Transition between regular and Mach shock reflections in plane overexpanded jets

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AIAA 2002-0977

Transition Between Regular and Mach ShockReflections in Plane Overexpanded Jets

A.N. Kudryavtsev, D.V. Khotyanovsky, and M.S. Ivanov

Institute of Theoretical and Applied Mechanics,

Russian Academy of Sciences, Siberian Branch

Novosibirsk 630090, Russia

A. Hadjadj and D.Vandromme

Institut National des Sciences Appliquées de Rouen

UMR CNRS 6614/ CORIA, St. Etienne du Rouvray

76801, France

40th Aerospace SciencesMeeting & Exhibit

14-17 January 2002 / Reno, NV

AIAA 2002-0977

TRANSITION BETWEEN REGULAR AND

MACH SHOCK REFLECTIONS IN PLANE

OVEREXPANDED JETS

A.N. Kudryavtsev

?

, D.V. Khotyanovsky

y

, and M.S. Ivanov

z

Institute of Theoreti al and Applied Me hani s

Russian A ademy of S ien es, Siberian Bran h

Novosibirsk 630090, Russia

A. Hadjadj

x

and D. Vandromme

{

Institut National des S ien es Appliqu�ees de Rouen

UMR CNRS 6614/ CORIA, St. Etienne du Rouvray

76801, Fran e

This paper presents omputational results on

the transition from regular to Ma h re e tion

in a plane supersoni jet operating under over-

expanded onditions. It is assumed that the jet

is exhausting from an idealized nozzle providing

a uniform ow at the nozzle exit. First, invis-

id simulatons are performed by solving numer-

i ally 2D Euler equations. The results demon-

strate that a hysteresis phenomenon is observed

as the jet/ambient pressure ratio de reases and

in reases ausing, at �rst, the transition from

regular to Ma h re e tion and, after that, the

ba k transition. The angles of forward and ba k

transitions are lose to the theoreti al deta h-

ment and von Neumann riteria, respe tively.

Further, Navier-Stokes omputations with

the k� " turbulen e model are ondu ted in or-

der to investigate the transition in a more re-

alisti situation. The omputations on�rm the

existen e of the hysteresis and give the transi-

tion angles in very good agreement with both

the theoreti al riteria and the results of invis-

id simulations. In fa t, turbulen e produ tion

is essentially on entrated in the jet shear layer

?

Senior Resear h S ientist, Computational Aero-

danami s Lab.

y

Junior Resear h S ientist, Computational Aerody-

nami s Lab.

z

Professor, Head of Computational Aerodynami s

Lab., Asso iate Fellow AIAA

x

Resear h S ientist, Laboratoire M�e anique des Flu-

ides Num�erique

{

Professor, Head of Laboratoire M�e anique des Fluides

Num�erique

Copyright

by the Ameri an Institute of Aeronauti s

and Astronauti s, In ., 2001. All rights reserved.

and seems to be negligible in the jet ore. It

justi�es the invis id simulations when studying

sho k wave re e tion in jets.

Introdu tion

The aerodynami study of supersoni jets ex-

hausting from onvergent-divergent nozzles is a

problem of great importan e in many spa e and

aeronauti al appli ations. Various physi al phe-

nomena involved in this problem are dire tly linked

to the performan e of jet engines. Though o�-

design operations with either overexpanded or

underexpanded exhaust ow indu e performan e

losses, in many ases su h regimes annot be

avoided. The imperfe t mat hing between the am-

bient pressure and the exit nozzle pressure leads to

the formation of a ompli ated sho k wave stru -

ture. Passing through the system of sho k waves,

the ow gradually adapts to the ambient ondi-

tions. For several de ades (see, for instan e, [1℄)

numerous experimental, numeri al and analyti al

investigations of the stru ture of supersoni jets

have been undertaken, but the subje t is quite

ompli ated and not yet learly understood.

In re ent years, an important progress was

a hieved in our understanding of fundamental as-

pe ts of the transition between regular and Ma h

sho k wave re e tions. It was revealed, both nu-

meri ally [2℄ and experimentally [3℄, that su h a

transition is a ompanied by a hysteresis. Namely,

if the in iden e angle of the sho k wave is varying,

the transition to Ma h re e tion and the ba k tran-

Ameri an Institute of Aeronauti s and Astronauti s

1

a) b) )

��

��

��

��

MS

RR

RR

SS

SS

IS

IS

M > 1

M > 1M < 1

EF

EF

��

��

��

��

MS

SS

SS

IS

IS

M > 1

M > 1M < 1

EFRR

RREF

��

��

��

��

IS

IS

M > 1

MS

RR

RR

M < 1 M > 1

SS

SS

JBJB

JBJB

EF

EF

Fig. 1 Three types of ows where the transition between regular and Ma h re e tions is observed:

a) ow between two wedges; b) ow in hannel with ramp; 3) overexpanded jet ow. Here IS is

in ident sho k wave, RS is re e ted sho k, JB is jet boundary, MS is Ma h stem, SS is slipstream

surfa e, and EF is expansion fan.

sition to regular re e tion are observed at di�erent

angles, as it was earlier onje tured in [4℄.

Some typi al on�gurations, where the transi-

tion between regular and Ma h re e tions is ob-

served, are shown in Fig. 1. The ow around two

symmetri al wedges (Fig. 1a) was used in most

part of experimental stidies on sho k wave re e -

tion transition in steady ows. The ow in a han-

nel with a ramp (Fig. 1b) an be onsidered as

a prototype of ows in real supesoni inlets. For

both these ows, it is well-do umented now [5℄ that

the transition to Ma h re e tion o urs at the in-

ident sho k wave angle � equal to the so- alled

deta hment angle �

d

and the ba k transition | at

� equal to the von Neumann angle �

N

. The an-

gles �

d

and �

N

are the theoreti al riteria dedu ed

from the analysis of sho k wave re e tion using

pressure-de e tion diagrams [6℄. Regular re e -

tion is theoreti ally impossible above �

d

, whereas

Ma h re e ton is not possible below �

N

. At high

Ma h numbers, these two angles bound an interval

of the in ident sho k wave angles (the dual solu-

tion domain, see Fig. 2), where both regular and

Ma h re e tions an exist. Numeri al experiments

give lear eviden e that the transition to Ma h re-

e tion o urs when, at in reasing the angle, the

upper boundary of the domain is rossed and the

reverse transition is observed when, at de reasing

the angle �, the lower boundary is rea hed. Thus,

two di�erent types of re e tion an be obtained

at the same angle within the dual solution domain

and the hange in the sho k wave on�guration is

a ompanied by a hysteresis phenomenon.

The sho k wave re e tion transition is also a

salient feature of sho k wave intera tions in su-

personi imperfe tly expanded jets. The variation

of the pressure ratio between the jet and ambient

28

30

32

34

36

38

40

42

44

2 2.5 3 3.5 4 4.5 5 5.5 6 6.5

Regular

reflection (RR)

Mach reflection (MR)

M

Dual solution domain

(RR + MR)

α

α

αΝ

d

Fig. 2 Theoreti al transition riteria; in ident

sho k wave angles as fun tions of Ma h number.

spa e hanges the in iden e angle of either the noz-

zle lip sho k (for overexpanded onditions) or the

barrel sho k (for underexpanded jets). It an be

assumed that the hysteresis phenomenon should

also o ur in su h ows. The simplest geometri al

on�guration for this ase is shown in Fig. 1 and

orresponds to a plane overexpanded jet. Here, in

ontrast to axisymmetri jets and a plane underex-

panded jet, both the in ident sho k wave IS and

the jet boundary JB (before its intera tion with

the re e ted sho k wave RS) are straight. Hen e,

the omparison with the theoreti al riteria an be

made dire tly.

Note, there exist some experimental [7,8℄ and

numeri al [9℄ eviden e of the hysteresis phenomena

in underexpanded jets, however, it is diÆ ult to say

if they refer to the same type of hysteresis.

The goal of the present paper is to investigate

numeri ally the sho k wave re e tion transition in


2

a plane overexpanded jet in order to establish def-

initely whether the transition is a ompanied by

the hysteresis. At �rst, the Euler simulations are

arried out with a high-order WENO (Weighted

Essentially Non-Os illatory) s heme. Further, a

se ond order TVD (total variation diminishing)

s heme is used to ondu t Navier-Stokes ompu-

tations with the k � " turbulen e model. It allows

us to investigate a more realisti ase of a high-

Reynolds-number turbulent jet and try to elu i-

date the impa t of vis osity and turbulen e on the

sho k wave re e tion transition.

Euler Simulations

Problem Formulation and Numeri al

Method

Sho k wave intera tion is primarily an invis id

phenomenon and the attempt to reprodu e them

on the basis of the Euler equations seems to be

natural. The e�e ts of vis osity and heat ondu -

tivity negle ted, the equations governing the ow

of a ompressible ideal uid are:

�

�t

�+

�

�x

j

�u

j

= 0 (1)

�

�t

�u

i

+

�

�x

j

(�u

i

u

j

+ pÆ

ij

) = 0 (2)

�

�t

�E +

�

�x

j

[(�E + p)u

j

℄ = 0: (3)

Here �, u

j

(j = 1; 2), p are the density, the ve-

lo ity omponents, and the pressure, respe tively;

the total energy per unit mass E for the perfe t

gas with the onstant spe i� heat ratio (= 1.4

for air) is the sum of internal and kineti energies:

E = e+

u

2

+ v

2

2

=

p

( � 1)�

+

u

2

+ v

2

2

: (4)

The 5th order �nite di�eren e WENO s heme

[10℄ was utilized to solve Eqs. (1{4) numeri ally.

The WENO s hemes are very appropriate for the

problem under onsideration be ause they have the

property of robust sho k apturing and provide

high a ura y in the regions where the solution

is smooth. The global Lax-Friedri hs splitting was

applied when al ulating numeri al uxes and the

3rd order TVD Runge-Kutta s heme was used to

advan e the solution in time.

The nozzle exit was taken as a part of the left

boundary of the omputational domain. All quan-

tities are �xed at the nozzle exit pres ribing a uni-

form supersoni ow with the jet Ma h number M

j

= 5 and the stagnation temperature equal to that

of the ambient uid. The jet pressure p

jet

was var-

ied in order to hange the angle of the nozzle lip

sho k �. The remaining part of the left boundary

was treated as a solid wall, and the re e tion pro e-

dure was used to spe ify the variables in the ghost

ells outside the omputational domain. On the far

�eld (upper) boundary, the density and the pres-

sure were taken to be equal to the given ambient

uid values, the streamwise velo ity was equal to

zero, and the normal-to-the-boundary velo ity was

not �xed but was determined by extrapolating the

Riemann invariant from the interior of the domain.

\Soft" boundary onditions were imposed on the

right (out ow) boundary putting the streamwise

derivatives of all quantities to zero. Owing to the

ow symmetry, only a half of the real jet ow was

omputed and the lower boundary was treated as

a symmetry line. The grid ell size in all omputa-

tions presented below was su h that 100 ells were

lo ated a ross the nozzle half-width h=2.

Results of Invis id Simulations

The omputations were started for the

jet/ambient pressure ratio p

jet

=p

ambient

or-

responding to the in ident sho k angle � = 41

Æ

.

It is noti eably higher than the deta hment angle,

whi h is equal to �

d

= 39:3

Æ

. Consequently,

Ma h re e tion is only possible in this ase. The

ambient onditions in the entire omputational

domain were taken as the initial ow�eld for this

omputation. After the omplex transient pro ess

of start-up of the jet ow, the ow began evolving

to the steady state, and a onverged solution was

�nally a hieved. It is shown in Fig. 3a. In ea h

subsequent omputation, the onvergent ow�eld

of the pre eding omputation was used as initial

data. The variation of jet pressure imposed as

a boundary onndition on the nozzle exit orre-

sponded to a 2

Æ

hange in the nozzle lip sho k

angle. The Ma h re e tion was preserved when

in reasing p

jet

until the value � = 31

Æ

was rea hed

(see Figs. 3 and 3d). It is in good agreement with

the theoreti ally predi ted von Neumann angle

�

N

= 30:8

Æ

. In numeri al simulation, an earlier

transition to regular re e tion an be expe ted

be ause it is impossible to resolve a very small

Ma h stem whose height is omparable with the

grid ell size.


3

a) � = 41

Æ

b) � = 39

Æ

) � = 33

Æ

d) � = 31

Æ

e) � = 33

Æ

f) � = 39

Æ

Fig. 3 The omputed ow�elds (numeri al s hlieren pi tures) of overexpanded jet ow at M

j

= 5.

After that, the pressure ratio was de reased.

The re e tion remained regular over all dual so-

lution domain, and the transition to Ma h re-

e tion o ured when the angle of in iden e was

hanged from 39

Æ

to 41

Æ

(Figs. 3f and 3a). It

again agrees with the theoreti al deta hment an-

gle. Thus, an evident hysteresis phenomenon was

observed: within the dual solution domain both

regular and Ma h re e tions an be obtained de-

pending on the initial onditions of the omputa-

tion. This hysteresis is very similar to that ob-

served earlier in numeri al simulations of the ow

around two symmetri al wedges and the ow in a

onverging hannel.

The omputed ow�elds show substantially dif-

ferent sho k wave stru tures for regular and Ma h

re e tions at the same value of p

jet

j=p

ambient

:

ompare Fig. 3b and 3f. In Fig. 4, the pressure dis-

tributions along the enterline for these two sho k

wave on�gurations are ompared. In the ase

of regular re e tion, the jet boundary is strongly

urved, and intensive ompression waves are fo-

used to the enterline, ausing the se ondary in-

rease in pressure, in addition to the primary peak

in the re e tion point of the nozzle lip sho k. For

Ma h re e tion, the only strong pressure rise is ob-

served just behind the Ma h stem. Further down-

stream, the pressure de reases as the ow a el-

erates again to supersoni velo ities and later os-

illates in a periodi al system of ompression and

rarefa tion waves within the jet ore bounded by

two slip surfa es emanating from the triple points.

Important quantities for pra ti al appli ations

are the position of the Ma h stem and its size.

Therefore, several semi-analyti al models were de-

veloped to predi t these quantities for overex-

panded jets [11,12℄. A tually, the Ma h stem

height is determined by the intera tion between

the expansion fan (EF in Fig. 1 ) and the slip


4

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5x/h

0.0

1.0

2.0

3.0

4.0

5.0

6.0p/p_ambient

regular reflection Mach reflection

Fig. 4 Pressure distributions along the enter-

line for regular and Ma h re e tions at � = 39

Æ

.

surfa e SS originated from the triple point. The

expansion fan, whi h is generated as a result of

the intera tion of the re e ted sho k wave RR

with the jet boundary JB, bends the slip surfa e

so that a virtual nozzle with liquid boundaries is

formed. The relations between quantities in the

entran e plane of the nozzle ( oin iding with the

Ma h stem) and the nozzle throat, where the ow

is soni , ontrol the position and the size of the

Ma h stem. This me hanism, responsible for the

existen e of the steady Ma h re e tion on�gura-

tion, is essentially the same that was learly de-

s ribed in [13℄ for the ase of re e tion of a wedge-

generated sho k wave.

The Ma h stem heights s measured from our nu-

meri al ow�elds are presented in Fig. 5. It is

worth noting a nonlinear behavior of s at angles

lose to �

N

. It seems that the extrapolation of the

urve should give a more a urate value for the

angle of transition to regular re e tion about 32

Æ

.

Also, a more a urate estimation of the transition

angles will be done in the next se tion using the

data from Navier-Stokes simulations.

Navier-Stokes Simulations

Governing equations

Let

�

f denote a ommon, Reynolds-averaged,

value of a quantity f , while

e

f is a mass-averaged

(Favre) value:

e

f =

�

�f=�� and, at last, f

00

= f �

e

f .

Then the Favre-averaged Navier-Stokes equations

for a ompressible turbulent uid ow an be writ-

ten as follows:

α = 39.7

30 32 34 36 38 40 420.00

0.10

0.20

0.30

0.40EulerNavier−Stokes

α

s/h

Fig. 5 Dependen e of nondimensional Ma h

stem height s=h on the in ident sho k angle

.

�

�t

��+

�

�x

j

��eu

j

= 0 (5)

�

�t

��eu

i

+

�

�x

j

(��eu

i

eu

j

+ �pÆ

ij

� ��

ij

) = 0 (6)

�

�t

��

e

E +

�

�x

j

h�

��

e

E + �p

�

eu

j

� ��

ij

eu

i

� �q

j

i

= 0: (7)

The tensor ��

ij

is a sum of vis ous and turbulent

stresses

��

ij

= (��+ �

t

)

�

�eu

i

�x

j

+

�eu

j

�x

i

�

2

3

�eu

k

�x

k

�

�

2

3

�kÆ

ij

;

(8)

and the turbulent heat ux is

�q

j

=

�

��

Pr

+

�

t

Pr

t

�

�e

�x

j

+

�

t

�

k

�k

�x

j

(9)

The two-equation k�" model is used to evaluate

the turbulent vis osity �

t

via the turbulent kineti

energy k and dissipation ". Details of the turbu-

len e model used are given in the next subse tion.

Turbulen e model

The k � " model is the most widely known

and extensively used two-equation eddy vis osity

model. It was originally developed to improve the

mixing-length model and to avoid the algebrai de-

s ription of the turbulent length s ale in omplex


5

ows [16℄. Di�erent versions of this model an

be found in the literature [18℄. This model gives

reasonably good results for free-shear-layer ows.

For wall bounded ows, the model provides good

agreement with experimental results for zero and

small mean pressure gradients but is less a urate

for large adverse pressure gradients [18℄.

When omputing turbulent free jet ows, the use

of the standard k� " model is an appropriate way

to orre tly predi t the mean and the turbulent

ow parameters. However, for high-speed ows,

it is important to in lude ompressible dissipation

and pressure-dilatation e�e ts in the two-equation

turbulen e models as suggested by Sarkar et al.

[20℄, Sarkar [19℄ and Vandromme [22℄.

In this study, we used an improved version of the

k�" turbulen e model to a ount for ompressibil-

ity e�e ts [17℄, [19℄, [20℄.

The modeled equations for k and " are written

as:

� Turbulen e energy transport equation:

D(�� k)

Dt

| {z }

Transport

=

�

�x

j

��

�+

�

t

�

k

�

�k

�x

j

�

| {z }

Diffusion

+ P

k

|{z}

Produ tion

� �� ("

s

+ "

)

| {z }

Destru tion

+ p

00

d

00

| {z }

Pressure�dilatation

(10)

where k is the turbulent kineti energy per unit

mass de�ned as k =

1

2

g

u

00

i

u

00

i

=

1

2

[

g

u

00

2

i

+

g

v

00

2

i

+

g

w

00

2

i

℄,

"

and p

00

d

00

represent the ontributions due to

ompressible dissipation and pressure-dilatation,

respe tively.

� Energy dissipation transport equation:

D(�� "

s

)

Dt

=

�

�x

j

��

�+

�

t

�

"

�

�"

s

�x

j

�

+ C

"1

"

s

k

P

k

�C

"2

��

"

2

s

k

(11)

in whi h the turbulent vis osity is expressed as

�

t

= C

�

��

k

2

"

and P

k

= ��

g

u

00

i

u

00

j

�

� eu

i

�x

j

�

is the exa t turbulent

kineti energy produ tion. Here, ��

g

u

00

i

u

00

j

is the

Reynolds stress, de�ned as follows:

��

g

u

00

i

u

00

j

= �

t

�

�eu

i

�x

j

+

�eu

j

�x

i

�

2

3

Æ

ij

�eu

l

�x

l

�

�

2

3

Æ

ij

�k

The model onstants are given by C

�

=0:09,

C

"1

=1:44, C

"2

=1:92, �

k

=1:0 and �

"

=1:3

Based on dire t numeri al simulation of isotropi

turbulen e and homogeneous shear ows, Sarkar et

al. [20℄ proposed the following model for pressure-

dilatation:

p

00

d

00

= ��

2

��P

k

M

2

t

+ �

3

�� "

s

M

2

t

:

Here M

t

=

p

k

is the turbulent Ma h number and

is the speed of sound.

The dissipation rate of turbulent kineti energy

ontains two parts, the solenoidal (in ompressible)

and dilatational ( ompressible) parts, "

s

and "

,

respe tively,

" = "

s

+ "

where "

s

is omputed from the in ompressible form

of " (equation 11) and "

is assumed to be a fun -

tion of "

s

and the turbulent Ma h number M

t

:

"

= �

1

M

2

t

"

s

:

Sarkar [20℄ re ommends �

1

= 0:5, �

2

= 0:4, and

�

3

=0:2.

Numeri al te hnique and boundary ondi-

tions

The turbulen e transport equations are similar

to the transport equations of the mean ow. For

the numeri al omputation of ea h term of the tur-

bulen e transport equations, it is re ommended to

use the same treatment as the one used for the

mean ow equations. In this study, the govern-

ing equations for the mean ow and turbulen e

quantities were integrated using an expli it se ond-

order �nite volume s heme. For the onve tive

terms, the upwind TVD s heme of Harten and Yee

[21℄ with van Leer's limiter was used and entral-

di�eren e methods were employed for the di�usion

terms of the momentum, energy, and turbulen e

equations. A se ond-order Runge-Kutta s heme

was used for the time mar hing.

The average omputational time for ea h simu-

lation was approximately 30 h CPU (global time)

using 10 pro essors on a parallel omputer (ORI-

GIN 2000). In all the simulations, the CFL number

was �xed at 0:8. The free-stream onditions were:

M

1

' 0; P

1

= 101322 Pa; T

1

= 300K

Low free-stream values (nearly zero) of the tur-

bulent kineti energy k and the dissipation rate "

were �xed near the free-stream at rest. These val-

ues were kept onstant for all the simulations. For


6

� = 31

Æ

� = 33

Æ

� = 33

Æ

� = 37

Æ

� = 37

Æ

� = 39

Æ

� = 39

Æ

� = 40

Æ

Fig. 6 Hysteresis phenomenon during the transition between regular and Ma h on�gurations in

a supersoni overexpanded turbulent jet. Ma h number ontours for M = 5 and for di�erent values

of �.


7

turbulent free-shear ows, the k� " model showed

no sensitivity to free-stream turbulen e [15℄.

The boundary onditions are of di�erent type.

Due to the symmetry nature of the problem, only

half of the jet was omputed. Symmetry bound-

ary onditions were applied along the axis of sym-

metry. Non-re e tive boundary onditions with a

�xed value of the stati pressure were used along

the outer boundary orresponding to the external

free-stream. A part of the inlet of the omputa-

tional domain oin ides with the nozzle exit. Su-

personi boundary onditions are pres ribed on it.

The rest of the inlet boundary is assumed to be

a solid wall. For the turbulen e transport equa-

tions, either zeroth-order extrapolation or free-

stream values are used for k and " along the outer

boundaries. If the ow is outgoing along the outer

boundary, zeroth-order extrapolation is used. If

there is ow entrainment, then free-stream values

are imposed along the outer boundaries. At the in-

ow, the nondimensional turbulent kineti energy

is de�ned as K

�

�

p

k=U

jet

' 2%, where U

jet

is

the velo ity of the jet. On e k is known, " is ob-

tained using the produ tion-equals-dissipation hy-

pothesis.

The size of the omputational domain is L

x

=2h

in the streamwise dire tion and L

y

=h in the ross-

streamwise dire tion (h is the nozzle width). The

omputations employed 500�250 equally spa ed

points. Grid-independent results were obtained us-

ing this mesh; the height of the Ma h stem was

used as a riterion to obtain grid-independent so-

lution for M=5 and �=42

Æ

.

Results of turbulent omputations

A series of turbulent jet omputations at M=5

were arried out for a set of pressure ratios orre-

sponding to in ident sho k wave angles � ranging

from 31

Æ

to 42

Æ

, whi h overs the whole dual solu-

tion domain.

Figure 6 shows the evolution of the jet stru ture

when in reasing and de reasing the jet/ambient

pressure ratio p

jet

=p

ambient

�

. The omputation

was started from the ase where only regular re-

e tion is possible (� = 31

Æ

). After onvergen e,

the pressure-ratio was progressively de reased step

by step until a sudden transition to Ma h re e -

tion observed at �=39:7

Æ

. This angle is very lose

to the theoreti al value of �= 39:3

Æ

obtained us-

ing the deta hment riterion. The ba k transition

was observed when hanging the sho k wave angle

�

in jet al ulations the pressure ratio is the most impor-

tant parameter ontrolling the ow-�eld.

from �=32

Æ

to �=31

Æ

. Note that for �=32

Æ

, we

still have MR with a small visible Ma h stem (see

Fig. 7). This angle slightly ex eeds the value of

�=30:9

Æ

theoreti ally predi ted by the von Neu-

mann riterion.

The dependen e of the normalized Ma h stem

height, s=h, on the in ident sho k wave angle, �,

for M=5 is shown in Fig. 5. The hysteresis e�e t

at in reasing and de reasing the jet/ambient pres-

sure ratio is evident. It should be noted that the

Ma h stem heights measured in the Navier-Stokes

and Euler simulation are surprisingly lose to ea h

other.

The normalized turbulent kineti energy k=U

2

jet

ow�eld is shown in Fig. 8 for both regular and

Ma h re e tion on�gurations. For both ases,

there is no turbulen e on the jet axis (potential

ore). Turbulen e produ tion o urs essentially in

the high shear region of the jet edge.

A detailed analysis of the ow-�eld during the

RR!MR transition reveals the existen e of a sub-

soni po ket downstream of the re e ted sho k

when � approa hes the value of 39

Æ

. This phe-

nomenon is depi ted in Fig. 9 for di�erent values

of �. Ea h pi ture orresponds to a steady state

solution. In fa t, when � is lose to 39

Æ

, a subsoni

po ket takes pla e just at the intera tion between

the re e ted sho k and the jet boundary. The size

of this po ket grows progressively towards the jet

symmetry line when � is in reased and ontami-

nates lo ally the ow downstream of the re e ted

sho k. The transition to Ma h re e tion o urs

when the downstream re e ted sho k ow be omes

entirely subsoni .

Figure 10 illustrates a high-shear layer instabil-

ity orresponding to the Kelvin-Helmholtz vorti es

emanating from the triple point. These vorti es

have a short life-time and a t essentially during

the transient pro ess when in reasing or de reas-

ing the pressure ratio. They ompletely disappear

when the steady state solution is rea hed. The

possible in uen e of these vorti es on the transi-

tion between regular and Ma h re e tions or on

the stability of either Ma h or regular on�gura-

tions is yet to be studied and understood.

Finally, some words should be said on ern-

ing the possibilty of experimental on�rmation of

the hysteresis phenomenon desribed above. As is

known [5℄, the observation of the hysteresis for the

two-wedge ow depends strongly on the level of

disturban es in the wind tunnel used. A high level

of disturban es an lead to an earlier transition to

Ma h re e tion and even to a omplete absen e of

the hysteresis phemonena. It is diÆ ult to predi t


8

Fig. 7 Numeri al al ulation of the Ma h number ontours of Ma h re e tion wave on�guration

for M =5 and �=32

Æ

. Note the existen e of a small visible Ma h stem with a normalized size of

s=h ' 0:015

Fig. 8 Turbulent kineti energy ow�eld normalized by the square of enterline jet velo ity U

jet

in the ase of regular re e tion at � = 31

Æ

(left) and Ma h re e tion at � = 42

Æ

(right).

Fig. 9 Detailed numeri al results for the ow Ma h number ontours of the turbulent jet and

for three di�erent values of the in ident sho k wave angle, �, at the beginning of the transition

RR!MR.

Fig. 10 Kelvin-Helmholtz instability along the slip surfa e emanating from the triple point at the

transient pro ess when hanging the value of � from 35

Æ

to 33

Æ

(Numeri al S hlieren pi ture).


9

a priori whether the overexpanded jet ow is a

more or less noisy obje t and, onsequently, more

or less suitable for experiments on the hysteresis

at the sho k wave re e tion transition. An addi-

tional problem an be that at high Ma h numbers

the nozzle lip sho k is rather strong and an indu e

a boundary layer separation inside the nozzle that

should destroy the gasdynami s heme onsidered

in this paper.

Con lusion

The sho k wave re e tion transition in an over-

expanded supersoni jet at the Ma h number M

j

=

5 has been numeri ally simulated. First, the Eu-

ler omputations have been performed, whi h show

that the transition from regular to Ma h re e tion

and the ba k transition o ur in agreement with

the theoreti al deta hment and the von Neumann

riteria, respe tively.Thus, the dependen e of the

sho k re e tion type on initial onditions and the

hysteresis phenomenon have been observed.

Further, numeri al investigations of the tran-

sition between regular and Ma h re e tions in

steady turbulent overexpanded jets have been per-

formed using the two-equation k � " model mod-

i�ed to a ount for ompressibility e�e ts. Very

good agreement of the omputational results with

the theoreti al riteria of transitions has been

found as well as with the results of the invis id

simulations. Turbulen e produ tion was on en-

trated within the jet boundary layer and did not

a�e t the jet ore ow substantially. As a result,

the hysteresis e�e t was also observed at in reasing

and de reasing the jet/ambient pressure ratio. The

numeri al study reveals that, when the jet pressure

ratio is suÆ iently low, subsoni onditions appear

downstream of the re e ted sho k for the regular

re e tion on�guration. To better understand this

phenomenon, whi h probably triggers the transi-

tion from regular to Ma h re e tion, further ana-

lyti al and numeri al studies for solving the prob-

lem of the intera tion between the re e ted sho k

and the jet boundary should be undertaken. Also,

further investigations should onsider the e�e t of

the nozzle boundary layer and its possible separa-

tion on the sho k wave on�guration.

A knowledgements

All turbulent omputations were performed on

the parallel omputer (ORIGIN 2000) of the CRI-

HAN (Centre de Ressour es Informatiques de

HAute Normandie, Rouen). The authors would

like to a knowledge CRIHAN for providing om-

putational resour es. The support of INTAS grant

N

Æ

99-0785 is gratefully a knowledged. The Rus-

sian authors are also grateful to the Russian Foun-

dation for Basi Resear h for support under Grants

No. 00-01-00824 and 01-07-90189.

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Transition between regular and Mach shock reflections in plane overexpanded jets

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Transcript of Transition between regular and Mach shock reflections in plane overexpanded jets