INVESTIGATION OF THE CONTINUOUS AND DISCRETE ADJOINT IN THE CONTROL OF PLANE JETS

Abstract

A comparison of the optimal control of twoand threedimensional plane jets using the continuous and discrete adjoint of the instationary Navier-Stokes equations was performed. The control aim was to reduce the sound emission in the near far-field by using a heat source actuation within the transitioning jet shear layers. The fully compressible Navier-Stokes equations were solved using dispersionrelation preserving spatial discretization schemes and a lowdissipation-dispersion Runge-Kutta scheme. The Reynolds number based on the slot diameter was set to 2000 and the Mach number to 0.9. Direct numerical as well as large-eddy simulations in two and three dimensions where performed to estimate the influence of modelling and resolution on the results. The results show a slight advantage of using the discrete adjoint, especially when handling boundary conditions, since the calculation of the gradient of the cost functional is more accurate. It is interesting, too, that the control efficiency reduces with increasing resolution and therefore dimension of the control. Reducing it by applying a selected interpolation in the control area shows an increase in efficiency and sound reduction. Introduction Optimal control of flows using the adjoint equations has become a valuable tool in fluid mechanics, with applications ranging from aerodynamic shape optimization (Kuruvila et al., 1994; Giles & Pierce, 2000; Brezillon & Gauger, 2004; Giering et al., 2005; Srinath & Mittal, 2010; Zymaris et al., 2010; Jameson & Ou, 2011) to sound reduction in compressible flows (Joslin et al., 2005; Wei & Freund, 2006; Spagnoli & Airiau, 2008; Freund, 2010; Kim et al., 2010; Rumpfkeil & Zingg, 2010; Marinc & Foysi, 2012). The principal task is to minimize a cost functional or objective (unwanted noise or drag, for example). Differential equation constraints are additionally imposed, which here consist of the primal flow equations. The minimization procedure requires the determination of the gradient of this cost functional (Gunzburger, 2002). Unfortunately, a finite difference approach requires O(n) solutions of our primal flow equations for n different design variables to obtain the gradient, which is impractical. Optimal control based on the adjoint equations on the other hand is independent of the number of design parameters. The adjoint may be calculated using two different routes. One possibility is to derive the adjoint equations analytically based on the problem describing partial differential equations (primal), before discretizing the resulting equations (“first optimize then discretize” (FOD). Alternatively, the primal flow equations are discretized first and these already discretized equations are used to determine the discrete adjoint equations (“first discretize then optimize” (FDO)). Both routes lead to different numerical results which are equal only in the limit of infinitely small grid and time steps. For a further discussion of disadvantages and advantages see Gunzburger (2002). For aeroacoustic sound reduction most authors used the continuous adjoint approach, so far (Joslin et al., 2005; Wei & Freund, 2006; Spagnoli & Airiau, 2008; Freund, 2010; Kim et al., 2010; Marinc & Foysi, 2012), making it possible to adjust or change the discretization or boundary treatment of the adjoined compared to the primal flow equations. However, Marinc & Foysi (2012) showed recently, that the gradient direction deviates from the exact gradient direction towards the end of the control horizon for instationary control simulations (figure 1, normalization was done by the corresponding values at the nozzle exit). It’s possible to even have opposite gradient directions rendering the minimization ineffective or even unsuccessful. Among the possible reasons are inconsistencies due to a different discretization of the adjoint compared to the primal flow equations, different boundary conditions, additional numerical filtering for stabilization in critical areas with large gradients or grid The perturbations are chosen to have Gaussian shapes in space a d time with an amplitude small enough to remain in the linear regime. Because of the localization in time the accuracy of our gradient can be estimated at several distinct times this way. When comparing FFD with Fgrad the case FFD · Fgrad < 0 can be considered as a worst case scenario as it implies that the calculated gradient isn’t an asc nt direction. Fig. 11 shows values of FFD and Fgrad for various perturbations and for different controls, obtained after different conjugate-gradient optimization steps for the DNS2D-case. The control-length was chosen to be ∆T ≈ 72. As the control require a finite tim to influenc the cost-functional, the control is not able to change the cost-functional at times near the end of the simulation. Thus, the gradient is zero for 52 ! t ≤ 72. For 20 ! t the values of FFD and Fgrad roughly agree, i.e. they show the same tendencies and the sign agrees, indicating that the calc lated gradient is acceptable during this period. For t ! 20 the functions have distinct values, which leads to the conclusion that the gradient contains no reliable information about the steepest descent direction and no significant reduction of the cost-functional can be expected from a control obtained by this signal. To summ rize, it can be stated that t calculated gradient is acceptable for a time-interval of approximately ∆t ≈ 50 for the DNS2D-case. This control-interval length is approximately twice as large than the time which is needed for the perturbations to influence the vortex pairing processes (∆t ≈ 7) and the subsequent time needed for the sound to propagate to the observation region Ω (∆t ≈ 19). Interestingly, the acceptable time-interval ∆t ≈ 50 is shorter than the interval of Tp = 79 used in section 3.3 for the control