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Velocity measurements for a turbulent nonseparated flow over solid waves

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Abstract

The laser-Doppler velocimeter was used to obtain measurements of the streamwise velocity over solid sinusoidal waves of small enough amplitude that a nonseparated flow existed. The measurements provide a critical test for Reynolds stress closure models since they are particularly sensitive to happenings in the viscous wall region (y + < 40), for which present theories are of uncertain accuracy. The results are compared with calculations that use an eddy viscosity model that successfully describes measurements of the wall shear stress along waves of small enough amplitude that a linear response is obtained. These calculations are in approximate agreement with measurements because they exactly account for inertia and viscous effects. However, there are significant differences which point to the inadequacy of turbulence models. In particular, non-linear effects and the amplitudes of the wave-induced velocity variations are underpredicted.

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Abbreviations

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 〈τ w 〉/ρU b 2

C p :

form drag coefficient associated with pressure forces on the wave surface

f :

Fanning friction factor defined by Eq. (7)

h :

half height of the channel

I :

intensity of turbulent velocity, equal to ū i 2

k 1, k 2 :

coefficients defined in Eq.(3)

k L :

lag parameter defined in Eq. (4)

p :

pressure

p + :

dimensionless pressure gradient, defined by Eq. (5)

p +eff :

dimensionless effective pressure gradient, defined by Eq. (4)

q :

root-mean of the sum of the squares of the turbulent velocity components, Eq. (9)

Re :

Reynolds number equal to hU b/v

U b :

bulk-average velocity, defined at the mid-section of the channel

U i :

velocity in the i-th direction

Ū i :

time averaged velocity component

u i :

velocity fluctuation, equal to U i − Ūi

¦Û i¦n :

magnitude of the amplitude of the n-th harmonic of the wave-induced variation of Ū i

u * :

friction velocity defined using τ w for a flat surface

x :

Cartesian coordinate in the horizontal direction

y :

Cartesian coordinate in the vertical direction

α:

wavenumber, equal to 2π/λ

θ:

phase lag

θ x :

phase lag of wave induced variation of variable x

θ x,n :

phase lag of n-th harmonic of the wave-induced variation of variable x

κ:

von Karman constant

λ:

wavelength

v :

kinematic viscosity

v t :

turbulent kinematic viscosity

ρ :

density of fluid

τ:

shear stress

τ w :

wall shear stress

〈 〉:

average along a wavelength

−:

time average

¦—¦ n :

amplitude of the n-th component of the wave-induced variation of X

&#x002B;:

as a superscript signifies a quantity made dimensionless using u * and v

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Authors and Affiliations

  1. Dept. of Chemical Engineering, University of Illinois, 61801, Urbana, IL, USA

    K. A. Frederick & T. J. Hanratty

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  1. K. A. Frederick
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  2. T. J. Hanratty
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Frederick, K.A., Hanratty, T.J. Velocity measurements for a turbulent nonseparated flow over solid waves. Experiments in Fluids 6, 477–486 (1988). https://doi.org/10.1007/BF00196509

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