Moukalled F., Mangani L., Darwish M. The finite volume method in computational fluid dynamics. An advanced introduction with OpenFOAM and Matlab (811443), страница 73
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(12.44) is considered. The OpenFOAM® definition can be found in“$FOAM_SRC/finiteVolume/interpolation/surfaceInterpolation/limited Schemes/SuperBee/SuperBee.H” file. In this case the script of the limiter function is given inListing (12.9) as48212 High Resolution Schemesscalar limiter(const scalar cdWeight,const scalar faceFlux,const typename LimiterFunc::phiType& phiP,const typename LimiterFunc::phiType& phiN,const typename LimiterFunc::gradPhiType& gradcP,const typename LimiterFunc::gradPhiType& gradcN,const vector& d) const{scalar r = LimiterFunc::r(faceFlux, phiP, phiN, gradcP, gradcN, d);return max(max(min(2*r, 1), min(r, 2)), 0);}Listing 12.9 The limiter function of the SuperBee schemewhere the arguments are as stated before including the gradients, the central differencing weights, etc., while the returned value follows exactly Eq.
(12.44).The r definition follows the same nested template class and the functionitself is defined in the file “$FOAM_SRC/src/finiteVolume/interpolation/surfaceInterpolation/ limitedSchemes/LimitedScheme/NVDTVD.H” according toEq. (12.66). The implementation details are given in Listing 12.10.scalar r(const scalar faceFlux,const scalar phiP,const scalar phiN,const vector& gradcP,const vector& gradcN,const vector& d) const{Listing 12.10 Script used to calculate r12.11Closure483scalar gradf = phiN - phiP;scalar gradcf;if (faceFlux > 0){gradcf = d & gradcP;}else{gradcf = d & gradcN;}{return 2*(gradcf/gradf) - 1;}}Listing 12.10 (continued)12.11ClosureThe chapter dealt with the bounding of HO convection schemes. This wasaccomplished by enforcing a convection boundedness criterion (CBC). Theresulting HO bounded schemes were denoted by HR schemes.
The NormalizedVariable Formulation (NVF) and Total Variation Diminishing (TVD) approacheswere introduced as frameworks for the development of HR schemes. Two techniques for the implementation of HO and HR schemes in structured and unstructured grids were introduced, namely the Downwind Weighing Factor(DWF) method and the Normalized Weighing Factor (NWF) method. The nextchapter is devoted to the discretization of the unsteady term.12.12ExercisesExercise 1a. Starting with the NVF form of the SMART scheme derive its equivalent TVDform.b. Starting with the TVD form of the OSHER scheme derive its equivalent NVFform.48412 High Resolution SchemesExercise 2For non-uniform grids the equations for the various schemes become geometrydependent. This is also true for the point Q through which schemes have to pass tobe second order accurate.Find the coordinates of Q in the general case of a non-uniform grid.Hint: define a normalized space variable as [33]~x ¼x xUxD xUExercise 3Derive the DWFf and NWFf relationships of the OSHER and SMART schemes.Exercise 4For the one dimensional uniform mesh shown in Fig.
12.22, use the NVF-SMART,NVF-OSHER, QUICK, and SOU schemes to compute /f for the followingsituations:a.b.c.d./U/U/U/U¼ 30;¼ 10;¼ 30;¼ 30;/C/C/C/C¼ 20;¼ 5;¼ 10;¼ 25;/D ¼ 10/D ¼ 15/D ¼ 5/D ¼ 5Exercise 5For the one dimensional mesh shown in Fig. 12.22, use the TVD-VanLeer andTVD-MINMOD schemes to compute /f for the following situations:a.b.c.d./U/U/U/U¼ 30;¼ 10;¼ 30;¼ 30;/C/C/C/C¼ 20; /D ¼ 10¼ 5; /D ¼ 15¼ 10; /D ¼ 5¼ 25; /D ¼ 5UCfDfxFig. 12.22 A one dimensional uniform gridExercise 6Consider the steady transport of a scalar / in the domain shown in Fig. 12.23. Thegoverning conservation equation is given by12.12Exercises485Fig.
12.23 Convection of atwo dimensional scalar field123456789outletinletr ðqv/Þ ¼ 0where q ¼ 1; v ¼ 2yx2 i 2xy2 j; and Dx ¼ Dy ¼ 1=3:a. Using the NVF-SMART scheme, applied via a deferred correction approach,discretize the equation over the computational domain and find the value of / ateach element centroid.b. Using the TVD-SMART scheme, applied via a deferred correction approach,discretize the equation over the computational domain and find the value of / ateach element centroid.c.
Using the NVF-SUPERBEE applied via the NVF-NWF method setup thesystem of equations over the domain.d. Using the TVD-MUSCL applied via the TVD-DWF method setup the system ofequations over the domain.Exercise 7The advection of a step profile in an oblique velocity field, v ¼ 2i þ j; shown inFig. 12.24 is governed byr ðqv/Þ ¼ 0Fig. 12.24 Advection of astep profile in an obliquevelocity field=1v=048612 High Resolution SchemesFor different grid sizes, setup the problem and solve it in OpenFOAM® anduFVM using the following HR advection schemes assuming unit dimensions inx and y directions, and compare results with the exact solution ðq ¼ 1Þ:a. MINMODb. OSHERc.
SMARTExercise 8The Smith-Hutton test governed byr ðqv/Þ ¼ 0and illustrated in Fig. 12.25, involves the pure advection of a step profile in arotational velocity field described asv ¼ 2y 1 x2 i 2x 1 y2 jFor different grid sizes, solve the test in OpenFOAM® and uFVM using thefollowing advection schemes, and compare results with the exact solution ðq ¼ 1Þ:a. Bounded CDb. MUSCLc. SUPERBFig.
12.25 Advection of astep profile in a twodimensional rotationalvelocity field=0= 10=0=0Exercise 9a. Using Doxygen [34] list all the derived classes of the classlimitedSurfaceInterpolationScheme<Type>.b. Verify the correct implementation of the derived upwind<Type> class: checkthe weights function.c. Find all OpenFOAM® limiter classes listed in Eq.
(12.44) (vanLeerLimiter).Compare the formula with the OpenFOAM® implementation.References487References1. Leonard BP (1988) Universal limiter for transient interpolation modeling of the advective transportequations: the ULTIMATE conservative difference. NASA TR-100916, ICOMP-88-112. Leonard BP (1987) SHARP simulation of discontinuities in highly convective steady flow.NASA TM-1002403. Leonard BP, Lock AP, MacVean MK (1995) Extended numerical integration for genuinelymultidimensional advective transport insuring conservation. In: Proceedings of the ninthinternational conference numerical methods in laminar and turbulent flows, vol 9(1), pp 1–124. Gaskell PH, Lau AKC (1988) Curvature compensated convective transport: SMART, a newboundedness preserving transport algorithm. Int J Numer Meth Fluids 8(6):617–6415.
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J Comput Phys 144(1):1–168. Leonard BP, Niknaffs HS (1991) Sharp monotonic resolution of discontinuities withoutclipping of narrow extrema. Comput Fluids 19:141–1549. Van Leer B (1977) Towards the ultimate conservation difference scheme V. A second ordersequel to Godunov’s method. J Comput Phys 23:101–13610. Zhu J, Rodi W (1991) A low-dispersion and bounded convection scheme. Comput MethodsAppl Mech Eng 92:87–9611. Yee HC, Warming RF, Harten A (1983) Implicit total variation diminishing (TVD) schemesfor steady-state calculations.
NASA Technical Memorandum 8483212. Sweby PK (1984) High resolution schemes using flux limiters for hyperbolic conservationlaws. SIAM J Numer Anal 21(5):995–101113. Chakravarth SR (1987) Development of upwind schemes for the Euler equations. NASAContractor Report 404314. VanderHeyden WB, Kashiwa BA (1998) Compatible fluxes for Van Leer advection. J ComputPhys 146:1–2815. Boris JP, Book DL (1973) Flux-corrected transport: I. SHASTA, a fluid transport algorithmthat works. J Comput Phys 11:38–6916. Kuzmin D, Turek S (2002) Flux correction tools for finite elements. J Comput Phys175:525–55817. Zalesak S (1979) Fully multidimensional flux corrected algorithm for fluid. J Comput Phys31:335–36218. Leonard BP (1988) Simple high-accuracy resolution program for convective modelling ofdiscontinuities.
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