Moukalled F., Mangani L., Darwish M. The finite volume method in computational fluid dynamics. An advanced introduction with OpenFOAM and Matlab (811443), страница 4
Текст из файла (страница 4)
. . . . . .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 .
![](https://s.studizba.com/z.php?f=/templates/si/i/noavatar.png&w=100&h=100&t=1"){kind=avatar}
