Convert the shear probe data

The processing of shear-probe data to derive the rate of dissipation of turbulence kinetic energy assumes that the data have been converted into physical units of shear [[math]\displaystyle{ \mathrm{s^{-1}} }[/math]]. The manufacturers of instruments provide guidance and software for the conversion of data into physical units. Therefore, conversion is discussed only briefly here. The main requirement is that you know the speed of profiling of your instrument, [math]\displaystyle{ U }[/math].

Instruments with a differentiator

Instruments that have a differentiator in their electronics are converted into physical units using

[math]\displaystyle{ s=\frac{N_s}{2\sqrt2SU^2G\gamma} }[/math]

where [math]\displaystyle{ s }[/math] is the shear signal in physical units of [math]\displaystyle{ \mathrm{s}^{-1} }[/math] , [math]\displaystyle{ N_s }[/math] are the raw numeric samples (the output of an analog-to-digital converter), [math]\displaystyle{ S }[/math] is the calibrated sensitivity of the shear-probe in units of [math]\displaystyle{ \mathrm{V\,(m\,s^{-1})^{-2}} }[/math], [math]\displaystyle{ U }[/math] is the speed of profiling in [math]\displaystyle{ \mathrm{m\,s^{-1}} }[/math], [math]\displaystyle{ G }[/math] is the gain of the differentiator in the electronics of the shear probe in units of [math]\displaystyle{ \mathrm{s} }[/math], and [math]\displaystyle{ \gamma }[/math] is the gain of the analog-to-digital converter used to create the samples in units of [math]\displaystyle{ \mathrm{counts\, V^{-1}} }[/math].

The sensitivity of the shear probe to shear is proportional to the square of the speed of profiling. Thus, one should set a minimum speed for the conversion of the shear-probe data into physical units. Otherwise, the conversion may produce enormously large and quite unrealistic values. Realistic minimum speeds for the conversion into physical units are [math]\displaystyle{ 0.05 }[/math] to [math]\displaystyle{ 0.1\, \mathrm{m\,s^{-1}} }[/math] because for slower speeds the shear-probe signal is likely to be dominated by electronic noise and the angle of attack will be large ([math]\displaystyle{ \gt 20^{\circ} }[/math]) even for low levels of dissipation. A discussion of the limitations of slow profiling is provided by .

Instruments without a differentiator

If your instrument does not have a differentiator, then the recorded signal is proportional to the cross-profile velocity fluctuations, say [math]\displaystyle{ u }[/math]. You must convert this signal into the time derivative of cross-profile velocity using the first difference operator

[math]\displaystyle{ s = f_s\, \left[ u(n) - u(n-1) \right] }[/math]

where [math]\displaystyle{ f_s }[/math] is the sampling rate of your data and [math]\displaystyle{ u }[/math] are the samples of the shear probe that have been converted into velocity fluctuations in units of [math]\displaystyle{ \mathrm{m\, s^{-1}} }[/math] following the recommendation of the manufacturer of your instrument.

The above equation is an approximation of a time derivative that deviates from a true (continuous-domain) derivative with increasing frequency. The square of the magnitude of the transfer function of a first difference operation is

[math]\displaystyle{ \left|𝐻_{\Delta}(𝑓)\right|^2 = \left[4𝑓_𝑁\, \sin\left(\frac{\pi}{2} \frac{𝑓}{𝑓_𝑁}\right) \right]^2 }[/math]

where [math]\displaystyle{ f_N = f_s/2 }[/math] is the Nyquist frequency. The square of the magnitude of the transfer function of a differentiator in the continuous domain is

[math]\displaystyle{ \left|H_c(f) \right| = \left[2\pi f\right]^2 \ \ }[/math].

Shear spectra computed using shear-probe data that has been processed by a first difference operator must be corrected by multiplying such spectra by the ratio of these two transfer functions. That is, the spectra are multiplied by

[math]\displaystyle{ \left| \frac{H_c}{H_{\Delta}} \right|^2 = \left[ \frac{2\pi f}{4𝑓_𝑁\, \sin\left(\frac{\pi}{2} \frac{𝑓}{𝑓_𝑁}\right)} \right]^2 \ \ . }[/math]

This correction factor is approximately [math]\displaystyle{ 1 }[/math] for [math]\displaystyle{ f\ll f_N }[/math] and reaches its maximum of [math]\displaystyle{ \left(\frac{\pi}{2}\right)^2 }[/math] at the Nyquist frequency of [math]\displaystyle{ f=f_N }[/math].

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