Non-normal effects on the horizontal shear layer with vertical stratification

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Non-modal instability mechanisms in plane shear flow

Here we briefly describe the Orr and lift up mechanisms, two well known mechanisms of non-modal energy growth of perturbations to parallel shear flow. These mechanisms occur in two limiting cases, the Orr mechanism for purely 2D flow (kz = 0) and the lift up for perturbations infinitely elongated in the streamwise direction (kx = 0). These introductory descriptions of the Orr and lift up mechanisms are placed here for consistency, most of their content can be also found later in the manuscript. After describing these well known mechanism, we end up by mentioning some insights on the more general oblique wave perturbations taking place for kx, kz 6= 0. These mechanisms are essentially inviscid so we set ν = 0 in this section.

Orr mechanism

The Orr mechanism (Orr [82], for more recent discussions see for example [92, 45]) is responsible for the possibility of transient growth of perturbation energy in the 2D case (kz = 0). This mechanism originates in the kinematic deformation of perturbation vorticity ωz = ∂xv − ∂yu by base flow advection, as exemplified in the figure 2.3 (same as figure 6.3). Figure 2.3 shows the evolution of the vorticity ωz of the optimal perturbation for U = tanh(y)ex, streamwise wavenumber kx = 3.77 and optimization time T = 7. Shown are the optimal initial perturbation (at time t = 0), the optimal response (at t = T = 7) and the optimal perturbation at the later time t = 14. The contours of the optimal perturbation are initially oriented against the base flow shear (figure 6.3a). As time evolves to t = 7 (figure 6.3b), the corresponding ωz is sheared to an almost cross-stream orientation, leading at this time to the maximum of energy amplification. As the optimal perturbation evolves in time until t = 14 (figure 6.3c), ωz is sheared further and the perturbation energy decreases back to a lower value. The energy amplification results from the kinematic deformation of ωz by the base flow. This kinematic deformation reduces the length of the ωz contours while leaving unchanged the integral of the ωz enclosed by the contours7. Stokes theorem implies that the velocity magnitude along the (reduced in length) contours must increase to keep the circulation along the contours equal to the (constant) integral of ωz. This mechanism produces a large increase in cross-stream velocity v. When time evolves further and ωz is sheared as in figure 6.3(c), the kinematic process just described is reversed and the energy goes to zero as t → ∞.

Lift-up mechanism

The lift-up mechanism was first reported as an algebraic instability by Ellingsen & Palm [44] in the simple case of streamwise independent perturbations to inviscid linear flow. It can be understood as the flow induced by streamwise vorticity that, superposed on positive shear, lifts up fluid at low velocity while pushing down high- velocity fluid. Here we show again results for the tanh profile. Figure 2.4 (same as 6.4) shows the streamwise vorticity ωx of the optimal initial perturbation (figure
6.4a) and the streamwise velocity u (figure 6.4b) of the optimal response, leading to the optimal gain at T = 7 for kz = 5.174 and kx = 0. The ωx and u fields in figure 2.4 are respectively normalized by the maximum total enstrophy at t = 0 and twice the maximum total energy at t = T = 7. Both fields are localized around y = 0, in the region with strong shear. The colorbar on figure 6.4(a) reflects the fact that at the initial time, 97.6% of the total enstrophy is given by ωx. As time evolves, ωx remains constant and induces a constant cross-stream velocity v. That v excites u through transport of base flow momentum, generating streamwise streaks. The colorbar on figure 6.4(b) reflects the fact that, after the perturbation evolves to t = 7, most of the perturbation velocity corresponds to u. As time evolves further, the forcing of u by v remains constant, implying that the energy of the perturbation grows unbounded as t → ∞.

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A look at the energy evolution of perturbations of unbounded constant shear flow

We are now familiar with the development of linear perturbations in two different cases, kz = 0 with Orr and kx = 0 in which there is lift-up. In a general case of oblique waves, the two mechanisms can be present in a non-trivial way. These are the types of perturbations that show the largest instantaneous growth rate, and so they are likely of importance for transition or turbulent structures.

Table of contents :

1 Introduction 
2 Some ideas on non-normality 
2.1 Mathematical framework
2.1.1 Eigenmode decomposition of linear dynamics
2.1.2 Normality
2.1.3 Optimal perturbations
2.2 A simple case study
2.2.1 Some possible origins of non-normality
2.2.2 Non-normality is normal
2.3 Probing non-normality without eigenmodes Some new ideas
2.3.1 A conserved quantity for normal systems
2.3.2 Manifestation of the conservation on trajectories
2.4 Some methodological essentials
2.4.1 Perturbative Navier-Stokes equations
2.4.2 Adjoint equations
2.4.3 Numerical methods
2.5 Non-modal instability mechanisms in plane shear flow
2.5.1 Orr mechanism
2.5.2 Lift-up mechanism
2.5.3 A look at the energy evolution of perturbations of unbounded constant shear flow
3 New results on inviscid lift-up 
3.1 Introduction
3.2 Formulation
3.2.1 Reformulating the optimization problem
3.3 Base flow examples: Couette and Poiseuille flow
3.3.1 Couette
3.3.2 Poiseuille
3.4 Large kz estimates
3.4.1 Inflectional shear flow in infinite domain
3.4.2 Bounded flow with maximum shear at a wall
3.5 Conclusion
4 Transient perturbation growth in time-dependent mixing layers 
4.1 Introduction
4.2 Mathematical formulation
4.3 Transient response of a frozen parallel hyperbolic tangent shear flow
4.3.1 Base flow and decomposition of perturbations
4.3.2 Optimal perturbations: OL-type and K-type
4.4 Transient response of K-type time-dependent flow
4.4.1 Base flow
4.4.2 Optimal perturbations from T0 = 0: E-type and H-type
4.4.3 E-type response with maximum growth rate for T0 = 0, T = 60 72
4.4.4 H-type response for large kz
4.5 Variation in optimization interval start time T0 6= 0
4.5.1 E-type response
4.5.2 H-type response
4.5.3 Anti-lift-up
4.6 Conclusions
5 Craya-Herring Dynamics 
5.1 Governing equations
5.2 Craya-Herring frame
5.3 Perturbation dynamics in the Craya-Herring frame
5.3.1 Rotating case and the Coriolis operator in L
5.4 Energy evolution equation
5.4.1 Parallel shear flow
5.5 Concluding remark
6 Non-normal effects on the horizontal shear layer with vertical stratification
6.1 Introduction
6.2 Problem formulation
6.2.1 Base state and governing equations
6.2.2 Computation of the optimal perturbations
6.3 Unstratified case (Fh = ∞)
6.3.1 2D case: shear instability and the Orr mechanism
6.3.2 Lift-Up mechanism (kx = 0)
6.3.3 General perturbations kx, kz 6= 0
6.4 Effect of stratification
6.4.1 Stratified lift-up (kx = 0)
6.4.2 Craya-Herring decomposition
6.5 Wave emission
6.5.1 Role of non-normality on wave-vortex interactions
6.6 Discussion
6.6.1 Possible relevance of the stratified lift-up
6.7 Summary and main conclusion
7 Conclusions and perspectives 
A Derivation of the Craya-Herring equations and some extra discussion
A.1 Vortical part
A.1.1 Effect of waves on the vortical part
A.1.2 Perturbative, two dimensional vortex dynamics
A.2 Wave part
A.2.1 Effects of the wave part on the wave part
A.2.2 Wave excitation by the vortex part
B Explicit role of pressure and the connection of the poloidal evolution with the vertical momentum equation 


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