Finite element solution techniques for large-scale problems in computational fluid dynamics

Element-by-element approximate factorization, implicit-explicit and adaptive implicit-explicit approximation procedures are presented for the finite-element formulations of large-scale fluid dynamics problems. The element-by-element approximation scheme totally eliminates the need for formation, storage and inversion of large global matrices. Implicit-explicit schemes, which are approximations to implicit schemes, substantially reduce the computational burden associated with large global matrices. In the adaptive implicit-explicit scheme, the implicit elements are selected dynamically based on element level stability and accuracy considerations. This scheme provides implicit refinement where it is needed. The methods are applied to various problems governed by the convection-diffusion and incompressible Navier-Stokes equations. In all cases studied, the results obtained are indistinguishable from those obtained by the implicit formulations

Finite element techniques for the Navier-Stokes equations in the primitive variable formulation and the vorticity stream-function formulation

Finite element procedures for the Navier-Stokes equations in the primitive variable formulation and the vorticity stream-function formulation have been implemented. For both formulations, streamline-upwind/Petrov-Galerkin techniques are used for the discretization of the transport equations. The main problem associated with the vorticity stream-function formulation is the lack of boundary conditions for vorticity at solid surfaces. Here an implicit treatment of the vorticity at no-slip boundaries is incorporated in a predictor-multicorrector time integration scheme. For the primitive variable formulation, mixed finite-element approximations are used. A nine-node element and a four-node + bubble element have been implemented. The latter is shown to exhibit a checkerboard pressure mode and a numerical treatment for this spurious pressure mode is proposed. The two methods are compared from the points of view of simulating internal and external flows and the possibilities of extensions to three dimensions

Numerical simulation of electrophoresis separation processes

A new Petrov-Galerkin finite element formulation has been proposed for transient convection-diffusion problems. Most Petrov-Galerkin formulations take into account the spatial discretization, and the weighting functions so developed give satisfactory solutions for steady state problems. Though these schemes can be used for transient problems, there is scope for improvement. The schemes proposed here, which consider temporal as well as spatial discretization, provide improved solutions. Electrophoresis, which involves the motion of charged entities under the influence of an applied electric field, is governed by equations similiar to those encountered in fluid flow problems, i.e., transient convection-diffusion equations. Test problems are solved in electrophoresis and fluid flow. The results obtained are satisfactory. It is also expected that these schemes, suitably adapted, will improve the numerical solutions of the compressible Euler and the Navier-Stokes equations

Parallel Three-Dimensional Computation of Fluid Dynamics and Fluid-Structure Interactions of Ram-Air Parachutes

This is a final report as far as our work at University of Minnesota is concerned. The report describes our research progress and accomplishments in development of high performance computing methods and tools for 3D finite element computation of aerodynamic characteristics and fluid-structure interactions (FSI) arising in airdrop systems, namely ram-air parachutes and round parachutes. This class of simulations involves complex geometries, flexible structural components, deforming fluid domains, and unsteady flow patterns. The key components of our simulation toolkit are a stabilized finite element flow solver, a nonlinear structural dynamics solver, an automatic mesh moving scheme, and an interface between the fluid and structural solvers; all of these have been developed within a parallel message-passing paradigm

Computational analysis of performance deterioration of a wind turbine blade strip subjected to environmental erosion

Wind-turbine blade rain and sand erosion, over long periods of time, can degrade the aerodynamic performance and therefore the power production. Computational analysis of the erosion can help engineers have a better understanding of the maintenance and protection requirements. We present an integrated method for this class of computational analysis. The main components of the method are the streamline-upwind/Petrov–Galerkin (SUPG) and pressure-stabilizing/Petrov–Galerkin (PSPG) stabilizations, a finite element particle-cloud tracking method, an erosion model based on two time scales, and the solid-extension mesh moving technique (SEMMT). The turbulent-flow nature of the analysis is handled with a Reynolds-averaged Navier–Stokes model and SUPG/PSPG stabilization, the particle-cloud trajectories are calculated based on the computed flow field and closure models defined for the turbulent dispersion of particles, and one-way dependence is assumed between the flow and particle dynamics. Because the geometry update due to the erosion has a very long time scale compared to the fluid–particle dynamics, the update takes place in a sequence of “evolution steps” representing the impact of the erosion. A scale-up factor, calculated in different ways depending on the update threshold criterion, relates the erosions and particle counts in the evolution steps to those in the fluid–particle simulation. As the blade geometry evolves, the mesh is updated with the SEMMT. We present computational analysis of rain and sand erosion for a wind-turbine blade strip, including a case with actual rainfall data and experimental aerodynamic data for eroded airfoil geometries

Finite element formulations for compressible flows

Researchers started their studies on the development and application of computational methods for compressible flows. Particular attention was given to proper numerical treatment of sharp layers occurring in such problems and to general mesh generation capabilities for intricate computational geometries. Mainly finite element methods enhanced with several state-of-the art techniques (such as the streamline-upwind/Petrov-Galerkin, discontinuity capturing, adaptive implicit-explicit, and trouped element-by-element approximate factorization schemes) were employed

Multiscale space–time computation techniques

A number of multiscale space–time techniques have been developed recently by the Team for Advanced Flow Simulation and Modeling (T*AFSM) for fluid–structure interaction computations. As part of that, we have introduced a space–time version of the residual-based variational multiscale method. It has been designed in the context of the Deforming-Spatial-Domain/Stabilized Space–Time formulation, which was developed earlier by the T*AFSM for computation of flow problems with moving boundaries and interfaces. We describe this multiscale space–time technique, and present results from test computations

Parallel finite element simulation of 3d incompressible flows: fluid-structure interactions

Massively parallel finite element computations of 3D, unsteady incompressible flows, including those involving fluid-structure interactions, are presented. The computation with time-varying spatial domains are based on the deforming spatial domain/stabilized space-time (DSD/SST) finite element formulation. The capability to solve 3D problems involving fluid-structure interactions is demonstrated by investigating the dynamics of a flexible cantilevered pipe conveying fluid. Computations of flow past a stationary rectangular wing at Reynolds number 1000, 2500 and 107 reveal interesting flow patterns. In these computations, at each time step approximately 3 × 106 non-linear equations are solved to update the flow field. Also, preliminary results are presented for flow past a wing in flapping motion. In this case a specially designed mesh moving scheme is employed to eliminate the need for remeshing. All these computations are carried out on the Army High Performance Computing Research Center supercomputers CM-200 and CM-5, with major speed-ups compared with traditional supercomputers. The coupled equation systems arising from the finite element discretizations of these large-scale problems are solved iteratively with diagonal preconditioners. In some cases, to reduce the memory requirements even further, these iterations are carried out with a matrix-free strategy. The finite element formulations and their parallel implementations assume unstructured meshes

A finite element study of incompressible flows past oscillating cylinders and aerofoils

We present our numerical results for certain unsteady flows past oscillating cylinders and aerofoils. The computations are based on the stabilized space-time finite element formulation. The implicit equation systems resulting from the space-time finite element discretizations are solved using iterative solution techniques. One of the problems studied is flow past a cylinder which is forced to oscillate in the horizontal direction. In this case we observe a change from an unsymmetric mode of vortex shedding to a symmetric one. An extensive study was carried out for the case in which a cylinder is mounted on lightly damped springs and allowed to oscillate in the vertical direction. In this case the motion of the cylinder needs to be determined as part of the solution, and under certain conditions this motion changes the vortex-shedding pattern of the flow field significantly. This non-linear fluid-structure interaction exhibits certain interesting behaviour such as 'lock-in' and 'hysteresis', which are in good agreement with the laboratory experiments carried out by other researchers in the past. Preliminary results for flow past a pitching aerofoil are also presented

Parallel computation of unsteady compressible flows with the EDICT

Recently, the Enhanced-Discretization Interface-Capturing Technique (EDICT) was introduced for simulation of unsteady flow problems with interfaces such as two-fluid and free-surface flows. The EDICT yields increased accuracy in representing the interface. Here we extend the EDICT to simulation of unsteady viscous compressible flows with boundary/shear layers and shock/expansion waves. The purpose is to increase the accuracy in selected regions of the computational domain. An error indicator is used to identify these regions that need enhanced discretization. Stabilized finite-element formulations are employed to solve the Navier-Stokes equations in their conservation law form. The finite element functions corresponding to enhanced discretization are designed to have two components, with each component coming from a different level of mesh refinement over the same computational domain. The primary component comes from a base mesh. A subset of the elements in this base mesh are identified for enhanced discretization by utilizing the error indicator. A secondary, more refined, mesh is constructed by patching together the second-level meshes generated over this subset of elements, and the second component of the functions comes from this mesh. The subset of elements in the base mesh that form the secondary mesh may change from one time level to other depending on the distribution of the error in the computations. Using a parallel implementation of this EDICT-based method, we apply it to test problems with shocks and boundary layers, and demonstrate that this method can be used very effectively to increase the accuracy of the base finite element formulation