Elements of Fluid Dynamics

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CFD Open Series Revision 1.85.2

Elements of Fluid Dynamics Ideen Sadrehaghighi,

Ph.D.

Flow instability

Votex Shedding of a Cylinder

Wing Tip Vortex

Great Wave by Kanagawa

ANNAPOLIS, MD

2

Contents 1 2

Introduction .................................................................................................................................. 9 Some Preliminary Concepts in Fluid Mechanics ............................................................ 12

2.1

2.2 2.3 2.4

2.5 2.6 2.7 2.8

2.9 2.10 2.11 2.12

2.13 2.14 2.15

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Linear and Non-Linear Systems .................................................................................................................... 12 Mathematical Definition.................................................................................................... 12 2.1.1.1 Linear Algebraic Equation ............................................................................................. 12 2.1.1.2 Nonlinear Algebraic Equations ..................................................................................... 12 Differential Equation ......................................................................................................... 13 2.1.2.1 Ordinary Differential Equation ..................................................................................... 13 2.1.2.2 Partial Differential Equation ......................................................................................... 13 Total Differential ................................................................................................................................................. 14 Lagrangian vs Eulerian Description ............................................................................................................ 14 Fluid Properties ................................................................................................................................................... 15 Kinematic Properties ......................................................................................................... 15 Thermodynamic Properties ............................................................................................... 15 Transport Properties.......................................................................................................... 15 Other Misc. Properties ...................................................................................................... 15 Stream Lines .......................................................................................................................................................... 16 Viscosity .................................................................................................................................................................. 16 Vorticity................................................................................................................................................................... 16 Vorticity vs Circulation ....................................................................................................... 17 Conservative and Non-Conservative forms of PDE............................................................................... 18 Physical .............................................................................................................................. 18 Mathematical..................................................................................................................... 18 How to choose which one to use?..................................................................................... 19 Divergence Theorem - Control Volume Formulation........................................................................... 19 General Transport Equation ..................................................................................................................... 19 Newtonian Fluid ............................................................................................................................................ 20 Some Flow Field Phenomena ................................................................................................................... 20 Viscous Dissipation ............................................................................................................ 20 Diffusion............................................................................................................................. 21 Convection ......................................................................................................................... 21 Dispersion .......................................................................................................................... 21 Advection ........................................................................................................................... 21 Inviscid vs. Viscous ....................................................................................................................................... 21 Steady-State vs. Transient ......................................................................................................................... 22 Flow Field Classification ............................................................................................................................ 22

Brief Review of Thermodynamics and Aerodynamics................................................. 24 3.1 3.2 3.3 3.4 3.5 3.6

Pressure .................................................................................................................................................................. 24 Perfect (Ideal) Gas .............................................................................................................................................. 24 Total Energy .......................................................................................................................................................... 24 Thermodynamic Process ................................................................................................................................. 24 First Law of Thermodynamics ....................................................................................................................... 24 Second Law of Thermodynamics.................................................................................................................. 25

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3.7 Isentropic Relation ............................................................................................................................................. 25 3.8 Static (Local) Condition .................................................................................................................................... 26 3.9 Stagnation (Total) Condition .......................................................................................................................... 26 3.10 Total Pressure (Incompressible) ............................................................................................................ 26 3.11 Pressure Coefficient ..................................................................................................................................... 27 3.12 Application of 1st Law to Turbomachinery......................................................................................... 27 Moment of Momentum .................................................................................................... 28 3.12.1.1 Euler‘s Pump & Turbine Equations .......................................................................... 28 Case Study - Application of 1st and 2nd Laws of Thermodynamics to Single Stage 3.12.1.2 Turbo Machines ........................................................................................................................... 29 3.13 Aerodynamics Distinction ......................................................................................................................... 30 Aerodynamic Practical Application.................................................................................... 31 Aerodynamic Forces and Moments................................................................................... 31 Leading-Edge Flow as a Governing Factor in Leading-Edge Vortex Initiation in Unsteady Airfoil Flows ...................................................................................................................................... 32 3.13.3.1 Identification of LEV initiation from CFD data ......................................................... 33 Measure of Compressibility & Compressible vs Incompressible Flows ............................ 34 Speed of Sound .................................................................................................................. 34 Mach Number .................................................................................................................... 34 Sonic Boom ........................................................................................................................ 35 Shock Waves ...................................................................................................................... 37 3.13.8.1 Compressible 1D Shock Waves Relations................................................................ 37 3.13.8.2 Quasi -1D Correlation Applied to Variable Area Ducts............................................ 38

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Viscous Flow ............................................................................................................................... 41 4.1

4.2

Qualitative Aspects of Viscous Flow ............................................................................................................ 41 No-Slip Wall Condition....................................................................................................... 41 Flow Separation ................................................................................................................. 41 Pressure Drag vs Skin Friction Drag ................................................................................... 42 Laminar vs Turbulent Flows ............................................................................................... 43 Skin Friction ....................................................................................................................... 44 Aerodynamic Heating ........................................................................................................ 44 Reynolds Number ............................................................................................................................................... 45 Reynolds Number Effects in Reduced Model .................................................................... 46 Case Study 1 - Scaling and Skin Friction Estimation in Flight using Reynold Number ....... 46 4.2.2.1 Interaction Between Shock Wave and Boundary Layer ............................................... 47 4.2.2.2 Reynolds Number Scaling ............................................................................................. 48 4.2.2.3 Discrepancy in Flight Performance and Wind Tunnel Testing...................................... 48 4.2.2.4 Flow Separation Type (A - B) ........................................................................................ 49 4.2.2.5 Over-Sensitive Prediction in Flight Performance ......................................................... 50 4.2.2.6 Aerodynamic Prediction ............................................................................................... 51 4.2.2.7 Skin Friction Estimation ................................................................................................ 51 Case Study 2 - Reynolds Number Effects Compared To Semi-Empirical Methods............ 53 4.2.3.1 Scaling Effects due to Reynolds Number ...................................................................... 54 4.2.3.2 Direct and Indirect Reynolds Number .......................................................................... 54 4.2.3.3 CFD Calculations ........................................................................................................... 55 4.2.3.4 Description of the CFD Code ........................................................................................ 55 4.2.3.5 Mesh Generation .......................................................................................................... 55

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4.2.3.6 4.2.3.7 4.2.3.8 4.2.3.9 4.2.3.10 4.2.3.11

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Boltzmann Method (LBM) ...................................................................................................... 66 5.1

5.2

5.3

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Residual & Mesh Dependence.................................................................................... 56 Results and Discussion.................................................................................................. 56 Reynolds Number Scaling ............................................................................................. 57 Reynolds Number Scaling using Semi Empirical Skin Friction Methods ....................... 59 Inspection of Local Boundary Layer Properties for Varying Reynolds Number ...... 63 Concluding Remarks ................................................................................................ 64

Preliminaries and Background ...................................................................................................................... 66 Approaches ........................................................................................................................ 66 5.1.1.1 Dilute Gas Regimes ....................................................................................................... 67 Book Keeping ..................................................................................................................... 67 Kinetic Theory .................................................................................................................... 68 Maxwell Distribution Function .......................................................................................... 69 Boltzmann Transport Equation.......................................................................................... 70 The BGKW Approximation ................................................................................................. 71 Lattices & DnQm Classification .......................................................................................... 72 Lattice Arrangements ........................................................................................................ 73 1D Lattice Boltzmann Method (D1O2) .............................................................................. 73 2D Lattice Boltzmann Method (D2Q9) .............................................................................. 74 The Navier-Stokes Equations (NS) .............................................................................................................. 75 NS Equation in Non-Inertial Frame of Reference .............................................................. 76 Some Basic Functional Analysis ......................................................................................... 76 5.2.2.1 Fourier Series and Hilbert Spaces ................................................................................. 77 5.2.2.2 Weak vs. Genuine (Strong) Solution............................................................................. 77 Boundary Layer Theory ................................................................................................................................... 77 Scaling Analysis for Boundary Layer Equation................................................................... 79 5.3.1.1 3D Boundary Layer ....................................................................................................... 79 5.3.1.2 Thermal Boundary Layer .............................................................................................. 80

Conservation Laws and Governing Equations ................................................................ 82

6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 6.10 6.11 6.12 6.13 6.14

Control Volume Approach ............................................................................................................................... 82 Integral Forms of Conservation Equations............................................................................................... 82 Mathematical Operators................................................................................................................................... 83 Conservation of Mass (Continuity Equation) .......................................................................................... 84 Centrifugal and Coriolis forces ...................................................................................................................... 84 Conservation of Momentum (Newton 2nd Law) ..................................................................................... 85 Conservation of Energy (1st Law of Thermodynamics) ..................................................................... 85 Scalar Transport Equation .............................................................................................................................. 86 Vector Form of N-S Equations........................................................................................................................ 86 Orthogonal Curvilinear Coordinate ....................................................................................................... 87 Cylindrical Coordinate for Governing Equation ................................................................. 88 Generalized Transformation to N-S Equation ................................................................................... 89 Coupled and Uncoupled (Segregated) Flows .................................................................................... 92 Simplification to N-S Equations (Parabolized) ................................................................................. 92 Non-dimensional Numbers in Fluid Dynamics ................................................................................. 93 Prandtl Number ................................................................................................................. 94 Nusset Number .................................................................................................................. 94

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6.15

6.16 6.17 6.18

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Linear PDEs and Model Equations ................................................................................... 110

7.1

7.2

7.3

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Rayleigh Number ............................................................................................................... 95 Other Dimensionless Number ........................................................................................... 95 Non-Dimensionalizing (Scaling) of Governing Equations .................................................. 95 Incompressible Navier-Stokes Equation ............................................................................................. 97 Porous Medium ................................................................................................................. 97 6.15.1.1 Literature Survey ..................................................................................................... 98 6.15.1.2 Some Insight into Physical Consideration of Porous Medium ................................ 99 Velocity–Pressure Formulation ....................................................................................... 101 6.15.2.1 Derivation of Volume Average N-S Equations (VANS) .......................................... 101 6.15.2.2 Discussion .............................................................................................................. 102 Vorticity Consideration in Incompressible Flow .......................................................................... 103 Inviscid Momentum Equation (Euler)............................................................................................... 105 Steady-Inviscid–Adiabatic Compressible Equations ........................................................ 105 Velocity Potential Equation ............................................................................................. 106 Hierarchy of Inviscid Fluid Equation ................................................................................................. 107

Mathematical Character of Basic Equations ......................................................................................... 110 Nonsingular Transformation............................................................................................ 111 The 'Par-Elliptic' problem ................................................................................................ 111 Exact (Closed Form) Solution Methods to Model Equations.......................................................... 112 Linear Wave Equation (1st Order) .................................................................................... 112 Inviscid Burgers Equation ................................................................................................ 112 Diffusion (Heat) Equation ................................................................................................ 113 Viscous Burgers Equation ................................................................................................ 113 Tricomi Equation.............................................................................................................. 114 2D Laplace Equation ........................................................................................................ 114 7.2.6.1 Boundary Conditions .................................................................................................. 114 Poisson’s Equation ........................................................................................................... 115 The Advection-Diffusion Equation ................................................................................... 115 The Korteweg-De Vries Equation..................................................................................... 115 Helmholtz Equation ......................................................................................................... 115 Exact Solution Methods ................................................................................................... 115 Solution Methods for In-Viscid (Euler) Equations ............................................................................. 116 Method of Characteristics ............................................................................................... 116 Linear Systems ................................................................................................................. 116 Non-Linear Systems ......................................................................................................... 118

Boundary Conditions ............................................................................................................ 121

8.1

Naming Convention for Different Types of Boundaries ................................................................... 121 Dirichlet Boundary Condition .......................................................................................... 121 Von Neumann Boundary Condition................................................................................. 121 Mixed or Combination of Dirichlet and von Neumann Boundary Condition .................. 121 Robin Boundary Condition .............................................................................................. 121 Cauchy Boundary Condition ............................................................................................ 121 Periodic (Cyclic Symmetry) Boundary Condition ............................................................. 122 Generic Boundary Conditions .......................................................................................... 122

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8.2

8.3 8.4

8.5

8.6 8.7 8.8 8.9 8.10 8.11 8.12 8.13 8.14

Wall Boundary Conditions ........................................................................................................................... 122 Velocity Field ................................................................................................................... 122 Pressure ........................................................................................................................... 123 Scalars/Temperature ....................................................................................................... 123 8.2.3.1 Common inputs for wall boundary condition ............................................................ 123 Symmetry Planes .............................................................................................................................................. 124 Inflow Boundaries ........................................................................................................................................... 124 Velocity Inlet .................................................................................................................... 124 Pressure Inlet ................................................................................................................... 125 Mass Flow Inlet ................................................................................................................ 125 Inlet Vent ......................................................................................................................... 125 Outflow Boundaries ........................................................................................................................................ 126 Pressure Outlet ................................................................................................................ 126 Pressure Far-Field ............................................................................................................ 126 Outflow ............................................................................................................................ 127 Outlet Vent ...................................................................................................................... 127 Exhaust Fan ...................................................................................................................... 127 Free Surface Boundaries ............................................................................................................................... 127 Velocity Field and Pressure.............................................................................................. 127 Scalars/Temperature ....................................................................................................... 128 Pole (Axis) Boundaries .................................................................................................................................. 128 Periodic Flow Boundaries ............................................................................................................................ 128 Non-Reflecting Boundary Conditions (NRBCs) ................................................................................... 129 Case Study 1 - Turbomachinery Application of 2D Subsonic Cascade............................. 129 Case Study 2 - CAA Application of Airfoil Turbulence Interaction Noise Simulation ...... 130 Turbulence Intensity Boundaries........................................................................................................ 131 Turbulence Intensity ........................................................................................................ 131 Immersed Boundaries.............................................................................................................................. 131 Free Surface Boundary ............................................................................................................................ 132 The Kinematic Boundary Condition ................................................................................. 132 The Dynamic Boundary Condition ................................................................................... 132 Other Boundary Conditions ................................................................................................................... 133 Further Remarks ........................................................................................................................................ 133

List of Tables Table 3.1 Classification of Mach number................................................................................................................. 35 Table 4.1 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).......................................................................................... 56 Table 4.2 Comparison of the Extrapolated Data and CFD in Drag Counts at Reynolds number 56 M ....................................................................................................................................................................................................... 61 Table 6.1 Classification of the Euler equation on different regimes ......................................................... 116

List of Figures Figure 1.1 Figure 2.1 Figure 2.2 Figure 2.3

Hierarchy of Basic Fluid Flow ................................................................................................................ 10 Description of Flow: Lagrangian (left) and Eulerian (right) .................................................... 14 Stream Lines around an Airfoil & Cylinder ....................................................................................... 16 Viscosity effects in parallel plate ........................................................................................................... 16

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Figure 2.4 A sink Vortex flow over a drain and history of a rolle up of a vortex over time................ 17 Figure 2.5 Circulation (Right) vs. Vorticity (Left) ................................................................................................ 17 Figure 2.6 A region V bounded by the surface S = ∂V with the surface normal n .................................. 19 Figure 2.7 Diffusion Process in Physics.................................................................................................................... 21 Figure 2.8 Transient test case of vortex shedding over a cylinder ............................................................... 22 Figure 2.9 Physical Aspects of a Typical Flow Field............................................................................................ 23 Figure 3.1 Control Volume showing sign convention for heat and work transfer ................................. 27 Figure 3.2 Control Volume for a Generalized Turbomachine ......................................................................... 28 Figure 3.3 Schematic section of Single Stage Turbomachine .......................................................................... 29 Figure 3.4 Schematic of Lift and Drag ....................................................................................................................... 32 Figure 3.5 Vorticity Plots from CFD (first row), Flow Visualization from Experiment (second row) ....................................................................................................................................................................................................... 33 Figure 3.6 An F/A-18 Hornet creating a vapor cone at Transonic speed .................................................. 35 Figure 3.7 Illustration of a sonic boom as received by human ears ............................................................. 36 Figure 3.8 Evolution of Shock Wave .......................................................................................................................... 36 Figure 3.9 Qualitative Depiction of 1D Flow Through Normal and Oblique Shocks ............................. 37 Figure 3.10 Oblique Shock Reflections on a Channel Flow (M=2 AoA=15˚) ........................................... 38 Figure 3.11 Comparison on Computation vs Theory for an Oblique Shock in 2D Channel Flow.... 39 Figure 3.12 Oblique Shock Relationship .................................................................................................................. 40 Figure 3.13 Compressible Fow in Converging-Diverging Ducts (Nozzles and Diffusers) ................... 40 Figure 4.1 Boundary Layer Flow along a Wall ...................................................................................................... 41 Figure 4.2 Airflow Separating from a Wing at a High Angle of Attack ........................................................ 41 Figure 4.3 Detached Flow induced by adverse pressure gradient................................................................ 42 Figure 4.4 Schematic of Velocity Profiles for Laminar vs Turbulent Flows .............................................. 43 Figure 4.5 Drag on Slender & Blunt Bodies ............................................................................................................ 43 Figure 4.6 Illustrating the calculation of Skin Friction ...................................................................................... 44 Figure 4.7 Quantitate Aspects of Viscous Flow ..................................................................................................... 45 Figure 4.8 Effects of Reynolds Number in Inertia vs Viscosity....................................................................... 45 Figure 4.9 Drag Coefficient versus Reynolds Number for a 1:5 Model and a Car (Courtesy of 35) .. 46 Figure 4.10 Flow features sensitive to Reynolds number for a cruise condition on a wing section ....................................................................................................................................................................................................... 47 Figure 4.11 Schematic representation of direct and indirect Reynolds number effects ..................... 48 Figure 4.12 Comparison of C-141 Wing Pressure Distributions Between Wind Tunnel and Flight ....................................................................................................................................................................................................... 49 Figure 4.13 Flat plate Skin Friction correlations comparison ........................................................................ 53 Figure 4.14 Cut of the volume mesh along the sweep........................................................................................ 55 Figure 4.15 Convergence history for the adapted mesh at Reynolds number 20 M - (Courtesy of Pettersson and Rizzi) ........................................................................................................................................................... 56 Figure 4.16 Wing Colored by Cp Contours - (Courtesy of Pettersson and Rizzi).................................... 57 Figure 4.17 Simulation Criterion as a Function of Reynolds Number for a Recrit at Reynolds Number 50 M - (Courtesy of Pettersson and Rizzi)................................................................................................. 58 Figure 4.18 Skin Friction Estimated with Karman-Shoenherr and Sommer-Short Methods anchored to CFD data at Reynolds Number 38 M - (Courtesy of Pettersson and Rizzi) .......................... 60 Figure 4.19 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) ........................................................................................................................................................... 60 Figure 4.20 Numerical Fit of Drag Due to Pressure - (Courtesy of Pettersson and Rizzi) .................. 61 Figure 4.21 HTP seen from above, positions where ........................................................................................... 64 Figure 5.1 Flow Regimes for Diluted Gas ................................................................................................................ 67 Figure 5.2 Simulations Spectrum of Rarefied Gas ............................................................................................... 68

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Figure 5.3 Position and velocity vector for a particle after and before applying a force, F ................ 70 Figure 5.4 Real Molecules versus LB particles ...................................................................................................... 72 Figure 5.5 Lattice arrangements for velocity vectors for typical 1D, 2D and 3D Discretization ...... 73 Figure 5.6 Schematics of solving 2D Lattice Boltzmann Model...................................................................... 75 Figure 5.7 The development of the boundary layer for flow over a flat plate ......................................... 78 Figure 5.8 Definition of Boundary Layer Thickness: (a) Standard Boundary Layer (u = 99%U), (b) ....................................................................................................................................................................................................... 79 Figure 5.9 Example of Subsonic 3D Boundary Layer ......................................................................................... 80 Figure 6.1 Control Volume bondcorresponding control surface S ............................................................... 82 Figure 6.2 Centrifugal and Coriolis force................................................................................................................. 85 Figure 6.3 Relation between Cartesian and Cylindrical; coordinate............................................................ 88 Figure 6.4 Conditions and Mathematical Character of N-S and its variation ........................................... 93 Figure 6.5 Some Methods for Simplifying Governing Equations ................................................................... 96 Figure 6.6 Earliest Forms of Porous .......................................................................................................................... 98 Figure 6.7 Effect of surface machining on the same numerically generated porous sample: ........... 99 Figure 6.8 Sketch of a Porous Medium, with l*f and l*s the Characteristic Lengths of the................ 100 Figure 6.9 Evolution of a Vortex Tube in pyroclastic flows .......................................................................... 104 Figure 6.10 Condition and Mathematical Character of Inviscid Flow ..................................................... 107 Figure 6.11 Hierarchy of Flow Equations............................................................................................................. 108 Figure 7.1 Two-way interchange of information between Parabolic and Elliptic flows .................. 111 Figure 7.2 Solution of linear Wave equation....................................................................................................... 112 Figure 7.3 Formulation of discontinuities in non-linear Burgers (wave) equation ........................... 113 Figure 7.4 Rate of Decay of solution to diffusion equation ........................................................................... 113 Figure 7.5 Solution to Laplace equation ............................................................................................................... 114 Figure 7.6 Solution to Poisson's equation ............................................................................................................ 115 Figure 7.7 Characteristics of Linear equation .................................................................................................... 117 Figure 7.8 Characteristics of nonlinear solution point ................................................................................... 119 Figure 8.1 Mixed Boundary Conditions................................................................................................................. 121 Figure 8.2 Symmetry Plane to Model one Quarter of a 3D Duct ................................................................ 124 Figure 8.3 Pole (Axis) Boundary .............................................................................................................................. 128 Figure 8.4 Periodic Boundary ................................................................................................................................... 128 Figure 8.5 Pressure contours plot for 2nd order spatial discretization scheme ................................... 130 Figure 8.6 Aero-Acoustics Application for NRBC’ ............................................................................................. 130 Figure 8.7 Immersed Boundaries ............................................................................................................................ 131 Figure 8.8 Sketch Exemplifying the conditions at a Free Surface Formed by the Interface Between Two Fluids ............................................................................................................................................................................. 132

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1 Introduction 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 notes from the textbooks, papers, and materials that I deemed to be important. At best, it could be used as a reference. 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 1-2. 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 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 Collis,, S,, “An Introduction to Numerical Analysis for Computational Fluid Mechanics”, Sandia National Laboratories Albuquerque, New Mexico 87185 and Livermore, California 94550. 2 Lomax, H., and Pulliam, T.,H., “Fundamentals of Computational Fluid Dynamics”, NASA Ames Research Center, 1999. 1

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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 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 volume3. Physically, this statement requires that 3

Wikipedia, “Fluid Dynamics “, the free encyclopedia

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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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2 Some Preliminary Concepts in Fluid Mechanics 2.1 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 input4. Nonlinear problems are of interest to engineers, physicists5 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 2.1.1.1 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. 2.1.1.2 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 4 5

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 them6. 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. 2.1.2.1 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).

2.1.2.2 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. 6

14

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.

2.2 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

2.3 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 equations7. 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. 7

15

rate-of-change of fluid parcel location in a neighborhood of the desired measurement points8. 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 respectfully9. 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.

2.4 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)10. 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 subsystems11. 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. 10 White, Frank M. 1974: “Viscous Fluid Flow”, McGraw-Hill Inc. 11 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. 8 9

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2.5 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

2.6 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.

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

17

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 present12.

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. 12

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

18

2.8 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 mathematical13. 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 13

Physics Stack Exchange.

19

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.

2.9 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 volume14.

2.10 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:

14

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

20

∂ ∂ ∂ ∂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

2.11 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( + ) ∂xj ∂xi ⏟ 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 shampoo15.

2.12 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 lost kinetic energy reappears in the form of internal energy of the fluid, hence causing the 15

From Wikipedia.

21

temperature to rise. This phenomenon is called viscous dissipation within fluid16. 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).17 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.

2.13 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.

16 17

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

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2.14 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 transient18.

Figure 2.8 Figure 1

2.15

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.

18

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

23

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 and Aerodynamics

3.1 Pressure Pressure is the limiting form of force per unit area19

d𝐅 p = lim ( ) as dA → 0 dA Eq. 3.1

3.2 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

3.3 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

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

3.5 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.

19

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

25

3.6 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

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

γ

p 2  ρ 2   T2  γ 1     p1  ρ1   T1 

(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 process20. 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 isentropic21.

20 21

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

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

3.9 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 equation22,

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

3.10 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

22

Same as above.

27

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.

3.11 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

3.12 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 volume23. 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. 23

28

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. 3.12.1.1 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.

29

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. 3.12.1.2 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

30

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.

3.13 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 deformation24. 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 24

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

31

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, certain differences 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 [J. D. Anderson]25, 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 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 25

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

32

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 parallel26. 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

Figure 3.4 Schematic of Lift and Drag Coefficients vs Angle of Attack on a Airfoil (Courtesy of Anderson)

Eq. 3.30 Where q∞ is the previously defined dynamic pressure, ½ ρV2 , and reference area S and reference length L are chosen to pertain to given geometric shape; for 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, can be obtained in any aerodynamic specific text books such as [J. D. Anerson]27. Generic variations for CL and CD versus angle of attack (α) are sketched in Figure 3.4. 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.]28. 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 Anderson, John D. 1984: “Fundamentals of Aerodynamics”, McGraw Hills Inc. John D. Anderson, Jr., “Fundamentals of Aerodynamics”, 5th Edition, McGraw-Hill Companies, 2011. 28 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. 26 27

33

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. 3.13.3.1 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 3.5 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: (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 (a)

Figure 3.5

(b)

(c )

(d)

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

34

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. 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. 3.31

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 not true for high speed flow when the density fluctuations are apparent and must be treated as compressible29. Speed of Sound The speed of sound, a molecular phenomenon, in a calorically perfect gas is given by

a  γRT

Eq. 3.32

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. 3.33

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 30-31,

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. 31 Graebel, W.P. (2001). Engineering Fluid Mechanics. Taylor & Francis. p. 16. 29 30

35

M

V c

Eq. 3.34

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 3.6 shows an F/A-18 creating a vapor cone at transonic speed just before reaching Mach 1 (By Ensign John Gay, U.S. Figure 3.6 An F/A-18 Hornet Creating a Navy). The Mach number is primarily used to Vapor Cone at Transonic Speed 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 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 3.1), 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. 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 3.1

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,

36

sounding much like an explosion to the human ear32. 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 3.7 Illustration of a sonic boom as received by human ears forced together, or compressed, 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 3.7).

Figure 3.8

32

Wikipedia.

Evolution of Shock Wave

37

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 3.8 displays shock wave for different flow regions as applicable to a jet fighter. 3.13.8.1 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 gradient33. 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.

Figure 3.9

Qualitative Depiction of 1D Flow Through Normal and Oblique Shocks

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 3.9, 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 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 values34. 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

33Anderson, 34

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

38

M12 sin 2β  1 tan θ  2 cot β 2 M1 (γ  cos 2 )  2 Eq. 3.35 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 ,

U12 U 22 p1  ρ1U  p 2  ρ 2 U , U t1  U t2 , h1   h2  2 2 2 1

2 2

Eq. 3.36 Other consideration in obtaining ϴ include:

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 3.10 exhibits the reflection of oblique shock wave on an in-viscid channel flow, generated by its edges with free-stream Mach number of 2.0, AOA = 15˚ and slip wall boundary conditions.

Figure 3.10

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

3.13.8.2 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

39

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. 3.37

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). These findings for converging-diverging duct, could best illustrated using the following Figure 3.13 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 Figure 3.11. The relationship for oblique shock and comparison with theory is given be where incident Mach number (M∞ = 3) is plotted against the theory35 as depicted in Figure 3.12.

Figure 3.11

35

Comparison on Computation vs Theory for an Oblique Shock in 2D Channel Flow

©2012 Mentor Graphics Corporation.

40

Figure 3.13

Compressible Fow in Converging-Diverging Ducts (Nozzles and Diffusers)

Figure 3.12

Oblique Shock Relationship

41

4 Viscous Flow 4.1 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 ignored36. 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. The general form on boundary layer equations, Figure 4.1 Boundary Layer Flow along a Wall shown in Figure 4.1, 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. (See Figure 4.2). Assuming that flow over a surface is produced by a pressure gradient where P3 > P2 > P1 along a surface as depicted in Figure 4.3. Following elements down-stream, 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 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 36

T fluid  T solid

Figure 4.2

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

Eq. 4.1

Airflow Separating from a Wing at a High Angle of Attack

42

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 4.3

Eq. 4.2

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 region37. 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. 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 37

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

43

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 4.4, 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:

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

Figure 4.4 Schematic of Velocity Profiles for Laminar vs Turbulent Flows

Eq. 4.3 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 4.5a, 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 4.5b, Dp is much greater than Df . For this case, because Dp is smaller for turbulent than for laminar flow, turbulent flow is desirable for blunt bodies. The above comments are not all-inclusive; they Figure 4.5 Drag on Slender & Blunt Bodies simply state general trends, and for any given body, the aerodynamic virtues of laminar versus turbulent flow must always be

44

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 present38. 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. 4.4

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 4.6. The dimensionless friction coefficient, Cf, is commonly referred to the free-stream dynamic pressure as:

Cf  Eq. 4.5

2τ w ρU 2

Figure 4.6

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 decreases39. 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 4.7, summarizes these finding for viscous flow.

John D. Anderson, Jr., “Fundamentals of Aerodynamics”, 5th Edition, McGraw-Hill Companies, ISBN 978-0-07339810-5, 2007. 39 Anderson, John D. 1984: “Fundamentals of Aerodynamics”, McGraw Hills Inc. 38

45

Figure 4.7

Quantitate Aspects of Viscous Flow

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

Re 

ρUL UL  μ ν

Eq. 4.6

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 4.8 and has great importance in Fluid Mechanics. It can be used to evaluate whether viscous or Figure 4.8 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 version40. 40

From Wikipedia, the free encyclopedia.

46

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. 4.7

To recognize Reynolds number effects a dependency test should be done41. Results from such a dependency study are presented in Figure 4.9. 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 4.9

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 41

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

47

design of an aircraft as derived from sub-scale wind tunnel testing as investigated by [Crook ]42. For a transport aircraft, the wing is the component most sensitive to Reynolds number change. Figure 4.10 shows the flow typically responsible for such sensitivity, which includes boundary layer transition, shock/boundary layer interaction and trailing-edge boundary layer. 4.2.2.1 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 5.8). 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 7.6

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. 7.8

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. 7.9

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. 7.10

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. 145 Example initial-boundary value problems using Laplace's equation from exampleproblems.com. 144

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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

7.3 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 7.1 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 6.1 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 gas146.

D. Anderson, J., Tannehill, R., Pletcher,”Computational Fluid Mechanics and Heat Transfer”, ISBN 0-89116471-5 – 1984. 146

117

(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. 7.11

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. 7.12

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 7.7

Characteristics of Linear equation

L  A  λ I  0 i T

i

1  1    β  l1  β2  1 L     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. 7.13

118

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. 7.14

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 properties147. 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. 7.15

The eigenvalues of [A] determine the characteristics direction and are148

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. 7.16

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. 147 148

119

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. 7.17

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. 7.18

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 7.8 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 solution149 D. Anderson, J., Tannehill, R., Pletcher,”Computational Fluid Mechanics and Heat Transfer”, ISBN 0-89116471-5 – 1984. 149

120

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8 Boundary Conditions 8.1 Naming Convention for Different Types of Boundaries

Boundary conditions and their correct implementation are among the most critical aspects of a correct CFD simulation150. 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 8.1).

Figure 8.1 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 domain151. 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. 8.1 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 constants152. 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. 152 Wikipedia, 150 151

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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 Cauchy153. 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.

8.2 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:

153

From Wikipedia, the free encyclopedia.

123

v t  v wall  0 ; v n  0

Eq. 8.2

Mass fluxes are zero and hence convective fluxes are zero.

Cwall = 𝑚̇ 𝜑 = 0 Eq. 8.3 Diffusive fluxes are non-zero and result in wall-shear stresses. 

D wall   τ n ds

Eq. 8.4

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. 8.5

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. 8.6

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. 8.7 wall

8.2.3.1 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).

124

     

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).

8.3 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 8.2). 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 8.2 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. 8.8

,

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. 8.9

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

8.4 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:

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        

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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8.5 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).

8.6 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

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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 tension154.

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.

8.7 Pole (Axis) Boundaries Used at the centerline (y = 0) of a 2D axisymmetric grid (Figure 8.3). 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 8.3). Figure 8.3

8.8 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 specified155. Used when physical geometry of interest and expected flow pattern and the thermal solution are of a periodically repeating nature (see Figure 8.4).

Figure 8.4

154 155

Periodic Boundary

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

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8.9 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 boundary156. 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 boundary157. 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 8.5 shows contour plot of the pressure of the flow. The field of interest is the flow-field close to the boundary158. 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 8.5, 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. 157 John R. Dea, “High-Order Non-Reflecting Boundary Conditions for the Linearized Euler Equations”, Monterey, California, 2008. 158 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. 156

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small.

Figure 8.5

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 8.6). 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 8.6

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 domain159.

8.10 Turbulence Intensity Boundaries When turbulent flow enters domain at inlet, outlet, or at a far-field boundary, boundary values are required for160:  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. 8.10

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).

8.11 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 8.7

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. 160 Bakker, Andre, ”Applied Computational Fluid Dynamics; Lecture 6 - Boundary Conditions”, 2002. 159

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its normal direction, so as to obtain fluid-cells projection points and solid-cell projection points, (see Figure 8.7).

8.12 Free Surface Boundary Free surfaces occur at the interface between two fluids. Such interfaces require two boundary conditions to be applied161: 



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 8.8 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 8.8 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. 8.11 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. 8.12 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 8.8) we obtain the dynamic boundary condition as

Eq. 8.13 161

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. 8.14 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. 8.15 where κ > 0 if the centers of curvature lie inside fluid (1).

8.13 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.

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

∮ ui ni dS = 0 ∂V

Eq. 8.16 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 equations162.

162

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

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