Abstract
The evolution of small perturbations in longitudinally nonuniform flows is studied with reference to the problem of the propagation of flow perturbations in a plane channel with walls of variable elasticity. Using the solution of the problem of the receptivity of the flow to local vibrations of the walls, the problem considered can be reduced to the solution of an integral equation for a single function, namely, the complex vibration amplitude of the walls. A numerical method for solving this equation on the basis of a piecewise-linear approximation of the unknown function is proposed. It is shown that the instability wave amplitude changes discontinuously at the junction of the rigid and elastic channel sections. A second method of investigating the process of propagation of perturbations in the flow considered is proposed. This method is based on laws of evolution of perturbations in nonuniform flows and an analytic solution of the problem of perturbation scattering on the junction of walls with different compliance. On the basis of this method the classical stability theory is generalized to include the case of nonuniform flows.
Similar content being viewed by others
REFERENCES
M. O. Kramer, "Boundary layer stabilization by distributed damping," J. Aeronaut. Sci., 24, 459 (1957).
R.W. Milling, "Tollmien-Schlichting wave cancellation," Phys. Fluid, 24, 979 (1981).
S.V. Manuilovich, "Propagation of an instability wave through part of a variable-width channel," Izv. Akad. Nauk SSSR, Mekh. Zhidk. Gaza, No. 2, 34 (1992).
E. D. Terent'ev, "A linear problem of a vibration in a boundary layer on a partially elastic surface," in: Intern. Workshop on Advances in Analytical Methods in Aerodynamics, Muedzyzdroje, Poland (1993), P. 23.
C. Davies and P. W. Carpenter, "Numerical simulation of the evolution of Tollmien-Schlichting waves over finite compliant panels," J. Fluid Mech., 335, 361 (1997).
S.V. Manuilovich, "Receptivity of plane Poiseuille flow to vibration of the channel walls," Izv. Ros. Akad. Nauk, Mekh. Zhidk. Gaza, No. 4, 12 (1992).
C. C. Lin, "Some mathematical problems in the theory of the stability of parallel flows," J. Fluid Mech., 10, 430 (1961).
H. Schlichting, Boundary Layer Theory, McGraw-Hill, New York (1968).
F. P. Bertolotti, T. Herbert, and P. R. Spalart, "Linear and nonlinear stability of the Blasius boundary layer," J. Fluid Mech., 242, 441 (1992).
Rights and permissions
About this article
Cite this article
Manuilovich, S.V. Propagation of Perturbations in Plane Poiseuille Flow between Walls of Nonuniform Compliance. Fluid Dynamics 38, 529–544 (2003). https://doi.org/10.1023/A:1026317710292
Issue Date:
DOI: https://doi.org/10.1023/A:1026317710292