Moukalled F., Mangani L., Darwish M. The finite volume method in computational fluid dynamics. An advanced introduction with OpenFOAM and Matlab (811443), страница 13
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(3.8) in the derivation yieldingZ D½qv þ ½qvr v f dV ¼ 0ð3:18ÞDtVAgain for the integral to be zero over any control volume, the integrand has to bezero. Thus,D½qv þ ½qvr v ¼ fDtð3:19ÞExpanding the material derivative of the momentum term and regrouping, thenon-conservative form is obtained asDvDqþvþ qr v ¼ fqð3:20ÞDtDt|fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl}Continuity523 Mathematical Description of Physical PhenomenaApplying the continuity constraint and expanding the material derivative, thenon-conservative form of the momentum equation reduces to@vþ ðv rÞv ¼ fq@tð3:21Þ3.5.2 Conservative FormThe conservative (or flux) version is obtained by applying the form of the Reynoldstransport theorem given by Eq.
(3.7) and is written asZ V@½qv þ r fqvvg f dV ¼ 0@tð3:22ÞBy setting the integrand to zero for the integral to be zero for any volume V, theconservative form of the momentum equation is obtained as@½qv þ r fqvvg ¼ f@tð3:23Þwhere qvv is the dyadic product, described in Chap. 2, which is a special case oftensor product with its divergence being a vector.Both forms will be used in this book for better describing the discretizationconcepts and for showing actual implementation details. In the derivations to followthe conservative form will be used.
The non-conservative form can be easilyobtained from the conservative form at any step by invoking the continuity constraint as explained above.The full form of the momentum equation is obtained once the external surfaceand body forces acting on the control volume are specified. The force f is split intotwo parts one denoted by f s representing the surface forces and the second by f brepresenting the body forces such thatf ¼ fs þ fbð3:24ÞThe details of these forces are given next.3.5.3 Surface ForcesFor the arbitrary macroscopic volume element depicted in Fig. 3.4, the forces actingon its surface are due to pressure and viscous stresses which can be expressed in3.5 Conservation of Linear Momentum53fs =Fig. 3.5 The surface forcesacting on a differential surfaceelement expressed in terms ofthe stress tensordS =ndSndSterm of the total stress tensor Σ, as shown in Fig.
3.5. In general there are ninecomponents of stress at any given point; one normal component and two shearcomponents (parallel to the surface that receives the stress) in each coordinateplane. Thus in Cartesian coordinates the stress tensor is given by0RxxR ¼ @ RyxRzxRxyRyyRzy1RxzRyz ARzzð3:25Þwhere terms of the form Rii represent normal stresses and Rij shear stresses.A normal stress can be either a compression, if Rii 0, or a tension, if Rii 0.
Themost important compressive normal stress is usually due to pressure rather than toviscous effects. The component Rij represents the stress acting on face i in thej direction with the direction of face i being positive if the outward normal to theface is in the positive direction.In practice the stress tensor is split into two terms such that00pR ¼ @ 001sxxzfflffl}|fflffl{1 B RxxþpB0 0Bp 0A þ BB syxB0 p@szxsxysyyzfflffl}|fflffl{Ryyþpszysxz CCCC ¼ pI þ ssyz CCszz Azffl}|ffl{Rzzþpð3:26Þwhere I is the identity tensor of size (3 × 3), p is the pressure, and s is the deviatoricor viscous stress tensor. The pressure is the negative of the mean of the normalstresses and is given byp¼1Rxx þ Ryy þ Rzz3ð3:27Þ543 Mathematical Description of Physical PhenomenaThe surface force acting on a differential surface element of area dS and orientation n, as illustrated in Fig.
3.5, is ðR nÞdS. Applying the divergence theorem,the total surface force acting on the control volume is given byZZZf s dV ¼ R n dS ¼ r R dV ) f s ¼ ½r R ¼ rp þ ½r s ð3:28ÞVSV3.5.4 Body ForcesBody forces, which are presented as forces per unit volume, may also arise due to avariety of effects. There are plenty of examples, but the predominant ones are givennext.3.5.4.1 Gravitational ForcesThe force representing the weight of the material volume per unit volume in thepresence of a gravitational field is denoted by gravitational force (Fig.
3.6) andgiven byf b ¼ qgð3:29Þwhere g is the gravitational acceleration vector.Fig. 3.6 Body forces actingon a differential elementzfb = gyx3.5 Conservation of Linear Momentum553.5.4.2 System RotationWhen solving fluid flow problems in a rotating frame of reference, the forces arisingas a result of the rigid body rotation of the reference frame should be accounted for.These can be viewed as body forces of the formf b ¼ 2q½- v q½- ½- r|fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl}Coriolis forcesð3:30ÞCentrifugal forceswhere - is the angular velocity of the rotating reference frame and r is the positionvector (Fig. 3.7).
Note that gravitational and centrifugal forces are dependent onposition but not on velocity. Thus they can be absorbed into a modified pressureand hence effectively ignored as a separate entity unless they appear in boundaryconditions. Coriolis forces however have to be treated explicitly. Other forces, suchas magnetic and electric, may be added depending on the particular situation. Dueto the many possible types of body forces, no specific type will be adopted in theequations to follow and the generic f b force is retained.Substituting the external force f in Eq. (3.23) by its equivalent expression, thegeneral conservative form of the momentum equation is obtained as@½qv þ r fqvvg ¼ rp þ ½r s þ f b@tð3:31ÞrxOFig.
3.7 Body forces due to a rigid body rotation in a rotating frame of reference3.5.5 Stress Tensor and the Momentum Equationfor Newtonian FluidsTo proceed further with the momentum equation, the type of fluid should be knownin order to relate the stress tensor s to the flow variables. For a Newtonian fluid, thestress tensor is a linear function of the strain rate [2] and is given by563 Mathematical Description of Physical Phenomenanos ¼ l rv þ ðrvÞT þ kðr vÞIð3:32Þwhere l is the molecular viscosity coefficient, k the bulk viscosity coefficientusually set equal to ð2=3Þl ðk ¼ ð2=3ÞlÞ, the superscript T refers to thetranspose of rv, and I is the unit or identity tensor of size (3 × 3) defined as21I ¼ 40030 01 050 1The expanded form of the stress tensor in adinate system can be written as2@u@v @u6 2l @x þ kr v l @x þ @y6 6 @v @u@v6þs ¼ 6l2l þ kr v6 @x @y @y4 @u @w@w @vþþll@z @x@y @zð3:33Þthree-dimensional Cartesian coor 3@u @wþl7 @z @x 77@w @v7lþ7@y @z75@wþ kr v2l@zð3:34ÞThe divergence of the stress tensor is a vector that can be expressed ash i½r s ¼ r l rv þ ðrvÞT þ rðkr vÞ 32 @@u@@v @u@@u @w2lþkrvþlþlþþ76 @x@x@y@x @y@z@z @x766 776@@v@u@@v@@w@v72l þ kr v þlþ¼6þ76 @x l @x þ @y@y@y@z@y @z766 7 54@@u @w@@w @v@@wlþlþ2lþ kr vþþ@x@z @x@y@y @z@z@zð3:35ÞSubstituting into Eq.
(3.31), the final conservative form of the momentumequation for Newtonian fluids becomesn hio@½qv þ r fqvvg ¼ rp þ r l rv þ ðrvÞT þ rðkr vÞ þ f b ð3:36Þ@tFor later reference the momentum equation is expanded intono@½qv þ r fqvvg ¼ r flrvg rp þ r lðrvÞT þ rðkr vÞ þ f b ð3:37Þ@t|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}Qv3.5 Conservation of Linear Momentum57and rewritten as@½qv þ r fqvvg ¼ r flrvg rp þ Qv@tð3:38ÞFor incompressible flows, the divergence of the velocity vector is zero, i.e.,r v ¼ 0, and the momentum equation reduces ton hio@½qv þ r fqvvg ¼ rp þ r l rv þ ðrvÞT þ f b@tð3:39ÞIf the viscosity is constant, the momentum equation can be further simplified.Taking just the first component of the vector equation [Eq.
(3.35)], and assuming lis constant, the following can be written:l@@u@@v @u@@u @w2þþþlþl@x @x@y @x @y@z @z @x 22222@ u @ u @ u @ v @ u @2wþ¼lþþþþ@x2 @x2 @y2 @yx @z2 @zx 2@ u @2u @2u@2u @2v @2wþ¼lþþþ 2þ@x2 @y2 @z2@x@yx @zx 2@ u @2u @2u@ @u @v @wþ þ¼lþþþ@x2 @y2 @z2@x @x @y @zð3:40ÞSubstitution in Eq. (3.37) yields after simplification@½qv þ r fqvvg ¼ rp þ lr2 v þ f b@tð3:41ÞFor inviscid flows the viscosity is zero and the momentum equation forincompressible and compressible inviscid flows becomes@½qv þ r fqvvg ¼ rp þ f b@tð3:42Þ3.6 Conservation of EnergyThe conservation of energy [6, 10] is governed by the first law of thermodynamicswhich states that energy can be neither created nor destroyed during a process; itcan only change from one form (mechanical, kinetic, chemical, etc.) into another.Consequently, the sum of all forms of energy in an isolated system remainsconstant.583 Mathematical Description of Physical PhenomenaFig.
3.8 A material volumemoves with the particles itencloseszMV(t+ t)MV(t)yxConsidering the material volume shown in Fig. 3.8, of mass m, density q, andmoving with a velocity v. Defining the total energy E of the material volume attime t as the sum of its internal and kinetic energies, then E can be written as1E ¼ m ^u þ v v2ð3:43Þwhere ^u is the fluid specific internal energy (internal energy per unit mass). The firstlaw of classical thermodynamics applied to the material volume states that the rateof change of the total energy of the material volume is equal to the rate of heataddition and work extraction through its boundaries.
Mathematically this is givenby dE¼ Q_ W_dt MVð3:44ÞThe adopted sign convention is such that heat added to the material volume andwork done by the material volume are positive. To apply the Reynolds transporttheorem on the material volume, B is set equal to E and b to e (the total energy perunit mass) such thatB¼E)b¼dE1¼ ^u þ v v ¼ edm2ð3:45ÞThe net rate of heat transferred to the material element Q_ is the sum of twocomponents. The first component is the rate transferred across the surface of theelement Q_ S and the second generated/destroyed (e.g., due to a chemical reaction)within the material volume Q_ V .
Moreover, the net rate of work done by the materialvolume W_ is due to the rate of work done by the surface forces W_ S and the rate of_ b . Thus the first law can be written aswork done by the body forces W3.6 Conservation of Energy59dEdt¼ Q_ V þ Q_ S W_ b W_ Sð3:46ÞMVBy definition, work is due to a force acting through a distance and power is therate at which work is done. Therefore the rate of work done by body and surfaceforces can be represented byW_ b ¼ Zðf b vÞdVVW_ S ¼ Zð3:47Þðf S vÞdSSThe rate of work due to surface forces can be expanded by replacing f S by itsequivalent expression as given in Eq.
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