3.2.4. Turbulence Modeling Overview

Turbulent reacting flows involve a very large range of length and time scales, requiring massive computational resources to directly resolve all of the physical processes for even the most simple problem. To be able to solve complex problems of interest in a reasonable amount of time, modeling approximations must be made. A filtered form of the time-dependent Navier-Stokes, energy, and species mass conservation equations presented in Laminar Flow Equations are used, and closure models are applied to the new terms that arise due to the filtering operation. Temporal filtering is used in the Reynolds-Averaged Navier-Stokes (RANS) method, and spatial filtering is used in the Large Eddy Simulation (LES) method. The form of the models are dependent on the type of filtering performed, and will be discussed for both the RANS and LES approaches in the following sections.

The length scales between the smallest control volume dimension and the largest mesh dimension are defined as being “resolved”, and the transport equations are used to solve the physics in this range. The effects of the resolved turbulent scales may be modeled for RANS closures or they may be directly solved for LES closures. Turbulence length scales can extend down many orders of magnitude beyond the smallest finite volume dimension to the Kolmogorov scales, and these subgrid scales must be modeled in either closure approach.

The output of the closure models is expressed as a source term in the conservation equations for the mean flow and as effective properties in the radiative transport equation. Hence, the output of the closure models can be interpreted as being cell-averaged values for the control volume for the appropriate time scale. For the RANS formulation used here, the time scale is long relative to the turbulence time scales (i.e., long time average). For LES, the time scale is the local advection time. For the current suite of models, the momentum closure model is of the lumped-parameter type; that is, it assumes homogeneity of the subgrid turbulence. The remaining closures, species and energy, are of the zone-model type; that is, they assume heterogeneity of the species and energy subgrid. Two zones (one combusting, one not) are used in the current zone models.

For length scales above the length scale of the mesh, the physics is modified via boundary and initial conditions. Momentum boundary conditions include specified velocity (wind, and mass sources), or constant pressure (inflow/outflow). Species boundary conditions include a mass source for the fuel (pool model). Thermal boundary conditions include flux and temperature conditions. The following sections provide details of the math models for conservation laws and fire physics models used in SIERRA/Fuego.

3.2.4.1. RANS Temporal Filtering

In many typical engineering applications, only time averages of physical quantities are of interest. Often, details of the turbulent fluctuations are of little concern. RANS formulations address this need by solving a temporally-filtered form of the transport equations, directly yielding the time-averaged variables of interest. For this reason, RANS approaches represent a relatively low-cost solution method at the expense of additional modeling complexity.

An independent variable $\phi$ can be temporally filtered to obtain its mean $\bar{\phi}$ with the mathematical form (Tennekes and Lumley [10])

(3.68) $\overline{\phi(\boldsymbol{x})} = \lim_{\tau \rightarrow \infty} {1 \over \tau} \int_{t_o}^{t_o + \tau} \phi(\boldsymbol{x},t)\, \mathrm{d}t.$

The original variable can be represented as the sum of its mean and fluctuating component, $\phi = \bar{\phi} + \phi'$ , with the properties that $\bar{\bar{\phi}} = \bar{\phi}$ and $\overline{\phi'} = 0$ . This is called the Reynolds decomposition of a variable.

In combustion problems, the overall exothermic process can result in large localized temperature increases and a correspondingly large density decrease in open systems where the molecular weight change from reactants to products is small. Allowing for turbulent fluctuations of density, the above temporal averaging procedure gives rise to additional terms involving time averages of products of density and other variable (e.g., velocity) fluctuations. An alternative approach to applying the Reynolds decomposition strictly to all independent variables is to consider a mass-weighted decomposition known as Favre averaging (Libby and Williams [11], p. 15; Kuo [12], p. 419). This simplifies all of the transport equations and eases modeling. A Favre-averaged variable $\tilde{\phi}$ is defined in terms of Reynolds averages as

(3.69) $\tilde{\phi} \equiv \frac{ \overline{\rho\phi} }{ \bar{\rho} }.$

A variable can then be decomposed into its Favre-mean and fluctuating component as

(3.70) $\phi = \tilde{\phi} + \phi'',$

where $\overline{\rho\phi''} = 0$ . Note that $\overline{\phi''} \ne 0$ . The relation between time averaged and Favre-averaged quantities is

(3.71) $\tilde{\phi} = \bar{\phi} \left(1 + {{\overline{\rho'\phi'}} \over {\bar{\rho}\bar{\phi}}}\right) .$

Favre averaging is used for all turbulent transport equations solved in SIERRA/Fuego.

For the RANS formulation used here, the laminar conservation equations of Laminar Flow Equations are first temporally filtered, revealing additional terms that can be simplified by substituting the Favre decomposition, resulting in the Favre-filtered equations that will be presented in Turbulent Flow Equations, Favre-Averaged. This procedure results in new terms in the equations that consist of time averages of products of fluctuating quantities, called Reynolds stresses. These moments must be modeled to close the system of equations.

The length of the time filter is typically much larger than the time scales of a turbulent flow, meaning that all time scales from the largest turbulence scale down to the minimum Kolmogorov scale are represented by these Reynolds stresses. In a strict sense, there can be no time dependence of a mean (time-averaged) quantity. However, if there are variations in mean quantities that occur on time intervals long compared to the averaging interval, then the transient terms for the mean quantities may be justified and required. For this reason, unsteady RANS simulations are possible with the present formulation. The available RANS turbulence closure models are discussed in Turbulence Closure Models.

3.2.4.2. LES Spatial Filtering

Unlike the RANS approach which models most or all of the turbulent fluctuations, LES directly solves for all resolved turbulent length scales and only models the smallest scales below the grid size. In this way, a majority of the problem-dependent, energy-containing turbulent structure is directly solved in a model-free fashion. The subgrid scales are closer to being isotropic than the resolved scales, and they generally act to dissipate turbulent kinetic energy cascaded down from the larger scales in momentum-driven turbulent flows. Modeling of these small scales is generally more straightforward than RANS approaches, and overall solutions are usually more tolerant to LES modeling errors because the subgrid scales comprise such a small portion of the overall turbulent structure. While LES is generally accepted to be much more accurate than RANS approaches for complex turbulent flows, it is also significantly more expensive than equivalent RANS simulations due to the finer grid resolution required. Additionally, since LES results in a full unsteady solution, the simulation must be run for a long time to gather any desired time-averaged statistics. The trade-off between accuracy and cost must be weighed before choosing one method over the other.

The separation of turbulent length scales required for LES is obtained by using a spatial filter rather than the RANS temporal filter. This filter has the mathematical form

(3.72) $\overline{\phi(\boldsymbol{x},t)} \equiv \int_{-\infty}^{+\infty} \phi(\boldsymbol{x}',t) G(\boldsymbol{x}' - \boldsymbol{x})\, \mathrm{d}\boldsymbol{x}',$

which is a convolution integral over physical space $\boldsymbol{x}$ with the spatially-varying filter function $G$ . The filter function has the normalization property $\int_{-\infty}^{+\infty} G(\boldsymbol{x})\, \mathrm{d}\boldsymbol{x} = 1$ , and it has a characteristic length scale $\Delta$ so that it filters out turbulent length scales smaller than this size. In the present formulation, a simple “box filter” is used for the filter function,

(3.73) $\begin{eqnarray} G(\boldsymbol{x}' - \boldsymbol{x}) &= 1/V (\boldsymbol{x}' - \boldsymbol{x}) & \in \mathcal{V} \\ &= 0 & \mathrm{otherwise} \end{eqnarray},$

where $V$ is the volume of control volume $\mathcal{V}$ whose central node is located at $\boldsymbol{x}$ . This is essentially an unweighted average over the control volume. The length scale of this filter is approximated by $\Delta = V^\frac{1}{3}$ . This is typically called the grid filter, as it filters out scales smaller than the computational grid size.

Similar to the RANS temporal filter, a variable can be represented in terms of its filtered and subgrid fluctuating components as

(3.74) $\phi = \bar{\phi} + \phi'.$

For most forms of the filter function $G(\boldsymbol{x})$ , repeated applications of the grid filter to a variable do not yield the same result. In other words, $\bar{\bar{\phi}} \ne \bar{\phi}$ and therefore $\overline{\phi'} \ne 0$ , unlike with the RANS temporal averages.

As with the RANS formulation, modeling is much simplified in the presence of large density variations if a Favre-filtered approach is used. A Favre-filtered variable $\tilde{\phi}$ is defined as

(3.75) $\tilde{\phi} \equiv \frac{ \overline{\rho\phi} }{ \bar{\rho} }$

and a variable can be decomposed in terms of its Favre-filtered and subgrid fluctuating component as

(3.76) $\phi = \tilde{\phi} + \phi''.$

Again, note that the useful identities for the Favre-filtered RANS variables do not apply, so that $\bar{\tilde{\phi}} \ne \tilde{\phi}$ and $\overline{\phi''} \ne 0$ . The Favre-filtered approach is used for all LES models in SIERRA/Fuego.