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April 6, 2000, 05:24 
secondary instability??

#1 
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thank you so much for explain me the KH instability in such a clear way!
I bet you have also clear idea on the socalled "secondary instability", with respect to the primary instability. let's say, if a 2D vortex roll twist to become Lamdaformed structures, it should be due to the primary instability; if later the Lamdaformed structures burst into a chaoslike turbulece, then it was due to "secondary instability", right? any logical relations between the "primary" and "secondary" instabilities? thanks a lot in advance!!! Anan 

April 6, 2000, 09:39 
Re: secondary instability??

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There are many 'pictures' of primary and secondary instability. The usual one is with the socalled Taylor vortices in the circular Couette flow (the flow between rotating cylinders). More precisely it is the circular flow between two concentric cylinders which rotate at different speeds. Before the onset of the instability the flow is laminar, axisymmetric and even cylindrical if you ignore the effect of the bottom boundary. The condition for (linear)instability depends on the viscosity and the angular velocities of the two cylinders, and their radii. In this case if w1*r1**2 is larger than w2*r2**2 (where w is the angular velocity and r the radius of the inner 1 and outer 2 cylinder), then the flow becomes unstable. As the inner cylinder rotation speed increases, the flow undergoes a (reversible) passage through a sequence of regular patterns (vortices, "Taylor" vortices). These are stationary vortices: a series of axisymmetric counterrotating cells, stacked above each other. This the first instability.
At higher rotation rates, the patterns become more complicated and other frequencies are excited, untill the flow is so irregular that it is defined as turbulent. This is the second instability. In the first instability the flow achieved a new state, where only a few modes (frequencies) are excited (i.e. vortices have a given size only, the "cells"). This flow is in quaisteady state, i.e. d/dt is not zero, but the the patterns are quite steady. In terms more analytical what happens is that the flow is unstable but only a few mode (lengthwaves) are excited. The larger waveslengths cannot be excited because the cylinder has a finite size, so the cells have the size of r2r1, while the small wavelengths (higher modes, high frequencies) are damped by the viscosity (at low speed of rotation the Reynolds number is small). When the velocity of the (inner) cylinder is increased the Reynolds number increases accordinly and more short wavelengths are excited (they are not damped any more by the viscosity). And this happens untill the flow become fully turbulent. This is the second instability. So the first and second instability can be understood in terms of the number of modes which can be excited. In the first instability only a few modes corresponding to the largest scale of the problem, while in the second instability all the other higher modes are also excited (corresponding to the samller scales). THe same happens also in convection, etc... Patrick 

April 6, 2000, 09:48 
Re: the first instability

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The first instability helps the flow to achieve a new state of equilibrium against the instability.
For example in the case of the rotating Couette flow the instability can be understood in terms of the inbalance in the flow due to the Centrifugal force who has a tendancy to pushe the inner flow outwards (this is why sometimes this instability is refered to as a centrifugal instability, under Rayleigh criterion for rotating flows). As the Taylors vortices are formed, they actually balance back the momenta in the flow to achieve a new (quasi) steady state. In convetion a similar thing happens where the first instability with the largest convective cells actually help the flow to balance back the temperature difference that produces the instability. So the first instability is a response of the flow to achieve a new state of equilibrium and balance the instability. 

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