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Behaviour of Turbulent Prandtl and Schmidt Numbers in A Technical Note Proc IMechE Part G: J Aerospace Engineering 2015, Vol. 229(7) 1349–1359 Behaviour of turbulent Prandtl/Schmidt ! IMechE 2014 Reprints and permissions: number in compressible mixing layer sagepub.co.uk/journalsPermissions.nav DOI: 10.1177/0954410014547441 uk.sagepub.com/jaero Afroz Javed1, NKS Rajan2 and Debasis Chakraborty1 Abstract The behaviour of turbulent Prandtl/Schmidt number is explored through the model-free simulation results. It has been observed that compressibility affects the Reynolds scalar flux vectors. Reduced peak values are also observed for compressible convective Mach number mixing layer as compared with the incompressible convective Mach number counterpart, indicating a reduction in the mixing of enthalpy and species. Prt and Sct variations also indicate a reduction in mixing. It is observed that unlike the incompressible case, it is difficult to assign a constant value to these numbers due to their continuous variation in space. Modelling of Prt and Sct would be necessary to cater for this continuous spatial variation. However, the turbulent Lewis number is evaluated to be near unity for the compressible case, making it necessary to model only one of the Prt and Sct.. Keywords Compressible mixing layer, turbulent Prandtl/Schmidt number, convective Mach number Date received: 16 March 2014; accepted: 23 July 2014 Introduction noted during testing and extract various measures of For all combustion processes, good mixing of fuel and engine efficiency. Massively parallel computing, oxidiser is a basic requirement. Owing to the lower together with the maturation of CFD codes, has pressure conditions combined with low residence made it possible to perform simulations of complete time, the mixing process becomes very critical for engine flow paths within a reasonable time using supersonic combustion ramjet engine applications. RANS methodology. The gaseous fuel and oxidiser streams eventually In order to model Reynolds stresses arising after mix along the length of the combustor after injection. the time averaging of Navier–Stokes equations, tur- In case of the liquid fuels, the atomised fuel droplets bulence models are used in RANS simulations with evaporate and the gaseous phase mixes and burns varying degree of complexity. Similar to Reynolds along the combustor length. The mixing between the stress in Navier–Stokes equation, Reynolds heat flux fuel and oxidiser occurs in a mixing layer formed vectors arise in time-averaged energy equation and between the two at high compressible speeds. In a Reynolds mass flux vectors in time-averaged species study carried out by Jackson and Grosch,1 it has equations. The turbulent transport of a scalar prop- been reported that the instabilities of the compressible erty has historically been modelled using the gradient shear flow are predominantly inviscid. In such a scen- diffusion hypothesis. This model choice assumes that ario, the mixings of mass, momentum and energy are the turbulent transport of the scalar is in the direction expected to be dominated by turbulent transport as against molecular transport. Thus, the ability to pre- 1Directorate of Computational Dynamics, Defence Research and dict turbulent mixing at high compressible speeds is Development Laboratory, Kanchanbagh, Hyderabad, India crucial in obtaining accurate numerical simulations 2Department of Aerospace Engineering, Indian Institute of Science, for supersonic combustor design and performance Bangalore, India evaluation. Computational fluid dynamics (CFD) simulations using Reynolds-averaged Navier–Stokes Corresponding author: Debasis Chakraborty, Directorate of Computational Dynamics, (RANS) methodology are now routinely used in the Defence Research and Development Laboratory, Kanchanbagh P.O., scramjet engine development cycle to determine opti- Hyderabad 500058, India. mal fuel injector arrangements, investigate trends Email: [email protected] Downloaded from pig.sagepub.com at DEFENCE RESEARCH DEV LAB on June 11, 2015 1350 Proc IMechE Part G: J Aerospace Engineering 229(7) of decreasing values for that scalar. This leads to the Schmidt number. At lower values of turbulent following expression for the Reynolds heat flux and Prandtl number, the increased turbulent thermal dif- mass flux vectors. fusion process allowed heat to be transferred away from the flame-holding (recirculation) zones at a ~ rate that was not sustainable causing flame blow- 00 00 t @h h u j ¼ ð1Þ out. Due to this kind of sensitivity of Scramjet per- Prt @xj formance parameters on the choice of values for turbulent Prandtl/Schmidt number, it would be t @Y~n Y00 u00 ¼ ð2Þ tempting to look for a direct solution of the n j Sc @x t j Reynolds heat and mass flux vectors transport equations with suitable modelling for unclose The diffusion rates are controlled by specifying the terms. However, even if suitable models were avail- turbulent Prandtl (Prt) and Schmidt (Sct) numbers. able to close each of the scalar flux equations, the The turbulent Prandtl number specifies the ratio of number of additional equations introduced would the rate of turbulent momentum transport to rate of greatly exceed the number of equations required turbulent energy transport, while the turbulent to solve the continuity and momentum equations Schmidt number defines the ratio of the turbulent with suitable modelling. Clearly, it would be momentum transport rate to turbulent mass transport impractical to include a full second-order closure rate. model in simulations of engineering interests. To It has been indicated by experimental measure- address this problem, many studies have been car- ments and direct numerical simulation (DNS) stu- ried out by coupling the gradient diffusion hypoth- dies that Prt and Sct numbers for averaged flow esis with models that allow the turbulent Prandtl fields can vary significantly in different regions of and/or Schmidt number to vary spatially. the flow even for relatively simple shear flows like Several authors10–13 have pursued the development boundary layers, jets and wakes.2–5 A review on the of models that allow for spatial variation of turbulent turbulent Prandtl/Schmidt number in several free Prandtl Schmidt numbers within the context of gradi- shear flows made by Reynolds3 gives following ent diffusion hypothesis. These models involve add- variations (from core to outer region): round jet itional transport equation for the scalar variance 0.73–1.7; round wake 0.8–0.3; plane jet 0.5–1.3; and its dissipation rate. Xiao et al.11 have compared plane wake 0.5–0.7. In another review made by some of the available experimental data for scramjet Baurle5 for high-speed reacting flows, the range of combustors, using their two-dimensional (2D) simu- turbulent Prandtl number variations for planar jets lation model for variable turbulent Prandtl and from 0.2 to 3.0, for round Jets 0.7 to 2.0 and for Schmidt numbers, and satisfactory matches are backward facing step from 0.7 to 3.0. In the same shown. Xiao et al.14 have also validated variable review, the variation of the turbulent Schmidt Schmidt number formulation for Scramjet applica- number is found for planer jets from 0.1 to 2.2, tions through 2D simulations. Keistler et al.15 have for round jets from 0.1 to 2.0, for jet into cross- used models for the prediction of turbulent Prandtl/ flow from 0.1 to 0.5, and for injection behind a Schmidt numbers for simulations of supersonic com- bluff body it is from 0.2 to 0.7. The variations bustors from two different experiments (SCHOLAR observed in the values of Prt and Sct in these and HyShot). Comparisons were made with available reviews indicate that these numbers are dependent measurements of pressure temperature and compos- on the type of flow as well as the spatial region in ition, and fair to good agreement was observed. the flow. However, it has been general practice to However, many authors have also shown that com- assume a constant value of Prt and Sct in RANS bustor flows could be modelled with a constant value computations. Choice of a unique value of these of turbulent Prandtl and Schmidt number. Jiang and parameters for a simulation of a complex flow, Campbell16 have shown through their 2D/axisymmet- especially in the case where no information is avail- ric simulations with the available experimental results able about the turbulent Prandtl/Schmidt number that an average value of Prt ¼ Sct ¼ 0.7 could be used variations can lead to incorrect predictions. Several for the numerical simulations. Also it is found that computations6–9 performed by different authors lower values have also been used by different authors have at times shown an extreme sensitivity to to get satisfactory results. Prt ¼ Sct ¼ 0.2 was used by values assumed for these parameters. For example, He et al.17 for mixing of a gaseous jet in air cross-flow, in the study carried out by Baurle and Eklund7 for and by Kaaling et al.18 for their studies on a low- a scramjet combustor at Mach 4.0 flight condition, emission combustor design. Both these studies a variation of turbulent Schmidt number from 0.25 showed good match with the available experimental to 0.75 resulted in unstart of the intake due to data. Star et al.19 performed simulations for HyShot intense heat release at lower turbulent Schmidt supersonic combustor experiments with different tur- number and unsustained combustion due to low bulent Prandtl/Schmidt numbers combinations. It turbulent mass transfer at higher turbulent was observed that the best pressure match with Downloaded from pig.sagepub.com at DEFENCE RESEARCH DEV LAB on June 11, 2015 Javed et al. 1351 experimental data for pressure was obtained with a flow problem of given length scale, the subgrid constant value of Sct ¼ 0.5 and Prt ¼ 0.7. model contribution is reduced and accuracy is Although the use of variable turbulent Prandtl/ increased. For some very high Reynolds number tur- Schmidt number formulation does not require a bulent flows, if Á is sufficiently reduced but is still priori knowledge of suitable values of these numbers, greater than the Kolmogorov scale, the subgrid the constant turbulent Prandtl/Schmidt number meth- model influence becomes approximately negligible if odology is computationally simple.
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