Shock waves are commonly observed in high-speed flows, and they often interact with the boundary layer close to the wall resulting in localized regions of high wall pressure and wall-heat flux. If the incoming boundary layer is turbulent in nature, it adds to the complexity of the interaction, resulting in an amplified turbulence kinetic energy and turbulent heat flux across the shock, which can significantly alter the flow topology. Additional instabilities developed due to shock/boundary-layer interaction (SBLI) also generate an unsteady motion of the shock wave. Considering, for example, a normal shock observed on an airfoil in transonic regime at two time instances, the significant shock oscillation observed on the suction side (buffet) is associated with a large breathing separation region, resulting in a rapid loss of lift and increase in drag, often hindering the maneuverability of the aerospace vehicle involved. Additional drawbacks of such an uncontrolled interaction include unsteady vortex shedding and shock/vortex interaction, which are the major causes of broadband noise emission. Also, a rapid generation of pressure fluctuations in the interaction region can be detrimental to the structural health of the wing. Such unsteady interactions are not only observed in external flows but also in internal flows such as a thrust generating supersonic nozzle of a rocket engine while operating at sea-level condition. The asymmetric nature of the flow structure and the shock unsteadiness leads to heavy side loads, which pose significant threat to the vehicle structure and its control. The main aim of the proposed project is to numerically simulate and characterize the shock/turbulent boundary layer interaction in a practical geometry of interest, overexpanded supersonic nozzles, using an innovative and efficient active control strategy, i.e. secondary jet, to mitigate the effects of SBLI governed by an optimal feedback mechanism to devise the control parameters.
The adjoint methods gives inaccurate sensitivities when directly applied to simulation techniques such as large eddy simulation (LES) and detached eddy simulation (DES). Although these high-fidelity simulations are essential in analyzing SBLI and reveal fundamental unsteady characteristics such as peak fluctuating wall pressure, heat load, wall pressure spectra and low frequency unsteadiness, their flow resolving nature inherits the chaotic dynamic characteristics of turbulence and prevents the application of adjoint method (Larsson & Wang, 2014). A small perturbation in the flow completely changes the time history of a simulation, which leads to a divergence in the adjoint solution when integrated over a long time period (Wang & Gao, 2013). This unphysical
and erroneous sensitivity prediction has resulted in exploration of other techniques suitable for chaotic flow simulations such as least-square shadowing method (Wang, 2014), which have exorbitant computational cost, especially for flows over practical geometries (Blonigan et al., 2016).
In spite of the relative simplicity of the unsteady RANS approach at a fraction of cost as compared to flow resolving the previous high-fidelity approaches (LES and DES), researchers (Memmolo et al. (2018)) have shown encouraging URANS results in case of buffet analysis over transonic airfoils. Application of AD technique to the URANS solver not only results in computing of primal function value (flow solution), but also can give the exact derivatives of the cost function with respect to a specific set of control parameters. The AD technique is applied to the primal fixed-point iterative solver such that the linearized residual has contributions from all aspects of the solution strategy such as spatial and temporal discretization, convergence acceleration techniques, etc. The exact differentiation of this primal iterator ensures that the resulting adjoint solver generates accurate sensitivities at any level of convergence established by the primal solver (Carnarius et al., 2010; Nemili et al., 2017). This approach however comes with an increased cost in memory and computational time. Note that the computation of adjoint in an unsteady simulation involves storing the entire flow history (at every iteration) during the forward-in-time integration of the governing equations, which are then retrieved to solve for the adjoint solution through the backward-in-time integration. To circumvent the huge cost associated with storing and retrieving the data, we will resort to checkpoint strategy, where the flow solutions at selective time interactions (checkpoints) are stored. These checkpoint solutions will then be used to recompute the intermediate state solutions for adjoint computation in the backward-in-time integration loop (Stumm & Walther, 2009). Such an approach has proven to be an effective compromise between store-all and recompute-all approaches, balancing out on the memory and computational time requirement for obtaining the adjoint solution (Nemili et al., 2017).
Considering the encouraging capability of URANS in predicting shock unsteadiness and other unsteady flow characteristics such as lift and drag coefficients and the advantages in computing the adjoint sensitivities, the idea is to extract the URANS-adjoint driven optimized SJ actuator values and implementing in a DDES simulation to mainly analyze two aspects of the problem: 1) how efficient are the URANS-adjoint driven optimized SJ control parameters when applied in a more advanced DDES framework in controlling the cost functions associated with mean characteristics (mean wall pressure, buffet amplitude, mean separation size, etc.); 2) what impact this control variables have on the unsteady turbulence characteristics of the flow (fluctuating pressure loads, instantaneous separation, peak heat load, etc.) gauged through the DDES methodology.
The choice of DDES approach is based on the efficient yet affordable alternative it provides in analyzing the unsteady turbulent flow features of SBLI, as compared to other turbulence resolving methods such as direct numerical simulation and LES, which require substantial computing power. DDES has been successfully applied in the past to study complex flow in practical geometries features associated with SBLI, both in internal flows (Martelli et al., 2017) and external flows (Memmolo et al., 2018). In DDES, while the regions of separated flow are tackled using the more sophisticated LES subgrid scale model which takes into account the varying spectrum of turbulence scales, the regions with attached boundary layer are simulated using well-established Reynolds-averaged Navier-Stokes (RANS) framework. Furthermore, the DDES approach is an improvement over the original DES methodology (Spalart, 2009), and is specifically tuned to prevent the depletion of eddy viscosity in the switch area between LES and RANS, which can further lead to an unphysical grid-induced flow separation.