Thompson, Warsi, Mastin - Numerical Grid Generation (523190), страница 13
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(34). Thus, using Eq. (81),(82)7. IntegralsExpressions for surface, volume, and line integrals are easily developed from the basevectors as follow.A. Surface integralReturning to the Divergence Theorem, Eq. (17), and its counterparts with the dot productreplaced by a cross product or simple operation (the latter with A replaced by a scalar), wehave approximate expressions for the surface integrals over the surface of the volumeelement given, using Eq.
(15), by(83)with the open circle indicating the product operation, and using the appropriate expressionfrom those given in the developments above. This emphasizes again that differenceforrepresentations based on integral formulations, e.g. finite volume, can be obtained by usingconservative expressions for the derivative operators directly in the partial differentialequations.B. Volume integralThe approximate expression for the volume integral over the volume element, againusing Eq.
(15), is simply(84)C. Line integralsUsing Eq. (3), the line integral on a coordinate line element on whichsimplyi variesis(85)where again the open circle indicates any type of operation, andis any tensor. Also sincewe have for a closed circuit lying on a coordinate surface on whichi isconstant,using Eq. (83), where (l,m,n) are cyclic. ButThen using the identity (13), we have(86)by using Eq. (34). With (1,j,k) cyclic we have for the circuit integral,(87)8. Two-Dimensional FormsIn two dimensions, let the x3 direction be the direction of invariance, and let the 3curvilinear coordinate be identical with x3. Also, for convenience of notation, let the othercoordinates be identified asA.
Metric elementsThenand the other base vectors are(88)The covariant metric components then are(89)From Eq. (16), the Jacobian is given by(90)The other contravariant base vectors are, from Eq. (33),(91)and the contravariant metric components are, from Eq. (37) or (38),(92)From Eq. (4) we have= 1 and= 2, so that by Eq. (91),(93)B. Transformation relationsDivergence (conservative), Eq.
(43):(94)(non-conservative), Eq. (67):(95)Gradient (conservative), Eq., (42):(96)(97)(non-conservative), Eq. (65):(98)(99)By Eq. (93), or directly from Eq. (76), these non-conservative forms may be given aswhich are the so-called "chain-rule conservative" forms. This form, however, is notconservative and the relations given by Eq. (93) must be substituted in the implementation inany case, since it is x , etc., rather than x, that is directly calculated from the grid pointlocations.Curl (conservative), Eq.
(44):(100)(non-conservative), Eq. (68):(101)Laplacian (conservative), Eq. (45):(102)(non-conservative), Eq. (65):(103)Second derivatives (non-conservative):(104)(105)(106)Normal derivative (conservative):(107)(108)(non-conservative):(109)(110)Tangential derivative (conservative):(111)(112)(non-conservative):(113)(114)Surface integral:(115)9. Time derivativesA. First DerivativeWith moving grids the time derivatives must be transformed also.
For the firstderivative we have(116)where here, and in Eq. (117) below, the subscripts indicate the variable being held constantin the partial differentiation. Here the time derivative on the left side is at a fixed position inthe transformed space, i.e., at a given grid point. The time derivative on the right is at a fixedposition in the physical space, i.e., the time derivative that appears in the physical equationsof motion.
The quantityis the grid point speed, to be written hereafter. Thus wehave, for substitution into the physical equations of motion, the relation(117)with to come from thetransformation relations given previously. With the time derivativestransformed, only time derivatives at fixed points in the transformed space will appear in theequations and, therefore, all computation can be done on the fixed uniform grid in thetransformed field without interpolation, even though the grid points are in motion in thephysical space. The last term in Eq. (117) resembles a convective term and accounts for themotion of the grid.B. Convective termsConsider the generic convective terms(118)is a velocity, which occur in many conservation equations.
Using Eq. (117) wewherehavewhere now the time derivative is understood to be at a fixed point in the transformed space.Then using Eq. (42) and (43) for the gradient and divergence, this becomes(119)By Eq. (16),by Eq. (33).
Butso that(120)We then can write(121)which is a conservative form of the generic convective terms with regard to the quantity,A. By Eq. (33), the quantity(122)is the contravariant velocity component in the i-direction, relative to the moving grid. ThusEq. (121) can be written in the conservative form,(123)Expanding the derivatives in Eq. (119) and using Eq. (40), we have(124)so that the non-conservative form(125). (Computationally,might beThe last summation is the divergence of the velocity,iincluded in the definition of U for use in the conservative form in the interest ofi can be evaluated directly ascomputational efficiency, since by Eq.
(33) the productthe cross product of the co-variant base vectors.)From Eq. (117) we have, with A taken asi,(126)by Eq. (4). Here the time derivative of i is, of course, at a fixed position in physical space.The quantity Ui introduced above in Eq. (122), thus could be written as(127)Here theare, of course, the contravariant velocity components.C.
Second derivativeThe second time derivative transforms as follows:(128)where the x,y subscripts on the left indicate the variables being held constant, and(129)(130)(131)(132)with (l,m,n) cyclic.Exercises1. Obtain the covariant and contravariant base vectors for cylindrical coordinates from Eq.(3) and (4). Show that Eq. (34) holds for this system.2. Obtain the elements of arc length, surface area, and volume for cylindrical coordinates.3. Obtain the relations for gradient, divergence, curl, and Laplacian for cylindricalcoordinates.4.
Demonstrate that the identity (21) holds for cylindrical coordinates.5. Demonstrate that Eq. (33), (38) and (39) hold for cylindrical coordinates.6. Repeat exercises 1 - 5 for spherical coordinates.7. Show that the covariant base vectors may be written in terms of the contravariant basevectors byHint: Cross k into Eq. (33) and use (13), rearranging k subscripts at the end. Recalling thatas can be expressed, this gives, a relation for (xl) i in terms of the derivatives( r)x .s8.
Show that the elements of the covariant metric tensor can be expressed in terms of thecontravariant elements byHint: Follow the development of Eq. (38), but with.9. Show that Eq. (65) is equivalent to the chain rule expression (1). Also show that the dotproduct of j with Eq. (65) leads, after interchange of indices, to the chain rule expression(4).10. Show thatHint: Sincewith respect to11. Show thatdepends oni.2only through the gij, differential g with respect to g jkRecall Eq.
(38).=0. Hint: Use cartesian coordinates.12. Obtain the two-dimensional relations in Section 6 from the general expression.13. Verify Eq. (74) for cylindrical and spherical coordinates.14. Obtain the normal and tangential derivatives (Section4) for cylindrical and sphericalcoordinates.IV. NUMERICAL IMPLEMENTATION1. Transformed EqationsIn order to make use of a general boundary-conforming curvilinear coordinate systemin the solution of partial differential equations, or of conservation equations in integral form,the equations must first be transformed to the curvilinear coordinates.
Such a transformationis accomplished by means of the relations developed in the previous chapter and produces aproblem for which the independent variables are time and the curvilinear coordinates. Theresulting equations are of the same type as the original ones, but are more complicated inthat they contain more terms and variable coefficients. The domain, on the other hand, isgreatly simplified since it is transformed to a fixed rectangular region regardless of its shapeand movement in physical space.
This facilitates the imposition of boundary conditions andis the primary feature which makes grid generation such a valuable and important tool in thenumerical solution of partial differential equations on arbitrary domains.A numerical solution of the transformed problem can be obtained using standardtechniques once the problem is discretized. Since the domain is stationary and rectangular,and since the increments of the curvilinear coordinates are arbitrary, the computation canalways be done on a fixed uniform square grid. Spatial derivatives at nearly all field points inthe transformed domain can therefore be represented by conventional finite-difference orfinite-volume expressions, as discussed in the next section.
In fact, the transformed problemhas the appearance of a problem on a uniform cartesian grid and thus may be treated as suchboth in the formation of the difference equations and in the solution thereof.The specific form of the transformed equations to be solved depends, of course, onwhich of the realtions in Chapter III are used, i.e., conservative or not. As an example,consider the generic convection-diffusion equation(1)Equations (III-123), (III-42), and Eq.
(III-43) may be used to transform the convective terms,the gradient, and the second divergence, respectively, and thereby yield the conservativeform:(2)where now the time derivative is understood to be at a fixed point in the transformed region,and the contravariant velocity components (relative to the moving grid) are given by Eq.(III-122). Eq. (2) can also be written in the form(3)which clearly shows the conservative form.
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