[american institute of aeronautics and astronautics 43rd aiaa fluid dynamics conference - san diego,...

American Institute of Aeronautics and Astronautics

1

Three-Equation K-ε-Vn Turbulence Model for High-Speed

Flows

Alexander M. Molchanov1 and Leonid V. Bykov

2

Moscow Aviation Institute (State University of Aerospace Technologies), Moscow, 125993, Russia

A turbulence model K-ε-Vn for high-speed compressible flows is developed. It is based on

modeling the rapid part of pressure-strain correlation depending on turbulent Mach number

and on the assumption that the velocity fluctuations normal to streamline play a key role in

turbulent mixing process. For the validation of the code the described numerical procedures

are applied to a series of high-speed jet flow problems. These include high-speed plane mixing

layers, fully expanded heated free jets, cold under-expanded and over-expanded air jets,

hypersonic compression ramp, base-bleed experiments with a cylindrical afterbody in

supersonic flow, dual-mode scramjet/ramjet combustor. Comparison of the simulation with

available experimental data showed a good agreement for the above problems.

Nomenclature

a Speed of sound ij

δ Kronecker delta tensor

b Mixing-layer thickness ηδ Spreading rate of the jet

K Turbulent kinetic energy ε Dissipation rate

M Mach number εij Dissipation rate tensor

MG Gradient Mach number µT Eddy (turbulent) viscosity

Mr Relative Mach number ν Kinematic viscosity

MT Turbulent Mach number Πij

Pressure-strain correlation tensor

VnM Turbulent Mach number based on

nV ′′

( )P

ijΠ

Incompressible rapid pressure-strain correlation

p Pressure ρ

Density

P Production of turbulent kinetic energy u

σ Deviation of streamwise Reynolds stress

Pij Production tensor v

σ Deviation of cross-stream Reynolds stress

ui Velocity components Subscripts

nV ′′ Velocity fluctuation normal to the streamlines

0 Total values of parameters

T Temperature a Nozzle exit

.ijk kT Turbulent transport tensor C Centerline value

xi Cartesian axes e External flow, ambient

y0 Mixing-layer Centerline

I. Introduction

T is well known that the growth rate of turbulent kinetic energy is critically reduced with the increase of turbulent

Mach number. Compressibility in high-speed flows has a stabilizing effect on turbulence so that the intensity of

turbulent mixing reduces as Mach number increases.

A typical supersonic effect is a drastic reduction in the growth rate of a plane free-shear layer flow. The reduction

was confirmed in a number of experiments under various conditions1-5

.

This effect plays an important role in present-day problems of rocket and airspace engineering. For example, in a

supersonic combustion ramjet reduced turbulence levels can be highly detrimental as they reduce the rate of fuel and

1 Associate Professor, Aerospace Heating Engineering Department, 4, Volokolamskoe Shosse, Moscow, Russia,

Email: [email protected], Senior Member AIAA 2 Senior Researcher, Aerospace Heating Engineering Department, 4, Volokolamskoe Shosse, Moscow, Russia.

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43rd Fluid Dynamics Conference

June 24-27, 2013, San Diego, CA

AIAA 2013-3181

Copyright © 2013 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.

Fluid Dynamics and Co-located Conferences


2

oxidizer mixing. Compressibility changes the nature of laminar-turbulent boundary-layer transition over hypersonic

vehicles during re-entry.

Compressible turbulence modeling is an essential element for calculations of many problems of practical

engineering interest, such as combustion, environment and aerodynamics.

However, it is not so unique. In some cases compressibility makes a very small impact on turbulence. For example,

the experimental data2,13,14

show that the axial turbulence intensity actually do not depend on Mach number.

Compressibility has a very little impact on the intensity of the turbulence just near the wall.

In particular, the local density extensions of standard incompressible turbulence models were found to be

inadequate in duplicating the experimentally observed reduction in growth rate of the mixing layer with increasing

Mach number.

The results of Sarkar6,11

, Simone et al.7, Zeman

8 and Molchanov

9,10 for compressible turbulent flows showed that

the compressibility effects are associated with the dilatational terms such as the dilatation dissipation and the pressure-

dilatation correlation.

But later the studies of Vreman et al.12

confirmed that the dilatational terms couldn’t be regarded as essential in

causing the reduced growth rate of turbulent kinetic energy. They pointed out that compressibility was found to affect

the production term via the pressure-strain correlation.

Besides, the models based on supplementing additional dissipation to the equation of turbulent kinetic energy are

unable to predict the increasing turbulence anisotropy. The experiments2,13,14

showed that with increase of Mach

number the cross-stream turbulence fluctuations are affected in a greater extent than the streamwise ones.

The increasing anisotropy of turbulence is confirmed by data of Blaisdell et al.15, Simone et al.7 and Freund et

al.16

.

Siyuan Huang and Song Fu17

showed that the main compressibility effect came from the reduced pressure–strain

term due to reduced pressure fluctuations. A damping function of turbulent Mach is used to take account of the

compressible effects. The pressure–strain correlation in compressible flows is corrected by the damping function of

turbulent Mach number.

Actually, the modeling of the pressure-strain correlation in compressible turbulence articulates on the simple

extension of models established in incompressible turbulence as mentioned in the works of Adumitroaie et al.18

,

Hamba19

and Marzougui et al.20

.

Adumitroaie et al.18

have developed a compressible correction depending on the magnitude of the turbulent Mach

number to the Launder, Reece and Rodi model21

. They proposed a model due to a conjunction between the traditional

techniques used by Launder, Reece and Rodi to integrate the Poisson’s equation for the pressure and the continuity

constraint.

Marzougui H.20

extended the incompressible Launder, Reece and Rodi model21

for the pressure-strain correlation

to compressible turbulent flow, in which the C1, C3 and C4 coefficients become dependent on the turbulent Mach

number.

Park C.H. and Park S.O.22

used the concept of moving equilibrium in homogeneous shear flow to modify the linear

pressure strain term part.

In Ref.23 the rapid and slow parts of pressure-strain are simulated separately depending on Mach number.

Gomez and Girimaji24

suggested that compressibility impact on turbulence only manifests itself via the rapid part

of pressure-strain. They use a rapid pressure-strain correlation model developed within their research group over the

last few years that is consistent with rapid distortion theory (RDT)25,26

. They developed an algebraic Reynolds stress

model for compressible flows by accounting for the changing nature of pressure at different gradient Mach number

MG regimes and showed that the major impact of compressibility on turbulence is implemented on large-scale levels

and is associated with the fact that the action of pressure is quite different at low and high-speed regimes. Pressure-

strain correlation scrambles the streamwise and stream-normal fluctuations leading to a low turbulent shear stress and

decreased production.

The formulas were obtained for pressure dilatation depending on gradient Mach number and production24

.

The results of rapid distortion theory (RDT) obtained by Girimaji et al.24-26

show that the effect of pressure has

three different regimes depending on Mach numbers:

1) In low speed flows pressure assumes the role of enforcing incompressibility and is governed by a Poisson

equation. A standard incompressible pressure-strain correlation without any modification can be used (denoted as ( )P

ijΠ ).

2) For an intermediate Mach numbers, both inertial and pressure terms are of the same order of magnitude:

ij ijPΠ ≈ . It is shown in Ref.24 that this regime leads to a stabilization of the turbulent kinetic energy growth rate.

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3) At very high Mach numbers pressure plays an entirely negligible effect compared to inertial terms such as ij

P .

Pantano and Sarkar27

showed that for high speeds the finite speed of sound causes a time delay in the transmission of

pressure signals in the flow. Therefore in this regime we may neglect the dominant terms in the pressure-strain

correlation model: ( )2

0ijΠ ≈ .

The present paper aims:

1) To develop a turbulence model for high-speed compressible flows on the basis of modeling of the rapid part of

pressure-strain correlation depending on turbulent Mach number

2) To obtain explicit formulas for all the components of Reynolds stress tensor assuming that velocity fluctuations

normal to the streamline play the key role in the mechanism of turbulent mixing process.

3) To come up with a versatile model that will be applicable to both free shear flows (jets, mixing layers, etc.) and

the near-wall ones.

II. Turbulence Model

IRST, let us consider a case of free-shear flows where we can neglect the impact of the wall.

The transport equations for the transport of the Reynolds stresses �

i ju u′′ ′′ at high Reynolds numbers may be written

as follows28

:

� �

.i j k i i ijk k ij ij ij

k

u u u u u T Pt x

ρ ρ ρε∂ ∂ ′′ ′′ ′′ ′′+ = + + Π − ∂ ∂

� , (1)

where:

( )., ,

,

j i

ijk k k i j i jk j ik kj i ki j ij i k j k

k k k

j ji i

ij ij jk ik

j i k k

u uT u u u u u p u p u P u u u u

x x x

u uu up p

x x x x

ρ τ τ δ δ ρ ρ

ρε τ τ

∂∂ ∂′′ ′′ ′′ ′′ ′′ ′′ ′′ ′′ ′′ ′′ ′′′ ′= − + + − − = − +

∂ ∂ ∂

′′ ′′′′ ′′∂ ∂∂ ∂′ ′′ ′Π = + = +

∂ ∂ ∂ ∂

� �

(2)

Dissipative termij

ρε , taking into account local isotropy, is modeled as

2

3ij ij

ε εδ= (3)

Pressure–strain correlation ij

Π is divided into the slow ( )1

ijΠ and rapid

( )2

ijΠ parts

( ) ( )1 2

ij ij ijΠ = Π + Π (4)

It is assumed that the slow part ( )1

ijΠ , which has the physical sense of the tendency towards isotropy and is

associated with small-scale turbulence, may be expressed by the formula, which is valid for an incompressible fluid28

:

( )

�1

1

2

3

i j

ij ij

u uC

Kρε δ

′′ ′′Π = − −

(5)

where constant 1

1.8C = (Ref.28).

The rapid part ( )2

ijΠ is associated with large-scale turbulence. This work is based on the supposition that the

largest contribution toward the growth inhibition of mixing layers comes from it.

The results of rapid distortion theory (RDT) obtained by Girimaji et al.24-26

show that the following formula can be

used for the rapid part of pressure-strain correlation ( )2

ijΠ

( ) ( )2

1 2,

P

ij ij ijC C P

Π ΠΠ = Π − (6)

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where 1 2,C C

Π Π are functions depending on the gradient Mach number:

1

2

1

2

1

2

1,

0

0,

1

0,

:

:

:0

g

g

g

C

C

C

C

C

C

M

M

M

α

α β

β

Π

Π

Π

Π

Π

Π

=

=

→

→

=

<

≤ ≤

<

=

(7)

where ,α β - some constants.

This work used the turbulent Mach number 2 /T

M K a= as a basis criterion for taking account of

compressibility influence on turbulence or, more precisely, the turbulent Mach number based only on one component

of turbulent kinetic energy:

�2 /

nVn VM a′′= (8)

(The reason for that will be given below)

That is why the functions 1 2,C C

Π Π presented by Girimaji et al.

24-26 were transformed to dependencies on MT and

adjusted according to the best agreement of the simulation results with the experimental data.

Functions that approximate these dependences on MT are shown in Figure 1.

Figure1. Dependence of coefficients 1CΠ and

2CΠ on turbulent Mach number.

For modeling ( )P

ijΠ the simple formula is used28

:

( )

2

2

3

P

ij ij ijC P PδΠ = − −

, (9)

where constant 2

0.6C = (Ref.28).

By taking the trace of equation (1) we obtain the transport equation for turbulent kinetic energy K:

( ) ( ) ( ). 2

1k k k

k

K u K T P Ct x

ρ ρ ρεΠ

∂ ∂+ = + − −

∂ ∂� , (10)

where 1

2

i

ii i k

k

uP P u u

xρ

∂′′ ′′= =

∂

�

- production on of turbulent kinetic energy.

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Thus, dilatational terms in the equation for turbulent kinetic energy are expressed via ( )2 TC M PΠ ⋅ - the product of

function of turbulent Mach number and production.

It is supposed that equation for turbulent dissipation rate ε has the standard form and is slightly modified from its

standard form to be consistent with considering a suppressing effect of compressibility on production:

( ) ( ) ( )[ ]1 2 2

1T

k

k k k

u C P C Ct x x x K

ε ε

ε

µ ε ερε ρ ε µ ρε

σΠ

∂ ∂ ∂ ∂+ = + + − −

∂ ∂ ∂ ∂

� (11)

Thus, a closed system of equations which allows for defining the Reynolds stress in high-speed flows by solving

the corresponding partial differential equations is obtained.

However, it is complicated to solve this set of equations. Six Reynolds stress transport equations have to be solved

simultaneously, which is a mathematical challenge. Besides, there is an obvious difficulty in specifying the boundary

conditions of the six Reynolds stress. Based on some approximation, Reynolds stress partial differential transport

equations may be simplified into algebraic expressions.

In this work explicit algebraic stress model (ASM) is built based on the following suppositions.

Main simplification is obtained by using the assumption that the velocity fluctuations normal to the streamline n

V ′′

play the key role in the mechanism of turbulent mixing process.

Let's consider a 2D flow in which the coordinate with a subscript "1" is directed along the streamline and the index

"2" denotes the coordinate normal to the streamline. Obviously, the following is true in this case:

2 1 1 2 3

1 2 3

, , , , 0, 0f f f

u u f u u ux x x

∂ ∂ ∂= = =

∂ ∂ ∂� �� (12)

Then, from (2):

1 1 1

12 2 2 22 11 1 2 1 2

2 2 2

, 0, 2 ,u u u

P u u P P u u P u ux x x

ρ ρ ρ∂ ∂ ∂

′′ ′′ ′′ ′′ ′′ ′′≈ − ≈ ≈ − ≈ −∂ ∂ ∂

� � �

(13)

Using formulas (1), (3), (5), (6), (9) and assumption (13), we obtain the following equation for � �2 2

2 nu V′′ ′′= :

�( ) �( )� �

( )2 2

2 2

1 2 1 1

2 21

3 3

T n

k K k

n

n k n

k

V

x x

VV u V C C P C C

t x K

µµ

σρ ρ ρε

′′

′′ ′′Π

∂ ∂+

∂ ∂

′′ ∂ ∂ + = + − + − ∂ ∂

� (14)

In the equation for the shear stress, the source in this case is equal to

( )� �

�

1 1 212 2 1 2 2 2 1

2

1u u u

S C C C u u Cx K

ρ ρεΠ Π

′′ ′′∂′′ ′′= − − − −∂

(15)

Hossain29

and Ljuboja, Rodi 30

showed that when making algebraic stress model we can neglect convective and

diffusive transfer in the differential equation for shear stress, i.e. 12 0S ≈ . Thus, from formula (15) we obtain an

explicit expression for the shear stress

� ( )�2

1 2 2 2 2 1

1 2

1 2

1 C C C u u uKu u

C K xε

Π Π′′ ′′− − ∂

′′ ′′ = −∂

�

(16)

From (16) we obtain a formula for the shear stress in the familiar form:

� 1

1 2

2

T

uu u

xρ µ

∂′′ ′′ = −

∂

�

, (17)

where the eddy viscosity is given by

( )�2 2

1 2 2

1

1n

T

C C C V K

C Kµ ρ

ε

Π Π′′− −

= , (18)

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It is assumed that the eddy viscosity obtained via formula (18) can be applied to all components of Reynolds stress,

and in general case the following formula is true:

� 2 2

3 3

ji m

i j T ij T ij

j i m

uu uu u K

x x xρ µ δ µ δ ρ

∂∂ ∂′′ ′′− = + − −

∂ ∂ ∂

�� , (19)

Following our suppositions, compressibility impact on turbulence is mainly caused by �2

nV ′′ rather than the whole

turbulent kinetic energy. Thus, it is reasonable to use as a basic criterion instead of the turbulent Мах 2 /T

M K a=

Mach number associated with the cross-stream velocity fluctuations:

�2 /

nVnVM a′′= (20)

The appropriate dependences ( ) ( )1 2

,Vn Vn

C M C MΠ Π

are similar to those in Fig. 1.

Thus, a three-parameter turbulence model which includes three partial differential equations for K , ε and �2

nV ′′ and

explicit formulas for Reynolds stress is obtained.

High-Reynolds-number turbulent viscosity models need strong corrections to be applicable down to solid

boundaries.

The condition of local isotropy is broken near the wall. The dissipation tensor is quite anisotropic near the wall, so

it is necessary to use Rotta’s model

�

i j

ij

u u

Kε ε

′′ ′′= (21)

instead of formula (3). Modeling of pressure-strain correlation also requires a near-wall correction.

Staying within K-ε turbulence model, the model of Durbin31

which used relaxing elliptical function to take account

of the wall influence seems most interesting.

However when using v2-f model for complex flows, there appears problems with boundary conditions on the wall.

Simulation results are sensitive to the near-wall grid clustering.

That is why this work applied the approach based on Elliptic Blending model (EBM) of Manceau and Hanjalic32

.

A single scalar equation for elliptic blending function α is solved

2

2

21

i

Lx

αα

∂− =

∂, (22)

and it is used for all stress components. Here

1/43/2 3

max ,L

KL C Cη

ν

ε ε

=

(23)

The pressure strain term and the stress dissipation are modeled by blending the “homogeneous” (away from the

wall) and the near-wall models32

( ) ( ) ( )2 21h w

ij ij ijα αΠ = Π + − Π , (24)

( )�

2 221

3

i j

ij ij

u u

Kε α εδ α ε

′′ ′′= + − (25)

On the wall α=0, far from the wall α=1.

For homogeneous pressure-strain correlation ( )h

ijΠ formulas (5), (6) and (9) are used

( )

�

1 1 2 2

2 2

3 3

h i j

ij ij ij ij ij

u uC C C P P C P

Kρε δ δΠ Π

′′ ′′ Π = − − − − −

(26)

The wall model for the pressure strain, satisfying the exact wall limit and stress budget is defined by33

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( ) � � �( )1

52

w

ij i k k j j k k i k l k l i j iju u n n u u n n u u n n n nK

ρεδ

′′ ′′ ′′ ′′ ′′ ′′Π = − + − +

(27)

where the unit wall-normal vector is evaluated from /α α= ∇ ∇n

Similar to the case of free shear flows let’s imagine that axis «1» is directed along the wall, and axis «2» is

directed normal to the wall. In this case

1 2 30, 1, 0n n n= = = (28)

and components ( )w

ijΠ are

( ) � ( ) � ( ) �

( ) �

11 2 2 22 2 2 33 2 2

12 1 2

5 5, 5 ,

2 2

5

w w w

w

u u u u u uK K K

u uK

ρε ρε ρε

ρε

′′ ′′ ′′ ′′ ′′ ′′Π = Π = − Π =

′′ ′′Π = −

(29)

Using the above formulas, we obtain the expression for sources in Reynolds stress equations

( )�

( )

�

2

2

1 1 2 2

1

2 2 2

3 3 3

i j w

ij ij ij ij ij ij

i j

ij ij ij ij ij

u uS P P

K

u uC C C P P C P

K

ρε α ρε

α ρε δ δ ρεδΠ Π

′′ ′′ = − + Π = − − − Π

′′ ′′ + − − − − − −

(30)

The trace of these sources equals

( ) ( ) ( )2 2

22 1 1 2

w

ii iiS P Cα α ρεΠ= − + − Π − , (31)

where trace

( ) � �( )

� �15 2 3 5 2 2 0

2

w

ii i k k i k l k l i i i k k i k l k lu u n n u u n n n n u u n n u u n nK K

ρε ρε ′′ ′′ ′′ ′′ ′′ ′′ ′′ ′′Π = − − + = − − =

(32)

So the source in turbulent kinetic energy equation equals

( )2

2

11

2K ii

S S P Cα ρεΠ= = − − (33)

Basing on (30) and considering that � �2 2

2 nu V′′ ′′= , we obtain the following formulas

�

( ) �2

2 2 2

22 1 2 1

2 2 26 1 ,

3 3 3

nn

VS C C P C V

K K

ρεα ρε α

′′′′

Π

= − − + − −

(34)

( ) ��

( )�

2 2 2 2 21 1 212 1 2 2 1

2

1 6 1n

u u uS C C C V C

x Kα α ρ α α ρε′′

Π Π

′′ ′′∂ = − − − − − + ∂

(35)

Considering assumptions of Hossain29

and Ljuboja, Rodi30

(12 0S ≈ ), we obtain the explicit formula for the shear

stress, similar to (16) with the wall corrections:

� ( )

( )� �

2 2

1 2 2 2 11 2 2 2

21

1

6 1n

C C C K uu u V

xC

α αρ ρ

εα α

Π Π− − ∂′′ ′′ ′′= −∂− +

(36)

Thus, the basis equatations are as follows:

( ) ( ) ( )2

21

T

k

k k K k

KK u K P C

t x x x

µρ ρ µ α ρε

σΠ

∂ ∂ ∂ ∂+ = + + − −

∂ ∂ ∂ ∂

� , (37)

( ) ( ) ( )2

2 21

11

T

k

k k k

u P C Ct x x x T

Cε

ε

ε

µ ερε ρ ε µ α ρε

σΠ

∂ ∂ ∂ ∂+ = + + − −

∂ ∂ ∂ ∂

� , (38)

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�( ) �( )� �

( )2 2

2 2

1 2 1 1

2 21

3 3

T n

k K k

nn k n

k

V

x x

VV u V C C P C C

t x K

µµ

σρ ρ ρε

′′

′′ ′′Π

∂ ∂+

∂ ∂

′′ ∂ ∂ + = + − + − ∂ ∂

� , (39)

Turbulent viscosity is determined by formula

( )

( )

�2 2 2 21 2 2

2 2

1

1

6 1

n

T

C C C V K

KC

α αµ ρ

εα α

′′Π Π− −=

− + (40)

As before, formula (19) is used for the components of Reynolds stress tensor.

The transport equation for ε (38) is also corrected considering the wall influence - coefficient 1Cε is function of

relation �2/n

K V ′′ 34

:

� ( )2

1 1.4 1 0.05 / 0.4exp 0.1n t

C K V Rε′′ = + + −

, (41)

where 2 /tR K εν= .

Furthermore, for time-scale we use the following expression34

1/ 2

max ,6 ,K

Tν

ε ε

=

(42)

instead of the ratio /K ε .

The other coefficients are equal to:

1 2 21.8, 0.6, , ,

0.23, 70

1 1.3 1.92,

L

KC C C

C Cη

ε εσ σ= =

= =

= = = (43)

Boundary conditions at the wall ( )0y → :

�2

20, 2n

KK V

yε ν′′= = → (44)

The analysis of the presented formulas shows that when approaching the wall the compressibility impact on

turbulent kinetic energy and dissipation rate weaken as 2 0α → . The major influence of compressibility on �2

nV ′′

appears much stronger, as the source in equation (39) is not dependent on α. Thus, �2

nV ′′ reduces when approaching the

wall much faster than K. Because the turbulent Mach number is calculated via �2

nV ′′ (formula (20)), compressibility

does not actually affect on turbulence in the close distance to the wall. This result is proved by the analysis conducted

by Wilcox35

. He showed that the use of traditional corrections on compressibility (Sarkar, etc.) causes an unwanted

decrease in skin friction for the flat-plate boundary layer.

III. Simulation Results

ESTING the model involved comparing the simulation results using this model with the available experimental data

for various types of flow. The computer program UNIVERSE-CFD, developed in Moscow Aviation Institute36

,

was used for most simulations. Some simulations were carried out using ANSYS FLUENT (with user defined

functions, UDF) and CFX (with CEL).

Test 1. High-speed plane mixing layers

This test involved the comparison with the experimental data of Goebel, Dutton2 and data from Ref. 24. The

experimental setup of a two-dimensional mixing layer consists of a channel with two incoming streams separated by a

T

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splitter plate. The top stream is labeled as primary and the lower as secondary. The primary stream is chosen as the

high-speed inlet.

Seven mixing layer cases were examined in Ref.2. For each case, static pressures were measured, schlieren

photographs were taken, and flowfield velocity measurements were obtained using an LDV system. These data were

used to obtain growth rates and to examine the development of the mean and turbulent velocity fields of compressible,

turbulent mixing layers. The operating conditions for the seven cases that were examined are listed in Table 1. As is

clear from this table, a wide variety of conditions were studied with freestream velocity ratios ranging from 0.16 to

0.79, freestream density ratios ranging from 0.57 to 1.55, and relative Mach numbers ranging from 0.40 to 1.97.

Table 1

Case 1 1d 2 3 3r 4 5

r=U2/U1 0.78 0.79 0.57 0.18 0.25 0.16 0.16

2 1/s ρ ρ= 0.76 0.76 1.55 0.57 0.58 0.6 1.14

M1 , M2 2.01, 1.38 2.02,1.39 1.91, 1.36 1.96, 0.27 2.22, 0.43 2.35, 0.3 2.27, 0.38

T1, T2 [K] 163, 214 151, 198 334, 215 161, 281 159, 275 171, 285 332, 292

U1, U2 [m/s] 515, 404 498, 392 700, 399 499, 92 561, 142 616, 100 830, 131

p [kPa] 46 55 49 53 53 36 32

To assess the relative performance of the new presented model (K-ε-Vn) against the calculations of the standard

models with the following three turbulence models were performed:

1. K-ε - standard two-equation model without the compressibility correction.

2. K-ε cc - standard two-equation model with the Sarkar et al.37

compressibility correction.

3. K-ε-Vn - new compressible model presented in this paper.

The supersonic mixing layer is characterized with relative Mach number:

( )

1 2

1 2/ 2

r

U U UM

a a a

− ∆= =

+ (45)

Figures 2-7 compare the normalized similarity profiles of Reynolds stress, cross-stream turbulence intensity and

streamwise turbulence intensity with the experimental data2 for cases 4 and 5. The mixing-layer thickness b was taken

to be the distance between transverse locations where the mean streamwise velocity was U1-0.1∆U and U2+0.1∆U.

The y coordinate of the mixing-layer centerline is y0. The standard deviations of the Reynolds stresses are defined as

� �1 2 21

,u v

u u u uσ σ′′ ′′ ′′ ′′= = .

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Figure 2. Similarity profiles of normalized Reynolds stress for Case 4.

Figure 3. Similarity profiles of normalized cross-stream turbulence intensity for Case 4.

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Figure 4. Similarity profiles of normalized streamwise turbulence intensity for Case 4.

Figure 5. Similarity profiles of normalized Reynolds stress for Case 5.

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Figure 6. Similarity profiles of normalized cross-stream turbulence intensity for Case 5.

Figure 7. Similarity profiles of normalized streamwise turbulence intensity for Case 5.

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Figures 8-11 present the mixing-layer growth rates ( )/C

db dx , Reynolds shear stress � ( )( )2/

Cu v U′′ ′′ ∆ and

turbulence intensities ( ) ( )/ , /v uC C

U Uσ σ∆ ∆ normalized by the corresponding value obtained without

compressibility correction in incompressible mixing layers at the same freestream velocity and density ratios, with

respect to the relative Mach number.

Figure 8. Normalized mixing-layer growth rates vs. relative Mach number.

Figure 9. Normalized Reynolds shear stress vs. relative Mach number.

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Figure 10. Normalized cross-stream turbulence intensity vs. relative Mach number.

Figure 11. Normalized streamwise turbulence intensity vs. relative Mach number.

The results clearly show:

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1) Mixing-layer growth rate reduces due to compressibility. K-ε-Vn yields reduced mixing-layer spreading rates at

high relative Mach number, consistent with the experimental data2,24 . Incompressible models do not capture the

mixing inhibition.

2) The increase of Mach number leads to the reduction of shear stress (Figure 9) and to a considerable reduction

of cross-stream turbulence intensity (Figure 10) as well as a very slight change of streamwise turbulence intensity

(Figure 11). All these mean that compressibility, in the first place, impacts on velocity fluctuations normal to

streamline, and impacts on shear stress via the value �2

nV ′′ in accordance with formula (16); the impact on velocity

fluctuations along the streamline is small. The presented model K-ε-Vn accounts for those facts quite well, however,

the model of Sarkar et al.37

does not.

3) The incompressible models grossly overpredict the magnitudes of shear stress cross-stream turbulence

intensity.

Test 2. Fully Expanded Heated Free Jets ( ,a e a e

p p T T= = ).

This test aimed at validating the presented turbulence model for the simulation of jets, whose temperature, density

and pressure at the nozzle exit are the same as those in the ambient, i.e.

, ,a e a e a e

T T p pρ ρ= = =

This condition makes an estimate of a pure effect of compressibility on jet parameters. Simulations with K-ε, K-ε

cc37

and K-ε-Vn turbulence models were performed and compared.

Simulation results were compared with the experimental data of Lau et al.4 and Krasotkin et al.

5 .

Figure 12 shows the axial distribution of the mean velocity on the jet axis for different Mach numbers ( 0.28,a

M =

1.37a

M = ). The results exhibit a consistent trend in which the curves move downstream as the Mach number is

increased. Simulations with K-ε-Vn turbulence model are in a good agreement with the experimental data and also

show the increase of the jet length with the increase of a

M .

Figure 12. Normalized centerline velocity /C au u vs. normalized distance from nozzle exit / ax R .

Calculation results (curves) compared to simulation data of Lau et al.4 (symbols).

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Figure 13. Radial distribution of normalized streamwise turbulence intensity for 1.37a

M = at / 16a

x R =

Calculation results (curves) compared to simulation data of Lau et al.4 (symbols).

Figure 13 shows the radial distributions of the axial turbulence intensity. Calculation results using K-ε cc and K-

ε-Vn turbulence models are in a very good agreement with the experimental data of Lau et al.4 .

Figure 14. Relative coordinate 0.75X vs. nozzle exit Mach number aM . Experimental data of Refs. 4,5.

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It is difficult to determine experimentally the length of the isentropic zone with sufficient accuracy.

To estimate the jet length it is more convenient to use non-dimensional coordinate 0.75 0.75 /a

X X R= - a

normalized distance from nozzle exit corresponding to the relative velocity / 0.75C au u = (Ref. 5). It should be

noted that in accordance with the data of Ref. 4 the maximum of the turbulent fluctuations and of the gradient of the

velocity Cu is observed in the section 0.75X .

Figure 14 shows dependence of the relative coordinate 0.75X on nozzle exit Mach number aM . The simulation

results (curves) were compared with the experimental data of Refs. 4,5.

Figure 15. Spreading rate of the jet ηδ vs. nozzle exit Mach number aM . Experimental data of Refs. 4,5.

Figure 15 shows the variation of spreading rate of the jet ηδ with Mach number. Simulations with K-ε cc37

and

K-ε-Vn turbulence models indicate that the spreading rate of the mixing layer decreases with increasing Mach number

and are in a good agreement with the experimental data.

Test 3. Cold under-expanded and over-expanded air jets at / 1a e

p p ≠

This test aimed at validating the presented model for under-expanded and over-expanded air jets.

The simulation was performed for air jets having total temperature 0

300T K= and nozzle exit Mach number

3.3aM = . The simulation results were compared with the experimental data of Safronov and Khotulev38

.

Figures 16, 17 present the simulation results and the experimental data for an under-expanded jet with static

pressure ratio / 1.5a ep p = , diameter of the profiled nozzle 53.7aD mm= and nozzle exit half cone angle 10aθ = ° .

The simulation was performed using a 400x100 trapezoidal grid. Various turbulence models were used:

1) standard -K ε model;

2) -K ccε turbulence model with compressibility correction of Sarkar et al.37

;

3) K-ε-Vn turbulence model presented in this paper.

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Figure 16. Centerline distribution of normalized relative pitot pressure. 1 – Safronov, Khotulev

experiment38

; 2 – simulation with standard -K ε model; 3 – simulation using present model.

Figure 17. Centerline distribution of Mach number. 1– Safronov, Khotulev experiment

38; 2 – simulation

with standard -K ε model; 3 – simulation using present model

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Using standard -K ε turbulence model considerably under-predicts the jet length if compared with the

experimental data, and also reduces a number of shock diamonds; all shock waves have significantly lower amplitude

than measured.

Using the other turbulence models (2,3) gave similar results, which are in a good agreement with the experimental

data of Safronov, Khotulev38

. The pictures only illustrate the simulation results using models 1) and 3).

Test 4. Hypersonic compression ramp test case This simulation case corresponds to the flow over the 34° compression ramp (see Figure 18) with input parameters

identical to those in the experiments of Ref.39. A fully contoured nozzle provides a Mach 9 nitrogen gas open-jet test

flow.

Specifically, the key parameters are defined as follows: free-stream Mach number Me=9.22, free-stream

temperature Te=64.5 K, the unit Reynolds number Re/m=4.7×107, the wall temperature, Tw=295 K, and flat-plate

length L=0.76 m.

Figure 18. Pictorial view of model in gun tunnel working section (Ref.39).

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Figure 19. Skin pressures in isothermal wall 34º turbulent compression ramp. Experimental data of Ref.39,

Me=9.22, Tw=295 K.

In Figure 19, the SST model over-predicts the peak pressure, over-predicts the length of separation region caused

by the shock wave/turbulent boundary layer interaction. The pressure distribution calculated by the present model

yields adequate prediction of the peak pressure compared with experimental results in the reattachment region, even

though it yields a separation extent smaller than the measured one.

Test 5. Base-Bleed Experiments with a Cylindrical Afterbody in Supersonic Flow

The flow field near a blunt cylindrical body with base bleed aligned in a supersonic flow (Figure 20) is

investigated in this test. The supersonic free stream expands as it turns the corner while the turbulent boundary layer

separates and then undergoes recompression, realignment, and redevelopment in the wake of the underbody13

. Fluid

from the region adjacent to the base is entrained and accelerated by the outer shear layer and then returned to the base

region by a recompression shock system. This region is referred to as the primary recirculation region. Introducing

base bleed, the primary recirculation region is moved downstream from the aftbody with a forward stagnation point

created, dependent on the relative strengths of the bleed jet and the recirculating region. Experiments performed by

several investigators (see Ref.14 for a complete list) demonstrated an important effect of such a shift in the location of

the primary recirculation region—a change in the base pressure ratio and, as a result, a change in the aftbody drag.

Base bleed then is an effective mechanism for reducing aftbody drag.

The experimental flow conditions are as follows. Free stream static pressure Pe=28700 Pa, free stream Mach

number Me=2.47, tunnel stagnation temperature 300 K, bleed flow mass flow rate ratio I=0.01, base radius=31.75 mm,

bleed orifice radius =2.7 mm, bleed flow stagnation temperature 300 K.

Here I - dimensionless injection parameter:

1 1

bleed

b

mI

U Aρ=�

, (46)

where bleed

m� - bleed mass flow rate, 1ρ - freestream density, 1U - freestream velocity, b

A - base area.

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(a)

(b)

Figure 20. (a) Schematic diagram of supersonic base flow (Ref.13) and (b) pathlines colored by axial velocity

magnitude(m/s) for I=0.01.

Calculations with the following three turbulence models were performed:

1. K-ε - standard two-equation model without the compressibility correction

2. SST - the Menter Shear Stress Transport Turbulence Model40

which proved to be the best for near-wall flows

3. K-ε-Vn – a new compressible model presented in this paper.

Figures 21-23 show axial distributions of velocity and turbulence intensities.

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Figure 21. Mean axial velocities along the centreline. Experimental data of Ref.13.

Figure 22. Axial turbulence intensity along the centerline. Experimental data of Ref.13.

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Figure 23. Radial turbulence intensity along the centerline. Experimental data of Ref.13.

Figure 24. Normalized Reynolds shear stress vs. radial coordinate for downstream location x/R0=1.9.

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Figure 25. Normalized Reynolds shear stress vs. radial coordinate for downstream location x/R0=2.52.

Figure 26. Normalized Reynolds shear stress vs. radial coordinate for downstream location x/R0=3.34

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Figure 27. Normalized Reynolds shear stress vs. radial coordinate for downstream location x/R0=4.08

Comparing the simulation results with experimental data shows that, as well as in test 1, the compressibility has

more impact on the cross-stream turbulence intensity than on the streamwise one. The agreement of the simulation

results using the presented K-ε-Vn model with the experiment is at least no worse than that obtained using the SST-

model.

Test 6. Dual-mode scramjet/ramjet combustor

The validation of presented model K-ε-Vn for scramjet applications was an objective of this test. Results obtained

from the simulations are compared with the experimental data from the UVa Supersonic Combustion Facility (SCF)41

.

Comparisons are made in terms of static pressures along the centerline of the experimental facility.

A schematic of the combustor configuration is presented in Fig. 28. The flowpath consisted of a two dimensional

Mach 2 nozzle, a constant-area rectangular isolator, and a rectangular combustion duct. An unswept 10-deg

compression ramp injector was located on one of the walls of the combustor (this wall is referred to here as the

injection wall). The ramp had a height-to-width ratio of 0.5 and, at its highest point, formed a duct blockage of

approximately 8%. Hydrogen was introduced into the combustor from a Mach 1.7 conical nozzle in the ramp base.

The centerline of the nozzle was parallel to the ramp 10-deg surface.

Linear dimensions of the combustor configuration are detailed in Table 2. Unless stated otherwise, all linear

dimensions reported here are normalized in terms of the normal height of the ramp H (perpendicular to the injection

wall).

Table 2. Combustor geometry dimensions

Ramp height H =6.4 mm Isolator height 4H

Isolator width 6H Isolator length 2H

Combustion duct inlet height 4H Combustion duct width 6H

X distance to 2.9-deg surface 10H X extent of combustion duct 58H

Ramp length 6H Ramp width 2H

Ramp compression angle 10 deg Injector exit Y location 0.6H

Injector port diameter 0.4H Injection angle 10 deg

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Figure 28. Schematic of combustor configuration.

For the purpose of this test, a total of two cases were identified for numerical simulation (see Table 3)

Table 3

Cases Air H2 Fuel plenum

conditions

Ambient conditions

Scan 1 P0=327.72 kPa

T0=1201 K

ma=0.202 kg/s

----------- Patm =99.1 kPa

Tatm = 294.4 K

Scan 4 P0=326.97 kPa

T0=1202 K

ma =0.201 kg/s

P0=527.14 kPa

T0=298.86 K

mfuel =1.01e-3 kg/s

Patm =99.1 kPa

Tatm = 294.4 K

Figure 29 shows the distribution of pressure along the fuel injector wall at symmetry plane in case of Scan 1 (air

without fuel). The simulation results are compared with that of the experiment and show a satisfactory match between

numerical and experimental pressure distributions when the K-ε-Vn turbulence model is used. The pressure peak

caused by the fuel injector ramp at x/H = –5 matches well with the experiment.

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Figure 29. Pressure along the fuel injector wall at symmetry plane. Scan1 - no fuel. Experimental data of

Ref.41; 1- simulation result, K-ε-Vn model

The shock produced by the compression ramp propagates downstream to the combustor and the extender-nozzle.

These shock structures cause the peaks and valleys in the pressure profile. Figure 30 shows the core flow through the

combustor, and the extender-nozzle remains supersonic in the absence of the combustion. There is also the presence of

a recirculation region in the combustor immediately aft of the compression ramp.

Figure 30. Mach number contour at symmetry plane. Scan1 - no fuel, K-ε-Vn model

Figure 31 shows the simulation results compared with the experimental data with burning (mfuel =1.01e-3 kg/s). The

results are provided using K-ε model (curve 1) and K-ε-Vn model ( curve 2).

To simulate hydrogen burning in air Eddy Dissipation model was used with coefficients: A=2.5, B=0.5.

Both the experimental data and the numerical results show considerable pressure increase in the combustor to

compare with the case without burning. The simulation results using the K-ε-Vn model match much better with the

experimental data as compared to K-ε model.

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Figure 31. Pressure along the fuel injector wall at symmetry plane. Scan 4 - mfuel =1.01e-3 kg/s.

Experimental data of Ref.41; 1- K-ε model, 2 - K-ε-Vn

IV. Conclusion

N explicit algebraic stress model (ASM) is developed. This model is based on modeling the rapid part of pressure-

strain correlation depending on turbulent Mach number and on the assumption that the velocity fluctuations

normal to streamline play a key role in turbulent mixing process.

Presented K-ε-Vn turbulence model yields reduced mixing-layer spreading rates at high Mach number, consistent

with the experimental data. Incompressible models do not capture the mixing inhibition.

The turbulence parameters (Reynolds stress, cross-stream and streamwise turbulence intensities) obtained from

the presented model are in a good agreement with the data.

The turbulence model developed in this paper allows to account for the fact that compressibility, in the first place,

makes a suppressive effect on velocity fluctuations normal to the streamline. It leads to the increase of turbulence

anisotropy with the increase of Mach number.

For the validation of the code the described numerical procedures are applied to a series of high-speed jet flow

problems. These include:

• High-speed plane mixing layers,

• Fully Expanded Heated Free Jets,

• Cold under-expanded and over-expanded air jets,

• Hypersonic compression ramp,

• Base-Bleed Experiments with a Cylindrical Afterbody in Supersonic Flow,

• Dual-mode scramjet/ramjet combustor

Comparison of the simulation with available experimental data showed a good agreement for the above problems.

A

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Acknowledgments

HIS work was supported by Russian Foundation for Basic Research under contract No.13-08-01328. Authors

would like to acknowledge Prof. Peter Nikitin for his support through this project. Authors express thanks to Anna

Arsentyeva for her help with translation.

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pp. 538-546. 3Papamoschou, D. and Roshko, A., "The compressible turbulent shear layer: an experimental study," Journal of Fluid

Mechanics, Vol. 197, 1988, pp. 453-477. 4Lau J.C., Morris P.J., Fisher M.J., "Measurements in subsonic and supersonic free jets using a laser velocimeter, " Journal of

Fluid Mechanics, vol. 63, part 1, 1979, pp. 1-27. 5Krasotkin, V.S., Myshanov, A.I., Shalaev, S.P., Shirokov, N.N. and Yudelovich, M.Ya., "Investigation of supersonic isobaric

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