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A New Look at the Dynamics of Vortices with Finite Cores

  • P. C. Parks

Abstract

The stability theory for wave-like disturbances in a pair of trailing line vortices developed by S. C. Crow has been modified to take account of finite core radii and appropriate distributions of vorticity within these cores. The difficulties encountered by Crow in calculating the self-induction effects of each vortex are avoided and, for a uniform distribution of vorticity within the cores, the self-induction function is expressible in terms of modified Bessel functions of the second kind. The essential features of Crow’s theory are, however, confirmed with small numerical changes — for example, the most unstable long waves which develop in the trail have a wavelength of 7.2b, where b is the separation distance of the two vortices (compared with Crow’s result of 8.4b).

The effect of axial flow within the cores may be considered by wrapping the core in a sheath of vortex rings. This model leads to a new stability criterion for a single “jet-vortex”. The stability diagrams for the trailing vortex pair are also modified by axial flows, the stable regions being reduced in size and the growth rate of the most unstable mode being increased.

The theory, which assumes the vortices extend from infinity to infinity, cannot be used to calculate the growth of perturba­tions deliberately introduced at the wing of the aircraft, but a modified discretised theory amenable to digital computation has been developed to investigate initial growth of these excited waves. Some results from digital computations are presented.

Keywords

Vortex Ring Stability Boundary Axial Flow Unstable Mode Stability Diagram 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Nomenclature

A

Subscript for anti-symmetric mode

a

Growth rate of unstable waves, or complex exponent

a′

M c k /4

aij

Matrix elements, defined after (9.2)

B1

1−(β2[ψ(γ)−1]/γ2)

B2

−(β2[ψ(γ)−1]/γ2)

b

Separation distance of undistrubed vortices

C

Subscript for center line

CL

Lift coefficient of wing

c

Radius of circular core

E

β22

e

Exponential function

H

−ψ(β)

I1

Modified Bessel function

i

√−1

i′,j′,k

Units vectors, defined in Section 3

J

−χ(β)

K0,K1

Modified Bessel functions of the second kind

k

Wave number

L

L Subscript for axial vortex lines

M

2πcU/Г0

M′

2πbU/Г0

M*

M′bk

n

Circumferential mode number of Batchelor and Gill11

q

Distance defined in Fig. 3

R

Subscript for vortex rings

r

Radial distance from center line

S

Subscript for symmetrical mode

s

Dummy variable in integrals in 8.1, also eigen-values 2πc2a/Гo in (8.23) and 2πb2a/Гo in (9.2)

t

Time

U

Axial velocity within core

u∿1,u∿2

Induced velocities, defined in (2.1)

u*,û*

Defined in and after (8.1.2)

Vo

Speed of aircraft

w*,ŵ*

Defined in and after (8.1.2)

x,y,z

Coordinate system, defined in Fig. 1

x′,y′,z′

Coordinate system, defined in Fig. 3

y1,z1,y2,z2

Displacements of vortex center lines, 1 for starboard, 2 for port

ŷ1,\(\hat{z}\)12,\(\hat{z}\)2

Amplitude of displacements, when y11 exp(ikx+at), etc.

α

Non-dimensional growth rate, 2πb2a/Гo

β

bk

Гo

Total strength of circulation of one vortex

γ

Non-dimensional core radius, ck

γ()

Strength of axial vorticity per unit area

γ(y′,z′)

Strength of axial vorticity per unit area

γ*

Euler′s constant, 0.5772157.....

Δx,Δz

Distances, defined after (8.1.2)

δ

Non-dimensional cut off distance used by Crow1

θ(x,t)

Angular perturbation of vortex ring, defined in Fig. 7

\(\hat{\theta }\)

Defined after (8.1.2)

κ

Strength of vortex ring, defined in (8.1.1)

μ

s + 1/2 ckMi, defined after (8.2.4)

μ′

(−s/M′bk)−i/2, defined after (9.4)

ξ

x/b

π

Pi, 3.1415926....

φ(x,t)

Angular perturbation of vortex ring, defined in Fig.7

χ(β)

βK1(ß)

Ψ(β)

β2Ko(β)+βK1(β)

Ψ(r,z)

Stokes’ stream function defined in (8.1.1)

ω(β,δ)

Integral defined in (2.3)

\({{\tilde{\omega }}_{\text{o}}}\)

Radius of vortex ring in (6.3)

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References

  1. 1.
    Crow, S. C., “Stability Theory for a Pair of Trailing Vortices,” AIAA Paper 70–53, Eighth AIAA Aerospace Sciences Meeting, New York, January 19–21, 1970. AIAA Journal (to appear)Google Scholar
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Copyright information

© Plenum Press, New York 1971

Authors and Affiliations

  • P. C. Parks
    • 1
  1. 1.NASA Langley Research Center and University of WarwickUSA

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