The framework for generating manoeuvres based on the solution of a time-optimal problem was exercised with the SDM model [23]. More details pertaining the model configuration, block structured grid and numerical simulations were presented in Chap- ter 3.2.1. To allow manoeuvring the aircraft, control surfaces were added to the com- putational mesh. Mesh block faces were placed on the control surfaces, and the mesh points on these faces were deflected to define the control surface mode shapes. After the surface grid point deflections are specified, transfinite interpolation is used to dis- tribute these deflections to the volume grid [104]. A view of the surface mesh for the deflected control surfaces is shown in Fig. A.20. An all-moving elevator, ailerons and rudder were used for longitudinal and lateral-directional control.
Figure A.20: Deflected control surfaces for the SDM model
Geometry reference values were given in Table 3.1. Mass and inertia properties of the SDM model are available for a free-flight model which represents a 1/72 scale aircraft [205]. These values were scaled-up to match the dimensions of the current computational model. The maximum total thrust force, Tm = 26.24kN, is assumed
to cross the centre of gravity. The direction of the thrust relative to the aircraft is assumed to remain unchanged with altitude and flight speed, and to vary linearly with the engine throttle.
To generate the aerodynamic database of forces and moments, four sub-tables were created, three were for the control surface deflections. The Mach number was varied be- tween 0.1 and 0.4, and all manoeuvres were simulated in the subsonic speed range. The angle of attack was varied between −14◦
and 28◦
Parameter Value xcg 9.91m mass 9295.44kg Ixx 12.874×103kg·m2 Iyy 75.673×103kg·m2 Izz 85.552×103kg·m2
Table A.4: Mass and inertia properties of the SDM model
was set to 20◦
. Around 6000 table entries were defined. The baseline table consisted of 156 flight conditions, and a brute force approach was considered for varying the Mach number and angle of attack, while keeping the sideslip angle to zero. For the lateral coefficients, a different approach was considered because all the lateral coefficients in the created baseline table were zero. Lateral coefficients obtained using DATCOM were used as low-fidelity data, and then co-kriging with a few Euler results was used to generate the updated baseline table. The dependence of the longitudinal forces and moments on other parameters was assumed to be an increment of the baseline table and the co-kriging data fusion method was used to include these variation in a compu- tationally efficient way. With fifteen additional samples the variation with the elevator, ailerons and rudder was included in the tables. Samples were located at the vertices of the parameter space and at a median value within the domain. The response of the aerodynamic coefficients to variations in angle of attack and elevator deflection at a Mach number of 0.4 is illustrated in Fig. A.21. Considering the non-linear features shown, it can be argued that the number of sample points used was small for an ade- quate representation of the aerodynamic loads. While static coefficients were adequate to represent aerodynamic loads for slow manoeuvres, the simulation of faster motions included dynamic dependencies. Dynamic derivatives were computed from forced mo- tions, and assumed to be independent of the Mach number and to vary with the angle of attack only. This assumption was demonstrated to be adequate in the speed range below a Mach number of 0.5, as given for example in [85] where dynamic derivatives from low-subsonic to high-transonic regime were compared to experimental results (see also Chapter 3.3.1).
Aerodynamic loads for a set of manoeuvres were predicted using the tabular model of forces and moments, and these were compared to the unsteady CFD solution, which is the reference solution because it is time-accurate. To guarantee a realism in the manoeuvres to be simulated, these were generated solving an optimal control problem. The problem of moving the aircraft from the initial state to the final state is rewritten as a control problem by minimizing a suitable cost function. Constraints are specified for the states, describing the aircraft position and attitude, and for the controls, re- alizing physical limitations on the use of control effectors. The resulting constrained
AoA [deg] -10 0 10 20 30 Elevator Deflectio n [deg] -20 -10 0 10 20 CL -1 -0.5 0 0.5 1 1.5
(a) Lift coefficient
AoA[deg ] -10 0 10 20 30 Elev ator Deflectio n [d eg] -20 -10 0 10 20 Cm -0.3 -0.2 -0.1 0 0.1 0.2
(b) Pitching moment coefficient
Figure A.21: Response of aerodynamic coefficients to angle of attach and elevator deflection at a Mach number of 0.4
optimization problem, with the six degrees of freedom equations of motion as a fur- ther constraint, can be solved using standard techniques, as in [206]. In the current framework, the DIDO code5 [207] is used for the solution of the optimal problem, with
aerodynamic forces and moments obtained from the look-up tables. Technical details can be found, for instance, in [208–210].
Two sets of manoeuvres were generated using the optimal control problem to demon- strate, first, a good comparison between the tabular model and the CFD solution for slow motions, and, then, to stress the limitations of the tabular model when confronted with flows exhibiting time-history effects. A variety of manoeuvres, two of which are illustrated in Fig. A.22, were simulated at low rate, and the solution of the optimal control problem was found using static tabular data. In all cases presented, the aero- dynamic loads from the tabular model compared well with the time-accurate solution. The assumption of quasi-steady aerodynamics describes well the flow around the mov- ing airframe, adapting instantaneously to changes in geometry attitude and without time-history effects influencing its development. The effect of increasing the angu- lar rate for a given manoeuvre was then investigated. The manoeuvre was a pull-up with time-varying angle of attack, and was simulated initially at a pitch rate as low as 2.0◦
/s. Whilst for 20.0◦
/s the inclusion of dynamic terms shifted the static pre- diction to match the time-accurate solution up to high angles of attack, discrepancies were observed at the higher end of the angle of attack range when the pitch rate was increased to 100.0◦
/s. Under these circumstances, significant history effects due to vortical interactions are present as illustrated inspecting the vortex surface footprint.
5
Turn 90deg
Wing−Over
Figure A.22: Wing-over and a 90-degree turn manoeuvres were simulated for the SDM model in [23]