# Measurement of turbulent wall velocity gradients using cylindrical waves of laser light

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## Abstract

A new optical instrument has been developed for direct measurement of instantaneous velocity gradients at the bounding wall. Light emerging from two tiny optical slits in the surface is used to form a “fan of fringes” in the region very near the wall. Doppler frequency of the light scattered by the seed particles is directly proportional to the velocity gradient. The system has been used to measure the statistics of the streamwise and spanwise velocity gradients in a turbulent boundary layer. The streamwise and spanwise rms fluctuations were found to be 38% and 11% of the mean streamwise value respectively. The latter result is subject to a large uncertainty.

### List of symbols

*a*slit width

*B*transfer function of the instrument

*B*^{*}normalized transfer function

- \(\overline B *\)
path-averaged value of the normalized transfer function

*c*constant in logarithmic velocity profile

*C*_{f}skin friction coefficient

*d*_{f}fringe spacing

*f*_{1},f_{2}frequencies at the downstream and upstream slits resp.

*f*_{d}heterodyne Doppler frequency of the signal

*g(t)*instantaneous wall velocity gradient

*G*Clauser shape factor

- \(\overline g \)
mean wall velocity gradient

*g′*rms value of the wall velocity gradient

*H*boundary layer shape factor

*i, j, k*unit vectors along

*x*,*y*,*z*axes*κ*wavenumber of laser light

*L*major axis of the elliptic cross-section of the laser sheet at the slit

*l*length of each slit

*N*number of cycles in a signal

*N*_{0}number of cycles without frequency-shifting

**n**difference of the unit vectors

**u**_{1}and**u**_{2}*P*power transmitted through a slit

*P*_{o}power incident on a slit

*Re*_{δ}_{1}Reynolds number based on displacement thickness and free-stream velocity

*Re*_{δ}_{2}Reynolds number based on momentum thickness and free-stream velocity

*S*spacing between the slits

*S*^{*}normalized spacing between the slits

*u*streamwise velocity

*u*_{1},u_{2}unit vectors along the local directions of propagation of the two cylindrical waves

*u*_{l}linear term in the streamwise velocity profile

*u*_{nl}nonlinear terms in the streamwise velocity

*u*_{nl}^{*}normalized value of nonlinear streamwise velocity

*u*_{nl}^{*}mean streamwise velocity

*u*_{τ}friction velocity

- u
^{+} mean velocity normalized with friction velocity

*v*velocity component normal to the wall

*v*^{*}normal velocity normalized with streamwise velocity

**V**velocity vector

*w*spanwise component of velocity

**W**minor axis of the elliptic cross-section of the laser sheet at the slit

*x*streamwise distance

- ±
*x*_{m} limiting values of streamwise distance for a signal

*x*^{*}normalized streamwise distance

*x*^{*}normalized value of

*x*_{m}*y*normal distance

*y*^{+}normal distance normalized with friction length scale

*z*spanwise distance

*z*^{+}spanwise distance normalized with friction length scale

*α*half-spreading angle of the cylindrical waves

*δ*boundary layer thickness in Coles' profile

*δ*_{1}displacement thickness of the boundary layer

*δ*_{2}momentum thickness of the boundary layer

*δ*_{3}energy thickness of the boundary layer

*ξ*constant in logarithmic velocity profile

*λ*wavelength of laser light

*ν*kinematic viscosity

*π*coefficient of wake function in Coles' profile

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