Adjoint-free method for mean resolvent analysis of periodic flows

TL;DR

This study distinguishes mean-flow resolvent and mean resolvent operators for periodically forced flows and introduces an adjoint-free projection-based method to approximate the mean resolvent in a low-rank form. By leveraging the relation between mean-flow resolvent modes and mean resolvent modes, the projection method reduces the high-dimensional eigenproblem to a small $d imes d$ problem, enabling efficient estimation of dominant input-output mechanisms without adjoint equations. Validation against an adjoint-based harmonic framework in a periodic axisymmetric jet reveals that the mean resolvent captures physically meaningful subharmonic receptivity and that the adjoint-free method converges rapidly for weakly unsteady bases (e.g., $duildrel< hinspace elax 10$), while requiring larger subspaces for strongly unsteady bases ($duildrel extstyle elax hicksim100$). The results demonstrate the potential of adjoint-free mean resolvent analysis as a practical tool for stable, low-cost linearization-based control design in statistically steady periodic flows and provide a foundation for extending to turbulent, stochastic, or chaotic regimes.

Abstract

The mean resolvent operator predicts, in the frequency domain, the mean linear response to forcing. As such, it provides the optimal linear time-invariant approximation of the input-output dynamics of time-varying flows in the statistically steady regime (Leclercq & Sipp 2023). In this paper, we introduce an adjoint-free projection-based method for mean resolvent analysis of periodic flows. To evaluate the convergence of the projection-based method against the subspace dimension, we also implement an adjoint-based approach based on the harmonic resolvent framework (Wereley & Hall 1990, 1991; Padovan et al. 2020). Both adjoint-free and adjoint-based approaches may also be implemented in a matrix-free paradigm, using a time-stepper. For a weakly unsteady base flow, the mean-flow resolvent qualitatively approximates the dominant receptivity peak of the mean resolvent but completely fails to capture a secondary receptivity peak. For a strongly unsteady base flow, even the dominant receptivity peak of the mean resolvent associated with vortex-pairing is incorrectly captured by the mean-flow resolvent. The projection method already converges for a subspace dimension of 10 in the weakly unsteady case, but requires at least 100 modes for quantitative predictions in the strongly unsteady case. However, even in this case, using a subspace dimension of 1 is already enough to correctly identify the dominant receptivity peak.

Figures (14)

• Figure 1: Two ways of defining linear time-invariant input-output operators for statistically steady flows. The symbols $E[.]$ and $\overline{(.)}$ respectively denote ensemble and time averages. In the mean-flow resolvent approach (a), the dynamics is linearised about the time-averaged mean flow. In the mean resolvent approach, the dynamics is linearised about unsteady trajectories modelled as realisations of a stochastic process, and the resolvent operator predicts, in the frequency domain, the ensemble-averaged response to the deterministic forcing $\mathbf{f}'$. For periodic flows, the ensemble average is replaced by a phase average; stochasticity arises from the random choice of a phase to start forcing leclercq2023mean.
• Figure 2: (a)-(b) Snapshots of the azimuthal vorticity field computed from the $T_0$-periodic base flow solution $\boldsymbol{\mathrm{Q}}\left(t\right)$ for (a) $\omega_0=6\pi/5$ and (b) $\omega_0=3\pi/5$. (c)-(d) Time-averaged azimuthal vorticity field from the corresponding mean state $\overline{\boldsymbol{\mathrm{Q}}}$. (e)-(f) First $n=1$, (g)-(h) second $n=2$ and (i)-(j) third $n=3$ harmonics extracted from the Fourier decomposition of the instantaneous flow, shown as the real part of the azimuthal vorticity field. The same mesh and Reynolds number $Re=1000$ were used in both computations.
• Figure 3: Radial vorticity magnitude profile associated with the mean flow $\overline{\boldsymbol{\mathrm{Q}}}=\hat{\boldsymbol{\mathrm{Q}}}_0$ and the first three harmonics $\hat{\boldsymbol{\mathrm{Q}}}_{n}$ ($n=1,2,3$) extracted from the Fourier decomposition of the instantaneous flow (see figure \ref{['fig:Fig12']}) for (a) $\omega_0=6\pi/5$ at an axial coordinate $z=1$ and (b) $\omega_0=3\pi/5$ at $z=2$.
• Figure 4: Energy norm of the various harmonics ($|n|\leq 6$) extracted from the Fourier decomposition of the two $T_0$-periodic base flow solution $\boldsymbol{\mathrm{Q}}\left(t\right)$ for $\omega_0=3\pi/5$ (empty circle) and $6\pi/5$ (black crosses).
• Figure 5: (a) Leading singular value $\lambda_1$, as a function of the forcing frequency $\omega$, for the weakly unsteady base flow oscillating at $\omega_0=6\pi/5$. Black line with white circles: resolvent analysis about the mean flow. Red filled squares: mean resolvent analysis computed for $\mathrm{N}_{\mathrm{h}}=1,2,3$ and $4$. A good convergence is achieved already for $\mathrm{N}_{\mathrm{h}}=2$. A mesh of size $\mathrm{N_r}\times \mathrm{N_z}=200\times 300$ was used for this calculation. (b) Same, but for the strongly unsteady base flow oscillating at $\omega_0=3\pi/5$, for which convergence is achieved starting from $\mathrm{N}_h\ge 5$, at least in the explored frequency range.