Turbulence Intensity and Turbulent Kinetic Energy (TKE)

• OK, thus far we have discussed how to separate the turbulent flow from the large-scale, uniform flow
• The next question is:
  • given a time series of wind speed observations, how can one quantitatively determine the intensity of the turbulence?
• Recall that u' is the turbulent part of the flow and represents the "deviation" of the total wind speed from the mean value:
• Q:  What variable can we quantitatively compute that would represent the intensity of turbulence?  HINT: it should have the same units as the variable being studied, so for wind speed, this variable should have units of LT^-1.  Answer

• Yes, we need to use statistics to study turbulence!!!!!!

Variance (s²) is one statistical measure of the dispersion of data about a mean value, and is defined as:

     (1)

where N is the total number of points in your data set.

Recall that for flows where you can separate the mean value from the turbulent part, you can write;

A_i = A + a'    (2)

where A_i is the instantaneous value of that variable.  Then you can write (2) as

a' = A_i - A     (3)

Substituting (3) into (1) gives:

       (4)   or

       (5)

Hence, the average of the square of the turbulent part of a variable can be interpreted as the variance of that variable.

The standard deviation (s) is related to the variance by

        (6)

So, as you can see from (6), if the standard deviation is large, then the turbulent part of the flow is large, i.e., the turbulence intensity is large.

A dimensionless parameter that is often used as a measure of the turbulence intensity (I) is given by:

 I = s_U / M          (7)

where M is the three-dimensional wind field.

Turbulent Kinetic Energy

• One of the more important variables used to study turbulence and it's evolution in the boundary layer is Turbulent Kinetic Energy (TKE).

• If one again assumes that the flow can be partitioned into mean and turbulent parts, then the total kinetic energy of the flow is simply the sum of the kinetic energy of the mean and turbulent flows.

• Per unit mass, the kinetic energy of the mean and turbulent parts of the flow can be expressed as:

     (8)

where MKE is the kinetic energy of the mean flow per unit mass, and e is the kinetic energy of the turbulent flow per unit mass.

• The instantaneous values of TKE can vary dramatically, so it is often useful to calculate a mean TKE value that is likely more representative of the overall flow:

   (9)

• TKE is generated by buoyant thermals and mechanically generated eddies
• TKE is suppressed or lost by layers of air that are becoming more stable and is also dissipated into heat by the effects of molecular viscosity.
• Hence, one can write a TKE budget equation that is a sum of the production and loss terms.
  • if the production terms are larger than the loss terms, TKE will increase and the boundary layer becomes more turbulent
  • if the loss terms are larger than the productions terms, TKE will decrease and the boundary layer becomes less turbulent
• Later on in the course, we will write out the TKE budget equation and physically interpret the various production and loss terms.

• In the mean time, a typical daytime diurnal variation of TKE for a convective boundary layer is shown in Fig. 2.8

• The vertical distribution of TKE for three boundary layer types is shown in Fig. 2.9.  Do the vertical distributions make sense to you? 

• Any questions???
• OK, let's do the TKE lab.