Moukalled F., Mangani L., Darwish M. The finite volume method in computational fluid dynamics. An advanced introduction with OpenFOAM and Matlab (Moukalled F., Mangani L., Darwish M. The finite volume method in computational fluid dynamics. An advanced introduction with OpenFOAM and Matlab.pdf), страница 4
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. . . . . .12.8 The DWF and NWF Methods. . . . . . . . . . . . .12.8.1The Downwind Weighing Factor(DWF) Method . . . . . . . . . . . . . . . .12.8.2The Normalized Weighing Factor(NWF) Method . . . . . . . . . . . . . . . .12.9 Boundary Conditions . . . . . . . . . .
. . . . . . . . .12.9.1Inlet Boundary Condition . . . . . . . . .12.9.2Outlet Boundary Condition . . . . . . . .12.9.3Wall Boundary Condition. . . . . . . . .12.9.4Symmetry Boundary Condition . . . . .......................................................................394395395396397397398........................................399400404406................................................................................407409411411413421422426................................................................................429429436438443450456456....................458459..........460......463467468470471472......................................................Contents12.10 Computational Pointers12.10.1 uFVM . .
. . .12.10.2 OpenFOAM®12.11 Closure . . . . . . . . . . .12.12 Exercises . . . . . . . . . .References. . . . . . . . . . . . . . .xvii..............................................................................................................................47247247548348348713 Temporal Discretization: The Transient Term . . . . . . . . . . .13.1 Introduction .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .13.2 The Finite Difference Approach . . . . . . . . . . . . . . . . .13.2.1Forward Euler Scheme . . . . . . . . . . . . . . . . .13.2.2Stability of the Forward Euler Scheme . . . . . .13.2.3Backward Euler Scheme. . . . . . .
. . . . . . . . .13.2.4Crank-Nicolson Scheme . . . . . . . . . . . . . . . .13.2.5Implementation Details. . . . . . . . . . . . . . . . .13.2.6Adams-Moulton Scheme . . . . . . . . . . . . . . .13.3 The Finite Volume Approach .
. . . . . . . . . . . . . . . . . .13.3.1First Order Transient Schemes . . . . . . . . . . .13.3.2First Order Implicit Euler Scheme . . . . . . . . .13.3.3First Order Explicit Euler Scheme . . . . . . . . .13.3.4Second Order Transient Euler Schemes . . . . .13.3.5Crank-Nicholson (Central Difference Profile) .13.3.6Second Order Upwind Euler (SOUE) Scheme .13.3.7Initial Condition for the FV Approach . . . . . .13.4 Non-Uniform Time Steps .
. . . . . . . . . . . . . . . . . . . . .13.4.1Non-Uniform Time Steps with the FiniteDifference Approach . . . . . . . . . . . . . . . . . .13.4.2Adams-Moulton (or SOUE) Scheme . . . . . . .13.4.3Non-Uniform Time Steps with the FiniteVolume Approach . . . . . . . . . . . . . .
. . . . . .13.4.4Crank-Nicolson Scheme . . . . . . . . . . . . . . . .13.4.5Adams-Moulton (or SOUE) Scheme . . . . . . .13.5 Computational Pointers . . . . . . . . . . . . . . . . . . . . . . .13.5.1uFVM . . . . . . . . . . . . . . . . . . . . . . . . . . . .13.5.2OpenFOAM® . . . . . . . . . . . . . . . . . . . . . .
.13.6 Closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .13.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .........................................................................489489492492494498500502503507508508510512512514515519........519521....................................52252352452552552652952953314 Discretization of the Source Term, Relaxation,and Other Details . . . . . .
. . . . . . . . . . . . . . . . . .14.1 Source Term Discretization. . . . . . . . . . . . .14.2 Under-Relaxation of the Algebraic Equations14.2.1Under-Relaxation Methods . . . . . .................535535538539....................................................................xviiiContents14.2.2Explicit Under-Relaxation. . .
. . . . . . . . . . . . .14.2.3Implicit Under-Relaxation Methods . . . . . . . . .14.3 Residual Form of the Equation . . . . . . . . . . . . . . . . . . .14.3.1Residual Form of Patankar’s Under-Relaxation .14.4 Residuals and Solution Convergence . . . . . . . . . . . .
. . .14.4.1Residuals . . . . . . . . . . . . . . . . . . . . . . . . . . .14.4.2Absolute Residual . . . . . . . . . . . . . . . . . . . . .14.4.3Maximum Residual . . . . . . . . . . . . . . . . . . . .14.4.4Root-Mean Square Residual . . . . . . . . . . . . . .14.4.5Normalization of the Residual . . . . . . .
. . . . . .14.5 Computational Pointers . . . . . . . . . . . . . . . . . . . . . . . .14.5.1uFVM . . . . . . . . . . . . . . . . . . . . . . . . . . . . .14.5.2OpenFOAM® . . . . . . . . . . . . . . . . . . . . . . . .14.6 Closure . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . .14.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Part III................................................540540544545546546547547547548549549550555555557...........................561561563564565565567569572.....578..................581582585588588592596597602.....603Algorithms15 Fluid Flow Computation: Incompressible Flows .
. . . . . . . .15.1 The Main Difficulty. . . . . . . . . . . . . . . . . . . . . . . . .15.2 A Preliminary Derivation . . . . . . . . . . . . . . . . . . . . .15.2.1Discretization of the Momentum Equation . .15.2.2Discretization of the Continuity Equation . . .15.2.3The Checkerboard Problem. . . . . . . . .
. . . .15.2.4The Staggered Grid . . . . . . . . . . . . . . . . . .15.2.5The Pressure Correction Equation . . . . . . . .15.2.6The SIMPLE Algorithm on Staggered Grid .15.2.7Pressure Correction Equation in TwoDimensional Staggered Cartesian Grids . .
. .15.2.8Pressure Correction Equation in ThreeDimensional Staggered Cartesian Grid . . . . .15.3 Disadvantages of the Staggered Grid . . . . . . . . . . . . .15.4 The Rhie-Chow Interpolation . . . . . . . . . . . . . . . . . .15.5 General Derivation .
. . . . . . . . . . . . . . . . . . . . . . . .15.5.1The Discretized Momentum Equation . . . . .15.5.2The Collocated Pressure Correction Equation15.5.3Calculation of the Df Term . . . . . . . . . . . .15.5.4The Collocated SIMPLE Algorithm . . . . . . .15.6 Boundary Conditions . . . . . . .
. . . . . . . . . . . . . . . . .15.6.1Boundary Conditions for the MomentumEquation. . . . . . . . . . . . . . . . . . . . . . . . . ..............................................Contentsxix15.6.2Boundary Conditions for the PressureCorrection Equation . . .
. . . . . . . . . . . . . . .15.7 The SIMPLE Family of Algorithms. . . . . . . . . . . . . .15.7.1The SIMPLEC Algorithm. . . . . . . . . . . . . .15.7.2The PRIME Algorithm. . . . . . . . . . . . . . . .15.7.3The PISO Algorithm . . . . . . .
. . . . . . . . . .15.8 Optimum Under-Relaxation Factor Values for v and p015.9 Treatment of Various Terms with the Rhie-ChowInterpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .15.9.1Treatment of the Under-Relaxation Term . . .15.9.2Treatment of the Transient Term . . . . . . . . .15.9.3Treatment of the Body Force Term . . .
. . . .15.9.4Combined Treatment of Under-Relaxation,Transient, and Body Force Terms . . . . . . . .15.10 Computational Pointers . . . . . . . . . . . . . . . . . . . . . .15.10.1 uFVM . . . . . . . . . . . . . . . . . . . . . . . . .
. .15.10.2 OpenFOAM® . . . . . . . . . . . . . . . . . . . . . .15.11 Closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .15.12 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .References. . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . .Flow Computation: Compressible Flows . . . . . . . . . .Historical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .The Conservation Equations . . . . . . . . . . . . . . . . . . .Discretization of the Momentum Equation . . . . . . . . .The Pressure Correction Equation .
. . . . . . . . . . . . . .Discretization of The Energy Equation. . . . . . . . . . . .16.6.1Discretization of the Extra Terms . . . . . . . .16.6.2The Algebraic Form of the Energy Equation .16.7 The Compressible SIMPLE Algorithm . . . . . . . . . . . .16.8 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . .
. .16.8.1Inlet Boundary Conditions . . . . . . . . . . . . .16.8.2Outlet Boundary Conditions . . . . . . . . . . . .16.9 Computational Pointers . . . . . . . . . . . . . . . . . . . . . .16.9.1uFVM . . . . . . . . . . . . . . . . . . . . . . . . . . .16.9.2OpenFOAM® . . .
. . . . . . . . . . . . . . . . . . .16.10 Closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .16.11 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .References. . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . .16 Fluid16.116.216.316.416.516.6..............................617621623624625628....................630630631632...................................636636636638649649653...............................................................................................655655656657658659663663665666667669672673673674687687689xxPart IVContentsApplications17 Turbulence Modeling .
