study with К־omega turbulence model for supersonique...

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STUDY WITH К־OMEGA TURBULENCE MODEL FOR SUPERSONIQUE FLOW

A. BEGHIDJA , H.GOUIDMI, R.BENDERRADJI

Laboratoire d’Energétique Appliquée et de Pollution Department of Mechanical Engineering, Faculté des sciences de l’ingénieur

Université Mentouri, Constantine ALGERIE

Abstract: Several numerical methods, built with implicit or explicit schemes, in curvilinear grid, were developed. They permit to solve the Navier-Stokes equations, in subsonic, supersonic and hypersonic regime, but with moderate Reynolds numbers. For the high Reynolds numbers, the LES with subgrid method appears to be best for these flows. In order to analyze the interaction phenomena boundary layers-Shock waves, and for to appears the effects of the numerical viscosity, some works were realized in Every-Val d’Essonne in France. The complex flows with high Reynolds numbers are now until predicted and analyzed with statistical models (RANS), and limited to the steady state. In this case, we use statistical K-ω turbulence model approach, for example. We analyze with this model, geometrics variations (Bérénice and Antarès), in order to determine there effects at the flow Finally, the results obtained from the energy balances seem to confirm some assumptions of the dissipative nature phenomena on the walls as well as the terms of production Keywords: k- ω turbulence model, compressible, nozzle, sub and supersonic, shock wave.

. 1. Introduction This work lies within the scope of a study led already to ONERA at ends of application in the aerospace one. We endeavour to treat the behaviour generated in the nozzle according to the various sonic modes of flow and varied geometries. Turbulence in compressible mode for example, was treating by J.Cousteix [1] to see the influence of compressibility on the turbulent boundary layer but in an analytical way. Favre and al.1976 [2] treated the compressible flows and the equations balanced by the density. Cline, Mr. C [3] who seized the development of data processing and the CFD (differences finished) and worked on the modelling of turbulence in the nozzle.

Morgans and its team [4] whose used the method of K-omega to study the turbulent compressible flow of a jet. Q.Xiao and H M. Tsai [5] studied the diffusion flow transonic by K-Omega Loh, C.Y. and Zaman [ 6] helped to develop the model of WILCOX (K – Omega) by their experimental study. Loh, C.Y.[7] numerically studied the compressible flow by using the finite element method. Anju K Bawa [8] used Fluent to study the reduction of the time of the design of a conduit aspiration industrialist has 50% of the real time. One uses the K-Omega of with dimensions and the experimental one of the other. Results obtained by simulation were more or less acceptable compared to other simulations made front.

Proceedings of the 8th WSEAS Int. Conference on Automatic Control, Modeling and Simulation, Prague, Czech Republic, March 12-14, 2006 (pp387-392)

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The objective of this study is to make the variation of the flowing in Bérénice and Antarès nozzle model of turbulence most used in simulation in CFD k–ω while applying for the nozzle, and in various modes (subsonic, supersonic, with shock wave) in the two-dimensional case and for two Cartesian frames of reference (x,y), axisymetric (x,r). we estimate that there are certain phenomena related to the behavior of the flow close to wall, in particular in the conduit collar. we planned to treat this case by an approach based on the function close to wall. Since the walls are the principal source of average vorticity and turbulence. This is possible with the model k–ω Indeed, the boundary layer can be subdivided in three zones. Internal the layer known as " under viscous layer " is almost laminar. The external layer fully turbulent logarithmic curve. Between the two a zone where the effects of molecular viscosity and turbulence are

significant. Consequently we introduced a +ω

term related to the height of the layer considered (y).

2.Transport equations The weighted averages by the mass, usually used in the studies of flow to several components, were particularly developed by FAVRE [ 2 ]. The mean velocity is defined by:

ρρ /~ii uu = Where the sign indicates a

conventional statistical average. In particular, the average value of ρ is: ρ . The fluctuations of density and speed are obtained by difference between the instantaneous values and their averages: ρρρ −=′ and

iii uuu ~−=′ Kinetic energy of turbulence k, and the specific rate of dissipation ω , is obtained following equations transport:

( ) ( ) ( ) kkkj

ki

ii

SYGxk

xuk

xk

t+−+

⎥⎥⎦

⎤

⎢⎢⎣

⎡

∂∂

Γ∂∂

=∂∂

+∂∂ ~ρρ (1)

( ) ( ) ( ) ωωωωρωωρ SYGxk

xu

xt jii

i

+−+⎥⎥⎦

⎤

⎢⎢⎣

⎡

∂∂

Γ∂∂

=∂∂

+∂∂ ~ 2)

Modeling of Effective Diffusivity

k

tk σ

µµ +=Γ ;

ωω σ

µµ t+=Γ ;

ωρ

αµk

t∗=

Modeling of the Production of Turbulence

i

jjik x

uuuG

∂

∂′′−=

~ρ To evaluate G K in

a way compatible with assumption of Boussinesq

2SG tk µ= ; ijij SSS ~~2≡ where

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛

∂

∂+

∂∂

=i

j

j

iij x

uxu

S~~~ ;

kGk

G ωαω =

It should be noted that, for large Reynolds numbers for model k-ω . 1=α Modeling of Turbulent Dissipation for the kinetic energy ωρβ β kfYk **= Modeling of Turbulent Dissipation for the rate of dissipation 2ωρβ βω kfY = It should

be noted that, for great Reynolds numbers for model k-ω . ∗

∞∗ = ββ i . Constants of the Model

3. Discretisation scheme: solver coupled model

The system of the equations for only one component of the fluid, written to show the average properties of the flow, for an arbitrary volume of control V with a surface and in Cartesian co-ordinates is :

[ ] ∫∫∫ =⋅−+∂∂

VV

HdVdAGFWdVt

(3)

Where the vectors W F and G are matrices: The equations of Navier-stokes expressed in the equation (3) are very difficult to solve undergo a transformation of variable making it possible to lead to the following form:

[ ] ∫∫∫ =⋅−+∂∂

∂∂

VV

HdVdAGFQdVtQ

W (4)

Where Q is the vector { }TTwvup ,,,, and

pT

Tp Tp ∂

∂=

∂∂

=ρρρρ ,

Expressed differently in the conservative form the system becomes:

kR wR *ζ 0tM kσ εσ 6 2,95 1,5 0,25 2,0 2,0

*∞α ∞α 0α *

∞β iβ βR

1,0 0,52 1/9 0,09 0,072 8


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[ ] ∫∫∫ =⋅−+∂∂Γ

VV

HdVdAGFQdVt (5)

Γ:stamp=f(u,v,w,Θ,ρ,ν) with ⎟⎠⎞

⎜⎝⎛ −=Θ

pr CT

U ρρ

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The vector of flow (5) is divided into two different flows [9]. This approach informs us on the characteristics of the eigenvalues of each part in each one of these directions. We obtain the following expression for the flow discretized with each face:

( ) QAFFF LR δˆ21

21

Γ−+= (6)

1ˆ −Λ= MMA ; maxλ

xCFLt ∆=∆ (7)

The step of time is calculated with condition CFL (Current-Friedrichs-Lewy). For the coupled method, it is proportional to the CFL, like defined in the equation (7). maxλ = the solution eigenvalue of the determinant 4. Results and discussion

4.1. On the quantitative level: within sight of the table: 1 it appears that: - Temperature of the supersonic speed range undergoes an increase between the values minimal and maximum and this evolution is followed systematically by the pressure. - the other paramatr remain unchanged some with the sonic mode and the geometry. - the kinetic energy turbulent (k) increases one decade of the subsonic mode towards the supersonic mode. - the term of production progresses two decades of the subsonic mode towards the supersonic mode. - In all the cases the Mach number remains unchanged from one geometry to another. 4.2. On the qualitative level: Contrary to what preceded, it appears certain variations locally. They are represented by contours of the curves fig 2. with fig 4. In the case of a supersonic mode we have: - the distribution of the Mach number fig 2. has a progression in normal plans with the axis (vertical for Bérénice and curvilinear for Antarès). - the energy production results with the throat for Antarès but upstream and with the throat for Bérénice

- the kinetic energy is concentrated just downstream from the throat for Bérénice and slimmer in the divergent one for Antarès. In the case of a subsonic mode we have:: - The energy production appears upstream in the throat, contrary to the supersonic case and always in a way more extended in the case of Antarès than that of Bérénice. - It is also identical being the term of production but more manifest in the zone surrounding all the throat

Fig.1. Diagram of nozzles: and Bérénice (BER) Antarès (ANT)


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Fig. 2. Contours of Mach number (supersonic case)

Fig. 3. Contours of the energy production

turbulent (subsonic case)

Fig. 4. Contours of the turbulent kinetic energy

(supersonic case)

Fig. 5. Contours of Mach number (subsonic case)

5. Conclusion The model k-w seems to answer certain requirements related to the treatments of turbulence meadows of the wall. The modeling of the terms of production , dissipation and diffusion deal with the transfer of the energy between the rollers of swirls. The geometry of the conduits highlights this phenomenon by the form of divergent downstream from the collar. The distribution of the isomach seems to be sensitive to the sonic mode and the geometrical form, but remains unchanged in average values. On the contrary, the terms of the kinetic energy, dissipation and production them vary in average and local values. Thus, we note that for the Antarès conduit thus for a low divergence the terms of energies are born upstream from the collar in any sonic mode. Only the temperature followed by the pressure seems to make a significant fall while passing a subsonic mode with the supersonic mode. This is obviously due to the shock wave incipient with the conduit collar. In conclusion the results obtained are very pertinent since they are added to those of other configurations whose whole delivers conclusive information to us.

Reférences: [1]. Cousteix, J " turbulence and boundary layer "

Edition Cepadues, 1989 [2]. Favre and al. «turbulence in mechanics of the

fluids " Edition Gautier Villars, 1976 [3].Cline, M.c , " Vnap2: With Computer Program

for Computation of Two-Dimensional,Time-Dependent, Compressible, Turbulent Flow, " Los Alamos National Laboratory carryforward, LA-8872 (July 1981).

[4].R.C. morgans and its team " application of the revised Wilcox (1998) K-ω model turbulence to has jet in Co flow " conference CSIRO: Melbourne, Austrlie, 1999.


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[5] Q.Xiao and H M. tsai and F Liu " computation of transonic to diffuse flow by has lagged K-ω turbulence model " newspaper of propulsion and power July 2003

[6] Loh, C.Y. and Zaman, K.B.M.Q.: Numerical Investigation of ' Transonic Resonance' With has Converge-Diverge Nozzle. Aiaa Paper 2002-0077, 2002.

[7] Loh, Ching There; Hultgren, Lennart S.; and Chang, Sin-Chung " Wave Computation in Compressible Flow Using the Space-Time

Conservation Element and Solution Method Element ". AIAA J, vol. 39, No 5, 2001, pp. 794-801.

[8] Anju K. Bawa, "First Successful Computer Simulation of Flow in Fluidic Windshield Washer Nozzle Shows Potential to Reduce Design Time 50%" Columbia, Maryland, 1999

[9] P.L.Roe. "Characteristic based schemes for the Euler equations". Annual Review of Fluid Mechanics, 18:337-365, 1986.

Table. 1 : Summary of the values of different the characteristics from nozzle

Supersonic

Supersonic with shock wave

Subsonic

Min

max

Min max Min max

ANT

BER

ANT

BER

ANT

BER

ANT

BER

ANT

BER

ANT

BER

Noz

zle

5.41

*10-0

2

4.61

*10-0

2

4.

00

4.58

2.48

*10-0

2

1.17

*10-0

2

1.79

1.06

4.02

*10-0

3

2.65

*10-0

3

0.60

5

0.21

7

Mac

h

1.13

*1003

9.79

*1002

4.75

*1007

3.23

*1008

49.9

418

2.50

*1008

1.09

*1008

8.59

34.9

5.07

*1006

4.69

*1006

Prod

uctio

n [k

g/m

s3 ]

1.00

1.00

4.17

*1004

6.32

*1004

1.00

1.00

3.57

*1004

2.74

*1004

1.00

1.00

3.21

*1003

2.14

*1003

Kin

etic

En

ergy

[m

2 /s2 ]

128

104

540

543

324

432

540

542

502

535

540

540

Tem

pér

atur

e [°

K]

6.51

*10-3

2.77

*10-3

1.50

1.50

0.19

1

0.40

9

1.50

1.50

1.12

1.35

1.49

1.50

Pres

sure

[a

tm]


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study with К־omega turbulence model for supersonique...

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