Peter W. Carpenter

Numerical simulation of the evolution of Tollmien–Schlichting waves over finite compliant panels

The evolution of two-dimensional Tollmien{Schlichting waves propagating along a wall shear layer as it passes over a compliant panel of nite 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 eect 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 suciently 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 indenitely 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 eect. 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 eect 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 nite-dierence/spectral spatial discretization. This approach facilitated the development of a highly ecient 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.

Instabilities in a plane channel flow between compliant walls

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.

Global behaviour corresponding to the absolute instability of the rotating-disc boundary layer

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,

A numerical simulation of the interaction of a compliant wall and inviscid flow

\theta

Boundary layer instability over compliant walls: Comparison between theory and experiment

. 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.

Modeling and Design of Microjet Actuators

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.

The stability of rotating-disc boundary-layer flow over a compliant wall. Part 1. Type I and II instabilities

Theoretical studies have shown that compliant walls are able to attenuate the Tollmien–Schlichting waves that lead to conventional two‐dimensional boundary‐layer transition. This phenomenon was demonstrated in towing‐tank tests conducted by Gaster et al. The results of these experiments also featured a different and very dramatic form of boundary‐layer breakdown. We contend that this type of breakdown was due to a hydroelastic mode of instability, namely traveling‐wave flutter. In this paper we model the two‐layer viscoelastic compliant wall of Gaster et al. and its interaction with the boundary‐layer flow using the asymptotic theory of Carpenter and Gajjar; en‐type calculations are carried out for the traveling‐wave flutter. Excellent agreement is found between the stability characteristics of the TWF mode and the measurements of the new form of breakdown found in the experiments; thus a complete understanding of the physical features found in the experiments is now available. Such understanding is essen...

The stability of rotating-disc boundary-layer flow over a compliant wall. Part 2. Absolute instability

A computational model is developed to aid the design of microelectromechanical systems (MEMS) for use in active turbulence control. The focus here is on micro-actuators and, in particular, a design employed by syntheticjet devices. This consists of a diaphragm within a cavity that, by its piezoinduced motion, creates an ejection of fluid through an orifice in the cavity’s lid. The diaphragm is modeled using classical thin-plate theory, with the stiffness of the attached piezodevice incorporated. For numerical economy, the fluid motion within the cavity is not modeled; instead, the pressure is calculated with the perfect gas law. However, in the orifice, where viscous forces are more dominant, one-dimensional Navier‐Stokes equations are solved. The actuator system is modeled in its entirety. All that is required to calculate the outlet jet velocity is the input voltage applied to the piezodevice. The numerical model is validated against experimental data for synthetic-jet devices and used to predict their optimal dimensions. An alternative mode of forcing the diaphragm is proposed that does not suffer from the drawbacks inherent in synthetic-jet operation at MEMS scale. This mode generates a jump in cavity pressure, creating a pufflike jet disturbance. This concept is explored with the aim of uncovering practical issues and simple design guidelines.

Progress on the Use of Compliant Walls for Laminar-Flow Control

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.

Optimization of Viscoelastic Compliant Walls for Transition Delay

A numerical study has been undertaken of the influence of a compliant boundary on absolute instability. In a certain parameter range absolute instability occurs in the boundary layer on a rotating disc, thereby instigating rapid transition to turbulence. The conventional use of wall compliance as a laminar-flow control technique has been to lower growth rates of convective instabilities. This has the effect of reducing amplification of disturbances as they propagate downstream. For absolute instability, however, only the suppression of its onset would be a significant gain. This paper addresses the question of whether passive wall compliance can be advantageous when absolute instability exists in a boundary layer. A theoretical model of a single-layer viscoelastic compliant wall was used in conjunction with the sixth-order system of differential equations which govern the stability of the boundary-layer flow over a rotating disc. The absolute/convective nature of the flow was ascertained by using a spatio-temporal analysis. Pinch-point singularities of the dispersion relation and a point of zero group velocity identify the presence of absolute instability. It was found that only a low level of wall compliance was enough to delay the appearance of absolute instability to higher Reynolds numbers. Beyond a critical level of wall compliance results suggest that complete suppression of absolute instability is possible. This would then remove a major route to transition in the rotating-disc boundary layer.