The hydrodynamic stability of flows over Kramer-type compliant surfaces is studied. Two main types of instability are considered. First, there are those which could not exist without viscosity, termed Tollmien–Schlichting Type Instabilities (TSI). Secondly, there are Flow-Induced Surface Instabilities (FISI), that depend fundamentally on surface flexibility and could exist with an inviscid fluid flow. Part 1, the present paper, deals mainly with the first type. The original Kramer experiments and the various subsequent attempts to confirm his results are reviewed, together with experimental studies of transition in flows over compliant surfaces and theoretical work concerned with the hydrodynamic stability of such flows.The Kramer-type compliant surface is assumed to be an elastic plate supported by springs which are modelled by an elastic foundation. It is also assumed that the plate is backed by a viscous fluid substrate having, in general, a density and viscosity different from the mainstream fluid. The motion of the substrate fluid is assumed to be unaffected by the presence of the springs and is determined by solving the linearized Navier–Stokes equations. The visco-elastic properties of the plate and springs are taken into account approximately by using a complex elastic modulus which leads to complex flexural rigidity and spring stiffness. Values for the various parameters characterizing the surface properties are estimated for the actual Kramer coatings.The boundary-layer stability problem for a flexible surface is formulated in a similar way to that of Landahl (1962) whereby the boundary condition at the surface is expressed in terms of an equality between the surface and boundary-layer admittances. This form of the boundary condition is exploited to develop an approximate theory which determines whether a particular change to the mechanical properties of the surface will be stabilizing or destabilizing with respect to the TSI. It is shown that a reduction in flexural rigidity and spring stiffness, an increase in plate mass, and the presence of an inviscid fluid substrate are all stabilizing, whereas viscous and visco-elastic damping are destabilizing.Numerical solutions to the Orr–Sommerfeld equation are also obtained. Apart from Kramer-type compliant surfaces, solutions are also presented for the rigid wall, for the spring-backed tensioned membrane with damping, previously considered by Landahl & Kaplan (1965), and for some of the compliant surfaces investigated experimentally by Babenko and his colleagues. The results for the Kramer-type compliant surfaces on the whole confirm the predictions of the simple theory. For a free-stream speed of 18 m/s the introduction of a viscous substrate leads to a complex modal interaction between the TSI and FISI. A single combined unstable mode is formed in the case of highly viscous substrate fluids and in this case increased damping has a stabilizing effect. When the free-stream speed is reduced to 15 m/s the modal interaction no longer occurs. In this case the eff...

A study is carried out of the linear global behaviour corresponding to the absolute instability of the rotating-disc boundary layer. It is based on direct numerical simulations of the complete linearized Navier–Stokes equations obtained with the novel velocity–vorticity method described in Davies & Carpenter (2001). As the equations are linear, they become separable with respect to the azimuthal coordinate, $\theta$. This permits us to simulate a single azimuthal mode. Impulse-like excitation is used throughout. This creates disturbances that take the form of wavepackets, initially containing a wide range of frequencies. When the real spatially inhomogeneous flow is approximated by a spatially homogeneous flow (the so-called parallel-flow approximation) the results ofthe simulations are fully in accordance with the theory of Lingwood (1995). If the flow parameters are such that her theory indicates convective behaviour the simulations clearly exhibit the same behaviour. And behaviour fully consistent with absolute instability is always found when the flow parameters lie within the theoretical absolutely unstable region. The numerical simulations of the actual inhomogeneous flow reproduce the behaviour seen in the experimental study of Lingwood (1996). In particular, there is close agreement between simulation and experiment for the ray paths traced out by the leading and trailing edges of the wavepackets. In absolutely unstable regions the short-term behaviour of the simulated disturbances exhibits strong temporal growth and upstream propagation. This is not sustained for longer times, however. The study suggests that convective behaviour eventually dominates at all the Reynolds numbers investigated, even for strongly absolutely unstable regions. Thus the absolute instability of the rotating-disc boundary layer does not produce a linear amplified global mode as observed in many other flows. Instead the absolute instability seems to be associated with transient temporal growth, much like an algebraically growing disturbance. There is no evidence of the absolute instability giving rise to a global oscillator. The maximum growth rates found for the simulated disturbances in the spatially inhomogeneous flow are determined by the convective components and are little different in the absolutely unstable cases from the purely convectively unstable ones. In addition to the study of the global behaviour for the usual rigid-walled rotating disc, we also investigated the effect of replacing an annular region of the disc surface with a compliant wall. It was found that the compliant annulus had the effect of suppressing the transient temporal growth in the inboard (i.e. upstream) absolutely unstable region. As time progressed the upstream influence of the compliant region became more extensive.

The stability of plane channel flow between compliant walls is investigated for disturbances which have the same symmetry, with respect to the channel centreline, as the Tollmien–Schlichting mode of instability. The interconnected behaviour of flow-induced surface waves and Tollmien–Schlichting waves is examined both by direct numerical solution of the Orr–Sommerfeld equation and by means of an analytic shear layer theory. We show that when the compliant wall properties are selected so as to give a significant stability effect on Tollmien–Schlichting waves, the onset of divergence instability can be severely disrupted. In addition, travelling wave flutter can interact with the Tollmien–Schlichting mode to generate a powerful instability which replaces the flutter instability identified in studies based on a potential mean-flow model. The behaviour found when the mean-flow shear layer is fully accounted for may be traced to singularities in the wave dispersion relation. These singularities can be attributed to solutions which represent Tollmien–Schlichting waves in rigid-walled channels. Such singularities will also be found in the dispersion relation for the case of Blasius flow. Thus, similar behaviour can be anticipated for Blasius flow, including the disruption of the onset of divergence instability. As a consequence, it seems likely that previous investigations for Blasius flow will have yielded very conservative estimates for the optimal stabilization that can be achieved for Tollmien–Schlichting waves for the purposes of laminar-flow control.

The flow-induced surface instabilities of Kramer-type compliant surfaces are investigated by a variety of theoretical approaches. This class of instability includes all those modes of instability for which the mechanism of generation involves essentially inviscid processes. The results should be applicable to all compliant surfaces that could be modelled theoretically by a thin elastic plate, with or without applied longitudinal tension, supported on a springy elastic foundation. with or without a viscous fluid substrate; material damping is also taken into account through the viscoelastic properties of the solid constituents of the coatings.The simple case of a potential main flow is studied first. The eigenmodes for this case are subjected to an energy analysis following the methods of Landahl (1962). Instabilities that grow both in space and time are then considered, and absolute and convective instabilities identified and analysed.The effects of irreversible processes on the flow-induced surface instabilities are investigated. The shear flow in the boundary layer gives rise to a fluctuating pressure component which is out of phase with the surface motion. This leads to an irreversible transfer of energy from the main stream to the compliant surface. This mechanism is studied in detail and is shown to be responsible for travelling-wave flutter. Simple results are obtained for the critical velocity, wavenumber and stability boundaries. These last are shown to be in good agreement with the results obtained by the numerical integration of the Orr–Sommerfeld equation. An analysis of the effects of a viscous fluid substrate and of material damping is then carried out. The simpler inviscid theory is shown to predict values of the maximum growth rate which are, again, in good agreement with the results obtained by the numerical integration of the Orr–Sommerfeld equation provided that the instability is fairly weak.Compliant surfaces of finite length are analysed in the limit as wave-length tends to zero. In this way the static-divergence instability is predicted. Simple formulae for critical velocity and wavenumber are derived. These are in exact agreement with the results of the simpler infinite-length theory. But, whereas a substantial level of damping is required for the instability on a surface of infinite length, static divergence grows fastest in the absence of damping on a surface of finite length.

The evolution of two-dimensional Tollmien–Schlichting waves propagating along a wall shear layer as it passes over a compliant panel of finite length is investigated by means of numerical simulation. It is shown that the interaction of such waves with the edges of the panel can lead to complex patterns of behaviour. The behaviour of the Tollmien–Schlichting waves in this situation, particularly the effect on their growth rate, is pertinent to the practical application of compliant walls for the delay of laminar–turbulent transition. If compliant panels could be made sufficiently short whilst retaining the capability to stabilize Tollmien–Schlichting waves, there is a good prospect that multiple-panel compliant walls could be used to maintain laminar flow at indefinitely high Reynolds numbers.We consider a model problem whereby a section of a plane channel is replaced with a compliant panel. A growing Tollmien–Schlichting wave is then introduced into the plane, rigid-walled, channel flow upstream of the compliant panel. The results obtained are very encouraging from the viewpoint of laminar-flow control. They indicate that compliant panels as short as a single Tollmien–Schlichting wavelength can have a strong stabilizing effect. In some cases the passage of the Tollmien–Schlichting wave over the panel edges leads to the excitation of stable flow-induced surface waves. The presence of these additional waves does not appear to be associated with any adverse effect on the stability of the Tollmien–Schlichting waves. Except very near the panel edges the panel response and flow perturbation can be represented by a superposition of the Tollmien–Schlichting wave and two other eigenmodes of the coupled Orr–Sommerfeld/compliant-wall eigensystem.The numerical scheme employed for the simulations is derived from a novel vorticity–velocity formulation of the linearized Navier–Stokes equations and uses a mixed finite-difference/spectral spatial discretization. This approach facilitated the development of a highly efficient solution procedure. Problems with numerical stability were overcome by combining the inertias of the compliant wall and fluid when imposing the boundary conditions. This allowed the interactively coupled fluid and wall motions to be computed without any prior restriction on the form taken by the disturbances.

No abstract

A theoretical study into the effects of wall compliance on the stability of the rotating-disc boundary layer is described. A single-layer viscoelastic wall model is coupled to a sixth-order system of fluid stability equations which take into account the effects of viscosity, Coriolis acceleration, and streamline curvature. The coupled system of equations is integrated numerically by a spectral Chebyshev-tau technique.Travelling and stationary modes are studied and wall compliance is found to greatly increase the complexity of the eigenmode spectrum. It is effective in stabilizing the inviscid Type I (or cross-flow) instability. The effect on the viscous (Type II) eigenmode is more complex and can be strongly destabilizing. An analysis of the energy flux indicates that this destabilization arises as a result of a large degree of energy production by viscous stresses at the wall/flow interface.The Type I and II instabilities are shown to be negative and positive energy waves respectively. The co-existence of eigenmodes of opposite energy type indicates the possibility of modal interaction and coalescence. It is found that, compared with the rigid disc, wall compliance promotes the interaction and coalescence of the Type I and II eigenmodes. There is an associated strong instability which appears to be characterized by marked horizontal motion of the compliant surface. Modal coalescence is interpreted physically as producing local algebraic growth which could advance the onset of nonlinear effects.

A method for numerically simulating the hydroelastic behaviour of a passive compliant wall of finite dimensions is presented. Using unsteady potential flow, the perturbation pressures which arise from wall disturbances of arbitrary form are calculated through a specially developed boundary-element method. These pressures may then be coupled to a suitable solution procedure for the wall mechanics to produce an interactive model for the wall/flow system. The method is used to study the two-dimensional disturbances which may occur on a Kramer-type compliant wall of finite length. Finite-difference methods are used to yield wall solutions driven by the fluid pressure after some perturbation from the equilibrium position. Thus, histories of surface deflection and wall energy are obtained. Such a modelling of the physics of the system requires no presupposition of disturbance form.A thorough investigation of divergence instability is carried out. Most of the results presented in this paper concern the response of the compliant wall while (and after) a point pressure pulse, carried in the applied flow, travels over the compliant panel. Above a critical flow speed and once sufficient time has passed, the compliant wall is shown to adopt the particular profile of an unstable mode. After this divergence mode has been established, instability is realized as a slowly travelling downstream wave. These features are in agreement with the findings of experimental studies. The role of wall damping is clarified: damping serves only to reduce the growth rate of the instability, leaving its onset flow speed unchanged. The present predictions provide an improvement upon some of the unrealistic aspects of predictions yielded by travelling-wave and standing-wave treatments of divergence instability.The response of a long compliant panel after a single-point pressure-pulse initiation, applied at its midpoint, is simulated. At flow speeds higher than a critical value, parts of the formerly (at subcritical flow speeds) upstream-travelling wave system change to travel downstream and show amplitude growth. The development of this ‘upstream-incoming’ wave illustrates how divergence instability can occur at locations upstream of the point of initial excitation. Faster flexural waves transmit energy upstream, thereafter these disturbances can evolve into slow downstreamtravelling divergence waves. The spread of the instability to locations both downstream and upstream of the point of initial excitation indicates that divergence is an absolute instability. This behaviour and the effects of wall damping clarified by the present work strongly suggest that divergence is a Class C instability.