While it is known that grid refinement affects the overall accuracy of numerical simulations, it has an even more marked impact on the solution of the species transport equations in reacting flow models. Unlike the standard mass, momentum, and energy conservation equations, the species conservation equations often feature stiff source terms representing chemical reactions that are sensitive to grid resolution and grid quality. Accurate prediction of species production impacts the prediction of pressure and temperature in turn. True assessments of reduced reaction mechanisms, turbulent combustion models, etc. are only valid once grid-induced errors are quantified and minimized. Thus, solution verification must go hand-in-hand with model validation, and effective tools for performing the former are needed.
Thermal and species turbulent transport in high-speed flows occur at different rates than turbulent momentum transport. These rates are often faster and the difference can be accounted for in traditional RANS simulations by using turbulent Prandtl and Schmidt numbers in the diffusion coefficient for the energy and species transport equations. Scalar fluctuations impact combustion by their effect on thermal/species transport and by their influence on the chemistry. The use of scalar fluctuation models is particularly important for combusting flows where the species and/or temperature gradients are more severe than in non-reacting mixing problems [11-14].
In non-reacting CFD simulations, the integration time step Δt generally scales with the fluid dynamic time scale. In a reacting simulation, the time scales associated with the chemistry also must be accounted for. Implicit single-step chemistry solution algorithms are often used, where the solution is advanced in time by the time step Δt and the instantaneous chemical source terms and the Jacobian matrix of the chemical source terms are evaluated only once, since only one unified time step is taken. However, under certain conditions, accurately capturing ignition phenomena and rapid intermediate species build-up can require very small time steps and thus be extremely computationally expensive, even for relatively small chemical systems. This occurs when the characteristic chemical time scale to be resolved is much smaller than the characteristic fluid-dynamic time scale, as shown in Figure 1.
Figure 1. Disparity of characteristic time scales in a chemically reacting system.
In order to allow for larger time steps that scale with the fluid-dynamic time scale while guaranteeing a solution that is accurate and mass-conservative, multi-dimensional CFD codes often decouple the solution of the chemistry part of the governing equations from the
convective-diffusive part based on an operator splitting numerical scheme. The resulting chemistry step entails solving a system of stiff ordinary differential equations (ODEs) for the chemical species, with the instantaneous species net production rates, ω!k, as the source terms. For a given Δt, a dedicated ODE solver may be employed that integrates the system of ODEs by automatically taking several sub-steps to accurately advance the thermo-chemical state in time, which is based on the required fractional time step dictated by the stiffness of the system of ODEs. These solution procedures are generally multi-step implicit methods with error correction and guarantees solution accuracy to a user-specified tolerance [15]. Ultimately, the chemical production term D(! !
Q) is dependent on the local temperature and species mass fractions.
D! !
( )
Q = f y(
i,T)
(1) Note that in some extended mechanisms, the rate coefficients and the production term may also be pressure dependent. Equation (1) may be evaluated numerically using a variety of techniques, including implicit treatment, stiff ODE solvers, In-Situ Adaptive Tabulation (ISAT) methods, etc., such as those found in Reference 16.As an illustrative example, the 3D SCHOLAR angled fuel injector mixing and combustion experiments conducted at NASA Langley Research Center provide data for CFD model validation for an environment that typifies the complex chemistry found in a scramjet combustor [17, 18]. Mean temperature, wall pressure, and species measurements are available, making these data sets useful for analyzing the capabilities of the scalar fluctuation model (SFM) to capture turbulent mixing effects in more complex flows involving hydrogen injection into three-dimensional cross-flows. Data was obtained at four stream-wise planes downstream of the injector. Computational analyses performed by Rodriguez and Cutler showed sensitivity to boundary conditions, including temperature, species composition, and inflow turbulence parameters [19]. They found their solutions to be most sensitive to the values used for the turbulent Prandtl and Schmidt numbers, Prt and Sct, and concluded that it is unlikely that constant values for these parameters can be calibrated to be valid for a wide range of flows. They suggest that the use of models providing for variable Prt and Sct values are necessary to properly simulate the SCHOLAR combustion experiment. Keistler, et al. proposed a model for variable Prt and Sct
values based on enthalpy and concentration variances [20], which was used to analyze the SCHOLAR combustor experiment, however, comparisons with the data have not shown overall consistency.
In Figures 2–4, we present a CFD analysis that clearly exhibits mesh-dependent behavior. The numerical analysis predicts variable Prt and Sct values using the SFM formulation, with compressibility corrections. The facility nozzle from the upstream heater was modeled to provide inflow boundary conditions for mean and turbulence quantities, to the test section. Near-wall modeling was used for no-slip walls, along with a constant wall temperature. Three levels of grid refinement were performed, denoted as Grid A, Grid B, and Grid C for subsequent levels of refinement, ranging from 868,000 to 6.4 million cells. A representative view of Grid C in the vicinity near the injector is provided in Figure 2. This domain was decomposed into an isolator section upstream of the injector, the injector section containing the angled fuel injector, and a downstream combustor section. The injector nozzle was included in a block that extended into the duct.
Figure 2. Schematic of SCHOLAR grid near injector with every other grid point shown for reference.
Figure 3. Lower (left) and upper (right) wall pressure grid sensitivity of the SCHOLAR combustion experiment.
Figure 4. Temperature grid sensitivity of the SCHOLAR combustion experiment.
Mixing and combustion cases were considered for all three grids using SFM. Wall pressure comparisons to data for all three grids for the combustion case are shown in Figure 3. Note that experimental error bars are not available. Substantial improvement downstream of x = 400mm is noted when going from Grid A to Grid C. Grid A was very coarse stream-wise in this section due to the long combustor, which caused a much more diffusive flow-field than that of more refined grids. The contours of mean temperature shown in Figure 4 indicate combustion in Grid A starting to take place well upstream of the other two grids. However, the amount of combustion taking place is much less, which coincides with the wall pressure result. The combustion onset location along the top wall for Grids B and C are in similar locations, providing some consistency in the results. Figure 5 shows mean temperature and N2 mole fraction contours compared to measured data for all three grids. As in the mixing case, the core jet mixing is under-predicted, even with the substantial increase in the number of grid points. More combustion appears to be taking place on planes 3 and 4 of Grid C compared to the experiment, but the overall contours provide a better match to data than coarser grids. These results clearly reveal an improved match to data with grid refinement going from Grid A to Grid C.
Figure 5. Grid-sensitivity comparison of the SCHOLAR combustion experiment.