Solving first-order hyperbolic problems for wave motion in nearly incompressible fluids, two-phase fluids, and viscoelastic media by the CESE method

This dissertation reports the development of generic first-order, hyperbolic partial differential equations and the associated numerical method for modeling wave motion in complex media, including propagating waves in liquids, in two-phase fluids, and in viscoelastic media. The model equations are cast into a set of first-order, fully coupled, hyperbolic, partial differential equations (pde's) with or without the dissipation terms. In this dissertation, the dissipation terms can be found with viscous terms and stiff source. For different types of dissipation terms, their treatment in the setting of the space-time Conservation Element and Solution Element (CESE) method is illustrated in this dissertation.;First, acoustic waves in low-speed, nearly incompressible flows are of interest. The Bulk modulus is employed to unify the constitutive equation for different media. As that in the most of the incompressible fluid flows, the energy equation is not included in the model equations, and thus the current model is suitable for isothermal flows only. The dependence of the Bulk modulus on the temperature of the medium is out of the scope of the present development.;Next, the constitutive model developed for the incompressible fluids is extended to that for wave motion in two-phase flows. Similar to that of incompressible fluids, the two-phase flow model assumes the fluid is nearly isothermal but is compressible to accommodate the disparity in density of the gas phase and the liquid phase where phase change is involved. To test the capabilities of the model, cavitation associated with the water hammer effect is modeled.;As the last example, a set of first-order, fully coupled, hyperbolic pde's have been derived for wave motion in viscoelastic media. In particular, wave propagation in soft tissues for biomedical applications is of interest. The model equations have been developed based on the fundamental description of the material, including linear theory of relaxation functions, the standard linear solid model, and the use of the internal variables. Small deformation and constant temperature are assumed in the present model.;For the hydro-acoustics model and for the wave in two-phase flow model, the Riemann invariants have been explicitly derived. Moreover, the Riemann invariants were obtained by using the three-dimensional equations instead of the usual one-dimensional approach. The Riemann invariants of the two-phase flow model are further analyzed to derive the Rankine-Hugoniot shock jump relationship for the water hammer effect in the two-phase flow. With the derived shock-jump condition, the required pressure ratio for creating the shock wave in two-phase flow has been obtained. The derived shock relation has also been used as the analytical solution to verify the numerical simulation.;The numerical solutions of all above three models have been obtained by using the space-time CESE method. The numerical results have been compared with the available experimental data and/or theoretical solutions. In this dissertation, the open-sourced software SOLVCON for generic hyperbolic pde's solver, developed at OSU, has been used. The CESE method is the default CFD method for SOLVCON although other solver kernel could be easily implemented.;For the hydro-acoustics model for wave motion in nearly incompressible flows, two sets of the numerical simulations have been performed. The first set involves air and water flows over a circular cylinder. The calculated Strouhal number was derived based on the numerical results and compared with the experimental data. The second case is an air flow over a rectangular cavity. The acoustic pressure is measured in the numerical simulation and is compared with the experimental data and the analytical solution.;Finally, the newly derived viscoelastic model for wave propagation in soft tissues has been solved. In particular, the ultrasound longitudinal waves in a soft tissue have been simulated. The focused longitudinal waves would cause a local shear deformation at the focal point.;This dissertation demonstrates that a wide range of wave phenomena can be accurately modeled by a set of coupled, first-order, hyperbolic pde's. Traditionally, wave motion in complex solids have been routinely modeled by second-order wave equations. The present dissertation shows that the first-order pde's represent a much more versatile and open theoretical platform to accommodate complex modeling requirements. The analyses of the first-order pde's are shown to be based on the eigen-structure of the Jacobian matrices. In the analyses, we strive to achieve genuine three-dimensional analyses instead of conventional one-dimensional approaches. Moreover, the present work also shows that SOLVCON is indeed a versatile solver for easy implementation for time accurate solutions of wave motion by using unstructured meshes and hybrid parallel computation. (Abstract shortened by UMI.).