative du) is associated with an increase in area (positive dA). Clearly for a subsonic compressible flow, to increase the velocity, we must have a convergent duct and to decrease, the velocity, must have a divergent duct. Similar to incompressible flow. 2. For supersonic flows (M > 1), the coefficient is positive. Therefore, an increase in velocity (positive du) is associated with an increase in area (positive dA). Inversely, a decrease in velocity (negative du), cause a decrease in area (negative dA). For supersonic flows, to increase the velocity we must have a divergent duct, and decease the velocity must have a convergent duct. 3. For M = 1 or sonic flow, Eq. (3.35) shows that dA = 0 which corresponds mathematically to local max/min in area. Physically, it represents the minimum area (throat).

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These findings for converging-diverging duct, could best illustrated using the following Error! Reference source not found. and by introduction to concepts of nozzle and diffuser. Where a nozzle is designed to achieve supersonic flow at is exit, conversely a diffuser tries to bring the flow down to subsonic. A prime example is 1D supersonic diffuser is in variable duck flow, where oblique shock relations is given by Error! Reference source not found.. The relationship for oblique shock and comparison with theory is given be where incident Mach number (M∞ = 3) is plotted against the theory58 as depicted in Figure 4.16.

A Cross Sectional Area C Coherence a f Figure 4.16 Frequency Oblique Shock Relationshipe h Nozzle Height o NPR Nozzle Pressure Ratio = pres/pa res P Pressure rms R Normalized Correlation t S Spectrum 1 U Velocity 2 σ Standard Deviation 3

Subscripts Ambient Exit Total Reservoir Root Mean Square Throat lower wall transducer upper wall transducer dynamic Pitot probe

Case Study – Unsteady Phenomena in Supersonic Nozzle Flow Separation Table 4.4 Nomenclature for Unsteady Phenomena in Supersonic Nozzle Flow This work by Separation [Papamoschou & Johnson]59 considers the instability of the jet plume from an over expanded, shock containing convergentdivergent nozzle and attempts to correlate this instability to internal shock-induced separation phenomena. Time resolved wall pressure measurements and Pitot measurements are used as primary diagnostics. For the conditions of this study flow separation is asymmetric resulting in a large separation zone on one wall and a small separation zone on the other wall. Correlations of wall pressures indicate a low-frequency, piston-like shock motion without any resonant tones. Correlations of Pitot pressure with wall pressures indicate strong coherence of shear-layer instability with the shock motion. The likely source of the plume instability is the interaction of unsteady waves generated past the main separation shock with the shear layer of the large separation region. In order to facilitate further, a nomenclature is given in Table 4.4. Background and Literature Survey Supersonic flow separation in a convergent-divergent nozzle results in instability of the plume exiting the nozzle. This can be used to enhance mixing of the nozzle flow. Alternatively, the instability ©2012 Mentor Graphics Corporation. Dimitri Papamoschou and Andrew Johnson, “Unsteady Phenomena in Supersonic Nozzle Flow Separation”, AIAA 2006-3360, 36th AIAA Fluid Dynamics Conference and Exhibit, 5 - 8 June 2006, San Francisco, California. 58 59

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can be used as an excitation means to destabilize a flow adjacent to the nozzle. Potential applications include fuel injection, ejectors, and thermal signature reduction from jet engines. The instability phenomenon was initially observed in coannular jet experiments at the University of California, Irvine60, where an arbitrary primary jet surrounded by a secondary jet from a convergent-divergent nozzle showed substantial improvements in mixing compared to the case where the secondary nozzle was simply convergent. Figure 4.17 presents a visual example of such instability. This has been investigated in round and rectangular jets at NASA Glenn Research Center61. A typical result is that the length of the potential core is reduced by 50% and the velocity past the potential core decays at a much faster rate than for the equivalent jet without MESPI. For a nozzle with a given expansion ratio, the range of nozzle pressure ratios over which the instability occurs coincides with the range of nozzle pressure ratios for which a shock was located inside the nozzle. Therefore, the phenomenon of supersonic nozzle flow separation was deemed responsible for the observed instability. Numerous past studies have investigated supersonic nozzle flow separation 6263, but their focus was on the internal flow phenomena and not so much on the unstable plume that emerges from the separation shock. A related effort has focused on the phenomenon of transonic resonance in convergent-divergent nozzles 64. Transonic resonance appears to occur in relatively small nozzles where the boundary layer before the shock is laminar. For large nozzles with a turbulent boundary layer, such as those investigated here, there is no evidence of ringing phenomena.

Figure 4.17 Primary Jet Flow at Mach 0.9 Surrounded by an annular secondary flow at Nozzle Pressure Ratio NPR =1.7 (a) Secondary Nozzle is Convergent; (b) Secondary Nozzle is ConvergentDivergent – (Courtesy of 36) Papamoschou, D., “Mixing Enhancement Using Axial Flow,” AIAA Paper 2000-0093, Jan 2000. Zaman, K.B.M.Q, and Papamoschou, D., “Study of Mixing Enhancement Observed with a Co-Annular Nozzle Configuration,” AIAA Paper 2000-0094, Jan. 2000. 62 Morrisette, E.L., and Goldberg, T.J., “Turbulent Flow Separation Criteria for Over expanded Supersonic Nozzles," NASA TP 1207, Aug. 1978. 63 Romine, G.L., “Nozzle Flow Separation,” AIAA Journal, Vol. 36, No.9, 1998, pp. 1618-1625. 64 Zaman, K.B.M.Q., Dahl, M.D., Bencic, T.J., and Loh, C.Y., “Investigation of a Transonic Resonance with Convergent- Divergent Nozzles,” Journal of Fluid Mechanics, Vol. 263, 2002, pp. 313-343. 60 61

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To better understand the phenomenon of nozzle flow separation and its connection to flow instability, a fundamental experimental effort was started at UCI using a specially designed facility, to be described later in this report. shows a picture of nozzle flow separation obtained in this facility 65. As is evident from the photograph, the phenomenon is very complex and much more intricate than one would expect from quasi-one dimensional theory. The illustration of Figure 4.18 (Top) highlights some key features of the flow, but it is by no means complete. The shock in the viscous case takes on a bifurcated structure consisting of an incident shock and a reflected shock merging into a Mach stem. This is commonly referred to as a lambda foot, and the point at which the three components meet is called the triple point. The Mach stem is essentially a normal shock producing subsonic outflow. For the range of conditions of interest here, the incident and reflected shocks are of the “weak” type resulting in supersonic outflow past both. The adverse pressure gradient of the incident shock causes the boundary layer to separate and detach from the wall as a shear layer that bounds the separation (recirculation) region. Emerging from the triple point is a slipstream forming a sonic throat that acts to reaccelerate the subsonic region. The reflected portion of the main shock structure will then emerge from the separation shear layer as an expansion fan that is then transmitted through the slipstream toward the other separation shear layer where it is reflected again into compression waves, this pattern repeating with downstream distance. Therefore the separation “jet” that emerges from the shock contains a series of alternating compression and expansion waves. In nozzles with straight or convex walls subjected to nozzle pressure ratios above about 1.4, separation is asymmetric wherein one lambda foot is larger than the other (see for example Figure 4.18 (Bottom)). The asymmetry does not flip during an experiment but may change sides from one experiment to the next. A recent Figure 4.18 Schematic of Supersonic Nozzle Flow Separation (Top), computational effort by [Xiao vs. schlieren Image (Bottom) , (Courtesy of Papamoschou, D., Zill) – (Courtesy of 36) et al.]66 also predicted asymmetric separation. This Papamoschou, D., Zill, A., “Fundamental Investigation of Supersonic Nozzle Flow Separation,” AIAA Paper 2004-1111. 66 Xiao, Q., Tsai, H.M., and Papamoschou, D., “Numerical Investigation of Supersonic Nozzle Flow Separation,” AIAA Paper 2005-4640, June 2005. 65

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asymmetry has been recognized as a key factor for mixing enhancement. [Papamoschou and Zill]67 discovered large eddies forming in the shear layer of the large separation region, sometimes occupying over half the test section height. It was suspected that these eddies were due to the unsteady nature of the main shock. The objective of this paper is to investigate possible connections between the oscillation of the main shock and the occurrence of large-scale turbulent fluctuations downstream of the shock. Experimental Setup for Flow Facility The experiments used a facility designed specifically for studying flow separation in nozzles of various shapes as described by [Papamoschou and Zill]68. The nozzle apparatus consists of two flexible plates that can be shaped using two sets of actuators to form the upper and lower walls. One set of actuators controls the transverse force applied to the plates and the other controls the moment applied, allowing variations in nozzle area ratio, nozzle contour and exit angle. The nominal test section dimensions are 22.9 mm in height, 63.5 mm in width, and 117 mm in length from throat to exit. The sidewalls of the nozzle incorporate large optical windows for visualization of the entire internal flow, from the subsonic converging section to the nozzle exit. The apparatus is connected to a system of pressure-regulated air capable of nozzle pressure ratios as high as 3.5. The nozzle pressure ratio (NPR) ranged from 1.2 to 1.8 resulting in ideally-expanded velocities Ue ranging from 170 m/s to 320 m/s. The Reynolds number prior to the shock, based on axial distance from the throat, was typically 2.5×106. This indicates a fully-turbulent boundary layer. Results of Plume Pitot Pressure A parametric investigation of the jet plume versus nozzle shape and pressure ratio has shown significant increase in turbulence fluctuations levels as the exit-to-throat area ratio increases. The fluctuations are quantified in terms of the pressure p3 measured by the Dynamic Pitot Probe (DPP), which for the experiments discussed here equals the local total pressure p0. Figure 4.19 shows the distribution of p3,rms a short distance from the nozzle exit and the threefold increase in rms fluctuation levels as the nozzle area ratio changes from Ae/At =1 (straight) to Ae/At =1.6 (converging-diverging).

Figure 4.19 RMS Total Pressure Profile of Jet Plume at x/he = 0.5 for Straight Nozzle (Ae/At =1) and Convergent-Divergent Nozzle (Ae/At =1.6) – (Courtesy of 36)

Wall Pressure Statistics Before attempting correlations of the shock motion with plume fluctuations, it is helpful to understand the behavior of the oscillating shock as well as the nature of the unsteady flow in its vicinity. To measure the fluctuations in the entire neighborhood of the shock, the nozzle was held at a fixed area ratio of Ae/At =1.6 and the nozzle pressure ratio was gradually increased pushing the shock from upstream to downstream of the wall transducers. Figure 4.20 shows the variation of p1, rms with nozzle pressure ratio. The resulting curve shows the relative magnitude of the wall pressure fluctuations in the various regions around the shock. At higher NPR, corresponding to when the wall transducers are measuring the attached boundary layer 67 68

Papamoschou, D., Zill, A., “Fundamental Investigation of Supersonic Nozzle Flow Separation,” AIAA 2004. See Previous.

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upstream of the shock, the fluctuations are comparatively small in magnitude. At the nozzle pressure ratio where the shock begins to influence the pressure transducer, there is a steep increase in p1, rms as a consequence of the large pressure jump across the shock. At lower nozzle pressure ratios, where the transducers are located in the separated region, the fluctuations are larger than in the attached boundary layer but significantly smaller than when the shock is over the transducer. The spectrum of the fluctuations in the attached boundary layer is significantly lower in intensity than the spectra in the two other regimes. It is reasonable to assume that the attached Figure 4.20 RMS Wall Static Pressure boundary layer plays little or no role on the Fluctuation vs. Nozzle Pressure Ratio Presenting shock motion. The spectral levels in the Different Flow Regimes – Courtesy of 36 separated region are substantial and match those of the shock motion for fhe/Ue> 0.3. Therefore, the fluctuations in the separation zone are likely to have an effect on the shock motion. Correlations between Wall Pressure Ports The cross correlation and coherence of the two wall transducers illuminate some important characteristics of the unsteady phenomena in supersonic nozzle flow separation. Figure 4.21 show the cross correlation and coherence, respectively, of the two wall transducers situated upstream, downstream and at the location of the shock. As one might expect, in the attached boundary layer there is no correlation between the upper and lower wall since the fluctuations are a result of random turbulent eddies. There is a significant correlation when measuring the shock itself, implying that the shock oscillates in a “piston-like” manner. The coherence plot confirms the relatively low frequency of the shock motion. Correlations Between Wall Pressure Ports and Dynamic Pitot Probe Several experiments were conducted at area ratio Ae/At = 1.6, taking simultaneous measurements of the DPP and the wall mounted transducers. Initially the DPP was held at fixed Figure 4.21 Cross Correlations of Upper and positions in both the large and small separation Lower Wall Transducers for Various Flow Regimes zones and the nozzle pressure ratio was varied. – Courtesy of 36 Later the NPR was fixed at 1.6 and the DPP was translated along certain paths inside and outside the nozzle. For studying the effect of NPR on the coherence between wall pressures and DPP, we consider the case of the DPP being situated near the upper wall where the large separation zone occurs for NPR > 1.4. Figure 4.22 plots the nces for NPR =1.2 and 1.6. For NPR =1.2, separation occurs fairly symmetrically and the wall probes are in the separated region. There is no significant

57

coherence between the DPP and either of the wall probes. At NPR =1.6, the shock sits over the wall probes and separation is asymmetric. We observe significant coherence between the DPP and the wall probes, the coherence between the DPP and lower wall (small separation zone) exceeding the coherence between the DPP and upper wall (large separation zone) at low frequencies. The coherences drop when the NPR increases further, pushing the shock downstream and locating the wall probes in the attached region. This experiment suggests that the best correlations between wall probes and DPP occur when the shock sits in the vicinity of the wall probes. It also suggests that

NPR=1.2

Figure 4.22

NPR=1.6

Coherence Between Upper and Lower Walls (C12), DPP and Lower Wall (C13), and DPP and Upper Wall (C23) – Courtesy of 36

asymmetric separation may amplify those correlations, although this is still speculative. also plots the coherence between the two wall probes, which remains large as the shock moves upstream of the probes. Having established that the best correlations between wall probes and DPP occur for NPR = 1.6 (shock sits over wall ports), the next step was to conduct a search for the locations of DPP where the correlations were maximized. The search pattern is shown in Figure 4.23. For a rapid assessment of the trends of the correlations

Figure 4.23

Translation Paths of Dynamic Pitot Probe. Red points Indicate Measurement Locations – Courtesy of 36

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versus DPP position, see69. Close to the shock, the correlations peak when the DPP is in the large separation zone (upper wall). There is consistently better correlation of DPP with the lower wall probe that with the upper wall probe. As we exit the nozzle, the DPP remains significantly correlated with the wall probes, and this correlation becomes rather insensitive with the transverse position of the DPP. This is probably because the instability excites the entire plume so it does not matter where the DPP sits. Interestingly, the better correlation of the DPP with the lower wall probe persists even as the DPP moves outside the nozzle. This suggests that instability eddies are created through an interaction between the expansion reflected from the smaller lambda foot and the shear layer of the larger separation. To provide further details, please consult the work by [Papamoschou & Johnson]70. Concluding Remarks An investigation has been conducted into the source of plume instability from over expanded convergent divergent nozzles. The effect of internal shock phenomena on the plume unsteadiness was a particular focus. Time resolved measurements of wall static pressures and total pressure in the plume were correlated. A summary of the key findings is as follows:  For nozzle pressure ratios that give rise to shock formation inside the nozzle, increasing the nozzle area ratio from 1 (straight nozzle) to 1.6 (convergent-divergent nozzle) results in a three-fold increase in the rms total pressure fluctuations near the nozzle exit. Spectra indicate that most of the instability energy is contained at low to moderate frequencies.  For the conditions of this study, the separation shock is asymmetric. This gives rise to a large separation region on one wall and a small separation region on the other wall.  The coherence and cross correlation of pressures measured on the upper and lower nozzle walls indicate that the shock oscillates in a piston-like manner with no noticeable rotational motion. The oscillation is a low-frequency phenomenon without any resonant tones.  There are substantial correlations between the wall pressures caused by the shock motion and the total pressure inside the large separation zone. The frequency content of the total pressure fluctuation is similar to that of the shock motion.  There is consistently better coherence between the total pressure in the large separation zone and the pressure on the wall opposite that zone. This suggests that the instability mechanism is due to an interaction between the expansion fan reflected from the smaller lambda foot with the shear layer of the larger separation zone.

Study of Aerothermal Loads in the Presence of Edney Type IV Interaction The simulation of supersonic flow over aircraft is usually focused on shockwave drag and aerothermal loads estimation71. For hypersonic vehicles, like space shuttle, where aerodynamic heating becomes more prominent, the accurate prediction of the latter is essential in order to design an efficient Thermal Protection System (TPS). If the analyzed geometry represents real prototype of the vehicle (rather than some simplified model) one can expect a highly complex flow structure that includes multiple shockwaves, expansion fans and contact discontinuities. Under these conditions the interaction between the shocks and boundary layer flow may lead to localized heat fluxes that are several times higher than the ones at the stagnation point. This article gives an account of the FloEFD© simulations, where such effects are known to occur. The investigation of shock/shock interaction in a 2D supersonic flow around a cylinder with an impinging shock generated by a wedge Dimitri Papamoschou and Andrew Johnson, “Unsteady Phenomena in Supersonic Nozzle Flow Separation”, AIAA 2006-3360, 36th AIAA Fluid Dynamics Conference and Exhibit, 5 - 8 June 2006, San Francisco, California. 70 Dimitri Papamoschou and Andrew Johnson, “Unsteady Phenomena in Supersonic Nozzle Flow Separation”, AIAA 2006-3360, 36th AIAA Fluid Dynamics Conference and Exhibit, 5 - 8 June 2006, San Francisco, California. 71 Dr. Leonid Gurov, Dr. Andrey Ivanov, Mentor, a Siemens Business. 69

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gives a good understanding of the processes affecting the surface heat transfer. Depending on the relative coordinates of the point where the wedge shock crosses the cylinder bow shock, six types of interaction are possible (Figure 4.24). Two cases that draw most attention are known as “Type III” and “Type IV” interactions, where the oblique shock crosses the normal shock. Such interactions should be avoided as they lead to the most significant increase of heat flux. A mixing layer impinges on the body surface in the first case, while the second case is notable for the formation of a small-scale supersonic jet that penetrates a region of low subsonic flow. The remaining four “types” have a minor effect. To demonstrate how the peak values of pressure and heat flux vary with the change of shock interaction type and freestream conditions, a set of experimental measurements is available. Some of these experiments were initiated after testing early layout of the Space Shuttle system (for its main elements, where extremely high heat fluxes were detected on the Orbiter Figure 4.24 Six types of shock/shock interaction as classified by nose. Obviously, External Tank Edney played the part of the wedge here and resulted in Type III/IV interaction near the nose. Ever since the 2D shock/shock interaction problem has become a wellknown The second part is focused on the study of Type IV interaction near the nose of the actual Space Shuttle Orbiter. It should be noted that the “Final” layout of this system minimizes the probability of such an incident. To be more specific, when the velocity of vehicle is about Mach 5 to 10 Figure 4.25 Dependencies of Space Shuttle Flight Altitude on and the pitch angle is close to Velocity zero, the oblique shock does

60

not cross the normal shock near the Orbiter nose, so one can expect a Type V/VI interaction at most. Further acceleration (up to M = 25) occurs at the altitude of approximately 100km (see Figure 4.25), where the mean free path of a particle is of the same scale ( ≈ 1 m) as the actual vehicle size, obviously such cases cannot be handled by Navier-Stokes equations. Moreover, the External Tank is discarded at the altitude of 113 km. As a compromise a special case is considered, where the pitch angle is assumed to be well below zero and the flight conditions correspond to the altitude of 65km (solid boosters are discarded at the altitude of 45km)72. Case Study 1 - Shock/Shock Interaction in a 2D Flow Around Circular Cylinder This validation case demonstrates FloEFD© capabilities to predict surface heat flux in the presence of shock/shock interaction. The input data for the calculation was taken from73 and corresponded to the experimental run #20. A uniform square mesh was used to discretize computational domain with the characteristic cell size being 1/150 of cylinder diameter. To attain stable shock structure, “timedependent” option was enabled. Let us study the flow near the cylinder in more detail by plotting pressure and Mach number iso-lines (Figure 4.26). According to these plots, the size of the produced jet in a transverse direction is about 1/20 of cylinder diameter. From the top and bottom sides it is ‘bounded’ by a contact discontinuity. Before reaching the cylinder jet passes through a series of oblique shocks and a terminal normal shock. The resulting flow splits into two halves; the one in the upper direction accelerates and becomes supersonic, while the flow in the lower direction remains subsonic. Figure 4.27-a shows the comparison between the calculated and measured pressure distributions around the cylinder. One can see a good agreement with experiment in terms of peak pressure relative coordinate, although the actual value is a bit over-predicted. The corresponding distribution of heat flux (Figure 4.27 - b) shows good agreement with experiment in terms of both, peak value and its relative coordinate. As a matter of fact, the obtained peak value (400 BTU/ ft2·s) is almost 13 times larger comparing to the estimated value at the stagnation point (30 BTU/ft2·s) in case of the symmetry flow around cylinder (no shock interaction).

Figure 4.26

Iso-lines Near Cylinder (a) Pressure (b) Mach Number

Dr. Leonid Gurov, Dr. Andrey Ivanov, Mentor, a Siemens Business. Holden, M. S. et. al., "Studies Of Aerothermal Loads Generated in Regions Of Shock/Shock Interaction In Hypersonic Flow", AIAA paper 88-0477, Jan 1988. 72 73

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The increase of heat flux at the angular position of -30˚ is due to laminar-turbulent transition in the boundary layer. Although minor, this effect was captured in the calculation. Similar transition is observed at the angular position of 25˚. One can notice that the predicted laminar-turbulent transition is shifted 5˚ towards the stagnation point. Such error can be considered insignificant74.

(a) Preesure

Figure 4.27

(b) Heat Flux

Comparison with Experimental Values on Cylinder Surface

Case Study 2 - Flow Over Space Shuttle Orbiter with External Tank While the preceding case was mainly focused on the quantitative flow analysis keeping the CAD model as simple as possible, this case is notable for the complex geometry analyzed. To perform the simulation it was convenient to use a CAD model of the Space Shuttle available online. In order to attain the angle of the oblique shock generated by the External Tank so that it could cross the normal shock near the Orbiter nose and, thus, result in type IV interaction the external flow Mach number was set to 5.6, while the pitch angle was adjusted to -23˚. The freestream conditions corresponded to the altitude of 65 km (p∞=9.922 Pa; T∞=231.45 K).

Figure 4.28

74

Mesh Generated after Solution-Adaptive Refinement

Dr. Leonid Gurov, Dr. Andrey Ivanov, Mentor, a Siemens Business.

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While preparing the geometry, specifying mesh settings and boundary conditions several assumptions were made that helped to reduce the overall CPU time for analysis. These had only minor effect on the accuracy of the solution in the regions of interest. A uniform half-symmetry mesh was used, where the characteristic cells size in the basic mesh was about 1/5 of the External Tank diameter. To refine mesh in the regions of interest (impinging shock, Orbiter bow shock, etc.), Solution-Adaptive Refinement (SAR) was used. The resulting mesh obtained after running SAR seven times comprised of about five million cells (Figure 4.28). The calculation was stopped after obtaining the converged values of the following surface goals Figure 4.30 Iso-lines of (a)Pressure and (b) Mach number Near the specified on the Orbiter nose Orbiter Nose min/max pressure, average surface heat flux. The pressure contours plotted over the whole Orbiter surface (Figure 4.29) give a good idea of the size of the region, where pressure increase caused by Type IV shock/ shock interaction is prominent. Comparing to the results obtained in the flow analysis around the single Orbiter , the presence of Type IV interaction leads to the increase of peak pressure and heat flux values by 5.5 times (Figure 4.31). A larger quantitative difference was observed in the preceding case. That was partly due to the larger Mach number of the external flow. Figure 4.29 Surface Pressure and Pressure iso-lines in the Concluding Remarks Symmetry Plane The effect of heat transfer rate increase caused by Edney type IV shock/shock interaction has been investigated. A well-known benchmark case was used for the initial test that showed good agreement with experimental data in terms of flow structure, surface pressure and heat flux distributions. The results obtained in Space Shuttle revealed a small-scale region on the Orbiter nose with a moderate increase of surface heat transfer rate caused by type IV interaction75.

75

Dr. Leonid Gurov, Dr. Andrey Ivanov, Mentor, a Siemens Business.

63

Figure 4.31

Distributions of (a) Pressure and (b) Heat Flux Along the Nose in the Symmetry Plane

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5 Viscous Flow Qualitative Aspects of Viscous Flow Viscous flow could be defined as a flow where the effects of viscous dissipation, thermal conductivity, and mass diffusion are important and could not be ignored76. All are consequence of assuming a viscous surface where the effects of friction, creating shear stress, on the surface are pronounced. There are number of interesting and important conditions associated with viscous effect that should be analyzed separately. In general, two regions to consider, even the divisions between not very sharp: 1. A very thin layer in the intermediate neighborhood of the body, δ, in which the velocity gradient normal to the wall, ∂u/∂y, is very large (Boundary Layer). In this region the very small viscosity of μ of the fluid exerts an essential influence in so far as the shearing stress τ = μ (∂u/∂y) may assume large value. 2. In the remaining region no such a large velocity gradient occurs and the influence of viscosity is unimportant. In this region the flow is frictionless and potential.

Figure 5.1

Boundary Layer Flow along a Wall

The general form on boundary layer equations, shown in Figure 5.1, and their characteristic will be discussed later. No-Slip Wall Condition Due to influence of friction, the velocity approaches zero on the surface and this is dominant factor in viscous flows which could easily be observed. Or more precisely

V fluid  V solid

and

Flow Separation Another contribution due to friction and shear stress is the effects of flow separation or adverse pressure gradient. Figure 5.2 displays a flow separation on the wing section at high angle of attack. Assuming that flow over a surface is produced by a pressure gradient where P3 > P2 > P1 along a surface as depicted in Figure 5.3. Following elements downstream, where the motion of elements is already retarded by friction. In addition, it must work its way along the flow against an increasing pressure, which tends to further 76

T fluid  T solid

Figure 5.2

White, Frank M. 1974: Viscous Fluid Flow, McGraw-Hill Inc.

Eq. 5.1

Airflow Separating from a Wing at a High Angle of Attack

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reduce its velocity. Consequently, at station 2, the velocity V2 is less than V1. As fluid elements continue to move down-stream, it may run out of stream and come to stop. And then under the action of the adverse pressure gradient, actually reverse its direction and start moving back upstream. The flow is now separated from the surface and creates a large wake of recalculating flow down-stream. It point of separation for a 2D flow is defined as

 u  Point of Seperation    0  y  y wall

Figure 5.3

Eq. 5.2

Detached Flow induced by adverse pressure gradient

Pressure Drag vs Skin Friction Drag As flow separates from the body down-stream, the pressure distribution over the body is greatly altered. In essence, the primary flow no longer sees the effective body, but rather the effective body up to separation point, and the deformed, separated region77. The pressure acting on the surface would be lower due to inverse (opposite) pressure. It could be visualized as if the pressure on the separated region has a tangential components acting opposite to drag direction. For viscous separated flows, p is reduced; hence, it could no longer fully cancel the pressure distribution over the reminder of body (d’Alembert paradox). Therefore, the net result is induction of a drag called pressure drag (Dp), beside the regular skin friction drag (Df) by shear stress. The occurrence of separated flow not only increases the drag but also results in substantial loss of lift. Therefore, it should be avoided on lifting surfaces, if possible. In summary, we see that the effects of viscosity are to produce two types of drag as follows:  Df is the skin friction drag, that is, the component in the drag direction of the integral of the shear stress τ over the body.  Dp is the pressure drag due to separation, that is, the component in the drag direction of the integral of the pressure distribution over the body. 77

White, Frank M. 1974: Viscous Fluid Flow, McGraw-Hill Inc.

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Dp is sometimes called form drag. The sum Df +Dp is called the profile drag of a two-dimensional body. For a three-dimensional body such as a complete airplane, the sum Df + Dp is frequently called parasite drag. Laminar vs Turbulent Flows Consider the viscous flow over a surface where path lines of various fluid elements are smooth and regular, as it called laminar flow. In contrast, if the motion of a fluid element is very irregular and tortuous, the flow is called turbulent flow. Because of the agitated motion in a turbulent flow, the higher-energy fluid elements from the outer regions of the flow are pumped close to the surface. Hence, the average flow velocity near a solid surface is larger for a turbulent flow in comparison with laminar flow. This comparison is shown in Figure 5.4, which gives velocity profiles for laminar and turbulent flow. Note that immediately above the surface, the turbulent flow velocities are much larger than the laminar values. If (∂V/∂n)n=0 denotes the velocity gradient at the surface, we have:

[(

Figure 5.4 Schematic of Velocity Profiles for Laminar vs Turbulent Flows

𝜕𝑉 ∂V > [( ) ] ) ] 𝜕𝑛 𝑛=0 Turbulent ∂n n=0 Laminar

Eq. 5.3 Because of this difference, the frictional effects are more severe for a turbulent flow; both the shear stress and aerodynamic heating are larger for the turbulent flow in comparison with laminar flow. However, turbulent flow has a major redeeming value; because the energy of the fluid elements close to the surface is larger in a turbulent flow, a turbulent flow does not separate from the surface as readily as a laminar flow. If the flow over a body is turbulent, it is less likely to separate from the body surface, and if flow separation does occur, the separated region will be smaller. As a result, the pressure drag due to flow separation Dp will be smaller for turbulent flow. This discussion points out one of the great compromises in aerodynamics. For the flow over a body, is laminar or turbulent flow preferable? There is no pat answer; it depends on the shape of the body. In general, if the body is slender, as sketched in Figure 5.5a, the friction drag Df is much greater than Dp. For this case, because Df is smaller for laminar than for turbulent flow, laminar flow is desirable for slender bodies. In contrast, if the body is blunt, as sketched in Figure 5.5b, Dp is much greater than Df . For this case, because Dp is smaller for turbulent than for Figure 5.5

Drag on Slender & Blunt Bodies

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laminar flow, turbulent flow is desirable for blunt bodies. The above comments are not all-inclusive; they simply state general trends, and for any given body, the aerodynamic virtues of laminar versus turbulent flow must always be assessed. Although, from the above discussion, laminar flow is preferable for some cases, and turbulent flow for other cases, in reality we have little control over what actually happens. Nature makes the ultimate decision as to whether a flow when left to itself, will always move toward its state of maximum disorder. To bring order to the system, we generally have to exert some work on the system or expend energy in some manner. (This analogy can be carried over to daily life; a room will soon become cluttered and disordered unless we exert some effort to keep it clean.) Since turbulent flow is much more “disordered” than laminar flow, nature will always favor the occurrence of turbulent flow. Indeed, in the vast majority of practical aerodynamic problems, turbulent flow is usually present78. Skin Friction When the boundary layer equations are integrated, the velocity distribution can be deduced, and point of separation can be determined. This in turn, permits us to calculate the viscous drag (skin friction) around a surface by a simple process of integrating the shearing stress at the wall and viscous drag for a 2D flow becomes:

 u  τ w  μ    y  y  0

L

Df  b  τ w cos φ ds

Eq. 5.4

so

Where b denotes the height of cylindrical body, φ is the angle between tangent to the surface and the free-stream velocity U∞, and s is the coordinate measured along the surface, as shown in Figure 5.6. The dimensionless friction coefficient, Cf, is commonly referred to the free-stream dynamic pressure as:

Cf  Eq. 5.5

2τ w ρU 2

Figure 5.6

Illustrating the calculation of Skin Friction

Aerodynamic Heating Another overall physical aspect of viscous flow is the influence of thermal conduction. On a fluid over a surface, the moving fluid elements have certain amount of kinetic energy. As the flow velocity decreases under influence of friction, the kinetic energy decreases79. This lost kinetic energy reappears in the form on internal energy of the fluid, hence, causing temperature to rise. This phenomenon is called viscous dissipation within the fluid. This temperature gradient between fluid and surface would cause the transfer of heat from fluid to surface. This is called Aerodynamic Heating of a body. Aerodynamic heating becomes more severe as the flow velocity increase, because more kinetic energy is dissipated by friction, and hence, the temperature gradient increases. In fact it is one of the dominant aspects of hypersonic flows. The block diagram of Figure 5.7, summarizes these finding for viscous flow. John D. Anderson, Jr., “Fundamentals of Aerodynamics”, 5th Edition, McGraw-Hill Companies, ISBN 978-0-07339810-5, 2007. 79 Anderson, John D. 1984: “Fundamentals of Aerodynamics”, McGraw Hills Inc. 78

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Figure 5.7

Quantitate Aspects of Viscous Flow

Reynolds Number The Reynolds number is a measure of ratio of inertia forces to viscous forces,

Re 

ρUL UL  μ ν

Eq. 5.6

0.05

Reynolds Number (Re) 10.0 200.0 3000.0

Where U and L are local velocity and characteristic length. This is a very important scaling tool for fluid flow equations as to be seen later. Additionally, it could be represents using dynamic viscosity ν = μ/ρ. This is a really is measure or scaling of inertia vs viscous forces as shown in Figure 5.8 and has great importance in Fluid Mechanics. It can be used to evaluate whether viscous or Figure 5.8 Effects of Reynolds Number in Inertia vs Viscosity inviscid equations are appropriate to the problem. The Reynolds Number is also valuable tool and guide to the in a particular flow situation, and for the scaling of similar but differentsized flow situations, such as between an aircraft model in a wind tunnel and the full size version80.

80

From Wikipedia, the free encyclopedia.

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Reynolds Number Effects in Reduced Model The kinematic similarity between full scale and scaled tests has to be maintained for reduced model testing (wind-tunnels). In order to maintain this kinematic similarity, all forces determining a flow field must be the same for both cases. For incompressible flow, only the forces from inertia and friction need to be considered (i.e., Reynold Number). Two flow fields are kinematically similar if the following condition is met

U1L1 U 2 L 2  ν1 ν2

Eq. 5.7

To recognize Reynolds number effects a dependency test should be done81. Results from such a dependency study are presented in Figure 5.9. At high Reynolds numbers, the drag coefficient is almost constant, and the values for the full scale vehicle are slightly lower than those for the scaled model. Below a certain Reynolds number, however, the drag coefficient from the scaled test noticeably deviates from the full scale results. That is due to the fact, that in this range, individual components of the car go through their critical Reynolds number. Violating Reynolds’ law of similarity can cause considerable error. On the other hand, for small scales, sometimes it is hard to maintain the same Reynolds number. That is for two main reasons. Wind tunnels have limited top speed. At the same time, increasing speed in model testing also has its limits in another perspective.

Figure 5.9

Drag Coefficient versus Reynolds Number for a 1:5 Model and a Car (Courtesy of 35)

Case Study 1 - Scaling and Skin Friction Estimation in Flight using Reynold Number Now that we familiar ourselves with some concepts if viscous flow, such as Reynolds Number, separation, boundary layer and skin friction, it is time to see their effects in real life situation. The purpose here is to conduct a brief review of skin-friction estimation over a range of Reynolds numbers, as this is one of the key parameters in performance estimation and Reynolds number scaling. These are among the most important in Aerodynamic performance. The flow around modern aircraft can be highly sensitive to Reynolds number and its effects when they move significantly the 81

Bc. Lukáš Fryšták, “Formula SAE Aerodynamic Optimization”, Master's Thesis, BRNO 2016.

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design of an aircraft as derived from sub-scale wind tunnel testing as investigated by [Crook ]82. For a transport aircraft, the wing is the component most sensitive to Reynolds number change. Figure 5.10 shows the flow typically responsible for such sensitivity, which includes boundary layer transition, shock/boundary layer interaction and trailing-edge boundary layer. Interaction Between Shock Wave and Boundary Layer The nature of the interaction between a shock wave and an attached boundary layer depends largely upon whether the boundary layer is laminar or turbulent at the foot of the shock. For a laminar boundary layer, separation of the boundary layer will occur for a relatively weak shock and upstream of the freestream position of the shock. The majority of the pressure rise in this type of shock /boundary layer interaction, generally described as a ¸ shock, occurs in the rear leg. The interaction of the rear leg with the separated boundary layer causes a fan of expansion waves that tend to turn the flow toward the wall, and hence re-attach the separated boundary layer. This is in contrast to the interaction between a turbulent boundary layer and a shock wave, in which the majority of the pressure rise occurs in the front leg of the shock wave. The expansion fan that causes reattachment of the laminar separated boundary layer is therefore not present, and the turbulent boundary layer has little tendency to re-attach. Here lies the problem of predicting the flight performance of an aircraft when the methods used to design the aircraft have historically relied upon wind tunnels operating below flight Reynolds number, together with other tools such as (CFD), empirical and semiempirical methods and previous experience of similar design aircraft. Industrial wind tunnels can only achieve a maximum chord Reynolds number of between 3 x 106 < Rec >1) where there is the thin layer of flow adjacent to a surface where the flow is retarded by influence of friction between a solid surface and fluid. (see Figure 6.8). Within this thin layer, known as Boundary Layer thickness, δ, the flow variables are influenced mostly on the normal direction to solid surface (ie., v 0. Figure 8.6

ξ ξ ξ a  υ 2 t x x the exact solution is : 2

(4.6)

Solution to Poisson's equation

u ( x, y ) 

Sin ( x ) Sin ( y )  2 2

u(x, t)  exp( kυυxt)sin(x  at) k  constant and u(x)  sin(kx) u  [0,1] Eq. 8.8

The Korteweg-De Vries Equation The motion of nonlinear dispersive wave is governed by this example.

u u  3 u u  3 0 t x x

Eq. 8.9

Helmholtz Equation This equation governs the motion of time dependent harmonic waves where k is a frequency parameter. Application includes the propagation of acoustics waves.

 2u 2u 2  k u 0 x 2 y 2

Eq. 8.10

Exact Solution Methods The solution is obtained from the list provided below. This list by no means exclusive and many more exists in literature. 1. Method of Characteristics Hazewinkel, Michiel, ed. (2001), "Laplace equation", Encyclopedia of Mathematics, Springer, ISBN 978-155608-010-4. 188 Example initial-boundary value problems using Laplace's equation from exampleproblems.com. 187

141

2. Shock Capturing Methods 3. Similarity Solutions 4. SCM (Split Coefficient Method) 5. Methods for solving Potential Equation 6. Methods for solving Laplace equation 7. Separation of Variable 8. Complex Variables 9. Superposition of Non-Linear Equation 10. Transformation of Variables 11. Manufacturing Solutions

Solution Methods for In-Viscid (Euler) Equations

The interest in Euler equations arises from the fact that in many primary design the information about the pressure alone is needed. In boundary layer where the skin friction and heat transfer is required, the outer edge condition using the Euler. The Euler equation is also of interest because of interest in major flow internal discontinuities such as shock wave or contact surfaces. Solutions relating to Rankine-Hugonist equations are embedded in Euler equation. The Euler equations govern the motion of an Inviscid, non-heat-conducting flow have different character in different regions. If the flow is time-dependent, the flow regimes is hyperbolic for all the Mach numbers and solution can be obtained using marching Subsonic Supersonic procedures. The situation is Flow Sonic M=1 M1 very different when a steady Steady Elliptic Parabolic Hyperbolic flow is assumed. In this case, Unsteady Hyperbolic Hyperbolic Hyperbolic Euler equations are elliptic when the flow is subsonic, and Table 8.1 Classification of the Euler equation on different regimes hyperbolic when the flow is supersonic. For transonic flows, has required research and development for many years. Table 8.1 shows the deferent flow regimes and corresponding mathematical character of the equations. Method of Characteristics Closed form solutions of non-linear hyperbolic partial differential equation do not exists for general cases. In order to obtain the solution to such an equations we are required to resort to numerical methods. The method of characteristics is the oldest and most nearly exact method in use to solve hyperbolic PDEs. Even though this technique is been replaced by newer finite difference method. A background in characteristic theory and its application is essential. The method of a characteristics is a technique which utilizes the known physical behavior of the solution in each point in the flow. Linear Systems Consider Steady Supersonic of Inviscid, Non-heat conducting of small perturbation for 2D perfect gas189.

D. Anderson, J., Tannehill, R., Pletcher,”Computational Fluid Mechanics and Heat Transfer”, ISBN 0-89116471-5 – 1984. 189

142

(1  M  ) xx   yy  0 β2

  ,v x y w w write in vector form  [ A] 0 x y

denoting (1  M  )  β 2 and u 

u v v u  0 ,  0 x y x y

Eq. 8.11

1  u  0  2  where w    and [ A]  β   v  1 0  The eigenvalues of this system are the eigenvalues of [A]. These are obtained by extracting the roots of characteristics equation of [A] as

[ A]  λ[I]  0 or

1 λ2 

1 β2  0 , λ

λ 

Eq. 8.12

1 1 1  0 , λ1  , λ2   2 β β β

This is pair of roots from the differential equation of characteristics. Next we determine the compatibility equation. These equations are obtained by pre-multiplying the system of equations by left eigenvectors of [A]. This effectively provides a method for writing the equations along the characteristics. Let L1 represents the left eigenvectors of [A] corresponding to λ1 and L2 represents the left eigenvectors corresponding to λ2. Drive the eigenvectors of [A]:

Figure 8.7

Characteristics of Linear equation

L  A  λ I  0 i T

i

1  1   2  l  β β L1   1   l1 , l 2   0    1 l 2  1  LT  β       A

The compatibility equations along λ1 is obtained from

 β  β  L1    , L2    1 1

Eq. 8.13

143

L  w i

T

x



  w

 [A]w y  0 or Li

T

x



 λi w y  0

(7.4)

1   u  uy x  β  compatabilty along λ 1 is obtained [-β 1]  0 v x  1 v y   β   βu  v   1  βu  v   0 in similar manar  βu  v   1  βu  v  x β y x β y

Eq. 8.14

It is expressed the fact that quantity (βu-v) is constant along λ1, and (βu+v) is constant along λ2. The quantities are called Riemann Invariants. Since these two quantities are constant and opposite pair of characteristics, it is easy to determine u and v at a point. If at a point we know (βu-v) and (βu+v), we can immediately compute both u and v. Non-Linear Systems The development presented so far is for a system linear equations for simplicity. In more complex nonlinear settings, the results are not as easily obtained. In the general case, the characteristics slopes are not constant and vary with fluid properties190. For a general nonlinear problem, the characteristics equation must be integrated numerically to obtain a complete flow field solutions. Consider a 2D supersonics flow of a perfect gas over a flat surface. The Euler equation governing this inviscid flow as a matrix form

w w  [ A] 0 x y u  v where w    p   e

and

(7.5)

  uv  1  0 [A]  2  u  a2  2  ρva   ρv 

 a2



v 2 u  a2 u ρua 2

v p 2 u  a2 ρu uv v u 



ρu

    0   0  v 2 2  u a  u 0





Eq. 8.15

The eigenvalues of [A] determine the characteristics direction and are191

v v uv  a u 2  v 2  a 2 uv  a u 2  v 2  a 2 λ1  , λ 2  , λ 3  , λ4  u u u2  a2 u2  a2

Eq. 8.16

The matrix of left eigenvectors associated with these values of λ may be written as See previous. D. Anderson, J., Tannehill, R., Pletcher,”Computational Fluid Mechanics and Heat Transfer”, ISBN 0-89116471-5 – 1984. 190 191

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ρu   a2  ρu  1 1 [ T]     u 2  v2  a 2  1  2 2 2  u  v  a

ρv a2 ρv u 1 v u 2  v2  a 2 u 1 v v u 2  v2  a 2 

0 1 1 ρva 1 ρva

 1 0  0   0 

Eq. 8.17

We obtain the compatibility relations by pre-multiplying the original system by [T]-1. These relations along the wave fronts are given by:

v

du dv β dp dy u   0 along  λ3 ds 3 ds 3 ρ ds 3 dx

du dv β dp dy v u   0 along  λ4 ds 4 ds 4 ρ ds 4 dx

Eq. 8.18

These are an ordinary differential equations which holds along the characteristic with slope λ 3, λ4, while arc length along this characteristics is denoted by s3, s4. In contrast to linear example, the analytical solution for characteristics is not known for the general nonlinear problem. It is clear that we must numerically integrate to determine the shape of the characteristics in step by step manner. Consider the characteristic defined by λ3. Stating at an initial data surface, the expression can be integrated to obtain the coordinates of next point at the curve. At the same time, the differentials equation defining the other wave front characteristics can be integrated. For a simple first-order integration this provide us with two equations for wave front characteristics. From this expressions, we determine the coordinate of their intersection, point A. Once the point A is known, the compatibility relations, (8.13), are integrated along the characteristics to this point. This Figure 8.8 Characteristics of nonlinear solution provide a system of equations at point A. This is a first-order estimate of the both the location of point point A and the associated flow variables. In the next step, the new intersection point B can be calculated which now includes the nonlinear nature of the characteristic curve. In a similar manner, the dependent variables at point B are computed. Since the problem is nonlinear, the final intersection point B does not necessary appear at the same value of x for all solution points. Consequently, the solution is usually interpolated onto an x=constant surface before the next integration step. This requires additional logic and added considerably to the difficulty in turning an accurate solution192 D. Anderson, J., Tannehill, R., Pletcher,”Computational Fluid Mechanics and Heat Transfer”, ISBN 0-89116471-5 – 1984. 192

145

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9 Boundary Conditions Naming Convention for Different Types of Boundaries

Boundary conditions and their correct implementation are among the most critical aspects of a correct CFD simulation193. Mathematically, there are four types of Dirichlet, Von Neumann, Mixed, Robin, Cauchy, and Periodic. Dirichlet Boundary Condition Direct specification of the variable value at the boundary. E.g. setting the distribution of a racer ϕi at a west boundary to zero: ϕw = 0. Von Neumann Boundary Condition Specification of the (normal) gradient of the variable at the boundary. E.g., setting a zero gradient ∂ϕ i /∂n=0 at a symmetry boundary. Mixed or Combination of Dirichlet and von Neumann Boundary Condition Direct specification of the variable value as well as its gradient. It is required to satisfy different boundary conditions on disjoint parts of the boundary of the domain where the condition is stated. (see Figure 9.1).

Figure 9.1 Mixed Boundary Conditions

Robin Boundary Condition It is similar to Mixed conditions except that a specification is a linear combination of the values of a function and the values of its derivative on the boundary of the domain194. Robin boundary conditions are a weighted combination of Dirichlet boundary conditions and Neumann boundary conditions. This contrasts to mixed boundary conditions, which are boundary conditions of different types specified on different subsets of the boundary. Robin boundary conditions are also called impedance boundary conditions, from their application in electromagnetic problems, or convective boundary conditions, from their application in heat transfer problems. If Ω is the domain on which the given equation is to be solved and ∂Ω denotes its boundary, the Robin boundary condition is:

au + b

∂u =g ∂n

on ∂Ω

Eq. 9.1 for some non-zero constants a and b and a given function g defined on ∂Ω. Here, u is the unknown solution defined on Ω and ∂u/∂n denotes the normal derivative at the boundary. More generally, a and b are allowed to be (given) functions, rather than constants195. Cauchy Boundary Condition In mathematics, a Cauchy boundary conditions augments an ordinary differential equation or a partial differential equation with conditions that the solution must satisfy on the boundary; ideally so to ensure that a unique solution exists. A Cauchy boundary condition specifies both the function Bakker André, Applied Computational Fluid Dynamics; Solution Methods; 2002. Gustafson, K., (1998). Domain Decomposition, Operator Trigonometry, Robin Condition, Contemporary Mathematics, 218. 432–437. 195 Wikipedia, 193 194

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value and normal derivative on the boundary of the domain. This corresponds to imposing both a Dirichlet and a Neumann boundary conditions. It is named after the prolific 19th-century French mathematical analyst Augustin Louis Cauchy196. Periodic (Cyclic Symmetry) Boundary Condition Two opposite boundaries are connected and their values are set equal when the physical flow problem can be considered to be periodic in space. They could be either physical or non-physical in nature. Among non-physical conditions, inflow, outflow, symmetry plane, pressure and for physical the wall (fixed, moving, impermeable, adiabatic, etc.). Some vendors choose their boundary to be reflected by above description, (OpenFOAM®); and some (i.e., CD-Adapco® and Fluent®) to use their own particular naming, depending to application in hand. Generic Boundary Conditions The most widely used generic B.C’s are:                 

Walls (fixed, moving, impermeable, adiabatic etc.) Symmetry planes Inflow Outflow Free surface Pressure Scalars (Temperature, Heat flux) Velocity Internal Pole Periodic Porous media Free-Stream Non-Reflecting Turbulence-Intensity Immersed Free Surface

Among others and excellent descriptive available through literature for each.

Wall Boundary Conditions All practically relevant flows situations are wall-bounded and near walls the exchange of mass, momentum and scalar quantities is largest. At a solid wall Stokes flow theory is valid i.e. the fluid adheres to the wall and moves with the wall velocity. Different treatment for the different variables in the Navier-Stokes equations is required. Velocity Field The fluid velocity components equal the velocity of the wall. The normal and tangential velocity components at an impermeable, non-moving wall are:

196

From Wikipedia, the free encyclopedia.

148

v t  v wall  0 ; v n  0

Eq. 9.2

Mass fluxes are zero and hence convective fluxes are zero.

Cwall = 𝑚̇ 𝜑 = 0 Eq. 9.3 Diffusive fluxes are non-zero and result in wall-shear stresses. 

D wall   τ n ds

Eq. 9.4

ij

Pressure The specification of wall boundary conditions for the pressure depends on the flow situation. In a parabolic or convection dominated flow a von Neumann boundary condition is used at the wall:

P n

0

Eq. 9.5

wall

In a flow with complex curvilinear boundaries, at moving walls, or in flows with considerably large external forces there may exist large pressure gradients towards the walls. The most common treatment of such boundaries is a linear extrapolation form the inner flow region. If the exact value of the pressure at the boundaries is not of interest no boundary conditions are needed when a staggered grid is used. When a pressure correction method is used, wall boundary conditions are also needed for pressure correction variable p’. Conservation of mass is only ensured when p’=0 at the walls. For the purpose of stability this is usually accomplished by a zero gradient condition. The boundary conditions for the pressure and for the velocity components are valid for both laminar and turbulent flows. In the case of a turbulent flow near wall gradients are significantly larger and a very high resolution is required particularly for high Reynolds number flows. Therefore, wall functions were invented that bridge the near wall flow with adequate (mostly empirical) relationships. Scalars/Temperature Direct specification of the scalar/temperature at the wall boundary (Dirichlet Boundary condition)

T(x, t)  Twall

Eq. 9.6

Specification of a scalar/temperature gradient i.e. specification of a scalar/temperature flux (von Neumann Boundary condition):

q wall ( x , t )   λ   

 T( x , t ) n

Eq. 9.7 wall

Common inputs for wall boundary condition Thermal boundary conditions (for heat transfer calculations). Wall motion conditions (for moving or rotating walls). Shear conditions (for slip walls, optional).

149

     

Wall roughness (for turbulent flows, optional). Species boundary conditions (for species calculations). Chemical reaction boundary conditions (for surface reactions). Radiation boundary conditions. Discrete phase boundary conditions (for discrete phase calculations). Wall adhesion contact angle (for VOF calculations, optional).

Symmetry Planes Used at the centerline (y= 0) of a 2-D axisymmetric grid. Can also be used where multiple grid lines meet at a point in a 3D O type grid. They used in CFD simulations to reduce the numerical effort (see Figure 9.2). Must be used carefully and only when both geometry and flow are symmetrical. Unsteady flows around symmetrical obstacles are always asymmetric: e.g. flow around a square obstacle. Steady flows in symmetrical diffusers or channel expansions can be asymmetric and symmetry conditions should only be used when an asymmetric flow can be excluded a priori. At a symmetry boundary the following conditions apply:

Figure 9.2 Symmetry Plane to Model one Quarter of a 3D Duct

 The boundary normal component of the velocity disappears and the flux through the boundary is zero:

⃗Vn = 0 Eq. 9.8

,

Csym = ṁ φ = 0

 Scalars have all zero gradients. Consequently the diffusive fluxes of the scalars are also zero:

φ  0 ; Dsym  0 n

Eq. 9.9

 The boundary normal gradient of tangential velocity components is also zero. As a result, the shear stresses disappear

Inflow Boundaries

An inflow boundary is an artificial boundary that is used in CFD simulations because the computational domain must be finite. Proper use of inflow boundary conditions can reduce the numerical effort and need to be selected carefully so that the flow physics is not altered. At the inflow usually variables are specified directly i.e. Dirichlet condition. The convective fluxes can be computed and are added to source term. Diffusive fluxes are computed and added to the central coefficient AP. Common inflow boundaries are: Pressure inlet, Velocity inlet, Mass flow inlet, among others. Velocity Inlet This types of boundary conditions are used to define the velocity and scalar properties of the flow at inlet boundaries. The contribution inputs usually includes:

150

        

Velocity magnitude and direction or velocity components Rotating (Swirl) velocity (for 2D axisymmetric problems with swirl) Temperature (for energy calculations) Turbulence parameters (for turbulent calculations) Radiation parameters Chemical species mass fractions (for species calculations) Mixture fraction and variance (for non-premixed or partially premixed combustion calculations) Discrete phase boundary conditions (for discrete phase calculations) Multiphase boundary conditions (for general multiphase calculations)

Pressure Inlet These boundary conditions are used to define the total pressure and other scalar quantities at flow inlets. Required inputs are:        

Total (stagnation) Pressure Total (stagnation) Temperature Flow direction Static pressure Turbulence parameters (for turbulent calculations) Radiation parameters Chemical species mass fractions (for species calculations) Mixture fraction and variance (for non-premixed or partially premixed combustion calculations)

Mass Flow Inlet These boundary conditions are used in compressible flows to prescribe a mass flow rate at an inlet. It is not necessary to use mass flow inlets in incompressible flows because when density is constant, velocity inlet boundary conditions will fix the mass flow. Some of the common inputs are:          

Mass flow rate, mass flux, or (primarily for the mixing plane model) mass flux with average mass flux Total (stagnation) temperature Static pressure Flow direction Turbulence parameters (for turbulent calculations) Radiation parameters Chemical species mass fractions (for species calculations) Mixture fraction and variance (for non-premixed or partially premixed combustion calculations) Discrete phase boundary conditions (for discrete phase calculations) Open channel flow parameters (for open channel flow calculations using the VOF multiphase model)

Inlet Vent boundary conditions are used to model an inlet vent with a specified loss coefficient, flow direction, and ambient (inlet) total pressure and temperature.

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Outflow Boundaries

An outflow boundary is also an artificial boundary that is used in CFD simulations because the computational domain must be finite. The location of the outflow boundary must be sufficiently downstream of the region of interest. At the outlet boundary recirculation zones may not be present and streamlines should be parallel. The mathematical formulation of the boundary condition may not influence the flow in the inner part of the domain. Zero gradient conditions are most widely used for all variables. The outlet boundary is usually used to check global mass conservation during an iterative process. Commonly used outflow boundaries include: Pressure outlet, Pressure far-field, Outlet vent, and Exhaust fan. Pressure Outlet These boundary conditions are used to define the static pressure at flow outlets (and also other scalar variables, in case of back flow). The use of a pressure outlet boundary condition instead of an out flow condition often results in a better rate of convergence when back flow occurs during iteration. The contributions inputs requires are:  

     

Static pressure Backflow conditions  Total (stagnation) Temperature (for energy calculations)  Backflow direction specification method  Turbulence parameters (for turbulent calculations)  Chemical species mass fractions (for species calculations)  Mixture fraction and variance (for non-premixed or partially premixed combustion calculations)  Multiphase boundary conditions (for general multiphase calculations) Radiation parameters Discrete phase boundary conditions (for discrete phase calculations) Open channel flow parameters (for open channel ow calculations using the VOF multiphase model) Non-reflecting boundary (for compressible density-based solver) Target mass flow rate (not available for multiphase flows)

Pressure Far-Field boundary conditions are used to model a free-stream compressible flow at in unity, with free-stream Mach number and static conditions specified. This boundary type is available only for compressible flows. Inputs are:        

Static pressure. Mach number. Temperature. Flow direction. Turbulence parameters (for turbulent calculations). Radiation parameters. Chemical species mass fractions (for species calculations). Discrete phase boundary conditions (for discrete phase calculations).

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Outflow Boundary conditions are used to model flow exits where the details of the flow velocity and pressure are not known prior to solution of the flow problem. They are appropriate where the exit flow is close to a fully developed condition, as the outflow boundary condition assumes a zero normal gradient for all flow variables except pressure. They are not appropriate for compressible flow calculations. Outlet Vent boundary conditions are used to model an outlet vent with a specified loss coefficient and ambient (discharge) static pressure and temperature. The inputs are:  

   

Static pressure Backflow conditions  Total (stagnation) temperature (for energy calculations)  Turbulence parameters (for turbulent calculations)  Chemical species mass fractions (for species calculations)  Mixture fraction and variance (for non-premixed or partially premixed combustion calculations)  Multiphase boundary conditions (for general multiphase calculations) Radiation parameters Discrete phase boundary conditions (for discrete phase calculations) Loss coefficient Open channel flow parameters (for open channel flow calculations using the VOF multiphase model)

Exhaust Fan Boundary conditions are used to model an external exhaust fan with a specified pressure jump and ambient (discharge) static pressure.  

   

Static pressure Backflow conditions  Total (stagnation) temperature (for energy calculations)  Turbulence parameters (for turbulent calculations)  Chemical species mass fractions (for species calculations)  Mixture fraction and variance (for non-premixed or partially premixed combustion calculations)  Multiphase boundary conditions (for general multiphase calculations)  User-defined scalar boundary conditions (for user-defined scalar calculations) Radiation parameters Discrete phase boundary conditions (for discrete phase calculations) Pressure jump Open channel flow parameters (for open channel ow calculations using the VOF multiphase model).

Free Surface Boundaries Velocity Field and Pressure Free surface boundaries can be rather complex and the location of the free surface is usually not known a-priori. E.g. the swash of a fluid in a tank, the pouring of liquid into a glass. Only at the initial time the position of the free surface is known and in the following an additional transport equation to determine the location of the free surface is needed. Two boundary conditions apply at the free

153

surface boundary:  

Kinematic boundary condition - Fluid cannot flow through the boundary. i.e. the normal component is equal to the surface velocity. Dynamic boundary condition - All forces that are acting on the free surface have to be in equilibrium. These include shear stresses from the fluid below the surface and possibly from a second fluid on the other side fluid and surface tension197.

In many CFD applications the free surface is treated as a flat plane where the symmetry condition is applied. Scalars/Temperature Treated in an analogue manner as the wall boundary condition. Direct specification i.e. Dirichlet boundary conditions or von Neumann boundary conditions or a combination of both.

Pole (Axis) Boundaries Used at the centerline (y = 0) of a 2D axisymmetric grid (Figure 9.3). It can also be used where multiple grid lines meet at a point in a 3-D O-type grid. No other inputs are required. (See Figure 9.3). Figure 9.3

Periodic Flow Boundaries

Pole (Axis) Boundary

Periodicity simply corresponds to matching conditions on the two boundaries. The velocity field is periodic BUT the pressure field is not. The pressure gradient drives the flow and is periodic. A pressure JUMP condition on the boundary must be specified198. Used when physical geometry of interest and expected flow pattern and the thermal solution are of a periodically repeating nature (see Figure 9.4).

Figure 9.4

197 198

Periodic Boundary

Georgia Tech Computational Fluid Dynamics Graduate Course; spring 2007. Solution methods for the Incompressible Navier-Stokes Equations.

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Non-Reflecting Boundary Conditions (NRBCs)

Many problems in computational fluid dynamics occur within a limited portion of a very large or infinite domain. Difficulties immediately arise when one attempts to define the boundary condition for such a system. Such boundary conditions are necessary for the problem to be well-posed, but the physical system under consideration has no boundary to model. One needs to define an artificial boundary whose behavior models the open edge of the physical system. Such a boundary definition is often called a non-reflecting boundary condition (NRBC), as its primary function is to permit wave phenomena to pass through the open boundary without reflection. The standard pressure boundary condition, imposed on the boundaries of artificially truncated domain, results in the reflection of the outgoing waves. As a consequence, the interior domain will contain spurious wave reflections. Many applications require precise control of the wave reflections from the domain boundaries to obtain accurate flow solutions. Non-reflecting boundary conditions provide a special treatment to the domain boundaries to control these spurious wave reflections. The method is based on the Fourier transformation of solution variables at the non-reflecting boundary199. The solution is rearranged as a sum of terms corresponding to different frequencies, and their contributions are calculated independently. While the method was originally designed for axial turbomachinery, it has been extended for use with radial turbomachinery. In many applications of CFD such as Turbomachinery because of close approximately of blades and the physical conditions, it is warranted to use NRBC’s. Another prime candidate is Computational Aero-Acoustics (CAA) which is concerns with propagation of traveling sound waves. In other word, by restricting our area of interest, we effectively create a boundary where none exists physically, dividing our com putational domain from the rest of the physical domain. The challenge we must overcome, then, is defining this boundary in such a way that it behaves computationally as if there were no physical boundary200. Case Study 1 - Turbomachinery Application of 2D Subsonic Cascade The first test case is an axial turbine blade where both the in- and outflow are subsonic and the NRBC will be compared to the Riemann boundary conditions. In the short flow-field simulations the in- and outflow boundaries are positioned at 0.4 times the chord from the airfoil. For the long flow-field simulation this distance becomes 1.5 times the chord. Figure 9.5 shows contour plot of the pressure of the flow. The field of interest is the flow-field close to the boundary201. To give a detailed look at that part of the flow, the pressure contours are put in close proximity. Unfortunately this means the flow-field at the suction side becomes less clear. The subsonic flow means that any reflections diffuse fairly quickly. Therefore there are almost no observable differences when the long flow-field is considered. For the short flow-field the reflections become more apparent when Riemann boundary conditions are used. At the outflow the pressure contours are clearly deflected away from the boundary and never cross it. At the inflow the opposite happens and the pressure contours are bend towards the boundary. This behavior is not observed when looking at the NRBC. Clearly these boundary conditions are successful in removing the reflections from the flow. One can have a closer look at the boundary itself to further clarify this comparison. The pressure at the outflow boundary presented in Figure 9.5, where we notice that the NRBC do a better job of simulating the pressure at the outflow, although it should be noted that on the absolute scale, all the differences are very

M. Giles, “Non-Reflecting Boundary Conditions for the Euler Equations.”, Technical Report TR 88-1-1988, Computational Fluid Dynamics Laboratory, Massachusetts Institute of Technology, Cambridge, MA. 200 John R. Dea, “High-Order Non-Reflecting Boundary Conditions for the Linearized Euler Equations”, Monterey, California, 2008. 201 F. De Raedt, “Non-Reflecting Boundary Conditions for non-ideal compressible fluid flows”, Master of Science at the Delft University of Technology, defended publicly on December 2015. 199

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small.

Figure 9.5

Pressure contours plot for 2nd order spatial discretization scheme

Case Study 2 - CAA Application of Airfoil Turbulence Interaction Noise Simulation The instantaneous contours of the non-dimensional pressure that is radiated from the airfoil due to the turbulence interaction mechanism (see Figure 9.6). In each case, the entire simulated domain is shown. It is qualitatively displays that the acoustic pressure waves do not appear to be acted by the edges of the domain, and are not acted by the changes in domain size between the two simulations. An exception to this is at the domain edge directly downstream of the airfoil. In this region, unphysical pressure disturbances can be seen that correspond to the vortical turbulence encountering the NRBC’s region. However, because these pressure disturbances appear inside the

Domain X Figure 9.6

Domain X/2 Aero-Acoustics Application for NRBC’

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zonal NRBC region, they are contained and do not radiate back into the domain202.

Turbulence Intensity Boundaries When turbulent flow enters domain at inlet, outlet, or at a far-field boundary, boundary values are required for203:  Turbulent kinetic energy k.  Turbulence dissipation rate ε. Four methods available for specifying turbulence parameters:    

Set k and ε explicitly. Set turbulence intensity and turbulence length scale. Set turbulence intensity and turbulent viscosity ratio. Set turbulence intensity and hydraulic diameter.

Turbulence Intensity The turbulence intensity I defined as:

I

2/3k u

Eq. 9.10

Here k is the turbulent kinetic energy and u is the local velocity magnitude. Intensity and length scale depend on conditions upstream:  Exhaust of a turbine. (Intensity = 20%. Length scale=1-10 % of blade span).  Downstream of perforated plate or screen (intensity=10%. Length scale = screen/hole size).  Fully-developed flow in a duct or pipe (intensity= 5%. Length scale = hydraulic diameter).

Immersed Boundaries The immersed boundaries (IB) method allows one to greatly simplify the grid generation and even to automate it completely. The governing equations are solved directly on a grid in their simplest form by means of very efficient numerical schemes. The grid generator detects the cell faces that are cut by the body surface and divides the cells into three types: solid and fluid cells, whose centers lie within the body and within the fluid, respectively; and fluid/solid interface cells, which have at least one of their neighbors inside the body/fluid. Then, the centers of the fluid and solid-interface cells are projected onto the body surface along

Fluid Cells

Interface Cells

Solid Cells

Figure 9.7

Immersed Boundaries

James Gill, Ryu Fattah, and Xin Zhangz, “Evaluation and Development of Non-Reactive Boundary Conditions for Aeroacoustics Simulations”, University of Southampton, Hampshire, SO16 7QF, UK. 203 Bakker, Andre, ”Applied Computational Fluid Dynamics; Lecture 6 - Boundary Conditions”, 2002. 202

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its normal direction, so as to obtain fluid-cells projection points and solid-cell projection points, (see Figure 9.7).

Free Surface Boundary Free surfaces occur at the interface between two fluids. Such interfaces require two boundary conditions to be applied204: 



A kinematic condition which relates the motion of the free interface to the fluid velocities at the free surface and A dynamic condition which is concerned with the force balance at the free surface.

Figure 9.8 Sketch Exemplifying the conditions at a Free Surface Formed by the Interface Between Two Fluids

The Kinematic Boundary Condition The position of a free surface can always be given in implicit form as F(xj , t) = 0. For instance, in Figure 9.8 the height of the free surface above the x-axis is specified as y = h(x, t) and an appropriate function F(x, y, t) would be given by F(x, y, t) = h(x, t) − y. Fluid particles on the free surface always remain part of the free surface, therefore we must have

DF ∂F ∂F = + uk Dt ∂t ∂xk

Eq. 9.11 This is the kinematic boundary condition. For surfaces whose position is described in the form z = h(x, y, t), the kinematic boundary condition becomes

w=

∂h ∂h ∂h +u +v ∂t ∂x ∂y

Eq. 9.12 where u, v, w, are the velocities in the x, y, z directions, respectively. For steady problems, we have DF/Dt = 0 and the kinematic boundary condition can be written as uini = 0 or symbolically u · n = 0, where n is the outer unit normal on the free surface. This condition implies that there is no flow through the free surface (but there can be a flow tangential to it!). The Dynamic Boundary Condition The dynamic boundary condition requires the stress to be continuous across the free surface which separates the two fluids (e.g., air and water). The traction exerted by fluid (1) onto fluid (2) is equal and opposite to the traction exerted by fluid (2) on fluid (1). Therefore we must have t(1) = −t(2). Since n(1) = −n(2) (see Figure 9.8) we obtain the dynamic boundary condition as

Eq. 9.13 204

τ1ij nj = τ2ij nj

“Viscous Fluid Flow: Boundary and initial conditions”, Lecture Series, Manchester, UK.

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where we can use either n(1) or n(2) as the unit normal. On curved surfaces, surface tension can create a pressure jump across the free surface. The surface tension induced pressure jump is given by

∆p = σκ

,

κ=

1 1 + R1 R 2

Eq. 9.14 In this expression σ is the surface tension of the fluid and κ is equal to twice the mean curvature of the free surface, where, R1 and R2 are the principal radii of curvature of the surface (for instance, κ = 2/a for a spherical drop of radius a and κ = 1/a for a circular jet of radius a). Surface tension acts like a tensioned membrane at the free surface and tries to minimize the surface area. Hence the pressure inside a spherical drop (or inside a circular liquid jet) tends to be higher than the pressure in the surrounding medium. If surface tension is important, the dynamic boundary condition has to be modified to

τ1ij n𝑗1 + σκn1𝑖 = τ2ij n𝑗1

Eq. 9.15 where κ > 0 if the centers of curvature lie inside fluid (1).

Other Boundary Conditions Other boundary conditions can occur in special applications. For instance, the presence of an elastic boundary leads to fluid-structure interaction problems in which the fluid velocity has to be equal to the velocity of the elastic wall, while the elastic wall deforms in response to the traction that the fluid exerts on it. At porous walls, the no-penetration condition no longer holds: the volume flux into the wall is often proportional to the pressure gradient at the porous surface. Non-uniformly distributed surfactants (substances which reduce the surface tension) can induce tangential stresses at free surfaces, etc.

Further Remarks For an incompressible fluid, the boundary conditions need to fulfill the overall consistency condition

∮ ui ni dS = 0 ∂V

Eq. 9.16 where ∂V is the surface of the spatially fixed volume in which the equations are solved. If there are no free surfaces (and associated dynamic boundary conditions), the pressure is only defined up to an arbitrary constant as only the pressure gradient (but not the pressure itself) appears in the NavierStokes equations. For initial value problems, the initial velocity field (at t = 0) already has to fulfill the incompressibility constraint. These remarks are particularly important for the numerical solution of the Navier-Stokes equations205.

205

“Viscous Fluid Flow: Boundary and initial conditions”, Lecture Series, Manchester, UK.

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