Trimmed Aerodynamics

Aerodynamics simulates the forces and moments that shape the flight trajectory. To model these effects, the designer can resort to many references, computer pre­diction codes, and wind-tunnel data. A two-volume set of missile aerodynamics,1 updated in 1992, is a compendium of experimental and theoretical results, quite suitable for aerodynamic analysis. Semi-empirical computer codes, like Missile DATCOM,2 can make your life much easier and generate aerodynamic tables quickly, but at the expense of insight into the physical underpinning of the data. If you venture into the hypersonic flight regime, the industry standard is the Su­personic Hypersonic Arbitrary Body Program (S/HABP)3 for missiles and reentry vehicles. For aircraft, the old faithful DATCOM is still available4 and made more

palatable by Roskam.5 Of recent vintage are two books by Stevens and Lewis6 and Pamadi7 that treat aerodynamics as part of the control problem. Finally, let us not forget the venerable book by Etkin8 that served two generations of engineers.

In missile and aircraft simulations the emphasis is more on performance rather than on stability and control. The autopilot, controlling the vehicle, is already designed before building the simulation and hopefully performs well throughout the flight regime. Therefore, the focus is on tabular modeling of the forces and moments and not on stability derivatives. Angle of attack and Mach number are the primary independent variables, sometimes supplemented by sideslip angle and altitude dependency (skin-friction effects).

Pseudo-five-DoF simulations are content with simple aerodynamic representa­tions. Because we assume that the moments are always balanced and that the trim drag of the control surfaces is included in the overall drag table, we need only two tables: normal and axial forces, or alternatively, lift and drag forces. If power on/off influences the drag, we have to double up the drag table, and, if the c. m. shifts significantly during the flight, we have to interpolate between changing trim conditions.

We shall proceed from general aerodynamic principles. Aerodynamic forces and moments are, in general, dependent on the following parameters:

aero forces and moments

M, Re, a, p,a,$, p, q,r, Sp, Sq. Sr, shape, scale, power

^ flow incidence angles body rates control surface

characteristics and rates deflections

where the Mach number is velocity/sonic speed and the Reynolds number is inertia forces/frictional forces.

The forces and moments are nondimensionalized by the parameters q (dynamic pressure), S (reference area), and l (reference length). The resulting coefficients are independent of the scale of the vehicle. If a missile flies a steady course, exhibiting only small perturbations, the dependence on the unsteady parameters a, 0, p, q, r may be neglected. For the trimmed approximation the moments are balanced, and their net effect is zero. Thus, only the lift and drag coefficients remain nonzero. With the effects of the trimmed control surface deflections implicitly included, the lift and drag coefficients are the following.

Lift coefficient:

Drag coefficient:

where L and D are the lift and drag forces, respectively. Their dependencies are reduced to

The Reynolds number primarily expresses the dependency of the size of the vehicle and skin friction as a function of altitude. With size and shape of a particular vehicle fixed and altitude dependency neglected, the coefficients simplify further:

CL or CD — f{M, a, power on/off}

Now let us treat skid-to-tum missiles and bank-to turn aircraft separately. By the way, I am using the term missile and aircraft somewhat loosely. A short-range air – to-air missile most likely will have tetragonal symmetry (configuration replicates every 90-deg rotation) and execute skid-to-tum maneuvers; but a cruise missile or a hypersonic vehicle, with planar symmetry, behaves like a bank-to-tum aircraft.