Numerically constrained one-dimensional interaction of a propagating planar shock wave

Department of Aerospace Engineering,Indian Institute of Technology, Bombay

Mumbai 400076, INDIA

A. Chatterjee

1D problem – numerically constrained interaction of a propagating planar shock wave

Shock Wave

x1 x2xsw

u(x, t) uc(x, t)

• Rightward planar propagating shock wave• uc(x, t) : arbitrary imposed flowfield downstream of shock wave constrains development of flowfield (u(x,t)) behind propagating shock wave

• xsw : current position of shock wave

Algorithm:

Proposed algorithm :

Unsteady 1D Euler equations in [x1 , x2] :

x1 < x < x2 x1 < xsw ( t + t ) < x2

(t + t )= Position of shock wave in [x1 , x2] at obtained from a pressure based sensor

xsw ( t + t )

H [u(x, t)]

= Explicit solution in [x1 , x2] at (t + t )

uc (x,t) = constraining flowfield downstream of moving shock wave

(3rd order ENO and 2nd order TVD Runge-Kutta)

u( x , t+ t) = H [u(x, t)] x < xsw ( t + t )

t + t )uc (x , x > xsw ( t + t )

(t + t )t

Test Case:

Constrained Interaction of Planar (rightward) Propagating Shock wave • Unsteady interaction of Mach 3 shock wave with Sine entropy wave

(Shu & Osher)

Initial Conditions:

V(x,0) =Vl x < - 4

Vr x - 4

Vl lul,pl)=(3.8571143, 2.629369, 10.333333)

Vr rur,pr)=(1+ 0.2sin(5x), 0, 1) x [-5 : 5]

Validation:

= H [u(x,

t)]

u( t + t )

Without Constrain (regular solution)

With Constrain

in [-5 : 5]

u( x , t+ t) = H [u(x, t)]

( t + t )x xsw

Solution Methodologies:

c(x, )uc(x, ), pc(x , ) ) = ( 1+0.2sin(5x), 0, 1 ) t + t t + t t + t

( t + t )x > xsw

Shock/entropy wave interaction( time=1.8)

Numerical Validation:

Shock/entropy wave interaction( time=1.8)

Numerical Validation:

Application: 2D Shock-vortex interaction problem

An initially planar shock wave interacts with a 2D compressible vortex superposed on ambient resulting in creation of acoustic waves and secondary shock structures.

U(r) = r1

BU(r) = Ar + r

0 < r < r1

rUmax

r1 r r2

Experimental Condition: (Dosanjh & Weeks, 1965)

Ms = 1.29Umax= 177 m/s (Mv=0.52)r1 = 0.277 cm

r2 = 1.75 cm

( Compressible vortex model )

Strong interaction with secondary shock formation

Application: a possible “constrained numerical experiment”:

• Solving numerically a reduced model of complex unsteady shock wave

phenomenon with appropriate constrains

• Demonstrate role of purely translational motion of an initially planar shock

wave in secondary shock structure formation when interacting with 2D

compressible vortex

• Planar shock wave interact with 1D flow field (uc)

• uc represents initial flowfield along vortex model normal to shock wave

.x1 x2

uc (x)

xsw

Computational Domain: [ 0, 20]

Initial position of shock = 8. 25 cm

Properties behind the shock : R-H condition

No. of cells : 900 equally spaced

uc constraining flowfield ahead of shock centered at 10.0 cm

• uc controls development of the flowfield behind shock wave (example of an

arbitrary constraining flowfield)

• Ignores shock wave (and vortex) deformation

Application …..

uc downstream of normal shock

Velocity distribution along horizontal lines (cases 1 to 4)

Case 1 & 2 : y 0.45

Case 3 & 4 : y 1.25

Vortex center y=0.0

Results:

Pressure Profiles (Case 1)

T1 = Start of the simulationT6 = Shock wave almost out of domain

Pressure Profiles (Case 2)

Case 1 & 2 : 1.25 (farther from vortex center)

generation of acoustic waves

Results: Case 3 & 4 : 0.45 (near vortex center)

Pressure Profiles (Case 3)

Pressure Profiles (Case 4)generation of upstream moving shocklet

• An algorithm proposed for constrained one dimensional interaction of a planar propagating shock wave.

• Validated for 1D shock-entropy wave interaction.

• Constraining flowfield can be “arbitrary”.

• Allows setting up a “constrained numerical experiment” otherwise not possible.

Conclusions: