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Numerical Simulation of Shock Wave and Boundary Layer Interactions on Premixed Combustible Gas Flows over a Compression Ramp

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International Journal of Emerging Technology and Advanced Engineering

Website: www.ijetae.com (ISSN 2250-2459,ISO 9001:2008 Certified Journal, Volume 4, Issue 12, December 2014)

293

Numerical Simulation of Shock Wave and Boundary Layer

Interactions on Premixed Combustible Gas Flows over a

Compression Ramp

S. Shyji

1

1S.C.T. College of Engineering, Trivandrum, Kerala, India Abstract- Numerical solution is a cost effective alternative

in various parametric studies including fluid dynamics problems. The interactions of a shock wave produced by the compression ramp on a premixed, chemically reacting combustible gases is presented here. This work is carried out using the development and testing of a solver, based on the solution of unsteady, compressible, turbulent standard Navier-Stokes equations using Unstructured Finite Volume Method (UFVM) with K- Renormalisation Group (RNG) Model. For the simple explicit scheme three stage Runge-Kutta time stepping scheme is employed. An eight-step hydrogen-air finite rate chemistry model is used to simulate the reacting flow field. A point Implicit method is employed here in which the simplicity of the explicit method is taken in fluid flow problem and the stiffness problem of the chemical combustion is avoided using the implicit treatment of the chemical terms. The present study gives fairly better agreement with the theoretical solutions.

Keywords—turbulence, reacting flows, stiffness, point implicit, compression ramp

I. INTRODUCTION

The flow field over a compression ramp is formed by the intersection of a flat plate with a wedge of angle 100 with the horizontal. This is a simplified model representing the flow around a deflected aerodynamic control surface. When a flow of combustible gases flows over the compression ramp, the interaction between them causes the initial pressure distributions to change, thus leading to a variation of overall forces loaded on the surface. The compression ramp configuration has been studied extensively, both experimentally and numerically, and there are numerous experimental data available for this configuration [1, 2]. The interactions of a shock wave produced by the compression Ramp on a premixed, chemically reacting combustible gases is studied here.

Simulations based on Navier-Stokes equations are now widely used in the determination of optimum performance evaluations and in arriving at optimum configurations. The supersonic reacting flow field over the compression ramp can be solved by adding finite rate chemistry to the standard compressible Navier-Stokes equation. The present solver used an Renormalization Group (RNG) based - two-equation turbulence model. The explicit treatment of all conservation terms, especially with the chemical source terms, results in stiffness and it degrades the overall performance of numerical method as the phenomenon of different time scales are solved simultaneously. Bussing and Murman [3] have introduced the method of preconditioning the conservation equations, wherein the treatment of chemical source terms alone are carried out implicitly and the remaining Navier-Stokes equations explicitly. This found to have the advantage of both explicit and implicit method and an even higher CFL almost equal to that of non-reacting situations can be achieved successfully. The developed solver is based on the two-dimensional Navier-Stokes equation governing compressible turbulent flows. The time integration is done using three- stage Runge-Kutta method. For modeling hydrogen-air reaction, an eight-step reaction mechanism proposed by Evans and Schexnayder [4], has been used.

II. GOVERNING EQUATIONS

The governing equations ofthe two-dimensional turbulent compressible flow in the conservation form can be expressed as,

 

S

dy

G

G

d

dx

F

F

d

dt

(2)

International Journal of Emerging Technology and Advanced Engineering

Website: www.ijetae.com (ISSN 2250-2459,ISO 9001:2008 Certified Journal, Volume 4, Issue 12, December 2014)

294 Where,                        i Y k E v u U



                         i uY u uk u P E uv P u u F

2

                         i vY v vk v P E P v vu v G

2                                        dx dY D dx d dx du Q F i k x xy xx v ' 0                                                dy dY D dy d dy dv Q G i k y yy xy v ' 0      

i k

H

H

G

0

0

0

0

i=1,2,….,Ns where Ns is the number of species considered.

dx

dT

k

v

u

Q

x

xx

xy

eff

dy

dT

k

v

u

Q

y

xy

yy

eff

x y

x

xx

u

u

v

3

2

2

y x

xy

u

v

x y

y

yy

u

u

v

3

2

2

t l

The turbulent viscosity µtcan be found from the selected turbulence model and the laminarviscosity µlis foundusing the Sutherland’s law of Laminar Viscosity

,

4

.

110

10

458

.

1

5 . 1 6

T

T

l

t l

eff

k

k

k

Where,

Pr

p l l

C

k

and

Pr

p t t

C

k

The Diffusion coefficient can be calculated from the equation, t t l l

Sc

Sc

D

'

III. TURBULENT MODELING

The modified k-ε model called renormalisation group (RNG) model proposed by Yakhot, et al[5].was used which removes all the small scales of turbulence motion from the governing equation by expressing their effects in terms of large scales and a modified viscosity.

The turbulent source terms are given by,

The turbulent viscosity is defined as,

The closure coefficients are,

 

 

k eff

k gradk H

div kU div dt

k

d

.    

 

 

      H grad div U div dt d eff k    .



t ij

k E H 2 k C E E k C

H t ij ij

2 2 * 1 2

    

t

C

k

2

68

.

1

,

42

.

1

,

39

.

1

,

08

.

0

1

2

C

C

C

k 3 0 1 * 1

1

1



 

 

C

C

.

,

0

4

.

37

,

0

.

01

k

(3)

International Journal of Emerging Technology and Advanced Engineering

Website: www.ijetae.com (ISSN 2250-2459,ISO 9001:2008 Certified Journal, Volume 4, Issue 12, December 2014)

295

All parameters exept the constant β are explicitely computed from the RNG calculations and β from the near wall turbulence data.

IV. CHEMISTRY MODELING

For modeling hydrogen-air reaction, an eight-step reaction mechanism proposed by Evans and

Schexnayder[4], has been used and is given in Table 1.

Reaction A N E

H2+M↔H+H+M 5.5 x 1018 -1.0 51987.0

O2+M↔O+O+M 7.2 x 1018 -1.0 59340.0

H2O+M↔H+OH+M 5.2 x 1021 -1.5 59386.0

OH+M↔O+H+M 8.5 x 1018 -1.0 50830.0

H2O+O↔OH+OH 5.8 x 1013 0.0 9059.0

H2O+H↔OH+ H2 8.4x 1013 0.0 10116.0

O2+H↔OH+O 2.2 x 1014 0.0 8455.0

H2+O↔OH+H 7.5 x 1013 0.0 5586.0

Reaction rates are expressed in Arrhenius law form

 

RT

E

AT

k

n

exp

and M is a third body.

 

N

i

N

i b f

ji ji

i ji

i ji

i

C

k

C

k

C

1 1

' ''

. ' ''

Where i = 1,2,3…. represents species and j = 1,2,3…

represents reactions.

The net change in concentration of any species can be found as,

N

i i

i

C

C

1 . .

and the net production of species is given by,

i i

i

W

.

C

The forward and backward reaction rates in the above equation are the functions of temperature, that can be calculated using the Arrhenius law,

 

RT

E

AT

k

n

exp

The reverse rate reaction is calculated using the equilibrium constant.

V. THERMODYNAMIC MODELING

The evaluation of thermodynamic properties such as specific heat at constant pressure, and enthalpy can be calculated from the standard thermodynamic data (McBride and Gordon) as,

4 3 2

T E T D T C T B A R C

i i i i i

pi

T

F

T

E

T

D

T

C

T

B

A

RT

H

i i i i i

i

i

2

3

4

5

4

3

2

Total energy of flow field is given by,

2 2

1

5

.

0

u

v

P

h

Y

E

i

N

i i

i

Temperature is worked out from the above equation using the Newton-Raphson method. The pressure is calculated from the resulting temperature as follows:

T

W

Y

R

p

i

N

i i

i

1

VI. NUMERICAL METHOD

Basically, finite volume technique is an integration of conservation laws. In other words, mass, momentum and energy should be conserved at the basic discrete level. The conservation equation applicable for a cell [6] is:

0

.

S

F

t

U

Applying Greens theorem,

0

.

 

d

S

dT

F

d

t

U

n

Where, is𝜞 the total surface area and Ω is the cell volume.

For the simple explicit scheme, the time stepping using Runge-Kutta method can be described as,

n i

i

U

U

0

0 0

1 0 1

i i i

i i

i

R

D

V

t

U

U

1 0

2 0 2

i i i

i i

i

R

D

V

t

U

(4)

International Journal of Emerging Technology and Advanced Engineering

Website: www.ijetae.com (ISSN 2250-2459,ISO 9001:2008 Certified Journal, Volume 4, Issue 12, December 2014)

296

n

i n i i i i

i i

i R D U U

V t U

U   3 2 0 1 0

3

6

.

0

,

6

.

0

2

1

and

3

1

Due to stiffness problem, a point implicit method is used in which, Governing equations of turbulent shear layer flows are treated explicitly whereas the equations involving the finite rate chemistry are solved implicitly. Thereby, higher time steps can be achieved. Thus the modified equation is,

S

y

G

x

F

t

U

SJ

The point implicit formulation of the time stepping can be written as,

n i i

U

U

0

0 0

1 0 1 0

i i i

i i

i

R

D

V

t

U

U

SJ

1 0

2 0 2 1

i i i

i i

i

R

D

V

t

U

U

SJ

2 0

3 0 3 2

i i i

i i

i

R

D

V

t

U

U

SJ

3 1

i n i n

i

U

U

U

6

.

0

,

6

.

0

2

1

and

3

1

In the point implicit scheme, all the six chemical species (H2, O2, H2O, OH, H, O) are treated implicitly, while all the

other species are treated explicitly.

VII. RESULTS AND DISCUSION

The flow field over a compression ramp is formed by the intersection of a flat plate with a wedge of angle 100 to the horizontal. The flat plate is given a length of 50 mm and the ramp of another 50mm with 100inclination to the horizontal. Typical Scramjet combustor operating conditions are given at the inlet. The premixed fuel (with 0.0285 mass fraction) and the oxidizer (with 0.226 mass fraction) are given with a Mach Number 4.0 at 900 K and 1 atmospheric pressure.

Free slip adiabatic wall conditions conditions are given on the lower wall surface up to the ramp and no slip in the lower ramp side. The upper side of the flow field is also given free slip adiabatic wall conditions.Supersonic exit conditions are given at the geometry exit.

The chemical reaction is set to start at 1000 K.Hence, the flow continues with the inlet conditions up to the leading edge of the ramp. Once the flow enters the leading edge, an oblique shock is formed from the tip of the leading edge and the temperature downstream of the leading edge starts increasing. The temperature now crosses 1000 K and the chemical reaction between the premixed fuel and the oxidizer begins. The temperature now grows further and reaches around 1200 K. A boundary layer is also formed in the slanting side of the wall, since the no-slip boundary condition is set there. Because of the boundary layer interactions, the oblique shock now get reflected and the turbulence in this region increases in a large extend. This will increase the chemical reaction rate and the temperature reaches of the order of 2200 K (Fig 1).

The Mach number downstream of the leading edge decreases because of the oblique shock. The reflection of the oblique shock and the intense turbulence thereafter decreases it further (Fig 2).

(5)

International Journal of Emerging Technology and Advanced Engineering

Website: www.ijetae.com (ISSN 2250-2459,ISO 9001:2008 Certified Journal, Volume 4, Issue 12, December 2014)

297 The Pressure downstream of the leading edge increases because of the oblique shock. The reflection of the oblique shock and the intense turbulence thereafter increases it further (Fig 3).

Till the leading edge of the ramp, there will not be any chemical reaction. As the reaction begins, the fuel and oxidizer get consumed (Fig.4) and the water and the hydroxyl mass fraction increases (Fig.5).

Fig.3Pressurealong the bottom wall of the ramp

Fig.4. Mass Fractions along the bottom wall of the ramp

(6)

International Journal of Emerging Technology and Advanced Engineering

Website: www.ijetae.com (ISSN 2250-2459,ISO 9001:2008 Certified Journal, Volume 4, Issue 12, December 2014)

298

VIII. CONCLUSION

The interactions of a shock wave produced by the compression ramp and its boundary layer on a premixed, chemically reacting combustible gases is investigated. The point implicit method avoids the stiffness problem and gives realistic results in standard supersonic combustor conditions.

REFERENCES

[1] YiguangJu, “Lower-Upper Scheme for Chemically Reacting Flow with Finite Rate Chemistry”,AIAA Journal, 1995, 33(8), 1418-1425. [2] ShayeYungster, “Numerical Study of Shock-Wave/Boundary-LayerInteractions in Premixed Combustible Gases”, AIAA Journal, 1992, 30(10), 2379-2387.

[3] Bussing, T. R. A., and Murman, E. M., “Finite Volume Method for the Calculation of Compressible Chemically Reacting Flows”, AIAA Journal, 26(4), 1988, 1070-1078.

[4] Evans, J. S., and Schexnayder, C. J., “Influence of Chemical Kinetics and Unmixedness on Burning Supersonic Hydrogen Flames”, AIAA Journal, 18(1), 1980¸ 188-193.

[5] Yakhot, V. et al. "Development of turbulence models for shear flows by a double expansion technique." Physics of Fluids A: Fluid Dynamics (1989-1993) 4.7 (1992): 1510-1520.

[6] Hirsch, Charles. Numerical computation of internal and external flows: the fundamentalsof computational fluid dynamics. Vol. 1. Butterworth-Heinemann, 2007.

[7] M. Deepu, S. S. Gokhale and S. Jayaraj,"Modeling of turbulent supersonic reacting shear flows using point implicit finite volume method".,16th international symposium on transport phenomena.,ISTP-16,2005,Prague.

Fig.6 Mach numbercontours along the wall surface

Fig.7 Temperature in Kcontours along the wall

Fig.8Pressure(MPa)contours along the wall surface

References

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