-t

Indian Journal of Engineering & Materials Sciences Vol. 9, April 2002, pp. 1 1 5- 1 27

A numerical study on the physics of mixing in two-dimensional supersonic stream Mohammad Alia, Toshi Fuj iwarab & Anwar Pervezc

" Department of Mechanical Engineering, Bangladesh University of Engineering and Technology, Dhaka 1 000, Bangladesh

b Department of Aerospace Engineering, Nagoya University, Japan

C Department of Mechanical Engineering, Bangladesh Institute of Technology, Khulna 9203, Bangladesh

Received 20 February 2001 .. accepted 6 December 200J

A numerical investigation has been performed on the physics of mixing for better understanding about the penetration and mixing mechanism, and eventually to find out the means of increasing the mixing efficiency. The two-dimensional full Navier-Stokes equations with an explicit Harten-Yee Mon-MUSeL Modified-flux-type Total Variation Diminishing (TVD) scheme have been used. A zero-equation algebraic turbulence model has been used to calculate the eddy viscosity coefficient. For this study the air of Mach number 5.0 is considered as main flow. Hydrogen gas at sonic condition is injected perpendicularly into it. The effects of molecular and turbulent diffusion coefficients, and that of boundary layer on mixing have been analyzed and discussed. The results show that upstream re-circulation plays an important role to increase the mixing of side jet with the main flow of high Mach number. I t has been found that the mixing is only possible by incorporating the molecular diffusion terms in the Navier-Stokes equations. The use of turbulence model increases the penetration height of hydrogen in the mixing field.

The study of the physics of mixing and combustion i s very important to find out the physical mechanisms that can increase their efficiencies. Particularly, the mixing of reactants and their complete combustion in Supersonic Combustion Ramjet (Scramjet) engines have drawn special attention of present scientists of Aerodynamics. In fact, in supersonic combustion, high penetration and mixing of injectant with main stream is difficult due to their short residence time in combustor. In an experimental study, Brown and Roshkol showed that the spreading of a plane turbulent mixing layer decreased remarkably with increasing the Mach number of supersonic jet. A similar conclusion was drawn by Papamoschou and Roshk02 on the basis of a theoretical analysis of shear-layers. These investigations showed that difficulty exists in achieving better mixing in high Mach number flows. Therefore, it is necessary to find out all the physical mechanisms that affect the mixing and combustion to analyze and optimize a supersonic combustor.

Several other investigations3." have been performed to analyze the mixing characteristics, and find out the means of increasing the mixing efficiency. In these investigations the authors showed a number of parameters that can affect on penetration and mixing. In an experiment, Rogers3 showed the effect of ratio between jet dynamic pressure and freestream dynamic pressure on the penetration and mixing of a sonic hydrogen jet injected normal to a Mach 4 airstream. In a

similar flow arrangement, Kraemer and Tiwari4 found that the relative change in jet momentum was directly proportional to the relative size between the flow field disturbance and the upstream separation distance. The downstream injectant penetration height i s directly proportional to the upstream separation distance, and, thus, the downstream mixing is dependent on the relative change i n jet momentum. S imilar conclusions were drawn by Thayer III and Corlett5 . They also found that the i njectant concentration in the separated flow region was high when the temperature of both the injectant and freestream was taken as investigation parameter. Yokota and Kaj i6.9 searched the enhancement of mixing by varying: (i) the angle of a finite length slit6 (when slit length i s smaller than the width of computational domain) and (ii) slit aspect ratios 7. Besides, they examined the effects of injection methodsB, and the existence of pressure wave in the injecting flow field9 on mixing and total pressure loss. In an experimental investigation, Moussa et al. I O

found that the geometric configuration of the boundaries of jet eX it plays an important role on the mixing and its development process. Andreopoulos and Rodi I I searched the penetration of jet into the cross flow for the jet to cross flow velocity ratio of 0.5, 1 .0 and 2.0. Also the authors discussed some other flow characteristics such as wakes, vortex motion, velocity gradient in shear layers near the jet exit.

1 1 6 INDIAN 1. ENG. MATER. SCI., APRIL 2002

_ _ _ . _ _ _ . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ .�?��:_ .I!

ALI et at. : PHYSICS OF MIXING IN TWO-DIMENSIONAL SUPERSONIC STREAM 1 1 7

The values of Cp and H are considered as functions of temperature and determined from the polynomial curve fitting developed by Moss l 7. Temperature IS calculated by Newton-Raphson method.

Transport Properties

The molecular viscosity coefficient 11 and thermal conductivity K of each species are determined by Sutherland formulae '8 as:

. . . (2)

. . . (3)

and those of gas mixture by Wilke's formulae ' 8 and Wassiljewa's equation l9 as:

and /IS /( .

/( - � I I - � 1 1/.\"-1 1 + - I, Aij zj Zi j=1 jof.i

where CPij [1 .0 + (,ui / ,u j ) 0.5 (Wj / W; ) 0.25 Y

(8 + 8 Wi / Wj )0.5

. . . (4)

. . . (5)

Aij = 1 .065¢,j' and Zj are the mole fractions, while Ilo, To and Ko are the reference values.

The effective molecular diffusion coefficient for each species is determined 1 9 as:

D _ 1 -2i il1ll - 11.\"-1 I, Z/Dij j=1 jof.i

( T ]--{).145 ( T ]

-2.0 Qo = - + - + 0.5

TEij TEij

. . . (6)

TEij = (TEi TEj f·5

a Eij = � (a Ei + a EJ T = absolute temperature (K) p = pressure (atm)

A zero-equation algebraic turbulence model developed by Baldwin and Lomax 16 is used to simulate boundary layer separation, re-circulation and shockexpansion regions near the injector. The primary advantage of using this model is that it does not need to calculate the boundary layer thickness, rather it calculates the eddy viscosity J-4 based on the local vorticity, w. This is very helpful because at the injection port and adjacent region it is difficult to define boundary layer thickness. Secondly, this model can successfully calculate the separated flows both over a flat plate and a compression corner. According to Baldwin and Lomax 16, the eddy viscosity ,ul is defined as :

{�I )il/I/er ,u = t �t )ouler

y � Y cros.\Over . . . (7) Y > Ycrossover

where y i s the normal distance from the wall and Ycros.\"over the smallest value of Y at which the value of viscosity in the outer region becomes less than or equal to the value of v iscosity in the inner region.

The viscosity in the inner region is given by:

. . . (8)

The mixing length in the inner region I is expressed as :

where

+ Pw ur Y � y Y = = ,uw ,uw

. . . (9)

( 1 0)

For two-dimensional flow� the magnitude of the vorticity is given by:

Iwl = ( au _ av )2 ay ax . . . ( 1 \ )

1 1 8 INDIAN 1. ENG. MATER. sel., APRIL 2002

For the outer region,

. . . ( 1 2)

where K is the Clauser constant, Ccp an additional constant, and

( 1 3 )

Here Flllax i s the maximum value of the function

. . . ( 1 4)

at each y station in the flow domain, and Ylllnx is the Y coordinate at which this maximum occurs. The function F KLEB(Y) is the Klebanoff intermittency factor given by

. . . ( 1 5)

V2dij is the difference between the magnitude of the maximum and minimum total velocity in the profile at a fixed x station, expressed as

. . . ( 1 6)

( � U 2 + v2 ) is taken to be zero along all x station. The following are the constants used for this model and are directly taken from Baldwin and Lomax 1 6 :

A+ = 26, C.vk = 0.25,

Ccp = 1 .6, k = 0 .4,

CKLEB = 0.3, K = 0.0 168

The values of the turbulent thermal conductivity of the mixture Kr and turbulent diffusion coefficient of ith species Dir are obtained from eddy viscosity coefficient Ilr by assuming a constant turbulent Prandtl and Lewis number equal to 0.9 1 and 1 .0, respectively. They can be expressed as

. . . ( 17 )

. . . ( 1 8)

The final values of Il, K and Dilll used in the governing equations are

( 19)

(20)

(2 1 )

Numerical Scheme The system of governing equations for non-reacting

flow is solved using an explicit Harten-Yee NonMUSCL Modified-flux-type TVD Scheme2o. The two-dimensional, rectangular physical coordinate system (x, y) i s transformed i nto the computational coordinate system (�, 11) in order to solve the problem on uniform grids. After applying the transformation, Eq. ( 1 ) can be expressed as

A 1\ A 1\ A a v a F a G a Fv a G v - + - + - = -- +--at a� a11 a� a11

where

A A V = j -IV , F = j -I (�x F + � , G), A G = rl (11x F + 11yG)

A A

. . . (22)

Fv = j -I (�xFv + � , G" ), Gv = j -I (11x Fv + 11yGv ) ·

The grid Jacobian j and metric terms are,

For the left hand side of Eq. (22), the explicit NonMUSCL TVD scheme can be written as

A 1/+ 1 A 1/ !1t [ A 1/ A 1/ 1 V i ,j = V i .j - ji.j !1� F i+1 / 2.j - F i-1 / 2.j

-+

ALI et al. : PHYSICS OF MIXING IN TWO-DIMENSIONAL SUPERSONIC STREAM 1 19

!1t l l\ /I 1\ /1 J - J; j - G ;,j+1 / 2 - G ;,j-1 / 2 . �T/ 1\

The variables F and G can be described as

1\ 1\

. . . (23)

. . . (24)

The R ;+1 /2 is an eigen vector matrix and ljJ ;+ 1 /2 is a vector with the elements q};+ 1 12 (l = 1 , 2, 3, 4, 5) . The variables used in the above equations are

a(z) = - lfI(Z) -- Z l { !1t 2 } 2 ��

1 = 1 -5 . . . . (25)

. . . (26)

. . . (27)

81 is a function that defines the range of entropy correction, and should be a function of the contravariant velocity and the corresponding sound speed for the computations. The form of the function used here is

. . . (28)

with a constant 8 set to 0. 1 5 . More details about the scheme can be found in Y ee2 1 •

Among the limiters22 for the second order TVD conditions, the minmod limiter was used to avoid the numerical oscillations at the discontinuity. Among the various approximate Riemann22, the Roe 's average was used which is the most common one due to its simplicity and ability to return to the exact solution whenever the variables lie on a shock or contact discontinuity.

The time step for calculation is determined by

�t -CFL

- max{IU I + lvl + c(� ; + �.; ) 1 I 2 + c(T/; + T/; ) 1 / 2} . . . (29)

where U and V are the contravariant velocities and can be expressed as

U = �xu + �yv. V = T/xu + T/yv . . . (30)

The Courant number CFL is chosen as 0.7 to obtain rapid convergence and avoid unsteadiness in calculation23 .

Boundary Conditions and Convergence Criterion The Navier-Stokes analysis imposes that the nor

mal and tangential velocity components are zero on the walls, The walls are assumed to be thermally adiabatic, so that (dTlan)w = O. For non-catalytic walls, the normal derivative of species mass fraction also vanishes, and consequently the gradient of total density becomes zero. The pressure is determined from the equation of state. The temperature, pressure and density at inflow boundary are assumed steady. At outflow boundary the variables are determined by first-order extrapolation due to supersonic character of flow. Throughout the present study, the following convergence criterion has been set on the variation of density:

JJ,KK L P/lew - Pold

J =I,K=! Pold JJ.KK

2

. . . (3 1 )

where JJ and KK are the total number of nodes i n the horizontal and vertical directions, respectively .

Program Verification To verify the present code, a comparison has been

made with the experimental data published by Weidner et al. 1 3 . The geometry of the experiment is shown in Fig. 2, where helium gas was injected at sonic condition from a 0.0559 cm slot into a rectangular duct of 25.4 cm long and 7.62 cm high. The slot was located 1 7 .8 cm downstream of the duct entrance. The flow conditions of helium at the slot exit were P = 1 .24 MPa, T = 2 17.0 K and M = 1 .0. At the entrance of the duct, the airstream conditions were P = 0.0663 MPa, T = 108,0 K and M = 2.9. Using the same geometry and flow conditions, the flow field was computed for

1 20 INDIAN 1. ENG. MATER. SCI., APRIL 2002

o

o

o

o

• w - . y AIR r - - - - - -" J H< o 0 Expclimcnt

0 0

Present calculation PlCr = 0.0663 MPa

O +-��--r-'--r-.--.-��� - 5 . 0 - 3 . 0 - 1 . 0 1 . 0 3 . 0 5 . 0 Dis tanc e f rom inj ec tor ( e m )

Fig. 2--Comparison between experimental and computed pressures along bottom wall.

Experiment

P/Pn:f

Prescnt calculation Pn:f = 2. 1 0 MPa x 3.8 1 em h = 7.02 em

o

Fig. 3--Comparison between experimental and computed static pressures at 3 .8 1 cm downstream of injector.

comparison taking a grid system consisting of 246x 165 nodes i n the horizontal and vertical directions, respectively.

Fig. 2 shows that the computed pressure along the bottom wall agrees well with the experiment in both the upstream and downstream of the injector. Both show a pressure rise in the upstream separated region and downstream reattachment region. An overprediction can be found at the immediate downstream of the injector where the turbulence is intensified by the disturbance caused by the injector. Fig. 3 gives the static pressure distribution along the vertical axis at 3 .8 1 cm downstream of the injector. Qualitatively, the computed pressure profile agrees with the experimental data. Small variation on the position of recompression shock and bow shock, and the pressures at

o

co Experiment o

o

Present calculation x = 3 . 8 l em h = 7.62 cm

o o ���������������� 0 . 0 0 0 . 0 4 0 . 0 8 0 . 1 2 0 . 1 6 0 . 20 0 . 2 4 0 . 2 8 0 . 3 2

Hel ium m a s s f rac t ion

Fig. 4--Comparison between experimental and computed mass fraction of helium at 3 . 8 1 cm downstream of injector.

these positions can be observed in the computation. In the experiment, the recompression shock occurs at ylh = 0.2 (y i s the vertical location and h is the height of domain), whereas in computation at 0. 16. After recompression, both show almost linear increase of pressure up to ylh = 0.6. The calculation determines the similar difference in the position of bow shock as that of recompression shock. In the experiment the position of bow shock is at ylh = 0.63, while in computation it is 0.59. Beyond the bow shock, the calculation shows the similar decreasing rate of pressure with experiment. Fig. 4 shows the comparison between the mass fraction profiles of injected helium along the same vertical axis at 3 .8 1 cm downstream of the i njector. The computed curve agrees with the experimental data at all points along this axis.

The computation shows that the overall computed results agree with the experiment in spite of the complexities of injected flow field. The code is, therefore, considered to be adequate for application to calculate the mixing flow fields.

Results and Discussion Study on numerical diffusion

The numerical diffusion, might be incorporated by the computer code, was checked by (i) reducing the mesh size, i .e. , grid refinement study, and (ii) using different kinds of limiters i n TVD scheme.

Grid refinement study The present i nvestigation adopted a grid system

consisted of 1 94 nodes along the horizontal direction and 1 2 1 nodes along the vertical direction. Using twice of this grid system, i .e . , 387 nodes along the horizontal direction and 24 1 nodes along the vertical direction, the flow field was calculated. The compari-

-+

-r

i

-4 \

ALI et al. : PHYSICS OF MIXING IN TWO-DIMENSIONAL SUPERSONIC STREAM 1 2 1

son of mixing between the two grid systems i s shown in Fig. Sea, b) where no significant difference can be found in penetration and distribution of hydrogen both in upstream and downstream of injector. Therefore, i t has been concluded that the grid system 194x 121 i s a reliable one for present analysis.

. . . : . . . . . : - . · r · · · ··

- · · · · _ ; _ · · · - : · · _ - : · · ..,; . • . . . • l . . • ' . , . . :' " " : " " " :

': : 0.05

Inje c lor

. . . .

Dis lance (a) Mole fraction contour ofH2 by grid system 1 94x 1 2 1 .

E �

o .,.; . . . . . . , . . . . . . . . . . . .. . . . .

.

0.4

. 0 � . O 5 . 0 6 . 0 7 . 0 8 . 0 9. 0 1 0. 0 Injec lor Di s tance (em)

(b) Mole fraction contour ofH2 hy grid system 387x24 1 .

Fig. 5--Effects of grid systems on mole fraction contour of hydrogen

' 20 lime stFlP:!O 1 0 1 poinls I ;>

--.... 1 .0 CIl ---a 0.8 (a) "--"'

.f' 0.6 () 0 0.4 -

tl) > 0.2 0.0

1 . 0 1 . 5 2 . 0 2 . 5 :1 . 0

X - Coordinate (m)

Effect of 'Limiters ' in TVD scheme Significant numerical diffusion can be found when

the l imiter i n TVD scheme has been changed from 'minmod' to 'superbee' . Fig. 6(a, b)22 compare the numerical diffusion between minmod and superbee limiters. The solid l ines are for exact solution and symbols for numerical solution. It is evident from these two figures that due to limitations of the Limiters numerical diffusion occurs in both cases. However, it is significantly lower i n the result computed by superbee limiter. Fig. 7(a, b) show the effects of

Injector Dis tance (em) (a) Mole fraction contour ofH2 using 'minmod limiter'.

.; .

�s 0

itj�� �.� .. . . . � . . . . . � . . ' ���� 0. 0

In jec tor Di s tance (em) (b) Mole fraction contour of H2 using 'superbee limiter'.

Fig. 7--Effects of l imiters on mole fraction contour of hydrogen.

CFI.,O.5 1 20 limo slnps 1 0 1 poinls 1 .2

--.... 1 . 0 (b) CIl E 0.8 "--"' .f' O.r, () 0 0 4 . Q) > 0.2

0.0

1 . 0 1 . 5 2 . 0 2 . � 3 . 0

X - Coordinate (m)

Fig. 6-Effects of l imiters on the li near convection of a square wave22.

1 22 INDIAN 1. ENG. MATER. SCI., APRIL 2002

limiters on mole fraction contour of hydrogen in present investigation. Fig. 7b shows that the use of superbee limiter in TVD scheme reduces the numerical diffusion significantly appeared by the use of minmod limiter, as shown in Fig. 7a. In upstream of injector, the penetration height of hydrogen is same for both limiters whereas i n downstream, penetration height of hydrogen is higher for minmod limiter caused by numerical diffusion. For both limiters, in upstream and on the top of injectors the contour l ines are concentrated and in downstream they are spreading in similar manner. Other characteristic phenomena of the flow field, such as the pattern of shock waves and their positions, pressure distributions, can be observed by pressure contours shown in Fig. 8(a, b). The separation shock of main air stream caused by upstream recirculation is clear to see in the flow field calculated by superbee limiter. By observing the separation shock of main flow it can be understood that in upstream of injector, the superbee limiter predicts higher pressure compared to the minmod limiter. In downstream both cases show the recompression shock

E �

o

� r--------------------------------'

Injector Distance (cm)

(a) Pressure contour using 'minmod l i miter'.

� r-----------------

Injector Distance (cm)

(b) Pressure contour using 'superbee l imiter'.

Fig. &---Effects of limiters on pressure [PaJ contour; (a) using minmod l imiter, (b) using superbee limiter. Only thick lines show the values. The difference in pressure between two successive l ines is 5x l04 Pa.

clearly, but the superbee limiter calculates slightly lower pressure than the minmod limiter. No signifi cant difference i n the pattern of shocks and their positions can be found between the two limiters, and for both cases similar trend of pressure distributions exists all over the flow fields. However, the significant difference in hydrogen penetration height leads us to perform the calculation using superbee l imiter as i t can reduce the numerical diffusion in present solution of mixing problem.

Effect of molecular diffusion coefficient on mixing

Fig. 9a shows that though the diffusion-flux terms in the governing equations [Eq. ( 1 )] are inactivated, hydrogen reaches to a certain distance along the vertical direction from the bottom wall in both upstream and downstream of i njector. This spreading of hydrogen is caused by stretching and folding of fluid layers in the flow field. Recal ling the species continuity equations in Eq. ( 1 ), we can see that even if the diffusion terms are inactivated, hydrogen can spread i n the flow field due to the flux terms. The non-mixing

� - - · - - - r · · · · · · · · · · · - -, . . . . . . . . . . . ; . _ -� . �

..; - . . - . ;- . - - . . -� - . . . . . � . . - . - . ; . . . . . . . -,- . - . . - � : :

f"') • • - - -:.. - - • - - -� • • • • • • : - - - - - � • - - • • • : • • • • • • � • - • • • • ; : : , . : : ; j -,- • • • • • � - - - • r - - • - -� • • - • • ", '

Injec tor Dis tance (cm) (a) H2 Mole fraction contour w ithout

diffusion coefficient.

. _ . _ . _ . _ . . , . _ . . . , . . . . . . :' . - - _ . ':

: : . . . . . . . . . . . .. - -

I niec tor Dis tance (cm)

(b) H2 Mole fraction contour with diffusion coefficient.

0. 0

0. 0

Fig. 9---Effeets of diffusion coefficient on mole fraction contour of hydrogen.

+

ALI et al. : PHYSICS OF MIXING IN TWO-DIMENSIONAL SUPERSONIC STREAM 1 23

�.---------------------------------------------------------�

C>

C> C"T:l

0 in S � u C> C'l

� C> 0 ..... .... .-<

Q) .S C :::E

C> O . 0

Solid line - for mole fraction Dotted line - for streamline

.. _ ------------------------

Injector

7 . 0 . 0 9 . 0 1 0 . 0 Distance (cm)

Fig. I O--Hydrogen mole fraction contour and streamline without diffusion coefficient.

/"'-. 6

�

1 24 INDIAN J. ENG. MATER. SCI., APRIL 2002

contour matches with it, which means that the mole fraction of hydrogen is approximately same along the 5th streamline. Again in downstream of injector the 7th

streamline shows that it is followed by one mole fraction contour line without any significant deviation and no other contour l ine crosses it. On the other hand, Fig. 1 1 shows that on left side of the domain the 4th

streamline is away from the topmost mole fraction contour l ine, whereas in downstream the topmost two contour l ines cross-over the 4th stream l ine. Besides in Fig. 1 0, the 5th streamline matches with the farthest line of hydrogen contours, whereas in Fig. 1 1 , the 5th

streamline is in deep of the mole fraction contours of hydrogen. Thus, the investigation shows that mixing between the ingredients, particularly hydrogen with other species, cannot occur without the action of molecular diffusion.

Effect of turbulent model on mixing

The use of turbulence model increases the transport coefficients, i .e . , viscosity coefficient, thermal conductivity and diffusion coefficient. The enhancement

o

· . , ' · ' · . · . '

. . . . . . . . . _ - _ ... . . . _ -, . : :

. � . . . . . , . . . . . � . . . - ; . . - . -:' . . . ..

. : . . . · . , .

- -:- - . . . . : . . . . . . ; . : 0.05

lnjec l o r D i s l a n c e ( c m )

(a) Mole fraction contour of H2 w i thout turbulence model .

.,.; . . . . - . � . . . . . . ;' . - . - . · · · i · - - · · -r · - - · ·T

· · · · · � · · · · · - f · · - - -T

- · · · ·

. . . � · · · · ·r · · · r · · · · · M · · · · · ·; · • • • • ·i· · · · · ·

:.!� 0 : : N - - - - - ',' . . . _ . �

. , ' . ' . . . � . . - . . -� . - . . -;. - . - - - : - - - - - -; - - . . . .: . . . . . -� � �

.

Di s lance f rom l e f t boundary

(b) Mole fraction contour of H2 w i th turbulence model .

0 . 0

Fig. 1 2-Effects of turbulence model on mole fraction contour of hydrogen.

of viscosity coefficient enlarges the size of recirculations as well as boundary layer thickness in both upstream and downstream of injector as they are directly proportional to the viscosity. The injection into the thick boundary layer enhances the mixing region. These phenomena can be understood by the comparison between the Figs 1 2a and 1 2b. I n Fig. 12a, we can see that in upstream of injector, the contour lines are accumulated on the edge of the separated turbulent boundary layer, whereas i n Fig. 1 2b, they are more uniformly distributed. In downstream, the turbulence model moves the corresponding contour lines further away from the bottom wal l , which means that the enhancement of diffusion coefficient causes higher penetration and mixing of hydrogen.

Characteristics of the growth of interaction and mixing

Fig. 1 3 shows the characteristics of the flow field when there is no injection. A turbulent boundary layer can be found in the flow field which starts from the upper end of backward-facing step. Besides, there exists a strong shock caused by the bottom wall and boundary layer separation. The side injection causes a hindrance of mainstream and consequently makes a thicker boundary layer and a steeper shock. In Fig. 14, it can be found that initially (t = 0.6 and 1 .0 IJ.s) the mixing layer of hydrogen is growing symmetrically. In fact, within turbulent boundary layer the mixing layer can grow symmetrically due to the lower velocity of main stream. As soon as the mixing layer interacts with the edge of the turbulent boundary layer it losses the symmetric nature and spreads more i n downstream due to the high momentum of main stream. At the interaction region the contour lines are concentrated but in both upstream and downstream

Separated turbulent boundary layer

Separation shock

is!i Fa . � t; :: .!: '2� ;����i����!������ � .� 0 c:.:::. ,,- .�. _ _ .� ..,:.:::- - -- --0 0. 0 1 . 0 2. 0 3. 0 4 . 0 5 . 0 6 . 0 7. 0 8 . 0 0 . 0 1 0 . 0

Di s tance f rom l e f t boundary (em) Fig. I 3--Temperature contour of the flow field (without injection).

ALI et al. : PHYSICS OF MIXING IN TWO-DIMENSIONAL SUPERSONIC STREAM 1 25

� t = 0.6 IlSee. t = 1 .0 "sec . ... . ; ....

0 : 01 ,

O. 0 1 . 0 ro-..---ro-�� t = 6.2 flSec.

�. 0 1�-dtf-4': O C--u, t = 1 0.4 "sec.

1 . 0 (a) Mole fraction contours of hydrogen.

,.' .--;// . Ii t = 10.4 "sec . ....... -l aycr interaeti?�. ,"

Dislance r r"om l�ft boundary (em) (b) Temperature contour of the growing flowfield.

Fig. 1 4--Temporal growth of side jet interaction and mixing.

they are scattered. The spread of hydrogen is the highest in downstream. The shock-front, caused by the expansion of injector, is evident in temperature contour of the growing flow field shown in Fig. 14b. When mixing layer grows sufficiently, the hindrance of side jet causes to generate an upstream recirculation and makes early separation of upstream boundary layer. This early separation thickens the turbulent boundary layer resulting rapid growth of hydrogen mixing layer. The growing shock-front interacts with the shock of main stream making the separation shock steeper and eventually helps thickening the upstream boundary layer as well as enlarging the upstream re-circulation region. Thus, the boundary layer increases the mixing of injectant in the flow field.

Mixing by now re-circulation

Fig. 1 5 shows the velocity vector near injector of the developed flow field. To analyze the mixing mechanisms, the flow field is divided into three sections; (i) upstream region of injector, (ii) top of in-

I njector Distance (em)

Fig. 1 5-Velocity vector near injector.

jector and the adjacent region, and (iii) downstream region. In upstream there are two re-circulations ; (a) one primary large re-circulation, elongated up to backward-facing step, and clockwise in direction, and (b) one secondary small re-circulation and anticlockwise in direction, caused by the primary recirculation and suction of side injection. Due to interaction between main and injecting flows, the air is slowed down and enters the re-circulation which creates large gradient of hydrogen mass concentration in re-circulation near side jet as well as causes easy diffusion of hydrogen. When this mixture, containing higher mass concentration of hydrogen, comes to the upper side of the re-circulation, it causes large gradient of hydrogen mass concentration which enhances the mixing with air. By this mechanism, hydrogen can reach up to the backward-facing step as well as surrounding regions of upstream primary re-circulation. As on the top of injector there exists a high mass concentration of hydrogen, therefore around the injector, mixing is caused by both diffusion and convection of injection. Immediately downstream of injector there is another re-circulation which enhances mixing by diffusion as described earlier. In far downstream where no re-circulation exists, mixing is occurred mainly by diffusion. In fact, due to re-circulation in upstream region, high amount of hydrogen can mix with air which is carried downstream by velocity of flow, mixes with more air and spreads in large region by expansion and diffusion phenomena.

Conclusions The physics of diffusion and mixing of hydrogen in

supersonic airstream has been studied using twodimensional full Navier-Stokes equations with zeroequation turbulence model. It has been found that the

126 INDIAN 1. ENG. MATER. SCI., APRIL 2002

high penetration and mixing of injectant is not easy due to short residence time of flow in combustor. The upstream re-circulation plays an important role for diffusion and mixing, specially when the Mach number of main stream is high. Investigation shows that the backward-facing step causes early separation of turbulent boundary layer and increases the upstream re-circulation region which can enhance mixing in upstream caused by increasing the mass concentration gradient of injectant. . On the other hand, around the injector, mixing is dominated by convection due to injection and re-circulation while at far downstream mixing is occurred mainly by diffusion. The boundary layer thickness has a significant effect on mixing and by thickening boundary layer penetration and mixing can be enhanced in both upstream and downstream of injector. The enhancement of mixing causes more uniform distribution of hydrogen in downstream of the injector.

Nomenclature

c = sound speed, mls Cp = specific heat at constant pressure, l/(kg.K) D, = turbulent diffusion coefficient, m2/s Dij = binary-diffusion coefficient for species i andj, m2/s E = total energy, 1/m3 F = flux vector in x-direction

A

F G A G

H J J K M

m

u U

A

U

U / v

v w x y y

= transformed flux vector in �-direction = flux vector in y-direction

= transformed flux vector in ll-direction = enthalpy, l/kg = transformation lacobian = number of grid points in x-direction = number of grid points in y-direction = Mach number = mass flux of species, kg/s = pressure, Pa = energy flux by conduction, W/m2

= universal gas constant, J/(kg.mol. K) = Sutherland constant for viscosity, K = Sutherland constant for thermal conductivity, K = temperature, K = effective temperature, K = physical time, s = horizontal velocity, mls = vector of conservative variables = transformed vector of conservative variables

= contravariant velocity in �-direction = vertical velocity, mls = contravariant velocity in ll-direction = molecular weight of species, gmlmol = horizontal cartesian coordinate, m = mass fraction of species = vertical cartesian coordinate, m

z = mole fraction of species � = transformed coordinate in horizontal direction

1) =; transformed coordinate in vertical direction p = mass density, kg/m3 a = normal stress, Pa a£ = effective collision diameter. A r = shear stress, Pa 11 = coefficient of dynamic viscosity, kg/(m.s) K = thermal conductivity, W/(m. K)

QD = diffusion collision integral ill = vorticity, S·I

Superscripts ns = number of species

Subscripts

i,j = index for species = laminar case

m = mixture = index for turbulence

v = viscous term

x = horizontal direction y = vertical direction

xy = reference plane 0 = reference value

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