FMM-based vortex method for simulation of isotropic turbulence on GPUs, compared with a spectral method

The Lagrangian vortex method offers an alternative numerical approach for direct numerical simulation of turbulence. The fact that it uses the fast multipole method (FMM)--a hierarchical algorithm for N-body problems with highly scalable parallel implementations--as numerical engine makes it a potentially good candidate for exascale systems. However, there have been few validation studies of Lagrangian vortex simulations and the insufficient comparisons against standard DNS codes has left ample room for skepticism. This paper presents a comparison between a Lagrangian vortex method and a pseudo-spectral method for the simulation of decaying homogeneous isotropic turbulence. This flow field is chosen despite the fact that it is not the most favorable flow problem for particle methods (which shine in wake flows or where vorticity is compact), due to the fact that it is ideal for the quantitative validation of DNS codes. We use a 256^3 grid with Re_lambda=50 and 100 and look at the turbulence statistics, including high-order moments. The focus is on the effect of the various parameters in the vortex method, e.g., order of FMM series expansion, frequency of reinitialization, overlap ratio and time step. The vortex method uses an FMM code (exaFMM) that runs on GPU hardware using CUDA, while the spectral code (hit3d) runs on CPU only. Results indicate that, for this application (and with the current code implementations), the spectral method is an order of magnitude faster than the vortex method when using a single GPU for the FMM and six CPU cores for the FFT

Vorticity structure and evolution in a transverse jet with new algorithms for scalable particle simulation

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mechanical Engineering, 2004.Includes bibliographical references (p. 188-200).Transverse jets arise in many applications, including propulsion, effluent dispersion, oil field flows, V/STOL aerodynamics, and drug delivery. Furthermore, they exemplify flows dominated by coherent structures that cascade into smaller scales, a source of many current challenges in fluid dynamics. This study seeks a fundamental, mechanistic understanding of the relationship between the dispersion of jet fluid and the underlying vortical structures of the transverse jet-and of how to develop actuation that optimally manipulates their dynamics to affect mixing. We develop a massively parallel 3-D vortex simulation of a high-momentum transverse jet at large Reynolds number, featuring a discrete filament representation of the vorticity field with local mesh refinement to capture stretching and folding and hair-pin removal to regularize the formation of small scales. A novel formulation of the vorticity flux boundary conditions rigorously accounts for the interaction of channel vorticity with the jet boundary layer. This formulation yields analytical expressions for vortex lines in near field of the jet and suggests effective modes of unsteady actuation at the nozzle. The present computational approach requires hierarchical N-body methods for velocity evaluation at each timestep, as direct summation is prohibitively expensive. We introduce new clustering algorithms for parallel domain decomposition of N-body interactions and demonstrate the optimality of the resulting cluster geometries. We also develop compatible techniques for dynamic load balancing, including adaptive scaling of cluster metrics and adaptive redistribution of their centroids. These tools extend to parallel hierarchical simulation of N-body problems in gravitational astrophysics,(cont.) molecular dynamics, and other fields. Simulations reveal the mechanisms by which vortical structures evolve; previous computational and experimental investigations of these processes have been incomplete at best, limited to low Reynolds numbers, transient early-stage dynamics, or Eulerian diagnostics of essentially Lagrangian phenomena. Transformation of the cylindrical shear layer emanating from the nozzle, initially dominated by azimuthal vorticity, begins with axial elongation of its lee side to form sections of counter-rotating vorticity aligned with the jet trajectory. Periodic rollup of the shear layer accompanies this deformation, creating arcs carrying azimuthal vorticity of alternating signs, curved toward the windward side of the jet. Following the pronounced bending of the trajectory into the crossflow, we observe a catastrophic breakdown of these sparse periodic structures into a dense distribution of smaller scales, with an attendant complexity of tangled vortex filaments. Nonetheless, spatial filtering of this region reveals the persistence of counter-rotating streamwise vorticity. We further characterize the flow by calculating maximum direct Lyapunov exponents of particle trajectories, identifying repelling material surfaces that organize finite-time mixing.by Youssef Mohamed Marzouk.Ph.D

Implementation and application of adaptive mesh refinement for thermochemical mantle convection studies

Numerical modeling of mantle convection is challenging. Owing to the multiscale nature of mantle dynamics, high resolution is often required in localized regions, with coarser resolution being sufficient elsewhere. When investigating thermochemical mantle convection, high resolution is required to resolve sharp and often discontinuous boundaries between distinct chemical components. In this paper, we present a 2-D finite element code with adaptive mesh refinement techniques for simulating compressible thermochemical mantle convection. By comparing model predictions with a range of analytical and previously published benchmark solutions, we demonstrate the accuracy of our code. By refining and coarsening the mesh according to certain criteria and dynamically adjusting the number of particles in each element, our code can simulate such problems efficiently, dramatically reducing the computational requirements (in terms of memory and CPU time) when compared to a fixed, uniform mesh simulation. The resolving capabilities of the technique are further highlighted by examining plume‐induced entrainment in a thermochemical mantle convection simulation

Development and validation of hybrid grid-based and grid-free computational VπLES method

A novel hybrid grid-based and grid-free computational method which is called VπLES is proposed and validated over several benchmark cases. VπLES splits the flow structures into large scale ones, resolved on the grid (Eulerian approach), and small scale ones, represented by vortex particles (Lagrangian approach). Two transport equations for grid and particle solution are derived which are dynamically coupled through the existence of coupling terms in each of them. The method resembles LES with an effort to directly reproduce the subgrid motion at least in the statistical sense

Integrating an N -Body Problem with SDC and PFASST

Vortex methods for the Navier–Stokes equations are based on a Lagrangian particle discretization, which reduces the governing equations to a first-order initial value system of ordinary differential equations for the position and vorticity of N particles. In this paper, the accuracy of solving this system by time-serial spectral deferred corrections (SDC) as well as by the time-parallel Parallel Full Approximation Scheme in Space and Time (PFASST) is investigated. PFASST is based on intertwining SDC iterations with differing resolution in a manner similar to the Parareal algorithm and uses a Full Approximation Scheme (FAS) correction to improve the accuracy of coarser SDC iterations. It is demonstrated that SDC and PFASST can generate highly accurate solutions, and the performance in terms of function evaluations required for a certain accuracy is analyzed and compared to a standard Runge–Kutta method

Lagrangian simulation of transverse jets with a distribution-based diffusion scheme

Thesis (Sc. D.)--Massachusetts Institute of Technology, Dept. of Mechanical Engineering, 2007.This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.Includes bibliographical references (leaves 248-255).Transverse jets form a dominant group of flow fields arising in many applications of modern energy utilization, including propulsion and effluent dispersion. Furthermore, they form canonical examples where the flow field is dominated by large-scale and small-scale vortical structures, whose inter-related dynamics is a challenging subject in modern fluid mechanics. This study seeks a mechanistic understanding of the vortical structures of the transverse jet and their evolution. A set of massively parallel three-dimensional vortex simulations of high-momentum transverse jets at intermediate Reynolds number, utilizing a discrete filament representation of the vorticity field to capture stretching and tilting of vorticity, is performed. A diffusion scheme to treat viscosity at intermediate Reynolds number is formulated and analyzed in a distribution-based description. The implementation of the diffusion scheme is achieved by performing interpolation, which is a process that has been widely used to regularize particle distributions in vortex simulations, with a new set of interpolation kernels. These kernels provide an accurate and efficient way to simulate vorticity diffusion in transverse jets. An improved formulation of the vorticity flux boundary conditions is rigorously derived.(cont.) This formulation includes separation of the wall boundary layer and feedback from the jet to the wall boundary layer, and describes detailed near-field jet structures. The results present the underlying mechanisms by which vortical structures evolve. Transformation of the jet shear layer emanating from the nozzle starts with jet streamwise lift-up of its lee side to form sections of counter-rotating vorticity aligned with the jet trajectory. Periodic rollup of the shear layer, which is similar to the Kelvin-Helmholtz instability in free shear layers, accompanies this deformation. A sudden breakdown of these coherent structures into dense vortical structures of smaller scales is observed. This breakdown to small-scale structures is due to the interaction of counter-rotating vortices and rolled-up shear layer. With a separated wall boundary layer, strong near-wall counter-rotating vortices are observed. This observation substantiates the importance of including the full interaction between the wall boundary layer and the jet shear layer in the investigation of transverse jet dynamics.by Daehyun Wee.Sc.D

The Immersed Body Method and Its Use in Modelling Vertical Axis Turbines

The focus of this thesis is on the development of a fluid–solid interaction (FSI) model, based on the idea of the immersed boundary method. The novelty of this approach is the combination of a two–fluid approach to represent the solid phase on a fluid finite–element mesh, with the conservative projection of data between two unrelated meshes. While this is an important feature for two–way coupled FSI models, this thesis analyses the outcome of this method based on one–way coupled FSI problems, in which the solid phase has a prescribed velocity. The presented FSI method is validated on several test cases with static solids as well as solids with a prescribed velocity. For complex computational fluid dynamic (CFD) problems, mesh adaptivity methods are used to reduce the computational effort while obtaining the same accuracy compared to fixed meshes. In this work mesh adaptivity is also used to increase the resolution of the fluid mesh near the solid boundary in order to obtain an accurate representation of the solid’s shape on the fluid mesh. However, spurious peaks in the pressure occur due to the projection of fields after adapting the mesh. This causes peaks in the drag force and results in a potential problem by decreasing the accuracy, especially for two–way coupled FSI problems. Since the FSI method was developed with two–way coupled FSI problems in mind, the occurrence of the spurious peaks was analysed and methods are shown to minimise the peaks in the drag force. Finally, the developed FSI method is applied to rotating vertical axis turbines and the results are compared to experimental results. This again shows the difficulties of applying the method and assesses how it can be used for turbine modelling, and furthermore used for analysing optimised turbine layouts.Open Acces

A surface vorticity method for wake–body interactions

The objective of this dissertation research is to develop a surface vorticity method for simulating high Reynolds number incompressible aerodynamic flows with strong unsteady interactions between wakes and lifting bodies. Examples of these types of flows include rotors in hover, propeller/wing installations, and impingement of vortex cores shed from wing strakes or flaps on downstream surfaces. Although higher-order panel codes provide good representation of potential flow around lifting bodies, their treatment of wakes is inadequate for our purpose. In the absence of significant boundary layer separation, the vorticity in these flows concentrates into thin shear layers. Therefore, vortex sheets are a natural mathematical representation of these flows. We leverage and extend rigorous methods from the vortex methods literature to model a wake as a free vortex sheet discretized as a triangulation of panels with linearly varying surface vorticity. The vorticity evolution equation is solved approximately by maintaining constant circulation along each half-edge in the triangulation, an approach that generalizes current methods for constant-strength elements. The vortex sheet is regularized with a smoothing parameter which provides an apparent thickness that mimics the limited viscous mixing in high Reynolds number flow. An adaptive paneling algorithm is implemented to maintain the desired level of detail as the wake triangulation stretches and deforms. The induced velocities from the wake vortex sheet are computed with a treecode implemented on a graphics processing unit (GPU) to allow computations with millions of panels. Lifting bodies are modeled with bound vortex sheets that are also triangulated with linear strength panels. These higher-order vorticity elements provide accurate velocity predictions on and near the surface, allowing for high resolution streamline tracing. Surface vorticity is determined by enforcing flow tangency constraints at each triangle centroid, zero circulation around each panel perimeter, and the unsteady pressure matching Kutta condition. These constraints result in an overdetermined system that is solved in a least squares formulation. Thus, our method is a second-order surface vorticity boundary element method that combines both solid bodies and wakes in a rigorous and consistent manner. The results of the method are shown to compare favorably to wind tunnel experimental results, including wake profiles, for a rectangular wing in a steady freestream, and for a horizontal axis wind turbine. Finally, we demonstrate the capabilities of our method in the context of strong wake–body interactions by simulating two flying wing aircraft in close formation, with the wake from the leading aircraft impacting the tailing aircraft.Ph.D

Mesh free method for the Poisson equation with 3D wall-bounded flow application

Thesis (S.M.)--Massachusetts Institute of Technology, Dept. of Aeronautics and Astronautics, 2010.Cataloged from PDF version of thesis.Includes bibliographical references (p. 58-60).The numerical approximation of the Poisson equation can often be found as a subproblem to many more complex computations. In the case of Lagrangian approaches of flow equations, the Poisson equation often needs to be solved on an irregular point distribution. Currently, mainly unstructured mesh-based approaches are used. Meshfree methods present a way to approximate differential operators on unstructured point clouds without the need for mesh generation. In this thesis, a 3d meshfree finite difference Poisson solver is presented. Its performance has been studies based on numerical convergence, parallel efficiency, and computational cost. Practical application of the solver is presented in a simulation of a potential flow field in a wall-bounded domain.by Anna Vasilyeva.S.M