Tabular Data Models

Often, it is not worth it to try to find a closed function modeling the aerodynamics. In such cases, the the simulator can obtain the forces and moments from a tabular data obtained from flight or wind tunnel tests.

Table 1 lists a simple example of a table. The table lists a drag polar typical of a wing with a laminar flow airfoil. To determine the drag, the simulator determines which two points in the table the angle of attack is between, and then calculates the drag by linear interpolation. To speed up the interpolation, the slope of each segment is often precomputed and stored in the table as well $^{\ref{ref:rolfe}}$.

  

Table: Sample table of C_X versus $\alpha$, with plot.

$\alpha$	CX
(deg)
-10	-0.0134
-8	-0.0108
-6	-0.0094
-4	-0.0082
-2	-0.0072
0	-0.0052
2	-0.0053
4	-0.0055
6	-0.0116
8	-0.0175
10	-0.0230

Aerodynamic force is generally a function of more the one variable; therefore, flight simulators use multidimensional tables. Theoretically, the table could have a dimension for each variable the force coefficient is a function of. For example, C_M may be expressed as

$$C_M = C_M[\alpha,\beta,\delta_e,\delta_f,\overline q,\mathbf{M}],$$ (63)

where the square bracket notation indicates a tabulated function. However, the memory and testing time required for a table with many dimensions is extremely large. (If each variable of the table in Equation 63 takes on 20 different values, which is a typical number, then the table will have 64 million data points.)

Fortunately, the effects of some variables are not influenced, or are influenced only weakly, by other variables. For example, the effect of $\delta_f$ on pitching moment is not influenced by $\overline q$, and vice versa. Thus, $\delta_f$ and $\overline q$ never need to appear in the same table. This allows the large table to be broken into several smaller tables, each limited to three or four dimensions. The tabular model sums the contributions from each table.

A tabular aerodynamic model has been used to simulate an F/A-18 $^{\ref{ref:butthril}}$. Equation 64 presents an example of an extremely simplified version of the model for the yawing moment:

 

$$C_{N_0}[\beta,\alpha,\mathbf{M}] + C_{N_{\delta_f}} [\beta,\alpha... ...t) + C_{N_{\delta_n}} [\beta,\alpha,\mathbf{M}]\left(\frac{\delta_n}{25}\right)$$ C_N =

$$+ \Delta C_{N_{\delta_a}}[\delta_a,\alpha,\mathbf{M}] + \Delta C_... ...+ C_{N_p}[\alpha,\mathbf{M}]\overline p + C_{N_r}[\alpha,\mathbf{M}]\overline r$$ (64)

The main problem with tabular models is the lack of smoothness. For example, the C_X curve from Table 1 is nondifferentiable at the tabulated points, and its derivative is discontinuous. Flight simulators can remedy this somewhat by using tables with smaller increments. Another technique is to use higher-order interpolation.