A framework for input–output analysis of wall-bounded shear flows

Abstract

We propose a framework to understand input-output amplification properties of nonlinear partial differential equation (PDE) models of wall-bounded shear flows, which are spatially invariant in one coordinate (e.g., streamwise-constant plane Couette flow). Our methodology is based on the notion of dissipation inequalities in control theory. In particular, we consider flows with body and other forcings, for which we study the inputto-output properties, including energy growth, worst-case disturbance amplification, and stability to persistent disturbances. The proposed method can be applied to a large class of flow configurations as long as the base flow is described by a polynomial. This includes many examples in both channel flows and pipe flows, e.g., plane Couette flow, and Hagen-Poiseuille flow. The methodology we use is numerically implemented as the solution of a (convex) optimization problem. We use the framework to study input-output amplification mechanisms in rotating Couette flow, plane Couette flow, plane Poiseuille flow, and Hagen-Poiseuille flow. In addition to showing that the application of the proposed framework leads to results that are consistent with theoretical and experimental amplification scalings obtained in the literature through linearization around the base flow, we demonstrate that the stability bounds to persistent forcings can be used as a means to predict transition to turbulence in wall-bounded shear flows.

Verification of some functional inequalities via polynomial optimization

“…for some polynomial φ. In such cases, a duality argument (Fantuzzi, 2019) shows that our sufficient conditions for the functional inequality in Problem 1 are equivalent to those proposed by Valmorbida et al (2015Valmorbida et al ( , 2016, Ahmadi et al (2019Ahmadi et al ( , 2016Ahmadi et al ( , 2017, and Chernyavsky et al (2021).…”

“…This is usually hard to do, even with computer assistance. Valmorbida et al (2015Valmorbida et al ( , 2016 and Ahmadi et al (2019Ahmadi et al ( , 2016Ahmadi et al ( , 2017 demonstrated that SOS polynomials and SDPs can be used to verify the nonnegativity of integral functionals with polynomial integrands in one or two spatial dimensions. This is achieved without discretization of the PDE state, but rather by requiring the polynomial integrand to be nonnegative pointwise after augmenting it by terms that integrate to zero.…”

Cause-and-effect of linear mechanisms sustaining wall turbulence

“…2017; Ahmadi et al. 2019; Jovanović 2021). The input–output approach combines the linearised Navier–Stokes equations with harmonic or stochastic forcing (white or coloured in time) to qualitatively predict structural features of turbulent shear flows.…”

Duality of convex relaxations for constrained variational problems

“…Replacing polynomial non-negativity with sum-of-squares (SOS) conditions enables one to maximize lower bounds on L * numerically by solving a hierarchy of semidefinite programs (SDPs), indexed by the degree of the Lagrange multipliers. This strategy was proposed by Valmorbida et al [16][17][18][19] and Ahmadi et al [1][2][3][4][5] in the context of stability analysis, input-output analysis, and safety verification for dynamical systems governed by polynomial PDEs, but applies equally well to constrained variational problems.…”

“…Moreover, many popular algorithms for solving SDPs require strong duality to guarantee convergence and avoid poor numerical conditioning. Consequently, the relaxations of (1.3) proposed by [1][2][3][4][5][16][17][18][19] and [9] are equivalent from the point of view of numerical computations.…”