Gas- Solid Cases:
Case 4.1 : This problem is of [S|EL]e type. A strong shock wave is generated at the interface and moves through the fluid to the left. In solid, the linear elastic model
Semi Infinite Elastic Solid
Water High pressure Gas
4.2.1.6 show the velocity, pressure and stress distribution in the gas and the solid medium, which are in good agreement with the analytical (exact) solution. The shear stress is zero everywhere in the solid medium, which agrees with the assumption of the 1D wave propagation in the solid.
Case 4.2 : This problem may be classified as [S|EL]e. A weak shock wave passes through the gas, and there is a elastic wave traveling to the right through the elastic media. Shear stress is zero everywhere in the solid. Figs. 4.2.2.1 to 4.2.2.6 show the velocity, pressure and stress distribution in the gas and the solid medium, which are in good agreement with the analytical (exact) solution.
Case 4.3 : The solution at the interface may be termed as [S|EL]e type as the pressure at the interface is larger than the initial pressure level on either medium. A shock wave travels through the gas to the left and an elastic wave to the right in the solid.
Figs. 4.2.3.1 to 4.2.3.6 show the velocity, pressure and stress distribution in the gas and the solid medium, which are in good agreement with the analytical (exact) solution. Shear stress is zero everywhere in the solid.
Case 4.4 : A rarefaction wave is generated at the interface and travels through the gas to the left. There is only a elastic wave traveling to the right through the solid. Hence, this is a [R|EL]e type problem. Figs. 4.2.4.1 to 4.2.4.6 show the velocity, pressure and stress distribution in the gas and the solid medium, which are in good agreement with the analytical (exact) solution. Shear stress is zero everywhere in the solid.
Case 4.5 : An unloading elastic wave propagates through the solid medium. A rarefaction wave travels through the gas to the left. Hence, this problem can be classified as [R|EUL]e. Shear stress is zero everywhere in the solid. Figs. 4.2.5.1 to 4.2.5.6 show the velocity, pressure and stress distribution in the gas and the solid medium, which are in good agreement with the analytical (exact) solution.
Water Solid Cases:
Case 4.6 : This is a [S|EL]e type problem. In this case a shock wave is generated that moves through the water to the left. Figure 4.2.6.3. indicates that the water- solid interface moves to the left for the elastic case. In the solid side, an elastic wave is identified. Shear stress is zero everywhere in the solid. Figures 4.2.6.1 to 4.2.6.6 show the velocity, pressure and stress distribution in the gas and the solid medium, which are in good agreement with the analytical (exact) solution.
Case 4.7 : The water-elastic solid interaction predicts a weak shock wave in the fluid side. It also predicts an elastic loading wave traveling to the right in the solid side.
Hence this problem may be classified as [S|EL]e. Shear stress is zero everywhere in the solid. Figures 4.2.7.1 to 4.2.7.6 show the velocity, pressure and stress distribution in the gas and the solid medium, which are in good agreement with the analytical (exact) solution.
Case 4.8 : A shock wave is generated and travels through the water to the left. This is [S|EL]e type problem. An elastic wave is found to travel towards right through the solid medium. Shear stress is zero everywhere in the solid. Figures 4.2.8.1 to 4.2.8.6 show
the velocity, pressure and stress distribution in the gas and the solid medium, which are in good agreement with the analytical (exact) solution.
Case 4.9 : There is a shock wave generated at the interface and travels to the left in the fluid medium. In the solid medium an elastic wave is observed. This is a [S|EL]e
type problem. Figures 4.2.9.1 to 4.2.9.6 show the velocity, pressure and stress distribution in the gas and the solid medium, which are in good agreement with the analytical (exact) solution. Shear stress is zero everywhere in the solid.
Case 4.10: This is a [R|EL]e type problem. A rarefaction wave is generated at the interface and travels through the water medium to the left. In the solid medium, the elastic model predicts the elastic wave traveling to the right. Shear stress is zero everywhere in the solid. Figures 4.2.10.1 to 4.2.10.6 show the velocity, pressure and stress distribution in the gas and the solid medium, which are in good agreement with the analytical (exact) solution.
Case 4.11: The elastic model predicts an unloading elastic wave in the solid medium.
Rarefaction wave is generated at the interface and moves to the left through water.
Hence, this problem can be classified as [R|EUL]e. Figures 4.2.11.1 to 4.2.11.6 show the velocity, pressure and stress distribution in the gas and the solid medium, which are in good agreement with the analytical (exact) solution. Shear stress is zero everywhere in the solid.
4.2.3.1. Shock-Tube Type Problems In Which The Wave Propagates In Directions Other Than Coordinate Direction:
Case 4.12 : In Fig. 4.2.12.1. and Fig. 4.2.12.2., the pressure and normal velocity distribution at a plane (y = 0.5) are presented. These show that the pressure (or negative normal stress) and the normal velocity are continuous across the interface. It suggests that the interface Riemann problem accurately predicts the interface pressure and normal velocity. However, the assumption made about the shear stress in the s-n coordinate and the tangential velocity to be zero in the solid is not satisfied accurately.
Figure 4.2.12.5 and Fig. 4.2.12.11 show that the tangential velocity component suffers from oscillation in both the fluid and the solid domains. Again, Fig. 4.2.12.9 shows that the shear stress in the s-n coordinate σsnis not zero in the solid side and suffers from oscillation. The nonzero value of the shear stress σsncan be a defect or limitation of the GSFM boundary condition that has been applied for σsnand the tangential velocity. However, the nonzero value for the shear stress is at most only about 4-5 % of the applied pressure. The direction cosines of the normal to the interface are calculated from the level set function at every time interval with re- initialization. Figures 4.2.12.1 to 4.2.12.18 show the different components of velocity, and stress in both x –y and s-n coordinate system which reveal the limitations and strength of the proposed GSFM for 2D wave propagation problem.
Case 4.13 : Figures 4.2.13.1 to 4.2.13.18 show the different components of velocity, and stress in both x –y and s-n coordinate system which reveal the limitations and strength of the proposed GSFM for 2D wave propagation problem. In Fig. 4.2.13.1.
0.5). These show that the pressure (or negative normal stress) and the normal velocity are continuous across the interface. It suggests that the interface Riemann problem accurately predicts the interface pressure and normal velocity. However, the assumption made about the shear stress in the s-n coordinate and the tangential velocity to be zero in the solid is not satisfied. Figure 4.2.13.5 and Fig. 4.2.13.11 show that the tangential velocity component suffers from oscillation in both the fluid and the solid domains. Again, Fig. 4.2.13.9 shows that the shear stress in the s-n coordinate σsnis not zero in the solid side and suffers from oscillation. The nonzero value of the shear stress σsncan be a defect or limitation of the GSFM boundary condition that has been applied for σsnand the tangential velocity.