Nonmodal instability of PDE discretizations  – Secondary Instability and Floquet Analysis

Velocity components of the Floquet mode.

Introduction

The main principle of secondary instability could be described as follows: when disturbances add up to a basic undisturbed flow, they grow such that their amplitude could be described as finite. As a result from the main instability they may achieve saturation characterized in a way as a steady-state, and by that present a new twisted base flow, mostly much more complicated. There is of course the issue of time for the changingof saturated state vs. that of the typical time for the flow to be receptive enough to allow such a state to even develope gently.
Assuming the latter is confirmed (e.g. a saturation characterized in a way as a steady-state developes) this new “base flow” may also be susceptible to disturbances. The linear stability procedure for such flows of a twisted base flow is called: “Secondary Instability Theory”.

Procedure

We shall begin with a twisted base flow of the following form:

As to point attention, we are starting with a flow of which the evolution in the longitudinal direction is neglected. Plain Y-Zis perpendicular to the axis of the vorticity structure.

Now if we shall use this flow in Linearized Navier-Stokes Equation (LNSE from now on…) we shall obtain the equations which governs the linear secondary instability with a profound structure in the horizontal direction (spanwise periodic):

In the above equation I’ve assumed a disturbance of the form :

Other assumptions may also be made and by that make the above equations look simpler.

Now if we decide to assume that this is more like a streak rather than a base flow with an apparent vortical structure, why then it could be proven that that the longitudinal component of the velocity is much higher than the horizontal or the vertical one.

If due to that claim we decide to neglect the last two we have at our disposal:

Now comes the punchline… We may set the equation (*)17 in terms of vertical velocity and vertical vorticity. This is the same procedure that I shall not repeat thoroughly: after taking the divergence to the momentum equation, and making use of the continuity equation to achieve an equation for the vertical velocity, we take derivatives of the momentum equations in X and Z according to Z and X to get the vertical vorticity (pay attention to the respectivenes of taking the derivativethan subtract, and we arrive to a set of equations in analogy to Orr-Sommerfeld\Squire:

and in a solid boundary or free flow we follow the boundary conditions:

The component of the horizontal velocity may be presented according to the identity:

and by that slip it out of (*)19

The streak (maybe the most known feature of transition by most… 😉) may be represented in the following form:

(It just needs the assumption that when Re—>0, then the wavenumber goes to zero)

The vertical velocity and the vertical vorticity are expressed in the following form:

when β`is the horizontal wave-number, and ϒ is termed “Floquet Detuning Parameter”

Velocity components of the Floquet mode.

What’s up next?…

next post shall deal with some excellent mathematical tools to decribe instability. We shall use terminology like Spectra, Pseudospectra,, pseudo eigenvalues and I promise you: this is going to be catchy and easy, whilst jumping to high level the understanding of on one of the most mysterious subjects in fluid dynamics.

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