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 15

(3.93) to the various conservation equations derived earlier, itcan be easily inferred that by assigning the right values for /; C/ , and Q/ ,Eq. (3.93) is a general equation that can represent any of the conservation equations. This is a very important observation that will reduce the necessary developments of the numerical techniques in the coming chapters by concentrating on thegeneral equation [Eq. (3.93)] rather than the individual conservation equations.3.8 Non-dimensionalization ProcedureThe differential equations representing conservation laws are rarely solved usingdimensional variables. The common practice is to write these equations in anon-dimensional form using dimensionless quantities that are obtained through theuse of proper characteristic scales. The use of non-dimensional variables has severaladvantages.

It allows reducing the number of appropriate parameters for theproblem considered and helps revealing the relative magnitude of the various termsin the conservation equation and consequently those that can be neglected.683 Mathematical Description of Physical PhenomenaThis simplifies the equation to be solved and leaves only terms of similar order ofmagnitude, which results in better numerical accuracy. In addition, the generatedsolution will be applicable to all dynamically similar problems.A dimensional variable is transformed into a non-dimensional one by dividingthe variable by a quantity (composed of one or more physical properties) that hasthe same dimension as the original variable.

For example spatial coordinates can bedivided by a characteristic length; velocity can be divided by a characteristicvelocity or a combination of quantities lref =qref lref that have collectively thesame units(m/s); pressure is usually divided by a reference dynamic as velocity2 pressure qref vref ; time can be divided by the ratio of a characteristic length to a reference velocity lref =vref , and so on.

The best way to fully understand how towrite equations in non-dimensional form is through an example. For that purpose,an incompressible viscous flow with constant viscosity and thermal conductivity,and with body forces acting in the y-direction (i.e., the gravitational acceleration isgiven by g ¼ ð0; g; 0Þ) is considered. The equations governing conservation ofmass, momentum, and energy in a three-dimensional Cartesian coordinate systemwith no heat generation are given by@u @v @wþ þ¼0@x @y @zð3:94Þ 2@@@@@p@ u @2u @2uðquÞ þ ðquuÞ þ ðqvuÞ þ ðqwuÞ ¼ þ lþþ@t@x@y@z@x@x2 @y2 @z2ð3:95Þ 2@@@@@p@ v @2v @2vðqvÞ þ ðquvÞ þ ðqvvÞ þ ðqwvÞ ¼ þ lþþ qg@t@x@y@z@y@x2 @y2 @z2ð3:96Þ 2@@@@@p@ w @2w @2wðqwÞ þ ðquwÞ þ ðqvwÞ þ ðqwwÞ ¼ þ lþþ 2@t@x@y@z@z@x2 @y2@zð3:97Þ 2@@@@@ T @2T @2TðqT Þ þ ðquT Þ þ ðqvT Þ þ ðqwT Þ ¼ kcpþþ@t@x@y@z@x2 @y2 @z2ð3:98ÞWhen body forces are of negligible magnitude in comparison with other forces,the term qg can be set to zero and removed from the equations.

In such situation,the flow field is independent of the temperature field and solution for the velocityfield can be established separately followed by the solution to the temperature field.However for a flow to exist the fluid should be forced through the domain.3.8 Non-dimensionalization Procedure69Therefore the fluid should possess an inlet velocity. This velocity becomes animportant parameter (the characteristic velocity) when writing the equations innon-dimensional forms and the heat transfer mechanism, if any, is said to occur byforced convection.On the other hand, when a flow field is naturally established due to a temperaturedifference in the domain, body forces cannot be neglected.

In such situations,variations in temperature cause variations in density (as mentioned earlier), whichgive rise to buoyancy forces that drive the flow. In this case the transfer of heat isstated to happen by natural convection. As the flow is initiated naturally, a characteristic velocity is not apparent and cannot be part of the dimensionless numbersince its scale is not known. Therefore in order to write the velocity in anon-dimensional form, a combination of physical quantities that has the samedimension as a velocity should be used.

The following discussion assumes a naturalconvection problem.If the difference in temperature DT ¼ T T1 (where T1 is a reference temperature between the minimum and maximum temperature in the domain, usuallytaken as the average value) is small such that terms of order DT 2 or higher can beneglected, then the value of density at any temperature T can be written as afunction of its value at the reference temperature T1 using a truncated Taylor seriesexpansion asq ¼ qjT¼T1 þdqð T T1 ÞdT T¼T1ð3:99Þwhere terms of order DT 2 or higher are omitted. Introducing the coefficient ofvolume expansion b defined as 1 @qb¼q @T pð3:100Þthe equation for density (or equation of state) becomesq ¼ q1 ½1 bðT T1 Þð3:101Þwhich is known in the literature by the Boussinesq approximation [11].

Using thisexpression for q in the body force term only and denoting the constant density valueby q to simplify the notation, the y-momentum equation is transformed to@@@@ðqvÞ þ ðquvÞ þ ðqvvÞ þ ðqwvÞ@t@x@y@z 2@@ v @2v @2v¼ ðp þ qgyÞ þ lþþþ qgbðT T1 Þ@y@x2 @y2 @z2ð3:102Þ703 Mathematical Description of Physical PhenomenaThis clearly shows that in solving natural convection problems the momentumand energy equations are coupled together necessitating a simultaneous solution ofboth equations.The non-dimensional forms of the conservation equations are obtained bydefining the following dimensionless parameters:xyzuvw^¼^x ¼ ; ^y ¼ ; ^z ¼ ; ^u ¼; ^v ¼;wLLLl=ðqLÞl=ðqLÞl=ðqLÞtp þ qgy ^T T1^t ¼; ^p ¼ 2;T ¼qL2 =ll =ðqL2 ÞTmax T1ð3:103Þwhere L is a characteristic length, l the dynamic viscosity of the fluid, Tmax themaximum temperature in the domain, and the over ^ is used to designatenon-dimensional quantities.

The various expressions in the conservation equationsare written in terms of the new variables as described next. The procedure isexplained by considering a typical term from each category.Typical term in the continuity equation:@u @ ½l^u=ðqLÞ l=ðqLÞ @^ul @^u¼¼¼ 2@x@ ðL^xÞL @^x qL @^xð3:104ÞTypical terms in the momentum equations:@@ ðl^u=LÞl=L @^ul2 @^uðquÞ ¼¼¼@t@ ðqL2^t=lÞ qL2 =l @^t qL3 @^t@@ ½l2 =ðqL2 Þ^u^u l2 =ðqL2 Þ @l2 @ðquuÞ ¼¼ð^u^uÞ ¼ 3 ð^u^uÞ@t@ ðL^xÞL@^xqL @^xp þ qgy@^p @ ðp þ qgyÞ=½l2 =ðqL2 Þ^¼)p¼ 2l =ðqL2 Þ@^x@ ðx=LÞqL3 @p@pl2 @^p)¼ 3¼ 2@x qL @^xl @xð3:105Þð3:106Þð3:107Þ@2u@ 2 ½l^u=ðqLÞl=ðqLÞ @ 2 ^ul2 @ 2 u^¼l¼l¼ 3 22222@xL@^xqL @^x@ ðL^xÞð3:108ÞqgbðT T1 Þ ¼ qgbðTmax T1 ÞT^ ¼ qgbðDT ÞT^ð3:109ÞlTypical terms in the energy equation: @ q T1 þ DT T^@lDT @ T^ðqT Þ ¼¼@tL2 @^t@ ðqL2^t=lÞð3:110Þ3.8 Non-dimensionalization Procedure71 @ l^u T1 þ DT T^ =L@lT1 @^u lDT @ ^ ^uTðquT Þ ¼þ 2¼ 2@xL @^xL @^x@ ðL^xÞ@ 2 T1 þ DT T^@2TkDT @ 2 T^k 2 ¼k¼ 22@xL @^x2@ ðL^xÞð3:111Þð3:112ÞSubstituting terms by their equivalent expressions, the non-dimensional forms ofthe continuity, momentum, and energy equations are obtained as^@^u @^v @ wþ þ¼0@^x @^y @^z 2@^u @@@@^p@ ^u @ 2 ^u @ 2 ^u^ ^uÞ ¼ þþ ð^þþu^uÞ þ ð^v^uÞ þ ðw@^t @^x@^y@^z@^x@^x2 @^y2 @^z2ð3:113Þð3:114Þ 2@^v @@@@^p@ ^v @ 2^v @ 2^v^ ^vÞ ¼ þþ ð^þþu^vÞ þ ð^v^vÞ þ ðwþ Gr T^ ð3:115Þ@^t @^x@^y@^z@^y@^x2 @^y2 @^z2 2^^ @2w^ @2w^@w@@@@^p@ w^ Þ þ ð^vw^ Þ þ ðw^w^Þ ¼ þþ ð^uwþþ 2@^y@^z@^z@^x2 @^y2@^z@^t @^xð3:116Þ@ T^@ ^ @ ^ @ ^1 @ 2 T^ @ 2 T^ @ 2 T^^^vT þ^T ¼þþþuT þw@^t @^x@^y@^zPr @^x2 @^y2 @^z2ð3:117Þwhere Gr is the Grashof number, Pr is the Prandtl number, and v the kinematicviscosity defined asGr ¼gbDTL3v2Pr ¼lcpkv¼lqð3:118ÞThe Grashof and Prandtl numbers [12–14] are dimensionless groups formed of acombination of the involved physical properties.

Therefore the number of parameters affecting the solution was reduced to two and solutions can be generated fordifferent values of these two parameters. Moreover any single solution will be validfor many combinations of the physical properties of which these two numbers arecomposed, as long as these combinations result in the Gr and Pr values for whichthe solution was obtained. The physical significance of these two dimensionlessnumbers and others that may arise when writing the conservation equations innon-dimensional forms using other dimensionless parameters under different conditions is discussed next.723 Mathematical Description of Physical Phenomena3.9 Dimensionless NumbersWriting the conservation equations in non-dimensional forms, results in dimensionless numbers that are very useful for performing parametric studies of engineering problems. For incompressible viscous flow, the dimensionless parametersgoverning natural convection heat transfer were reduced to the two dimensionlessnumbers Gr [12–14] and Pr [12–14].

Under different conditions (e.g., compressibleflows, Porous flows, etc.) other types of fluid forces and dissipation terms may beincluded in the governing equations resulting in different non-dimensional groups.For flow in porous media, for example, Darcy number ðDaÞ [15, 16] emerges as animportant parameter, for a free surface flow the Weber number ðWeÞ [17, 18], for anopen channel flow the Froude number ðFr Þ [19], for a compressible flow the Machnumber ðM Þ [20], and so on. Some of the most important dimensionless groups arediscussed below.3.9.1 Reynolds NumberThe Reynolds number ðReÞ [12, 13] is defined asRe ¼qULlð3:119Þand may be interpreted as a measure of the relative importance of advection (inertia)to diffusion (viscous) momentum fluxes.