MODELLING OF TURBULENT ENERGY FLUX IN CANONICAL SHOCK-TURBULENCE INTERACTION

Transcription

1 MODELLING OF TURBULENT ENERGY FLUX IN CANONICAL SHOCK-TURBULENCE INTERACTION Rssell Qadros, Krishnend Sinha Department of Aerospace Engineering Indian Institte of Technology Bombay Mmbai, India 476 Johan Larsson Department of Mechanical Engineering University of Maryland, College Park MD 74, USA ABSTRACT Spersonic and hypersonic flows often enconter high heat loads de to presence of shock waves. The nclosed trblent energy flx correlation in the mean energy conservation eqation significantly affects the heat transfer rate. In conventional RANS models, which are based on gradient diffsion hypothesis, it is modelled in terms of trblent condctivity and gradient of mean temperatre. This modelling approach often overpredicts the vale of energy flx in the region of the shock. In the crrent work, we first apply the conventional and realizable model to predict the trblent energy flx in canonical shock-trblence interaction. The shortcomings of these models in predicting the energy flx across the shock are highlighted, and a differential eqation model is proposed based on linear theory. The reslts obtained are compared with available direct nmerical simlation data and a good match is fond for trblent energy flx generation across a shock. Finally, for its ease of implementation, an algebraic model is proposed with the aid of linear theory that predicts the trblent energy flx correctly across the shock for a range of pstream Mach nmbers. INTRODUCTION High-speed flows in aerospace applications have shock waves interacting with bondary layers in different parts of the vehicle srface and in engine components. Sch shock/bondary-layer interactions (SBLI) are often marked by high localized srface pressre and heat transfer rates. Predicting the heating loads is especially important in spersonic and hypersonic applications with trblent bondary layers. Majority of the existing trblence models for heat flx prediction yield acceptable reslts in bondarylayer flows (Bowersox, 9), bt their predictive capability is severely limited in shock-dominated flows (Roy & Blottner, ). An important nclosed term in the mean energy conservation eqation, which governs the heat transfer rate, is the trblent energy flx vector ũ j e. Here, j represents the velocity flctation in the j th direction, e represents the internal energy flctation and tilde represents Favre averaging. Conventionally, this term is modelled as a prodct of trblent condctivity and gradient of mean temperatre. Trblent condctivity is related to the eddy viscosity µ T via a trblent Prandtl nmber Pr t. A constant vale of Pr t =.89 gives satisfactory reslt in flat plate bondary layers and is often sed in SBLI configrations. This modelling approach however overpredicts the actal wall heat transfer rate significantly in SBLI. An alternate model based on variable Pr t approach is proposed by Xiao et al. (7). It solves additional differential eqations for enthalpy variance and its dissipation rate, and employs these two qantities in the formlation of trblent heat flx vector. In shock-dominated flows, this approach leads to improved wall heat flx predictions, yet it overpredicts experimental data. To the best of or knowledge, there is no direct stdy which involves modelling of the trblent energy flx across the shock. An effort in this direction is the work by Bowersox (9) which proposes an algebraic model for the trblent energy flx in spersonic flows, bt is limited to zero-pressre-gradient bondary layers withot shock waves. In a recent work, Qadros et al. (5) present a detailed stdy of the trblent energy flx at a shock wave. They investigate the physical processes that govern the generation of energy flx correlation in a canonical shocktrblence interaction (STI). This model problem consists of a homogeneos isotropic trblence which is prely vortical in natre being carried by a one-dimensional niform mean flow passing throgh a nominally normal shock wave. The trblence is amplified by the shock, and the shock in trn gets distorted. Schematic of this problem is shown in figre. This is possibly the simplest configration that isolates the effect of shock on trblence withot other effects sch as bondary-layer gradient, flow separation and streamline crvatre. Inspite of its geometrical simplicity, STI displays a range of physical effects sch as trb-

2 Uniform mean flow U Distorted shock.5 Vortical trblence Figre. Schematic showing a shock wave distorted pon interaction with trblent flctations. lence anisotropy, generation of acostic waves, baroclinic torqes, and n-steady shock oscillations. The work presented in Qadros et al. (5) relies primarily on linear interaction analysis (LIA), a theoretical approach to analyse canonical STI. The analysis involves solving the fndamental interaction of a single two-dimensional plane wave with a shock which generates downstream distrbances that can be characterised in terms of Kovasznay modes of vorticity, entropy and acostic. Integration over a specified pstream trblence spectrm yields threedimensional trblence statistics downstream of the shock. The LIA reslts obtained for energy flx are compared with direct nmerical simlation (DNS) data available from Larsson et al. (3). A good match is fond between LIA and DNS data in predicting some of the key physics of the interaction. As part of modelling the trblent energy flx in shock-trblence interaction, Qadros et al. (5) identify the mechanisms contribting to trblent energy flx generation based on linearised Rankine-Hgoniot conditions which are valid for small pstream flctations. The dominant terms are frther modelled based on LIA reslts. The reslting differential eqation for trblent energy flx is solved along with the k and ε eqation to give the jmp across the shock as well as the downstream decay. A good match in the streamwise evoltion is fond pon comparison with the DNS data for a range of pstream Mach nmbers. However, it is to be noted that inclding an additional differential eqation for energy flx may reqire significant modification in the existing comptational flid dynamics (CFD) codes. For their ease of implementation, algebraic heat flx models are preferred in CFD, and are in se in most Reynolds-averaged Navier-Stokes (RANS) simlations. In this paper, we apply existing trblence models to predict the trblent heat flx generated at a shock wave. These inclde conventional eddy viscosity model and realizable model. A variant of the previosly modelled differential eqation for the trblent energy flx is also presented. These model predictions are evalated against linear theory reslts and available DNS data. Finally, we develop a new algebraic model to accrately predict the energy flx generated in shock-trblence interaction. TURBULENT ENERGY FLUX IN SHOCK- TURBULENCE INTERACTION Qadros et al. (5) stdied the generation of trblent energy flx in the canonical STI problem sing LIA, y x M Figre. Variation of e with pstream Mach nmber. The velocity flctation is normalized by the pstream mean velocity and the energy flctation is normalized by the downstream mean temperatre and the specific gas constant R. All correlations are frther normalized by the pstream TKE. M M t Re λ R kk /(U ) Table. List of the DNS cases from Larsson et al. (3) with the listed parameters corresponding to the location jst pstream of the shock. which models the pstream trblence as a collection of planar waves. Each of these waves is considered to independently interact with the shock. The governing eqations downstream of the shock are linearized Eler eqations and the jmp in flctations across the shock is determined by linearized Rankine-Hgoniot conditions. A set of linear algebraic eqations are obtained by sbstitting the planar waveforms into the governing eqations. Solving these eqations yields the distrbance field downstream of the shock for a given pstream vortical wave. The downstream trblent field for a fll three-dimensional pstream trblence is obtained by integrating the two-dimensional planar wave reslts over a specified energy spectrm. Details of this analysis can be fond in Mahesh et al. (996). DNS of canonical STI was carried ot by Larsson et al. (3) for a vortical trblence passing throgh a normal shock. Table shows the cases from the DNS analysis whose data is tilized in the crrent stdy. The Mach nmbers range from low spersonic to hypersonic limit. The trblent Mach nmber M t and the Reynolds nmber based on

3 Taylor scale Re λ for each of the cases are also listed. Here, trblent Mach nmber is defined as M t = R kk /ã, where R kk represents twice the trblent kinetic energy, and ã represents the Favre averaged speed of sond. Re λ is given by ρ R kk /3λ/µ, where λ is the Taylor scale and µ is the average dynamic viscosity. The trblent energy flx vales pstream of the shock are zero as the trblence is vortical in natre and void of temperatre flctations. However, jst downstream of the shock, both LIA and DNS predict a peak positive energy flx. Frther downstream, DNS shows a steep decrease in the energy flx vales to negligible levels whereas LIA predicts a constant far-field vale. The peak vales of energy flx as predicted by both DNS and LIA for varying Mach nmbers are shown in figre (reprodced from Qadros et al. (5)). Also shown is the far-field vale of LIA, and at high Mach nmbers, it is almost eqal to the LIA peak energy flx. Note that conventional Reynolds averaging/flctation is sed instead of its Favre conterpart, as negligible difference is observed pon comparison sing the DNS data set. The peak energy flx as per theory matches well with DNS for low Mach nmbers pto M <, with the theory overpredicting the vales at higher shock strengths. The LIA prediction reaches an asymptotic vale of.3 at high Mach nmbers, while DNS shows a limiting vale less than. Overall, a good qalitative match is observed between LIA and DNS indicating that the key physical mechanisms governing the energy flx transport are captred by the theory. It is important for any model that aims to predict the trblent energy flx reslts, to captre this peak positive vale. Qadros et al. (5) formlated a differential eqation-based model, that for a given Mach nmber, captres the corresponding peak positive vale and the acostic decay that follows. A good match with DNS data was obtained for a range of pstream Mach nmbers, and a slight variant of this model is discssed in this work. However, greater emphasis is laid in the crrent stdy on the development of an algebraic model de to its ease of implementation in comparison with the differential eqation-based model. 3 MODELLING OF TURBULENT ENERGY FLUX 3. Conventional Modelling The trblent energy flx is conventionally modelled sing gradient diffsion hypothesis as e = κ T T ρ, () where ρ and T are the mean density and temperatre. Thermal condctivity κ T is given by κ T = µ T C v /Pr T, where C v is the specific heat at constant volme and Pr T represents the trblent Prandtl nmber having a constant vale of.89. Eddy viscosity is given by µ T = C µρk /ε, where C µ =.9, k is the trblent kinetic energy (TKE) and ε is the dissipation rate of TKE. In order to compte for the trblent energy flx in the given model problem, the mean flow variables are prescribed as hyperbolic tangent profiles across the shock, with the shock thickness obtained from DNS data. The vales of k and ε sed in expression for eddy viscosity are obtained by solving the corresponding differential eqations (Sinha et al., 3) sing a forth order M Figre 3. Peak e vales for varying pstream Mach nmbers as predicted by conventional model with the shock-thickness taken from the DNS data. Note that the peak e increases withot bond as the shock-thickness is decreased, as it wold be in an actal CFD simlation dring grid-refinement. The reslts correspond to trblent Mach nmber M t =.5 jst before the shock. Normalisation as described in figre. Rnge-Ktta method with the inlet bondary vales specified from the DNS data. The reslting vales for k and ε match well with DNS as reported in earlier works (Sinha et al., 3; Sinha, ). The vale of energy flx obtained sing () is zero in the region pstream and downstream of the shock de to niform mean flow temperatre on either sides of the normal shock. However, the energy flx assmes a peak negative vale in the region of the shock, and the peak vales obtained for a range of pstream Mach nmber are highlighted in figre 3. In the limit of M, the vale of energy flx predicted by the model redces to zero, and with increasing Mach nmber, the magnitde of the negative peak rises to a vale that is almost two orders of magnitde higher than the post-shock DNS predictions. Also, in a CFD framework, this formlation yields a grid-dependent vale of energy flx in the region of the shock, and increasing the grid point density frther increases the magnitde of the negative peak. 3. REALIZABLE MODEL In the canonical shock-trblence interaction, the Reynolds stress as predicted by the conventional model is given by (Sinha et al., 3) ρ = 4 3 µ T + ρk. () 3 This modelling approach leads to physically high vales of TKE in the region of shock de to its proportionality with the mean velocity gradient. The realizability correction limits the vale of eddy viscosity in the region of shock throgh the form C µ = min(cµ, o Cµ/s) o (Thivet et al., ), where s=s/(ε/k), S = (S i j S ji (/3)Skk ) and 3

4 (9) LIA M.5 DNS -.5 Realizable Model M M Figre 4. Variation of peak e with pstream Mach nmber as predicted by realizable model with shock thickness taken from the DNS data. The peak vale is independent of the shock thickness, which is indicative of a gridindependent soltion in a CFD simlation. Normalisation as described in figre. S i j = /( i / j + j / i ). In the region of high gradients sch as shock C µ =( c o µε)/(sk) and therefore 3C o µ ρk µ T =. (3) Ths the expression for energy flx given by () redces to e = 3C µ kc v Pr T T in the shock region. The vale of TKE as well as the mean variables reqired for the above formlation are obtained as described in the previos section. Similar to the conventional model, the realizable model yields a zero vale of energy flx otside the region of shock. However, inside the shock region, a peak negative vale is obtained as per (4). This peak energy flx vale for varying Mach nmber is plotted in figre 4. For almost all Mach nmbers, the realizable model predicts a negative energy flx in the shock region bt the magnitde is restricted de to the realizability constraint, and is of the same order as the post-shock DNS vale. Contrary to the eddy viscosity formlation, the realizability limiter yields a peak e which is independent of the shock thickness assmed i.e., in a real CFD simlation, this peak vale in the shock region will be insensitive to grid refinement. In the limit of M, the realizable model switches to the conventional model formlation and predicts a vale of zero energy flx as shown in the inset figre. In the limit of M, the model satrates to a negative limiting vale, a trend similar to the DNS data bt opposite in sign. (4) Figre 5. Variation of e with pstream Mach nmber as predicted by (8). Also shown are the corresponding DNS vales extrapolated to shock center and far-field LIA reslts. Normalisation as described in figre. 3.3 Differential Eqation-Based Model A differential eqation-based model was proposed by Qadros et al. (5) to model the jmp in trblent energy flx across the shock as well as the downstream decay. This fomlation was based on the linearized Rankine-Hgoniot conditions across the shock, and LIA was sed to obtain the appropriate model parameters. The model eqation is given by ( e ) = c k c k where c = (/γ)α( b ), b =.4( exp( M )) and c = /(γ)α. Here α = /(+/AF), where AF is the anisotropy factor (ratio of streamwise to transverse Reynolds stress) jst behind the shock. The soltion was obtained by nmerically integrating the above differential eqation as mentioned in the previos section. A good match was fond pon comparison with DNS data for a range of Mach nmbers. Now consider the differential eqation governing the jmp in TKE at the shock is given by (Sinha et al., 3) (5) k ρũ i ρk 3 ( b ) (6) Also, from the mean energy conservation eqation across the shock, we can write = C T p. (7) Using the above two eqations, (5) can be written as ( ) ( e ) k T = c e C p where c e = c o (/3)c ( b ). AF is assigned a vale of.6 based on LIA reslts. This eqation is compact with (8) 4

5 M = 3.5 accont this viscos decay reslting in the mismatch in the far-field vale. 3.4 Proposed Algebraic Model To overcome complexities involved in implementing an additional differential eqation in an existing CFD solver, we propose an algebraic model for the trblent energy flx at the shock as e = β kc v T () x/l ε Figre 6. Variation of e along streamwise direction as per (9) (line) and DNS (symbol) for the case of M = 3.5. The vertical lines near x/l ε = represent the mean shock thickness. Normalisation as described in figre. only one modelled term governing the vale of energy flx across the shock. Moreover, similar to conventional form, the energy flx is expressed in terms of gradient in mean temperatre. Figre 5 shows the energy flx vales as predicted by (8) for varying pstream Mach nmbers. Also shown are the downstream DNS vales of energy flx extrapolated to the shock center and the far-field LIA reslts. e attains a vale of zero as M and satrates to a positive limiting vale at high Mach nmbers similar to the DNS and LIA reslts. A good match is seen with the LIA reslts, and the model slightly overpredicts as compared to the DNS data. The acostic mode is dominant in the region jst behind the shock (Qadros et al., 5). The decay of the energy flx following the positive peak is largely affected by this mode. This decay can therefore modelled in terms of the dissipation length scale L ε = k (3/) /ε, which is representative of the large acostic scales in the trblence field. Therefore, the fll transport eqation for energy flx can be written as ( ) ( e ) k T = c e C p c e (9) L ε where c =.3+3exp( M ) is a Mach nmber dependent modelling constant which is fond to match the DNS data well. Figre 6 shows the streamwise variation of energy flx for the case of M = 3.5 obtained by nmerically integrating (9). The shock is located at x = and the corresponding DNS mean shock thickness is shown sing two vertical lines. The model predicts a zero vale of energy flx pstream of the shock with a peak positive vale obtained across the shock. This peak vale matches with the downstream DNS vale extrapolated to the shock center at x=. The downstream acostic decay is captred well pto abot x/l ε =.5 beyond which the model nderpredicts the energy flx vales. The acostic decay is dominated in the region x L ε beyond which vicos stresses contribte to the energy flx decay. The crrent model does not take into This form is similar to the realizablity constraint shown in (3) with β here as an nknown modelling parameter. Using (7), the above eqation can be written as e = βk γ, () where the trblent energy correlation is seen to be proportional to the trblent kinetic energy. This is physically consistent with linear theory reslts that show all correlations generated across the shock to be directly proportional to the pstream TKE. Frther, the normalised expression for energy flx can be written as e Norm. = e R T k. () Using () with the pstream vales of k and U along with (), the expression can be simplified to e Norm. = β M T r, (3) where T r is the ratio between the downstream and the pstream mean temperatre. In the high Mach nmber regime, this ratio is proportional to M, and the above expression satrates to a limiting vale e (γ+ ) Norm. = β γ(γ ). (4) A similar trend is seen in both the DNS and LIA reslts at high Mach nmbers as explained earlier in figre. In order to find ot the vale of modelling parameter β, we eqate (4) to the LIA far-field limiting vale of.38.this gives a vale of β =.7 for γ =.4. However, in the limit of M, the energy flx attains this vale of β which is physically incorrect. On the contrary, the energy flx predicted by the conventional model in the M limit is zero (as e is proportional to the mean temperatre gradient) which is consistent with the DNS and LIA predictions. We therefore propose a low Mach nmber correction of the form β =( e M )β, (5) 5

6 PROPOSED MODEL decay. Finally, for its ease of implementation, an algebraic model is developed based on linear theory, and it gives a good match for amplification in energy flx across the shock for varying Mach nmbers. DNS LIA ACKNOWLEDGEMENTS The first athor wishes to acknowledge Vishn Goodwill, a gradate stdent nder the spervision of the second athor, for sefl discssions. The first and second athors wold like to thank Indian Space Research Organisation- Space Technology Cell (ISRO-STC) and Aeronatics Research and Development Board (ARDB) for spporting this research M Figre 7. Variation of e with pstream Mach nmber as per new algebraic formlation. Also seen are the DNS vales of energy flx extrapolated to the shock center and the far-field LIA reslts. which yields β = as M and β β as M. Figre 7 shows the model predictions of energy flx for a range of pstream Mach nmbers. Also shown in the figre are the downstream DNS vales of energy flx extrapolated to the shock center and the far-field LIA vales. The model matches well with DNS for M <, and for higher Mach nmbers, it is close to LIA and slightly overpredicts the DNS vales. 4 CONCLUSION In this paper, we look at varios modelling strategies to predict the trblent energy flx generated in a canonical shock-trblence interaction. Application of conventional k ε models based on gradient diffsion hypothesis yields large negative vale of energy flx in the shock region, which rises drastically with increasing Mach nmbers. These vales are mch higher in magnitde than the postshock DNS predictions which yield a low positive energy flx. Realizable model also predicts peak negative vales of the energy flx in the shock region bt the peak vale is limited by the realizability constraint. A differential eqation model based on linear interaction analysis is formlated that provides a good estimate of trblent energy flx generated across the shock as well as the downstream acostic REFERENCES Bowersox, R. D. W. 9 Extension of eqilibrim trblent heat flx models to high-speed shear flows. J. Flid Mech. 633, 6 7. Larsson, J., Bermejo-Moreno, I. & Lele, S. K. 3 Reynolds- and Mach nmber effects in canonical shocktrblence interaction. J. Flid Mech. 77, Mahesh, K., Lele, S. K. & Moin, P. 996 The interaction of a shock wave with a trblent shear flow. Tech. Rep. 69. Thermosciences division, Department of Mechanical Engineering, Stanford University, Stanford, CA. Qadros, R., Sinha, K. & Larsson, J. 5 Trblent energy flx generated by shock/vortical-trblence interaction. To be sbmitted to J. Flid Mech.. Roy, C. J. & Blottner, F. G. Review and assessment of trblence models for hypersonic flows. Progress in Aerospace Sciences 4 (7), Sinha, K. Evoltion of enstrophy in shock/homogeneos trblence interaction. J. Flid Mech. 77, 74. Sinha, K., Mahesh, K. & Candler, G. V. 3 Modeling shock-nsteadiness in shock-trblence interaction. Phys. Flids 5, Thivet, F., Knight, D. D., Zheltovodov, A. A. & Maksimov, A. I. Importance of limiting the trblent stresses to predict 3D shock-wave/bondary-layer interactions. 3rd International Symposim on Shock Waves, Fort Worth, TX. Xiao, X., Hassan, H. A., Edwards, J. R. & Jr., R. L. Gaffney 7 Role of trblent Prandtl nmbers on heat flx at hypersonic Mach nmbers. AIAA Jornal 45 (4),