Turbulence and Taylors Hypothesis

• When studying turbulence (the spectrum of eddy sizes, for example), it is difficult to obtain an instantaneous snapshot of the boundary layer and all eddies formed within it.
• It is often easier and cheaper to make measurements of eddies in the boundary layer at one point over a long period of time (i.e., a weather station versus a Doppler radar)
• To study turbulence from a continuous record of measurements from a single point, we need to assume that the turbulence is frozen.
• What does this mean?
• As the mean flow advects the eddies past your sensor, the fundamental properties of the eddies remain unchanged, or frozen. -->
• Mathematically, how can be express Taylors hypothesis?

For any variable, z, that can be used to study turbulence, then the total derivative is equal to zero, i.e.,  dz/dt = 0 if it is "frozen".  We can write the total derivative as:

      (1)

or since dz/dt = 0 , (1) can be written as:

    (2)

Considering the example in the figure above, assuming that the flow is entirely in the x direction, then then local change of the thermal's temperature will be given by:

       (3)