Thompson, Warsi, Mastin - Numerical Grid Generation (523190), страница 33
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(With more distorted boundariesthe Laplace solution might be more reliable than the interpolation.) Different forms ofinterpolation, or an equation other than the Laplace, for the determination of the controlfunction in the field would allow some control of coordinate line spacing in the field.However, since only a single control function is involved, it is not possible to exercisecontrol of the coordinate line spacing in the field in both directions.Another approach in which the boundary point distribution can only be fixed in aspecified manner is to take the basic generation equation to be Eq. (7c) which for conformalcoordinates (h2=h1) takes the form(25)where P=2 ln(h1).
An exact solution of Eq. (25) can be obtained if appropriate values of Pare known at the boundaries. The important problem then becomes the choice of those pointsat the inner and outer boundaries which can be put in orthogonal correspondence with oneanother. This can be accomplished if the 1-coordinate, both at the inner and outerboundaries is selected to satisfy the Laplace equation 2 1=0. This condition can besatisfied by taking 1 as the angle traced out by the common radii of those concentric circleswhich are the conformal maps of the contours in the physical plane.
The solution of Eq. (25)under these conditions then can be used to generate non-conformal coordinates by acoordinate transformation of the other coordinate 2.An orthogonal grid can be generated by solving the Laplace equations (21a) providedthat the boundary point distribution is compatible. Since a conformal mapping generates anorthogonal grid, a compatible boundary point distribution can be obtained by conformallymapping the boundary contour as follows (cf. Ref. [43]): Consider an open physicalboundary contourwhereandare to be lines of constantgenerated are to be lines of constant 1.2,whileand a connecting lineto beEach point of the set that defines this contour is successively mapped onto the real axisin the complex plane by a hinge point transformation (such a transformation has the effect ofmapping one point onto the real axis while points already on the real axis remain there):The straight linein the complex plane:on the real axis is then mapped conformally onto an open rectanglePoints are then placed as desired along the sidesandof this rectangle, these pointsandbeing assigned successive integer values of 1 and 2, respectively.
(Thisonplacement of points on these two sides is arbitrary and may be done by any distributionfunction desired.) The key to the construction of a compatible boundary point distribution isthen that the points on the other sides of the rectangle, i.e.,and, are placed with thesame distributions chosen forand. The points in the physical plane that correspondto these boundary points on the rectangle in the complex plane are then determined byexponential spline interpolation among the values at the original set of points defining thecontour, except for the open side of the rectangle where the points in the conformaltransformations.
Finally the orthogonal grid is generated by solving the Laplace equations(21a) with this fixed boundary point distribution.C. Systems based on first-order equationsEquations (10) are formally related to the Cauchy-Riemann equations (with F=1), butotherwise form a set of first order nonlinear partial differential equations. In order topreserve the orientation of coordinates, the sign of F is taken to be positive throughout thedomain.
For certain choices of the function F the system is hyperbolic, and the completeinitial-value problem is then(26)Here = o is the given body contour, and, unlike the elliptic problem, the data on anotherboundary cannot be specified.This system may be shown to exhibit the following important properties:(i). First, g22 in principle can be expressed as a function of g11. <.p> (ii) Because of (i), F > 0is a function of1, 2,and g11, i.e.,For brevity, writingwe have(iii). For a well-posed initial value problem the system of equations in (26) must behyperbolic.A test for the well-posedness is that small perturbations produce small effects.
Usingthis test, for Eqs. (26) to be hyperbolic, the function f(z), defined asmust be a strictly decreasing function of z.3. Three-Dimensional Orthogonal CoordinatesThe problem of three-dimensional orthogonal coordinate generation, though of muchimportance in many practical problems, has received little attention in comparison to itstwo-dimensional counterpart. The reason is not so much in the complicated form of thegoverning equations but rather in the prescription of the boundary conditions and in theirnumerical implementation.Orthogonality in three dimensions is difficult to achieve, and only exists when thecoordinate lines on the bounding surfaces follow lines of curvature, i.e., lines in the directionof maximum or minimum curvature of the surface.
Therefore, three dimensional orthogonalcoordinates will not be available in most cases with nontrivial geometry. It is possible,however, to have the system locally orthogonal at boundaries, and/or to have orthogonalityof surface coordinates.The governing equations for generation of orthogonal coordinates are obtained in astraightforward manner and have been listed above as Eq. (4) - (6).
The set of equationswhich are to be solved for x1,x2,x3 and h1,h2,h3 has Eq. (4) and (6). The set (6) has sixequations for the three unknowns. On the other hand, without imposing the orthogonalitycondition, gij = 0 (i j), there are six equations for the determination of six unknowns. Thusthe orthogonality does not reduce the number of the equations which govern the distributionof the metric coefficients, and it would be wrong to try to select a set of three equations outof the available six.4. Nearly-Orthogonal SystemsSince a part of the truncation error is decreased as the grid becomes more orthogonal,it is of interest to generate grids which are "nearly-orthogonal".
Such grids do notapproximate orthogonality sufficiently well, however, for the terms arising fromnonorthogonality in transformation relations to be dropped. The generation ofnearly-orthogonal grids naturally follows some of the procedures discussed above in thischapter, but with the conditions for orthogonality only partially satisfied. Several proceduresare discussed in Ref. [1] and Ref.
[42].A simple procedure for generating a nearly-orthogonal system from a nonorthogonalsystem is to first generate curves of a nonorthogonal system by connecting points obtainedby any specified distribution function along straight lines connecting boundary points on twoarbitrary closed boundaries. Coordinate lines connecting points on each succeeding pair ofcurves from the original coordinate system then are constructed as follows: At selectedpoints on the inner curve, normals are constructed, and the points of intersection with thenext curve outward are determined. Normal directions form the intersection point aredetermined and translated to the original point in the inner curve.
Then a second point on theouter curve is determined as before. Finally, the new coordinate lines are constructed asstraight lines joining the selected points on the inner curve with points located halfwaybetween the corresponding pair of points on the outer curve located as described above. Theresulting lines will not actually be orthogonal to either the inner or outer curve, and theslopes of these lines will, in fact, be discontinuous at each curve. The observed departuresfrom orthgonality, however, have been small and the departure may be made arbitrarilysmall by the addition of more curves. Since the procedure is applied successively betweenpairs of coordinate lines, concave bodies can be treated as well.Exercises1.
The unit tangent vector on a curve C defined in the parametric form = (s), with s asthe arc length along C, is given by =d /ds. Let C be a plane curve in the xy-plane havingas the unit normal vector. Using the conditionand the convention that ( , ,), in the order shown, form a right-handed triad of vectors, find the components of . Hereis the constant unit vector along the z-axis.2. Let (x,y) and (x,y) be the conformal coordinates in the xy-plane so that theCauchy-Riemann equationsare satisfied.
Consider the curve C defined in excercise 1 and the normal derivative operatorand show that the Cauchy-Riemann equations in the natural coordinates (s,n) are3. Let F( i) be a scalar function of position and(a) Show that the unit normal vectoris given by( i)=constant be a surface.to the surface(b) Prove that the normal derivative of F on the surface=constant in curvilinear coordinates=constant is(c) In particular, for two-dimensional curvilinear coordinates show that(d) Particularize the results in (c) for orthogonal curvilinear coordinates. Write the partialderivative operatorfor orthogonal coordinates.4.
Consider Eq. (26) of this chapter, which form a system of first-order partial differentialequations for two-dimensional orthogonal coordinates. It was stated subsequently that theseequations form a hyperbolic system if the initial value problem is well-posed. To prove thisassertion consider the perturbed state x+ x, y+ y, F( , ,z+ z), where.Retaining only the first order terms, develop a system of algebraic equations in ( x) ,( y), ( x) , ( y) , and show that the resulting matrix has eigenvalues given by2= -F(F + zFz)Show from the preceding result that the eigenvalues are real only when zF is a strictlydecreasing function of z.X. CONFORMAL MAPPINGInnovations in conformal mapping continue to extend this classical technique to morecomplicated configurations, and surveys of the various techniques available are given in Ref.[7] and Ref.
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