Converging Shocks: Experiments and Simulations J. Cummings 1, P. E. Dimotakis 2, A. K-W. Lam 2, G. P. Katzenstein 2, R. Samtaney 3 1 Center for Advanced Computing Research, California Institute of Technology 2 Graduate Aeronautical Laboratories, California Institute of Technology 3 Princeton Plasma Physics Laboratory, Princeton University Introduction A collaboration between experiments and simulations has been established with the long-term goal of providing validation for the Virtual Test Facility (VTF). The short-term goals of this collaborative effort are to use the VTF simulations in designing the validation experiments. We seek to provide a test bed for exploration/validation of shock focusing, converging shocks and unsteady converging flows. Design of Gas Interface The initial configuration, shown in Figure 5(a), is that of an incident shock ‘I’ of Mach number upstream of a density interface ‘C’ separating two gases of density  1 =1,  1 and sound speed c 1 (left) and density  2 = , specific heat ratio  2 and sound speed c 2 (right). The final desired configuration, shown in Figure 5(b), shows that the incident shock has completely traversed the interface ‘C’ and results in a transmitted shock ‘T’ which is perfectly circular. We compute the shape of ‘C’ assuming regular refraction of the shock wherein all the shocks and the contact meet at a single point. This configuration, depicted in Figure 5(c), occurs upstream of the final configuration. When ‘I’ refracts at ‘C’ the transmitted shock moves along rays emanating from the wedge apex. Validation Experiments The first set of experiments will be in gases in two dimensional geometries (wedges) using suitably modified shock-tube test-section. Phase 0 experiments will be performed with the planar shock entering the test section with no interface. Since the hinged plates are adjustable, the angle of the upper plate to the horizontal and the angle of the lower plate to the horizontal may differ. In this manner, two experiments may be performed at once. Phase 0 Experiments – Objectives: The first documented shock wave reflection phenomenon was reported by Ernst Mach in the late 19th century. It was then investigated some 60 years later by von Neumann and Bleakney. Experiments were performed the 60’s and 70’s by Glass and other groups. Mach reflections are now a well understood phenomenon. In order to validate the code used for the parallel simulations and as a first step towards producing cylindrical converging shocks, experiments will be performed without a membrane interface. Consequently, a planar shock will propagate through the hinge assembly. In this situation, the hinge plates act as two wedges. We will focus on the transition from Regular Reflection (RR) to Mach Reflection (MR) and vice versa. These RR->MR and MR->RR transitions will investigated using the VTF as well. Phase 1 Experiments: These will be performed with a 2-D gas lens, embodied in an elliptic interface separating gases with differing ratios of specific heat. A thin membrane will be used as an interface. The upper and lower hinged plates will have the same angle to the horizontal. These experiments rely on refraction of finite amplitude planar shock to generate a cylindrical shock conforming to wedge geometry. Prior to shock incidence, the 2D interface is formed and maintained by a thin membrane (few microns thick) that ruptures on interaction with the shock. A shock incidence, refraction and convergence sequence is sketched in Figure 1a. Phase 2 Experiments: These will be performed in the same manner as Phase 1 experiments, The test section will be modified to introduce a second interface part way along the wedge, nominally tangent to a circular arc at that station as shown in Figure 1b. Cylindrical spatial-perturbations will be imposed on the second interface. The interaction of this interface with the cylindrically converging shock will generate a combination of RM and RT (Rayleigh-Taylor) instabilities. For all phases, schlieren will be used as a visualization method, but for Phase 2, laser-induced scattering (LIS) from a vertical laser sheet will also be utilized. Significance of the experiments: The converging shock experiments will be significant in several ways. First, these experiments will demonstrate a methodology for the generation of converging shocks. Secondly, these experiments will be a means to address the issue of converging shock stability. Thirdly, if these shocks are stable, the experiments will permit the study of RM instability in converging geometries. Description of the experimental facility: The experimental facility is the GALCIT 17-inch shock-tube shown in Figure 2. A planar shock is produced in a 17” shock tube of length 24 m (79.3’). At the end of the tube, a box-shaped test section with dimensions 60 cm x 30 cm x 25 cm (24” x 12” x 10”) is attached by a flange aligned to a cookie cutter insert, which removes the boundary layer of the incoming shock. The test section itself has vertical cookie cutters which remove the boundary layer from the sides. It contains a hinge assembly consisting of two plates with sharp leading edges joined by an adjustable hinge. With a range of 5º – 15º to the horizontal, the plates are in full view of the 37 cm (14.5”) wide x 16.5 cm (6.5”) high viewing area. The hinge assembly is accessible from the rear of the test section and may be removed while keeping the plate angles fixed. The cookie cutters, hinge assembly and an exploded view of the test section is shown in Figure 3. Current status: 17” Shock Tube Facility has been upgraded and preparations have been made for converging shock experiments. Phase 0 test section design has been completed. Phase 0 test section fabrication is in progress. Phase 1 test section modifications are presently in detailed design stages. Phase 2 test section modifications are presently in conceptual stage. Simulations of Converging Shocks The simulation effort is involved with (1) the design the shock-lens to achieve a cylindrically converging shocks, (2) to simulations of the converging shocks using the designed lens using nonlinear Euler codes and the fluid-solid coupling algorithms in the VTF to handle the complex experimental geometry, (3) address issues of shock stability to the with the caveat that Euler codes may be imperfect tools to understand shock stability. Verification Test: Guderley Similarity Solution: Imploding cylindrical and spherical shocks collapse self-similarly (Guderley 1942). The radius of the shock was derived as r(t)=C(T c -t) n where T c is the time to collapse. For  =1,4 and cylindrical shocks, n= We simulated a radially inwards moving cylindrical piston. A time sequence of the density field is shown in Figure 4(a) where we observe that the circular shock remains circular in our Cartesian mesh level-set based simulation. Furthermore, from a fit to the numerical simulations in Figure 4(b), we obtain n= which is close to the theoretically predicted value. At any time t, let  (t) be the angle between ‘I’ and ‘C’, and let ‘T’ make an angle  (t) to I. At t=0,  (0)=  (0)=0 and at time t i,  (t i )=  0 and  (t i )=  0 where t i is the time taken by the incident shock to traverse ‘C’ completely. In the equation for t_i, M_{t0} is the transmitted shock strength at the foot of the interface. M t0 can be computed from a one dimensional shock contact interaction. If we neglect local curvature effects, then using local shock polar analysis and assuming the transmitted shock front moves along rays emanating from the wedge apex, a nonlinear system of algebraic equations can be derived. These equations can be expressed as a nonlinear function of  (t), given below, which needs to be solved for every time instant t 2 [0,t i ]. We solve the nonlinear function of  (t) by an iterative bisection method. The above computed interface is used in nonlinear Euler simulations. At time t i the results of the simulations are shown in Figure 7, in which we observe that the transmitted shock is indeed congruent with a circular arc. However, we note that the pressure distribution behind the transmitted shock is not uniform. In fact the pressure varies as p t = p t0 +  sin 2  + O(sin 4  ) where p t0 is the pressure behind the transmitted shock at the foot of the interface. In the simulation the azimuthal pressure variation is about 5%. A time sequence of the density field is shown in Figure 8 in which we have superimposed a circular arc in each frame, whose radius is the distance from the wedge apex to the point where the shock intersects the axes of symmetry. The largest departure of the shock front from circularity, denoted as  /R is at most during the simulation. This simulation essentially verifies that we have obtained a circular shock. In the above equation M tn is the normal Mach number of the transmitted shock. M tn is a function of the local angle  (t) between ‘I’ and ‘C’, and M, ,  1 and  2 ; and L t =Mc 1 t i and L t =M t0 c 2 t i are the distances traveled by the incident and transmitted shocks, respectively. Once  (t) is determined, the local coordinates of the interface are computed as follows. The locus of the above computed points is shown in Figure 6.. It turns out for the proposed experimental parameters of M=1.3122,  =1.4,  1 =1.5,  2 =1.4,  0 = deg., the interface is well approximated by an ellipse of aspect ratio Conclusion and Future Work The GALCIT 17” shock-tube facility will be used for experimental investigations of converging shocks in 2D wedge geometry. The fabrication for Phase 0 experiments is and detailed design of Phase 1 experiments is in progress. These experiments will be used to validate the VTF. We have computed a gas interface which leads to a circular transmitted shock using local shock polar analysis which has been verified using Euler simulations. Thus, the VTF simulations will help design Phase 1 experiments. Our future objective include comparison of VTF simulations with Phase 0 and Phase 1 experiments. Figure 1: (a) Schematic depicting shock incidence, reflection and refraction sequence in Phase 1 experiments. (b) Schematic depicting setup of RM instability experiments in a wedge 1 (a) 1 (b) Figure 2: The GALCIT 17” shock-tube facility. Figure 3: (a) The cookie cutters in the test section. (b) The hinge assembly (c) Exploded view of the test section 3 (a) 3 (c)3 (b) Figure 4: (a) Time sequence of density field in a impulsively started cylindrical piston problem. The shock collapses self-similarly (b) Radius of the shock as a function of time. The slope in this log-log plot is the Guderley similarity exponent. 4 (a) 4 (b) Figure 5: (a) Initial configuration. (b) Final configuration in which ‘T’ is circular. (c) Regular refraction of shock ‘I’ refraction at the contact ‘C’ leads to a reflected shock ‘R’ and transmitted shock ‘T’ in which all waves meet at a single point. 5 (a) 5(b) 5(c) Figure 6: Locus of computed interface points using local shock polar analysis. Figure 7: Pressure (left) and density (right) at t i from Euler simulations of the shock interaction with the gas interface computed using local shock polar analysis. The pressure (density) images are superimposed with circular arc shown in red (black) which shows the transmitted shock has uniform curvature. Figure 8: Time sequence of density field at times t = 3.3t i, 4.9 t i, 6.6 t i, 8.3 t i as the shock travels down the converging section of the wedge. For each frame a circular arc (black) is superimposed. Figure 9: Time sequence of density (left) and pressure (right) in a slice from a 3D shock – interface simulation. In a real experiment, there will be three dimensional perturbations. To simulate these, we performed three dimensional simulations of a shock interaction with a nominally two dimensional elliptic interface. A prescribed spectrum of three dimensional disturbances with randomized phases were superimposed on the gas interface. The simulation was performed on ASCI White (Frost) on 1024 processors, with a resolution of 2000x400x400 mesh points, resulting in over a 1 Tb of data. Preliminary results of this simulation are shown in Figure 9. We observe that while density interface shows the growth of RM-type modes, the transmitted shock front appears smooth and stable. and the shock appears to have achieved the desired shape.