Hi all.
I have just begun with the field of instability and going through texts of Francois Charru.
While going through Rayleigh Benard thermal instability (where, to best of what I have understood, the condition when due to temperature difference the advection will be prominent i.e. movement to fluid will start) I came across the following attached plot. I was unable to get what the wave number represents here?
Like in Taylor stability, it represents the wavelength of the perturbation, but what does k stand for here?
If it stands for perturbation wavelength, then what is perturbation here?
Is it different diameter particles/fluid elements exposed to heat surface?
Hello,
The wavenumber in the Rayleigh-Benard instability could be viewed as the inverse of the size of the rouleaux (rolls) which are formed as a result of the non-uniform temperature distribution in a plane horizontal fluid layer heated from below. As you mentioned, the wavenumber ( = modulus of the horizontal wavevector) is indeed related to the perturbation solution when you solve the linearised equation of motion. The instability occurs when the growth rate (often noted sigma) is strictly positive, corresponding to the region above the curve in attach here. For sigma = 0, you have the so-called curve of the marginal instability. Hope this could help a bit.
Cheers,
Hi,
If I understood you correctly, then perturbation in this case are the rolls (or fluid volume) which start to rise from the bottom (heated region) to colder region. So wavenumber here is inverse of the size of these fluid volumes (assumed spherical and hence rolls).
Kindly correct if I am wrong.
Thanks
Dear Deewakar,
the wave number is defined as k=2 Pi/lamda where lambda is the wave length of the pattern. In case of Rayleigh Bernard convection we seek solution in form of roles (assumed to be alligned with the y-axis) and the temperatur perturbation is theta=a(t) cos(k x) sin (pi z). The wavelength corresponding to these roles is the width of a pair of roles and not a single role, since successive roles rotate in opposite direction. Correspondingly the width of on role is lambda/2=Pi / k.
By the way these roles are not spherical.
Regards Andreas
Hi Andreas
If I got it correctly, you mean the perturbation is the temperature perturbation to the already stable temperature (at-least to some extent) profile that is existing due to conduction only.
The temperature has effect to effect the density and hence it acts as a perpetration. When the wave number of perturbation (temperature disturbance in a given length and related to width of pair of roles) becomes large enough that roles start to rise we say that instability has occurred. Kindly correct me if I am wrong.
The roles here means fluid volume acted upon temperature effects. May I know what you mean by width of pair of roles. Is it their separation ??
Thanks
deewakar
Dear Deewakar,
Pl try to read Lord Rayleigh's 1916 original paper, a classic available free on internet, and also Chandrashekar's book for more modern treatment, then go back to the definition for the dimensionless wave number and Lord Rayleigh's original derivation, you will have a better understanding of its mathematical and true physical sense.
kk
L. Rayleigh, On convection currents in a horizontal layer of fluid, when the higher temperature is on the under side, Phil. Mag., 32:529–46, 1916.
Subrahmanyan Chandrasekhar (1982). Hydrodynamic and Hydromagnetic Stability (Dover). ISBN 0-486-64071-X
P.G. Drazin and W.H. Reid (2004). Hydrodynamic Stability, second edition (Cambridge University Press).
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