I have two questions relating to turbulence. Let us suppose we have decaying turbulence How mush time does it takes to dissipate all energy and get pure laminar flow and what is the temeparture increase during the disspiation. How can one calculate such a time? 2)Does turbulence depends upon the initial condition.

I think:

Turbulence is no different from other flows except that in our opinion it's got lots of time and space randomness to it.

Energy cascades from larger eddies to smaller ones, until they are smaller than our interest. Actually the molecular motion of heat is just a terminally-cascaded collection of the smallest possible eddies.

If energy is taken from an eddy by viscous losses, these are proportional to the velocity gradient, so it's a simple diffusion equation like Fourier's law describes. Eddy velocities of a certain scale or wavelength would decay exponentially with a time constant proportional to the square of the wavelength. Like all exponential decays, this is never complete. The temperature increase during this dissipation would approach an enthalpy increase that matches the kinetic energy lost from all the motion that has disappeared - that is, the sum of kinetic and enthalpic energies would stay constant.

Of course turbulence depends upon the initial condition. Certainly the initial conditions determine the energy that starts cascading down through the smaller and smaller eddy wavelengths. They also determine the initial velocity field, though that field's relation to the evolving chaotic velocity field over time grows more and more obscure.

At least, like I said, "I think". I'm a physicist with no course work at all in fluid mechanics, though I'm certainly enjoying Anderson's wonderful introductory CFD book right now.