J. S. B. Gajjar

A general theory for two- and three-dimensional wall-mode instabilities in boundary layers over isotropic and anisotropic compliant walls

An asymptotic theory is developed for two- and three-dimensional disturbances growing in a two-dimensional boundary layer over a compliant wall. The theory exploits the multideck structure of the boundary layer to derive asymptotic approximations at a high Reynolds number for the perturbation wall pressure and viscous stresses. These quantities can be regarded as driving the wall and, accordingly, the equation(s) of motion for the wall is (are) used as the characteristic equation(s) for finding the eigenvalue(s). The main assumptions are that the amplitude of the disturbance is sufficiently small for linear theory to hold, the Reynolds number is large, the disturbance wavelength is long compared with the boundary-layer thickness, and the critical and viscous wall layers are well separated. The theory was developed to study the travelling-wave flutter instability discussed by Carpenter and Garrad, i.e., the Class B instability of Benjamin and Landahl. Under certain limiting processes both the upper-branch and conventional triple-deck scalings for the Tollmien-Schlichting instability can be obtained with the present approach. Accordingly, the theory also gives a reliable qualitative guide to the effect of anisotropic wall compliance on the Tollmien-Schlichting instability.The theory is applied to various cases including two- and three-dimensional disturbances, developing in boundary layers over isotropic and anisotropic compliant walls. The disturbances can be treated as either temporally or spatially growing. Eigenvalues are very accurately predicted by means of the theory, especially near points of neutral stability. The computational requirements are trivial compared with those required for full numerical solution of the Orr-Sommerfeld equation. For isotropic compliant walls the theory confirms the earlier result of Miles and Benjamin that the phase shift in the disturbance velocity across the critical layer plays a dominant role in destabilization of the Class B travelling-wave flutter through making irreversible energy transfer possible due to the work done by the fluctuating pressure at the wall. The theory elucidates the secondary role played by the phase shift occurring across the wall layer. Viscous effects are much more important for anisotropic compliant walls which admit substantial horizontal, as well as vertical, displacement. For these walls an important mechanism for irreversible energy transfer is the work done by fluctuating shear stress. This almost invariably has a stabilizing effect on the travelling-wave flutter. In addition there is a weaker effect arising from the effect of anisotropic wall compliance on the phase shift across the wall layer. This may be stabilizing or destabilizing.

On hypersonic self-induced separation, hydraulic jumps and boundary layers with algebraic growth

Analytical and numerical properties are described for the free interaction and separation arising when the induced pressure and local displacement are equal, in reduced terms, for large Reynolds number flow. The interaction, known to apply to hypersonic flow, is shown to have possible relevance also to the origins of supercritical (Froude number > 1) hydraulic jumps in liquid layers flowing along horizontal walls. The main theoretical task is to obtain the ultimate behaviour far beyond the separation. An unusual structure is found to emerge there, involving a backward-moving wall layer with algebraically growing velocity at its outer edge, a detached shear layer moving forward and, in between, reversed inertial flow uninfluenced directly by the adverse pressure gradient. As a result the pressure then increases like (distance) m , with m = 2((7) — 2)/3 (= 043050…), and does not approach a plateau. Some more general properties of (Falkner-Skan) boundary layers with algebraic growth are also described.

Once again on the supersonic flow separation near a corner

Laminar boundary-layer separation in the supersonic flow past a corner point on a rigid body contour, also termed the compression ramp, is considered based on the viscous–inviscid interaction concept. The ‘triple-deck model’ is used to describe the interaction process. The governing equations of the interaction may be formally derived from the Navier–Stokes equations if the ramp angle [theta] is represented as [theta] = [theta]0Re[minus sign]1/4, where [theta]0 is an order-one quantity and Re is the Reynolds number, assumed large. To solve the interaction problem two numerical methods have been used. The first method employs a finite-difference approximation of the governing equations with respect to both the streamwise and wall-normal coordinates. The resulting algebraic equations are linearized using a Newton–Raphson strategy and then solved with the Thomas-matrix technique. The second method uses finite differences in the streamwise direction in combination with Chebychev collocation in the normal direction and Newton–Raphson linearization. Our main concern is with the flow behaviour at large values of [theta]0. The calculations show that as the ramp angle [theta]0 increases, additional eddies form near the corner point inside the separation region. The behaviour of the solution does not give any indication that there exists a critical value [theta]*0 of the ramp angle [theta]0, as suggested by Smith & Khorrami (1991) who claimed that as [theta]0 approaches [theta]*0, a singularity develops near the reattachment point, preventing the continuation of the solution beyond [theta]*0. Instead we find that the numerical solution agrees with Neilands (1970) theory of reattachment, which does not involve any restriction upon the ramp angle.

The hydrodynamic stability of channel flow with compliant boundaries

An asymptotic theory is developed for the hydrodynamic stability of an incompressible fluid flowing in a channel in which one wall is rigid and the other is compliant. We exploit the multideck structure of the flow to investigate theoretically the development of disturbances to the flow in the limit of large Reynolds numbers. A simple spring-plate model is used to describe the motion of the compliant wall, and this study considers the effect of the various wall parameters, such as tension, inertia, and damping, on the stability properties. An amplitude equation for a modulated wavetrain is derived and the properties of this equation are studied for a number of cases including linear and nonlinear theory. It is shown that in general the effect of viscoelastic damping is destabilizing. In particular, for large damping, the analysis points to a fast travelling wave, short-scale instability, which may be related to a flutter instability observed in some experiments. This work also demonstrates that the conclusions obtained by previous investigators in which the effect of tension, inertia, and other parameters is neglected, may be misleading. Finally it is shown that a set of compliant-wall parameters exists for which the Haberman type of critical layer analysis leads to stable equilibrium amplitudes, in contrast to many other stability problems where such equilibrium amplitudes are unstable.

Numerical solution of the Navier-Stokes equations for the flow in a cylinder cascade

A numerical study of the steady, two-dimensional incompressible flow past a cascade of circular cylinders is presented. The Navier–Stokes equations are written in terms of the streamfunction and vorticity and solved using a novel numerical technique based on using the Chebychev collocation method in one direction and high-order finite differences in the other direction. A direct solver combined with Newton–Raphson linearization is used to solve the discrete equations. Steady flow solutions have been obtained for large Reynolds numbers, far higher than those obtained previously, and for varying gap widths between the cylinders. Three distinct types of solutions, dependent on the gap width, have been found. Comparisons with theoretical predictions for various flow quantities show good agreement, especially for the narrow gap width case. However, existing theories are unable to explain the solution properties which exist for intermediate gap widths.

An Analytic Approach for Calculating Absolutely Unstable Inviscid Modes of the Boundary Layer on a Rotating Disk

An analytical treatment of inviscidly absolutely unstable modes is pursued using the long-wavelength asymptotic approach. It is shown using the inviscid Rayleigh scalings in conjunction with the linear critical layer theory that the rotating-disk boundary layer flow undergoes a region of absolute instability for some small azimuthal wave numbers. The analytically calculated branch points for the absolute instability are found to be in good agreement with those obtained via a numerical solution of the inviscid Rayleigh equation.

Flow past wing-body junctions

The three-dimensional laminar flow past a junction formed by a thin wing protruding normally from a locally flat body surface is considered for wings of finite span but short or long chord. The Reynolds number is taken to be large. The leading-edge interaction for a long wing has the triple-deck form, with the pressure due to the wing thickness forcing a three-dimensional flow response on the body surface alone. The same interaction describes the flow past an entire short wing. Linearized solutions are presented and discussed for long and short two-dimensional wings and for certain three-dimensional wings of interest. The trailing-edge interaction for a long wing is different, however, in that the three-dimensional motions on the wing and on the body are coupled together and in general the coupling is nonlinear. Linearized properties are retrieved only for reduced chord lengths. The overall flow structure for a long wing is also discussed, including the traditional three-dimensional corner layer, which is shown to have an unusual singular starting form near the leading edge. Qualitative comparisons with experiments are made.

Direct spatial resonance in the laminar boundary layer due to a rotating-disk

Numerical treatment of the linear stability equations is undertaken to investigate the occurrence of direct spatial resonance events in the boundary layer flow due to a rotating-disk. A spectral solution of the eigenvalue problem indicates that algebraic growth of the perturbations shows up, prior to the amplification of exponentially growing instability waves. This phenomenon takes place while the flow is still in the laminar state and it also tends to persist further even if the non-parallelism is taken into account. As a result, there exists the high possibility of this instability mechanism giving rise to nonlinearity and transition, long before the unboundedly growing time-amplified waves.

On the Nonlinear Evolution of a Stationary Cross-Flow Vortex in a Fully Three-Dimensional Boundary Layer Flow

One of the earliest experimental and theoretical investigations of the stability of three-dimensional (3D) boundary layers was conducted by Gregory, Stuart & Walker (1955), (hereafter referred to as GSW). The boundary layer flows studied were the flow over a rotating disk, and the flow over swept wings. Using a china-clay visualisation technique they were able to demonstrate the presence of a highly regular, stationary, pattern of vortices spaced equally around the disk, or along the surface of the wing. In addition, with the aid of a microphone probe, they were able to detect travelling waves close to the surface of the disk. Stuart in GSW suggested that the instabilities could be explained in terms of the inflexional character of the effective mean velocity profile in certain directions, (the term effective mean velocity profile refers to a certain linear combination of the stream-wise and spanwise velocity components). According to his suggestion the stationary pattern was that associated with the inviscid instability of the velocity profile which had a zero at a point of inflexion. The non-stationary, or travelling wave pattern, could also be explained in terms of the inviscid instabilty of the mean flow.

Absolute instability of the von Karman, Bodewadt and Ekman flows between a rotating disc and a stationary lid

The stability of a family of boundary-layer flows, which includes the von Kármán, Bödewadt and Ekman flows for a rotating incompressible fluid between a rotating disc and a stationary lid, is investigated. Numerical computations with the use of a spectral method are carried out to analyse absolute and convective instability. It is shown that the stability of the system is enhanced with a decrease in distance between the disc and the lid.