# 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.

### Operator splitting and Adaptive Mesh Refinement

Numerical Combustion with detailed chemical kinetics is a very challenging problem. The set of the species conservation equations is mathematically very stiff. The stability of the explicit projection schemes for such problems requires a very strict CFL condition, limiting the maximum timestep to a few nanoseconds. To overcome this limitation, semi-implicit stiff integration is carried out in an operator-split projection scheme. A set of ODEs is solved implicitly using commercially available stiff ODE solvers during the process. The next limiting CFL condition, that does not allow the full potential usage of semi-implicit scheme, is the diffusion of intermediate species like H ion. Diffusion sub-stepping with Operator-Splitting is performed to get around this problem. As a result, stable time stepping increases from few nanoseconds to hundreds of nanoseconds.

These schemes are currently being used to investigate the impact of thermal interactions between the heat-conducting burner plates (flame-holders) and premixed flames. Their role in static and dynamic stability is being investigated. These are direct simulations with all the time and length scales resolved. Adaptive Mesh Refinement procedures are being incorporated in the codes to be able to increase the size of the simulation domain.

### Large Eddy Simulations

Large Eddy Simulations (LES) is considered as one of the more promising numerical approaches for the analysis of turbulent combustion, balancing computational complexity and predictive accuracy. While DNS resolves all the turbulent scales, it is computationally expensive and impractical for high Reynolds number large scale applications. RANS, on the other hand, models the influence of turbulence on the mean flow and hence can not capture the unsteady flow. In LES, rather than averaging the effect of turbulence, the equations are filtered, enabling the larger scales of turbulence to be explicitly resolved and computed (as with DNS), while the smallest ones are modeled (as with RANS modeling). This enables capturing the unsteadiness in the flow and results in better predictions as compared to RANS technique, because the effect of turbulence is represented more accurately due to the explicit computation of the large eddies. Modeling the sub-grid scale effects on the other hand ensures that the approach is computationally manageable.

An integral component of LES is the turbulent combustion sub-grid model, which is necessary to incorporate the effect of turbulence-chemistry interactions at the under-resolved scales on the reaction rate. The reaction mechanism incorporated is also important, particularly when studying unconventional combustion (e.g. oxy-fuel combustion conditions). Our study involves developing high fidelity LES solvers in OpenFOAM, focusing on the implementation of turbulent combustion models (such as the thickened flame model) and chemistry integration approaches (eg flamelet generated manifolds). These will be used to study the dynamic response of lean premixed flames in step and swirl systems, while also focusing on instability mitigation approaches.