Abstract
Measurements of the streamwise velocity over solid sinusoidal waves with height to wavelength ratios of 2a/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.
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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 \(\left\langle {\bar \tau _w } \right\rangle\)/ϱU 2b
- 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 ith 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 nth 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 2π/λ
- γ:
-
per cent of the time the velocity has a positive value
- θ:
-
phase lag
- θn :
-
phase lag of nth harmonic
- κ:
-
von Karman constant
- λ:
-
wavelength
- ν:
-
kinematic viscosity
- νt :
-
turbulent kinematic viscosity
- ρ:
-
density of fluid
- τ:
-
shear stress
- τw :
-
wall shear stress
- ψ:
-
streamfunction
- 〈 〉:
-
average along a wavelength
- −:
-
time average
- +:
-
as a superscript signifies a quantity made dimensionless using u * and ν
References
Abrams, J.; Hanratty, T. J. 1985: Relaxation effects over a wavy surface. J. Fluid Mech. 151, 443–455
Buckles, J. J. 1983: Turbulent separated flow over wavy surface. Ph.D. thesis, Dept. of Chemical Engineering, University of Illinois, Urbana
Buckles, J. J.; Hanratty, T. J.; Adrian, R. J. 1984: Turbulent flow over large-amplitude wavy surfaces. J. Fluid Mech. 140, 27–44
Buckles, J. J.; Hanratty, T. J.; Adrian, R. J. 1985: Separated turbulent flow over a small amplitude wave. Laser anemometry in fluid mechanics, Vol. 2 (eds. Adrian, R. J.; Durao, D. F. G.; Durst, F.; Asanuma, T.; Whitelaw, J. H.) Lisboa: Ladoan
Frederick, K. A. 1986: Velocity measurements for turbulent non- separated flow over solid waves. Ph.D. thesis, Dept. of Chemical Engineering, University of Illinois, Urbana
Frederick, K. A.; Hanratty, T. J. 1988: Velocity measurements for a turbulent nonseparated flow over solid waves. Exp. Fluids 6, 477–486
Hanratty, T. J.; Abrams, J. A.; Frederick, K. A. 1983: Flow over solid wavy surfaces. In: Structure of complex turbulent shear flows, pp. 78–88 (eds. Dumas, R.; Fulachier, L.) Berlin, Heidelberg, New York: Springer
Kuzan, J. D. 1983: Separated flow over a large amplitude wavy surface. M.S. thesis, Dept. of Chemical Engineering, University of Illinois, Urbana
Kuzan, J. D. 1986: Velocity measurement for turbulent separated and near separated flow over solid waves. Ph.D. thesis, Dept. of Chemical Engineering, University of Illinois, Urbana
McLean, J. W. 1983: Computation of turbulent flow over a moving wavy boundary. Phys. Fluids 26, 2065–2073
Reynolds, W. C.; Tiederman, W. G. 1967: Stability of turbulent channel flow, with application to Malkus's theory. J. Fluid Mech. 27, 253–272
Thorsness, C. B.; Morrisroe, P. E.; Hanratty, T. J. 1978: A comparison of linear theory with measurements of the variation of shear stress along a solid wave. Chem. Eng. Sci. 33, 579–592
Zilker, D. P. 1976: Flow over wavy surfaces. Ph.D. thesis, Dept. of Chemical Engineering, University of Illinois, Urbana
Zilker, D. P.; Hanratty, T. J. 1979: Influence of the amplitude of a solid wavy wall on a turbulent fluid, part 2: Separated flows. J. Fluid Mech. 90, 257–271
Zilker, D. P.; Cook, G. W.; Hanratty, T. J. 1977: Influence of the amplitude of a solid wavy wall on a turbulent flow, part. 1: Non-separated flows. J. Fluid Mech. 82, 29–51
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Kuzan, J.D., Hanratty, T.J. & Adrian, R.J. Turbulent flows with incipient separation over solid waves. Experiments in Fluids 7, 88–98 (1989). https://doi.org/10.1007/BF00207300
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DOI: https://doi.org/10.1007/BF00207300