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How can I prove that assuming a power law forcing to drive incompressible turbulent flows leads to a scale invariant turbulent energy spectrum?

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To maintain incompressible turbulence, the incormpressible Navier Stokes equations

$$ \frac{\partial u_i}{\partial x_i} = 0 $$

$$ \frac{\partial u_i}{\partial t} + u_j\frac{\partial u_i}{\partial x_j} = -\frac{\partial p}{\partial x_i} + \nu_0 \frac{\partial ^2 u_i}{\partial x_j \partial x_j} $$

have to be suplemented by some kind of an energy source acting at large scales. Often, a Gaussian, white in time forcing is assumed such that its two-point correlator is given by

$$ \langle \hat{f}_i(\hat{k})\hat{f}_j(\hat{k}') \rangle = 2D(k)(2\pi)^{d+1}\delta(\hat{k}+\hat{k}') $$

How can I generally prove that assuming a power law for the energy input spectrum of the forcing D(k) leads to a scale invariant turbulent energy spectrum? Is assuming a power law like this sufficient and necessary to guarantee scale invariance?

asked Oct 12, 2013 in Theoretical Physics by Dilaton (4,175 points) [ revision history ]
edited May 1, 2014 by Dilaton

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