Scientific Computing

Transport Element Methods

A motivation for the development of the multi-purpose treecode was the possibility to perform more efficient simulations using Passive Scalar Transport for two reasons.

First, we observe in the curl-form of Navier-Stokes equation using Boussinesq approximation , that we do not need the value of the temperature field, but only the value of the gradient of the temperature, in order to compute the baroclinic generation of vorticity. In terms of Passive Scalar Transport, this means that we can use elements carrying the gradient of the temperature, instead of Conserved Scalar Elements, carrying the temperature information. In fact, with Conserved Scalar Elements, we solve the Energy equation, then we need to differentiate the temperature field to obtain the temperature gradient, using finite differences for example. This takes more CPU time and we may loose accuracy. Now that we developed the adaptive treecode algorithm, we are able to solve the energy equation in its gradient form (which requires the gradient of the velocity for each particle). In other words, we will compute the evolution of the temperature gradient, instead of the temperature field.

The second reason is that the support may be smaller since we only need to cover the support of gradients, and not the whole field as it is done with Conserved Scalar Elements, which results in less elements. As a consequence, we have a faster simulation.

Lagrangian Simulation of Combustion

Accurate and efficient computational algorithms for the simulation of high Reynolds number turbulent reacting flows with fast chemical reactions are valuable for the study of turbulence-combustion interactions in engineering systems utilized in automotive, aerospace and utility industries, as well as in problems related to safety and environmental concerns.

As the first step, we develop a Lagrangian method for the accurate simulation of low-Mach number, variable-density, diffusion-controlled combustion. Our previous axisymmetric implementation [1] was used to model fire plum rise and dispersion. Such a model plays an essential role to assess the environmental damage from large fires. Results include the rate of burning, fire dynamics, emissions and temperature field. Our current efforts are concentrated on the creation of an equivalent 3D simulation tool for investigating diffusion-controlled combustion. A new method is currently being developed by using a distribution-based treatment of diffusion and a transport element scheme.

Multipurpose Adaptive Tree Code

Fluid simulations using Lagrangian vortex methods are interesting in many ways. Since they are grid-free methods, the distribution of computational elements is adaptive, and the simulation is performed only over the support covered by vorticity. The vortical structures, which are important for understanging the dynamics of many interesting fluid systems, are readily identified, since the computational elements represent vorticity. The mechanical deformation of each vortical structures can be easily correlated to the important phenomena such as mixing and transition.

Recently, these methods become even more efficient by implementing fast-multipole type approaches to compute pairwise interactions of vortex elements. Our parallel adaptive tree-code has provided an efficient way to deal these pairwise interactions, for computing the local velocity induced by vortex elements. However, velocity evaluation is not the only place where pairwise particle interaction occurs. For many applications, we need velocity gradients from vortex elements, expansion velocity from a nontrivial divergence field, and recovery of scalar properties from distributed particles.

In this study, an extension of our previous tree-code to a multipurpose tree-code is made. A single universal set of expansion coefficients is recombined in a different ways to compute expansion for various quantities. Our multipurpose tree-code forms an essential part for multiphysics simulations, such as reacting flow simulations.