Elements of Fluid Dynamics

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CFD Open Series Revision 1.75

Elements of Fluid Dynamics Ideen Sadrehaghighi

Flow instability

Votex Shedding of a Cylinder

Wing Tip Vortex

Great Wave by Kanagawa

ANNAPOLIS, MD

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Contents 1

Introduction .................................................................................................................................. 9

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

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 known Flow Field Phenomena.................................................................................................... 20 Viscous Dissipation ............................................................................................................ 20 Diffusion............................................................................................................................. 20 Convection ......................................................................................................................... 20 Dispersion .......................................................................................................................... 21 Advection ........................................................................................................................... 21 Inviscid vs. Viscous ....................................................................................................................................... 21 Steady-State vs. Transient ......................................................................................................................... 21 Flow Field Classification ............................................................................................................................ 22 Boltzmann Method (LBM) ......................................................................................................................... 22 Introduction & Background ............................................................................................... 23 Kinetic Theory .................................................................................................................... 24 Maxwell Distribution Function .......................................................................................... 24 Boltzmann Transport Equation.......................................................................................... 25 The BGKW Approximation ................................................................................................. 26 Lattices and the DnQm Classification ................................................................................ 27 Lattice Arrangements ........................................................................................................ 28

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1D Lattice Boltzmann Method (D1O2) .............................................................................. 28 2D Lattice Boltzmann Method (D2Q9) .............................................................................. 29 Navier-Stokes Equations (NS) ................................................................................................................. 31 Some Basic Functional Analysis ......................................................................................... 31 2.17.1.1 Fourier series and Hilbert Spaces ............................................................................ 31 2.17.1.2 Weak vs. Genuine (Strong) Solution........................................................................ 32 Qualitative Aspects of Viscous Flow...................................................................................................... 32 No-slip wall condition ........................................................................................................ 33 Flow Separation ................................................................................................................. 33 Pressure Drag..................................................................................................................... 33 Skin Friction ....................................................................................................................... 34 Aerodynamic Heating ........................................................................................................ 35 Shock Wave ..................................................................................................................................................... 35 Reynolds Number ......................................................................................................................................... 36 Reynolds Number Effects in Reduced Model .................................................................... 37 Case Study 1 - Scaling and Skin Friction Estimation in Flight using Reynold No................ 38 2.20.2.1 Interaction between Shock Wave and Boundary Layer .......................................... 38 2.20.2.2 Reynolds Number Scaling ........................................................................................ 39 2.20.2.3 Discrepancy in Flight Performance and Wind Tunnel Testing ................................ 40 2.20.2.4 Flow Separation Type (A - B) ................................................................................... 41 2.20.2.5 Over-Sensitive Prediction in Flight Performance .................................................... 41 2.20.2.6 Aerodynamic Prediction .......................................................................................... 42 2.20.2.7 Skin Friction Estimation ........................................................................................... 43 Case Study 2 - Reynolds Number Effects Compared To Semi-Empirical Methods............ 45 2.20.3.1 Scaling Effects due to Reynolds Number................................................................. 45 2.20.3.2 Direct and Indirect Reynolds Number ..................................................................... 45 2.20.3.3 CFD Calculations ...................................................................................................... 46 2.20.3.4 Description of the CFD Code ................................................................................... 46 2.20.3.5 Mesh Generation..................................................................................................... 46 2.20.3.6 Residual and Mesh Dependence ............................................................................. 47 2.20.3.7 Results and Discussion ............................................................................................ 48 2.20.3.8 Reynolds Number Scaling ........................................................................................ 49 2.20.3.9 Reynolds Number Scaling using Semi Empirical Skin Friction Methods.................. 51 2.20.3.10 Inspection of Local Boundary Layer Properties for Varying Reynolds Number ...... 54 2.20.3.11 Concluding Remarks ................................................................................................ 56

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A Brief Review of Thermodynamics and Aerodynamics ............................................. 57

Pressure .................................................................................................................................................................. 57 Perfect (Ideal) Gas .............................................................................................................................................. 57 Total Energy .......................................................................................................................................................... 57 Thermodynamic Process ................................................................................................................................. 57 First Law of Thermodynamics ....................................................................................................................... 57 Second Law of Thermodynamics .................................................................................................................. 58 Isentropic Relation ............................................................................................................................................. 58 Static (Local) Condition .................................................................................................................................... 59 Stagnation (Total) Condition .......................................................................................................................... 59 Total Pressure (Incompressible) ............................................................................................................ 59 Pressure Coefficient ..................................................................................................................................... 60

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Application of 1st law to Turbomachinery .......................................................................................... 60 1.1.1 Moment of Momentum .................................................................................................... 61 3.12.1.1 Euler‘s Pump & Turbine Equations .......................................................................... 61 Case Study - Application of 1st and 2nd Laws of Thermodynamics to Single Stage 3.12.1.2 Turbo Machines ........................................................................................................................... 62 Speed of Sound ............................................................................................................................................... 63 Mach Number ................................................................................................................................................. 64 Sonic Boom ...................................................................................................................................................... 65 Aerodynamic Forces and Moments ....................................................................................................... 65

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Conservation Laws and Governing Equations ................................................................ 67

Control Volume Approach ............................................................................................................................... 67 Integral Forms of Conservation Equations............................................................................................... 67 Mathematical Operators................................................................................................................................... 68 Conservation of Mass (Continuity Equation) .......................................................................................... 69 Centrifugal and Coriolis forces ...................................................................................................................... 69 Conservation of Momentum (Newton 2nd Law) ..................................................................................... 70 Conservation of Energy (1st Law of Thermodynamics) ..................................................................... 70 Scalar Transport Equation .............................................................................................................................. 71 Vector Form of N-S Equations........................................................................................................................ 71 Orthogonal Curvilinear Coordinate ....................................................................................................... 72 Cylindrical Coordinate for Governing Equation ................................................................. 73 Generalized Transformation to N-S Equation ................................................................................... 74 Coupled and Uncoupled (Segregated) Flows .................................................................................... 77 Simplification to N-S Equations (Parabolized) ................................................................................. 77 Non-dimensional Numbers in Fluid Dynamics ................................................................................. 78 Prandtl Number ................................................................................................................. 79 Nusset Number .................................................................................................................. 79 Rayleigh Number ............................................................................................................... 80 Other Dimensionless Number ........................................................................................... 80 Non-Dimensionalizing (Scaling) of Governing Equations .................................................. 80 Measure of Compressibility & Compressible vs Incompressible Flows ................................. 82 Incompressible Navier-Stokes Equation ............................................................................................. 82 Porous Medium ................................................................................................................. 83 4.16.1.1 Literature Survey ..................................................................................................... 83 4.16.1.2 Some Insight into Physical Consideration of Porous Medium ................................ 84 Velocity–Pressure Formulation ......................................................................................... 86 4.16.2.1 Derivation of Volume Average N-S Equations (VANS) ............................................ 87 4.16.2.2 Discussion ................................................................................................................ 88 Boundary Layer Theory ............................................................................................................................. 89 Scaling Analysis for Boundary Layer Equation................................................................... 90 4.17.1.1 3D Boundary Layer .................................................................................................. 91 4.17.1.2 Thermal Boundary Layer ......................................................................................... 91 Vorticity Consideration in Incompressible Flow ............................................................................. 92 Inviscid Momentum Equation (Euler).................................................................................................. 93 Steady-Inviscid–Adiabatic Compressible Equations .......................................................... 94 Compressible 1D Shock Waves Relations .......................................................................... 94 Quasi 1D Correlation Applied to Variable Area Ducts ....................................................... 96

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Velocity Potential Equation ............................................................................................. 98 Hierarchy of Fluid Governing Equation ............................................................................................... 99

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Linear PDEs and Model Equations ................................................................................... 103 Mathematical Character of Basic Equations ......................................................................................... 103 Nonsingular Transformation............................................................................................ 104 The 'Par-Elliptic' problem ................................................................................................ 104 Exact (Closed Form) Solution Methods to Model Equations.......................................................... 105 Linear Wave Equation (1st Order) .................................................................................... 105 Inviscid Burgers Equation ................................................................................................ 105 Diffusion (Heat) Equation ................................................................................................ 106 Viscous Burgers Equation ................................................................................................ 106 Tricomi Equation.............................................................................................................. 107 2D Laplace Equation ........................................................................................................ 107 5.2.6.1 Boundary Conditions .................................................................................................. 107 Poisson’s Equation ........................................................................................................... 108 The Advection-Diffusion Equation ................................................................................... 108 The Korteweg-De Vries Equation..................................................................................... 108 Helmholtz Equation ......................................................................................................... 108 Exact Solution Methods ................................................................................................... 108 Solution Methods for In-Viscid (Euler) Equations ............................................................................. 109 Method of Characteristics ............................................................................................... 109 Linear Systems ................................................................................................................. 109 Non-Linear Systems ......................................................................................................... 111

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Boundary Conditions ............................................................................................................ 114

Naming Convention for Different Types of Boundaries ................................................................... 114 Dirichlet Boundary Condition: ......................................................................................... 114 Von Neumann Boundary Condition:................................................................................ 114 Mixed or Combination of Dirichlet and von Neumann Boundary Condition: ................. 114 Robin Boundary Condition: ............................................................................................. 114 Cauchy Boundary Condition: ........................................................................................... 114 Periodic (cyclic symmetry) Boundary Condition: ............................................................ 114 Wall Boundary Conditions ........................................................................................................................... 115 Velocity Field ................................................................................................................... 115 Pressure ........................................................................................................................... 115 Scalars/Temperature ....................................................................................................... 116 6.2.3.1 Common inputs for wall boundary condition ............................................................ 116 Symmetry Planes .............................................................................................................................................. 116 Inflow Boundaries ........................................................................................................................................... 117 Velocity Inlet .................................................................................................................... 117 Pressure Inlet ................................................................................................................... 117 Mass Flow Inlet ................................................................................................................ 118 Inlet Vent ......................................................................................................................... 118 Outflow Boundaries ........................................................................................................................................ 118 Pressure Outlet ................................................................................................................ 118 Pressure Far-Field ............................................................................................................ 119

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Outflow ............................................................................................................................ 119 Outlet Vent ...................................................................................................................... 119 Exhaust Fan ...................................................................................................................... 120 Free Surface Boundaries ............................................................................................................................... 120 Velocity Field and Pressure.............................................................................................. 120 Scalars/Temperature ....................................................................................................... 120 Pole (Axis) Boundaries .................................................................................................................................. 121 Periodic Flow Boundaries ............................................................................................................................ 121 Non-Reflecting Boundary Conditions (NRBCs) ................................................................................... 121 Case Study 1 - Turbomachinery application of 2D Subsonic Cascade ............................. 122 Case Study 2 - CAA Application of Airfoil Turbulence Interaction Noise Simulation ...... 123 Turbulence Intensity Boundaries........................................................................................................ 123 Turbulence Intensity ........................................................................................................ 124 Immersed Boundaries.............................................................................................................................. 124

List of Tables Table 1 Mesh and Residual Dependence on CD in Drag counts relative to the baseline mesh with a Residual of -5.5. ...................................................................................................................................................................... 47 Table 2 Comparison of the Extrapolated data and CFD in Drag Counts at Reynolds number 56 M ....................................................................................................................................................................................................... 52 Table 3 Classification of Mach number .................................................................................................................... 64 Table 4 Classification of the Euler equation on different regimes ............................................................. 109

List of Figures Figure 1 Hierarchy of Basic Fluid Flow .................................................................................................................... 10 Figure 2 Description of flow: Lagrangian (left) and Eulerian (right) ......................................................... 14 Figure 3 Stream Lines around an Airfoil & Cylinder........................................................................................... 16 Figure 4 Viscosity effects in parallel plate .............................................................................................................. 16 Figure 5 A sink Vortex flow over a drain and history of a rolle up of a vortex over time ................... 17 Figure 6 Circulation (Right) vs. Vorticity (Left).................................................................................................... 17 Figure 7 Diffusion Process in Physics ....................................................................................................................... 20 Figure 8 Transient test case of vortex shedding over a cylinder................................................................... 21 Figure 9 Physical aspects of a typical flow field ................................................................................................... 22 Figure 10 Simulations Spectrum................................................................................................................................. 24 Figure 11 Position and velocity vector for a particle after and before applying a force, F ................. 25 Figure 12 Real molecules versus LB particles ....................................................................................................... 27 Figure 13 Lattice arrangements for velocity vectors for typical 1D, 2D and 3D Discretization ....... 28 Figure 14 Schematics of solving 2D Lattice Boltzmann Model....................................................................... 30 Figure 15 Boundary Layer flow along a wall ......................................................................................................... 33 Figure 16 Detached Flow induced by adverse pressure gradient................................................................. 34 Figure 17 Illustrating the calculation of Skin Friction ....................................................................................... 34 Figure 18 Quantitate Aspects of Viscous Flow ...................................................................................................... 35 Figure 19 Evolution of Shock Wave ........................................................................................................................... 36 Figure 20 Effects of Reynolds Number in Inertia vs Viscosity........................................................................ 36 Figure 21 Drag coefficient versus Reynolds number for a 1:5 model and a real car (Courtesy of 35) ....................................................................................................................................................................................................... 37 Figure 22 Flow features sensitive to Reynolds number for a cruise condition on a wing section .. 38 Figure 23 Schematic representation of direct and indirect Reynolds number effects ......................... 39

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Figure 24 Comparison of C-141 wing pressure distributions between wind tunnel and flight ..... 40 Figure 25 Flat plate Skin Friction correlations comparison ............................................................................ 43 Figure 26 Cut of the volume mesh along the sweep ........................................................................................... 46 Figure 27 Convergence history for the adapted mesh....................................................................................... 47 Figure 28 Wing colored by Cp contours .................................................................................................................. 48 Figure 29 Simulation criterion as a function of Reynolds Number for a Recrit at Reynolds Number 50 million .................................................................................................................................................................................. 50 Figure 30 Skin Friction Estimated with Karman- ................................................................................................ 51 Figure 31 Skin Friction Estimated with Karman- ................................................................................................ 52 Figure 32 Numerical fit of Drag due to Pressure.................................................................................................. 53 Figure 33 HTP seen from above, positions where ............................................................................................... 55 Figure 34 Reθ as function of Re infinity, extrapolated and .................................................................................... 56 Figure 35 Control Volume showing sign convention for heat and work transfer .................................. 60 Figure 36 Control volume for a generalized Turbomachine ........................................................................... 61 Figure 37 Schematic section of single stage Turbomachine............................................................................ 62 Figure 38 An F/A-18 Hornet creating a vapor cone at transonic speed..................................................... 64 Figure 39 Illustration of a sonic boom as received by human ears .............................................................. 65 Figure 40 Control Volume bondcorresponding control surface S ................................................................ 67 Figure 41 Centrifugal and Coriolis force.................................................................................................................. 70 Figure 42 Relation between Cartesian and Cylindrical; coordinate ............................................................ 73 Figure 43 Conditions and Mathematical character of N-S and its variation equations........................ 78 Figure 44 Some Methods for Simplifying Governing Equations .................................................................... 81 Figure 45 Earliest forms of porous ............................................................................................................................ 83 Figure 46 Effect of surface machining on the same numerically generated porous sample: ............ 85 Figure 47 Sketch of a porous medium, with l*f and l*s the characteristic lengths of the ..................... 86 Figure 48 The development of the boundary layer for flow over a flat plate .......................................... 89 Figure 49 Definition of boundary layer thickness: (a) standard boundary layer (u = 99%U), (b) . 90 Figure 50 An example of subsonic 3D boundary Layer..................................................................................... 91 Figure 51 Evolution of a Vortex Tube in pyroclastic flows .............................................................................. 92 Figure 52 Qualitative pictures of 1D flow through Normal and Oblique shocks ................................... 95 Figure 53 Oblique shock reflections on a channel flow (M=2 AOA=15˚) ................................................. 96 Figure 54 Comparison on Computation Vs Theory for an oblique shock in 2D channel flow ......... 97 Figure 55 Compressible flow in converging-diverging ducts (Nozzles and Diffusers) ........................ 97 Figure 56 Oblique Shock Relationship ..................................................................................................................... 98 Figure 57 Condition and Mathematical Character of Inviscid (except Boundary Layer) equation 99 Figure 58 Hierarchy of Equations according to T.J, Chung ........................................................................... 100 Figure 59 Hierarchy of flow equations .................................................................................................................. 101 Figure 60 Two-way interchange of information between Parabolic and Elliptic flows ................... 104 Figure 61 Solution of linear Wave equation........................................................................................................ 105 Figure 62 Formulation of discontinuities in non-linear Burgers (wave) equation ............................ 106 Figure 63 Rate of Decay of solution to diffusion equation ............................................................................ 106 Figure 64 Solution to Laplace equation ................................................................................................................ 107 Figure 65 Solution to Poisson's equation ............................................................................................................. 108 Figure 66 Characteristics of Linear equation ..................................................................................................... 110 Figure 67 Characteristics of nonlinear solution point .................................................................................... 112 Figure 68 Mixed Boundary Conditions.................................................................................................................. 114 Figure 69 Symmetry Plane to Model one Quarter of a 3D Duct ................................................................. 116 Figure 70 Pole (Axis) Boundary .............................................................................................................................. 121 Figure 71 Periodic Boundary .................................................................................................................................... 121 Figure 72 Pressure contours plot for 2nd order spatial discretization scheme .................................... 122

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Figure 73 Figure 74

Aeroacoustics Application for NRBC’ ................................................................................................ 123 Immersed Boundaries ............................................................................................................................. 124

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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, 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

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 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

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

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

DQ x, y,z, t  

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

Eq. 2.1

Lagrangian vs Eulerian Description The Lagrangian specification of flow field is a way of looking a fluid motion where observer follows an individual parcel as it moves through space and time. This could be analog as sitting in a boat and drifting down the river. The equations of motion that arise from this approach are relatively simple because they result from direct application of Newton’s second law. But their solutions consist merely of the fluid particle spatial location at each instant of time, as depicted in Figure 2 (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

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 (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 rate-of-

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

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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.

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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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 3 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 3

Stream Lines around an Airfoil & Cylinder

Viscosity A measure of the importance of friction in fluid flow. Viscosity is a fluid property by virtue of which a fluid offers resistance to shear stresses. Consider a fluid in 2-D steady shear between two infinite plates h apart, as shown in the Figure 4. 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 4 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

 T  2 T0  S μ   μ 0  T0  T  S

Eq. 2.3

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

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

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of velocity as

ω   V

Eq. 2.4 In 3D flow, vorticity (ω) is in plane of flow and perpendicular to stream lines as depicted in Figure 5 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 5

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 6 Circulation (Right) vs. Vorticity (Left) however has nothing to do with a path, it is defined at a point and would indicate the rotation in the

12

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

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flow field at that point (see Figure 6). So, if an infinitesimal paddle wheel is imagined in the flow, it would rotate due to non-zero vorticity.

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 And corresponding non-conservative (Eq. 2.6)

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Physics Stack Exchange.

Eq. 2.7

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

Divergence Theorem - Control Volume Formulation 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 Eq. 2.9

CS

or

CV

⃗⃗)ds ∭(∇. 𝐅⃗) dV = ∯(𝐅⃗. 𝐧 V

S

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.

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.

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∂ ∂ ∂ ∂Q (ρQ) + (ρUj Q) = (ΓQ ) + S⏟ Q ⏟ ∂t ∂x ∂xj ∂xj ⏟j ⏟ Source Transient ⏟ Diffusion Eq. 2.10

Transport

1 Ui where Q = e φ Γ { Q

Mass Conservation Momentum Conservation Energy Conservation Scalar Diffusion Coefficient

Newtonian Fluid Cases where the shear forces, τ, are proportional to transverse local velocity gradient by means of coefficient of viscosity μ

τμ

u y

Eq. 2.11

Some known 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 temperature to rise. This phenomenon is called viscous dissipation within fluid15. 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 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 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.). 15

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

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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).16 Advection Refers to the convection of a scalar concentration and very significant. Examples include 1st order linear wave equation. In other word, advection is the transport mechanism of a fluid from one location to another, and is dependent on motion and momentum of that fluid.

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

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 8 shows a test case of vortex shedding for flow over a cylinder which is inherently transient17.

Figure 8 Figure 1 Figure 1.1 16 17

Transient test case of vortex shedding over a cylinder

Transient test case of vortex shedding over a cylinder Transient test case of Vortex shedding over a cylinder

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

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Flow Field Classification

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 as illustrated in Figure 9.

Boltzmann Method (LBM)

Internal Vs External Laminar Vs Turbulent

Vicous Vs Invscid

Physical Flow Field

Newtonian Vs NonNewtonian

Attached Vs Detached

Compressible vs Incompressible

Steady Vs Unsteady

Lattice Boltzmann Methods (LBM) (or Figure 9 Physical aspects of a typical flow field thermal lattice Boltzmann methods (TLBM)) is a class of Computational Fluid Dynamics (CFD) methods for fluid simulation. Instead of solving the Navier–Stokes equations, the discrete Boltzmann equation is solved to simulate the flow of a Newtonian fluid with collision models such as Bhatnagar–Gross–Krook (BGK). By simulating streaming and collision processes across a limited number of particles, the intrinsic particle interactions evince a microcosm of viscous flow behavior applicable across the greater mass18. It is a modern approach in Computational Fluid Dynamics and often used to solve the incompressible, timedependent Navier-Stokes equations numerically. Its strength lies however in the ability to easily represent complex physical phenomena, ranging from multiphase flows to chemical interactions between the fluid and the surroundings. The method finds its origin in a molecular description of a fluid and can directly incorporate physical terms stemming from a knowledge of the interaction between molecules. For this reason, it is an invaluable tool in fundamental research, as it keeps the cycle between the elaboration of a theory and the formulation of a corresponding numerical model short. At the same time, it has proven to be an efficient and convenient alternative to traditional solvers for a large variety of industrial problems19. In LBM, the fluid is replaced by fractious particles. These particles stream along given directions (lattice links) and collide at the lattice sites. The LBM can be considered as an explicit method. The collision and streaming processes are local. Hence, it can be programmed naturally for parallel processing machines. Another beauty of the LBM is handling complex phenomena such as moving boundaries (multiphase, solidification, and melting problems), naturally, without a need for face tracing method as it is in the traditional CFD. 18 19

CFD online NIST is an agency of the U.S. Commerce Department, “Lattice Boltzmann Methods”, 2002.

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Introduction & Background There are two main approaches in simulating the transport equations (heat, mass, and momentum), continuum and discrete20. In continuum approach, ordinary or partial differential equations can be achieved by applying conservation of energy, mass, and momentum for an infinitesimal control volume. Since it is difficult to solve the governing differential equations for many reasons (nonlinearity, complex boundary conditions, complex geometry, etc.), therefore finite difference, finite volume, finite element, etc., schemes are used to convert the differential equations with a given boundary and initial conditions into a system of algebraic equations. The algebraic equations can be solved iteratively until convergence is insured. Let us discuss the procedure in more detail, first the governing equations are identified (mainly partial differential equation). The next step is to discretize the domain into volume, girds, or elements depending on the method of solution. We can look at this step as each volume or node or element contains a collection of particles (huge number, order of 1016). The scale is macroscopic. The velocity, pressure, temperature of all those particles represented by a nodal value, or averaged over a finite volume, or simply assumed linearly or bi-linearly varied from one node to another. The phenomenological properties such as viscosity, thermal conductivity, heat capacity, etc. are in general known parameters (input parameters, except for inverse problems). For inverse problems, one or more thermos-physical properties may be unknown. On the other extreme, the medium can be considered made of small particles (atom, molecule) and these particles collide with each other. This scale is microscale. Hence, we need to identify the inter-particle (inter-molecular) forces and solve ordinary differential equation of Newton’s second law (momentum conservation). At each time step, we need to identify location and velocity of each particle, i.e, trajectory of the particles. At this level, there is no definition of temperature, pressure, and thermo-physical properties, such as viscosity, thermal conductivity, heat capacity, etc. For instance, temperature and pressure are related to the kinetic energy of the particles (mass and velocity) and frequency of particles bombardment on the boundaries, respectively. This method is called Molecular Dynamics (MD) simulations. In bookkeeping process we need to identify location (x, y, z) and velocity (cx, cy, and cz are velocity components in x; y and z direction, respectively) of each particle. Also, the simulation time step should be less than the particles collision time, which is in the order of fero-seconds (10-12 s). Hence, it is impossible to solve large size problems (order of cm) by MD method. At this scale, there is no definition of viscosity, thermal conductivity, temperature, pressure, and other phenomenological properties. Statistical mechanics need to be used as a translator between the molecular world and the macroscopic world. The question is, is the velocity and location of each particle important for us? For instance, in this room there are billions of molecules traveling at high speed order of 400 m/s; like rockets, hitting us. But, we do not feel them, because their mass (momentum) is so small. The resultant effect of such a ‘‘chaotic’’ motion is almost nil, where the air in the room is almost stagnant (i.e., velocity in the room is almost zero). Hence, the behavior of the individual particles is not an important issue on the macroscopic scale, the important thing is the resultant effects. What about a middle man, sitting at the middle of both mentioned techniques? the lattice Boltzmann method (LBM). The main idea of Boltzmann is to bridge the gap between micro-scale and macroscale by not considering each particle behavior alone but behavior of a collection of particles as a unit, Figure 10. The property of the collection of particles is represented by a distribution function. The keyword is the distribution function. The distribution function acts as a representative for collection of particles. This scale is called meso-scale. The mentioned methods are illustrated in Figure 10. LBM enjoys advantages of both macroscopic and microscopic approaches, with manageable computer resources. LBM has many advantages. It is easy to apply for complex domains, A. A. Mohamad, “Lattice Boltzmann Method”, Fundamentals and Engineering Applications with Computer Codes, ISBN 978-0-85729-454-8. 20

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easy to treat multi-phase and multi-component flows without a need to trace the interfaces between different phases. Furthermore, it can be naturally adapted to parallel processes computing. Moreover, there is no need to solve Laplace equation at each time step to satisfy continuity equation of incompressible, unsteady flows, as it is in solving Navier–Stokes (NS) equation. However, it needs more computer memory compared with NS solver, which is not a big constraint. Also, it can handle a problem in micro and macro scales with reliable accuracy. Kinetic Theory It is necessary to be familiar with the concepts and terminology of kinetic theory before proceeding to LBM. The following sections are intended to introduce the reader to the basics and fundamentals of kinetic theory of particles. I tried to avoid the detail of mathematics; however more emphasis is given to the physics. For detail information, see [A. A. Mohamad]21.

Molecular Dynamics • Microscopic scale • Hamilton s Equation

Continuum Lattice Boltzmann Method • Mesoscopic scale • Boltzman Equatio Figure 10

• Macroscopic Scale • Finite Volume, Finite Element, etc • Navier-Stokes Equations

Simulations Spectrum

Maxwell Distribution Function In 1859, Maxwell (1831–1879) recognized that dealing with a huge number of molecules is difficult to formulate, even though the governing equation (Newton’s second law) is known. As mentioned before, tracing the trajectory of each molecule is out of hand for a macroscopic system. Then, the idea of averaging came into picture. The idea of Maxwell is that the knowledge of velocity and position of each molecule at every instant of time is not important. The distribution function is the important parameter to characterize the effect of the molecules; what percentage of the molecules in a certain location of a container have velocities within a certain range, at a given instant of time. The molecules of a gas have a wide range of velocities colliding with each other’s, the fast molecules transfer momentum to the slow molecule. The result of the collision is that the momentum is conserved. For a gas in thermal equilibrium, the distribution function is not a function of time, where the gas is distributed uniformly in the container; the only unknown is the velocity distribution function. For a gas of N particles, the number of particles having velocities in the x-direction between cx and cx + dcx is Nf (cx)dcx. The function f(dcx) is the fraction of the particles having velocities in the interval cx and cx dcx; in the x-direction. Similarly, for other directions, the probability distribution function can be defined as before. Then, the probability for the velocity to lie down between cx and cxdcx; cy and cy dcy; and cz and cz dcz will be N f(cx) f(cy) f(cz) dcx dcy dcz: It is important to mention that if the above equation is integrated (summed) over all possible values of the velocities, yields the total number of particles to be N,

A. A. Mohamad, “Lattice Boltzmann Method”, Fundamentals and Engineering Applications with Computer Codes, ISBN 978-0-85729-454-8. 21

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 f(c

x

) f(c y ) f(cz ) dc x dc y dc z  1

Eq. 2.12

Since any direction can be x, or y or z, the distribution function should not depend on the direction, but only on the speed of the particles. Therefore,

f(cx ) f(c y ) f(cz )   (c2x  c2y  c2z ) Eq. 2.13 where φ is another unknown function, that need to be determined. The value of distribution function should be positive (between zero and unity). Hence, in Eq. 2.13, velocity is squared to avoid negative magnitude. The possible function that has property of Eq. 2.13 is logarithmic or exponential function. After some manipulation and math, as well as help from kinetic theory, the final form of distribution is 3 2

 mc 2      2  2kT 

 m  f(c)  4π   ce 2ππk  

Eq. 2.14

Note that this function increases parabolically from zero for low speeds, reaches a maximum value and then decreases exponentially. As the temperature increases, the position of the maximum shifts to the right. The total area under the curve is always one, by definition. This equation called Maxwell or Maxwell– Boltzmann Distribution function. For detail information. see 21. Figure 11 Position and velocity vector for a particle after and Boltzmann Transport before applying a force, F Equation A statistical description of a system can be explained by distribution function f (r, c, t) ; where f (c, r, t) ; c is the number of molecules at time t positioned between r and dr which have velocities between c and cdc; as mentioned before. An external force F acting on a gas molecule of unit mass will change the velocity of the molecule from c to c +F dt and its position from r to r +c dt . (see Figure 11). The number of molecules, f (r, c, t) before applying the external force is equal to the number of molecules after the disturbance, f (r + cdt, c + Fdt, t + dt); if no collisions take place between the molecules. Hence,

f(r  cdt , c  Fdt , t  dt) dr dc  f(r, c, t) dr dc  0

Eq. 2.15

However, if collisions take place between the molecules there will be a net difference between the numbers of molecules in the interval drdc: The rate of change between final and initial status of the distribution function is called collision operator, Ώ. Hence, the equation for evolution of the number

26

of the molecules can be written as,

f(r  cdt , c  Fdt , t  dt) dr dc  f(r, c, t) dr dc  Ω(f) dr dc dt

Eq. 2.16

Dividing the above equation by dt dr dc and as the limit dt → 0; yields

df (r, c, t)  Ω(f) dt

Eq. 2.17

The above equation states that the total rate of change of the distribution function is equal to the rate of the collision. The Ώ is a function of f and need to be determined to solve the Boltzmann equation. For system without an external force, the Boltzmann equation can be written as,

f (r, c, t)  c.f  Ω(f) t

where c and f are vectors

Eq. 2.18 Eq. 2.18 is an advection equation with a source term (Ώ), or advection with a reaction term, which can be solved exactly along the characteristic lines that is tangent to the vector c, if Ώ is explicitly known. The problem is that Ώ is a function of f and Eq. 2.18 is an integral-differential equation, which is difficult to solve. Therefore, there are several approximations available for Ώ. The relation between the above equation and macroscopic quantities such as fluid density, q; fluid velocity vector u, and internal energy e, is as follows

ρ(r, t)   m f(r, c, t) dc ρ(r, t) u(r, t)   m c f(r, c, t) dc ρ(r, t) e(r, t) 

1 m u a2 f(r, c, t) dc  2

and e 

3 k BT 2m

Eq. 2.19 where m is the molecular mass and ua the particle velocity relative to the fluid velocity, the peculiar velocity, ua = c – u. Eq. 2.19 are conservation of mass, momentum, and energy, respectively. The BGKW Approximation It is difficult to solve Boltzmann equation because the collision term is very complicated. The outcome of two body collisions is not likely to influence significantly, the values of many measured quantities (Cercignani, 1990). Hence, it is possible to approximate the collision operator with simple operator without introducing significant error to the outcome of the solution. Bhatnagar, Gross and Krook (BGK) in 1954 introduced a simplified model for collision operator. At the same time Welander (1954), independently, introduced similar operator. The collision operator is replaced as,

Ω(f)  ω(f eq - f)  Eq. 2.20

1 eq (f - f) τ

27

The coefficient ω is called the collision frequency and τ is called relaxation factor (ω=1/τ). The local equilibrium distribution function is denoted by feq; which is Maxwell–Boltzmann distribution function. After introducing BGKW approximation, the Boltzmann equation (Eq. 2.20), without external forces) can be approximated as,

f 1  c.f  (f eq - f) t τ f i (r  ci Δt , t  Δt)  f i (r, t) 





Δt eq f i (r, t)  f i (r, t) τ

Eq. 2.21 The above equation is the working horse of the lattice Boltzmann method and replaces Navier–Stokes equation in CFD simulations. It is possible to derive Navier–Stokes equation from Boltzmann equation. The local equilibrium distribution function with a relaxation time determine the type of problem needed to be solved. The beauty of this equation lies in its simplicity and can be applied for many physics by simply specifying a different equilibrium distribution function and source term (external force). Adding a source term (force term) to the above equation is straightforward. However, there are a few concerns, which will be discussed in the following chapters. Also, the details of implementing the above equation for different problems, such as momentum, heat and mass diffusion, advection–diffusion without and with external forces. It is possible to use finite difference or finite volume to solve partial differential Eq. 2.21. Some authors used this approach to solve fluid dynamic problems on non-uniform grids. The main focus of the book is to solve Eq. 2.21 in two steps, collision and streaming. Lattices and the DnQm Classification Lattice Boltzmann models can be operated on a number of different lattices, both cubic and triangular, and with or without rest particles in the discrete distribution function22.A popular way of classifying the different methods by lattice is the DnQm scheme. Here "Dn" stands for "n dimensions", while "Qm" stands for "m speeds". For example, D3Q15 is a 3-Dimensional Lattice Boltzmann model on a cubic grid, with rest particles present. Each node has a crystal shape and can deliver particles to

Figure 12 22

From Wikipedia, the free encyclopedia.

Real molecules versus LB particles

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15 nodes: each of the 6 neighboring nodes that share a surface, the 8 neighboring nodes sharing a corner, and itself (The D3Q15 model does not contain particles moving to the 12 neighboring nodes that share an edge; adding those would create a "D3Q27" model). Real quantities as space and time need to be converted to lattice units prior to simulation. Non-dimensional quantities, like the Reynolds number, remain the same. Lattice Arrangements In general, two models can be used for lattice arrangements, called D1Q3Q and D1Q5Q, as shown in Figure 13. D1Q3 is the most popular one. The black nodes are the central node, while the gray nodes are neighboring nodes. The factitious particles stream from the central node to neighboring nodes through linkages with a specified speed, called lattice speed.

Figure 13

Lattice arrangements for velocity vectors for typical 1D, 2D and 3D Discretization

1D Lattice Boltzmann Method (D1O2) The kinetic equation for the distribution function (temperature distribution, species distributions, 12, etc.), fk(x,t) can be written as:

f k (r, t) f (r, t)  ck . k Ω k t x

ck 

x t

Eq. 2.22

where k = 1,2 (for one dimensional problem, D1Q2). The left-hand side terms represent the streaming process, where the distribution function streams (advects) along the lattice link. The right-

29

hand term, Xk; represents the rate of change of distribution function, fk; in the collision process. BGK approximation for the collision operator can be approximated a

Ωk 





1 (f k (x, t) - f keq (x, t) τ

Eq. 2.23

The term s represents a relaxation time toward the equilibrium distribution (f eq k ), which is related to the diffusion coefficient on the macroscopic scale. The above equation is the working horse for the diffusion problem in one dimensional space, which can be reformulated as,

f k (x  Δx, t  Δt)  f k (x, t)[1- ω]  ω f keq (x, t) Eq. 2.24 where ω=Δt/τ is the relaxation time. Eq. 2.24 represents a number of equations for different k values (k ¼ 1 and 2), in each direction. The schematic diagram for three nodes with necessary linkages, central and neighboring nodes, where c1 = c, c2= -c. The dependent variable can be related to the distribution function fi, as, 2

 ( x, t )   f k (x, t) , f k 1

eq k

 w k (x, t)

2

and

w k 1

k

1

Eq. 2.25

2D Lattice Boltzmann Method (D2Q9) The process can be modelled using the Boltzmann transport equation, which is

f (x, t)  u.f  Ω t

Eq. 2.26

where f(x, t) is the particle distribution function, u is the particle velocity, and is the collision operator23. The LBM simplifies Boltzmann's original idea of gas dynamics by reducing the number of particles and confining them to the nodes of a lattice. For a two dimensional model, a particle is restricted to stream in a possible of 9 directions, including the one staying at rest. These velocities are referred to as the microscopic velocities and denoted by ei, where i = 0, , , , 8. This model is commonly known as the D2Q9 model as it is two dimensional and involves 9 velocity vectors. For each particle on the lattice, we associate a discrete probability distribution function fi(x, ei, t) or simply fi(x, t), i = 0 , , , 8, which describes the probability of streaming in one particular direction. The macroscopic fluid density and velocity can be defined as a summation of microscopic particle distribution function, 8

 (x, t)   f i (x, t) , i 0

u(x, t) 

1



8

 f (x, t) i 0

i

Eq. 2.27

The key steps in LBM are the streaming and collision processes which are given by

23

Yuanxun Bill Bao & Justin Meskas,” Lattice Boltzmann Method for Fluid Simulations”, April 14, 2011.

30

Equilibrium

[fi (x, t) − fi

f⏟ i (x + ∆x, t + ∆t) − fi (x, t) = − ⏟ Steaming

(x, t)]

τ Collision

Eq. 2.28

In the actual implementation of the model, streaming and collision are computed separately, and special attention is given to these when dealing with boundary lattice nodes. In the collision term of (Eq. 2.28), feq i (x; t) is the equilibrium distribution, and _ is considered as the relaxation time towards local equilibrium. For simulating single phase owes, it success to use Bhatnagar-Gross-Krook (BGK) collision, whose equilibrium distribution feqi is defined by

f ieq (x, t)  w i ρ  ρs i (u(x, t)) i0  4/9  w i   1/9 i  1,2,3,4 1/36 i  5,6,7,8 

,

,

 e .u 9 (e i .u) 2 3 u.u  s(u)  w i 3 i    2 c 2 c 2 c2   c

Δx is the lattice speed Δt

Eq. 2.29

The fluid kinematic viscosity in the D2Q9 model is relate d to the relaxation time by

 

2 - 1 (x) 2 6 t

Eq. 2.30

The algorithm can be summarized in Figure 14. Notice that numerical issues can arise as τ → 1/2. During the streaming and collision step, the boundary nodes require some special treatments on the distribution functions in order to satisfy the imposed macroscopic boundary conditions.

1 - Initialize ρ, u, fi and feq

2 - Streaming step: move fi → fi* in the direction of ei

3 - Compute macroscopic ρ and u from f*i using Eq. (1.19)

6 - Repeat step 2 to 5

5 - Collision step: calculate the updated distribution function fi = f*i - 1/τ (f*i-feqi) using Eq. (1.20)

4 - Compute feq i using Eq. (1.20)

Figure 14

Schematics of solving 2D Lattice Boltzmann Model

31

The Navier-Stokes Equations (NS)

The Navier-Stokes equations are the basic governing equations for a viscous, heat conducting fluid. It was originally developed by the Frenchman (Claude Louis Marie Henri Navier) and Englishman (George Gabriel Stokes) who proposed them in the early to mid-19th century. It is a vector equation obtained by applying Newton's Law of Motion to a fluid element and is also called the momentum equation. It is supplemented by the mass conservation equation, also called continuity equation and the energy equation. Usually, the term Navier-Stokes equations is used to refer to all of these equations, illustrated by (constant properties) as:

.u  0

,

ρ

Du  p  μΔu  FB Dt

Eq. 2.31

The equations can be written in initial notation as well, and will be discussed in detail later on. where ρ is density of the fluid (taken to be a known constant); u ≡ (u, v, w)T is the velocity vector; p is fluid pressure; μ is viscosity, and FB is a body force. D/Dt is the substantial derivative expressing the Lagrangian, or total, acceleration of a fluid parcel in terms of a convenient laboratory-fixed Eulerian reference frame; ∇ is the gradient operator; Δ is the Laplacian, and ∇· is the divergence operator. We remind the reader that the first of these equations (which is a three component vector equation) is just Newton’s second law of motion applied to a fluid parcel; the left-hand side is mass (per unit volume) times acceleration, while the right-hand side is the sum of forces acting on the fluid element. While in LBM, the fluid is replaced by fractious particles, Navier–Stokes equations (NS) solve mass, momentum and energy conservation equations on discrete nodes, elements, or volumes. In other words, the nonlinear partial differential equations convert into a set of nonlinear algebraic equations, which are solved iteratively. The primary reason why LBM can serve as a method for fluid simulations is that the Navier-Stokes equations can be recovered from the discrete equations through the Chapman-Enskog procedure, a multi-scaling expansion technique 24-25-26. Some Basic Functional Analysis Here, we will introduce some basic notions from fairly advanced mathematics that are essential to anything beyond a superficial understanding of the Navier–Stokes equations. There are numerous key definitions and basic ideas needed for analysis of the N-S equations. These include Fourier series, Hilbert (and Sobolev) spaces, the Galerkin procedure, weak and strong solutions, and various notions such as completeness, compactness and convergence27. 2.17.1.1 Fourier series and Hilbert Spaces In this subsection we will briefly introduce several topics required for the study of PDEs in general, and the Navier–Stokes equations in particular, from a modern viewpoint. These will include Fourier series, Hilbert spaces and some basic ideas associated with function spaces, generally. Fourier Series We begin by noting that there has long been a tendency, especially among engineers, to believe that only periodic functions can be represented by Fourier series. This misconception apparently arises Z. Guo, B. Shi, and N. Wang, “Lattice BGK Model for Incompressible Navier-Stokes Equation”, J. Computational Phys. 165, 288-306 (2000). 25 M. Sukop and D.T. Thorne, “Lattice Boltzmann Modeling: an introduction for geoscientists and engineers”. Springer Verlag, 1st edition. (2006). 26 R. Begum, and M.A. Basit, “Lattice Boltzmann Method and its Applications to Fluid Flow Problems”, Euro. J. Sci. Research 22, 216-231 (2008). 27 J. M. McDonough, “Lectures in Computational Fluid Dynamics of Incompressible Flow: Mathematics, Algorithms and Implementations”, Departments of Mechanical Engineering and Mathematics University of Kentucky, 1991, 2003, 2007. 24

32

from insufficient understanding of convergence of series (of functions), and associated with this, the fact that periodic functions are often used in constructing Fourier series. Thus, if we demand uniform convergence some sense could be made of the claim that only periodic functions can be represented. But, indeed, it is not possible to impose such a stringent requirement; if it were the case that only periodic functions could be represented (and only uniform convergence accepted), there would be no modern theory of PDEs. Recall for a function, say f(x), defined for x ∈ [0,L], that formally its Fourier series is of the form 

f(x)   a k k (x)

Eq. 2.32

k 1

where {ϕk} is a complete set of basis functions, and L

a k  f,  k   f(x) k (x)dx

Eq. 2.33

0

The integral on the right-hand side is a linear functional termed the inner product when f and ϕk are in appropriate function spaces. The requirements on f for existence of such a representation can be found28. 2.17.1.2 Weak vs. Genuine (Strong) Solution A genuine solution is the one in which function is continuous but bounded discontinuities in the derivative of function may occur. A weak solution is the solution which is genuine except along a surface space across which the function may be discontinuous. A constraint is placed upon the jump in function across the discontinuity in domain of interest. Clearly the existence of shock waves in inviscid supersonic flow is an example of a weak solution. Therefore, genuine solution is a weak solution and a weak solution which is contentious is a genuine solution29. In numerical solutions, the differential form is used together with difference approximations (FDM), where integral form is used with the finite volume method, FVM. These are equivalent in uniform grids. The differential form does not have a solution in the classical sense in presence of discontinuities (e.g., compressible flows with shocks), hence, one uses the weak form of the integral equations30.

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 ignored31. 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:

J. M. McDonough, “Lectures in Computational Fluid Dynamics of Incompressible Flow: Mathematics, Algorithms and Implementations”, Departments of Mechanical Engineering and Mathematics University of Kentucky, 1991, 2003, 2007. 29 Anderson, Dale A; Tannehill, John C; Plecher Richard H; 1984:”Computational Fluid Mechanics and Heat Transfer”, Hemisphere Publishing Corporation. 30 V. Viitanen, VTT Technical Research Centre of Finland, ResearchGate. 31 White, Frank M. 1974: Viscous Fluid Flow, McGraw-Hill Inc. 28

33

1. A very thin layer in the intermediate neighborhood of the body, δ, in which the velocity gradient normal to the wall, ∂u/∂y, is very large (Boundary Layer). In this region the very small viscosity of μ of the fluid exerts an essential influence in so far as the shearing stress τ = μ (∂u/∂y) may assume large value. 2. In the remaining region no such a large velocity gradient occurs and the influence of viscosity is unimportant. In this region the flow is frictionless and potential. Figure 15

The general form on boundary layer equations, shown in Figure 15, and their characteristic will be discussed later.

Boundary Layer flow along a wall

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

T fluid  T solid

Eq. 2.34

Flow Separation Another contribution due to friction and shear stress is the effects of flow separation or adverse pressure gradient. Assuming that flow over a surface is produced by a pressure gradient where P 3 > P2 > P1 along a surface as depicted in Figure 16. 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 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

Eq. 2.35

Pressure 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 region32. 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 32

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

34

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. 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

Figure 16

L

Df  b  τ w cos φ ds

Eq. 2.36

so

Detached Flow induced by adverse pressure gradient

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

Cf  Eq. 2.37

2τ w ρU 2

Figure 17

Illustrating the calculation of Skin Friction

35

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 decreases33. 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 18, summarizes these finding for viscous flow.

Figure 18

Quantitate Aspects of Viscous Flow

Shock Waves

A shock wave is a very thin region in a supersonic 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 19 displays shock wave for different flow regions as applicable to a jet fighter.

33

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

36

Figure 19

Evolution of Shock Wave

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

Re 

ρUL UL  μ ν

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 20 and has great importance in Fluid Mechanics. It can be used to evaluate whether viscous or 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 different-sized flow situations, such as between an aircraft model in a wind tunnel and the full size version34.

34

From Wikipedia, the free encyclopedia.

0.05

Figure 20

Eq. 2.38

Reynolds Number (Re) 10.0 200.0 3000.0

Effects of Reynolds Number in Inertia vs Viscosity

37

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. 2.39

To recognize Reynolds number effects a dependency test should be done35. Results from such a dependency study are presented in Figure 21. 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 21

35

Drag coefficient versus Reynolds number for a 1:5 model and a real car (Courtesy of 35)

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

38

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 design of an aircraft as derived from sub-scale wind tunnel testing as investigated by [Crook ]36. For a transport aircraft, the wing is the component most sensitive to Reynolds number change. Figure 22 shows the flow typically responsible for such sensitivity, which includes boundary layer transition, shock/boundary layer interaction and trailing-edge boundary layer.

Figure 22

Flow features sensitive to Reynolds number for a cruise condition on a wing section

2.20.2.1 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 0. Figure 65

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. 5.8

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. 5.9

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. 5.10

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. 134 Example initial-boundary value problems using Laplace's equation from exampleproblems.com. 133

109

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

Solution Methods for In-Viscid (Euler) Equations

The interest in Euler equations arises from the fact that in many primary design the information about the pressure alone is needed. In boundary layer where the skin friction and heat transfer is required, the outer edge condition using the Euler. The Euler equation is also of interest because of interest in major flow internal discontinuities such as shock wave or contact surfaces. Solutions relating to Rankine-Hugonist equations are embedded in Euler equation. The Euler equations govern the motion of an Inviscid, non-heat-conducting flow have different character in different regions. If the flow is time-dependent, the flow regimes is hyperbolic for all the Mach numbers and solution can be obtained using marching Subsonic Supersonic procedures. The situation is Flow Sonic M=1 M1 very different when a steady Steady Elliptic Parabolic Hyperbolic flow is assumed. In this case, Unsteady Hyperbolic Hyperbolic Hyperbolic Euler equations are elliptic when the flow is subsonic, and Table 4 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 4 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 2-D perfect gas135.

D. Anderson, J., Tannehill, R., Pletcher,”Computational Fluid Mechanics and Heat Transfer”, ISBN 0-89116471-5 – 1984. 135

110

(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. 5.11

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. 5.12

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 66

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. 5.13

111

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. 5.14

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 properties136. 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. 5.15

The eigenvalues of [A] determine the characteristics direction and are137

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. 5.16

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. 136 137

112

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 2 v u  v2  a 2 

0 1 1 ρva 1 ρva

 1 0  0   0 

Eq. 5.17

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. 5.18

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 67 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

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difficulty in turning an accurate solution138

D. Anderson, J., Tannehill, R., Pletcher,”Computational Fluid Mechanics and Heat Transfer”, ISBN 0-89116471-5 – 1984. 138

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6 Boundary Conditions Naming Convention for Different Types of Boundaries Boundary conditions and their correct implementation are among the most critical aspects of a correct CFD simulation139. 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 68).

Figure 68

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 domain140.

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 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 Cauchy141. 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. The most widely used generic B.C’s are: 

Walls (fixed, moving, impermeable, adiabatic etc.)

Bakker André, Applied Computational Fluid Dynamics; Solution Methods; 2002. Gustafson, K., (1998). Domain Decomposition, Operator Trigonometry, Robin Condition, Contemporary Mathematics, 218. 432–437. 141 From Wikipedia, the free encyclopedia. 139 140

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              

Symmetry planes Inflow Outflow Free surface Pressure Scalars (Temperature, Heat flux) Velocity Internal Pole Periodic Porous media Free-Stream Non-Reflecting Turbulence-Intensity Immersed

Among others and excellent descriptive available through literature for each.

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

v t  v wall  0 ; v n  0

Eq. 6.1

Mass fluxes are zero and hence convective fluxes are zero.

Cwall = 𝑚̇ 𝜑 = 0 Eq. 6.2 Diffusive fluxes are non-zero and result in wall-shear stresses. 

D wall   τ n ds ij

Eq. 6.3

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:

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P n

0

Eq. 6.4

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. 6.5

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

q wall ( x , t )   λ 6.2.3.1         

 T( x , t ) n

Eq. 6.6 wall

Common inputs for wall boundary condition Thermal boundary conditions (for heat transfer calculations). Wall motion conditions (for moving or rotating walls). Shear conditions (for slip walls, optional). Wall roughness (for turbulent flows, optional). Species boundary conditions (for species calculations). Chemical reaction boundary conditions (for surface reactions). Radiation boundary conditions. Discrete phase boundary conditions (for discrete phase calculations). Wall adhesion contact angle (for VOF calculations, optional).

Symmetry Planes Used at the centerline (y= 0) of a 2-D axisymmetric grid. Can also be used where multiple grid lines meet at a point in a 3D O type grid. They used in CFD simulations to reduce the numerical effort (see Figure 69). Must be used carefully and only when both geometry and flow are symmetrical. Unsteady flows

Figure 69

Symmetry Plane to Model one Quarter of a 3D Duct

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

⃗V⃗n = 0

Eq. 6.7

,

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. 6.8

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

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

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

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    

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.

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)

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   

     

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).

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

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

Open channel flow parameters (for open channel flow calculations using the VOF multiphase model)

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

   

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

Free Surface Boundaries Velocity Field and Pressure Free surface boundaries can be rather complex and the location of the free surface is usually not known a-priori. E.g. the swash of a fluid in a tank, the pouring of liquid into a glass. Only at the initial time the position of the free surface is known and in the following an additional transport equation to determine the location of the free surface is needed. Two boundary conditions apply at the free 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 tension142.

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.

142

Georgia Tech Computational Fluid Dynamics Graduate Course; spring 2007.

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Pole (Axis) Boundaries Used at the centerline (y = 0) of a 2D axisymmetric grid (Figure 70). 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.

Periodic Flow Boundaries

Figure 70

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 specified143. Used when physical geometry of interest and expected flow pattern and the thermal solution are of a periodically repeating nature (see Figure 71).

Figure 71

Periodic Boundary

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 boundary144. The solution is rearranged as Solution methods for the Incompressible Navier-Stokes Equations. 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. 143 144

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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 Aeroacoustics (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 computational 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 boundary145. 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 72 shows contour plot of the pressure

Figure 72

Pressure contours plot for 2nd order spatial discretization scheme

of the flow. The field of interest is the flow-field close to the boundary146. 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 John R. Dea, “High-Order Non-Reflecting Boundary Conditions for the Linearized Euler Equations”, Monterey, California, 2008. 146 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. 145

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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 72, 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 small. Case Study 2 - CAA Application of Airfoil Turbulence Interaction Noise Simulation The instantaneous contours of the nondimensional pressure that is radiated from the airfoil due to the turbulence interaction mechanism (see Figure 73). 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 zonal NRBC region, they are contained and do not radiate back into the domain147.

Domain X

Domain X/2

Turbulence Intensity Boundaries When turbulent flow enters domain at inlet, outlet, or at a far-field boundary, boundary values are required for148:  Turbulent kinetic energy k.  Turbulence dissipation rate ε.

Figure 73

Aeroacoustics Application for NRBC’

Four methods available for specifying turbulence parameters:  Set k and ε explicitly.  Set turbulence intensity and turbulence length scale. 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. 148 Bakker, Andre, ”Applied Computational Fluid Dynamics; Lecture 6 - Boundary Conditions”, 2002. 147

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 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. 6.9

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

Immersed Boundaries The immersed boundaries (IB) method allows one to greatly simplify the grid generation and even to automate it completely. The governing equations are solved directly on a grid in their simplest form by means of very efficient numerical schemes. The grid generator detects the cell faces that are cut by the body surface and divides the cells into three types: solid and fluid cells, whose centers lie within the body and within the fluid, respectively; and fluid/solid interface cells, which have at least one of their neighbors inside the body/fluid. Then, the centers of the fluid and solid-interface cells are projected onto the body surface along its normal direction, so as to obtain fluid-cells projection points and solid-cell projection points, (see Figure 74).

Fluid Cells

Interface Cells

Solid Cells

Figure 74

Immersed Boundaries