Hostname: page-component-84b7d79bbc-g78kv Total loading time: 0 Render date: 2024-07-26T03:15:28.221Z Has data issue: false hasContentIssue false
  • English
  • Français

Reynolds stress anisotropy in shock/isotropic turbulence interactions

Published online by Cambridge University Press:  26 February 2021

Nathan E. Grube  and
M. Pino Martín Open the ORCID record for M. Pino Martín [Opens in a new window]
Nathan E. Grube
Affiliation:
Department of Aerospace Engineering, University of Maryland, Glenn L. Martin Hall, 4298 Campus Drive, College Park, MD20742, USA
M. Pino Martín*
Affiliation:
Department of Aerospace Engineering, University of Maryland, Glenn L. Martin Hall, 4298 Campus Drive, College Park, MD20742, USA
*
†Email address for correspondence: mpmartin@umd.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

Linear interaction analysis (LIA) predicts that isotropic vortical turbulence interacting with a normal shock wave at Mach 2 or greater produces a post-shock Reynolds stress tensor whose transverse components $R_{22}$ and $R_{33}$ are greater than the streamwise component $R_{11}$. However, computational and experimental studies have consistently reported Reynolds stress tensors with $R_{11}>R_{22}$ in this regime. This discrepancy is primarily due to nonlinear terms which have been neglected in the linear analysis. However, if the effect of the nonlinear terms is to move the anisotropy from a state with $R_{22}>R_{11}$, through isotropy, to a state of qualitatively opposite anisotropy, this would seem contrary to the usual return-to-isotropy effect. Here, we explain the redistribution of energy between the transverse and streamwise Reynolds stresses by considering the wavelengths of the vortical waves emitted by the interaction. Our study shows that the transverse Reynolds stresses are concentrated at smaller length scales than the streamwise stress. In turn, the evolution of the modes associated with $R_{22}$ occurs on a faster time scale, and their energy is more quickly transferred away, allowing $R_{11}$ to dominate. The overall effect is that the global anisotropy quickly attains a state where $R_{11}>R_{22}$. Furthermore, a quantitative model is developed based on this understanding of the flow physics. The model uses LIA along with an eddy viscosity approximation for the nonlinear interactions in order to quantify the redistribution of energy between the transverse and streamwise Reynolds stresses. The model predicts anisotropy levels approximately matching those observed in direct numerical simulation data.

JFM classification

Turbulent Flows: Isotropic turbulence Compressible Flows: Shock waves Turbulent Flows: Turbulence modelling
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 913 , 25 April 2021 , A19
Check for updates
Copyright
© The Author(s), 2021. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Agui, J.H., Briassulis, G. & Andreopoulos, Y. 2005 Studies of interactions of a propagating shock wave with decaying grid turbulence: velocity and vorticity fields. J. Fluid Mech. 524, 143–195.CrossRefGoogle Scholar
Andreopoulos, Y., Agui, J.H. & Briassulis, G. 2000 Shock wave–turbulence interactions. Annu. Rev. Fluid Mech. 32, 309–345.CrossRefGoogle Scholar
Barre, S., Alem, D. & Bonnet, J.P. 1996 Experimental study of a normal shock/homogeneous turbulence interaction. AIAA J. 34, 968–974.CrossRefGoogle Scholar
Boukharfane, R., Bouali, Z. & Mura, A. 2018 Evolution of scalar and velocity dynamics in planar shock-turbulence interaction. Shock Waves 28, 1117–1141.CrossRefGoogle Scholar
Champagne, F.H., Harris, V.G. & Corrsin, S. 1970 Experiments on nearly homogeneous shear flow. J. Fluid Mech. 41 (01), 81–139.CrossRefGoogle Scholar
Chen, C.H. & Donzis, D.A. 2019 Shock–turbulence interactions at high turbulence intensities. J. Fluid Mech. 870, 813–847.CrossRefGoogle Scholar
Gao, X., Bermejo-Moreno, I. & Larsson, J. 2020 Parametric numerical study of passive scalar mixing in shock turbulence interaction. J. Fluid Mech. 895, A21.CrossRefGoogle Scholar
Grube, N.E. & Martín, M.P. 2021 Direct numerical simulation of shock/highly compressible isotropic turbulence interactions. Phys. Rev. Fluids (submitted).Google Scholar
Hannappel, R. & Friedrich, R. 1995 Direct numerical simulation of a Mach 2 shock interacting with isotropic turbulence. Appl. Sci. Res. 54 (3), 205–221.CrossRefGoogle Scholar
Huete, C., Jin, T., Martínez-Ruiz, D. & Luo, K. 2017 Interaction of a planar reacting shock wave with an isotropic turbulent vorticity field. Phys. Rev. E 96, 053104.CrossRefGoogle ScholarPubMed
Jamme, S., Cazalbou, J.-B., Torres, F. & Chassaing, P. 2002 Direct numerical simulation of the interaction between a shock wave and various types of isotropic turbulence. Flow Turbul. Combust. 68 (3), 227–268.CrossRefGoogle Scholar
Jones, W.P. & Launder, B.E. 1972 The prediction of laminarization with a two-equation model of turbulence. Intl J. Heat Mass Transfer 15 (2), 301–314.CrossRefGoogle Scholar
Kida, S. & Orszag, S.A. 1990 Energy and spectral dynamics in forced compressible turbulence. J. Sci. Comput. 5 (2), 85–125.CrossRefGoogle Scholar
Larsson, J., Bermejo-Moreno, I. & Lele, S.K. 2013 Reynolds- and mach-number effects in canonical shock–turbulence interaction. J. Fluid Mech. 717, 293–321.CrossRefGoogle Scholar
Larsson, J. & Lele, S.K. 2009 Direct numerical simulation of canonical shock/turbulence interaction. Phys. Fluids 21, 126101.CrossRefGoogle Scholar
Launder, B.E. & Sharma, B.I. 1974 Application of the energy-dissipation model of turbulence to the calculation of flow near a spinning disc. Lett. Heat Mass Transfer 1 (2), 131–137.CrossRefGoogle Scholar
Lee, S., Lele, S.K. & Moin, P. 1991 Direct numerical simulation and analysis of shock turbulence interaction. In In 29th Aerospace Sciences Meeting, Reno, NV, USA. AIAA Paper 1991-523.Google Scholar
Lee, S., Lele, S.K. & Moin, P. 1993 Direct numerical simulation of isotropic turbulence interacting with a weak shock wave. J. Fluid Mech. 251, 533–562.CrossRefGoogle Scholar
Lee, S., Lele, S.K. & Moin, P. 1997 Interaction of isotropic turbulence with shock waves: effect of shock strength. J. Fluid Mech. 340, 225–247.CrossRefGoogle Scholar
Mahesh, K., Lele, S.K. & Moin, P. 1997 The influence of entropy fluctuations on the interaction of turbulence with a shock wave. J. Fluid Mech. 334, 353–379.CrossRefGoogle Scholar
McKenzie, J.F. & Westphal, K.O. 1968 Interaction of linear waves with oblique shock waves. Phys. Fluids 11 (11), 2350–2362.CrossRefGoogle Scholar
Moyal, J.E. 1952 The spectra of turbulence in a compressible fluid; eddy turbulence and random noise. Math. Proc. Camb. Phil. Soc. 48 (2), 329–344.CrossRefGoogle Scholar
Quadros, R., Sinha, K. & Larsson, J. 2016 a Kovasznay mode decomposition of velocity-temperature correlation in canonical shock-turbulence interaction. Flow Turbul. Combust. 97, 787–810.CrossRefGoogle Scholar
Quadros, R., Sinha, K. & Larsson, J. 2016 b Turbulent energy flux generated by shock/homogeneous- turbulence interaction. J. Fluid Mech. 796, 113–157.CrossRefGoogle Scholar
Ribner, H.S. 1954 a Convection of a pattern of vorticity through a shock wave. Tech. Rep. NACA-TR-1164. National Advisory Committee for Aeronautics. Lewis Flight Propulsion Lab.; Cleveland, OH, United States.Google Scholar
Ribner, H.S. 1954 b Shock-turbulence interaction and the generation of noise. Tech. Rep. NACA-TR-1233. National Advisory Committee for Aeronautics. Lewis Flight Propulsion Lab.; Cleveland, OH, United States.Google Scholar
Ribner, H.S. 1998 Comment on “Experimental study of a normal shock/homogeneous turbulence interaction”. AIAA J. 36 (3), 494.CrossRefGoogle Scholar
Rotta, J.C. 1951 Statistische theorie nichthomogener turbulenz. Z. Phys. 129, 547–572.CrossRefGoogle Scholar
Ryu, J. & Livescu, D. 2014 Turbulence structure behind the shock in canonical shock–vortical turbulence interaction. J. Fluid Mech. 756, R1.CrossRefGoogle Scholar
Sethuraman, Y.P.M. & Sinha, K. 2020 a Effect of turbulent mach number on the thermodynamic fluctuations in canonical shock–turbulence interaction. Comput. Fluids 197, 104354.CrossRefGoogle Scholar
Sethuraman, Y.P.M. & Sinha, K. 2020 b Modeling of thermodynamic fluctuations in canonical shock–turbulence interaction. AIAA J. 58 (7), 3076–3089.CrossRefGoogle Scholar
Sethuraman, Y.P.M., Sinha, K. & Larsson, J. 2018 Thermodynamic fluctuations in canonical shock–turbulence interaction: effect of shock strength. Theor. Comput. Fluid Dyn. 32 (4), 629–654.CrossRefGoogle Scholar
Sinha, K., Mahesh, K. & Candler, G.V. 2003 Modeling shock unsteadiness in shock/turbulence interaction. Phys. Fluids 15 (8), 2290–2297.CrossRefGoogle Scholar
Smits, A.J. & Dussauge, J.-P. 2006 Turbulent Shear Layers in Supersonic Flow, 2nd edn. Springer Science & Business Media.Google Scholar
Sreenivasan, K.R. 1984 On the scaling of the turbulence energy dissipation rate. Phys. Fluids 27 (5), 1048–1051.CrossRefGoogle Scholar
Tennekes, H. & Lumley, J.L. 1972 A First Course in Turbulence. MIT Press.CrossRefGoogle Scholar
Tian, Y., Jaberi, F.A., Li, Z. & Livescu, D. 2017 Numerical study of variable density turbulence interaction with a normal shock wave. J. Fluid Mech. 829, 551–588.CrossRefGoogle Scholar
Tian, Y., Jaberi, F.A. & Livescu, D. 2019 Density effects on post-shock turbulence structure and dynamics. J. Fluid Mech. 880, 935–968.CrossRefGoogle Scholar
Uberoi, M.S. 1956 Effect of wind-tunnel contraction on free-stream turbulence. J. Aeronaut. Sci. 23 (8), 754–764.CrossRefGoogle Scholar
Wilcox, D.C. 1998 Turbulence Modeling for CFD, 2nd edn. DCW Industries.Google Scholar