Phenomena involving interactions between liquids and solids are widely observed in natural and industrial applications. Interactions between different phases of fluids and solid structures have great importance in engineering applications such as aircraft and submarine designs, multiphase liquid flow analyses, aerodynamics, vehicle dynamics and in the fields of aeronautic, civil and mechanical engineering. The research in this dissertation focuses on numerical investigations of the effects of liquid-solid interactions on structures found in engineering systems and the nonlinear dynamic behaviors involved. The effects of liquid sloshing on a container mounted on a carrier are studied from an equivalent mechanical model of liquid sloshing in the container. The nonlinear mechanical model is presented and inviscid and viscous liquids are considered and compared for their effects on sloshing. The influence of gravitational acceleration on 3D nonlinear sloshing of the liquid in the carrier is studied in detail with a variety of system parameters. The motion and mobilization of multiphase fluid in porous media is analyzed, with a model in which oil slugs are trapped in an axisymmetric capillary tube saturated with water. Governing equations are derived for incompressible two-phase core-annular flow in the capillary model. Numerical solutions for capturing the evolution of the interface between oil and water are developed using a level set approach. The development of a water film surrounding the oil slug shows a significant effect on mobilizing the oil slug. The fluttering and oscillation of a panel structure is studied to investigate fluid-structure interactions and their effects on the structure. The interactions between fluid and structure are incorporated into the governing equations. A new approach based on the Periodicity Ratio method is developed in this study so the characteristics of a nonlinear system, subjected to non-periodic excitations, can be diagnosed. Vortex control of fluid flow over circular cylinders with detached plates are conducted numerically to develop a comprehensive understanding of the complex interactions a between fluid and structure, which is significant in aeronautic engineering. The uniform and linear shear intake flow are taken into consideration. The position and thickness of the detached plate are investigated. A thin plate separated from the cylinder plays an important role on the flow phenomenon in the vicinities of the cylinder and on the exertions applying on the cylinder subjected to the uniform and linear shear flow. Accuracy and reliability of the numerical calculations used in the engineering analyses are also investigated. The 4th-order Runge-Kutta method and a newly developed P-T method are studied and compared for their characteristics. Due to its inherent drawbacks found in the research, the Runge-Kutta method may cause information loss and lead to incorrect conclusions in comparison with the P-T method.