Structured Input-Output Modeling and Robust Stability Analysis of Compressible Flows

TL;DR

The paper advances robust stability analysis for compressible flows by reformulating the compressible Navier–Stokes equations into a quadratic perturbation framework and embedding the nonlinear terms as a structured uncertainty. It leverages the structured singular value ($\mu$) to compute input-output gains, upper and lower bounds, and structured I/O modes, offering potentially less conservative stability margins than unstructured resolvent analyses. Applied to compressible plane Couette flow across subsonic, sonic, and supersonic regimes, the approach reveals Mach-number–dependent instability features and emphasizes thermodynamic versus momentum mechanisms through distinct forcing/response modes. The work demonstrates that incorporating nonlinearity structure can filter non-physical instabilities and yield more physically interpretable flow dynamics, with implications for analyzing compressible turbulence and broader nonlinear flows.

Abstract

The recently introduced structured input-output analysis is a powerful method for capturing nonlinear phenomena associated with incompressible flows, and this paper extends that method to the compressible regime. The proposed method relies upon a reformulation of the compressible Navier-Stokes equations, which allows for an exact quadratic formulation of the dynamics of perturbations about a steady base flow. To facilitate the structured input-output analysis, a pseudo-linear model for the quadratic nonlinearity is proposed and the structural information of the nonlinearity is embedded into a structured uncertainty comprising unknown `perturbations'. The structured singular value framework is employed to compute the input-output gain, which provides an estimate of the robust stability margin of the flow perturbations, as well as the forcing and response modes that are consistent with the nonlinearity structure. The analysis is then carried out on a plane, laminar compressible Couette flow over a range of Mach numbers. The structured input-output gains identify an instability mechanism, characterized by a spanwise elongated structure in the streamwise-spanwise wavenumber space at a subsonic Mach number, that takes the form of an oblique structure at sonic and supersonic Mach numbers. In addition, the structured input-output forcing and response modes provide insight into the thermodynamic and momentum characteristics associated with a source of instability. Comparisons with a resolvent/unstructured analysis reveal discrepancies in the distribution of input-output gains over the wavenumber space as well as in the modal behavior of an instability, thus highlighting the strong correlation between the structural information of the nonlinearity and the underlying flow physics.

Figures (16)

• Figure 1: The perturbation dynamics expressed in feedback forms: (a) the nonlinear system in \ref{['eq:perturbation_dynamics_1']}; (b) the system in \ref{['eq:structuredI/O_continuous_form']} obtained after the structured I/O modeling; (c) the system in \ref{['eq:IO-operator_discretized_perturbation_dynamics']} resulting from spectral discretization of the structured I/O system in \ref{['eq:structuredI/O_continuous_form']}. Note that the dashed box in (b), which represents the dynamic (linear) map that takes the modeled forcing/inputs $\mathbf{f}_\chi$ to the corresponding modeled outputs $\mathbf{y}_\chi$, becomes the frequency response operator $\mathcal{H}(k_x, k_z, \omega)$ in (c) after discretization.
• Figure 2: Steady base flow profiles of compressible plane Couette flow at different Mach numbers.
• Figure 3: Eigenvalue spectra of the linear operator $\hat{\mathbf{L}}(k_x,k_z)$---with and without accounting for the perturbations in viscosity about base flow---for $k_x = k_z = 0.1$, $Re = 2 \times 10^5$, $M_r = 2$, $Pr = 0.72$, $N_y = 200$. The eigenvalues $\omega^{(e)}$ are plotted in terms of complex wavespeeds $c_w = c_r + \mathbf{i} c_i = \omega^{(e)}/k_x$ with $\omega^{(e)}$ satisfying $\mathbf{\hat{L}} (k_x,k_z)\mathbf{q}^{(e)} = -\mathbf{i} \omega^{(e)} \mathbf{q}^{(e)}$, where the negative sign is utilized to be consistent with the temporal frequency sign convention in the existing literature.
• Figure 4: Distributions of the $\mu$ upper and lower bounds (log-scaled), percentage gap between the $\mu$ bounds, and the resolvent gain (log-scaled) over the wavenumber pair $(k_x, k_z)$ grid for $M_r=0.5$. The circle and the asterisk denote the $(k_x, k_z)$ values associated with the largest computed resolvent gain and $\mu$ bounds, respectively. Moreover, the dashed line in resolvent gain plot denotes $k_x=k_z$.
• Figure 5: The $\mu$ bounds and resolvent gain as functions of the temporal frequency for $M_r=0.5$ and $(k_x,k_z)=(0.01,11.24)$, which corresponds to the maximum resolvent gain on the wavenumber pair grid considered for the results in Fig. \ref{['fig:Resolvent_Mach_half']}.

Theorems & Definitions (1)

• Definition 1: packard1993zhou1996robust