Numerical grid generation

4.1

Introduction

The analysis in the preceding chapter suggests that if a finite-time break-up of the mid-scale system (2.2.4a-c,e) exists, it is distinct from that of the related large-scale system (3.2.1), (3.2.3), (3.2.4). Hence we now turn to a numerical treatment of the

unsteady nonlinear governing equations (2.2.4a-c, e ) . As it will

emerge, the mid-scale system (2.2.4a-c,e) can admit a finite-time break-up solution which is distinct from that for the large-scale flow. In this chapter we describe in detail a method of numerical

adaptive-gridding which was developed in order to capture this

break-up solution in the numerical calculations.

In Computational Fluid Dynamics the provision of a ‘g o o d ’ computational grid is essential for the production of a high quality flow solution. The precise quantities which make a particular grid a good one are not always clear and could be dependent upon the flow situation in question. However, it is believed that such a grid should be smooth and, in order to avoid large solution errors, the grid nodes should be close together where rapid solution variation occurs. Also, in regions of slow solution variation, the grid nodes may be more widely spaced in order to economize on computing time and storage.

Computational grids are often generated with these simple principles in mind, and nodes are clustered where rapid solution

variation may be expected, such as (in full Navier-Stokes

computations) in the vicinity of aerofoil leading edges. However, certain features of the solution, such as the location of shock waves, are in general unknown at the time the grid is generated and before the solution is obtained. In these regions the solution may be varying rapidly, but the grid may not be particularly dense; hence truncation errors will probably be large there. In order to overcome such problems much effort has been devoted in recent years to developing procedures which adapt the grid to the solution in some way, permitting a better solution to be obtained. Examples of such procedures are given by Catherall (1991), Eiseman (1987) and H a w k e n , Hansen k Gottlieb (1991).

redistribution of grid points by application of the principle of equidistribution, whereby the product of a local weight function, w, and the grid spacing, dx, is made constant;

w d x = constant , (4.1.1)

where w is some appropriate measure of local solution activity

(usually indicated by high solution gradients). The aim is to

achieve an approximate equidistribution of the truncation error, which is normally some power of the product of the grid spacing and

a function of solution gradients, so minimizing the maximum

truncation error. After solving (4.1.1) for x it may be seen that the node spacing dx will be small where solution activity is high, and large where there is little activity. A deficiency of this

method lies in the fact that the resulting grid spacings are

nonuniform and so the truncation error is only of first order in the

grid spacing dx. Also, for unsteady systems involving time

derivatives, interpolation of the solution onto the new grid is

necessary^ toiUoU Aurvier icct^

Hawken, Hansen & Gottlieb (1991) overcome this obstacle by transforming to a numerical coordinate system that moves with the nodes and in which the nodes are equidistributed such that the new coordinate system is uniformly distributed. This idea is used here to develop the current grid generating procedure, although we should remark that the current work was carried out independently of the p re v i o u s .

4.2

A grid-transformation procedure

To fix matters we consider systems of unsteady, o n e ­

dimensional partial differential equations in which the coordinate system, (x,t), is transformed to a new coordinate system, (x,t), via the relation

X = g(x, t) , (4.2.1)

for some smooth function g(x,t) which we may choose in order to meet

our particular requirements. This is different from the method of

Hawken, Hansen & Gottlieb (1991) who seek to minimize an error measure to effect the grid adaptation.

The partial derivatives, namely ^ and combinations of

these, are also transformed via (4.2.1) and the transformed versions must be obtained. To this end, we note that differentiation with respect to t, keeping x fixed, transforms in the following way:

H +

i â

Similarly, for differentiation with respect to x, keeping t fixed, we have

à - Ë à -

4

2

3

Expressions for ^ and — in (4.2.2) and (4.2.3) are needed and are

found by performing the same operations on (4.2.1) to obtain

and

respectively. Substitution for ^ and ^ from the above into (4.2.2) and (4.2.3) yields

dt

dt

for the transformed derivatives. So an extra factor proportional to

the grid velocity will appear in the transformed partial

differential equations.

It now remains to choose a suitable function g(x,t). As

(4.2.1) defines a transformation of the coordinate system we suppose

that the function g(x,t) should be smooth, strictly monotonie

increasing and, for our purposes of adaptive gridding, should be sensitive to the local solution activity. The last requirement may be interpreted as requiring the function g(x, t) to have a small

slope in regions of high solution activity. Also, in order to

economize on computation, the slope may be allowed to become

relatively large in regions of slow solution variation. Herein we

discuss the option of g(x,t) as a function of solution gradients (cf. the notion of equidistribution) and, for the sake of argument, regard A(x,t), in the untransformed coordinate system, to be the dependent variable in question. Clearly there are any number of

possibilities for the description of this grid-transformation

equation. Below we study three examples of such a grid-

transformation equation and investigate for the desired properties

of g(x, t) mentioned in the above paragraph. Ve also consider the

possible consequences of each of these grid-transformation equations on the solution of the transformed numerical system of equations. For the analytic investigation we characterize the regions of high solution activity by

d x

d x oo and those of slow solution variation by

0.

Firstly, perhaps the candidate for the grid-transformation equation which first comes to mind is given by

Ve see immediately that the right-hand side of (4.2.5) is strictly positive so that g(x,t) is monotonie. Also it is obvious that

d k -2 d k

solution activity. It is also important to investigate how ^ varies with 0^. To do this we use (4.2.4b) to find that

d k