Numerical Simulation of a Pulsatile Flow through a Flexible Channel

An algorithm for approximation of an unsteady fluid-structure interaction problem is proposed. The fluid is governed by the Navier-Stokes equations with boundary conditions on pressure, while for the structure a particular plate model is used. The algorithm is based on the modal decomposition and the Newmark Method for the structure and on the Arbitrary Lagrangian Eulerian coordinates and the Finite Element Method for the fluid. In this paper, the continuity of the stresses at the interface was treated by the Least Squares Method. At each time step we have to solve an optimization problem which permits us to use moderate time step. This is the main advantage of this approach. In order to solve the optimization problem, we have employed the Broyden, Fletcher, Goldforb, Shano Method where the gradient of the cost function was approached by the Finite Difference Method. Numerical results are presented. Mathematics Subject Classification. 74F10, 75D05, 65M60. Received: January 15, 2005. Revised: June 1 and November 15, 2006. Introduction We consider a pulsatile incompressible flow through a channel with elastic walls. Following [29], the therm pulsatility means the rapid increase and decrease of the flow rate in a first phase, followed by a longer phase where the flow rate is small. This kind of fluid-structure interaction arises in car industries, for example the dynamic behavior of a hydraulic shock absorber [22] or in the design of sensors subject to large acceleration during impact [20] or in bio-mechanics, for example, the interaction between a bio-fluid and a living tissue [28]. The mathematical model which governs the fluid is the unsteady Navier-Stokes equations with boundary condition on the pressure. For the structure, a particular plate model is used. The most frequently, the fluid-structure interaction problems are solved numerically by partitioned procedures, i.e. the fluid and the structure equations are solved separately. There are different strategies to discretize in time the unsteady fluid-structure interaction problem. A family of explicit algorithms known also as staggered was successfully employed for the aeroelastic applications [11]. As it shown in [22] and [26], the staggered algorithms are unstable when the structure is light and its density is comparable to that of its fluid, such in the bio-mechanics applications. For a simplified fluid-structure problem, the unconditionally instability of the explicit algorithms is proved in [3].