Abstract
In order to clarify the behavior of solutions of the Lagrangian-averaged Navier–Stokes-α (LANS-α) equations in the presence of solid walls, we identify a variety of exact solutions of the full equations and their boundary layer approximations. The solutions demonstrate that boundary conditions suggested for the LANS-α equations in the literature(1) for a bounded domain do not apply in a semi-infinite domain. The convergence to solutions of the Navier–Stokes equations as α → 0 is elucidated for infinite-energy solutions in a semi-infinite domain, and non-uniqueness of these solutions is discussed. We also study the boundary layer approximation of LANS-α equations, denoted the Prandtl-α equations, and report solutions for turbulent jets and wakes. Our version of the Prandtl-α equations includes an extra term necessary to conserve linear momentum and corrects an earlier result of Cheskidov.(2)
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
REFERENCES
J. E. Marsden and S. Shkoller, Proc. Roy. Soc. London 359:1449-1468 (2001).
A. Cheskidov, C. R. Acad. Sci. Paris, Ser. I 334:423-427 (2002).
S. Chen, C. Foias, D. D. Holm, E. Olson, E. S. Titi, and S. Wynne, Phys. Rev. Lett. 81:5338(1998); S. Chen, C. Foias, D. D. Holm, E. Olson, E. S. Titi, and S. WynnePhys. Fluids 11:2343(1999); S. Chen, C. Foias, D. D. Holm, E. Olson, E. S. Titi, and S. WynnePhysica D 133:49(1999).
T. Wei and W. W. Willmarth, J. Fluid Mech. 65:439(1974).
M. V. Zaragola, Ph.D. thesis (Princeton University, 1996).
V. Putkaradze and P. D. Weidman, Turbulent wake solutions of the Prandtl-Ά equations, Phys. Rev. E 67:036304(2003).
J. Leray, Acta Math. 63:193(1934).
D. D. Holm, J. E. Marsden, and T. S. Ratiu, Phys. Rev. Lett. 349:4173-4177 (1998).
D. D. Holm, J. E. Marsden, and T. S. Ratiu, Adv. in Math. 137:1-81 (1998).
C. Foias, D. D. Holm, and E. S. Titi, Physica D 152-153:505(2001); C. Foias, D. D. Holm, and E. S. TitiIbid, Dynam. and Differential Equations, 14:1(2002).
H. Schlichting, Boundary Layer Theory, 7th ed. (McGraw-Hill, 2000).
K. Heimenz, Dingl. Polytech. J. 326:321(1911).
J. T. Stuart, J. Aero/Space Sci. 26:124(1959).
K. Tamada, J. Phys. Soc. Japan 46:310(1979).
J. M. Dorrepaal, J. Fluid Mech. 163:141(1986).
F. Homann, ZAMM 16:153(1936).
T. von KÁrmÁn, ZAMM 1:233(1921).
U. T. BÖdewadt, ZAMM 20:241(1940).
L. Prandtl, in Ayerodynamic Theory III, W. F. Durand, ed. (1936), p. 16.
D. D. Holm, Physica D 133:215(1999).
J. E. Marsden and S. Shkoller, Arch. Rational Mech. Anal. 166:27-46 (2003).
S. Goldstein, Modern Developments in Fluid Dynamics (Dover, 1965).
I. Wygnanski, F. Champagne, and B. Marasli, J. Fluid Mech. 168:31(1988).
A. Townsend, Proc. Roy. Soc A 197:124(1949).
E. Gutmark and I. Wygnanski, J. Fluid Mech. 73:465(1976).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Holm, D.D., Putkaradze, V., Weidman, P.D. et al. Boundary Effects on Exact Solutions of the Lagrangian-Averaged Navier–Stokes-α Equations. Journal of Statistical Physics 113, 841–854 (2003). https://doi.org/10.1023/A:1027364720090
Issue Date:
DOI: https://doi.org/10.1023/A:1027364720090