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Periodic vorticity and its effect on flow over a flat plate

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Abstract

An approximate solution is derived for the mean and oscillatory velocity profiles in the boundary layer on a flat plate, with a periodic source of vorticity near the leading edge.

It is predicted that the mean velocity profile slope near the plate is increased and that the skin friction is slightly increased. The oscillatory boundary layer width is an appreciable fraction of the normal layer width and the oscillatory boundary layer persists close to the plate, even after the boundary layer flow becomes turbulent.

A stability analysis based on the vortical instability criterion leads to an upper bound for the disturbance amplitude of the velocity component in the free-stream direction.

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Abbreviations

A :

amplitude of the fluctuating component of vorticity

C :

integration constant

F(Y) :

function defined by (4.5)

vibration “g”:

acceleration of 981 cm s−2

K :

constant

n :

frequency in cycles per sec

t :

time

x :

ordinate along the plate

X :

dimensionless variable defined as 2x/U

y :

ordinate perpendicular to the plate

Y :

dimensionless variable defined in (3.9)

u(x, y, t):

total instantaneous velocity component in the x direction

ū(x, y):

mean velocity component in the x direction

u 1(x, y, t):

instantaneous oscillatory velocity component in the x direction

v(x, y, t):

total instantaneous velocity component in the y direction

v(x, y):

mean velocity component in the y direction

v 1(x, y, t):

instantaneous oscillatory velocity component in the y direction

U(x, t):

total instantaneous free-stream velocity

Ū(x):

mean free-stream velocity

U 1(x, t):

instantaneous oscillatory free-stream component

U :

constant free-stream velocity outside the boundary layer

δ :

steady boundary layer thickness

δ o :

oscillatory boundary layer thickness

ν :

kinematic viscosity of the fluid

λ :

velocity ratio (U 1U 2)/(U 1+U 2)

ω(x, y, t):

total instantaneous vorticity component

ū(x, y):

mean vorticity component

Ω 1(x, y, t):

instantaneous oscillatory vorticity component

Ψ(x, y):

source function for the vorticity

ψ(x, y):

stream function

τ x :

mean shear stress in the boundary layer at a point x on the plate

(τ x )0 :

local skin friction on the plate

′:

differentiation with respect to Y

-:

average over one cycle of the oscillation

References

  1. [1]

    McCormack, P. D., D. Cochran, and L. Crane, Phys. Fluids 9 (1966) 1555.

  2. [2]

    Lin, C. C., Proc. Ninth Intern. Congress Appl. Mech., Vol. IV, p. 155, Elsevier, Amsterdam 1957.

  3. [3]

    Sewell, K. G., Proc. Heat Transf. and Fluid Mech. Inst., p. 106, Stanford University Press, Stanford (Calif.) 1960.

  4. [4]

    Schubauer, G. B. and P.S. Klebanoff, Contribution on the Mechanics of Boundary Layer Transition, NACA Report TR 1289, NACA, Washington (D.C.) 1956.

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Additional information

This work was supported by a grant from the Propulsion Division, Air Force Office of Scientific Research, United States Air Force (Dr. A.F. Wolfson).

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McCormack, P.D. Periodic vorticity and its effect on flow over a flat plate. Appl. Sci. Res. 20, 444–464 (1969). https://doi.org/10.1007/BF00382414

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Keywords

  • Boundary Layer
  • Vorticity
  • Velocity Profile
  • Stability Analysis
  • Velocity Component