Indicative Results and Progress on the Development of the Unified Single Solution Method for Fluid-structure Interaction Problems

This paper presents the progress on the development of a novel unified solution method for solving strongly coupled fluid-structure interaction problems. The method has been developed and fully tested for solids in [1]. The new approach is based on continuum mechanics formulation for fluids and structures where both continua can be solved using the momentum and continuity equation. The difference between the two continua lies in the constitutive equations. In this framework a single set of equations is used for the simultaneous solution of both fluid and solid. The common equations are written such that velocity and pressure are unknown variables for both continua. The discretisation method used for the solution of the problems is finite volumes. The physical interface between the two continua is treated as an internal part of the computational domain and no explicit exchange of information is needed. The test case used to demonstrate the idea is wave propagation in liquid filled flexible vessels. The solution is fully implicit and transient. Results regarding pressure, velocity and wall distention at different times and various locations along the tube are presented. The method is stable and robust and can be used for the next step of development and validation against classical analytical and numerical models. NOMENCLATURE Co Courant number c m/sec wave speed D m diameter f Hz frequency h m thickness I unit tensor K Pa bulk modulus L m length F fluid p Pa pressure S solid t sec time U m/sec velocity ∆t s time step ∆x m grid interval η m/s dynamic viscosity λ Pa Lamé’s coefficient μ Pa Lamé’s coefficient ε strain tensor Ε Pa Young’s modulus ν Poisson ratio ρ kg/m density σ Pa Cauchy stress tensor τ Pa applied end shear ω Hz frequency of undamped oscillations INTRODUCTION Fluid structure interaction occurs in many areas of engineering (aerospace, civil or mechanical) as well as other scientific disciplines including medicine, biomechanics etc. During this interaction, the normal and shear stress due to the fluid flow act on the structure and cause it to deform which in turn affects the fluid flow and consequently the stress of the fluid. Thus, the response of the system can be determined only if the coupled problem is solved. In the conventional approach for fluid-structure interaction problems, the fluid and solid components are treated separately and information is exchanged at their interface. According to the conventional terminology, the current numerical methods can be grouped in two major categories: Partitioned methods and monolithic methods. Both methods use two separate sets of equations for fluid and solid. In Figure 1 a schematic description of these methods is depicted.