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 7

This isthe subject of Chaps. 15 and 16.81IntroductionChapter 15 is concerned with the prediction of incompressible flows. The difficulties associated with resolving the strong coupling between pressure andvelocity, with the absence of an equation for pressure, are overcome by theSIMPLE algorithm with the derivation of a pressure correction equation. The RhieChow interpolation is then introduced to allow realizing solutions to flow problemson collocated grids.

Finally the implementation of a number of frequentlyencountered boundary conditions is detailed.Chapter 16 extends the SIMPLE algorithm into compressible flows. Thedependence of density on pressure and temperature is accounted for in the pressurecorrection equation through a density correction, giving rise to a convection-liketerm that transforms the mathematical nature of the equation from elliptic (forincompressible flows) to hyperbolic. Implementation details for a number ofboundary conditions are also provided.1.3.4ApplicationsThis part describes the implementation and application of the numerical techniquesdeveloped in the previous chapters.Chapter 17 applies these numerical techniques to address some of the challengesfaced when solving turbulent flow problems.

It introduces several two-equationturbulence models and details the treatment of the near wall region.Chapter 18 reviews the implementation of boundary conditions in OpenFOAM®and provides the needed information for adding new boundary conditions in thecode. The no-slip wall boundary condition is described in some details.Chapter 19 outlines the solution procedure of a reference test case in whichsolvers and boundary conditions are applied to solve a turbulent incompressibleflow problem.Finally, Chapter 20 presents some closing remarks.1.4ClosureThe chapter discussed the growing role played by Computational Fluid Dynamics(CFD) as a core design tool in a whole class of applications and provided a generaloverview of the Finite Volume Method (FVM).

It also clarified the purpose of thebook, its intended use, and a summary of its content. The next chapter will give abrief review of the mathematical operations that will be used throughout the book.Chapter 2Review of Vector CalculusAbstract This chapter sets the ground for the derivation of the conservationequations by providing a brief review of the continuum mechanics tools needed forthat purpose while establishing some of the mathematical notations and proceduresthat will be used throughout the book.

The review is by no mean comprehensiveand assumes a basic knowledge of the fundamentals of continuum mechanics.A short introduction of the elements of linear algebra including vectors, matrices,tensors, and their practices is given. The chapter ends with an examination of thefundamental theorems of vector calculus, which constitute the elementary buildingblocks needed for manipulating and solving these conservation equations eitheranalytically or numerically using computational fluid dynamics.2.1 IntroductionThe transfer phenomena of interest here can be mathematically represented byequations involving physical variables that fall under three categories: scalars,vectors, and tensors [1–3].

Throughout this book scalars are designated by lightfaceitalic, vectors by lower boldface Roman, and tensors by boldface Greek letters. Inaddition, matrices are identified by upper boldface Roman letters.A scalar represents a quantity that has magnitude such as volume V, pressure p,temperature T, time t, mass m, and density q. A vector represents a quantity of agiven magnitude and direction such as velocity v, momentum L ¼ mv, and force F.A matrix is a rectangular array of quantities ordered along rows and columns.A tensor is a mathematical object analogous to but more general than a vector,represented by an array of components, such as the shear stress tensor.

Moreover,the conservation equations are composed of terms that represent the product of twoor more variables. The multiplication involved may be of various types to bedetailed later and the variables could be a combination of the three types describedabove. Whenever the multiplication results in a scalar, the product will be enclosed© Springer International Publishing Switzerland 2016F.

Moukalled et al., The Finite Volume Method in Computational Fluid Dynamics,Fluid Mechanics and Its Applications 113, DOI 10.1007/978-3-319-16874-6_29102 Review of Vector Calculusby parentheses “(product)”, if it results in a vector it will be enclosed by squarebrackets “[product]”, and if it results in a tensor it will be enclosed by curly brackets“{product}”.2.2 Vectors and Vector OperationsThe most frequently used vector in fluid dynamics is the velocity vector that will bedesignated by v. The components of the velocity vector in a three-dimensionalCartesian coordinate system will be denoted by u; v; and w in the x; y; and zdirection, respectively (Fig. 2.1). In Cartesian coordinates, v is written asv ¼ ui þ vj þ wkð2:1Þwhere i; j; and k are unit vectors in the x; y; and z direction, respectively.

A vector isusually presented in a column format with its transpose, denoted with a superscript T,in a row format as23uv ¼ 4 v 5 vT ¼ ½ uwvwð2:2ÞThe magnitude of a vector is given bykvk ¼pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiu2 þ v2 þ w2ð2:3ÞThe sum of two vectors v1 and v2 is the sum of their components, i.e.,v 1 ¼ u 1 i þ v1 j þ w 1 kv 2 ¼ u 2 i þ v2 j þ w 2 k) v1 þ v2 ¼ ðu1 þ u2 Þi þ ðv1 þ v2 Þj þ ðw1 þ w2 ÞkFig. 2.1 The components ofa vector v in athree-dimensional Cartesiancoordinate systemð2:4Þzvwuyxv2.2 Vectors and Vector Operations11or232 323u1u2u1 þ u2v1 ¼ 4 v 1 5 v 2 ¼ 4 v 2 5 ) v1 þ v2 ¼ 4 v 1 þ v 2 5w1w2w1 þ w2ð2:5ÞThe multiplication of a vector v by a scalar s results in the vector sv such thatsv ¼ sðui þ vj þ wkÞ23su6 7¼ sui þ svj þ swk ¼ 4 sv 5ð2:6ÞswThe product of two vectors is not as straightforward.

When multiplying a vector v1 byanother vector v2 two types of multiplications arise [4–6]. The first is denoted by thescalar or dot product, ðv1 v2 Þ, and the second by vector or cross product ½v1 v2 .2.2.1 The Dot Product of Two VectorsBy definition, the dot product of two vectors v1 and v2 is a scalar quantity given byv1 v2 ¼ kv1 kkv2 kcosðv1 ; v2 Þð2:7Þwhere cosðv1 ; v2 Þ denotes the cosine of the angle between v1 and v2 . From thedefinition of the vector dot product, it follows thatii¼jj¼kk¼1ij¼ik¼ji¼jk¼ki¼kj¼0ð2:8ÞIn terms of orthonormal Cartesian components, the dot product of the two vectorsv1 and v2 can be calculated asv1 v2 ¼ ðu1 i þ v1 j þ w1 kÞ ðu2 i þ v2 j þ w2 kÞ¼ u1 u2 þ v1 v2 þ w1 w2ð2:9Þ2.2.2 Vector MagnitudeFrom Eq.

(2.9) it follows that the magnitude of a vector v can be obtained askvk ¼ffipffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiv v ¼ u2 þ v2 þ w2ð2:10Þ122 Review of Vector Calculus(b))v11v1v1v 2 = v 1 cos (v2)v2v1vv2=v2cos((a)v2Fig. 2.2 a Projection of vector v1 onto the unit direction of vector v2 ; b Projection of vector v2onto the unit direction of vector v12.2.3 The Unit Direction VectorA unit vector ev in the direction of v can be derived from the definition of the dotproduct as9¼1>zfflfflfflfflffl}|fflfflfflfflffl{>>v=¼ kvk >v v ¼ kvkkvk cosðv; vÞ ¼ kvk2 ) v vkvk) ev ¼>kvkv ev ¼ kvk kev k cosðv; ev Þ ¼ kvk ) v ev ¼ kvk >>>|{z} |fflfflfflfflffl{zfflfflfflfflffl};¼1ð2:11Þ¼1Therefore the component of a vector in the direction of another vector (i.e., magnitude of the projected length) can be viewed as the dot product of the vector to beprojected with the unit direction of the other vector as shown in Fig.

2.2a, b.2.2.4 The Cross Product of Two VectorsWhereas the dot product of two vectors v1 and v2 is a scalar quantity, their cross orvector product is a vector v3 normal to the plane formed by the vectors v1 and v2 , ofmagnitude calculated askv3 k ¼ kv1 v2 k ¼ kv1 kkv2 kjsinðv1 ; v2 Þj;ð2:12Þand of direction given by the right hand rule. As shown in Fig. 2.3, the magnitudeof the cross product of two vectors represents the area of the parallelogram spannedby the two vectors.

Since, in addition, the resulting vector is normal to the plane2.2 Vectors and Vector Operations13v 3 = v1 × v 2areav 3 = v1 v 2 sin ( )v2v1Fig. 2.3 The cross product of two vectorsformed by the vectors, the cross product of two vectors represents their surfacevector.It is then clear that the cross product of two collinear vectors is zero as theydefine no area, and that the cross product of two orthogonal unit vectors is a unitvector perpendicular to the two unit vectors. Adopting the right hand rule to definethe direction of the resulting vector, the following cross product operations hold:ii¼jj¼kk¼0j k ¼ i ¼ k ji j ¼ k ¼ j ik i ¼ j ¼ i kð2:13ÞUsing the above relations, the cross product of two vectors in terms of theirCartesian components is given byv1 v2 ¼ ðu1 i þ v1 j þ w1 kÞ ðu2 i þ v2 j þ w2 kÞ¼ u1 u2 i i þ u1 v 2 i j þ u1 w 2 i kþ v1 u2 j i þ v1 v2 j j þ v1 w2 j kþ w1 u2 k i þ w1 v2 k j þ w1 w2 k k¼ u1 u2 0 þ u1 v2 k þ u1 w2 ðjÞþ v1 u2 ðkÞ þ v1 v2 0 þ v1 w2 iþ w1 u2 j þ w1 v2 ðiÞ þ w1 w2 0¼ ðv1 w2 v2 w1 Þi ðu1 w2 u2 w1 Þj þ ðu1 v2 u2 v1 Þkð2:14Þwhich can be written using determinant notation as iv1 v2 ¼ u1 u2jv1v2 23v1 w2 v2 w1k w1 ¼ 4 u2 w1 u1 w2 5w2 u1 v 2 u2 v 1ð2:15Þ142 Review of Vector CalculusExample 1Compute the area of the triangle formed by points (Fig.

2.4):P1 ð0; 0; 0Þ; P2 ð1; 0; 0Þ and P3 ð0:5; 1; 0Þ:SolutionThe surface defined by the triangle ðP1 ; P2 ; P3 Þ can be computed using thecross product of two sides as! !S123 ¼ 0:5 P1 P2 P1 P3S123!P1 P2 ¼ ðx2 x1 Þi þ ðy2 y1 Þj þ ðz2 z1 Þk ¼ i!P1 P3 ¼ ðx3 x1 Þi þ ðy3 y1 Þj þ ðz3 z1 Þk ¼ 0:5i þ jP3P1P2Fig. 2.4 Example 1S123 ¼ 0:5i ð0:5i þ jÞ ¼ 0:5k ) kS123 k ¼ 0:52.2.5 The Scalar Triple ProductIn addition, combined products of three vectors v1 , v2 , and v3 may arise such asðv1 ½v2 v3 Þ, which can be calculated using the following determinant (to beexplained later): u1ðv1 ½v2 v3 Þ ¼ u2 u3v1v2v3w1 w2 w3 ð2:16ÞAs shown in Fig.

2.5, the absolute value of the scalar triple product representsthe volume of the parallelepiped formed by the vectors v1 , v2 , and v3 .Fig. 2.5 Geometricrepresentation of scalar tripleproductvolumev 3 ( v1 v 2 )v1 v 2v3v2v12.2 Vectors and Vector Operations15Example 2Compute the volume of the pyramid defined by the points:P1 ð0; 0; 0Þ; P2 ð1; 0; 0Þ; P3 ð0:5; 1; 0Þ, and P4 ð0:5; 0:5; 1Þshown in Fig. 2.6.SolutionThe volume of the pyramid can be computed usingthe scalar triple product asP4! ! !V ¼ 0:25 P1 P4 P1 P2 P1 P3P3C¼ 0:25ð0:5i þ 0:5j þ kÞ k¼ 0:25P1P2Fig.