Eddy viscosity modeling in the prediction of turbulent, backmixed combustion performance

The turbulent flowfield in a backmixed combustor is modeled analytically by simultaneously solving the governing partial differential conservation equations. An algebraic and a two-equation eddy viscosity model are employed in the numerical simulation to account for the turbulence transport processes. Predicted distributions of isothermal flow properties are systematically compared to experimentally obtained data to appraise the eddy viscosity models. In addition, predicted distributions of reacting flow properties are presented to illustrate the applicability of current numerical methods to the prediction of continuous combustion flows. The turbulent momentum and mass transport properties are evaluated in isothermal flow for a range of mixing conditions. Both eddy viscosity models qualitatively describe the system hydrodynamics, but the detailed flow structure is inadequately represented. The mass transport predictions from the algebraic viscosity model agree favorably with experiment. The inferior performance of the two-equation viscosity model is remedied by refining the boundary condition specification for the turbulence energy dissipation rate. It is shown that the turbulence energy dissipation rate adjacent to critical solid walls strongly influences the overall mixing characteristics of the two-equation model. The isothermal flow results indicate that the algebraic eddy viscosity model provides cost-effective predictions of the general fluid flow patterns and mass transport trends in confined flows exhibiting strong recirculation. The two-equation eddy viscosity model provides better resolution of the small-scale turbulence processes but requires careful testing to ensure realistic predictions. The hot flow calculations accentuate the inadequate transport characteristics identified in the isothermal flow analysis and verify that considerable testing of the numerical model is required before proceeding to the complicating conditions of combustion.


Introduction
The design and operation of continuous combustion devices such as gas turbines, |)oilers, and furnaces may be assisted by the development and application of suitable predictive models that account for pollutant production, combustion efficiency, and heat release behavior.The emergence of continuum flow models formulated from governing conservation equations represents a significant advance toward the mmaerical prediction of condmstion-chamber performance.The ap-plication of continuum models to the design of combustion hardware is attractive because the point-by-point prediction of veh)city, temperature, and composition is an integral part of the computational procedure.The quantitative prediction"of turbulent, baekmixed combustion has not yet been achieved be(rause of uncertainties linked to the submodels of fui~darnental combustion processes.The present study addresses an essential element of continuum flow modeling--the use of eddy viscosity submodels to account for turbulence transport processes.The objective is to determine the utility and applicability of eddy viscosity submodels in the nnmerical simula-1675 TURBULENT FLAMES tion of flows exhibiting strong recirculation or backrnixing.

Approach
Selected eddy viscosity submodels are systematically evaluatdd in a combined analytical and experimental investigation.A backmixed laboratory combustor is employed to conduct tests for both isothermal (nonreacting) and hot (reacting) flow conditions.For the case of isothermal flow, the eddy viscosity submodels are first evaluated for the prediction of system hydrodynamics.A tracer gas is subsequently introduced to evaluate the eddy viscosity submodels for nonuniform, multicomponent flow where mass as well as momentum exchange is important.Experimental measurements and numerical predictions of velocity and tracer isopleths for a range of approach velocities and backmix characteristics provide the critical tests.
The results reported here emphasize the case of isothermal flow.A satisfactory assessment and understanding of the isothermal case is a prerequisite to engaging the hot flow case.Combustion imposes the additional requirement for submodels of chemistry and turbulence/chemistry interaction that may mask fundamental deficiencies in the eddy viscosity submodel.However, hot flow results are presented as an indication of the suitability of the two eddy viscosity submodels in the prediction of reacting flows.
The results provide useful information for the development of applied computational tools for analyzing the performance of continuous combustion systems.In particular, the results identify strengths and weaknesses in the use of two eddy viscosity submodels, and demonstrate the need to thoroughly test predictive methods for the ease of isothermal flow before proceeding to the ease of hot flow.

Combustion System
The experimental configuration is a 51 mm I.D. x 457 mm cylindrical Vyeor combustion chamber containing an aerodynamic (opposed-jet) flameholder as shown in Fig. 1.The incoming flow of premixed methane and air is opposed by a high velocity jet ( r h J r h m < < 1) issuing from a 1.The opposed-jet combustor (OJC) exhibits essential features found in practical continuous combustion systems (backmixing, high intensity combustion) but avoids the attendant complexities (nonpremixed, two phase combustion) appropriate for subsequent studies.The separation of the critical backmixing regime from influential boundary conditions is important to testing the eddy viscosity submodels.In addition, the OJC provides a versatile flowfield to challenge the predictive method.For example, the rate and intensity of backmixing in the recirculation zone may be altered experimentally and numerically over a broad range by varying the ratio of approach velocity to jet velocity.This approach is standard practice in engineering calculations of turbulent flows and encompasses the introduction of turbulent exchange coefficients that imply Newtonian stress-strain relationships, Fourier heat conduction, and Fickian diffusion.A turbulent "eddy" viscosity, ~tt, is used to prescribe the turbulent momentum flux, i.e.

-pu'v'=Ix, ~+ (2)
The turbulent transport of scalar quantities is related to the eddy viscosity through the appropriate Prandtl or Schmidt numbers as follows:

Numerical Method
The theoretical analysis of the opposed-jet combustor is accomplished by solving numerically the time-averaged conservation equations for turbulent, backmixed flow with provision for chemical reaction.The backmixing requires that the governing mass, momentum, energy, and species conservation equations be elliptic.The computational procedure used to solve simultaneously the elliptic partial differential equations for the baekmixed flowfield is based upon the TEACH method described by Gosman et al. 1 Equations containing the primitive (p -1)) variables are solved simultaneously to generate an isothermal flow solution.Equations for stagnation entbalpy (/~) and species mass fraction (Y~) are then added to obtain a solution for hot flow conditions.The dependent variables (~b) are cast into a generalized equation of the form: --(oU~b)+----r(oVd~)=--F,,,

Turbulence Submodels
Turbulent transport properties are evaluated using time-mean or "effective" quantities. ~t,~ The present study explores specification of the eddy viscosity, Ixt, and turbulent mass transport, rti , in confined flows exhibiting strong backmixing or recirculation.Comparative calculations are performed using two distinct turbulence submodels to define the eddy viscosity--an algebraic model and a more sophisticated two-equation submodel: The algebraic formulation is commonly used in predictive modeling.The variation in eddy viscosity depends only upon local density (~patially invariant for isothermal conditions) for a given geometry and inlet flow rate.The two-equation submodel introduces additional dependent variables that complicate the numerics and add computational cost.However, the two-equation submodel accounts for spatial variations in turbulent kinetic energy and length scale, and has been tested satisfactorily in a number of fluid dynamic encounters.~

Results (Isothermal Flow)
To effectively test the submodels of eddy viscosity over a range of turbulent conditions, experimental data were obtained for approach velocities of 15.24 and 7.62 m/see and jet velocities of 30.5, 61 and 130 m/see.Velocity profiles were mapped throughout the flowfield using a 1 mm O.D. pitot tube.Tracer profiles were generated by introducing alternatively pure carbon monoxide (CO) in the jet and pure carbon dioxide (CO2) in the mainstream.The mixing between the jet and mainstream was assessed by mapping CO and CO 2 profiles throughout the flowfield using a 3 mm O.D. stainless steel tube and Beckman 315B NDIR instruments for the gas sampling and analysis.The probe dimensions are small (probe O.D./OJC I.D. = .06)to minimize hydrodynamic interferences.The Reynolds number of the approach stream ranged from 2.5 x 104 (at 7.62 m/see) to 5.0 x 104 (at 15.24 m/see).

Momentum Transport
The eddy viscosity submodels were first tested for a homogeneous, isothermal flowfield to explore the effectiveness of the submodels in predicting momentum transport.Experimental and predicted profiles of the axial component of mean velocity are presented in Fig. 2 for an average inlet velocity of 15.24 m/sec.The selected profiles emphasize the flow regime dominated by recirculation.The velocity data are subject to experimental errors on the order of five percent in the bulk flow region, but in the recirculation zone the results are compromised by pitot tube errors of twenty-five percent or more.The calculated velocity profiles for both turbulence submodels assume an experimentally determined velocity profile as an inlet condition.
The trend of the velocity profiles agree qualitatively with the experimental results.The two-equation submodel retains the inlet turbulent velocity profile in the approach section while the velocity profiles predicted by the algebraic submodel flatten out due to the spatially uniform viscosity.The presence of the opposing-jet is communicated further upstream for the two-equation submodel than for the algebraic eddy viscosity, and the twoequation submodel demonstrates better correlation with the experimentally observed stagnation point.
The mean velocity profiles in the recirculation zone are also better represented by the two-equation turbulence submodel, although both submodels over-predict the jet expansion.In addition, the two-equation velocity profiles decrease too rapidly near the chamber wall.Downstream from the backmixed zone both submodels fail to adjust for the redistribution of the bulk flow in the turbulent wake of the recirculation zone and under-predict the velocity adjacent to the jet wall.Both submodels display good agreement in the bulk flow regime in the absence of recirculation or wall effects.
The recirculation zone strength was examined by reducing the approach velocity by a factor of two while retaining the same jet velocity.The experimental and predicted velocity profiles are presented in Fig. 3.The calculated results demonstrate a significant departure from experiment in the region of strong backmixing.The reverse flow region again penetrates further upstream in the twoequation submodel prediction.
The acceleration of the bulk flow around the recirculation zone is under-predicted in contrast to the over-predicted jet expansion for both submodels.This result is especially evident near the plane of the jet exit where the redistribution of the bulk flow with the jet discharge is restricted.This effect is conveyed downstream from the recirculation zone where the two-equation velocity profiles correlate well with experiment near the chamber wall but deviate near the jet wall.

Mass Transport
The two eddy viscosity submodels were secondly tested in conjunction with the turbulent mass exchange submodel (Eq. 3) for the conditions of an isothermal, nonhomogeneous flowfield.Spatial distributions of the predicted tracer concentrations are compared to experimental measurements in Figs. 4 and 5 for the baseline conditions outlined previously.The tracer gas concentration data were obtained at radial increments of 1.3 mm and axial stations every 13 mm.The concentration isopleths (experimental) were constructed from values measured at the discrete data points in the experimental flowfield and compared to concentration isopleths (predicted) constructed from values predicted at the discrete grid points in the computational field.stream is 100 percent carbon monoxide.)Immediately evident is the contrasting axial and radial spread of the predicted tracer concentration profiles.Differences in axial transport of CO are consistent with the observations forwarded in the previous section on momentum transport.More notable are the diverse radial transport predictions.The spreading of the jet discharge into the mainstream for the algebraic eddy viscosity conforms to the experimental results.In contrast, the two-equation submodel limits the radial mixing of the jet stream (tracer) with the bulk flow.The segregation of the jet and the mainstream is verified by the high CO concentrations along thejet wall, in contrast to the absence of CO near the chamber wall.This result was further substantiated by introducing a carbon dioxide (COz) tracer in the approach stream.The numerical analysis of CO 2 concentrations for the two-equation submodel predict negligible mass transport of CO 2 from the mainstream into the recirculation zone.Downstream, the predicted mixing of CO 2 from the bulk stream into the flow associated with the jet stream is minor.Experimentally CO 2 was observed to be well mixed into the jet.The results for the strong recirculation zone (Fig. 5) demonstrate similar trends.Although the velocity data presented earlier in Fig. 3 gave poorer agreement for conditions of strong recirculation, the mixing is improved because of the enhanced backmixing.

Results (Hot Flow)
A two-step global reaction mechanism for methane oxidation was adopted for the present study to illustrate the applicability of current numerical methods to the prediction of continuous combustion flows, and to briefly explore the suitability of the eddy viscosity submodels for the case of hot flow: A detailed description of the numerical formulation of the reaction rate expressions and boundary conditions may be found elsewhere .6 The dependent variables for the hydrocarbon system include the mass fractions YcH4 and Yc02" Distributions of other major species-water (H20), oxygen (O2) carbon monoxide (CO)--are related to YCH, and Ycos by elemental mass conservation.An initial solution for the reacting flowfield was obtained using published reaction rates 7 which were later refined to more closely resemble the experimental data.This approach was justified on the basis that the reaction rate data were obtained from a well-stirred reactor study and may not apply to the current study, which deviates substantially from well-stirred conditions.Global reaction rates were judged acceptable for the present study because of the uncertainties encompassing the specification of the system aerodynamics.
Since the aim of the study is to predict the performance of baekmixed combustion systems, the hot flow calculations were extended to include the prediction of the pollutant species, nitric oxide (NO).Nitric oxide kinetics were based on the familiar Zeldovieh 8 mechanism: Noting that reaction ( 8) is the rate limiting step and adopting the simplifying assumption of 0 / 0 2 equilibrium results in the reaction rate expression: (10)   The temperature and 0 2 distributions obtained from the solution of the hydrocarbon systems were used as a basis for the NO kinetic calculations.
The experimental tests conducted to complement the numerical predictions of hot flow properties utilized stoiehiometric proportions of premixed methane and air in both the main and jet streams.The reactants were initially at ambient temperature and the combustion was completed at atmospheric pressure.The test matrix included the same approach velocities selected for the isothermal eases--7.62and 15.24 m / s e e --a n d a jet velocity of 130 m/sec.
Selected results of hot flow properties are presented in Figs. 6, 7, and 8 for the conditions indicated.Velocity, temperature, and NO data are presented as indicators of useful design information that may be derived from the numerical simulation of backmixed combustion processes.A photograph of the opposed-jet combustor is shown in Fig. 6 with predicted velocity vectors superimposed.The detailed maps of flowfield properties presented in Figs.7 and 8 were obtained by conventional probing techniques.Temperature measurements were made using an uncoated Pt/Pt-13% Rh finewire thermocouple.The data are not corrected for radiation losses, but provide an adequate description of the OJC heat release distribution.A Scott Model 125 chemiluminescent analyzer was used for oxides of nitrogen measurements.Gas samples were extracted via a moveable 3.2 mm O.D. watercooled, 316 stainless steel probe having a tubular inlet, and conveyed through a heated teflon sample line to a packaged exhaust gas analysis system.The measurement of species susceptible to sample transformations (e.g.NO, NO2) must be regarded as qualitative estimates of the actual local concentrations.

Isothermal Flow
The results of the momentum and mass transport studies identify several important characteristics of the turbulence submodels.The turbulent viscosity from the two-equation submodel varies through the flowfield as is physically expected.The two-equation submodel also demonstrates the generation and dissipation of turbulence kinetic energy near the jet exit, the stagnation point and near the boundaries consistent with other numerical investigations.9 In contrast, the turbulent viscosity from the algebraic submodel is spatially uniform throughout the flowfield for isothermal flow, and varies only with a change in the kinetic energy of the inlet streams.The impact of these differences between the two eddy viscosity submodels is evidenced by the velocity profiles, the predicted location of the stagnation point, the radial mass transport, and the dissipation of the kinetic energy of turbulence in the wake region.
Velocit!r Profiles.The predicted velocity profiles agree qualitatively with experimental results.The two-equation submodel more closely approximates the experimental trends although the predicted velocity contours for conditions of strong recirculation are quantitatively incorrect throughout the flowfield for both turbulence submodels.The departure from experiment is especially evident near walls, and suggests the need to refine the wall functions used to specify boundary conditions for turbulent kinetic energy or dissipation rate, and/or to decrease the grid spacing near the wall.
Stagnation Point.The location of the stagnation point as indicated by the velocity profiles and upstream extent of tracer measurements is better described by the two-equation submodel.The effective viscosity predicted by the algebraic submodel is spatially uniform through the flowfield for given inlet conditions.No account is made for changes in the kinetic energy of turbulence or energy dissipation rate in the region of encounter between the main and jet flows.The lower viscosity predicted in the reeireulation zone by the two-equation submodel is more effective in describing the actual axial momentum and mass exchange.
Radial Mass Transport.Although the experimentally determined tracer isopleths are in closest agreement in the reeirculation zone for the two-equation submodel, the downstream radial spread is more effectively described by the algebraic submodel.The uniformly high viscosity predicted by the algebraic turbulence submodel aids the overall mass transport throughout the flowfield, i.e. axial and radial diffusion from the jet is greater for the algebraic rather than for the two-equation submodel.
The radial mass transfer predicted by the two-equation turbulence submodel suggests that the principal mechanism of mass transport is by large-scale convection rather than by small scale gradient diffusion processes, m The tracer concentration varies gradually through the recirculation zone.The downstream mixing layer is bounded by steep concentration gradients that enclose the jet mixing region.The numerical predictions indicate that at high approach velocities the bulk flow detours around the reeirculation zone, retaining upstream flow properties (low viscosity and small-scale mixing) and effectively precludes radially-directed mass transport.Conversely, the experimental trend reinforces the conjecture that turbulent exchange processes should be active throughout the annulus downstream from the recirculation zone.
Wake Region.The kinetic energy of turbulence predicted by the two-equation submodel is quickly dissipated in the wake region downstream of the recirculation zone.The resulting low viscosity impedes turbulent mixing throughout the wake region.In contrast, the uniformly high viscosity of the algebraic submodel is effective in predicting the radial spread of tracer observed experimentally.

Diagnosis
The results and subsequent discussion have exposed deficiencies in both of the eddy vis-cosity submodels that were tested and have identified directions for possible refinements.A supplementary investigation was initiated to determine the origin of the shortcomings of the two-equation eddy viscosity submodel in the present application and to improve the performance of the submodel in strongly recirculating flows.
A series of numerical experiments were conducted to examine the effect of boundary condition specifications on the turbulence transport characteristics of the two-equation eddy viscosity submodel for isothermal flow conditions.Turbulence properties at the jet and main stream inlets and adjacent to solid walls were systematically varied to improve numerical/experimental correlation and to determine the sensitivity of the solution to inlet and solid boundary conditions.
Modifications to the main inlet conditions, i.e. increased turbulence kinetic energy ( k ) and decreased turbulence energy dissipation (e), improved the radial spread of tracer concentration but did not overcome the experimental discrepancies.The increased viscosity in the main stream also tended to move the predicted stagnation point downstream.Similar variations at the jet inlet had little impact on the radial mass transport but had a significant influence on the predicted stagnation point.Parametric changes reducing the viscosity in the vicinity of the jet exit tended to extend the stagnation point upstream.
The principal outcome of the parametric studies was the observation that in confined flows, where the recirculation zone may contact solid boundaries, the specification of near-wall turbulence energy dissipation rate may significantly influence the mixing characteristics of the two-equation turbulence submodel.These results are consistent with the findings of earlier studies 11 where a reduction of the energy dissipation rate adjacent to a downstream facing wall benefited numerical/experimental correlation.Similarly, the reduction of the energy dissipation rate on the upstream facing step of the jet tube was found to radically alter the predicted mass transport behavior of the two-equation submodel.The improved radial mass transport is evidenced by the tracer concentration profiles in Fig. 9.The stagnation point correlation may be restored by modifying the turbulence kinetic energy or the rate of turbulence energy dissipation at the jet inlet.
The analysis has shown that the turbulence energy dissipation rate in the flowfield is highly dependent on the shear stress along critical solid walls.The initially poor perform-Flc.9. Isothermal flow tracer concentration profiles for refined jet boundary conditions (Ix: two-equation).D m = 15.24m/sec; Dj = 130m/see; q~,~ = 0.; 100% CO Jet.
ance of the two-equation eddy viscosity submodel was remedied by refining the wall boundary conditions for the turbu.lenceenergy dissipation rate.The supplementary investigation has also demonstrated the need to test the two-equation turbulence submodel for isothermal flow conditions prior to simulating hot flows.

Hot Flow
The predicted hot flow results for the baseline eddy viscosity submodels qualitatively describe the bulk convective flowfield and the flameholding characteristics of the OJC, but the extent of chemical reaction is generally overestimated.The accelerated consumption of fuel and oxygen and the assumed adiabatic boundary conditions precipitate elevated flame temperatures and excess nitric oxide production.The spatial distributions of hot flow properties substantiate the inherent limitations of the eddy viscosity submodels that emerged in the isothermal flow analysis.The poor numerical simulation of the detailed flame structure is attributed to deficiencies in the coupled turbulence/chemistry submodels and the boundary condition specifications.

Conclusions
The relative merits of an algebraic and a two-equation eddy viscosity submodel for simulating turbulence transport properties in recirculating flows have been assessed by comparing continuum model predictions of the turbulent, backmixed flowfield in an opposed-jet combustor to experimental measurements.
Both eddy viscosity models qualitatively describe the bulk hydrodynamic flowfield, but the detailed flow structure is inadequately represented.Mass transport estimates based on the algebraic viscosity model conform favorably to experiment.The two-equation turbulence submodel yields consistently poor correlation.The deficiency of the two-equation eddy viscosity model predictions is attributed to the boundary condition specification for the turbulence energy dissipation rate.It is shown that the turbulence energy dissipation rate adjacent to critical solid walls strongly influences the overall mixing characteristics of the two-equation submodel.
The results indicate that the algebraic eddy viscosity model provides cost-effective predictions of the general fluid flow patterns and mass transport trends in turbulent, confined flows exhibiting strong recirculation.The two-equation eddy viscosity submodel provides better resolution of small-scale turbulence proeesses but requires careful testing to ensure realistic predictions.
It is also shown that considerable testing of continuum flow models in isothermal flow is required before proceeding to the complicating conditions of combustion.The hot flow calculations accentuate the inadequate transport characteristics identified in the isothermal flow analysis.Attempts to refine the chemistry submodel or to quantify the interaction of fluid motions on chemical reaction rates are dependent on the correct turbulence submodel formulation.