Derivation and analysis of the analytical velocity and vortex stretching expressions for an O (N log N)-FMM

In the current paper, a method for deriving the analytical expressions for the velocity and vortex stretching terms as a function of the spherical multipole expansion approximation of the vector potential is presented. These terms are essential in the context of 3D Lagrangian vortex particle methods combined with fast summation techniques. The convergence and computational efficiency of this approach is assessed in the framework of an O(N log N)-type Fast Multipole Method (FMM), by using vorticity particles to simulate a system of coaxial vortex rings for which also the exact results are known. It is found that the current implementation converges rapidly to the exact solution with increasing expansion order and acceptance factor. An investigation into the computational efficiency demonstrated that the O(N log N)-type FMM is already viable for a particle size of only several thousands and that this speedup increases significantly with the number of particles. Finally, i t is shown that the implementation of the FMM with the current analytical expressions is at least twice as fast as when opting for using even the simplest implementation of finite differences instead

Petascale turbulence simulation using a highly parallel fast multipole method on GPUs

This paper reports large-scale direct numerical simulations of homogeneous-isotropic fluid turbulence, achieving sustained performance of 1.08 petaflop/s on gpu hardware using single precision. The simulations use a vortex particle method to solve the Navier-Stokes equations, with a highly parallel fast multipole method (FMM) as numerical engine, and match the current record in mesh size for this application, a cube of 4096^3 computational points solved with a spectral method. The standard numerical approach used in this field is the pseudo-spectral method, relying on the FFT algorithm as numerical engine. The particle-based simulations presented in this paper quantitatively match the kinetic energy spectrum obtained with a pseudo-spectral method, using a trusted code. In terms of parallel performance, weak scaling results show the fmm-based vortex method achieving 74% parallel efficiency on 4096 processes (one gpu per mpi process, 3 gpus per node of the TSUBAME-2.0 system). The FFT-based spectral method is able to achieve just 14% parallel efficiency on the same number of mpi processes (using only cpu cores), due to the all-to-all communication pattern of the FFT algorithm. The calculation time for one time step was 108 seconds for the vortex method and 154 seconds for the spectral method, under these conditions. Computing with 69 billion particles, this work exceeds by an order of magnitude the largest vortex method calculations to date

3D Lagrangian VPM: simulations of the near-wake of an actuator disc and horizontal axis wind turbine

The application of a 3-dimensional Lagrangian vortex particle method has been assessed for modelling the near-wake of an axisymmetrical actuator disc and 3-bladed horizontal axis wind turbine with prescribed circulation from the MEXICO (Model EXperiments In COntrolled conditions) experiment. The method was developed in the framework of the open- source Parallel Particle-Mesh library for handling the efficient data-parallelism on a CPU (Central Processing Unit) cluster, and utilized a O(N log N)-type fast multipole method for computational acceleration. Simulations with the actuator disc resulted in a wake expansion, velocity deficit profile, and induction factor that showed a close agreement with theoretical, numerical, and experimental results from literature. Also the shear layer expansion was present; the Kelvin-Helmholtz instability in the shear layer was triggered due to the round-off limitations of a numerical method, but this instability was delayed to beyond 1 diameter downstream due to the particle smoothing. Simulations with the 3-bladed turbine demonstrated that a purely 3-dimensional flow representation is challenging to model with particles. The manifestation of local complex flow structures of highly stretched vortices made the simulation unstable, but this was successfully counteracted by the application of a particle strength exchange scheme. The axial and radial velocity profile over the near wake have been compared to that of the original MEXICO experiment, which showed close agreement between results

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

Exploring Fundamental Turbulent Physics Using Direct Numerical Simulation

It has been shown in many studies that turbulent flows are highly dependent on their initial conditions. This thesis explores turbulent flow using direct numerical simulation (DNS) in a variety of situations, and culminates in the development of physically realizable initial conditions. The reaction of isotropic homogeneous turbulent flow to the instantaneous insertion of a wall is investigated using two-point correlations. A model with which to predict the behavior of the two-point correlations is also proposed. The proposed model utilizes a reflection technique that with a linear operation, it accurately predicts the behavior of the non-linear two point correlations. The model works exceedingly well for correlations involving wall-perpendicular velocities, but does not predict correlations involving only wall-parallel velocities as well. A vorticity approach is covered, in an effort to highlight which parts of the correlation decomposition are important to the prediction of the correlations after wall imposition. The vorticity study also helps highlight why the proposed linear model predicts the flow. The impact of the initial conditions on axisymmetric contraction flow of turbulent flow is examined, and as a consequence new initial conditions are developed based off of a physically realizable flow condition. The development of the new-initial conditions and the resulting fields are covered, as well as a study on the value of the turbulent decay exponent associated with decay of isotropic turbulent velocity fields

Physics Based Washing Machine Simulations.

This thesis describes the development of a simulation of the interaction of cloth and water that takes place inside a washing machine. The simulation consists of four basic parts: a large deformation elastic thin plate model for the cloth based on Love (1944), a rectangular-Cartesian-mesh solver for the Navier-Stokes equations based on Brown et al. (2001), the Immersed Boundary method of Peskin (1972) for cloth/fluid interaction, and a domain-mapping technique for representing irregular domain boundaries on Cartesian grids. Although the lack of an accompanying experimental effort prevented its thorough validation, the final simulation was subjected to a variety of validation tests involving analytical solutions and experimental measurements in simple geometries. The implementation of the thin plate model combined with the Immersed Boundary method was able to match the natural frequencies of a vibrating plate within +/- 1%, and was able to predict large deformation beam shapes with similar accuracy. In addition, this validation effort suggests that the ratio between the Immersed Boundary method’s Lagrangian and Eulerian point-spacings should be approximately unity for better accuracy, when accounting for finite bending stiffness. Furthermore, it was found that the Immersed Boundary method formulation may provide better results with a narrow desingularization of the two-dimensional cloth onto the three-dimensional Cartesian mesh while sacrificing numerical stability. Complicated moving boundaries are handled by a domain-mapping technique that uses a Heaviside function to switch between solving the equations for the cloth/fluid mixture and specifying the velocity field for the washing machine’s solid boundaries. This boundary-condition formulation was benchmarked against well-known steady and unsteady flow fields: circular Couette flow, and a uniform flow past a cylinder. Using these individually verified basic components together, two and three-dimensional simulations of the washing machine processes are created. A selection of studies involving the effect of different numerical and physical parameters on the kinematics of cloth motion and the statistics of the cloth stresses in a vertical-axis washing machine are reported. In particular, the coarse grid simulations predicted a realistic and qualitatively correct pattern for the motion of the cloth pieces.Ph.D.Mechanical EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/57704/2/dakcabay_1.pd