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 10

(2.8), Eq. (2.72) reduces to ½s v ¼ sxx u þ sxy v þ sxz w i þ syx u þ syy v þ syz w j þ szx u þ szy v þ szz w kð2:73ÞThe above equation can be derived using matrix multiplication as2sxx½s v ¼ 4 syxszxsxysyyszy32 3 23sxx u þ sxy v þ sxz wsxzusyz 54 v 5 ¼ 4 syx u þ syy v þ syz w 5szx u þ szy v þ szz wszzwð2:74ÞIn a similar way the divergence of a tensor s is found to be a vector given by @sxx @syx @szx@sxy @syy @szyþþþþ½ r s ¼iþj@x@y@z @x@y@z@sxz @syz @szzþþkþ@x@y@zð2:75ÞThe double dot product of two tensors s and frvg is a scalar computed as1@u@v@wþ ij þ ikþ01 B @x@x@x CCBiisxx þ ijsxy þ iksxz þCB @u@v@wCjiþjjþjkþðs : rvÞ ¼ @ jisyx þ jjsyy þ jksyz þ A : BB @y@y@y CCBkiszx þ kjszy þ kkszz@ @u@v@w Aki þ kj þ kk@z@z@z0iið2:76ÞThe final value is obtained by expanding the above product and performing thedouble dot product on the various terms.

For example,ijsxy : ji@u@u@u@u¼ i j : j isxy¼ |{z}i i sxy¼ sxy|{z}@y@y@y@y¼1¼1ð2:77Þ322 Review of Vector CalculusPerforming the same steps on every term in the expanded product, the final formof ðs : rvÞ is obtained asðs : rvÞ ¼ sxx@u@u@u@vþ sxyþ sxzþ syx@x@y@z@x@v@v@w@w@wþ szyþ szzþ syy þ syz þ szx@y@z@x@y@zð2:78Þ2.5 Fundamental Theorems of Vector CalculusAll mathematical formulations presented in this book will be performed usingvectors. Therefore a good knowledge of the fundamental theorems of vector calculus is helpful.

A brief review of some of these theorems is presented next.2.5.1 Gradient Theorem for Line IntegralsThe gradient theorem for line integrals relates a line integral to the values of afunction at its endpoints [22]. It states that if C is a smooth curve, as shown inFig.

2.13, described by the vector rðtÞ ¼ r½xðtÞ; yðtÞ; zðtÞ for a t b, and s is ascalar function whose gradient, rs, is continuous on C, thenZrs dr ¼ sðr ðbÞÞ sðr ðaÞÞð2:79ÞCwhere a and b are the endpoints of C. It follows that the value of the integral over aclosed contour is zero.Fig. 2.13 A schematicdepiction of a curve C of ascalar function s showing itsend points and the positionvector rðtÞzCr(t)r(a)r(b)yx2.5 Fundamental Theorems of Vector Calculus332.5.2 Green’s TheoremGreen’s theorem expresses the contour integral of a simple closed curve C in termsof the double integral of the two dimensional region R bounded by C [23–26].Let C denotes the closed contour (Fig. 2.14) of a two dimensional region R. Ifuðx; yÞ and vðx; yÞ are functions of continuous partial derivatives defined on R, thenIZZ @v @uðudx þ vdyÞ ¼dxdyð2:80Þ@yCR @xIn Eq.

(2.80) the contour integral along C is taken positive in the counterclockwise direction.CRdSdrFig. 2.14 Schematic of a region R and its closed contour CGreen’s theorem can be written in a more compact form using vectors. For thatpurpose defining dr; v and the area vector dS asdr ¼ dxi þ dyjv ¼ ui þ vjdS ¼ dxdykthen the vector form of Green’s theorem is given byIZZv dr ¼½r v dSCð2:81Þð2:82ÞRGreen’s theorem is helpful for computing line integrals arising in two-dimensional flows.Example 7HCompute 2y3 dx þ 3xy2 dy where C is the CCW-oriented boundary of theCregion R shown in Fig.

2.15.The vector field in the above integral is ðu; vÞ ¼ ð2y3 ; 3xy2 Þ. The line integralcan be computed directly. But, it is more easily computed using Green’stheorem using a double integral. Applying Green’s theorem the integrand isobtained as342 Review of Vector Calculus@v @u¼ 3y2 6y2 ¼ 3y2@x @ySince the line integral is over a semi circle, the region R is mathematicallygiven by 1x1pffiffiffiffiffiffiffiffiffiffiffiffiffi0 y 1 x2RFig. 2.15 Example 7The value of the integral is obtained asZZ I2y3 dx þ 3xy2 dy ¼pffiffiffiffiffiffiffiffiZ1 Z1x2@v @uy2 dydxdA ¼ 3@x @y1DCCZ1¼ 310ffi!y¼pffiffiffiffiffiffiffiZ11x23=2y3 1 x2dxdx ¼ 3 y¼01Let x ¼ cos h ) dx ¼ sin hdhThusIZp2y dx þ 3xy dy ¼ 3CZpsin hdh þ2sin2 h cos2 hdh200 h sin 2h p h sin 4h p¼ þ 24 0 832 03p¼82.5.3 Stokes’ TheoremStokes’ theorem is a higher dimensional version of Green’s theorem [27–29].Whereas Green’s theorem relates a line integral to a double integral, Stokes theoremrelates a line integral to a surface integral. Let v be a vector field, S an orientedsurface, and C the boundary curve of S, oriented using the right-hand rule, asdepicted in Fig.

2.16. Stokes’ theorem states the following:2.5 Fundamental Theorems of Vector CalculusFig. 2.16 A surface S in athree-dimensional space ofcontour C35zSdSvxCZI½r v dS ¼v drð2:83ÞCSwhere r is such that dr=ds is the unit tangent vector and s the arc length of C. Thecurve of the line integral, C, must have positive orientation, meaning that dr pointscounterclockwise when the surface normal, dS, points toward the viewer, followingthe right-hand rule.2.5.4 Divergence TheoremLet V represents a volume in three-dimensional space (Fig. 2.17) of boundary S.

Letn be the outward pointing unit vector normal to S. If v is a vector field defined on V,then the divergence theorem [30, 31] (also known as Gauss’ theorem) states thatIZðr vÞdV ¼Vv n dSð2:84ÞSThe divergence theorem implies that the net flux of a vector field through aclosed surface is equal to the total volume of all sources and sinks (i.e., the volumeintegral of its divergence) over the region inside the surface.

It is an importanttheorem for fluid dynamics.362 Review of Vector CalculusFig. 2.17 A volume inthree-dimensional space witha piecewise smoothboundary SzSdSdS = ndSyvVxThe divergence theorem can be used in different contexts to derive many otheruseful identities (corollaries) [32].

In specific it can be applied to the product of a scalarfunction, s, and a non-zero constant vector, to derive the following important relation:ZI½rsdV ¼VsdSð2:85ÞSThe divergence theorem is equally applicable to tensors, in which case it iswritten asZI½r sdV ¼ ½s ndSð2:86ÞVSExample 8Use the divergence theorem to evaluateZZ F dS@V5where F ¼ 3x þ z5 i þ ðy2 sinðx2 zÞÞj þ xz þ yex kand V is a box defined by0x1with an outward pointing surface0y30z22.5 Fundamental Theorems of Vector Calculus37SolutionThis is a difficult field to integrate however using the divergence theorem itcan be transformed toZZZZZ F dS ¼ðr FÞdV@VVwhere the divergence of F is obtained asr F ¼ 3 þ 2y þ xintegrating over the box, the integral is evaluated asZ1Z3Z2Z1Z3ZZð3 þ 2y þ xÞdzdydx ¼ð6 þ 4y þ 2xÞdydx F dS ¼@V0 0 00 0Z1ð18 þ 18 þ 6xÞdx ¼ 36 þ 3 ¼ 39¼02.5.5 Leibniz Integral RuleThe Leibniz integral rule gives a formula for differentiating a definite integralwhose limits are functions of the differential variable [33–36].

Let /ðx; tÞ representsa function that depends on a space variable x and time t. Then Leibniz integral rulecan be stated as followsddtZbðtÞZbðtÞ/ðx; tÞdx ¼að t Þ@/@b@adx þ /ðbðtÞ; tÞ /ðaðtÞ; tÞ@t@t@t|fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl}aðtÞTermIITermIII|fflfflfflfflfflffl{zfflfflfflfflfflffl}ð2:87ÞTerm IThe meaning of the various terms in Eq. (2.87) can be inferred from Fig. 2.18.The first term on the right side gives the change in the integral because / ischanging with time t, while the second and third terms accounts for the gain andloss in area as the upper and lower bounds are moved, respectively.382 Review of Vector Calculus( x,t + t )Term I( x,t )Term IIITerm IIa (t ) a (t + t )b(t ) b(t + t )tFig.

2.18 Curves showing the spatial distribution of a function at times t and t þ DtThe three-dimensional form of this formula applied to a volume V ðtÞ enclosed bya surface SðtÞ with its surface elements moving with a velocity vs can be written asddtZZ/dV ¼V ðt ÞV ðtÞ@/dV þ@tZ/ðvs nÞdSð2:88ÞSðtÞwhere /ðt; xÞ is a scalar function of space and time. For a non-moving volume V,the equation reduces toddtZZ/dV ¼VV@/dV@tð2:89ÞThe above equations are also applicable to vectors and tensors.2.6 ClosureThe chapter offered a brief review of vector and tensor operations. In addition thefundamental theorems of vector calculus were presented.

The next chapter will relyon information presented in this chapter to derive the conservation equationsgoverning the transfer phenomena of interest in this book.2.7 Exercises392.7 ExercisesExercise 1Let v1 ; v2 and v3 be three vectors given by231v1 ¼ 4 2 55231v2 ¼ 4 1 510238v3 ¼ 4 5 52Find:a.b.c.d.v1 þ v2 ; v1 þ 2v2 ; 3v2 4v3jv1 j; jv2 j; jv3 jv1 v2 ; v3 v2 ; v2 ð v1 v3 ÞA unit vector in the direction of ðv1 þ v2 þ v3 ÞExercise 2Let i; j and k be unit vectors in the x; y; and z direction, respectively, and let v beany vector, which in a Cartesian coordinate system is given byv ¼ ui þ vj þ wkProve thatv ¼ C ½i ðv iÞ þ j ðv jÞ þ k ðv kÞwhere C is a constant to be determined.Exercise 3Find rs if s is the scalar function given bya.

s ¼ y2 e2x3zb. s ¼ Lnðx þ y2 þ z3 Þc. s ¼ tan1xyzExercise 4If s is a scalar function and v is a vector function, prove the following identities:a.b.c.d.r ðrsÞ ¼ 0r ðsvÞ ¼ sr v þ v rsr ðsvÞ ¼ sr v þ rs vr ðv1 v2 Þ ¼ v2 ðr v1 Þ v1 ðr v2 Þ402 Review of Vector CalculusExercise 5Use Green’s theorem to calculate the area enclosed by an ellipse of semi-major andsemi-minor axes a and b, respectively.Exercise 6Find the Laplacian of the scalar s ðr2 sÞ for the cases when s is given by:a.

s ¼ x3 þ z2 e2y3xb. s ¼ z þ Lnðx þ yÞc. s ¼ sin1 ðx þ y þ zÞExercise 7Verify the divergence theorem for the parallelepiped with centre at the origin andfaces in the planes x ¼ 2; y ¼ 1; z ¼ 4 and v given bya. v ¼ 5i þ 7j 3kb. v ¼ iðy zÞ þ jðx zÞ þ kðx yÞc.

v ¼ iy2 z þ jxz2 þ kx2 yExercise 8For a surface S representing the upper half of a cube centered at the origin, with oneof its vertices at ð1; 1; 1Þ, and with edges parallel to the axes, verify Stokes’stheorem for the case when the curve C is the intersection of S with the xy plane andthe vector v is given byv ¼ iðy þ zÞ þ jðx þ zÞ þ kðx þ yÞExercise 9Find a function F for which the divergence is the given function K in the followingcases:a. K ðx; y; zÞ ¼ p.b. K ðx; y; zÞ ¼ z2 x.pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffic. K ðx; y; zÞ ¼ ðx2 þ z2 ÞExercise 10RRUse the divergence theorem to evaluate the integralð6xi þ 4yjÞ dF where the@Fsurface is a sphere defined as @F ! x2 þ y2 þ z2 ¼ 10.2.7 Exercises41Exercise 11Let F be a radial vector field defined as F ¼ xi þ yj þ zk and let C to be a solidcylinder of radius r and height h with its axis coinciding with the x-axis and itsbottom and top faces located along the x ¼ 0 and x ¼ b plane, respectively. VerifyGauss theorem in both flux and divergence forms.Exercise 12Given a square matrix A defined as0a11BA ¼ @ a21a12...1..