Ye. Belyayev , A. Kaltayev

(1)

UDK 532.5; 544.452.42

Ye. Belyayev^*, A. Kaltayev

Faculty of Mechanics and Mathematics, al-Farabi Kazakh National University, Almaty, Kazakhstan

Numerical study of supersonic turbulent free shear layer

Abstract. Numerical study of two-dimensional supersonic turbulent free shear layer is performed. The system of Fa- vre-Averaged Naveir-Stokes equations for multispecies flow is solved using ENO scheme of third-order in accuracy.

The k-ε two-equation turbulence models with compressibility correction are applied to calculate the eddy viscosity coefficient. In order to produce the roll-up and pairing of vortex rings, an unsteady boundary condition is applied at the inlet plane. At the outflow, the non-reflecting boundary condition is taken. The obtained results are compared with available experimental data.

Keywords: supersonic free shear flow, mixing layer, ENO-scheme, k-ε model, compressibility.

Introduction

Compressible mixing layer is an important flow in extensive engineering applications. In par- ticular, the shear layer configuration is a simple and yet fundamental to understand how fuel flow will mix and combust with supersonic oxidizer flow in SCRAM jet engines combustion chambers of hypersonic vehicles. As is well known the main objectives of investigating the physical processes in combustion chamber of these engines is aimed to maximize thrust by enhancing the fuel-air mixing and combustion.

It is necessary to take into account the influence of gas-dynamical structure, turbulence effects and chemical reactions for understanding physical structure of fuel-air mixture combustion in numerical model. Studying combustion in shear layer requires accurate predictions of mixing and combustion efficiency to which special attention should be paid to simulation the unsteady behavior of mixing layer roll-up and vortex formation. The gas- dynamical structure of mixing between two paral- lel super-subsonic flows has been comprehensively studied by many investigators. Nowadays, there are a large number of works on experimental [1-9], analytical [10-11] and numerical [12-27] study of this problem in the view of above physical effects as separately as with including all of them. Expe- rimental efforts investigating the roll of large scale structures and growth mechanisms in compressible

mixing layer have been done in sufficient details by researchers [1-6]. There are a great deal of re- searches devoted to the turbulence problem and influence of turbulence quantities on the mixing and vorticity formation [7-9].

The behavior of shear layers of perfect gases have been entirely realized in mathematical models, but the practical design of supersonic ramjet (scram- jet) engines requires the shear layer growth enhance- ments for multispecies gases. Successful numerical models of such flows with the detail flow physics represent a difficult problem. Therefore the investigators studied some physical phenomena separately or proposed the numerical method, which are important for solution of this complex system. In [12, 15-16]

have been modeled the free shear layer flowfield structures using the system of compressible Euler equations. For example, in [12] have been numerical- ly studied the supersonic-subsonic free shear layer applying high order WENO scheme to the system of 2D axisymmetric Euler equations and numerical turbulence model taken as a SGS model. During numerical experiment revealed that at high-convective Mach number turbulence mixing rates reduces and vortex roll-up and pairing suppresses. In [13-14, 17]

have been performed numerical experiment based on the system of Navier-Stokes equations for monatom- ic (air) gas to study the growth of instabilities in supersonic free shear layers. Xiao-Tian Shi et al. [17]

conducted numerical simulations of compressible mixing layers based on discontinuous Galerkin me-

(2)

International Journal of mathematics and physics 5, №1, 4 (2014)

Figure 3 – Comparison of present calculation with experimental data by longitudinal mean velocity profiles at five longitudinal positions in the shear layer

a) x=6, b) x=12, c) x=15, d) x=18, e) x=21 cm.

a) b)

c) d)

e)

(3)

Figure 4 – Comparison of present calculation with experimental data by Reynolds stress at five longitudinal positions in the shear layer

a) x=6, b) x=12, c) x=15, d) x=18, e) x=21 cm.

a) b)

c) d)

e)

(4)

Figure 5 – Comparison of present calculation with experimental data by turbulent intensity at five longitudinal positions in the shear layer

a) x=6, b) x=12, c) x=15, d) x=18, e) x=21 cm.

a) b)

c) d)

e)

(5)

thod with inflow perturbation for prediction of the flowfield structures obtained in experiments. Numer- ical experiments of influence of unsteady inflow per- turbations on the mixing in supersonic free shear layers on the basis of second and fourth order MacCor- mack scheme have been performed by authors [13- 14]. Their studies revealed that normal velocity perturbation is more efficient than streamwise and spanwise. To date rarely performed the numerical investigation of growth of instabilities in shear layer using unsteady disturbances for multispecies gas mixture. In these works have accurately predicted the gas-dynamical structure of shear layers by ad- vanced numerical methods without chemical reactions terms.

In the present study, the third order essentially non-oscillatory (ENO) finite difference scheme is adopted to solve the system of Favre-averaged Navier-Stokes equations to supersonic planar shear layer. The k-ε two-equation turbulence model with compressibility correction is used to predict the turbulence characteristics. To verify the mathematical model and numerical algorithm obtained results compared with experimental study of Samimy and Elliot [8-9] for supersonic-subsonic free shear layer.

The inflow physical parameters profile across the non-premixed hydrogen (fuel) and air stream at the splitter plate leading edge is assumed to vary smoothly according to a hyperbolic-tangent function (Fig. 1).

Figure 1 – An illustration of the flow configuration

Mathematical model

The two-dimensional Favre-averaged Navier- Stokes equations for multi-species flow with chemical reactions is:

   

₀_,

∂

∂   

z x

t ^v ^v

F F E E

U    

-

- (1)

where the vector of the dependent variables and the vector fluxes are given as























k Y E

w u

U

k t

 ,

 





















u uk uY

u p E

uw p u u

E

k t

2

 ,

 





















w wk wY

w p E

p w uw w

F

k t

2

 ,

(6)

t T k l

l t kx x xz xx xz xx

v x

J k q w u

E 















 



 







 



 



 x

∂ Re

, 1 Re

, 1 , - ,

, ,

0 



 



 





 ,

T l t

k l t kz

z zz xz zz xz

v z z

J k q w u

F 















 



 







 



 





 



 



 





∂ Re

, 1 Re

, 1 , ,

, ,

 0 ,

 

  



Pk Mt D C Pk k C f k



^T

W  0,0,0,0,0,  1 ²  , _₁  /  _₂ _₂²/

Here, the viscous stresses, thermal conduction, and diffusion flux of species are:

 

;

3 -2 Re 2

1 



 



 



 



 

 _x _x _z

k l t

xx u u w



 



 

;

3 -2 Re 2

1 



 



 



 



 

 _z _x _z

k l t

zz w u w



 



 

;

Re

1 z x

k l t zx

xz u w



 



 



 

 



1 ; Re

Pr 1

2 1 xk

N

k k

k l t

x h J

x M

q T



 



 





 



 

  

  1 ;

Re Pr

1

2 1 zk

N

k k

k l t

z h J

z M

q T



 



 





 



 

  

 

x Y

J Sc ^k

k l t

kx 





 



 



 

  Re

1 ,

z Y

J Sc ^k

k l t

kz 





 



 



 

  Re

1 .

Parameters of the turbulence are:

z w x

w z

u x

P_k _txx u _txz _tzx _tzz



 



 



 



     ;

3 ; 2 2

Re 



 



 



 











 



 

z w x u x

t u

txx 

 ;

Re 



 











 

x w z

t u

txz 



3 ; 2 2

Re 

 



 



 











 



 

z w x u z

t w

tzz 



T k M

M_t² 2 _²  / ;



^Re



^;

exp 3 . 0

1 ²

2 t

f_     _Re _Re ² _;











 







t kl

3 1 0 1 92 1 44

1 ₂

1 . ;C . ; . ; .

C_  _  _k  _  ,

where k, – turbulent kinetic energy, rate of dissi- pation of turbulent kinetic energy. P_k – is turbulence production term, M_t – is the turbulence Mach number.

Yk – is the mass fraction of k^th species, N

k 1... , with N -number a components in a gas mixture. The thermal equation for multi-species gas is:

(7)

W M p T₂





 _, ¹

1



 











 ^N

k k

Wk

W Y ^,



 N 

k Yk

1 1 (2) where W_k is the molecular weight of the species.

The equation for a total energy is given by



² ²



2 1

2 p u w

E h

t  M   



 



(3) The enthalpy of the gas mixture is calculated

according to





 ^N

k Ykhk

h

1

, with specific enthalpy of k^th species evaluated suing ^ ^



^T

T pk k

k h c dT

h

0

0 .

The specific heat at constant pressure for each component c_pk is:

W C

c_pk  _pk / ,



 



 ⁵  1

1 i

ki i

pk a T

C ,

a_jk a_jkT_^j^¹

where the molar specific heat C_pk is given in terms of the fourth degree polynomial with respect

to temperature, consistent with the JANAF Ther- mochemical Tables [28].

The system of the equations (1) is written in the conservative, dimensionless form. The air flow parameters are ,u,w,T,h,W,R, hydrogen jet parameters are ₀,u₀,w₀,T₀,h₀,W₀,R₀. The go- verning parameters are the air flow parameters, the pressure and total energy are normalized by _u_², the enthalpy by R0T_ /W_, the molar specific heat by R₀ and the spatial distances by the thickness of the splitter plate



_.

The coefficient of viscosity is represented in the form of the sum of _l - molecular viscosity and

t - turbulent viscosity: _l_t, where _t is defined according to k-ε model with compressibility correction. The mixture averaged molecular viscosity is evaluated using from Wilke's formula.

Initial and boundary conditions At the entrance:

0 0 0 0 0

1 W

T M R

u 

 , w₁0, p₁p₀, T₁T₀, Y_k₁Y_k₀, kk₀,  ₀ at x0, 0≤zH₁.

∞ 0 ∞

∞ ∞

2 W

T M R

u _  , w₂ 0, p₂  p_∞, T2 T∞, Y_k₂ Y_k_∞, k k_,  _ at x0, H₁ ≤z ≤H₂. In the region of H₁≤z≤H₁ all physical variables are varied smoothly from fuel flow to air flow using a hyperbolic-tangent function of any variable φ, so the inflow profiles are defined by



 

 

 

 





_(z)₀_.₅ ₂ ₁ ₀_.₅ ₂_- ₁ _tanh₀_.₅z_/ at x 0, 0 zH .

(8)

International Journal of mathematics and physics 5, №1, 4 (2014) where  (_u,_v,_p,_T,_Y_k,_k,),  - is the mo-

mentum thickness. The pressure is assumed to be uniform across the shear layer. On the lower and upper boundary the condition of symmetry are im- posed. At the outflow, the non-reflecting boundary condition is used [29].

In order to produce the roll-up and pairing of vortex rings, an unsteady boundary condition is also applied at the inlet plane, i.e.

) cos(

) ( )

(z A UGaussian z t u

u    

) sin(

)

(z t

UGaussian A

w

w _factor  

), 2 / - exp(

)

(z z² ²

Gaussian 

where u

 

z - velocity profile at the entrance corresponding to hyperbolic-tangent function. The unsteady part of the condition is about 0.2-0.3 percent of the Favre-averaged velocity.

) (u u₀ U  

 _ - the difference of two stream velocities which measures the strength of shearing.

) (z

Gaussian - is a Gaussian function which has a peak value of unity at z=0 and the 2 width was matched to the vorticity layer thickness at the entrance. Coefficient A is the disturbance ampli- tude which is defined from the AU , where this product was specified to be 0.2-0.3 percent of the higher inflow speed. The w_factor was set at 0.7 (70%) of AU based on [16].

The



























 



w

W T R W

T R







 2

2 /

2 ^∞

∞

∞ 0

0 0 0

is the frequency of perturbation.

 



u/ z



_max⁰

u

w u 

 ^ 



- is the vor-

ticity thickness.

Method of solution

To take into account the flow in the shear (at the entrance) and mixing layer, i.e., in regions of

high gradients, more accurately, we refine the grid in the longitudinal and transverse direc- tions by the transformations







 













 











 _v _vm F_v F_vm

t

~

~ E F E 2 E 2

U , (4)

where U U J

~  _, _J

xE

E 



~  , F _zF J



~ , E_v2 _{x v2}E J



~  , E_vm _{x vm}E J



~  , F_v2 _{z v2}F J



~  ,

z vm J

vm F

F 



~  _{, and}J 

   

, / x,z is the Jacobian of mapping.

System (4) linearized with respect to the vector U in form:





































  ^ ^ ^ ^ ^ ^











2 1 1 21

1 1

1 1 n

n v n vm n v

n vm

n t En F E E F F

U ~ ~ ~ ~ ~ ~

~ Δ

^^U^~ⁿ^^O

 

^^t² . (5) Here,

1

1 ~

~ⁿ A Uⁿ ⁿ

E _ , F~ⁿ¹B U_ⁿ~ⁿ¹, (6) A

A_  _x ,B_ 



_z B_,A E U



 / , B F U



 / - are the Jacobian matrices [32-33].

The terms containing the second derivatives are presented as sums of two vectors:

(9)

~ⁿ₂¹ ~ⁿ₂₁¹ ~ⁿ₂₂

v v

v E E

E ^  ^  , F~_v F~_v_n ₁ F~_vⁿ₂₂

1 21

n2^  ^  , (7) where the vectors ~ⁿ₂₁¹

Ev , ~ⁿ₂₁¹

Fv are written in the following form:

n T n t

x n

v t u w E

E J















 







 



 







 



 







 



 1 1 1

1

n21 , ,Pr

3 ,4 Re 0

~

















,

n T n t

z n

v t u w E

F J















 







 



 







 



 







 



 1 1 1

1

n21 ,Pr

3 ,4 ,

Re 0

~

















,

and the vectors ~ⁿ₁₂

Ev ,F~ contain the remaining dissipative functions of the form: _vⁿ₂₂ Pr ,

3 4 , Pr

0 , 0 , Re 0

~_n ²

12

n T

v x w w u u

E J















 







 



 



 





 







 



 

 

 

 



 



Pr , 3 4 , Pr

0 , 0 , Re 0

~_n ²

22

n T

v z u u w w

F J















 







 



 



 





 







 



 

 

 

 



 



,

3 , 2 3 ,

, 2 ReJ 0

~







 







 



 





 







 



 



 



 



 







 



 



 

 

 

 

 

 





v u w u

u w u v

u v

w

y z

z y

t z vm x

E

.

3 . 2

3 , , 2 ,

ReJ 0

~







 







 



 





 







 



 







 







 







 

 

 

 

 

 

 

 





w v w u

u w u w

w u

w

y x

t z vm z

F

According to a principle of construction ENO scheme [30-31] the system (5) for integration on time is formally represented as:

   



²



2 1 2 1

1

12

~ ) (~

~

t F O

F E

E

B F A E

A t U

vmn v n

m

n m B



 























 







 







  



 



























(8)

Here E^m , F^m

is called the modified flux vector. It consists from the original flux vector (E~, F~and ad- ditional terms of third-order accuracy E_,D_

, E_,D_ ) :

1 n

n

m E~ (E D )

E _ _



 ^ , (9)

(10)

International Journal of mathematics and physics 5, №1, 4 (2014) modified flux

F 

m

is written similarly and A^ A^ I, A^ A^A^¹, B^ B^B^¹,

I

- unity matrix.

Applying factorization to (8), we obtain two one-dimensional operators, which are resolved by matrix sweep:

Step 1:

n n

n x t i n

i n RHS RHS

J U A

A A A

t

I _ _   _ _



 























     



 ^_ _ ^_ _ ^*

1 2 2

/ 1 2

/

1 1 U

) Re (

Step 2:

* 1 1

2 2

/ 1 2

/

1 1 U~ U

) Re

( 























     



 ^_j_ _ ⁿ ^_j_ _ ⁿ ^t ^z _n ⁿ^

J U B

B B B

t

I 



 , (10)

n

RHS_

     

n i j

   

ij



i j



ⁿ

j i j

i j

i E D E D A E D E D

A^_₁_/₂ _  _ _₁  _  _  ^_₁_/₂ _  _  _ _ _₁ ,

 



^



^

 n

j i j

i E D

A  _



2 /

1 minmod



E__i_₁_/₂_j E__i_₁_/₂_j





 

   

















 











ij ij

- j

i j

i

ij ij

- j

i j

i

U U

D D

U U

D D

~ ~

if ,

~ ~

if ,

2 / 1 2

/ 1

2 / 1 2

/ 1



 - -

m m



 ,

 



^



^

 n

j i j

i E D

A _  _

2 /

1 ^ 

  

 



^

 

j i j i j

i E E

R

R ¹₁_/₂ minmod _ ₁_/₂ _ ₁_/₂

 

   

















 

















ij ij

- j

i j

i

ij ij

- j

i j

i

U U

D D

U U

D D

~ ~

if ,

~ ~

if ,

2 / 1 2

/ 1

2 / 1 2

/ 1



 - -

m m



 ,



 



i / j

 

i / ij

j /

i Rsign R I t R R E~

E _ ^ _ ^ _ _



 



 



 ¹ ₁ ₂ ¹ ₁ ₂

2

1 2

1

  ,

   

i j

 

ij

j

i Rsign R t R R I E

D ~

6

1 1 ²

2 2 2

/ 1 1 2

/

1   

 

 











  



 

 

 ,

 j 

D_i ₁_/₂ E__i_₁_/₂_j  D__i_₁_/₂_j,

where

     

other ,

0

   

 s min a,b signa signb s

b

a, if

) minmod(









 

b a b

b a b a

a if

) if ,

m( .

The second term ^RHSⁿ^ is written similarly.

In approximation of derivatives in convective and diffusion terms, we use second-order central-difference operators.

The numerical solution of the system (5) is calculated in two steps. The first determines the dy- namic parameters and second determines the mass species.

(11)

Then it is necessary to define Jacobian matrix which in a case of the thermally perfect gas repre- sents difficult task. This problem is connected with the explicit representation of pressure through the unknown parameters. Here pressure is determined by introducing an effective adiabatic parameter of the gas mixture [34].

sm sm

e

 h



_, ₍₁₁₎

where  ^ 

 N i

T

T p

i

sm Y c dT

h _i

1 ₀ , ^^ ^

 N i

T T v i

sm Y c dT

e _i

1 ₀ – is

the enthalpy and internal energy of the mixture minus the heat and energy of formation; ^T0 ²⁹³^K-

is the standard temperature of formation, which allows to write an expression for the pressure

 

W M

T M

w h u E

p _t ² ² ⁰ ₂ ₂⁰

2 1 1













   



 





 .

The temperature is found from the Newton- Raphson iteration [32-33, 35].

The equations for species are solved by the scalar sweep, where in the first-step convection and diffusion terms are included and calculated using ENO scheme [30-31]. In the second-step, the matrix equation with terms (w_k W_k_k) is solved implicitly. These source terms

W 

_k_{are li-}

nearized by expansion in a Taylor series,



 



 



 

 

 

 

 

  

 _m ^k ^k

m k kn

kn T W

T Y W Y W W

W   



 ¹

Results and discussion

The parameters of coordinate transformation have the form:

     

















 



 



























 







 







1 1 1 1

1 1 - 1

1

1 L

x L

x

H

x 



 



 ,

 











 



 



 

 1 -1 sh( K)

z arsh z K

z

с

 

 ,



 



 



 

 



 

 

 H

e z H

e z

K ln 1 ( -1) ^c 1 ( -1) ^c 2

1  

 ^,





, are refinement factors (>1 and



>1), L - is the length of the computational do- main in the generalized coordinates, and z_c- is the point with respect to which grid refinement is performed.

Previously the shear layer problem for mona- tomic (air) gas has been tested by the following parameters:

K T

M₀ 0.51, ₀ 285.07 , P₀56088.91 Pa, K

T

M∞1.8, ∞176.58 , P∞54648.65 Pa. The computational grid is 526x201. The channel height and length were 8 cm and 50 cm, re- spectively. The splitter plate thickness is 0.3175

cm, and at the trailing edge is 0.05 cm. The initial momentum thickness ^

_  



  

 



 

 u^*1 u^*dz



  is

0.05 cm. The geometrical parameters above are taken from experimental work of Samimy and El- liot [8-9]. Experiment was conducted in tunnel, present calculation performed for planar channel to estimate the behavior of turbulence quantities. Fig- ures 2-4 shows (M_c =0.51) the comparison of the calculated distributions of longitudinal (axial) mean velocity, variation of the momentum () and vorticity (_w) thicknesses, and turbulence quantities with the experimental data [8-9]. The non-

(12)

International Journal of mathematics and physics 5, №1, 4 (2014) dimensional variables

 



∞ ₀



0

*

-u u

u

u  u ,

 

w

zc

z z



 

* ,

  





  



 



 

 u^*1 u^*dz



  ,

 



u/ z



_max⁰

u

w u 

 ^

 are de

fined as in the experiments [8-9]. Figure 2 indi- cates that the shear layer growth in terms of momentum and vorticity thickness is predicted rea- sonably accurate by the present algorithm, as compared to experimental data.

Figure 2 – Comparison of present calculation with experimental data by the growth of momentum and vorticity thickness.

The comparison of calculated transverse distri- bution of the normalized streamwise mean velocity at five longitudinal positions with experimental measurements as shown in Figure 3 suggest that in the fully developed region for x≥12 cm the mean flow is self-similar.

Further comparison of the calculated results with experimental data are shown for the develop- ment of the Reynolds stress uv/



u∞-u_o



² in Figures 4 and streamwise turbulence intensity



u u_o



k/ - 3 /

2 ∞ in Figure 5. The contribution of transverse velocity fluctuating component to turbulent kinetic energy was neglected. It is visible from figures that the calculated turbulence quantities are distorted at x≥15 cm, which shows that the turbulence similarity is achieved further downstream than the mean flow similarity. The preliminary test shown that the mean and turbulence quantities are in a good agreement with experimental data.