Small Disturbance Method

The small disturbance method was presented to illustrate the use of the Euler equa- tions applied to several two-dimensional test cases and an initial extension to a three- dimensional case [162]. Beside the sub- and supersonic regime, the capabilities of the implemented approach were demonstrated in the transonic speed range for a NACA 64A010 aerofoil and LANN wing. The small disturbance Euler solution was compared to the underlying nonlinear Euler solution. For the aerofoil case, the first harmonic unsteady pressure coefficient distribution displayed a good agreement upstream and downstream of the shock region. Deviations were observed around the shock, with spikes detected in the small disturbance solution. Although differences in the shape of the shock impulse, the load contribution of the shock impulses can be considered equal, and the first harmonic lift coefficient compared well between the two data sets. This is quite remarkable because the shock moves up to 20% of the aerofoil chord depending on the reduced frequency. This asserts the validity of shock capturing in a perturba- tion method applied to the transonic regime, as originally introduced by Lindquist and Giles [163]. It is argued that the load contribution to the pitching moment coefficient is not considered, which typically exhibits higher nonlinearities than the force coefficient because of the leverage arm between the shock impulse and the reference point. With the use of different nonlinear Euler codes in addition to the underlying CFD code, the application of the LANN wing corroborates the capability of the small disturbance Eu- ler equations for a more complex configuration. It is demonstrated that variations in

the unsteady pressure coefficient obtained using different CFD solvers are more signifi- cant than those detected between the small disturbance Euler code and the underlying nonlinear CFD code. This confirms the capabilities of the small disturbance Euler method to predict unsteady loads even with a complex shock structure as observed for the LANN wing.

A 53◦

low aspect ratio cropped delta wing was then tested in the transonic regime as a frequent high manoeuvrable aircraft wing [164]. The flow topology consists of leading-edge vortices forming at higher angle of attack and the application of the small disturbance Euler method was aimed to simulate unsteady aerodynamic loads due to rigid body, flap and elastic harmonic motions. The important concern to guarantee flap efficiency during the entire flight envelope calls for an accurate prediction of control derivatives, which was demonstrated to be possible with the small disturbance method. An aeroelastic-like example, using an equation with polynomial coefficients for the local amplitudes for the elastic eigenmode, provided an additional test of the Euler method. In all cases, a reduction of one order in computational time was achieved using the small disturbance method with respect to the nonlinear counterpart.

Inviscid methods reach their limitations with flows where viscous effects are a dom- inant feature (separation, shock/boundary layer interaction), and the extension of the small disturbance method to viscous flows is an attractive alternative to time marching the RANS equations. A small disturbance Navier-Stokes method was developed from the existing inviscid solver supplementing the viscous algorithms and incorporating turbulence models in an appropriate formulation [165]. Within a linearized framework, higher harmonics in the aerodynamic response are considered to be negligible. With the use of a triple decomposition of the flow development [166], an arbitrary instan- taneous flow quantity is constructed as sum of a steady mean component, a periodic perturbation and a turbulent fluctuation. The simulation process consists, first, of a turbulent steady state solution on the reference grid using the nonlinear Navier-Stokes equations, which provide the prerequisite mean flow values contained in the source term and the convective flux Jacobian. Then, the small disturbance equations are solved for the complex amplitude of the unsteady flow solution. Two two-dimensional test cases were considered and the Spalart-Allmaras turbulence model used in all calculations. For the NACA 64A010 aerofoil featuring a weak shock, the viscous solutions are com- pared to the small disturbance Euler solution. The inviscid solution, not including the influence of the boundary layer in the shock formation, predicts a shock located fur- ther downstream with a stronger gradient, and an overextended recompression before merging into the pressure recovery curve close to the trailing edge. Varying the re- duced frequency, the computational speedup of the small disturbance viscous solution over the underlying nonlinear viscous solution is between a factor of 5 and 28. The perturbation method requires more than three times the working memory computed with the time-accurate method. The second test case is the NLR 7301 aerofoil fea-

turing a strong shock. When including the viscous effects, the shock location in the zeroth harmonic component is in good agreement with experimental data whereas the inviscid solution predicts a shock situated 20% further downstream due to the neglect of the strong shock/boundary layer interaction. Furthermore, contribution of higher harmonics in the shock region was detected in the nonlinear solutions, resulting in a poor prediction of the unsteady pitching moment coefficient obtained using the small disturbance Navier-Stokes equations.

Derived from the wing of a supersonic transport aircraft configuration, the 50◦ NASA clipped delta wing was considered in conditions featuring varying shock strength and leading-edge vortex formation [157]. Inviscid and viscous solutions were compared to available experimental data. For the weak shock case, the experimental data of first harmonic unsteady pressure coefficient are best reproduced using the inviscid methods. Surprisingly, the small disturbance Euler method performs better in terms of unsteady aerodynamic loads than the nonlinear counterpart and, in particular, predicts the very similar damping term compared with the nonlinear Navier-Stokes calculations. The inclusion of the viscous effects has a significant and consistent improvement on the moment damping with respect to the inviscid methods, and is of paramount importance when free pitching oscillations are considered. The second case is a medium strength shock extending from the wing root to the tip featuring a leading edge vortex formation. The interaction of the shock with the vortical flow in the outer wing section close to the wing tip results in deviations between the two viscous solutions in the unsteady pressure distribution. The dynamic shock/vortex interaction introduces in this limited wing region higher harmonics into the flow solution, which are beyond the assumptions of the small disturbance method. A strong shock is then considered, extending beyond the wing tip. In these circumstances, the inclusion of the viscous terms improves the predictions of the small disturbance method compared to the inviscid counterpart. Depending on the flow conditions, the computational efficiency increase varied between a factor of 10 and 20 in all cases.