Power spectra of fluid velocities measured by laser Doppler velocimetry

The power spectrum and the correlation of the laser Doppler velocimeter velocity signal obtained by sampling and holding the velocity at each new Doppler burst are studied. Theory valid for low fluctuation intensity flows shows that the measured spectrum is filtered at the mean sample rate and that it contains a filtered white noise spectrum caused by the steps in the sample and hold signal. In the limit of high data density, the step noise vanishes and the sample and hold signal is statistically unbiased for any turbulence intensity.List of symbols A cross-section of the LDV measurement volume, m² - A empirical constant - B bandwidth of velocity spectrum, Hz - C concentration of particles that produce valid signals, number/m³ - d _m diameter of LDV measurement volume, m - f(₁, ₂ | u) probability density of t _i; and t _j given (t) for all t, Hz² - probability density for t _j-t_i, Hz - n (t, t) number of valid bursts in (t, t) = N + n - N (t, t) mean number of valid bursts in (t, t) - N _e mean number of particles in LDV measurement volume - valid signal arrival rate, Hz - mean valid signal arrival rate, Hz - R _uu time delayed autocorrelation of velocity, m²/s² - S _u power spectrum of velocity, m²/s²/Hz - t ₁, t ₂ times at which velocity is correlated, s - t _i, t _j arrival times of the bursts that immediately precede t ₁ and t ₂, respectively, s - t _ij t _j–t _i s - T averaging time for spectral estimator, s - T _u integral time scale of u (t), s - T Taylor's microscale for u (t), s - u velocity vector = U + u, m/s - u fluctuating component of velocity, m/s - U mean velocity, m/s - u _m sampled and held signal, m/s Greek symbols (t) noise signal, m/s - _m (t) sampled and held noise signal, m/s - bandwidth of spectral estimator window, radians/s - time between arrivals in pdf, s - Taylor's microscale of length = UT m - kinematic viscosity - ₁, ₂ arrival times in pdf, s - root mean square of noise signal, m/s - _u root mean square of u, m/s - delay time = t ₂ - t ₁ s - B duration of a Doppler burst, s - circular frequency, radians/s - _c low pass frequency of signal spectrum radians/s Other symbols ensemble average - conditional average - ^ estimate