Hydrodynamic Stability: Non-Normality and Transient Energy Growth

Welcome back. Last week, we derived the Orr-Sommerfeld equation, which governs the evolution of small disturbances in 2D parallel shear flows. This equation was cast as an eigenvalue problem, L𝜑 = c𝜑, that determined stream function amplitudes, 𝜑, and their associated complex wave speeds, c, that characterized the behavior of small perturbations. The governing Orr-Sommerfeld operator was defined as L = B^-1A, where 

A = i/(k Re) (D²-k²)² + U (D²-k²) – U’’,

B = D²-k²,

and D was the derivative operator d/dy. Each eigenvalue-eigenfunction pair (c, 𝜑) determined a characteristic instability mode with growth rate depending on the imaginary part of c. At the very end of the previous blog, we pointed out that L is non-normal, which essentially means that L has non-zero off-diagonal terms when it is discretized onto a grid. This can cause coupling between different instability modes that can lead to transient energy growth. In this blog, we will discuss this energy growth and briefly explore how it can be quantified using pseudospectra. 

Transient energy growth is a finite-time increase in disturbance energy in a flow that is linearly stable and has eigenfunctions that all decay to zero in the limit t → ∞. The growth in perturbation energy can be several orders of magnitude, which can trigger nonlinearities and cause a flow to become turbulent. This occurs because eigenfunctions are not orthogonal to each other, which causes two decaying exponential terms to compete so that their difference becomes larger before it ultimately decays to zero. 

Although the spectrum of L cannot capture transient growth, the pseudospectra of L can provide insights into this phenomenon. The pseudospectra depend on a small parameter 𝜀, and can be defined in several equivalent ways. In one formulation, the pseudospectrum at a particular value of 𝜀 is the set of all complex numbers, z, that are eigenvalues of L or eigenvalues of a small perturbation to L with norm less than 𝜀. Alternatively, the pseudospectrum can be defined using the resolvent operator, (zI – L)^-1, as the set of all complex numbers, z, such that 

||(zI – L)^-1|| > 𝜀^-1. 

This definition can make intuitive sense because the resolvent blows up to infinity at an eigenvalue, but has a very large, but still finite, norm at complex numbers that are close to being eigenvalues. For non-normal systems, the pseudospectra at a variety of 𝜀 can extend into the unstable half plane even when all the eigenvalues are stable. This fact can be leveraged to compute lower bounds on transient energy growth. One estimate is the Kreiss constant, defined by

K = sup_{𝜀 > 0} 𝜂(𝜀)/𝜀, 

where 𝜂(𝜀) is the maximum protrusion of the pseudospectrum into the unstable half plane at a particular value of 𝜀. In turns out that the state-transition operator e^Lt must exceed K at some positive time t > 0, so 

sup_{t > 0} ||e^Lt|| > K. 

This growth in energy is proportional to ||e^Lt||², and the transient energy growth is bounded below by K². In the next blog post, we will explore pseudospectra further as we cover some basic principles in resolvent analysis. Until then, please take care. 

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