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 8

2.6 Example 22.2.6 Gradient of a Scalar and Directional DerivativesAn important vector operator, which arises frequently in fluid dynamics, is the “del”(or “nabla”) operator defined asr¼@@@iþ jþ k@x@y@zð2:17ÞWhen the “del” operator is applied on a scalar variable s it results in the gradient of s[7, 8] given byrs ¼@s@s@siþ jþ k@x@y@zð2:18ÞThus the gradient of a scalar field is a vector field indicating that the value of schanges with position in both magnitude and direction.The projection of rs in a certain direction of unit vector el is given byds¼ rs el ¼ krsk cosðrs; el Þdlð2:19Þand is called the directional derivative of s along the direction of the unit vector el ,as schematically depicted in Fig. 2.7.

The maximum value of the directionalderivative is krsk and is obtained when cosðrs; el Þ ¼ 1, that is in the direction ofrs. Therefore, it can be stated that the gradient of a scalar field s indicates thedirection and magnitude of the largest change in s at every point in space.Moreover, rs is normal to the constant s surface that passes through that point.162 Review of Vector Calculuszlsdirectional ds= s el=derivative dls = s ( x, y, z )elCkijyxFig. 2.7 The rate of change of sðx; y; zÞ in the direction of vector elExample 3Let f ðx; y; zÞ ¼ x2 y þ y2 z þ z2 x(a) find rf at point ð3; 2; 0Þ.(b) find the derivative at point ð3; 2; 0Þ along the direction ð1; 2; 2Þ:Solution(a)@f¼ 2xy þ z2@x@f¼ x2 þ 2yz@y@f¼ y2 þ 2xz@z rf ¼ 2xy þ z2 i þ x2 þ 2yz j þ y2 þ 2xz kThusrf jð3;2;0Þ ¼ 12i þ 9j þ 4k(b) The unit vector along direction ð1; 2; 2Þ is1i þ 2j þ 2k1i þ 2j þ 2kel ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼312 þ 2 2 þ 222.2 Vectors and Vector Operations17The derivative along the direction ð1; 2; 2Þ isdf ¼ rf jð3;2;0Þ eldl ð3;2;0Þ1i þ 2j þ 2k3¼ ð12 þ 18 þ 8Þ=3 ¼ 38=3¼ ð12i þ 9j þ 4kÞ 2.2.7 Operations on the Nabla OperatorThe dot product of the del operator with a vector v of components u; v; and w in thex; y; and z direction, respectively, results in the divergence of the vector [7, 8],which is a scalar quantity written asrv¼@u @v @wþ þ@x @y @zð2:20ÞPhysically the divergence of a vector field over a region is a measure of how muchthe vector field points into or out of the region.The divergence of the gradient of a scalar variable s is denoted by the Laplacianof s and is a scalar given byr ðrsÞ ¼ r2 s ¼@2s @2s @2sþþ@x2 @y2 @z2ð2:21ÞThe Laplacian of a vector follows from the above definition of the Laplacianoperator and is a vector computed as ð2:22Þr2 v ¼ r2 u i þ r2 v j þ r2 w kExample 4Find the divergence of v ¼ ðu; v; wÞ ¼ ð3x; 2xy; 4zÞSolutionThen divergence of v is obtained as@u @v @wþ þ@x @y @z¼ 3 þ 2x þ 4¼ 7 þ 2xrv¼182 Review of Vector CalculusAnother quantity of interest is the curl of a vector field [7, 8] formed between the“del” operator and the vector v, resulting in the following vector:@@@i þ j þ k ðui þ vj þ wkÞ@x@y@z ijk @ @ @ ¼ @w @v i þ @u @w j þ @v @u k¼ @y @z@z @x@x @y @x @y @z uv wrv¼ð2:23ÞExamples of the divergence and curl of a vector field are schematically displayedin Fig.

2.8. The radial vector field shown in Fig. 2.8a has only divergence with zerocurl. In fluid mechanics this vector field represents the velocity field of a sink/sourceflow. On the other hand Fig. 2.8b depicts a rotational vector field which has onlycurl with zero divergence (i.e., a divergence free vector field). Such a field corresponds to the velocity field of a vortex flow.The divergence of a vector v with its gradient also arises in the equations ofinterest in this book and is computed as@@@i þ j þ k ðui þ vj þ wkÞ½ðv:rÞv ¼ ðui þ vj þ wkÞ @x@y@z@@@¼ u þv þwðui þ vj þ wkÞ@x@y@z @u@u@u@v@v@v@w@w@wþvþwiþ u þv þwjþ uk¼ u þv þw@x@y@z@x@y@z@x@y@zð2:24Þ(a)(b)Fig.

2.8 a A radial vector field, b a solenoidal vector field2.2 Vectors and Vector Operations19Example 5Determine for the flow fields shown in Fig. 2.9a, b, c which is divergence free(i.e., neither expanding nor compressing) and which is irrotational (i.e., doesnot undergo a rotation)(a)(b)(c)Fig. 2.9 Example 5a rF¼0b rF¼0c rF¼2þ2¼4r F ¼ 0i þ 0j þ 2kr F ¼ 0i þ 0j þ 0kr F ¼ 0i þ 0j þ 0k2.2.8 Additional Vector OperationsIf s is a scalar function, and v1 ; v2 and v3 are vector fields, then the followingrelations, which are listed without proof, apply:r ð r vÞ ¼ 0ð2:25Þr ðrsÞ ¼ 0ð2:26Þr ðsvÞ ¼ sr v þ v rsð2:27Þr ðsvÞ ¼ sr v þ rs vð2:28Þrðv1 v2 Þ ¼ v1 ðr v2 Þ þ v2 ðr v1 Þ þ ðv1 rÞv2 þ ðv2 rÞv1 ð2:29Þr ð v1 v2 Þ ¼ v2 ð r v1 Þ v1 ð r v2 Þr ðv1 v2 Þ ¼ v1 ðr v2 Þ v2 ðr v1 Þ þ ðv2 rÞv1 ðv1 rÞv2ð2:30Þð2:31Þr ð r vÞ ¼ r ð r vÞ r 2 vð2:32Þðr vÞ v ¼ v ðrvÞ rðv vÞð2:33Þ202 Review of Vector Calculus2.3 Matrices and Matrix OperationsA matrix A of order M N is a rectangular array of quantities (numbers orexpressions) arranged in M rows and N columns [9–11].

An element of A locatedon the ith row and jth column is denoted by aij . For example, element a32 of the4 3 matrix shown in Fig. 2.10 is 12.Based on this definition it follows that a column vector v of dimensionality N isa matrix of order N 1 and a scalar s is a matrix of order 1 1.The transpose of a matrix A of order M N is another matrix denoted by AT oforder N M for which the rows of A are the columns of AT and the columns of Aare the rows of AT . Mathematically, this can be written as A ¼ aij ) AT ¼ ajið2:34ÞTwo matrices of the same order are equal if their corresponding elements areequal.

Two matrices of the same order can be added or subtracted element byelement. For example, if A and B are given by1A¼321472B¼31 41 6then A þ B and A B are found to beAþB¼1 30 0813AB¼361201If a matrix is multiplied by a scalar s than all its elements are multiplied by s.Mathematically this is written as A ¼ aij ) sA ¼ saijð2:35ÞTo multiply two matrices A and B, the number of columns of A must be equal tothe number of rows of B. Therefore, if A is of size M X for the product P ¼ ABi1234j1231 25 40 123 64723Fig. 2.10 Example of a 4 3 matrix=a11a12a13a21a22a23a31a32a33a41a42a43= aij2.3 Matrices and Matrix Operations21to be possible, B must be of size X N. The size of P will be M N with itselement pij obtained aspij ¼XXð2:36Þaik bkjk¼1If A is a 3 2 matrix and B a 2 4 matrix given by21A ¼ 4 12323 552B¼3100342then P ¼ AB will be a 3 4 matrix computed as216P ¼ 4 122373 535210 403 29p11 ¼ 1 2 þ 2 ð3Þ ¼ 4 >>>=p12 ¼ 1 ð1Þ þ 2 0 ¼ 1 >p13 ¼ .

. ....>>>>;2p116¼ 4 p21p3123p12p13p22p32p23p3316) P ¼ 4 11 119 2p1437p24 5p34683792 515 22.3.1 Square MatricesIf the number of columns N of matrix A is equal to its number of rows, then A is asquare matrix of order N. The elements aii of a square matrix A form its maindiagonal which stretches from top left to bottom right. The diagonal composed ofelements aij for which i þ j ¼ N þ 1 is called the cross diagonal and it extends fromthe bottom left to top right.Square matrices possess properties that are not applicable to other types ofmatrices such as symmetry and antisymmetry. In addition, many operations such astaking determinants and calculating eigenvalues are only defined for square matrices.The result of multiplying a square matrix of order N by itself is a square matrixof order N.

Therefore a square matrix can be multiplied by itself as many times asneeded and the notation Ak designates A multiplied by itself k times, i.e.,Ak ¼ |fflfflfflfflfflfflfflfflfflfflfflfflfflfflA A ffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflA. . . Affl}k timesð2:37Þ222 Review of Vector CalculusA square matrix A is symmetric if aij ¼ aji i.e., AT ¼ A , and antisymmetric ifaij ¼ aji . An example of a symmetric square matrix of order 3 is254 32327327 51and of an antisymmetric square matrix of order 4 is2036 3 064 2 14 332 41 3 770 2 520A diagonal square matrix D is one for which all elements off the main diagonal arezero while elements on the main diagonal are arbitrary.

An example of a squarediagonal matrix of order 3 is235 0 040 0 0 50 0 2A diagonal matrix of order N for which all elements on the main diagonal are 1 (i.e.,aii ¼ 1) is called an identity matrix of order N and is designated by I. An identitymatrix of order 4 is given by321 0 0 060 1 0 077I¼640 0 1 050 0 0 1The inverse of a square matrix A of order N is the square matrix A1 of order NsatisfyingA1 A ¼ AA1 ¼ Ið2:38ÞAn upper triangular matrix U is a square matrix in which all elements below themain diagonal are zero. Mathematically this can be expressed asuij i jU¼ð2:39Þ0 i[j2.3 Matrices and Matrix Operations23A lower triangular matrix L is a square matrix in which all elements above themain diagonal are zero.

Using mathematical notation, this is written as‘ij i jL¼ð2:40Þ0 i\jExamples of upper and lower triangular square matrices of order 3 are21U ¼ 40032 64 5 50 723L ¼ 4 19022300542.3.2 Using Matrices to Describe Systems of EquationsMatrices can be used to compactly describe systems of equations [12]. A system ofN equations in N unknowns can be written asa11 /1 þ a12 /2 þ a13 /3 þ .