A data-driven approach to guide supersonic impinging jet control

Active flow control has the potential to alleviate many aerodynamic problems; some objectives include mitigation of thermomechanical loads, separation and unsteadiness, or flow-acoustic interactions due to supersonic jets impinging on a ground plane, of interest in this work. Optimal control inputs are predicated on identifying responsiveness to external forcing (receptivity to perturbations) that depend on the underlying flow mechanisms, which are often examined with the linearized frequency-domain Navier-Stokes equations applied to steady basic states. Classical linear stability extracts such information from the laminar state (Theofilis, 2011; Juniper et al., 2014), or, the time-averaged turbulent state, whose modes are related to larger turbulent coherent structures (Crighton & Gaster, 1976). More recently, other manners of analyzing the mean flow have been employed with overlapping objectives to extract singular or eigenvalue spectra with corresponding modes (Schmid, 2007; Luchini & Bottaro, 2014; Ranjan et al., 2020). From a control standpoint, optimal forcing-response relationships may be derived from the mean flow through resolvent or input-output modes that maximize linear energy growth. Example applications of control strategies derived from perturbed steady basic states may be found in Herrmann et al. (2021, 2023).

When unsteady data is available in the form of snapshots, additional information may be derived to discern the key mechanisms as well as to propose control inptus. Modal decomposition techniques (Taira et al., 2020) have become increasingly popular to represent the dynamics (model order reduction), and their potential implications for control have also been elaborated on (Rowley & Dawson, 2017; Brunton & Noack, 2015). In the present work, we propose a data-driven approach (§2) to derive a forcing-response relationship using perturbations in the time-varying flow with snapshot-based modal decomposition serving as a linear surrogate for the dynamics. Dynamic Mode Decomposition (DMD) provides such a representation (Schmid, 2010) and is thus used to generate an efficient reduced order model (DMD-ROM) for the nonlinear behavior contained in the snapshots. The method also provides a reduced basis for projection of the input control forcing and the evolution in time of the forced system. The procedure may be viewed as a much simplified data-driven analog of the synchronized Large Eddy Simulation (LES) method of Adler & Gaitonde (2018) to determine time-local perturbation growth in the turbulent flow.

Several advantages may be delineated. Since it is data-driven, the DMD-ROM contains the main dynamics encapsulated in the snapshots, including configuration complexity and flow parameters such as Mach and Reynolds numbers, large values of which can often impede more difficult linearized Navier-Stokes driven approaches because of spurious modes. In addition, the time-domain nature of the method eases specification of realizable actuator-specific inputs, including the nature of perturbations, localization in space, and phase variations in time, this last advantage also enables control assessments of time-local events. The approach may also be applied only with select snapshot variables; this is advantageous, for example, when data is obtained from experiments. More crucially, even when all data is available, say, from uncontrolled scale-resolved simulations, only a subset of variables may be selected to construct the DMD-ROM. In the impinging jet problem of interest, the hydrodynamic-acoustic interaction is well encapsulated in the scale-resolved pressure field, and is therefore the only variable used.

For computational economy – an important consideration for time-domain methods – we use a reduced subspace, constructed from projection of flow and forcing snapshots onto a Proper Orthogonal Decomposition (POD) basis. This significantly speeds up parametric studies of actuator locations and forcing properties. Evaluation metrics such as energy gain for asymptotic or short-time events are also computed in the reduced space; projection back to the full physical space may be performed only when needed.

The capabilities of the DMD-ROM approach are highlighted in the context of a complex supersonic impinging jet containing interacting shocks and nonlinear instabilities. The resonant dynamics have been extensively reviewed by Edgington-Mitchell (2019). Briefly, Kelvin-Helmholtz structures in the shear-layer emanating from the nozzle exit impinge on the ground plane to generate acoustic waves that travel back to the nozzle exit to complete the loop. Two applications of the DMD-ROM are considered to address asymptotic and short-time phenomena, respectively. The first examines the tonal behavior of the impinging jet. For this, we employ harmonic pressure forcing at the nozzle receptivity region to interfere with the aeroacoustic feedback resonance and modify the most amplified tones. The forced linear response is confirmed with post facto scale-resolved simulations featuring realizable blowing-suction actuators with the same spectral forcing properties.

The second aspect concerns transient or time-local control to address the problem of energy growth intrinsic to conditionally selected events occurring over finite time. This example targets the vital acoustic feedback receptivity events in the resonance cycle which initiate the convective shear-layer instability. Perturbation-based insights into such transient structures may be found in Karami et al. (2018). Here we use conditional space-time proper orthogonal decomposition (CST-POD) method to isolate an ensemble of shear-layer events directly from the unsteady flow (Schmidt & Schmid, 2019; Stahl et al., 2023) for perturbation-based receptivity analysis, with the objective of determining the optimal forcing phase to mitigate the convective instability that perpetuates the resonance cycle.

The proposed method, summarized in the schematic of Fig. 1, is implemented in four steps: (1) construction of the DMD-ROM from snapshots of the unforced flow, (2) creation and projection on the reduced space of the forcing snapshot time-sequence, (3) time advancement of the linearized forcing response and gain evaluation in the reduced space and (4) expansion of the forced response back to the full-space when desired.

The first step (1), uses the unforced sequence of $N$ flow snapshots, $\textbf{Q}=[q_{1},...,q_{N}]$, to model the linear operator A that marches the system from one snapshot to the next, $q_{n+1}=\textbf{A}q_{n}$. Since Q and A are full sized, $M\times N$ and $M\times M$ matrices, where $M$ is the number of spatial degrees of freedom, the classical DMD approach (Schmid, 2010) is employed to obtain the reduced $N\times N$ operator, $\tilde{\textbf{A}}$, by projection onto the POD modes, U, obtained from the singular value decomposition of Q.

The second step (2), constructs the $M\times N$-sized forcing snapshot matrix, $\textbf{F}=[f_{1},...,f_{N}]$, based on the chosen actuator location and perturbation properties. For the jet examples, the most natural location for actuators is the region around the nozzle exit. Although the forcing function may be an arbitrary function of time, here we consider harmonic forcing $f(t)=A\sin(\omega t+\phi)$, with $A$, $\omega$ and $\phi$ being the amplitude, angular frequency and phase shift in time, respectively. The reduced flow and forcing snapshots are projected onto the POD modes of the flow, $\tilde{q}_{n}=\textbf{U}^{-1}q_{n}$ and $\tilde{f_{n}}=\textbf{U}^{-1}f_{n}$, respectively.

The third step (3), shown in the overall scheme of Fig. 1, evolves the reduced forced system in time,

$$\tilde{q}^{\prime}_{n+1}=\tilde{\textbf{A}}\tilde{q}^{\prime}_{n}+\tilde{f}_{n},\hskip 21.68121ptn=0,1,\ldots,N$$

(1)

where $\tilde{q}^{\prime}_{n+1}$ represents the linear response of the system and $n=0$ represents the initial condition. Since interpreting the forced results directly from the reduced coordinates can be ambiguous, in the fourth step (4), the full space equivalent of $q^{\prime}_{n+1}$ may be reassembled when desired by projection back to the full-space domain to obtain the forced solution ($\textbf{Q}^{\prime}=\textbf{U}\tilde{\textbf{Q}}^{\prime}$). For the present feed-forward study, projection back to the full-space is only necessary after the time-iteration stage is complete. Full-space expansion within each iteration may become necessary in feedback control to inform changes in the forcing.

The procedure to obtain gain in the reduced space exploits features that are consistent between it and the full-sized system. Specifically, we use the $L2$-norm ratio of the forced and unforced solutions $\sigma=||\tilde{\textbf{Q}}^{\prime}||_{2}/||\tilde{\textbf{Q}}||_{2}$ to obtain a relative sense of gain. This provides an adequate assessment for the asymptotic forcing analysis (§3.1). For the transient problem (§3.2), we use the ratio of the $L2$-norm of individual snapshots to the initial condition snapshot: $\sigma(t)=||\tilde{q}(t)||_{2}/||\tilde{q}_{o}||_{2}$. Since this transient norm applies to both unforced and forced systems, the value may then be assessed to understand the relative effects of the forcing over time. Operations in the reduced space assure computational economy and enable testing of a wide range of forcing parameters.

Finally, we note that the approach differs from the DMD Control (DMDC) method of Rosenfeld & Kamalapurkar (2021), which uses forced data for system identification to distinguish the dynamic and forcing operators. The current DMD-ROM approach could potentially be enriched with such data when combined with a learned control operator; however the focus here is on obtaining the linear response of the unforced dataset to different forcing parameters, optimal results from which may be refined with limited computationally intense nonlinear simulations or experimental testing.

We consider the aeroacoustic resonance of a planar (spanwise homogeneous), Mach 1.27, underexpanded jet (nozzle-width, $D=0.0254m$) impinging on a plate surface located $4D$ away, as shown in Fig. 2(a) which also displays domain dimensions.

The uncontrolled (baseline) data, containing multiple resonant tones, is acquired from an LES at a Reynolds number of $Re=5.8\times 10^{5}$ and comprises $3{,}855$ pressure snapshots from the mid-plane of the domain sampled at a nondimensional frequency $St=fD/a=10$, where $f$ is the frequency (Hz) and $a=343m/s$ is the ambient speed of sound. The methodology for the LES, including numerical scheme and mesh, follow those established in Stahl et al. (2022).

As noted earlier, the pressure variable is suitable for the dynamics of interest since it encompasses the hydrodynamic and acoustic phenomena that dictate the feedback loop. A DMD-ROM based on mean-subtracted pressure fluctuations aids in stability by ensuring all DMD eigenvalues are neutrally stable (Towne et al., 2018). A representative baseline pressure snapshot, shown in Fig. 2(b), illustrates the natural asymmetric (at this time instant, predominantly blue on the right, red on the left outside the stagnation region) side-to-side flapping of the jet and feedback acoustics. The probe location in Fig. 2(b) monitors the acoustic feedback tones used in the rest of the paper. The forcing is applied on rectangular regions of size $0.2D\times 0.07D$ centered at the nozzle lip exit, displayed in the inset of Fig. 2(c), which shows both pressure forcing for the DMD-ROM and blowing-suction actuation for the LES.

A data-driven framework is presented to discern the linear response of a turbulent flow with a view towards control analyses. The nonlinear system is modeled with a dynamic mode decomposition reduced order model (DMD-ROM). Prescribed forcing conditions are projected onto the same reduced space as the unsteady flow, and its effects obtained in the time-domain. Computational efficiency is ensured by calculating the gain due to forcing in the reduced subspace, which enables rapid scanning of large ranges of forcing parameters, from which candidates may be expanded to the physical space for further study and verified with scale-resolved methods.

The key advantages of the method are that, being data-driven, it may be applied to any flowfield for which time-resolved snapshots are available, regardless of geometry and flow parameters, without need to solve the linearized Navier-Stokes equations. Furthermore, the time-domain nature of the technique facilitates examination of the forced evolution in the unsteady flow itself, eases actuator perturbation inputs and examination of transient events where phase becomes important.

Application of the model to the complex physics associated with a resonating supersonic impinging jet forced at the nozzle receptivity region, successfully predicts the switch from flapping to symmetric resonance modes, as validated with a full three-dimensional, nonlinear simulation. For transient events, the effects of forcing at different phases within the resonant feedback cycle provides insight into means of diminshing or amplifying convective instabilities with respect to the critical events within the cycle. In both examples, thousands of forcing parameters were tested across amplitude, frequency, and phase variables, highlighting the performance and utility of the method.

Acknowledgments: The authors acknowledge support from the Collaborative Center for Aeronautical Sciences and the Office of Naval Research.

Declaration of interest: The authors declare no conflict of interest.