PowerFLOW®’s underlying technology can be broken down into two major components: fluids solver and turbulence modeling. Each of these areas will be described in conceptual form below.

### MOTIVATION & THEORETICAL OVERVIEW OF DIGITAL PHYSICS®

The traditional approach in CFD is to start with the Navier-Stokes (N-S) equations. The N-S equations are a set of partial differential equations that statistically describe a real fluid. These equations are fundamentally sound (at least for the majority of practical fluid flow problems of interest). The problem is that for most cases of interest, the solutions to these very complex and highly non-linear equations are characterized by many degrees of freedom. Analytical solutions cannot be obtained except for the simplest of problems. The problem with the traditional approach employed over the past years lies of course not in the N-S equations themselves but in the analytical and numerical techniques employed to provide approximate solutions to them.

The conceptual approach used to solve fluid flow problems in Exa's PowerFLOW software is significantly different from the approach used in all other commercial CFD codes. The main difference is that while traditional CFD approaches start with a mathematical description of a fluid at the continuum level (the N-S equations), Exa’s Digital Physics simulates fluids at a more fundamental, or kinetic, level via a discrete Boltzmann equation. The Boltzmann equation governs the dynamics of particle distribution functions.

There is a strong motivation to simulate a fluid this way. First, by using a kinetic description, the modeled physics is simpler and more general than what is captured in the N-S equations. The physics is simpler since it is restricted to capturing the kinetic behavior of particles or collections of particles as opposed to attempting to solve non-linear PDEs, which is very difficult. This mesoscopic description is also more general since by augmenting the particle interactions at this level, more complex fluid physics, valid for a much wider range of spatial and time scales, as well as for multi-phase or multi-species flows, can be modeled more accurately. It is theoretically known that N-S is an approximation to Boltzmann kinetic theory at some specific limit of time and spatial scales.

However, a complete reproduction of all aspects of microscopic particle dynamics would be prohibitively expensive to achieve computationally. Similarly, simulation of continuum Boltzmann equation is also prohibitive. The desire to resolve this problem has led to the investigation of a new approach whereby the goal is to construct a simplified form of the mesoscopic dynamics that still contains sufficient physics to allow accurate recovery of the desired macroscopic behavior. This is the idea underlying the Digital Physics method used in PowerFLOW; it is also referred to as the Lattice-Boltzmann Method (LBM).

To get a deeper sense of how the two approaches to simulating fluid flow compare, we can examine the theoretical framework underlying both of them.

First, both methods begin with what you might call microscopic physical reality. At this level, we find a very large number of discrete particles moving about in continuous space. The particles are free to move about in any direction, occupy any location in space and have any speed. The collection of speeds and locations particles may have is called their "phase space".

We first examine the theoretical steps taken to solve fluid flows using the traditional CFD approach. In this approach, the first step is to apply the powerful and well-known methods of kinetic theory and statistical physics to this microscopic description, along with some assumptions about the nature of the collision process, to derive the macroscopic, continuum hydrodynamic equations. These are the non-linear Navier-Stokes equations.

In order to solve these equations computationally, this continuous representation must now be discretized. This means that instead of solving for the fluid variables at all locations in time and space, only properties of the fluid at discrete spatial and temporal locations may be computed numerically. This discrete set of locations is called the computational grid. One of the requirements to achieve accurate solutions is to use an adequately resolved grid. This means that there are enough grid points throughout a simulation volume to capture all of the relevant physics. This discretization step introduces errors into the solution of the desired continuous equations which, when coupled with an inadequately resolved grid, can potentially lead to divergent or unstable solutions, especially when complex models of unresolved physics, such as turbulence, are introduced. This is because the non-linear nature of these equations makes mathematical analysis of the discretization step difficult, and sometimes even impossible. There are also serious issues associated with the realization of physical boundary conditions. Lastly, in order to avoid numerical instability and to reduce the computational cost of these solutions, a steady-state solver is often employed.

To contrast with this, we now turn to the Lattice Boltzmann approach. With this method, the necessary discretization step occurs at the microscopic level where phase space itself is restricted to discrete values. This means that particles are restricted to discrete locations in space, and to discrete velocities at discrete moments in time. This method is inherently time-dependent. The set of particles that have a given discrete velocity at a given spatial location is called the particle distribution at that grid point. The particle distribution is the main computational element of this method. All fluid properties and their evolution can be derived from it. A desirable consequence of this discretization methodology is an exact realization of the most fundamental properties in particle dynamics in terms of exact conservation of mass, momentum, and energy.

On the numerical method front, fluid convection is realized in LBM via particle microscopic velocities. The latter consists of a set of constant values, forming exact links at each lattice site to its neighboring sites. As a consequence, the convection step is exact (corresponding to a CFL# = 1 everywhere at all times). This minimizes numerical dissipation while maximizing time step size. In comparison, fluid convection in N-S is a nonlinear process that is a function of both space and time. It is relatively much more difficult to deal with the convection process with N-S, particularly with time-explicit schemes. Microscopic velocity based particle representation allows boundary condition be realized on a more fundamental physics level via a particle scattering process as in reality, resulting in stable and more precise computations of surface fluid dynamic properties. The constant microscopic velocities also enable better handling of boundary conditions involving complex geometry. Furthermore, nonlinear fluid physics is fully contained in the local collision process in LBM. All these factors allow for efficient and parallel computation of time dependent flows with complex geometry.

The next step is to apply a discrete form of kinetic theory, an analogue to the continuous theory used in the derivation of the N-S equations, to the discrete microscopic description. It can be shown that when certain requirements on the microscopic dynamics are satisfied the macroscopic equations derived are identical to the time-dependent N-S equations. This means that the method accurately provides the desired fluid flow behavior, since the same macroscopic equations of traditional CFD (the N-S equations) are solved, but by solving these very difficult equations indirectly and at a more fundamental level, rather than directly as is the traditional approach.

The Lattice Boltzmann Method and its advanced extensions provide a solution for the resolved fluid in PowerFLOW. This is then coupled to boundary conditions for inlets, outlets and walls. If the Reynolds number is low enough (< 10,000) then this is all that is necessary to build a very accurate direct simulation of the fluid flow. However, it is required that all relevant scales of motion are resolved in this direct simulation approach – thus the computational power limits the Reynolds number that can be accurately simulated. PowerFLOW has been proven to be highly accurate in this mode. (Examples such as complex flow as a turbulent jet and around a circular cylinder can be described). The major difficulty for PDE based CFD solvers for the relatively low Reynolds number regimes is due to uncontrolled numerical dissipation. PowerFLOW has substantial advantages in this area.

### TURBULENCE MODELING USING CFD

For higher Reynolds number flows it is not computationally practical to perform direct simulations by resolving all of the scales, thus it becomes necessary to incorporate turbulence models to account for the unresolved turbulent flow structures. Here again, PowerFLOW takes a different approach that is more fundamentally sound. The approach is based on the fact that PowerFLOW is a time-dependent solver and is thus capable of directly simulating turbulent flow structures.

There are three basic categories of turbulent scales of motion: the dissipative range, the inertial range, and the anisotropic range. The dissipative and inertial ranges of turbulence are universal in nature lending themselves to a possible theoretical description. Turbulence theory is based on describing these universal aspects. The anisotropic turbulence contains the largest scales of turbulent motion and is not universal in nature – turbulence theory is much harder to apply to this range.

The traditional approach to this problem has been to employ standard turbulence theory to account for the effective eddy dissipation across the entire spectrum of turbulence including the anisotropic turbulence. These are often applied to steady-state solutions where the time dependent nature of the flow is ignored. This approach is known as Reynolds Average Navier Stokes or RANS. Here turbulence theory is being applied to define the eddy viscosity on all scales of turbulent motion, even the anisotropic scales. This necessarily leads to empirical tuning of the turbulence model parameters in an attempt to account for the effect of anisotropic scales, as well as to higher turbulent dissipation resulting in time-steady behavior.

PowerFLOW leverages the fact that it is a transient solver with high 3D resolution. This allows PowerFLOW to directly resolve the anisotropic turbulent scales (or very large eddies) directly. PowerFLOW uses turbulence theory to model only what it naturally applies to – the universal scales of turbulence in the dissipative and inertial ranges. The dynamics of the sub-grid scale turbulence is represented by two additional equations derived from an extended Renormalization Group theory. This approach is known as Very Large Eddy Simulations or VLES. Finally, VLES is self-consistently coupled to the LBM for the resolved flow dynamics.

Turbulence models are employed within PowerFLOW as follows. Based on local and instantaneous flow field information, a local effective relaxation time is introduced in the Lattice Boltzmann collision operator to account for the unresolved universal scales of turbulent motion, similar to the concept of eddy viscosity in a Navier-Stokes solver. This relaxation time is locally determined for each cell in the simulation domain for each time-step via the supplementary two equations. The LBM VLES model is more applicable to situations encompassing turbulence, due to the fact that large and small (sub-grid) scales are not well separable.

### SUMMARY

This brief overview has described, at a very high level, the fundamental approach used in PowerFLOW for solving complex fluid dynamic problems - from the basic approach in the fluid momentum solver, to the incorporation of modeling the dissipative effects of unresolved turbulence. Exa’s philosophy is to guide each step in the evolution of this technology by our research in the fundamental physics being modeled. Accuracy is our dominant concern as without it engineering decisions cannot be made with confidence. The PowerFLOW product suite is then built around this core to provide its users with the ability to solve real world problems in a time frame that meets their requirements.