1. 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

2. 1D problem – numerically constrained interaction of a propagating planar shock wave Shock Wave x1x1 x2x2 x sw u(x, t)u c (x, t) Rightward planar propagating shock wave u c (x, t) : arbitrary imposed flowfield downstream of shock wave constrains development of flowfield (u(x,t)) behind propagating shock wave x sw : current position of shock wave

3. Algorithm: Proposed algorithm : Unsteady 1D Euler equations in [x 1, x 2 ] : x 1 < x < x 2 x 1 < x sw ( t + t )< x 2 (t + t ) = Position of shock wave in [x 1, x 2 ] at obtained from a pressure based sensor x sw ( t + t ) H [u(x, t)] = Explicit solution in [x 1, x 2 ] at (t + t ) u c (x,t) = constraining flowfield downstream of moving shock wave (3 rd order ENO and 2 nd order TVD Runge-Kutta) u( x, t+ t) = H [u(x, t)] x < x sw ( t + t ) u c (x, x > x sw ( t + t ) (t + t )t

4. 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) = V l x < - 4 Vr x  - 4 V l  l  u l,p l )=(3.8571143, 2.629369, 10.333333) Vr  r  u r,p r )=(1+ 0.2sin(5x), 0, 1)  x [-5 : 5] Validation:

5. = 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  x sw Solution Methodologies:  c (x, )  u c (x, ), p c (x, ) ) = ( 1+0.2sin(5x), 0, 1 ) t + t ( t + t ) x > x sw

6. Shock/entropy wave interaction( time=1.8) Numerical Validation:

7. Shock/entropy wave interaction( time=1.8) Numerical Validation:

8. 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) = r1r1 B Ar + r 0 < r < r 1 r U max r 1  r  r 2 Experimental Condition: (Dosanjh & Weeks, 1965) M s = 1.29 U max = 177 m/s (M v =0.52) r 1 = 0.277 cm r 2 = 1.75 cm ( Compressible vortex model ) Strong interaction with secondary shock formation

9. 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 (u c ) u c represents initial flowfield along vortex model normal to shock wave. x1x1 x2x2 u c (x) x sw

10. Computational Domain : [ 0, 20] Initial position of shock = 8. 25 cm Properties behind the shock : R-H condition No. of cells : 900 equally spaced u c constraining flowfield ahead of shock centered at 10.0 cm u c controls development of the flowfield behind shock wave (example of an arbitrary constraining flowfield) Ignores shock wave (and vortex) deformation Application …..

11. u c 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

12. Results: Pressure Profiles (Case 1) T 1 = Start of the simulation T 6 = Shock wave almost out of domain Pressure Profiles (Case 2) Case 1 & 2 :  1.25 (farther from vortex center) generation of acoustic waves

13. Results: Case 3 & 4 :  0.45 (near vortex center) Pressure Profiles (Case 3) Pressure Profiles (Case 4) generation of upstream moving shocklet

14. 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: