Moukalled F., Mangani L., Darwish M. The finite volume method in computational fluid dynamics. An advanced introduction with OpenFOAM and Matlab (811443), страница 67
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Values of /C between 0 and 1 0\/C~~~ [1while values of /C that are less than 0 /C \0 or greater than 1 /C~ 0 or /~ 1 indicate aindicate an extremum at C. In addition, values of /CCgradient jump. These configurations are illustrated in Fig. 12.2.Normalization is also useful for transforming the functional relationships of HO~ and /~ : For example, the normalizedschemes into linear relations between /fCfunctional relationships of the HO schemes presented in the previous chapter are asfollows:Upwind:/f ¼ /C~ ¼/~)/fCCentral difference:1/f ¼ ð/C þ /D Þ2~ ¼)/fSecond order upwind:31/f ¼ /C /U22~~ ¼ 3/)/f2 CFROMM:/f ¼ /C þQUICK:331/f ¼ /D þ /C /U848Downwind:/f ¼ /D/D /U41~1þ/C2~ ¼/~ þ1)/fC4~~ ¼ 3 þ 3/)/f8 4 C~ ¼1)/fð12:5Þð12:6Þð12:7Þð12:8Þð12:9Þð12:10Þ~ can always beThus, for all HO schemes that are based on three nodal values, /f~~~expressed as a linear function of / ; i.e., / ¼ ‘/ þ k; where the values of ‘ andCfC~ in the~ is plotted as a function of /k depend on the scheme.
Therefore, if /fC~~/C ; /f plane, then the functional relationships of these schemes will appear asstraight lines on the plot. The resultant plot is denoted by the Normalized VariableDiagram (NVD). An NVD on which the functional relationships of the aboveschemes are plotted is displayed in Fig. 12.3.43212 High Resolution Schemes(a)ffDUUCDUC=1=0DfC<0C>0(b)fCDUC=1D=1=0UUDffD(c)fC=1ffUUCDUC(d)=0DffDUUCDUC=0D=1fC=0(e)ffCUUCDUCfD=1=0D~ \0; b /~ [ 1; c /~ ¼ 1; d /~ ¼ 0; andFig. 12.2 Schematics of the situations when a /CCCC~ \1e 0\/CThe Normalized Variable Formulation (NVF)433KSOU12.1f1ICQUDOWNWINDUPWIND3/41/2LRANTCE3/8FROMM1/401/21CFig.
12.3 Some HO schemes written in normalized form and plotted on a Normalized VariableDiagram (NVD)Example 1Derive the NVF form of the QUICK scheme.SolutionStarting with331/f ¼ /D þ /C /U848Applying normalization to both sides yields331/D þ /C /U /U/f /U848¼/D /U/D /Unoting that3 3 1 3 6 1 8þ ¼ þ ¼ ¼18 4 8 8 8 8 8the following is obtained:43412 High Resolution Schemes331ð/D /U Þ þ ð/C /U Þ ð/U /U Þ/f /U48¼8ð/D /U Þð/D /U Þ3 3 /C /U¼ þ8 4 /D /Uthus~ ¼ 3 þ 3/~/f8 4 CThe NVD reveals that except for the first order upwind and downwind schemes,all second order and third order schemes pass through the point Qð0:5; 0:75Þ (foruniform grids). In fact, it can be shown that for a scheme to be second orderaccurate it has to pass through Q.
If, in addition, its slope at Q is 0.75 then it will bethird order accurate (e.g., QUICK). The upwind scheme was shown to be verydiffusive, while the downwind scheme very compressive (anti-diffusive). Therefore,from the NVD it can graphically be deduced that any scheme whose functionalrelationship is close to the upwind scheme is diffusive while anyone close to thedownwind scheme is compressive.Example 2Show that for schemes developed over uniform cartesian grids to be secondorder accurate their functional relationships should pass through the pointQð1=2; 3=4Þ in the NVD.SolutionSecond order schemes involve three points.
Expanding /C ; /U ; and /D interms of /f ; the following is obtained:1/C ¼ /f Dx/0f þ O Dx223/U ¼ /f Dx/0f þ O Dx221/D ¼ /f þ Dx/0f þ O Dx22The value of /f is generally obtained as a combination of the values at thethree locations as131a/C þ b/U þ c/D ¼ ða þ b þ cÞ/f þ a b þ c Dx/0f þ O Dx222212.1The Normalized Variable Formulation (NVF)435The value of /f will be second order accurate if131 a bþ c222¼0)b¼ca3A first order approximation of /f is obtained asða þ b þ cÞ/f ¼ a/C þ b/U þ c/D ) /fab/ þ/¼ð a þ b þ cÞ C ð a þ b þ cÞ Uc/þð a þ b þ cÞ D~ ¼ a/~ þc) ða þ b þ cÞ/fCFor the above approximation to be second order accurate the followingshould be true:ca~ ¼ a/~ ¼ a/~ þ c ) 2a þ 4c /~ þcþc /aþfCfC33The above equality will be satisfied for any value of a and c, and conse~ ¼ 1=2; in which~ ¼ 3=4 and /quently any second order scheme, when /fCcase2a þ 4c ~2a þ 4c 3 111~¼ aþc) aþc¼ aþc/f ¼ a/C þ c )334 222Therefore all second order schemes pass through the point Qð1=2; 3=4Þ:The HO schemes presented in the previous chapter were shown to drasticallydecrease the truncation error suffered by the first order upwind scheme, whileremaining stable.
Still, one of the main shortcomings of these schemes is theirunboundedness, i.e., their tendency to produce under/overshoots and even oscillations near sudden jumps or steep gradients in the convected variable (see Figs. 11.14band 11.17). While in some applications small overshoots and/or oscillations may betolerable, in others, they can lead to catastrophic results, such as in turbulent flowcalculations where the convected variable can be the viscosity coefficient.This oscillatory behavior near steep gradients characterizes all HO linear convective schemes.
In fact these schemes are not monotonous in the sense that theyproduce local maxima and/or minima, i.e., they are not extrema preserving. For ascheme to be extrema preserving, maxima in the solution must be non-increasing andminima non-decreasing (the scheme should not produce over/under shoots). In fact it43612 High Resolution Schemeswas demonstrated by Godunov and Ryabenki [5] that any linear numerical schemethat is monotone can be at most first-order accurate. This implies that all higher orderlinear schemes cannot be monotonicity preserving, and that to construct monotonicitypreserving schemes, non-linear limiter functions should be used.
With this understanding, work on developing high order oscillation-free convection schemesresulted in several techniques [6–10] that can be grouped under two categories. In thefirst approach [11–13] a limited anti-diffusive flux is added to a first-order upwindscheme in such a way that the resulting scheme is capable of resolving sharp gradientswithout oscillations. In the second category, a smoothing diffusive flux is introducedinto an unbounded HO scheme to damp unphysical oscillations [14–17].Due to their multi-step nature and the difficulty in balancing the two fluxes, fluxblending techniques tend to be very expensive numerically. This is why in this booktwo approaches for developing HR schemes falling under the flux limiter methodwill be presented. The first follows a composite procedure whereby high orderschemes are combined with bounded low order ones, with the switch between thembeing controlled by a certain criterion [18].
The second method is based on addingto a diffusive first order upwind term an anti-diffusive flux multiplied by a fluxlimiter. In this case, the resulting HR schemes are also denoted by Total VariationalDiminishing (TVD) schemes as explained in a later section.The composite schemes approach will be presented first within the framework ofthe Normalized Variable Formulation (NVF) and will be visualized on aNormalized Variable Diagram (NVD). Therefore the NVF and NVD are firstdescribed. The use of the NVD will be instrumental for the definition of a criterionthat ensures the boundedness of any high order interpolation scheme.12.2The Convection Boundedness Criterion (CBC)A numerical scheme is expected to preserve the physical properties of the phenomenon it is trying to describe or approximate.
Therefore the conditions that abounded convection scheme should satisfy can best be understood by analyzing thephysical properties of convection. Since convection transports fluid properties fromupstream to downstream, then approximation to convection should possess thistransportive attribute. Thus, numerical convection schemes should be upwindbiased or else they will lack the convective stability. Therefore in addition to thevalues at the nodes straddling the interface /C and /D , the value at the far upwindnode, i.e., /U , is also important in analyzing advective schemes.
Values at nodesfarther away are less important. In the NVF presented above, values are normalizedsuch that the effect of /U is also considered. This is extremely useful as it helpsidentifying the conditions for which the numerical convection scheme is monotone.Whereas Spekreise’s [19] and Barth and Jespersen [20] definition of a monotonescheme (or bounded scheme) involves all neighbors surrounding the face, Leonard[21] and Gaskel and Lau [4] based their definition of monotonicity only on theneighboring points along the local coordinate system such that12.2The Convection Boundedness Criterion (CBC)437minð/C ; /D Þ /f maxð/C ; /D Þð12:11ÞNormalizing, the above condition becomes~ ;1 /~ max /~ ;1min /CfCð12:12ÞThe Convection Boundedness Criterion (CBC) for implicit steady state flowcalculation developed by Gaskell and Lau states that for a scheme to have theboundedness property its functional relationship should be continuous, should be~ and from above by unity, and should pass through thebounded from below by /C~ [ 1 or~ \1 ; and for /points (0, 0) and (1, 1), in the monotonic range 0\/CC ~~~/C \0; the functional relationship f /C should be equal to /C : The aboveconditions illustrated on an NVD in Fig.
12.4, can be mathematically formulated as8 >~>> f /C >>>>~ ¼1>f />C>< ~~ \1~~ \f //f ¼ f /C with /CC>>>>~>f /C ¼ 0>> >>>~ ¼/~:f /CCcontinuous~ ¼1if /C~ \1if 0\/Cð12:13Þ~ ¼0if /C~ \0 or /~ [1if /CCThe Convection Boundedness Criterion is quite intuitive and can be interpretedby referring to Figs. 12.4 and 12.5. When /C is in a monotonic profile the interpolation profile at the cell surface should not yield any new extremum. Thus it isconstrained by the / values at the nodes straddling the face. As the value of /C getcloser to /D while still in the monotonic regime, the value of /f will also tendtoward /D : When /C becomes equal to /D ; then /f also becomes equal to /D andFig.
12.4 The ConvectionBoundedness Criterion(CBC) on an NVD Diagramshowing the region~ is boundedwhere /ff1DOWNWIND3/ 4DINPWU01/ 21C43812 High Resolution Schemes~ andFig. 12.5 Values of /fthe Convection BoundednessCriterion1fC0vfuC01fuvf1vf0uCf~ ;/~ passes through the point (1, 1). When the value ofthus the condition that /Cf~ [ 1; / is assigned the upwind value, i.e., / : This has the/C is such that /CfCeffect of yielding the largest outflow condition possible while fulfilling the condition that /f is bounded by the nodes straddling the cell face. This behavior meansthat any undue oscillation will be damped since /C will tend to a lower valuebecause outflow is larger than inflow in these conditions.Therefore if there is no external physical mechanism to yield the extrema(a source term for example) the extrema will die out. A similar mechanism takes~ \0: However as / gets closer to / coming from theplace when /CCUnon-monotonic region, /f will be equal to the upwind value /C until /C ¼ /Uimplying the condition that the profile of /f passes through the point (0, 0).~ [ 1 the solution will be in a region where convection is~ \0 or /When /CCdominant and the upwind approximation will be an excellent one.12.3High Resolution (HR) SchemesConstructing a bounded HO scheme, i.e., a HR scheme, using the NVD is relativelya simple exercise.
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