A new analytical solution for self-similar compressible vortices is derived in this paper. Based on the previous incompressible formulation of intense vortices, we derived a theoretical model that includes density and temperature variations. The governing equations are simplified assuming strong vortex conditions. Part of the hydrodynamic problem (mass and momentum) is shown to be analogous to the incompressible kind and as such the velocity is obtained through a straightforward variable transformation. Since all the velocity components are bounded in the radial direction, the density and pressure are then determined by standard numerical integration without the usual stringent simplification for the radial velocity. While compressibility is shown not to affect the tangential velocity, it influences only the meridional flow (radial and axial velocities). The temperature, pressure, and density are found to decrease along the converging flow direction. The traditional homentropic flow hypothesis, often employed in vortex stability and optical studies, is shown to undervalue the density and greatly overestimate the temperature. Comparable to vorticity diffusion balance for the incompressible case, the incoming flow carries the required energy to offset the contributions of conduction, viscous dissipation, and material expansion, thus keeping the temperature steady. This model is general and can be used to obtain a compressible version for all classical previous incompressible analysis from the literature such as Rankine, Burgers, Taylor, and Sullivan vortices.

1.
Rankine
,
W. J. M.
, 1858,
Manual of Applied Mechanics
,
C. Griffen Co.
,
London, England
.
2.
Helmholtz
,
H.
, 1858, “
Uber die Integrale der Hydrodynamischen Gleichungen, Welche den Wirbelbewegungen Entsprechen
,”
J. Reine Angew. Math.
0075-4102,
55
, pp.
25
55
.
3.
Taylor
,
G. I.
, 1930, “
Recent Work on the Flow of Compressible Fluids
,”
J. Lond. Math. Soc.
0024-6107,
5
, pp.
224
240
.
4.
Hollingsworth
,
M. A.
, and
Richards
,
E. J.
, 1955, “
A Schlieren Study of the Interaction Between a Vortex and a Shockwave in a Shock Tube
,” Aeronautical Research Council Technical Report No. 17985.
5.
Howard
,
L. N.
, and
Matthews
,
D. L.
, 1956, “
On the Vortices Produced in Shock Diffraction
,”
J. Appl. Phys.
0021-8979,
27
, pp.
223
231
.
6.
Ragsdale
,
R. G.
, 1960, “
NASA Research on the Hydrodynamics of the Gaseous Vortex Reactor
,” NASA Technical Report No. D-288.
7.
Sibulking
,
M.
, 1962, “
Unsteady, Viscous, Circular Flow Part 3: Application to the Ranque-Hilsh Vortex Tube
,”
J. Fluid Mech.
0022-1120,
12
, pp.
289
298
.
8.
Meager
,
A.
, 1961, “
Approximate Solution of Isentropic Swirling Flow Through a Nozzle
,”
ARS Journal
,
32
, pp.
1140
1148
.
9.
Rott
,
N.
, 1959, “
On The Viscous Core of a Line Vortex
,”
Z. Angew. Math. Phys.
0044-2275,
10
, pp.
73
81
.
10.
Mack
,
L. M.
, 1960, “
The Compressible Viscous Heat-Conducting Vortex
,”
J. Fluid Mech.
0022-1120,
8
, pp.
284
292
.
11.
Bellamy-Knights
,
P. G.
, 1980, “
Viscous Compressible Heat Conducting Spiraling Flow
,”
Q. J. Mech. Appl. Math.
0033-5614,
33
, pp.
321
336
.
12.
Sibulking
,
M.
, 1962, “
Unsteady, Viscous, Circular Flow Part 3: Application to the Ranque-Hilsh Vortex Tube
,”
J. Fluid Mech.
0022-1120,
12
, pp.
289
298
.
13.
Brown
,
S. N.
, 1965, “
The Compressible Inviscid Leading Edge Vortex
,”
J. Fluid Mech.
0022-1120,
22
, pp.
17
32
.
14.
Hall
,
M. G.
, 1965, “
The Structure of Concentrated Vortex Cores
,”
Prog. Aeronaut. Sci.
0079-6026,
7
, pp.
53
100
.
15.
Colonius
,
T.
,
Lele
,
S. K.
, and
Moin
,
P.
, 1991, “
The Free Compressible Viscous Vortex
,”
J. Fluid Mech.
0022-1120,
230
, pp.
45
73
.
16.
Bagai
,
A.
, and
Leishman
,
J. G.
, 1993, “
Flow Visualization of Compressible Vortex Structures Using Density Gradient Techniques
,”
Exp. Fluids
0723-4864,
15
, pp.
431
442
.
17.
Chiocchia
,
G.
, 1989, “
A Hodograph Approach to the Rotational Compressible Flow of an Ideal Fluid
,”
Q. Appl. Math.
0033-569X,
47
, pp.
513
528
.
18.
Ardalan
,
K.
,
Meiron
,
D. I.
, and
Pullin
,
D. I.
, 1995, “
Steady Compressible Vortex Flows: The Hollow-Core Vortex Array
,”
J. Fluid Mech.
0022-1120,
301
, pp.
1
17
.
19.
von Ellenrieder
,
K.
, and
Cantwell
,
B. J.
, 2000, “
Self-Similar Slightly Compressible Free Vortices
,”
J. Fluid Mech.
0022-1120,
423
, pp.
293
315
.
20.
Perez-Saborid
,
M.
,
,
M. A.
,
Gomez-Barea
,
A.
, and
Barrero
,
A.
, 2002, “
Downstream Evolution of Unconfined Vortices: Mechanical and Thermal Aspects
,”
J. Fluid Mech.
0022-1120,
471
, pp.
51
70
.
21.
Rusak
,
Z.
, and
Lee
,
J. H.
, 2002, “
The Effects of Compressibility on the Critical Swirl of Vortex Flows in a Pipe
,”
J. Fluid Mech.
0022-1120,
461
, pp.
301
319
.
22.
Mandella
,
M. J.
, 1987, “
Experimental and Analytical Studies of Compressible Vortices
,” Ph.D. thesis, Department of Applied Physics, Stanford University.
23.
Kalkhoran
,
I. M.
, and
Smart
,
M. K.
, 2000, “
Aspects of Shock Wave-induced Vortex Breakdown
,”
Prog. Aerosp. Sci.
0376-0421,
36
, pp.
63
95
.
24.
Cattafesta
,
L. N.
, and
Settles
,
G. S.
, 1992, “
Experiments on Shock/Vortex Interaction
,” AIAA Paper No. 92-0315.
25.
Taylor
,
G. I.
, 1918, “
On the Dissipation of Eddies
,”
The Scientific Papers of Sir Geoffrey Ingram Taylor
(
Meteorology, Oceanography and Turbulent Flow
),
G. K.
Batchelor
, ed.,
Cambridge University Press
,
Cambridge
, Vol.
2
, pp.
96
101
.
26.
Burgers
,
J. M.
, 1948, “
A Mathematical Model Illustrating the Theory of Turbulence
,”
0065-2156,
1
, pp.
171
199
.
27.
Sullivan
,
R. D.
, 1959, “
A Two-Cell Vortex Solution of the Navier-Stokes Equations
,”
J. Aerosp. Sci.
0095-9820,
26
(
11
), pp.
767
768
.
28.
Vatistas
,
G. H.
,
Kozel
,
V.
, and
Minh
,
W.
, 1991, “
A Simpler Model for Concentrated Vortices
,”
Exp. Fluids
0723-4864,
11
, pp.
73
76
.
29.
Han
,
Y. Q.
,
Leishman
,
J. G.
, and
Coyone
,
A. J.
, 1997, “
Measurements of the Velocity and Turbulence Structure of a Rotor Rip Vortex
,”
AIAA J.
0001-1452,
35
(
3
), pp.
477
485
.
30.
Jeng-Lih
,
H.
,
Jing-Fa
,
T.
,
Chye-Horng
,
C.
, and
Cheng-Tsair
,
Y.
, 1999, “
Measurements of Propeller Tip Vortices Around Near Wake Zone
,”
Ship Technology Research
,
46
(
2
), pp.
93
110
.
31.
Leishman
,
J. G.
, 1998, “
Measurements of a Periodic Wake of a Hovering Rotor
,”
Exp. Fluids
0723-4864,
125
, pp.
252
361
.
32.
Coton
,
F. N.
,
Copland
,
C. M.
, and
Galbraith
,
A. D.
, 1998, “
An Experimental Study of the Idealized Vortex System on a Novel Rotor Blade Tip
,”
Aeronaut. J.
0001-9240,
102
, pp.
385
392
.
33.
Neuwerth
,
S. B.
, and
Jacob
,
D.
, 2000, “
The Inlet-Vortex System of Jet Engines Operating Near the Ground
,”
18th AIAA Applied Aerodynamics Conference
,
Denver, CO
, Aug. Paper No. AIAA-2000-3998, pp.
14
17
.
34.
Leishman
,
J. G.
, 2002,
Principles of Helicopter Aerodynamics
,
Cambridge University Press
,
Cambridge, U.K
.
35.
Lan
,
N.
, 2003, “
Mathematical Analysis of Dust Devils
,” M.S. thesis School of Engineering, Cranfield University.
36.
Mcalister
,
K. W.
, 2004, “
Rotor Wake Development During the First Revolution
,”
J. Am. Helicopter Soc.
0002-8711,
49
(
4
), pp.
371
390
.
37.
Dumitrescu
,
H.
, and
Frunzulica
,
F. A.
, 2004, “
Free-Wake Aerodynamic Model for Helicopter Rotors
,”
,
5
(
3
), pp.
1
9
.
38.
Malovrh
,
B.
, and
Gandhi
,
F.
, 2005, “
Sensitivity of Helicopter Blade-Vortex-Interaction Noise and Interaction Parameters
,”
J. Aircr.
0021-8669,
42
(
3
), pp.
685
697
.
39.
Vatistas
,
G. H.
, and
Aboelkassem
,
Y.
, 2006, “
Extension of the Incompressible ṉ2 Vortex Into Compressible
,”
AIAA J.
0001-1452,
44
(
8
),
1912
1915
.
40.
Vatistas
,
G. H.
, 1998, “
New Model for Intense Self-Similar Vortices
,”
J. Propul. Power
0748-4658,
14
(
4
), pp.
462
469
.
41.
Scully
,
M. P.
, 1975, “
Computation of Helicopter Rotor Wake Geometry and Its Influence on Rotor Harmonic Airloads
,” Aeroelastic and Structures Research Laboratory, Massachusetts Institute of Technology, Report No. ASRL TR 178–1.
42.
Vatistas
,
G. H.
, 2004, “
The Fundamental Properties of the n=2 Vortex Model
,”
Transactions CSME
,
28
(
1
), pp.
53
58
.
43.
Oswatitsch
,
K.
, 1945, “
Der Luftwiderstand als Integral des Entropiestromes
,”
Nachr. Ges. Wiss. Goettingen, Math.-Phys. Kl.
0369-6650, pp.
88
90
.