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Numerical investigation of incompressible flow in grooved channels. Part 1. Stability and self-sustained oscillations

Published online by Cambridge University Press:  21 April 2006

N. K. Ghaddar ,
K. Z. Korczak ,
B. B. Mikic  and
A. T. Patera
N. K. Ghaddar
Affiliation:
Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139
K. Z. Korczak
Affiliation:
Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139
B. B. Mikic
Affiliation:
Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139
A. T. Patera
Affiliation:
Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139
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Abstract

Incompressible moderate-Reynolds-number flow in periodically grooved channels is investigated by direct numerical simulation using the spectral element method. For Reynolds numbers less than a critical value Rc the flow is found to approach a stable steady state, comprising an ‘outer’ channel flow, a shear layer at the groove lip, and a weak re-circulating vortex in the groove proper. The linear stability of this flow is then analysed, and it is found that the least stable modes closely resemble Tollmien–Schlichting channel waves, forced by Kelvin–Helmholtz shear-layer instability at the cavity edge. A theory for frequency prediction based on the Orr–Sommerfeld dispersion relation is presented, and verified by variation of the geometric parameters of the problem. The accuracy of the theory, and the fact that it predicts many qualitative features of low-speed groove experiments, suggests that the frequency-selection process in these flows is largely governed by the outer, more stable flow (here a channel), in contrast to most current theories based solely on shear-layer considerations. The instability of the linear mode for R > Rc is shown to result in self-sustained flow oscillations (at frequencies only slightly shifted from the originating linear modes), which again resemble (finite-amplitude) Tollmien-Schlichting modes driven by an unstable groove vortex sheet. Analysis of the amplitude dependence of the oscillations on degree of criticality reveals the transition to oscillatory flow to be a regular Hopf bifurcation.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 163 , February 1986 , pp. 99 - 127
Copyright
© 1986 Cambridge University Press

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References

Arvizu, D. E. & Moffatt, R. J. 1982 Experimental heat transfer from an array of heated cubical elements on an adiabatic channel wall. Rep. No. HMT-33, Thermosciences Div. Stanford University, Stanford, California.
Babuska, I. & Dorr, M. R. 1980 Error estimates for the combined h and p versions of the finite element method. Numer. Maths 37, 257.Google Scholar
Basdevant, C., Deville, M. O., Haldenwang, P., Lacroix, J. M., Orlandi, P., Ouazzini, J., Patera, A. T. & Peyret, R. 1985 Spectral and finite difference solutions of the Burgers equation. Computers and Fluids (to appear).Google Scholar
Delves, L. M. & Hall, C. A. 1979 An implicit matching procedure for global element calculations. J. Inst. Maths Applies 23, 223.Google Scholar
Drazin, P. G. & Reid, W. H. 1981 Hydrodynamic Stability. Cambridge University Press.
Durbin, P. A. 1984 Resonance in flows with vortex sheets and edges. J. Fluid Mech. 145, 275.Google Scholar
Gatski, T. B. & Grosch, C. E. 1984 Embedded cavity drag in steady and unsteady flows. AIAA Paper No. 84–0436.Google Scholar
Ghaddar, N. K., Greiner, M., Patera, A. T. & Mikic, B. B. 1985 Heat transfer enhancement by oscillatory perturbation of a stable separated flow. InJtl Comm. Heat Mass Transfer 12, 369.Google Scholar
Ghaddar, N. K., Karniadakis, G. E. & Patera, A. T. 1985 A conservative isoparametric spectral element method for forced convection; application to fully-developed flow in periodic geometries. Numer. Heat Transfer (to appear).Google Scholar
Ghaddar, N. K., Magen, M., Mikic, B. B. & Patera, A. T. 1986 Numerical investigation of incompressible flow in grooved channels. Part 2: Resonance and heat transfer enhancement. J. Fluid Mech (to appear).Google Scholar
Ghadda, R. N. K. & Patera, A. T. 1985 Stability and resonance in grooved-channel flows. In Proc. 1CASE/NASA Langley Workshop on Stability of Time Dependent and Spatially Varying Flows, Hampton, Virginia, to appear.
Gharib, M. 1983 The effect of flow oscillations on cavity drag, and a technique for their control. Ph.D. thesis, California Institute of Technology, Pasadena, California
Gottlieb, D. O. & Orszag, S. A. 1977 Numerical Analysis of Spectral Methods: Theory and Application. SIAM.
Greiner, M. 1985 Experimental investigation of flow and transport enhancement in grooved channels. Ph.D. thesis, Massachusetts Institute of Technology, Cambridge, Massachusetts
Herbert, T. 1976 Finite amplitude stability of plane parallel flows. In Proc. 5th Intl Conf. on Numerical Methods in Fluid Dynamics (ed. A. I. Van de Vooren & P. J. Zandbergen). Lecture Notes in Physics, vol. 59, p. 235. Springer.
Karachmeti, K. 1956 Sound radiated from surface cutouts in high-speed flows. Ph.D. thesis, California Institute of Technology, Pasadena, California
Karniadakis, G. E., Bullister, E. T. & Patera, A. T. 1985 A spectral element method for solution of the two- and three-dimensional time-dependent incompressible Navier—Stokes equations. In Proc. Intl Symposium on Finite Element Methods for Nonlinear Problems, Trondheim, Norway, to appear.
Korczak, K. Z. & Patera, A. T. 1985 An isoparametric spectral element method for solution of the Navier—Stokes equations in complex geometry. J. Camp. Phys. (to appear).Google Scholar
Landau, L. D. & Lifshitz, E. M. 1959 Fluid Mechanics. Pergamon.
Marcus, P. S. 1984 Simulation of Taylor-Couette flow. Part 1. Numerical methods and comparison with experiment. J. Fluid Mech. 146n, 45.Google Scholar
Michalke, A. 1964 On the inviscid instability of the hyperbolic-tangent velocity profile. J. Fluid Mech. 19, 543.Google Scholar
Mehta, U. B. & Lavan, Z. 1969 Flow in a two-dimensional channel with a rectangular cavity. Trans. ASME E: J. Appl. Mech. December.Google Scholar
Moffatt, H. K. 1964 Viscous and resistive eddies near a sharp corner J. Fluid Mech. 18, 1.Google Scholar
Osswald, G. A., Ghia, K. N. & Ghia, U. 1983 Study of incompressible separated flow using an implicit time-dependent technique. In Proc. AIAA Computational Fluid Dynamics Conference, Danvers, Massachusetts.
Orszag, S. A. 1971 Accurate solution of the Orr—Sommerfeld stability equation. J. Fluid Mech. 50, 689.Google Scholar
Orszag, S. A. & Patera, A. T. 1983 Secondary instability of wall-bounded shear flows. J. Fluid Mech. 128, 347.Google Scholar
Pan, F. & Acrivos, A. 1967 Steady flows in rectangular cavities. J. Fluid Mech. 28, 643.Google Scholar
Patera, A. T. 1984 A spectral element method for fluid dynamics; laminar flow in a channel expansion. J. Comp. Phys. 54, 468.Google Scholar
Patera, A. T. 1985 Fast direct Poisson solvers for high-order finite element discretizations in rectangularly-decomposable domains. Submitted to J. Comp. Phys.Google Scholar
Przmieniecki, J. S. 1963 Matrix structural analysis of sub-structures. AIAA J. 1, 138.Google Scholar
Reihman, T. C. 1967 Laminar flow over transverse rectangular cavities. Ph.D. thesis, California Institute of Technology, Pasadena, California
Rockwell, D. 1982 Oscillations of impinging shear layers. AIAA Paper No. 82–0047.Google Scholar
Rockwell, D. & Naudascher, E. 1979 Self-sustained oscillations of impinging free shear layers. Ann. Rev. Fluid Mech. 11, 67.Google Scholar
Sarohia, V. 1977 Experimental investigation of oscillations in flows over shallow cavities. AIAA J. 15, 984.Google Scholar
Strang, G. & Fix, G. J. 1973 An Analysis of the Finite Element Method. Prentice-Hall.