# Symmetries and Invariant Solutions of Turbulent Flows and their Implications for Turbulence Modelling

• Martin Oberlack
Part of the International Centre for Mechanical Sciences book series (CISM, volume 442)

## Abstract

First a short introduction to the notion of symmetries of differential equations is given including infinitesimal transformations, invariant functions and invariant solutions. Then it is shown that the symmetry properties i.e. invariant transformations of the Navier-Stokes equations are pivotal to understand the physics of fluid flow. We demonstrate that all common symmetries “transfer” to the statistical equations such as the Reynolds stress transport equations or the multi-point correlation equations. From the knowledge of the symmetries we derive from the latter equations a broad variety of invariant solutions (scaling laws) using only first principles. These solutions comprise classical results such as the logarithmic-law-of-the-wall and other wall bounded shear flows. Also homogeneous and inhomogeneous time-dependent flows are analyzed and solutions are discussed. Since the symmetries of fluid motion are admitted by all statistical quantities of turbulent flows we give necessary conditions on turbulence models such that they “capture” the proper physics i.e. the symmetries and their corresponding invariant solutions. Particularly we will investigate two-equation models such as the κ-ε model as well as Reynolds stress transport models with respect to their symmetry properties. Finally we give conditions for the sub-grid scale model in large-eddy simulation of turbulence to obey the proper symmetries. For all of the latter turbulence models it is demonstrated that symmetry violation gives rather disadvantageous prediction capabilities of the model under investigation.

## Keywords

Symmetry Breaking Reynolds Stress Isotropic Turbulence Partial Differential Equation Invariant Solution

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## References

1. Andreev, V. K., and Rodionov, A. A. (1988). Group analysis of the equations for plane flows of an ideal fluid in the lagrange variables. Dokl. Akad. Nauk SSSR 298 (6): 1358–1361.
2. Andreev, V. K., Kaptsov, O. V., Pukhnachov, V. V., and Rodionov, A. A. (1998). Applications of Group Theoretical Methods in Hydrodynamics. Kluwer Academic Press.Google Scholar
3. Bardina, J., Ferziger, J. H., and Reynolds, W. C. (1980). Improved sub-grid scale models for large eddy simulation. AIAA-paper 80 (1357).Google Scholar
4. Barenblatt, G. I. (1993). Scaling laws for fully developed turbulent shear flows. part 1. basic hypotheses and analysis. J. Fluid Mech. 248: 513–520.
5. Batchelor, G. K. (1946). The theory of axisymmetric turbulence. Proc. Roy. Soc. A 186: 480–502.
6. Batchelor, G. K. (1967). An Introduction to Fluid Dynamics. Cambridge University Press.Google Scholar
7. Biringen, S., and Reynolds, W. C. (1981). Large-eddy simulation of the shear-free turbulent boundary layer. J. Fluid Mech. 103: 53–63.
8. Birkhoff, G. (1954). Fourier synthesis of homogeneous turbulence. Comm. Pure Appl. Math. 7: 19–44.
9. Bluman, G. W., and Kumei, S. (1989). Symmetries and Differential Equations. Applied Mathematical Sciences 81. Springer-Verlag.Google Scholar
10. Cantwell, B. J. (1978). Similarity transformations for the two-dimensional, unsteady, stream-function equation. J. Fluid Mech. 85: 257–271.
11. Cantwell, B. J. (1997). Introduction to symmetry analysis. Course Notes prepared for AA 218 — Similitude in Engineering Mechanics. Stanford University.Google Scholar
12. Chandrasekhar, S. (1950). The theory of axisymmetric turbulence. Phil. Trans. Roy. Soc. Lond. A 242: 557–577.
13. DeGraaff, D. B., Webster, D. R., and Eaton, J. K. (1999). The effect of Reynolds number on boundary layer turbulence. Experimental Thermal and Fluid Science 18 (4): 341–346.
14. Durbin, P. (1991). Near-wall turbulence closure modeling without ‘damping functions’. Theoret. Comput. Fluid Dyn. 3: 1–13.
15. El Telbany, M. M. M., and Reynolds, A. J. (1980). Velocity distributions in plane turbulent channel flows. J. Fluid Mech. 100: 1–29.
16. Fernholz, H. H., Krause, E., Nockemann, M., and Schober, M. (1995). Comparative measurements in the canonical boundary layer at Re52 lt 6 × 104 on the wall of the german-dutch windtunnel. Phys. Fluids 7 (6): 1275–1281.
17. Ferziger, J. H., and Peric, M. (1996). Computational Methods for Fluid Dynamics. Springer-Verlag.Google Scholar
18. Ferziger, J. H. (1996). Large eddy simulation. In Gatski, T. B., Hussaini, M. Y., and Lumley, J. L., eds., Simulation and Modeling of Turbulent Flows. Oxford University Press. 109–154.Google Scholar
19. Fischer, M., Durst, F., and Jovanovié, J. (1999). Reynolds number dependence of near wall turbulent statistics in channel flows. In Laser Techniques Applied to Fluid Mechanics (Selected Papers from the 9th International Symposium, Lissabon, Portugal, 13–16 Juli 1998). Springer-Verlag.Google Scholar
20. Fureby, C., Tabor, G., Weller, H. G., and Gosman, A. D. (1997). A comparative study of subgrid scale models in homogeneous isotropic turbulence. Phys. Fluids 9 (5): 1416–1429.
21. Gatski, T. B., and Speziale, C. G. (1993). On explicit algebraic stress models for complex turbulent flows. J. Fluid Mech. 254: 59–78.
22. George, W. K., Castillo, L., and Knecht, P. (1996). The zero pressure-gradient turbulent boundary layer. Report TRL-153, Turbulence Research Laboratory, School of Engineering and Applied Sciences, SUNY Bufallo, NY.Google Scholar
23. Germano, M., Piomelli, U., Moin, P., and Cabot, W. (1990). A dynamic subgrid-scale eddy viscosity model. In Proceedings of the Summer Program 1990. Center for Turbulence Research.Google Scholar
24. Germano, M., Piomelli, U., Moin, P., and Cabot, W. (1991). A dynamic subgrid-scale eddy viscosity model. Phys. Fluids A 3: 1760–1765.
25. Germano, M. (1986). A proposal for a redefinition of the turbulent stresses in the filtered Navier-Stokes equations. Phys. Fluids 29: 2323–2324.
26. Greenspan, H. P. (1990). The Theory of Rotating Fluids. Breukelen.Google Scholar
27. Hanjalic, K., and Launder, B. E. (1976). Contribution towards a Reynolds stress closure for low Reynolds number turbulence. J. Fluid Mech. 74: 593–610.
28. Härtel, C., and Kleiser, L. (1997). Galilean invariance and filtering dependence of near-wall gridscale/subgrid-scale interactions in large-eddy simulation. Phys. Fluids 9: 473–475.
29. Hirai, S., Takagi, T., and Matsumoto, M. (1988). Predictions of the laminarization phenomena in an axially rotating pipe flow. J. Fluids Eng. 110: 424–430.
30. Ibragimov, N. H. (1995a). CRC Handbook of Lie Group Analysis of Differential Equations, volume 1: Symmetries, Exact Solutions, and Conservation Laws. CRC Press.
31. Ibragimov, N. H. (1995b). CRC Handbook of Lie Group Analysis of Differential Equations, volume 2: Applications in Engineering and Physical Sciences. CRC Press.
32. Ibragimov, N. H. (1996). CRC Handbook of Lie Group Analysis of Differential Equations, volume 3: New Trends in Theoretical Developments and Computational Methods. CRC Press.
33. Iida, O., and Kasagi, N. (1993). Redistribution of the Reynolds stresses and destruction of the turbulent heat flux in homogeneous decaying turbulence. In 9th. Symp. on Turb. Shear Flows, Kyoto, Japan, 24.4.1–24. 4. 6.Google Scholar
34. Johnston, J. P., Halleen, R. M., and Lazius, D. K. (1972). Effects of spanwise rotation on the structure of two-dimensional fully developed turbulent channel flow. J. Fluid Mech. 56: 533–557.
35. Khor’kova, N. G., and Verbovetsky, A. M. (1995). On symmetry subalgebras and conservation laws for the k-e turbulence model and the Navier-Stokes equations. Amer. Math. Soc. Transi. 167 (2): 61–90.
36. Kikuyama, K., Murakami, M., Nishibori, K., and Maeda, K. (1983). Flow in axially rotating pipe. Bulletin JSME 26 (214).Google Scholar
37. Kim, J., Moin, P., and Moser, R. (1987). Turbulence statistics in fully developed channel flow at low Reynolds number. J. Fluid Mech. 177: 133–166.
38. Kolmogorov, A. N. (1941a). Decay of isotropic turbulence in incompressible viscous fluids. Dokl. Akad. Nauk SSSR A 31: 538.
39. Kolmogorov, A. N. (1941b). Dissipation of energy in the locally isotropic turbulence. C.R. Acad. Sci. SSSR 32: 16–18.
40. Kolmogorov, A. N. (1941c). The local structure of turbulence in incompressible viscous fluids for very large Reynolds numbers. Dokl. Akad. Nauk SSSR 30.Google Scholar
41. Kristoffersen, R., and Andersson, H. I. (1993). Direct simulations of low-Reynolds-number turbulent flow in a rotating channel. J. Fluid Mech. 256: 163–197.
42. Launder, B. E., Reece, G. E., and Rodi, W. (1975). Progress in the development of Reynolds-stress turbulence closure. J. Fluid Mech. 68: 537–566.
43. Lee, M. J., and Kim, J. (1991). The structure of turbulence in a simulated plane couette flow. In 8th Symp. Turb. Shear Flows, München, 5.3.1–5. 3. 6.Google Scholar
44. Lilly, D. K. (1992). A proposed modification of the germano subgrid-scale closure method. Phys. Fluids 4: 633–635.
45. Loitsyansky, L. (1939). Some basic laws of isotropic turbulent flow. Technical Report 440, Centr. Aero. Hydrodyn. Inst. Moscow. (Trans. NACA Tech. Memo. 1079 ).Google Scholar
46. Métais, O., and Lesieur, M. (1992). Spectral large eddy simulation of isotropic and stably-stratified turbulence. J. Fluid Mech. 239: 157–194.
47. Mohamed, M. S., and LaRue, J. C. (1990). The decay power law in grid-generated turbulence. J. Fluid Mech. 219: 195–214.
48. Moin, P., and Kim, J. (1982). Numerical investigation of turbulent channel flow. J. Fluid Mech. 118: 341–377.
49. Niederschulte, G. L. (1996). Turbulent Flow through a Rectangular Channel. Dissertation, University of Illinois, Department of Theoretical and Applied Mechanics.Google Scholar
50. Oberlack, M., and Peters, N. (1993). Closure of the two-point correlation equation as a basis of Reynolds stress models. In So, R., Speziale, C., and Launder, B., eds., Near-Wall Turbulent Flows, 85–94. Elsevier Science Publisher.Google Scholar
51. Oberlack, M., Cabot, W., and Rogers, M. M. (1998). Group analysis, DNS and modeling of a turbulent channel flow with streamwise rotation. In Moin, P., ed., Proceedings of the Center for Turbulence Summer Program 1998. Center for Turbulence Research, Stanford University/NASA Ames, CA, USA. 221–242.Google Scholar
52. Oberlack, M. (1994). Herleitung und Lösung einer Längenmass-und Dissipations-Tensorgleichung für turbulente Strömungen. Dissertation, Inst. f. Techn. Mechanik, RWTH Aachen.Google Scholar
53. Oberlack, M. (1997). Non-isotropic dissipation in non-homogeneous turbulence. J. Fluid Mech. 350: 351–374.
54. Oberlack, M. (1999). Similarity in non-rotating and rotating turbulent pipe flows. J. Fluid Mech. 379: 1–22.
55. Oberlack, M. (2000a). On symmetries and invariant solutions of laminar and turbulent wall-bounded flows. Zeitschrift für Angewandte Mathematik und Mechanik 80 (11–12): 791–800.
56. Oberlack, M. (2000b). Symmetrie, Invarianz und Selbstähnlichkeit in der Turbulenz. Habilitation, Inst. f. Techn. Mechanik, RWTH Aachen.Google Scholar
57. Oberlack, M. (2001). On the decay exponent of isotropic turbulence. In Proceedings of the Annual GAMM Meeting, ETH Zurich.Google Scholar
58. Olver, P. J. (1986). Applications of Lie Groups to Differential Equations. Graduate Texts in Mathematics. Springer-Verlag.
59. Orlandi, P., and Fatica, M. (1997). Direct simulations of turbulent flow in a pipe rotating about its axis. J. Fluid Mech. 343: 43–72.
60. Pope, S. B. (1975). A more general effective-viscosity hypothesis. J. Fluid Mech. 72: 331–340.
61. Pukhnachev, V. V. (1972). Invariant solutions of Navier-Stokes equations describing motions with free boundary. Dokl. Akad. Nauk 202: 302.Google Scholar
62. Reich, G. (1988). Strömung und Wärmeübertragung in einem axial rotierenden Rohr. Dissertation, Technische Universität Darmstadt.Google Scholar
63. Rogers, M. M., and Moin, P. (1987). The structure of the vorticity field in homogeneous turbulent flows. J. Fluid Mech. 243: 33–66.
64. Rotta, J. C. (1972). Turbulente Strömungen. Teubner, Stuttgart.
65. Saddoughi, S. G., and Veeravalli, S. V. (1994). Local isotropy in turbulent boundary layers at high Reynolds number. J. Fluid Mech. 268: 333–372.
66. Saffman, P. G. (1967). The large-scale structure of homogeneous turbulence. J. Fluid Mech. 27: 581–593.
67. Schlichting, H. (1982). Grenzschicht-Theorie. Verlag G.Braun, Karlsruhe, 8. edition.Google Scholar
68. Smagorinsky, J. (1963). General circulation experiments with the primitive equations. Mon. Weath. Rev. 91: 99–164.
69. Speziale, C. G. (1981). Some interesting properties of two-dimensional turbulence. Phys. Fluids A 28 (8): 1425–1427.
70. Speziale, C. G. (1985). Galilean invariance of subgrid-scale stress models in the large-eddy simulation of turbulence. J. Fluid Mech. 156: 55–62.
71. Tagawa, M., Nagano, Y., and Tsuji, T. (1991). Turbulence model for the dissipation components of Reynolds stresses. In 8th Symp. Turb. Shear Flows, München, 29.3.1–29. 3. 6.Google Scholar
72. von Kármán, T., and Howarth, L. (1938). On the statistical theory of isotropic turbulence. Proc. Roy. Soc. A 164: 192–215.
73. Wagner, C., and Friedrich, R. (1998). On the turbulence structure in solid and permeable pipes. Int. J. Heat Fluid Flow 19: 459–469.
74. Yoshizawa, A., Tsubokura, M., Kobayashi, T., and Taniguchi, N. (1996). Modeling of the dynamic subgridscale viscosity in large eddy simulation. Phys. Fluids 8: 2254–2256.
75. Zagarola, M. V., Smits, A. J., Orszag, S. A., and V., Y. (1997). Scaling of the mean velocity profile for turbulent pipe flow. Phys. Rev. Letters 78(2).Google Scholar
76. Zagarola, M. V. (1996). Mean flow Scaling of Turbulent Pipe Flow. Dissertation, Princeton University.Google Scholar
77. Zang, Y., Street, R. L., and Koseff, J. R. (1993). A dynamic mixed subgrid-scale model and its application to turbulent recirculating flows. Phys. Fluids 8: 3186–3196.