Linear Stability Of Boundary-layer Flow Research Articles

The linear stability of flat-plate boundary-layer flow of fluid with temperature-dependent viscosity

The linear stability of boundary-layer flow of fluid with temperature-dependent viscosity over a heated or cooled flat-plate is investigated. Decomposition of the disturbance into normal temporal modes leads to a sixth-order “modified” eigenvalue problem. Making the additional ad hoc assumption of parallel flow leads to a simpler sixth-order “parallel” eigenvalue problem which, unlike the modified problem, reduces to the classical Orr–Sommerfeld problem in the isothermal case. Two viscosity models are considered, and for both models numerically-calculated stability results for both the modified and parallel eigenvalue problems are obtained. For both viscosity models it is, perhaps surprisingly, found that for both eigenvalue problems a non-uniform decrease in viscosity across the layer stabilizes the flow while a non-uniform increase in viscosity across the layer destabilizes the flow. Results for the two eigenvalue problems are shown to be quantitatively similar with, however, the parallel problem always over-predicting the critical Reynolds number in comparison to the modified problem. Finally, we discuss the physical interpretation of our results in terms of velocity–profile shape and thin-layer effects.

The absolute instability of boundary-layer flow over viscoelastic walls

The linear stability of boundary-layer flow over a viscoelastic-layer wall is considered. A companion matrix technique is used to formulate the stability problem as a linear matrix eigenvalue problem for complex frequency and all the eigenvalues may be determined without any initial guess values. The eigenvalues are compared with those obtained with an accurate shooting method. The instability character of the boundary-layer flow is further investigated with the purpose of finding the conditions under which the instability of the flow could become absolute. The mapping technique of Kupferet al. (1987) is used to identify the occurrence of absolute instability eigenvalues. Absolute instabilities are discovered for cases of soft damped wall over certain ranges of Reynolds number. The effects of wall material stiffness, damping coefficient, thickness of layer, and Reynolds number on the occurrence of absolute instability are examined and presented.

The linear stability of boundary-layer flow over compliant walls: effects of boundary-layer growth

The linear stability of boundary-layer flow over compliant or flexible surfaces has been studied by Carpenter & Garrad (1985), Yeo (1988) and others on the assumption of local flow parallelism. This assumption is valid at large Reynolds numbers. Non-parallel effects due to growth of the boundary layer gain in significance and importance as one gets to lower Reynolds number. This is especially so for a compliant surface, which can sustain a variety of wall-related instabilities in addition to the Tollmien—Schlichting instabilities (TSI) that are found over rigid surfaces. The present paper investigates the influence of boundary-layer non-parallelism on the TSI and wall-related travelling-wave flutter (TWF) on compliant layers. Corrections to the growth rate of locally parallel theory for boundary-layer non-parallelism are obtained through a multiple-scale analysis. The results indicate that flow non-parallelism has an overall destabilizing influence on the TSI and TWF. Flow non-parallelism is also found to have a very strong destabilizing effect on the branch of TWF that stretches to low Reynolds number. The results obtained have important implications for the design and use of compliant layers at low Reynolds numbers.

The Linear Stability of Boundary-Layer Flow over Compliant Walls—The Effects of the Wall Mean State, Induced by Flow Loading

This paper is concerned with the linear hydrodynamic stability of boundary-layer flow over a compliant wall subjected to steady-state stresses on its surface. Surface stress loading induces an equilibrium state of stress and strain in the compliant wall, termed the wall mean (basic) state. Flow stability is affected by the surface loading, as a result of an interaction between the propagating wall perturbation and the induced wall mean state. This perturbation-mean state interaction in the wall is accounted for in this paper by the application of a nonlinear elasticity theory, suitably linearized about the mean state, in which the compliant material is assumed to obey a linear isotropic (stress-strain) law between the second Piola-Kirchoff stress tensor and the Green strain tensor. Surface loading due to combine hydrostatic pressure and boundary-layer viscous shear is considered. It is found that hydrostatic pressure can strongly affect the stability of flow over compliant walls; with a stabilizing influence on the Tollmen-Schlichting instabilities and a destabilizing influence on the compliance-related instabilities. The underlying mechanics for both incompressible compliant layers are explained. The present study shows that the pressure of the operational environment should be taken into account when evaluating the performance of compliant coatings as transition-delaying or noise-reducing devices in underwater applications.