Spectral analysis of turbulent flow I have the "energy spectral density" curve for a specific location in a turbulent flow. My curve is based on turbulence frequency (w) rather than wavenumber (k).
The mean kinetic energy can be calculated based on the equation (Bkf4) in (http://brennen.caltech.edu/fluidbook/basicfluiddynamics/turbulence/turbulencescales.pdf). However, in this pdf transformation from the wavenumber domain to frequency domain is done simply by substituting wavenumber (k) by frequency (w). On the other hand, the dissipation of kinetic energy in turbulent flow can be defined as equation 7.14 in (http://www.astronomy.ohio-state.edu/~ryden/ast825/ch7.pdf). So, if I want to transform this equation (7.14) from wavenumber domain to frequency domain, can I simply replace wavenumber (k) by frequency (w), or there is more complexity to it.
 A: In turbulence, going from frequency space ($\omega$) to wavenumber space ($k$) or vice versa is highly non-trivial. I don't know which turbulent flow you are dealing with. If it is has a large value of mean flow compared to turbulent fluctuation velocity (for example in a wind tunnel) and if the turbulence is homogeneous in the direction of mean flow, then as a first approximation you may use Taylor's frozen turbulence hypothesis. This hypothesis simply states that a frozen field of turbulence is advected past the fixed sensor with mean velocity of the flow, and therefore measurements in time and space are related by that mean velocity. 
If the turbulence has weak or zero mean flow (for example, in turbulent natural convection) then matter becomes complicated. Then you may have to use some other heuristic model such as large eddies sweeping smaller eddies past the fixed sensor (before they undergo significant change) so that you can at least inter-convert spectrum data at the higher frequency/wavenumber range. In such cases, best thing would be to modify the experiment itself to obtain spatial data: simply translate the sensor in a direction (in which turbulence is homogeneous) at high speed (ensuring that it doesn't disturb the flow).
