3.2. Case Study (1D Fluid Structure Interaction)
3.2.3.1. Shock Tube Type Problems
Gas- Solid Cases:
Case 3.1. : Figs. 3.2.1.1. to 3.2.1.3. show the pressure (p), velocity (u) and density (d) distribution, respectively. This problem falls in the category of [S|EL]e / [S|EL-PL]ep. The interface moves to the right. Shock wave is generated at the interface and moves through the fluid to the right. The load on the solid is higher than the yield strengthκ0 of the solid. In the solid, the linear elastic model predicts only the fast elastic wave (P- wave) moving to the right. The linear elastic plastic model captures the presence and movement of a slow plastic wave through the solid to the right. The predicted pressure with the Elastic Plastic model is slightly lower than that predicted by the
elastic solid model. Hence, the predicted velocity with the Elastic Plastic model is slightly larger than that predicted by the elastic solid model. The density is same for both of these models.
Case 3.2. : This problem falls in the category of [–|EL]e / [–|EL-PL]ep. This case is a non-reflection case when we use the elastic model. No shock or rarefaction wave passes through the gas. In order to prove the robustness and accuracy of the GSFM this non-reflection or shock impedance matching case is computed. When the interface algorithm under-predicts or over-predicts any physical quantities, non- physical humps appear at the interface location as can be seen in Liu et al (2003). The linear elastic model predicts only the fast elastic wave (P-wave) moving to the right.
But the load on the solid is higher than the yield strengthκ0 of the solid and there is a faster elastic wave and a slower plastic wave traveling to the right through the solid as predicted by the elastic-plastic model. Figs. 3.2.2.1. to 3.2.2.3. show the pressure (p), velocity (u) and density (d) distribution, respectively. For the elastic plastic model, the predicted interface velocity is larger than that by the elastic model. No non-physical hump is visible at the interface which indicates the robustness of the algorithm. The interface has moved to the right.
Case 3.3. : This problem falls in the category of [S|EL]e / [S|EL]ep. Both the elastic and the elastic plastic model predictions are the same, except for the elastic plastic model predicts an unphysical kink near the elastic wave (Fig. 3.2.3.1). In this case, the loading is below the tensile strength κ0of the solid and hence, there is no plastic wave visible using the elastic-plastic model. The linear elastic model can be regarded to
solution (Rebecca (2005)). The solution at the interface may be termed as Shock- Shock 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. Figs. 3.2.3.1. to 3.2.3.3. show the pressure (p), velocity (u) and density (d) distribution, respectively.
The interface has moved to the right.
Case 3.4. : This is a [R|EL]e /[R|EL]ep type problem. A rarefaction wave is generated at the interface and travels through the fluid to the left. The fluid-solid interface moved to the right. There is only an elastic wave traveling to the right through the solid. The stress level in the solid does not exceed the tensile strength and hence there is no plastic wave present. The elastic-plastic model predicts an unphysical kink in the stress profile near the elastic wave front (Fig. 3.2.4.1). The linear elastic model can be regarded to give better result in this case, as it can be proved to match well with the analytical solution (Rebecca (2005)). Figs. 3.2.4.1. to 3.2.4.3. show the pressure (p), velocity (u) and density (d) distribution, respectively.
Case 3.5. : This is a [R|EUL]e / [R|EUL]ep type problem. An unloading elastic wave propagates through the solid medium. As the loading does not exceed the tensile strength κ0of the medium, there is no plastic wave present. The solution near the unloading elastic wave front suffers from oscillation and the oscillation is higher for the elastic-plastic model (Fig. 3.2.5.1). The linear elastic model can be regarded to give better result in this case, as it can be proved to match better with the analytical solution (Rebecca (2005)). A rarefaction wave travels through the gas to the left. Figs.
3.2.5.1. to 3.2.5.3. show the pressure (p), velocity (u) and density (d) distribution, respectively. Due to unloading at the interface, the interface moved to the left.
Water Solid Cases:
Case 3.6. : This is a [S|EL]e / [S|EL-PL]ep type problem. In this case a shock wave is generated that moves through the water to the left. The load on the solid is higher than the yield strengthκ0 of the solid. The plastic medium is supposed to exert lesser resistance to the fluid pressure than the elastic medium. The magnitude of the pressure at the interface predicted via the linear elastic model is higher than that using linear elastic-plastic model (Fig. 3.2.6.1). The velocity predicted at the interface by the elastic plastic model is larger than that by the elastic solver (Fig.3.2.6.2). On the solid side, the elastic model predicts only the elastic wave. On the other hand, a fast elastic wave and a slow plastic wave are identified for the elastic-plastic model. From Fig.
3.2.6.1 it can be seen that the location of the shock wave is different for the two solid models. This may be due to the movement of the interface predicted by the two models. Fig. 3.2.6.3. shows the density (d) distribution and indicates that the water- solid interface moves to the right for the elastic plastic case and to the left for the elastic case.
Case 3.7. : This problem falls in the category of [–|EL]e / [R|EL-PL]ep. The load on the solid is higher than the yield strengthκ0 of the solid. This is a case where there is no wave moving through the water (non-reflection case) for the water-elastic solid model. But the water-elastic plastic solid interaction predicts a rarefaction wave in the fluid side. It also predicts a fast elastic and a slow plastic loading wave traveling to the right in the solid side (Fig.3.2.7.1). The velocity at the interface predicted by the elastic-plastic model is larger than that predicted by the elastic model (Fig.3.2.7.2).
The movement of the interface is to the right and the density distribution predicted by
the two solid model are almost similar except for the position of the interface which can be seen from the density (d) distribution (Fig.3.2.7.3).
Case 3.8. : This is a [S|EL]e / [S|EL-PL]ep type problem. A shock wave is generated and travels through the water to the left (Fig.3.2.8.1). The plastic medium is supposed to exert lesser resistance to the fluid pressure than the elastic medium. The magnitude of the pressure at the interface predicted via the linear elastic model is higher than that using linear elastic-plastic model (Fig. 3.2.8.1). The interface velocity predicted for the elastic plastic case is larger than for the elastic case (Fig. 3.2.8.2). The density distribution for the two solid models is almost same except for the position of the interface. The position of the shock wave predicted by the two solid solvers is different. This is because the interface for the elastic plastic model moves to the right to a greater extent than for the elastic model as can be seen from the density (d) plot (Fig. 3.2.8.3).
Case 3.9. : This is case of type [S|EL]e / [S|EL]ep. The loading in the solid medium is well below the tensile strength κ0for the steel. Hence, there is no plastic wave in the solid predicted by the elastic plastic model. The elastic-plastic model seems to have unphysical overshot in the stress level near the elastic wave front (Fig.3.2.9.1). The linear elastic model in this case performs better as it can be proved to match well with the analytical solution (Rebecca(2005)). There is a shock wave generated at the interface and travels to the left in the fluid medium. Figs. 3.2.9.1 to Fig. 3.2.9.3 show the pressure (p), velocity (u) and density (d) distribution, respectively.
Case 3.10. : This case can be classified as [R|EL]e / [R|EL-PL]ep. A rarefaction wave is generated at the interface and travels through the water medium to the left (Fig.
3.2.10.1). The load on the solid is higher than the yield strengthκ0 of the solid. In the solid medium, the elastic model predicts the elastic wave only and the elastic-plastic model predicts a slow plastic wave along with the fast elastic wave (Fig.3.2.10.1). The plastic medium is supposed to exert lesser resistance to the fluid pressure than the elastic medium. The magnitude of the pressure at the interface predicted via the linear elastic model is higher than that using linear elastic-plastic model (Fig. 3.2.10.1). The interface velocity for the elastic-plastic model is larger than that for the elastic model (Fig.3.2.10.2). Fig. 3.2.10.3. show the density (d) distribution and the two solid models agrees well.
Case 3.11. : This problem is of type [R|EL]e / [R|EL-PL]ep. The load on the solid is higher than the yield strengthκ0 of the solid. The elastic model only predicts an unloading elastic wave in the solid medium. On the other hand, the elastic plastic model predicts a loading plastic wave and a very weak unloading elastic wave (Fig.3.2.11.1). In the fluid medium, a rarefaction wave travels to the left. The velocity at the interface is larger for the elastic-plastic model than for the elastic model (Fig.
3.2.11.2). The interface movement to the right is found to be more for the elastic plastic case than for the elastic case. This can be seen from the density (d) plot (Fig.
3.2.11.3).
Effect of ζ (material constant for the elastic-plastic model (Eqn 3.1.2.12)) on the interface status:
Case 3.12. : This case can be classified as [S|EL]e / [S|EL-PL]ep. In this case, the effect of using 3 different values of ζ in the elastic plastic model has been investigated for the gas-solid case under the same initial conditions as of case 3.1. Three different values of ζ means 3 different solid materials. As ζ increases, the material becomes more and more plastic and it is evident from Figs. 3.2.12.1 to 3.2.12.3. The interface pressure drops and the interface velocity grows as the plasticity effect increases.
Case 3.13. : This is a [S|EL]e / [S|EL-PL]ep type problem. In this case, the effect of using 3 different values of ζ is investigated for the water-solid mediums. The other initial conditions are same as of case 3.6. Similar information as found for Case 3.12 is evident here. As ζ increases, the material becomes more and more plastic and it is evident from Figs. 3.2.13.1 to 3.2.13.3. The interface deformation is more and its moves further to the right. As a result, the position of the shock wave is to the right to that obtained by the elastic solver. Again the interface pressure decreases as
ζ increases. The opposite is true for the velocity at the interface.