# Turbulent flows with incipient separation over solid waves

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## Abstract

Measurements of the streamwise velocity over solid sinusoidal waves with height to wavelength ratios of 2*a*/λ=0.05, 0.125 and with dimensionless wave numbers α^{+}=0.00624, 0.00135 have been made. For these conditions the instantaneous flow reverses direction, but the time-averaged flow is non-separated. Many features of the flow are similar to those reported in previous papers for a time-averaged flow that is separated. Approximate agreement is obtained from an eddy viscosity model derived for flow over small amplitude waves. However, the differences are more interesting than the agreement in that they point out shortcomings of present Reynolds stress models. Comparison with other measurements in the literature shows how increasing Reynolds number decreases the size of the separated region. Measurements of pressure profiles and of drag are interpreted in terms of measured flow patterns.

### List of symbols

*A*van Driest parameter which is a measure of the extent of the viscous wall region

*Ā*value of van Driest constant for flow over a flat plate

*a*amplitude of the wave, defined by Eq. (1)

*C*_{s}skin friction drag coefficient equal to \(\left\langle {\bar \tau _w } \right\rangle\)/ϱ

*U*_{b}^{2}*C*_{p}form drag coefficient associated with pressure forces on the wave surface

*f*fanning friction factor defined by Eq. (5)

*h*half height of the channel

*k*_{1},*k*_{2}coefficients defined in Eq. (8)

*k*_{L}lag parameter defined in Eq. (9)

*p*pressure

*p*_{w}pressure at the wall

*p*^{+}dimensionless pressure gradient, defined by Eq. (10)

*g*^{+}dimensionless effective pressure gradient, defined by Eq. (9)

*q*root-mean of the sum of the squares of the turbulent velocity components, Eq. (12)

*Re*Reynolds number equal to

*h U*_{B}/v*U*_{i}velocity in the

*i*th direction*Ū*_{i}time averaged velocity component

*u*_{i}velocity fluctuation, equal to

*U*_{i}−*Ū*_{i}- ¦
*Û*_{i}¦_{n} magnitude of the amplitude of the

*n*th harmonic of the wave-induced variation of*Ū*_{i}*U*_{b}bulk-average velocity

*u*^{*}friction velocity defined using \(\bar \tau _w\) for a flat surface

*u*_{t}^{*}friction velocity defined using the actual drag on the wave surface

*x*Cartesian coordinate in the horizontal direction

*y*Cartesian coordinate in the vertical direction

*y*_{s}location of wave surface

- α
wavenumber, equal to 2

*π*/*λ*- γ
per cent of the time the velocity has a positive value

- θ
phase lag

- θ
_{n} phase lag of

*n*th harmonic- κ
von Karman constant

- λ
wavelength

- ν
kinematic viscosity

- ν
_{t} turbulent kinematic viscosity

- ρ
density of fluid

- τ
shear stress

- τ
_{w} wall shear stress

- ψ
streamfunction

- 〈 〉
average along a wavelength

- −
time average

- +
as a superscript signifies a quantity made dimensionless using

*u*^{*}and*ν*

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### References

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