A number of grid girls have contacted me recently, complaining that the

*F1 Show* on SkySports fails to provide the level of informative technical discussion they seek in a programme nominally targeted at the motorsport enthusiast.

In particular, they've asked to understand a little more about the correlation issues which crop up between Computational Fluid Dynamics (CFD), wind-tunnel testing, and full-scale track testing.

Perhaps the best way to begin such an explanation is to introduce the concept of a commutative diagram, familiar to all mathematically inclined grid girls.

The idea here is that two operations can be applied to object A. One operation is depicted as a horizontal path, the other as a vertical path. If the same result is obtained irrespective of the order in which the operations are applied, then the operations are commutative. In terms of the labelling in this particular version of the diagram, it is written that:

Now, in the motorsport arena, aerodynamic data can be acquired by four distinct means: (A) Scale-model CFD simulation; (B) Scale-model testing in the wind-tunnel; (C) Full-scale CFD simulation; and (D) Full-scale track testing.

Whilst the data acquired from full-scale track testing can be treated as veridical, it has increasingly been considered to be an expensive means of generating such information, and has therefore become a severely limited form of data acquisition. This has placed greater emphasis on CFD and wind-tunnel testing. However, both of the latter techniques have systematic errors associated with them, and to go from one data set to another requires the application of correction factors and more general mathematical transformations. For example, scale-model testing in the wind-tunnel can only be mapped to full-scale data if correction factors are applied for the blockage imposed by the walls of the wind-tunnel. Moreover, it is impossible to replicate both the Reynolds number and the Mach number of the full-scale flow with a scale model, hence Reynolds number corrections must be introduced.

Given any pair of data-sets, if one of them can be treated as veridical, then a regression analysis can establish the corrections which must be applied to compensate for the bias of the non-veridical data-set. One might, for example, have CFD and wind-tunnel coefficient-of-lift (C_{L}) values for each combination of ride-height and angle-of-attack. By allowing a single parameter to vary (e.g. angle of attack), a particular set of paired C_{L}-values can be isolated and represented as a scatter-plot, the x-coordinate of each data-point being the wind-tunnel C_{L}-value, the y-coordinate being the CFD C_{L}-value. A regression analysis then establishes a functional relationship between the CFD coefficients and the wind-tunnel coefficients.

A vital test to ensure that one has a self-consistent scheme of correctional transformations is to test for the commutativity of these relationships. Thus, for example, if one begins with a set of half-scale wind-tunnel data, one should be able to: (i) map the half-scale wind-tunnel data to scale-model CFD data, then map the scale-model CFD data to full-scale CFD data, and then map the full-scale CFD data to full-scale track testing data; (ii) map the half-scale wind-tunnel data directly to full-scale track testing data; and (iii) the results of these two transformations should be in agreement, within some reasonable approximation. If there is any doubt, the results should be statistically tested with something like Analysis of Variance (ANOVA) to determine if the variation is simply the result of random sampling error.

Similarly, one should be able to start with half-scale CFD data, and map the results to full-scale track testing data by the two possible routes, without getting different results. In terms of the commutative diagram we started off with, there actually needs to be a bi-directional arrow between objects A and B.

As ever, the generalities are simple, the implementation difficult and messy.