Could any one help the theory behind Taylor-Gortler vortex

What exactly do you want to know? You may want to have a look at Herbert Oertels "Prandtl's Essentials of Fluid Mechanics" which has a good introduction to this topic.

Thank you very much for the book suggested. I just want to know Is there any Taylor-Gortler like vortices for a lid-driven cubic cavity at Re 400 and Re 1000. Koseff and Street 1984 stated these vortices are formed at high Re.

I'll try to throw in my 2 cents...

A classic example of Taylor-Gortler vortices appear in the case of a spin down of a cylinder. See:

http://www.ae.gatech.edu/labs/windtu...on1/taylor.JPG http://www.kgroesner.de/forschung/experiment/ZYL-14.GIF

This is quite easy to understand. Rotating mass (consider the fluid column to be rigid) needs a centripetal force in order to rotate otherwise rotation would not be possible.

So there is a pressure rise towards the cylinder wall (low pressure in the center, high pressure at the wall). BUT there is a cylinder wall... and at the surface of wall the velocity is zero. And thus there is a boundary layer from the wall to the (say still rigid) rest of the rotating fluid. It is in this boundary layer where the instability will occur: the (high) pressure rise is imposed on the boundary layer but the corresponding velocities are absent. Thus the balancing of forces is disturbed.

Now viscosity is able to overcome this by some extent. But when the differences are high enough (the Reynolds number or rotation is increased) a tiny disturbance will bring up a small radial motion, mass is conserved so next to this motion, both sides, mass, fluid, liquid, will move towards the opposite radial direction. This will pull, because there is shear, along the layers next to them so motions towards the initial radial direction will appear. Etcetera etcetera. In no time you'll see a series of your Taylor-Gortler vortices.

Now coming back the (cubic) lid driven cavity. Especially below Re of less than 1000 I'll doubt Taylor-Gortler instabilities. For one thing they are classically associated with a concave wall (which is absent). Secondly, at least for the 2d ldc cases I know, instabilities, vortices, which is the onset towards turbulence, will occur at higher Re numbers (depending on your definition of RE of course).