Moukalled F., Mangani L., Darwish M. The finite volume method in computational fluid dynamics. An advanced introduction with OpenFOAM and Matlab (811443), страница 66
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11.22.(b) Derive a high-order scheme using a linear combination ða/U þ b/C þ c/D Þof these expressions in such a way as to eliminate the first and second orderderivatives from the final expression for /f with the additional condition thata + b + c = 1.(c) Prove that the scheme you just derived is third order accurate in its representation of the convection operator.xyfUUUCDDDfxFlow directionFig. 11.22 A uniform one dimensional grid systemExercise 2The QUICK scheme fits a quadratic function to three nodal values to estimate thevalue of a scalar at a cell face, according to133/f ¼ /U þ /C þ /D848(a) For a uniform cartesian two dimensional grid write down the expressions for/e , /w , /n and /s in terms of the values at neighboring nodes, assuming thatthe velocity components u and v are known, constant, and positive.(b) Neglecting diffusion and assuming a uniform source Q/ per unit volume,derive an algebraic discretization of the / scalar conservation equation givenbyr:ðqv/Þ ¼ Q/11.11Exercises423in the formaC /C þXaF /F ¼ bCF NBðCÞassuming that a cell is of unit depth, with the area of its e, w, n, and s facesdenoted by Se, Sw, Sn, and Ss, respectively, and its volume by VC.(c) Splitting the QUICK expression for /f into the form “Upwind differencing” + “deferred correction” so that the coefficients become similar to those ofthe upwind scheme, then moving the deferred correction to the source term,write an expression for this source term.Exercise 3In a steady two dimensional situation, the variable / is governed byr:ðqv/Þ ¼ Q/where q ¼ 1, Q/ ¼ 15 3/.The flow field is such that v = li + 4j everywhere and Dx ¼ Dy ¼ 1.
The domainis discretized using the orthogonal grid shown in Fig. 11.23 with the values of /given at the inlet boundaries as shown in the figure. Using the finite volumeapproach and the Second Order Upwind convection scheme to(a) derive the algebraic equations for the four control volumes, and(b) compute the values of /1 , /2 , /3 and /4 using 3 iterations of a simpleGauss-Siedel type solver.x2 x1= 1022 yv = 1i + 4 j34y= 50Fig. 11.23 A two dimensional configuration discretized using a non-uniform Cartesian grid forthe advection of a scalar / in the presence of a source term42411 Discretization of the Convection Termy=0xy123=0=0456789outlet= 10=0=0xinletFig.
11.24 Convection of a two dimensional scalar fieldExercise 4Consider the steady transport of a scalar / in the domain shown in Fig. 11.24. Thegoverning conservation equation is given byr:ðqv/Þ ¼ 0where q ¼ 1, v = 2yx2i − 2xy2j, and Dx ¼ Dy ¼ 1=3.(a) Using the UPWIND scheme, discretize the equation over the computationaldomain and find the value of / at each element centroid.(b) Using the QUICK scheme, applied via a deferred correction approach, discretize the equation over the computational domain and find the value of / ateach element centroid.Exercise 5 (OpenFOAM®)(a) UsingDoxygen[19],listallderivedclassesofthesurfaceInterpolationScheme<Type> class (these classes implement the virtualinterpolate function).(b) Describe the weights function of the class midPoint<Type>, downwind<Type> and linear<Type>.(c) Write a class that inherits the surfaceInterpolationScheme<Type> class andimplements the interpolate function using a geometric average.11.11Exercises425=1v=0Fig.
11.25 Advection of a step profile in an oblique velocity fieldExercise 6 (OpenFOAM®, uFVM)The advection of a step profile in an oblique velocity field, v = 2i + j, shown inFig. 11.25 is governed byr : ðqv/Þ ¼ 0For different grid sizes, setup the problem and solve it in OpenFOAM® and uFVMusing the following advection schemes assuming unit dimensions in x andy directions, and compare results with the exact solution ðq ¼ 1Þ:(a) UPWIND(b) QUICK(c) SOUExercise 7 (OpenFOAM®, uFVM)The Smith-Hutton test governed byr : ðqv/Þ ¼ 0and illustrated in Fig. 11.26, involves the pure advection of a step profile in arotational velocity field described asv ¼ 2y 1 x2 i 2x 1 y2 j42611 Discretization of the Convection Term=0= 10(-1,0)(0,1)=0inlet(0,0)outlet=0(1,0)Fig. 11.26 Advection of a step profile in a two dimensional rotational velocity fieldFor 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) UPWIND(b) QUICK(c) SOUReferences1.
Spalding DB (1972) A novel finite difference formulation for differential expressionsinvolving both first and second derivatives. Int J Numer Meth Eng 4:551–5592. Leonard BP (1979) A stable and accurate convective modelling procedure based on quadraticupstream interpolation. Comput Methods Appl Mech Eng 19:59–983. Raithby GD (1976) A critical evaluation of upstream differencing applied to problemsinvolving fluid flow. Comput Methods Appl Mech Eng 9:75–1034. Patankar SV (1980) Numerical heat transfer and fluid flow.
Hemisphere PublishingCorporation, McGraw-Hill5. Patankar SV, Baliga BR (1978) A new finite-difference scheme for parabolic differentialequations. Numer Heat Transf 1:27–376. Courant R, Isaacson E, Rees M (1952) On the solution of nonlinear hyperbolic differentialequations by finite differences. Commun Pure Appl Math 5:243–2557. Moukalled F, Darwish M (2012) Transient schemes for capturing interfaces of free-surfaceflows. Numer Heat Transf Part B Fundam 61(3):171–2038.
Darwish M, Moukalled F (2006) Convective schemes for capturing interfaces of free-surfaceflows on unstructured grids. Numer Heat Transf Part B Fundam 49(1):19–429. Leonard BP (1979) A survey of finite difference of opinion on numerical muddling of theincomprehensible defective confusion equation. In: Hughes TJR (ed) Finite element methodsfor convection dominated flows, AMD-34, ASMEReferences42710. Shyy W (1985) A study of finite difference approximations to steady state convectiondominate flow problems. J Comput Phys 57:415–43811.
Leonard BP, Leschziner MA, McGuirk J (1978) Third order finite-difference method forsteady two-dimensional convection. In: Taylor C, Morgan K, Brebbia CA (eds) Numericalmethods in laminar and turbulent flows. Pentech Press, London, pp 807–81912. Fromm JE (1968) A method for reducing dispersion in convective difference schemes.J Comput Phys 3:176–18913. Stubley GD, Raithby GD, Strong AB (1980) Proposal for a new discrete method based on anassessment of discretization errors. Numerical Heat Transfer 3:411–42814.
de Vahl Davis G, Mallinson GD (1972) False diffusion in numerical fluid mechanics.University of New South Wales, School of Mechanical and Industrial Engineering (Report1972/FMT/1)15. Raithby GD (1976) Skew upstream differencing schemes for problems involving fluid flow.Comput Methods Appl Mech Eng 9:153–16416. Darwish M, Moukalled F (1996) A new route for building bounded skew-upwind schemes.Comput Methods Appl Mech Eng 129:221–23317. Khosla PK, Rubin SG (1974) A diagonally dominant second-order accurate implicit scheme.Comput Fluids 2:207–20918. OpenFOAM (2015) Version 2.3.x. http://www.openfoam.org19. OpenFOAM Doxygen (2015) Version 2.3.x. http://www.openfoam.org/docs/cpp/Chapter 12High Resolution SchemesAbstract This chapter continues the development of convection schemes anddiscusses approaches by which boundedness is enforced on High Order(HO) convection schemes to produce High Resolution (HR) schemes.
The recipefor a HR scheme is shown to involve a combination of a HO profile and aConvection Boundedness Criterion (CBC) ensuring that no oscillatory behavior isexperienced in the solution. The Normalized Variable Formulation (NVF) and theTotal Variation Diminishing (TVD) frameworks for developing HR schemes areintroduced. Even though the two approaches look very different they are shown tobe almost identical.
The Normalized Variable Diagram (NVD) and Sweby’s (orr w) diagram for visualizing HR schemes in the NVF and TVD formulation,respectively, are presented. The functional relationships for several HR schemes arespecified in the context of both the NVF and TVD formulations. In addition to theDeferred Correction (DC) procedure discussed in the previous chapter, two additional techniques for the implementation of HO and HR schemes in structured andunstructured grids are introduced, namely the Downwind Weighing Factor(DWF) method and the Normalized Weighing Factor (NWF) method.12.1The Normalized Variable Formulation (NVF)The Normalized Variable Formulation (NVF) is a framework for the descriptionand analysis of High Resolution (HR) schemes.
It was introduced by Leonard [1–3]and gained popularity with the Gaskell and Lau simplified ConvectionBoundedness Criterion (CBC) [4]. The Normalized Variable Diagram (NVD) is auseful tool for the development and analysis of HO and HR schemes.The NVF is a face formulation procedure based on locally normalizing thedependent variable for which the value /f at face f is to be constructed.
Theapproach relies on the upwind ð/C Þ; downwind ð/D Þ; and far upwind ð/U Þ nodevalues, illustrated in Fig. 12.1, to express the normalized variable as© Springer International Publishing Switzerland 2016F. Moukalled et al., The Finite Volume Method in Computational Fluid Dynamics,Fluid Mechanics and Its Applications 113, DOI 10.1007/978-3-319-16874-6_1242943012 High Resolution Schemes(a)DCDDvfUUUUUUfCDDDCDUvfDDUUDDfDC(b)UUDCUUUCvffDFig. 12.1 A schematic showing a the U, C, and D node locations used in describing convectionschemes on structured grids b the C, D and extrapolated U nodes for an unstructured grid~ ¼ / /U//D /Uð12:1ÞWith this normalization the relation/f ¼ f ð/U ; /C ; /D Þð12:2Þ12.1The Normalized Variable Formulation (NVF)431is transformed to ~ ¼f /~/fCð12:3Þsince the normalized values of /D and /U become equal to~ ¼ 0 and /~ ¼1/UDð12:4Þ ~ becomes an indicator of smoothness for thewhile the normalized value of /C /C~ \1 ; represent a monotonic profile~/ field.
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