Mechanical picture of the linear transient growth of vortical perturbations in incompressible smooth shear flows

The linear dynamics of perturbations in smooth shear flows covers the transient exchange of energies between (1) the perturbations and the basic flow and (2) different perturbations modes. Canonically, the linear exchange of energies between the perturbations and the basic flow can be described in terms of the Orr and the lift-up mechanisms, correspondingly for two-dimensional (2D) and three-dimensional (3D) perturbations. In this paper the mechanical basis of the linear transient dynamics is introduced and analyzed for incompressible plane constant shear flows, where we consider the dynamics of virtual fluid particles in the framework of plane perturbations (i.e., perturbations with plane surfaces of constant phase) for the 2D and 3D case. It is shown that (1) the formation of a pressure perturbation field is the result of countermoving neighboring sets of incompressible fluid particles in the flow, (2) the keystone of the energy exchange mechanism between the basic flow and perturbations is the collision of fluid particles with the planes of constant pressure in accordance with the classical theory of elastic collision of particles with a rigid wall, making the pressure field the key player in this process, (3) the interplay of the collision process and the shear flow kinematics describes the transient growth of plane perturbations and captures the physics of the growth, and (4) the proposed mechanical picture allows us to reconstruct the linearized Euler equations in spectral space with a time-dependent shearwise wave number, the linearized Euler equations for Kelvin modes. This confirms the rigor of the presented analysis, which, moreover, yields a natural generalization of the proposed mechanical picture of the transient growth to the well-established linear phenomenon of vortex–wave-mode coupling.

Figure 1

A sketch illustrating the initially introduced velocity perturbation field, which is tilted opposite the shear. The solid lines indicate the lines of constant phase.

Figure 2

Qualitative sketches illustrating (a) the induction of the pressure perturbation and (b) the mechanical picture of energy transfer from a shear flow to a single SFH, initially located at $k_y/k_x\gg 1$. (a) The dynamical interplay of a multitude of fluid particles from both sides of the zero perturbation velocity lines, $k_{x}x+k_{y}y=2\pi m$, i.e., from both sides of the pressure wall ($p=\max$), is found to be the basis of the of the pressure induction. (b) The circles $1,1^{\prime }$ and $1^{″}$, or equally $2,2^{\prime }$ and $2^{″}$, indicate arbitrarily chosen virtual particles at different times. The lines $k_{x}x+k_{y}y=2m\pi ,(2m+2)\pi$ and $k_{x}x+k_{y}y=(2m+1)\pi$ respectively represent the intersections of the maximum and minimum pressure planes with the $z=0$-plane.

Figure 3

Simplified sketch pointing out the mechanical feature of the reflection at the pressure wall ($p=\max$).

Figure 4

Sketch of the elastic reflection of fluid particle from the “pressure wall.”

Figure 5

Evolution of (a) the 2D and 3D Kelvin mode normalized energy $E_{\mathbf{k}}\left(t\right)/E_{\mathbf{k}}\left(0\right)$ and (b) the angle $\theta \left(t\right)=\measuredangle (\mathbf{u},\delta \mathbf{u})$ for the following sets of parameters: $A=0.05,k_x=1,k_y\left(0\right)=10$, and $k_z=0$ (solid lines), $k_z=1$ (thin dashed lines), $k_z=0.25$ (thick dashed lines); $C/u_y\left(0\right)=0.1$ for the latter two.