Comparative benchmark results for different solution methods for fluidstructure interaction problems are given which have been developed as collaborative project in the DFG Research Unit 493. The configuration consists of a laminar incompressible channel flow around an elastic object. Based on this benchmark configuration the numerical behavior of different approaches is analyzed exemplarily. The methods considered range from decoupled approaches which combine Lattice Boltzmann methods with hp-FEM techniques, up to strongly coupled and even fully monolithic approaches which treat the fluid and structure simultaneously.

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In his seminal paper on fluid motion, Stokes developed a general constitutive relation which admitted the possibility that the viscosity could depend on the pressure. Such an assumption is particularly well suited to modelling flows of many fluids at high pressures and is relevant to several flow situations involving lubricants. Fluid models in which the viscosity depends on the pressure have not received the attention that is due to them, and we consider unidirectional and two-dimensional flows of such fluids here. We note that solutions can have markedly different characteristics than the corresponding solutions for the classical Navier-Stokes fluid. It is shown that unidirectional flows corresponding to Couette or Poiseuille flow are possible only for special forms of the viscosity. Furthermore, we show that interesting non-unique solutions are possible for flow between moving plates, which has no counterpart in the classical Navier-Stokes theory. We also study, numerically, two two-dimensional flows that are technologically significant: that between rotating, coaxial, eccentric cylinders and a flow across a slot. The solutions are found to provide interesting departures from those for the classical Navier-Stokes fluid.

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The goal of this article is to contribute to the discussion of the efficiency of lattice-Boltzmann (LB) methods as CFD solvers. After a short review of the basic model and extensions, we compare the accuracy and computational efficiency of two research simulation codes based on the LB and the finite-element method (FEM) for two-dimensional incompressible laminar flow problems with complex geometries. We also study the influence of the Mach number on the solution, since LB methods are weakly compressible by nature, by comparing compressible and incompressible results obtained from the LB code and the commercial code CFX. Our results indicate, that for the quantities studied (lift, drag, pressure drop) our LB prototype is competitive for incompressible transient problems, but asymptotically slower for steady-state Stokes flow because the asymptotic algorithmic complexity of the classical LB-method is not optimal compared to the multigrid solvers incorporated in the FEM and CFX code. For the weakly compressible case, the LB approach has a significant wall clock time advantage as compared to CFX. In addition, we demonstrate that the influence of the finite Mach number in LB simulations of incompressible flow is easily underestimated.

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