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STUDY OF COMPRESSIBLE SHOCK FOR A STUDY OF COMPRESSIBLE SHOCK FOR A TWO DIMENSIONAL MODEL OF A TWO DIMENSIONAL MODEL OF A

DELAVAL NOZZLEDELAVAL NOZZLE

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What is Computational Fluid What is Computational Fluid Dynamics?Dynamics?

CFD is a method for solving complex fluid flow and heat CFD is a method for solving complex fluid flow and heat transfer problemstransfer problems

STEP 1:-STEP 1:-CFD divides a flow area into a large number of CFD divides a flow area into a large number of cells or control volumes, collectively referred to as the cells or control volumes, collectively referred to as the “mesh” or “grid”.“mesh” or “grid”.

STEP 2:-STEP 2:-In each of the cells ,the Navier Stokes In each of the cells ,the Navier Stokes Equations ,i.e. the partial differential equations that Equations ,i.e. the partial differential equations that describe fluid flow are rewritten algebraically, to relate describe fluid flow are rewritten algebraically, to relate such variables as pressure, velocity and temperature in such variables as pressure, velocity and temperature in neighboring cells.neighboring cells.

STEP 3:-STEP 3:-The equations are then solved numerically The equations are then solved numerically yielding a picture of the flow corresponding to the level of yielding a picture of the flow corresponding to the level of resolution of the mesh.resolution of the mesh.

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Overview of the ProblemOverview of the Problem Description of the problem.Description of the problem. Governing Equations and their dicretised finite volume Governing Equations and their dicretised finite volume

formform Shock in a Delaval nozzle and its importance.Shock in a Delaval nozzle and its importance.

Developing the two dimensional model according to the Developing the two dimensional model according to the geometric equations.geometric equations.

Applying Boundary conditions and initial conditions.Applying Boundary conditions and initial conditions. Modification of the model in order to visualize the shockModification of the model in order to visualize the shock Analysis of the results from the contours and plots for both Analysis of the results from the contours and plots for both

viscid and inviscid cases.viscid and inviscid cases. ConclusionsConclusions

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( ) 1.75 0.75cos[2( 0.5) ] 0.0 0.5( ) 1.25 0.25cos[2( 0.5) ] 0.5 1

A x x for xA x x for x

The Equations that describe the nozzle geometry are

Description of the problem

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Governing Equations for Inviscid Form of NS Equations

22

( ) ( )

u vuvu u pU E Fv uv v p

e u e p v e p

Governing Equations for the K-Epsilon form of the compressible NS Equations

21 3 2

2

ij

(Pr )modulus of mean strain tensor= 2S

Tmk b

i ik

Tk b

j i

tk

ij

Dk k G G YDt x x

D C G C G CDt x x k k

G oduction of turbulent kinetic energy Swhere S S

S

2

1 2 3

12

( ) Pr2

, , tan ,

jiij

j i

tib

t i

m t

k

uumean strain rate x x

pG Generation of turbulence due to buoyancy g xY Dilation dissipation term MC C C are cons ts and are turbulent prandtl numbers

0U E Ft x y

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FINITE VOLUME DISCRETISATION OF THE GENERAL CONSERVATION FORM

The conservation principles are applied to a fixed region in space knownAs a control volume.Cell centered Finite Volume scheme.We assume that all properties in the center are averaged over each cell We calculate the values of the conserved variables averaged across the volumes.

0U E Ft x y

a

b

c

j+1

i,j

i+1

i-1

j-1d

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abcd

Integrating this equation over the finite volume method abcd from the figure,

0

where dV=dxdy(1).After applying Green's theorum,we get

(1) . 0t abcd abcd

U E F dVt x y

U dxdy H ndS

H

1, ,

1 1 1 1, , , ,2 2 2 2

1 1 1 1, , , ,2 2 2 2

. ( )(1)

( ) 0

( )

( ) 0

abcd abcd

n ni j i j

abcd ab bc cd dai j i j i j i j

ab bc cd dai j i j i j i j

Ei Fj H nds Edy Fdx

Substituting we get Udxdy Edy Fdxt

U US E y E y E y E yt

F x F x F x F x

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, ,1 , 1 1 , 1, ,2 2

, ,1 1, 1 1,, ,2 2

, ,1 , 1 1 , 1, ,2 2

, ,1 1, 1 1,, ,2 2

0.5( ) 0.5( )

0.5( ) 0.5( )

0.5( ) 0.5( )

0.5( ) 0.5( )

i j i ji j i ji j i j

i j i ji j i ji j i j

i j i ji j i ji j i j

i j i ji j i ji j i j

E E E F F F

E E E F F F

E E E F F F

E E E F F F

Substitu

1, , 1, 1, , 1 , 1

exp ,

02 2

n ni j i j i j i j i j i j

ting these ressions

E E F FU Uwe finally get t x x

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What is What is SHOCKSHOCK??• A surface or sheet of discontinuity (i.e., of abrupt A surface or sheet of discontinuity (i.e., of abrupt

changes in conditions) set up in a changes in conditions) set up in a supersonicsupersonic field field or or flowflow, through which the fluid undergoes a finite , through which the fluid undergoes a finite decrease in velocity accompanied by a marked decrease in velocity accompanied by a marked increase in pressure, density, temperature, and increase in pressure, density, temperature, and entropy.entropy.

• In other words ,Shock is a thin section of a control In other words ,Shock is a thin section of a control volume where there are abrupt changes in fluid volume where there are abrupt changes in fluid properties.properties.

• Formed due to the compressibility effects of fluids.Formed due to the compressibility effects of fluids.• Normal shock occurs in a plane perpendicular to the Normal shock occurs in a plane perpendicular to the

flow.flow.

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What happens when there is a What happens when there is a shock?shock?

Fluid velocity increases in converging part Fluid velocity increases in converging part lowering pressure until max velocity reaches lowering pressure until max velocity reaches at throat (Mach=1).at throat (Mach=1).

Diverging section continues to accelerate Diverging section continues to accelerate fluid to supersonic speed with continued fluid to supersonic speed with continued drop in pressure until we hit region of normal drop in pressure until we hit region of normal shock in nozzle.shock in nozzle.

At normal shock region pressure and At normal shock region pressure and temperature increase across the wave and temperature increase across the wave and velocity drops.velocity drops.

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POST PROCESSINGOrganization and interpretation

Of the data and imagesUsing FLUENT

PRE-PROCESSING1)Building the model and applying the mesh(HYPERMESH)2)Where to apply the boundary conditions(GAMBIT)

SOLVINGCalculating and producing the resultsUsing FLUENT

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CONDITIONS FOR THE VISCOUS MODEL

SOLVER-SEGREGATED

MODEL- K-epsilon

Operating Pressure- 101325 Pascal

Inlet Conditions- Mass Flow Inlet Mass flow rate-1.3 kg/sec Total temperature-300 KTurbulence Specification method-Intensity and Hydraulic Diameter

Outlet conditions - Pressure Outlet Outlet Pressure-88000 Backflow total Temperature-300K

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CONDITIONS FOR THE INVISCID MODEL

SOLVER-COUPLED

MODEL- INVISCIDOperating Pressure- 101325 Pascal

Inlet Conditions- Mass Flow Inlet Mass flow rate-1 kg/sec Total temperature-300 KTurbulence Specification method-Intensity and Hydraulic Diameter

Outlet conditions - Pressure Outlet Outlet Pressure-88000 Backflow total Temperature-300K

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Mach number contours Pressure contours

CONTOURS FOR THE ACTUAL GEOMETRIC MODEL

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MODIFIED MODEL OF DELAVAL NOZZLEMODIFIED MODEL OF DELAVAL NOZZLE

THROAT

CONVERGING REGION

DIVERGING REGION

GRID HAS BEEN STRETCHED IN THE X DIRECTION BY FIVE TIMES TOTAL LENGTH OF MODEL IN X DIRECTION IS 5 mmTOTAL LENGTH OF MODEL IN Y DIRECTION IS 5 mm

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Regions of shock

VELOCITY CONTOURS FOR THE VISCOUS CASE

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MACH NUMBER CONTOURS FOR THE VISCOUS CASEMACH NUMBER CONTOURS FOR THE VISCOUS CASE

Shock Mach Number

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TEMPERATURE CONTOURS FOR THE VISCOUS CASE

Temperature increases

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PRESSURE CONTOURS FOR THE VISCOUS CASE

Effect of Back Pressure

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VELOCITY CONTOURS FOR THE INVISCID CASE-COUPLED SOLVERVELOCITY CONTOURS FOR THE INVISCID CASE-COUPLED SOLVERSharp change in velocity

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MACH NUMBER CONTOURS FOR THE INVISCID CASE-COUPLED MACH NUMBER CONTOURS FOR THE INVISCID CASE-COUPLED SOLVERSOLVER

Sharp change in mach number

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TEMPERATURE CONTOURS FOR THE INVISCID CASE-COUPLED TEMPERATURE CONTOURS FOR THE INVISCID CASE-COUPLED SOLVERSOLVER

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PRESSURE CONTOURS FOR THE INVISCID CASE-COUPLED PRESSURE CONTOURS FOR THE INVISCID CASE-COUPLED SOLVERSOLVER

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X Vs Mach number

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

-1 0 1 2 3 4 5 6

x in mm

mach nu

mbe

r

Y=0

Y=0.001

X Vs mach number

0

0.2

0.4

0.6

0.8

1

1.2

1.4

-1 0 1 2 3 4 5 6

x in mm

mac

h nu

mbe

r

Y=0

Y=0.001

COMPARISON OF MACH NUMBER FOR VISCOUS AND INVISCID CASESVISCOUS INVISCID

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x Vs Velocity

0

50

100

150

200

250

300

350

400

450

-1 0 1 2 3 4 5 6

x in mm

Vel

ocity

in m

/sec

Y=0

Y=0.001

X Vs Velocity

0

50

100

150

200

250

300

350

400

450

-1 0 1 2 3 4 5 6

X in mm

Veloc

ity in

m/sec

Y=0

Y=0.001

COMPARISON OF VELOCITY FOR VISCOUS AND INVISCID CASES

VISCOUS INVISCID

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X Vs Pressure

-60000

-40000

-20000

0

20000

40000

60000

80000

100000

120000

-1 0 1 2 3 4 5 6

x in mm

Press

ure in Pas

cals

Y=0

Y=0.001

X Vs pressure

-20000

0

20000

40000

60000

80000

100000

120000

140000

160000

180000

-1 0 1 2 3 4 5 6

x in mm

pres

sure in

pas

cals

Y=0

Y=0.001

COMPARISON OF PRESSURE FOR VISCOUS AND INVISICID CASES

VISCOUS INVISCID

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X Vs temperature

200

210

220

230

240

250

260

270

280

290

300

-1 0 1 2 3 4 5 6

x in mm

tempe

rature in

Kelvin

Y=0

Y=0.001

X Vs temperature

220

230

240

250

260

270

280

290

300

310

-1 0 1 2 3 4 5 6

x in mm

tempe

rature in

Kelvin

Y=0

Y=0.001

COMPARISON OF TEMPERATURE FOR VISCOUS AND INVISCID CASES

VISCOUS INVISCID

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Purpose of Shock Purpose of Shock Analysis?Analysis?

Finding fluid properties before and after a Finding fluid properties before and after a shockwave.shockwave.

Understanding of normal shock is vital for Understanding of normal shock is vital for effects of high speed nozzle flow.effects of high speed nozzle flow.

Allows us to find the best fluids to use in Allows us to find the best fluids to use in an application.an application.

Setting the back pressure( Pressure Setting the back pressure( Pressure applied at the nozzle discharge region)applied at the nozzle discharge region)

CONCLUSIONS

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