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Effect of roughness on the velocity profile of a laminar boundary layer

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Abstract

Flow past a rough wall is examined. Calculations are made to find the roughness-induced mean velocity which is expressed in an integral form in terms of the spectral density of the roughness and an influence function. Values of the influence function are tabulated using the known values of the modified Hankel functions of order 1/3 and their integrals. The first order change in lower critical Reynolds number due to the roughness-induced change in profile is calculated; the stability of the profile is increased due to the presence of roughness.

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Abbreviations

A 1, A 2, A 3 :

functions of ũ

c :

velocity of wave propagation in the boundary layer

Z 1i (α, β, y):

Fourier-Stieltjes transform of w 1i

E(ã):

two-dimensional spectral density of roughness distribution

E, F, ψ, T :

various combinations of X 3 and X 4 defined by (67)

F(ã):

(isotropic) spectral density of roughness distribution

G :

auxiliary function introduced to solve (53)

h(x, z):

roughness height

H(α, β):

Fourier-Stieltjes transform of h(z, x)

H 3/(1)1 , H 3/(2)1 :

Hankel function of order 1/3

O 1 :

an integral involving E, F, T defined by (71)

O 2 :

an integral involving F, G defined by (74)

O 3 :

integral of O 1

P :

pressure over rough wall

p :

roughness induced pressure

p 1, p 2 :

first and second order (in ε/λ) contribution to roughness induced pressure

R :

flow Reynolds number

ũ i :

velocity past smooth wall

u i :

roughness induced velocity

U i :

velocity past rough wall

w 1i , w 2i :

first and second order (in ɛ/λ) contributions to roughness induced velocity

X (0), X (1) :

first and second order (in ε +) contribution to inner solution for dz 12

ψ 3, ψ 4 :

integrals of modified Hankel function of order 1/3

α, β :

streamwise, transverse wavenumbers

ã :

wavenumber magnitude

ε :

R.M.S. roughness height

ε + :

wall layer stretching parameter

η :

stretched variable normal to the wall

λ :

Taylor microscale of roughness distribution

ν :

kinematic viscosity of fluid

π:

pressure over smooth wall

ρ :

density of fluid

ψ 3, ψ 4 :

integrals of modified Hankel function of order 1/3

References

  1. Singh, K., J. L. Lumley and R. Betchov, Tables of the Modified Hankel Function of Order 1/3 and its Integrals to Argument 10, Engineering Bulletin B-87, The Pennsylvania State University, 1963.

  2. Lin, C. C., Theory of Hydrodynamic Stability, Cambridge University Press, Cambridge 1955.

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Authors and Affiliations

  1. Dept. of Mathematics, Saint Mary's University, Halifax, Nova Scotia, Canada

    K. Singh

  2. Dept. of Aerospace Engineering, The Pennsylvania State University, 16802, University Park, Pa., USA

    J. L. Lumley

Authors
  1. K. Singh
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  2. J. L. Lumley
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Additional information

This work was supported by U.S. Naval Ordnance Systems Command under Contract NOw 65-0123-d through the Garfield Thomas Water Tunnel, Ordnance Research Laboratory.

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Singh, K., Lumley, J.L. Effect of roughness on the velocity profile of a laminar boundary layer. Appl. Sci. Res. 24, 168–186 (1971). https://doi.org/10.1007/BF00411712

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  • DOI: https://doi.org/10.1007/BF00411712

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