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Theoretical comments on the paper of Mr. E.N. Pales
THEORETICAL
COMMENTS
ON THE PAPER
OF MR. E.N. FALES
BY
ti~AN1
m.voN
AND
C.C.LIN*
Mr. Fales’ experiments revealed a new interesting case of vortex When a flat plate moving in water was more or less abruptly formation. decelerated, a row of parallel vortex columns appeared. We believe that the vortex formation can be traced to instability of the boundary layer during and after the deceleration. It is known that unstable oscillations develop in most shear layers, when the speed of motion is sufficiently high. In the case of a shear layer between two parallel streams of different velocity for example, Lessen3 found that the flow becomes definitely unstable when the Reynolds number based on the thickness of the mixing region is of the order of 20. In this case, it is also known that the vortex sheet eventually rolls up into a periodic It may be expected, then, that the spacformation of discrete eddies. ing between the individual vortices in the final stage will be approsimately equal to the wave length of the initial unstable oscillations of the shear zone. An analogous situation may exist in the present case, in which the During the uniform motion boundary layer represents the shear layer. the velocity distribution on the boundary layer is stable at the Reynolds U
Y FIG.
1.
A typical
velocity
profile.
’Scientific Advisory Board to Chief of Staff, U. S. Air Force, Pasadena, Calif. ’ Department of Mathematics, Massachusetts Institute of Technology, Cambridge, At present, holder of Guggenheim Fellowship at California Institute of Technology. (1948); NACA TR NO. 979 (1950). 3 M. LESSEN, Sc.D. Thesis, M.I.T.
Mass.
517
518
TH.
VON KARMAN
[J. F. 1.
AND C. C. LIN
numbers prevailing in the experiments. However after the sudden stoppage of the plate the velocity distribution is changing with time, and is of the general form shown in Fig. 1. This flow is certainly less stable than the profile corresponding to steady flow. John E. Plapp recently computed the stability characteristics of a similar velocity profile given by the equation u = Av(l
- 7$,
?j = y/6.
He found that the flow becomes unstable if the Reynolds number R 6-- g
Y
> 408,
where U = 4A/27 is the maximum speed. At Ra = 408, the oscillation has a frequency f, wave length X, and wave speed c as follows : f = 0.207 U/6, x = 2.10 6, c = 0.43 u. In the present experiments, the plate is not suddenly stopped, so Howthat it still has a finite velocity at the time of vortex formation. Typical values of ever, a qualitative correlation may still be expected. f, X, and c, according to the experiments of Mr. Fales, are f = 13.0 cps. x = 0.04 ft. c = 0.52 fps. Based on these values one obtains for the corresponding and 6 u = 1.2 fps.
values of U
6 = 0.23 in. In the case considered, the plate was moving at 1.55 fps. before stoppage, and at 0.72 fps. at the moment of “vortex birth.” Thus the value of U = 1.2 fps. is not unreasonable. There is no conclusive value for 6 from the experiments. Estimates based on boundary-layer theory yield a value of 0.14 in. which is somewhat too small. However, in view of the substantial differences of velocity profiles between theory and Thus, the above experiment, this is not an unreasonable discrepancy. analysis tends to support the point of view that the vortices are actually traceable to the instability of the flow after the stoppage of the plate. The writers wish to thank Dr. W. D. Rannie and Mr. John E. Plapp for permission to use their unpublished results.
COMMENTS
ON THE PAPER
OF MR. E.N. FALES
BY
ti~AN1
m.voN
AND
C.C.LIN*
Mr. Fales’ experiments revealed a new interesting case of vortex When a flat plate moving in water was more or less abruptly formation. decelerated, a row of parallel vortex columns appeared. We believe that the vortex formation can be traced to instability of the boundary layer during and after the deceleration. It is known that unstable oscillations develop in most shear layers, when the speed of motion is sufficiently high. In the case of a shear layer between two parallel streams of different velocity for example, Lessen3 found that the flow becomes definitely unstable when the Reynolds number based on the thickness of the mixing region is of the order of 20. In this case, it is also known that the vortex sheet eventually rolls up into a periodic It may be expected, then, that the spacformation of discrete eddies. ing between the individual vortices in the final stage will be approsimately equal to the wave length of the initial unstable oscillations of the shear zone. An analogous situation may exist in the present case, in which the During the uniform motion boundary layer represents the shear layer. the velocity distribution on the boundary layer is stable at the Reynolds U
Y FIG.
1.
A typical
velocity
profile.
’Scientific Advisory Board to Chief of Staff, U. S. Air Force, Pasadena, Calif. ’ Department of Mathematics, Massachusetts Institute of Technology, Cambridge, At present, holder of Guggenheim Fellowship at California Institute of Technology. (1948); NACA TR NO. 979 (1950). 3 M. LESSEN, Sc.D. Thesis, M.I.T.
Mass.
517
518
TH.
VON KARMAN
[J. F. 1.
AND C. C. LIN
numbers prevailing in the experiments. However after the sudden stoppage of the plate the velocity distribution is changing with time, and is of the general form shown in Fig. 1. This flow is certainly less stable than the profile corresponding to steady flow. John E. Plapp recently computed the stability characteristics of a similar velocity profile given by the equation u = Av(l
- 7$,
?j = y/6.
He found that the flow becomes unstable if the Reynolds number R 6-- g
Y
> 408,
where U = 4A/27 is the maximum speed. At Ra = 408, the oscillation has a frequency f, wave length X, and wave speed c as follows : f = 0.207 U/6, x = 2.10 6, c = 0.43 u. In the present experiments, the plate is not suddenly stopped, so Howthat it still has a finite velocity at the time of vortex formation. Typical values of ever, a qualitative correlation may still be expected. f, X, and c, according to the experiments of Mr. Fales, are f = 13.0 cps. x = 0.04 ft. c = 0.52 fps. Based on these values one obtains for the corresponding and 6 u = 1.2 fps.
values of U
6 = 0.23 in. In the case considered, the plate was moving at 1.55 fps. before stoppage, and at 0.72 fps. at the moment of “vortex birth.” Thus the value of U = 1.2 fps. is not unreasonable. There is no conclusive value for 6 from the experiments. Estimates based on boundary-layer theory yield a value of 0.14 in. which is somewhat too small. However, in view of the substantial differences of velocity profiles between theory and Thus, the above experiment, this is not an unreasonable discrepancy. analysis tends to support the point of view that the vortices are actually traceable to the instability of the flow after the stoppage of the plate. The writers wish to thank Dr. W. D. Rannie and Mr. John E. Plapp for permission to use their unpublished results.