NASA Technical Reports Server (NTRS) 19910020768: Computation of compressible quasi-axisymmetric slender vortex flow and breakdown

NORTH-HOLLAND

PHYSICS

PUBLISHING

v 91-80082

Computation of compressible quasi-axisymmetric slender vortex
flow and breakdown

Osama A. Kandil 1 and Hamdy A. Kandil 2

Department of Mechanical Engineering and Mechanics , Old Dominion University , Norfolk , VA 23529-0247, USA

Analysis and computation of steady, compressible, quasi-axisymmetric flow of an isolated, slender vortex are considered.
The compressible Navier-Stokes equations are reduced to a simpler set by using the slenderness and quasi-axi symmetry
assumptions. The resulting set along with a compatibility equation are transformed from the diverging physical domain to a
rectangular computational domain. Solving for a compatible set of initial profiles and specifying a compatible set of boundary
conditions, the equations are solved using a type-differencing scheme. Vortex breakdown locations are detected by the failure
of the scheme to converge. Computational examples include isolated vortex flows at different Mach numbers, external
axial-pressure gradients and swirl ratios. Excellent agreement is shown for a bench-mark case between the computed results
using the slender vortex equations and those of a full Navier-Stokes solver.

Reprinted from COMPUTER PHYSICS COMMUNICATIONS

V

164

Computer Physics Communications 65 {1991 ) 164-172

North-Holland

Computation of compressible quasi-axisymmetric slender vortex
flow and breakdown

Osama A. Kandil 1 and Hamdy A. Kandil 2

Department of Mechanical Engineering and Mechanics, Old Dominion University, Norfolk, VA 23529-0247. USA

Analysis and computation of steady, compressible, quasi-axisymmetric flow of an isolated, slender vortex are considered.
The compressible Navier— Stokes equations are reduced to a simpler set by using the slenderness and quasi-axisymmetry
assumptions. The resulting set along with a compatibility equation are transformed from the diverging physical domain to a
rectangular computational domain. Solving for a compatible set of initial profiles and specifying a compatible set of boundary
conditions, the equations are solved using a tvpe-differencing scheme. Vortex breakdown locations are detected by the failure
of the scheme to converge. Computational examples include isolated vortex flows at different Mach numbers, external
axial-pressure gradients and swirl ratios. Excellent agreement is shown for a bench-mark case between the computed results
using the slender vortex equations and those of a full Navier-Stokes solver.

1. Introduction

The phenomenon of vortex breakdown or
bursting was observed in the water vapor con-
densation trails along the leading-edge vortex cores
of a gothic wing. Two forms of the leading-edge
vortex breakdown, a bubble type and a spiral
type, have been documented experimentally [1],
The bubble type shows an almost axisymmetric
sudden swelling of the core into a bubble, and the
spiral type shows an asymmetric spiral filament
followed by a rapidly spreading turbulent flow.
Both types are characterized by an axial stagna-
tion point and a limited region of reversed axial
flow. Much of our knowledge of vortex break-
down has been obtained from experimental stud-
ies in tubes where both types of breakdown and
other types as well were generated [2-4],

The major effort of numerical simulation of
vortex breakdown flows has been focused on in-
compressible, quasi-axisymmetric isolated vortices.
Grabowski and Berger [5] used the incompressible,
quasi-axisymmetric Navier-Stokes equations. Ha-

1 Professor and Eminent Scholar.

Graduate Research Assistant.

fez et. al [6] solved the incompressible, steady,
quasi-axisymmetric Euler and Navier-Stokes
equations using the stream function -vorticity for-
mulation and predicted vortex breakdown flows
similar to those of Garbowski and Berger. Spall,
Gatski and Grosch [7] used the vorticity-velocity
formulation to solve the three-dimensional, in-
compressible, unsteady Navier-Stokes equations.

Flows around highly swept wings and slender
wing-body configurations at transsonic and super-
sonic speeds and at moderate to high angles of
attack are characterized by vortical regions and
shock waves, which interact with each other. Other
applications which encounter vortex-shock inter-
action include a supersonic inlet ingesting a vortex
and injection into a supersonic combustor to en-
hance the mixing process, see Delery et. al [8] and
Metwally, Settles and Horstman [9], These prob-
lems and others call for developing computational
schemes to predict, study and control com-
pressible vortex flows and their interaction with
shock waves. Unfortunately, the literature lacks
this type of analysis with the exception of the
preliminary work of Liu, Krause and Menne [10]
and Copening and Anderson [11],

In this paper, the steady, compressible Navier-
Stokes equations are simplified using the quasi-

0010-4655/91 /$0?. 50 C 1991 - Elsevier Science Publishers B.V. (North-Holland)

O.A Kandil and H.A. Kandil / Compressible quasi-axisymmetric slender vortex flow

165

axisymmetry and slenderness assumptions. A
compatibility equation [10] has been used and the
governing equations are transformed to a rectan-
gular computational domain by using a Levey-
Lee-type transformation. A compatible set of ini-
tial conditions and boundary conditions is ob-
tained and the problem is solved using a type-dif-
ferencing scheme. The numerical results show the
effects of compressibility, external axial pressure
gradients and the swirl ratio on the vortex break-
down location. A bench-mark flow case has been
solved using these equations and the full Navier-
Stokes equations. The results are in excellent
agreement with each other.

2. Highlights of the formulation and computational
scheme

Starting with the steady, compressible Navier-
Stokes equations which are expressed in the cylin-
drical coordinates (x, r and <£), assuming the
isolated vortex flow to be slender [r/l = O
(1/v/jRc), v/U rx: = 0 (1 / {Re ), where / is a char-
acteristic length, v the radial velocity, V r x the
freestream velocity and Re the freestream Reyn-
olds number] and quasi-axisymmetric [9/9<f>( ) =
0], and performing an order-of-magnitude analy-
sis, the equations are reduced to a compressible,
quasi-axisymmetric, boundary-layer-like set. The
dimensionless flow variables p, p, u, v, w, T and
p are non-dimensionalized by p w , p*a^, a
alc/Cp and p^ for the density, pressure, velocity,
temperature and viscosity, respectively, where C p
is the specific heat at constant pressure. Next, we
introduce a Levey- Lee-type transformation which
is given by

dx,

7J =

Pc f r P,

Hi) 1 Pc

r.

( 1 )

where A is given by

MSF =

Mi) ',«)

fir)

= modified shape factor characterizing
the growth of vortex-flow boundary (2)

and /(p) is a function relating the density integral
at any axial station to that at the initial station. It
is equal to 1 for incompressible flow. The sub-
script e refers to external conditions and the sub-
script i refers to initial location.

The governing equations become

W 19 A

^ + 7\M {Xur) + Vr v = 0 '

l r/ j

where v = V 77 r , and

P x P

9tj

^=97’

( 3 )

9 u 9 u 19 p A .. w 2

3 T 6 —

otj p of p r

M 9 / cr 9w \
+ \r drj \ A drj y

(4a)

where

6 = 17 and c = — — ,

PcPe X PePe

(4b)

A 2 9 p

— w = V-,
r 9tj

(5)

9w 9w A ,

u ~ + V l^ + — (r-0M)w

9£ 1 ” 9tj
M 9

9T

pr

A 2

■r 2 dv 1

9tj

1

V K

r

9 7]

u

dp <

\ Vw 1

p

9£ + i

P r

1

Me (1

Qji. ) 2 1

~r

*"\l

9i) )

'Ut)

( 6 )

(7)

where Pr = Prandtl number = 0.72.

p = :L 7^P r ’ ( 8 )

where y = ratio of specific heats.

The viscosity p is related to the temperature
through the Sutherland law. At the initial
boundary, £ = £ jt we specify

u, = u(t)), w, = w(tj) and T t =T( tj). (9)

The other compatible initial conditions are ob-
tained from a compatibility equation and eqs. (5)

166

O.A, Kandil and H.A. Kandil / Compressible quasi -axi symmetric slender vortex flow

and (8). At the vortex axis, r/ = 0, we specify

w =

3 T
8 ?]

= 0 .

( 10 )

At the outer boundary, tj = rj e , we assume the
boundary to be a stream surface, specify the axial

pressure gradient (dp/d£) e and use the Euler
equations to match the outer profiles to those of
the viscous core to obtain the conditions on u e ,

w e> T e , p e .

Eqs. (3)-(7) are solved using an implicit, type-
differencing scheme. The computational proce-

AXIAL Ot STANCE . X

h -a.se af -a.a2 dP/d»-a.i25 b*u-e.4

AXIAL DISTANCE . X

H -0.75 d{ -0.04 dP/d*-0.125 b#t«-0.4

AXIAL DISTANCE . X

H -0.9 d$ -0.02 dP/d.-0.25 b«t*-0.4

AXIAL DISTANCE . X

H -0.75 dj -0.04 dP/dx-0. 25 b#U-0.4

A = MSF, B = u a> C = p a , D = T a

Fig. 1. Slender quasi- axisymmetric flow solutions for the effect of Mach number, external axial pressure gradient and swirl ratio.

O.A. Kandil and H. A. Kandil / Compressible quasi -axisymme trie slender vortex flow

167

dure consists of two parts. In the first part a
compatible set of initial profiles are obtained at
£ = £, and in the second part we use eqs. (4)-(8)
and the compatibility equation to obtain p , u, w,
p, T and V (or v).

N >6.9 df -0.02 dP/dx-0. 129 b*U-0.2

AXIAL OtSTANCE . X

H 9 df >0.02 dP/dx-0. 29 b»t»-0.2

3. Numerical examples

In the present numerical examples, the outer
edge of the vortex, rj e , is taken as 10, and 1000
grid points are used and hence Arj e = 0.01. The

N -0.79 df -0.04 dP/dx-0. 129 b*U-0.2

AXIAL 01 STANCE . X

N -0.79 df -0.04 dP/dx-0. 29 b*U-0.2

Fig. 1 (continued).

168

O.A. Kandil and H A . Kandil / Compressible quasi-axisymmetric slender vortex flow

results are shown for two Mach numbers: M = 0.5
and 0.75. The step size in the axial direction is
0.02 for M = 0.5 and 0.04 for M = 0.75. For each
Mach-number case, we solve for two external axial
pressure gradients; (3 p/dx) e = 0.125 and 0.25 and
two swirl ratios; /? = (w/u) r=l = 0.2 and 0.4. The
initial profiles for u,, w, and T ■ are u, = constant,
Wj = /?u,r(2 - r 2 ) for r< 1 and w t = j3u t /r for

r > 1 and T t = 2.5, respectively. Fig. 1 shows MSF, “

u a , p a and T a which are referred to by curves A. W

B, C and D; respectively. The results show that
the breakdown length is more than doubled when =

the Mach number increases from 0.5 to 0.75. They 9
also show that while the outer boundary continu-
ously increases for M = 0.5, it initially decreases —

and then increases for M = 0.75; see the A curves. w

AXIAL VELOCITY DISTRIBUTION CIRCUH. VELOCITY DISTRIBUTION

PRESSURE DISTRIBUTION DENSITY DISTRIBUTION

PRESSURE . p „■ ■ _• „• „•

H ■ 0.5 df • 9.02 4p/4m » 9.25 b«ti * 0.4 DENSITY , ^

H • 0.5 d? ■ 0.02 4p/4* • 0.25 b«U - 0.4

Fig. 2. Flow profiles for slender quasi-axisymmetric flows at M - 0.5 and 0.75, ft = 0.4, (d p/dx) c = 0.25.

in in ii

O.A. Kandil and H.A. Kandil / Compressible quasi -axisymmei he slender vortex flow

169

The adverse pressure gradient at the vortex axis
decreases faster for M = 0.75 than for M = 0.5.
The results also show that the external axial pres-
sure gradient is a dominant parameter on the
breakdown length. As the external axial pressure
gradient is doubled, the breakdown length sub-
stantially decreases. Doubling the swirl ratio
slightly decreases the breakdown length.

AXIAL VELOCITY DISTRIBUTION

AXIAL VELOCITY . u

H - 0.75 df • 0.04 dp/d* - 0.39 b.tt -0.A
PRESSURE DISTRIBUTION

PRESSURE . p

N • 0.75 df • 0.04 dp/d« > 0.35 b.U *0.4

Fig. 2 shows the profiles of u , w, p and p
across r at axial stations until the breakdown
location for M = 0.5 and 0.75 for the cases of
(d/?/d.x) e = 0.25 and /? = 0.4. The initial profiles
are indicated by the number 1 and the next shown
station is indicated by 3. At M = 0.75. it is noticed
that the pressure and density gradients in the axial
direction decrease faster than those at M = 0.5.

— — ^rvr4f\irs*cvono

CIRCUH. VELOCITY . «

X - 0.79 df - 0.04 dp/d* - 0.29 b»U -0.4

.V? l.W I.W I.IV I. I? I

OEKSITY .y

H • 0.79 df - 0.04 dp/d* - 0.25 b.U -0 4

Fig. 2 (continued).

170

O.A. Kandil and H A. Kandil / Compressible quasi -axisymmetric slender vortex flow

The profiles show that the viscous diffusion at
M = 0.75 is larger than that at M = 0.5.

Fig. 3 shows the profiles of w, w, v and p
which has been computed by the present method
and by an upwind Navier-Stokes solver for the
case of M = 0.5, (8 = 0.6 and (dp/dx),. = 0. For
the Navier-Stokes solver a rectangular grid of
100x51 x 51 in the axial direction and cross- flow

plane is used. The curves are labeled by the capital
letters A, B, ... etc. Comparing the curves of the
two sets, a remarkable agreement is seen.

It is concluded from the given numerical exam-
ples that increasing the flow Mach number has a
favorable effect on the vortex breakdown location.
The external axial pressure gradient is a dominant
parameter on the vortex breakdown. Its effect

AXIAL VELOCITY DISTRIBUTION

CIRCUH. VELOCITY DISTRIBUTION

CIRCUM. vaOCITY . •
M«».S d*«8,2 S-VorU*

RAO I AL VELOCITY DISTRIBUTION

PRESSURE DISTRIBUTION

PRESSURE . p
H>M d»*8.2 S-VoH**

Fig. 3. Flow profiles for slender quasi-axisymmetric flows using the present method and the full Navier-Stokes equations,

j8 - 0.6, (dp/dx) e = 0.0.

M = 0.5,

O.A. Kandil and H A. Kandil / Compressible quasi -axisymmeiric slender cortex flow

171

Ax14l Velocity Distribution

Am I 4 1 Velocity , u
M-0.5 b«U-9.6 dx-0 . 2 Full N-S

-.29 - .18 - .16 - .14 - .12 - .10 - .00 - .06 - .§4 - .02 0

Horlzontsl Cosp. , v
H-0.5 bott-0.6 dx-0.2 Full N-$

Vsrtlctl Velocity Distribution
-i — i — i — i — i — i — i — i — i — r

rmrsi

Vsrtlcsl Velocity t p
M*0,5 bsU>0.6 dx-0.2 Full N-$

Prsssur* Distribution

Pr.iaur. . p

M«i,5 b.U-1.6 <!««•. 2 Full N-S

Fig. 3 (continued).

decreases as the Mach number is increased. Com-
parison of the present results with the full Navier-
Stokes results gives a strong confidence in the
present analysis. The present formulation and re-
sults are used to generate compatible initial pro-
files for the full Navier- Stokes solutions, and to
provide data for breakdown-potential cases for
accurate computations using the full Navier-
Stokes equations. The full Navier- Stokes equa-

tions are currently applied to these cases, so that
we can solve for the flow in the breakdown region.

Acknowledgement

This research work is supported by the NASA
Langley Research Center under Grant No. NAG-
1 - 994 .

172

0,A Kandil and H. A . Kandil / Compressible quasi -axisymmetric slender cortex flow

References

[1] N.C. Lamboume and D.W. Bryer, Bursting of leading-edge
vortices: Some observations and discussion of the phe-
nomenon, Aeronautical Research Council, R&M 3282
(1961).

[2] T. Sarpkaya, Vortex breakdown in swirling conical flows,
AIAA J. 9 (1971) 1791.

[3] S. Leibovich, Vortex stability and breakdown survey and
extension, AIAA J. 23 (1984) 1194.

[4] M.P. Escudier and N. Zender, Vortex flow regimes, J.
Fluid Mech. 115 (1982) 105.

[5] W.J. Grabowskj and S.A. Berger, Solutions of the
Navier-Stokes equations for vortex breakdown, J. Fluid
Mech. 75 (1976) 525.

[6] M. Hafez, G. Kuruvila and M.D. Salas, Numerical studv
of vortex breakdown, J. Appl. Num. 2 (1987) 291.

[7] R.E. Spall, T. Gatski and C.E. Grosch, A criterion for
vortex breakdown, ICASE Report 87-3 (January 1987).

[8] J. Delery, E. Horowitz, O. Leuchter and J.L. Solignac.
Fundamental studies of vortex flows, Rech. Aerosp. No.
(1984) 1.

[9] O. Metwally, G. Settles and C. Horstman, An experimen-
tal study of shock wave/vortex interaction, AIAA Paper
89-0082 (January 1989).

[10] C.H. Liu, E. Krause and S. Menne, Admissible upstream
conditions for slender compressible vortices, AIAA Paper
86-1093 (July 1986).

[11] G. Copening and J. Anderson, Numerical solutions to
three-dimensional shock/vortex interaction at hypersonic
speeds, AIAA Paper 89-0674 (January 1989).

AIAA 91-0752

COMPUTATION OF STEADY AND
UNSTEADY COMPRESSIBLE
QUASI-AXISYMMETRIC VORTEX
FLOW AND BREAKDOWN

O. A. KANDIL AND H. A. KANDIL
Old Dominion University, Norfolk, VA

C. H. LIU

NASA Langley Research Center, Hampton, VA

29th Aerospace Sciences Meeting

January 7-10, 1991 /Reno, Nevada

For permission to copy or republish, contact the American Institute of Aeronautics and Astronautics
370 L’Enfant Promenade, S.W., Washington, D.C. 20024

COMPUTATION OF STEADY AND UNSTEADY COMPRESSIBLE
QUASI-AXISYMMETRIC VORTEX FLOW AND BREAKDOWN

Osama A. Kandil* and Hamdy A. Kandil**

Old Dominion University, Norfolk, VA

C. H. Liu***

NASA Langley Research Center, Hampton, VA

Abstract

The unsteady, compressible Navier-Stokes equations
are used to compute and analyze compressible quasi-
axisymmetric isolated vortices. The Navier-Stokes equa-
tions are solved using an implicit, upwind, flux -difference
splitting finite-volume scheme. The developed three-
dimensional solver has been verified by comparing its so-
lution profiles with those of a slender, quasi-axisymmetric
vortex solver for a subsonic, isolated quasi-axisymmetric
vortex in an unbounded domain. The Navier-Stokes
solver is then used to solve for a supersonic quasi-
axisymmetric vortex flow in a configured circular duct.
Steady and unsteady vortex-shock interactions and break-
down have been captured. The problem has also been
calculated using the Euler solver of the same code and
the results are compared with those of the Navier-Stokes
solver. The effect of the initial swirl has been tentatively
studied.

Introduction

The phenomenon of vortex breakdown or bursting
was observed in the water vapor condensation trails
along the leading-edge vortex cores of a gothic wing.
Two forms of the leading-edge vortex breakdown, a
bubble type and a spiral type, have been documented
experimentally 1 . The bubble type of vortex breakdown
shows an almost axisymmetric sudden swelling of the
core into a bubble, and the spiral type of vortex break-
down shows an asymmetric spiral filament followed by a
rapidly spreading turbulent flow. Both types are char-
acterized by an axial stagnation point and a limited
region of reversed axial flow. Much of our knowl-
edge of vortex breakdown has been obtained from ex-
perimental studies of pipe flows where both types of
breakdown and other types as well were generated and
documented 2 ' 4 . The major effort of numerical sim-
ulation of vortex breakdown flows has been focused
on incompressible, quasi-axisymmetric isolated vortices.
Grabowski and Berger 5 used the incompressible, quasi-
axisymmetric Navier-Stokes equations to study isolated
vortex flow in an unbounded region. Hafez, et al 4 solved
the incompressible, steady, quasi-axisymmetric Euler and

• Pro/e* »or tod Eminent Scfaoiv, OepMDcm of Medunical Eoftneering
tod Mechanics, Associate Fellow, ALAA.

"Graduate Research Aitiasnf, Same Dept., Member AIAA
"•Group Leader, Theoretical Bow Physics Branch. Senior Member,
AIAA.

This paper is declared a work of the U.S. Government and
is not subject to copyright protection in the United States.

Navier-Stokes equations using the stream funoion-
vorticity formulation for isolated vortex flows. They pre-
dicted vortex breakdown flows similar to those of Gar-
bowski and Berger. Menne 7 has also used the stream
function vorticity formulation for unsteady, incompress-
ible quasi-axisymmetric isolated vortex flow. Menne and
Liu 1 used the Navier-Stokes equations to study three-
dimensional incompressible flows in a tube. Spall, et al 9 ,
presented a study of the structure and dynamics of bubble-
type vortex breakdown in incompressible flows using the
vorticity-velocity formulation. For more information on
the physical and computational aspects of the incompress-
ible vortex breakdown, the reader can refer to the paper
by Krause 10 .

Flows around highly swept wings and slender wing-
body configurations at transonic and supersonic speeds
and at moderate to high angles of attack are character-
ized by vortical regions and shock waves, which inter-
act with each other. Other applications which encounter
vortex-shock interaction include a supersonic inlet ingest-
ing a vortex and injection into a supersonic combustor to
enhance the mixing process, see Delery, et. al 11 and Met-
wally, settles, and Horseman 12 . Figure 1 shows these ex-
amples, where Fig. l.a and l.b are taken from ref. 1 1 and
Fig.l.c is taken from ref. 12. These problems and oth-
ers call for developing computational schemes to predict,
study and control compressible vortex flows and their in-
teraction with shock waves. Unfortunately, the literature
lacks this type of analysis with the exception of the pre-
liminary work of Liu, Krause and Menne 13 , Copening and
Anderson 14 , Delery, et al 11 , Kandil and Kandil 15 , Mead-
ows, Kumar and Hussaini 14 .

In this paper, we use the unsteady, compressible full
Navier-Stokes equations to compute and analyze com-
pressible and supersonic quasi-axisymmetric isolated vor-
tices. An implicit upwind, flux-difference splitting finite-
volume scheme, which is based on the Roe scheme, has
been used to solve the full Navier-Stokes equations. The
three-dimensional solver, which is called “FTNS3D", has
been used to solve two problems of isolated vortex flows.
The first problem is that of a subsonic, isolated quasi-
axisymmetric vortex in an unbounded domain. This case
has been verified by comparing the flow-profiles solutions
with those of the slender vortex solver of ref. 15. Next,
the three-dimensional Navier-Stokes equations are used
to solve for a supersonic quasi-axisymmetric vortex flow

in a configured circular duct 11 . Since the flow is quasi-
axisymmetric, the solution is obtained by forcing the flow
variables to be equal on two axial (meridian) planes. So-
lutions for steady and unsteady vortex-shock interactions
and breakdown have been obtained. These solutions are
compared with those of the Euler equations using the in-
viscid version of same solver. Effects of the initial swirl
ratio has been tentatively investigated.

Formulation

The conservative form of the dimensionless, unsteady,
compressible, full Navier- Stokes equations in terms of
time-independent, body-conformed coordinates and
£ 3 is given by

d 0

dt m

cKEv),

0;m=l-3, s=l-3

5T +

(1)

where

£m _ £17

t (x 1 ,x 2l x 3 )

(2)

Q =

11

pu,,pu2,pu 3l pe]‘

(3)

E m = inviscid flux

= j[^U m ,pU 1 U m + ^“p.pUjU.

+^ m P> pu 3 U m + a 3 * m p, (pe + p)U m ]‘ (4)

(E») < = viscous and heat— conduction flux in
direction

= 3^2,0^,

- qk)] 1 ; k = 1 - 3, n = 1 - 3 (5)

U B = 0k{“u k (6)

The first element of the three momentum elements of Eq.
(5) is given by

+«'W|pj (7 )

The second and third elements of the momentum elements
are obtained by replacing the subscript 1, everywhere in

Eq. (7), with 2 and 3, respectively. The last element of
Eq. (5) is given by

^r(upT\p — qij = [(^’a p ^ n

<5u,

L

(7 - l)Pr^ d? J

+Wupg

; p = 1 - 3 (.3)

The reference parameters for the dimensionless form
of the equations are L, and for the

length, velocity, time, density and molecular viscosity,
respectively. The Reynolds number is defined as Re =
P<x VocL/poc, where L is the initial radius of the Vor-
tex or the duct inlet radius. The pressure, p, is related
to the total energy per unit mass and density by the gas
equation

P = (7 - 1 )P

j( u ? + u? + u|)

(9)

The viscosity is calculated from the Sutherland law

P = T 3 ' 2 (I±^),C = 0.4317 (10)

and the Prandtl number P r » 0.72. In Eqs. (l)-(8), the
indicial notation is used for convenience.

Computational Scheme

The computational scheme used to solve the full
Navier-Stokes equations is an implicit, upwind, flux-
difference splitting, finite-volume scheme. It employs
the flux -difference splitting scheme of Roe. The Jaco-
bian matrices of the inviscid fluxes. A, = s = 1-3,
are split into backward and forward fluxes according to
the signs of the eigenvalues of the inviscid Jacobian ma-
trices. Flux limiters are used to eliminate oscillations in
the shock region. The viscous and heat-flux terms are
centrally differenced. The resulting difference equation
is solved using approximate factorization in the £ l ,
and if 3 directions. In addition to the three-dimensional
flows, the present computer program can solve for ax-
isymmetric and quasi-axisymmetric flows. The resulting
computer program can also be used to solve the Euler
equations. Thi s c ode is a modified version of the CFL3D
which is currently called “FTNS3D”. The modifications
have been developed by the present authors.

2

Oil

Computational Applications

In this section, two computational applications are
presented. The first application is that of a steady, sub-
sonic quasi-axisymmetric vortex flow in an unbounded
domain. The purpose of this application is to verify
the Navier-Stokes solver by comparing the results of this
case with those of a previously developed slender vortex
solver, see ref. 15. The second application is that of a
steady and unsteady, supersonic quasi-asymmetric vortex
flow in a configured circular duct. This application is
solved by using pseudo-time stepping and accurate-time
stepping. The results are compared with those of the Eu-
ler equations solver of the same computer program. Next,
we consider each application and discuss its results.

Steady Subsonic Quasi- Axisymmetric Vortex
Flow in an Unbounded Domain

Here, the three-dimensional Navier-Stokes is used
to solve for an isolated quasi-axisymmetric flow. The
computational domain for the Navier-Stokes equations is
a parallelopiped rectangular domain with a square cross-
section of 10x10 units. The downstream length is 10
units. The rectangular grid consists of51x51xl00 points
in the two directions of the square section and in the axial
direction, respectively. The grid is clustered algebraically
at the axis of the parallelopiped domain. The step size in
the axial direction is 0.1.

For the slender-vortex solver 15 , the computational do-
main is a cylindrical one and the solution is obtained on
one meridian plane having a radius of 10 and a length
of 10. The number of grid points in the radial direction
is 1000. The step size in the axial direction is 0.1. The
initial profile for the slender vortex solver are given by
u, = axial velocity a constant, w, = tangential velocity ■
0 Ui r^-r 2 ) for r 5 1 and Wj • 0 Uj/r for r £ 1 and T, =
temperature = 2.5, where 0 * 0.6. The Mach number at
the outer radius of the initial station, M. * 0.5. The other
compatible initial profiles for pi, v ; and p\ (pressure, ra-
dial velocity and density; respectively) are obtained from
the radial momentum equation, a compatibility equation 13
and the equation of state. The external axial pressure gra-
dient is selected as (jfcj =0. The external boundary
conditions on the cylindrical outer boundary are obtained
by using the Euler equations to match the outer profiles to
those of the viscous core in order to obtain the conditions
on u«, w„ T, and p % .

For the Navier-Stokes solver, the initial profiles are
obtained from the previous initial profiles by interpolating
the slender vortex profiles on the rectangular grid at the
initial station. The Reynolds number of the Navier-Stokes
solver for this case is set at 100.

Figure 2 shows the Navier-Stokes solutions on the
left and the slender-vortex solutions on the right. The
figure shows comparison of the profiles of axial velocity
u, tangential velocity w, radial velocity v, pressure p and
density p at the same axial stations which are marked

by A, B, C It is remarkable to see the excellent

agreement between the Navier-Stokes solutions and the
slender-vortex solutions at every axial station. It should
be emphasized here that the Navier-Stokes solutions for
this quasi-axisymmetric flow have been obtained by using
the three-dimensional solver on a three-dimensional grid.

Having verified the Navier-Stokes solver, the next
problem to consider is the supersonic vortex flow in a
configured circular duct

Supersonic Quasi-Axisymmetric Vortex in
a Configured Circular Duct

Figure 3 shows a configured circular duct which con-
sists of a straight cylindrical part at the inlet that is fol-
lowed by a short, diverging cylindrical part At x =
0.75 and beyond, the duct radius is kept constant and a
convergent-divergent nozzle with a throat radius of 0.95
is attached. The overall dimensions of the duct is 1 x 2.90.
This configured duct ensures that the inlet supersonic flow
will becomes supersonic at the exit Moreover, the con-
vergent part near the inlet ensures the stability of the
formed shock in the inlet region. This configured duct has
also been used by Delery, et al l 1 for their Euler equations
computations in an attempt to computationally model an
experimental set up. It should be pointed here that the Eu-
ler equations, used by Delery, et. al, assume isenthalpic
flow in order to drop the energy equation. This is a seri-
ous approximation since the upstream flow is a rotational
flow. Moreover, as our present calculations show, the
flow is actually unsteady and hence, the isenthalpic as-
sumption is not valid.

The Navier-Stokes solver is used to compute this flow
case by using a grid of 200 x 51 on two meridian planes,
where the 200 points are in the axial direction and the 5 1
points are in the radial direction. The grid is clustered at
the center line (CL) and at the wall. It is also clustered
in the diverging part near the inlet The two meridian
planes are spaced circumferentially at a certain angle so
that the aspect ratio of the minimum grid size will be less
than 2. The upstream Mach number is M* = 1.75 and
the Reynolds number for the Navier-Stokes computations
is 10*. The initial profile for the tangential velocity is
given by

U 0

1 — exp

- 0 ]

(ID

where Uoc ■ 1.74, r* ■ 0.2 and k, = 0.1. The maximum
^5- is at r a 0.224 and it is equal to 0.32. The radial
velocity, v, at the initial station is set equal to zero and
the radial momentum equation is integrated to obtain the
initial pressure profile. Finally, the density p is obtained
from the definition of the speed of sound for the inlet
flow. With these compatible set of profiles, the computa-
tions for both the Navier-Stokes equations and the Euler
equations start The exit boundary conditions are obtained
by extrapolation from the interior since the flow is super-
sonic at the exit The wall boundary conditions follow the

3

typical Navier-Stokes and Euler equations solid-boundary
conditions. These computations have been carried out on
the CRAY YMP of the NAS-Ames computational facil-
ities. The CPU time is 30 ps/grid point/iteration for the
Navier-Stokes calculation and 20 ^s/grid point/iteration
for the Euler equations.

a. Pseudo-Time-Stepping Solutions

Figure 4 shows the pseudo-time stepping solutions of
the Navier-Stokes (NS) equations on the left and the Euler
(E) equations on the right Each column in the figure
shows flow properties at the center line, the total Mach
contours, the streamlines throughout the duct and a blow
up of the streamlines in the vortex breakdown region.

The figure of properties along the center line of the
NS solution shows a strong shock at the inlet. Behind
the shock, the pressure and density sharply increase and
the axial velocity decreases to a negative value (upstream
flow) at x = 0.10. The axial velocity becomes more
negative one more time at x = 0.4 indicating the formation
of another bubble. At x = 0.7, the axial velocity becomes
positive and it continuously increases till the duct exit.
The Mach number contours of the NS solutions shows
the shock system near the inlet. The shock at the center
line is a normal strong shock, then it becomes an oblique
strong shock, again it becomes a normal strong shock, and
at the wall it becomes an oblique weak shock (supersonic-
supersonic flows upstream and downstream of the shock).
It is seen that the oblique shock at the wall is followed
by a separation bubble (see the streamline figure) which
is due to the shock and the divergence of the duct at
this location. The streamlines figure of the NS solution
shows a very large vortex-breakdown bubble and the
blow-up figure of the streamlines shows another small
bubble upstream of the large one.

The figure of Euler solution shows similar vortex
breakdown features with a few differences from the NS
solution. These differences are due to the absence of vis-
cous forces. The figure of properties along the center line
of the E solution shows a strong shock at the inlet. Behind
the shock, the pressure and density sharply increase to a
level higher than that of the NS solution. The veloc-
ity decreases to a negative value which is less than that
of the NS solution. The Mach-number contours of the E
solution show the stock system near the inlet. The shock
at the center line is a normal strong system, then it be-
comes an oblique weak shock and at the wall it becomes
a strong normal shock where there is no stock induced
separation. Another normal stock develops at x » 0.61,
where the axial velocity becomes substantially negative.
The streamline figure stows three vortex-breakdown bub-
bles; two small counter rotating bubbles and a third large
bubble. The size of vortex-breakdown bubbles of the E
solutions is larger than those of the NS solution.

It should be stressed here that this is the first time,
that we know of, such solutions have been presented for
supersonic vortex breakdown.

b. Time-Accurate-Stepping Solutions

It has been noticed during the pseudo-ume stepping
solutions that the residual-error dropped two orders of
magnitude and then it went through oscillations. It is
then decided that time-accurate-stepping solutions must
be checked. The same problem was recalculated using the
NS equations and E equations with At = 0.005. Figures
5—8 show snap shots of the time accurate solutions of
the NS equations on the left and the E equations on the
right. The snap shots are shown every 400 time steps.
The figures show the streamlines (Fig. 5), blow-up of
streamlines in the breakdown region (Fig. 6), total Mach-
number contours (Fig. 7) and flow properties at the center
line (Fig. 8).

Following the snap shots of NS streamlines (Figs. 5
and 6), we see a large bubble forming at the center line
at the time t = 4. At t = 6, the bubble expands in the
upstream and lateral directions. During this time period t
= 4-6, the Mach contours (Fig. 7) show the shock system
at the inlet moving in the upstream direction. At t = 8 ,
two bubbles appear and are convected in the downstream
direction. The Mach contours show that the shock moves
upstream and reaches the inlet as a normal strong shock.
At t = 10, a new vortex breakdown occurs producing
new small bubbles which combine to form a large bubble
at t * 12. It should be noted here that the bubble at
t = 12 resembles that at t = 4. This suggests that the
vortex breakdown process is almost periodic. To show
the periodicity of the breakdown, one has to pick up
the exact corresponding snap shots which are one period
apart This search is underway and it will be shown in
the near future. The solutions at t * 12, 14 and 16 show a
trend of repetition of the breakdown process. It should be
noted that the separation bubble at the wall goes through
a periodic process of convection and reproduction. The
Mach contours in the period of t = 10-16 show that the
shock system moves in the downstream direction again.
Figure 8 shows snap shots of the corresponding properties
variations along the center line. It shows the shock
motion and the motion of the negative values of the axial
velocity.

Following the snap shots of the E equations, we see
that a large vortex appears at the center line at t = 4.
At = 6, the vortex grows up and extends laterally and
upstreams. At t * 8, it is convected downstreams and
another vortex appears behind the shock near the inlet.
Figure 7 stows that the shock system near the inlet moves
upstreams in the period of t * 4-8. The convection
process and production of new vortices behind the shock
continue thereafter (Figs. 5 and 6) while the shock system
moves downstreams. It should be noted that the motion
of the shock system of the E solutions is larger than that
of the NS solutions. The reason is the absence of the
viscosity and hence the flow slips at the wall. Moreover,
there is no separation bubble at the divergent part of the
channel. The flow properties at the center line show the

4

motion of the shock system and the motion of the negative
values of the axial velocity.

Effect of Increasing Swirl Ratio

Next, the flow conditions and_duct dimensions are
kept constant while the initial swirTratio is increased to
3 = 0.38. The pseudo-time-stepping results are shown in
Fig. 9. It is seen that the number of vortex breakdown
bubbles increases to three instead of the two bubbles of
the previous case, Fig. 4. Moreover, we notice that the
shock system of the present case is nearer to the inlet in
comparison with the shock system of the previous case.
Figure 10 show snap shots of the time-accurate-stepping
solutions of this case at t * 5, 12.5 and 15.5. Again,
we see the vortex breakdown process of production and
convection and the associated oscillation of shock sys-
tem. The solution shows larger size and more number
of bubbles in comparison with those of the previous case
(Figs. 5-7). The time step of this case is the same as that
of the previous one. At = 0.005.

Concluding Remarks

The unsteady, compressible full Navier-Stokes equa-
tions are used to compute and analyze compressible and
supersonic quasi-axisymmetric isolated vortices. First,
the three-dimensional Navier-Stokes solver has been
verified by solving for a subsonic, isolated quasi-
axisymmetric vortex in an unbounded domain. The re-
sults have been compared with those of a slender-vortex
solver and they are in excellent agreement Second, the
three-dimensional Navier-Stokes and Euler solvers are
used to solve for a supersonic quasi-axisymmetric vor-
tex in a configured circular duct The duct is designed
such that the inflow and outflow conditions are super-
sonic. The quasi-axisymmetric solution is obtained by
forcing the flowfield vector to be equal on two merid-
ian planes in close proximity of each other. For the first
time, we have obtained supersonic vortex breakdown so-
lutions behind a shock. The time-accurate solution of
the problem shows that the vortex breakdown bubbles
and the shock system ahead of them are time dependent
The solution strongly indicates that the vortex breakdown
process and the motion of the shock system are periodic.
The Euler solution shows larger size and more number
of bubbles than those of the Navier-Stokes solution. The
Euler solution also shows that the amplitude of the shock
oscillation is larger than that of the Navier-Stokes solu-
tion. Increasing the initial swirl ratio shows that the size
and number of vortex-breakdown bubbles increase. These
results are vital for the mixing process in scramjets and
their design for the best performance and efficiency. A
very careful parametric study is underway to show the ef-
fects of the swirl ratio, Mach number, Reynolds number
and relative dimensions of the duct. Three-dimensional
solutions are currently being developed.

Acknowledgements

This research work is supported by the NASA Lang-
ley Research Center under Gram No. NAG- 1-994. The
authors would like also to acknowledge the computational
time provided through a grant from the NAS-Ames com-
putational facilities.

References

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Leading-Edge Vortices: Some Observations and Dis-
cussion of the Phenomenon,” Aeronautical Research
Council, RAM 3282, 1961.

2. Sarpkaya, T., “Vortex Breakdown in Swirling Conical
Flows,” AIAA Journal, Vol. 9, No. 9, Sept. 1971,
pp. 1791-1799.

3. Leibovich, S., "Vortex Stability and Breakdown Sur-
vey and Extension,” AIAA Journal, Vol. 23, No. 9,
Sept 1984, pp. 1194-1206.

4. Escudier, M. P. and Zender, N., “Vortex Flow
Regimes,” Journal of Fluid Mechanics, Vol. 115,
1982, pp. 105-122.

5. Grabowslri, W. J. and Berger, S. A., “Solutions of
the Navier-Stokes Equations for Vortex Breakdown,”
Journal of Fluid Mechanics, Vol. 75, Part 3, 1976,
pp. 525-544.

6. Hafez, M., Kuruvila, G. and Salas, M. D„ “Numeri-
cal Study of Vortex Breakdown," Journal of Applied
Numerical Mathematics, No. 2, 1987, pp. 291-302.

7. Menne, S., “Vortex Breakdown in an Axisymmetric
Flow,” AIAA 88-0506, January 1988.

8. Menne, S. and Liu, C. H„ “Numerical Simulation
of a Three-Dimensional Vortex Breakdown,” Z. Flug-
wiss.Welttraumforsch.14, 1990, pp. 301-308.

9. Spall, R. E., Gatski, T. B. and Ash, R. L., “The Struc-
ture and Dynamics of Bubble-Type Vortex Break-
down,” Proc. R. Soc., London, A429, 1990, pp.
613-637.

10. Krause, E., “Vortex Breakdown: Physical Issue and
Computational Simulation,” Third International Con-
gress of Fluid Mechanics, Cairo, Egypt, January 1990,
VoL 1. pp. 335-344.

11. Delery, J., Horowitz, E., Leuchter, O. and Solignac,
J. L., “Fundamental Studies of Vortex Flows,” La
Recherche Aerospatiale, No. 1984-2, 1984, pp.
1-24.

12. Metwally, O.. Settles, G. and Horstman, C., “An Ex-
perimental Study of Shock Wave/Vortex Interaction,”
AIAA 89-0082, January 1989.

13. Liu, C. H., Krause, E. and Menne, S., “Admissible
Upstream Conditions for Slender Compressible Vor-
tices,” AIAA 86-1093, 1986.

5

14 . Copening, G. and Anderson, J., "Numerical Solutions
to Three-Dimensional Shock/Vortex Interaction at Hy-
personic Speeds,” AIAA 89-0674, January 1989.

15. Kandil, O. A. and Kandil, H. A., “Computation
of Compressible Quasi-Axisymmetric Slender Vortex
Flow and Breakdown,” IMACS 1st International Con-
ference on Computational Physics, University of Col-
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in the Journal of Computer Physics Communications,
North Holland Publishing Co., 1991.

16. Meadows, K., Kumar, A. and Hussaini, M., “A Com-
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6

Vartleil OlvUnca

Figure 4. Comparison of steady flow solutions using Navier-Stokes and Euler Eqs.; properties
variation at centerline, Mach contours, streamlines, blow-up at breakdown, M * 1.75,
0 = 0.32, R, = 10 4 .

9

I

Navier-Stokcs solutions Euler solutions

Figure 5. Comparison of unsteady flow streamlines using Navier-Stokes and Euler Eqs., M =
1.75, 0 * 0.32, R 3 = 10 4 , At » 0.005.

10

Navier-Stokes solutions Euler s,,^^

Figure 6. Comparison of unsteady flow blow-up of streamlines using Navier-Stokes and Euler
Eqs., M * 1.75, 0 = 0.32, R, =■ 10 4 , At » 0.005.

11

Navicr-Stokes solutions Euler solutions

Figure 7. Comparison of unsteady flow Mach contours using Navier-Stokes and Euler Eqs., M
* 1.75, $ = 0.32, R« * 10 4 , At = 0.005.

12

Navter-Stokes solutions

Euler solutions

n = 800, t = 4

n = 1200, t = 6

n = 1600, t = 8

n = 800, t

n = 1200, t = 6

n = 1600,1 = 8 '*

n = 2000, t = 10

n = 2000, t = 10

n = 2400, t = 12

2400, t = 12 ’ *

Figure 8. Comparison of unsteady flow f 4
properties at centerline using Navier-Stokes
and Euler Eqs., M = 1.75, 0 = 0.32, *

R« = 10 4 , At = 0.005.

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