Transient perturbation growth in time-dependent mixing layers

Abstract

We investigate numerically the transient linear growth of three-dimensional (3D) perturbations in a homogeneous time-evolving mixing layer in order to identify which perturbations are optimal in terms of their kinetic energy gain over a finite, predetermined time interval. We model the mixing layer with an initial parallel velocity distribution U(y) = U 0 tanh(y/d)e x with Reynolds number Re = U 0 d/ν = 1000, where ν is the kinematic viscosity of the fluid. We consider a range of time intervals on both a constant 'frozen' base flow and a time-dependent two-dimensional (2D) flow associated with the growth and nonlinear saturation of two wavelengths of the most-unstable eigenmode of linear theory of the initial parallel velocity distribution, which rolls up into two classical Rayleigh instabilities commonly referred to as Kelvin-Helmholtz (KH) billows, which eventually pair to form a larger vortex. For short times, the most-amplified perturbations on the frozen tanh profile are inherently 3D, and are most appropriately described as oblique wave 'OL' perturbations which grow through a combination of the Orr and lift-up mechanisms, while for longer times, the optimal perturbations are 2D and similar to the KH normal mode, with a slight enhancement of gain. For the time-evolving KH base flow, OL perturbations continue to dominate over sufficiently short time intervals. However, for longer time intervals which involve substantial evolution of the primary KH billows, two broad classes of inherently 3D linear optimal perturbation arise, associated at low wavenumbers with the well-known core-centred elliptical translative instability, and at higher wavenumbers with the braid-centred hyperbolic instability. The hyperbolic perturbation is relatively inefficient in exploiting the gain of the OL perturbations, and so only dominates the smaller wavenumber (ultimately) core-centred perturbations when the time evolution of the base flow or the start time of the optimization interval does not allow the OL perturbations much opportunity to grow. When the OL perturbations can grow, they initially grow in the braid, and then trigger an elliptical core-centred perturbation by a strong coupling with the primary KH billow. If the optimization time interval includes pairing of the primary billows, the secondary elliptical perturbations are strongly suppressed during the pairing event, due to the significant disruption of the primary billow cores during pairing.

On the longitudinal optimal perturbations to inviscid plane shear flow: formal solution and asymptotic approximation

“…In both cases, asymptotic estimates of the optimal gain are given as a function of the maximum shear and its first (respectively second) derivative at the wall (respectively at the inflection point). Both asymptotic predictions have been tested: in the free shear layer case by comparing to the directly computed optimal gain and optimal perturbations to a tanh profile using the direct-adjoint method (Arratia et al 2013), and by comparing to the optimal perturbations and to the slope of the algebraic growth rate for Poiseuille flow, derived in the present paper, in the case of wall-bounded shear flow.…”

“…For this, a pseudospectral code for the direct numerical simulation of the Navier-Stokes equations (NSE) linearized around a tanh profile has been used to compute the evolution of infinitesimal perturbations in the inviscid limit. The continuous adjoint NSEs have been implemented on the same code in order to retrieve the optimal perturbations through a power iteration algorithm that consists in solving the direct NSEs followed by the adjoint NSEs backwards in time (code and methodology as in Arratia et al 2013).…”

“…The dashed grey line shows the corresponding estimate given by (4.5) ( ) perturbations to a tanh profile, numerically computed using a direct-adjoint technique implemented in a pseudospectral code (Arratia, Caulfield & Chomaz 2013). The dashed grey line shows the estimate from (4.5) and the dashed black line shows the approximation of the viscous optimal gain obtained from (5.2) with the inviscid optimal gain for a tanh profile as G inv .…”

“…As for the free shear flow, the present local estimate may be turned into a full asymptotic solution by introducing an inner layer where the wall-normal FIGURE 8. Instantaneous growth rate σ (t) of the optimal perturbation to a tanh profile for T = 7 and k z = 4π, computed with the direct-adjoint method described in Arratia et al (2013). coordinate is rescaled by k −2/3 z , in which the inner layer solution is the Airy function vanishing at the wall and evanescent at infinity, as imposed by the matching condition with the outer layer.…”

Transient growth in strongly stratified shear layers

“…However, due to the non-normality of the Navier-Stokes operator, it is possible that, over a finite horizon time T, other perturbations may exhibit larger energy gain (see the review of Schmid 2007). As demonstrated by Arratia, Caulfield & Chomaz (2013), the optimal perturbation with largest gain in an unstratified shear layer tends towards the KH instability as T → ∞. It is thus natural to ask how stratification might modify this observation, building on the previous calculations by Tearle (2004), who considered optimal perturbation growth in stratified flow using an approach based on linear algebra.…”