Engineering topics

Hall, Philip

Publications and source records attributed to Hall, Philip.

At least 37 records · Page 2

NASA NTRS1992

On the spatial evolution of long-wavelength Goertler vortices governed by a viscous-inviscid interaction

The generation of long-wavelength, viscous-inviscid interactive Goertler vortices is studied in the linear regime by numerically solving the time-dependent governing equations. It is found that time-dependent surface deformations, which assume a fixed nonzero shape at large times, generate steady Goertler vortices that amplify in the downstream direction. Thus, the Goertler instability in this regime is shown to be convective in nature, contrary to the earlier findings of Ruban and Savenkov. The disturbance pattern created by steady and streamwise-elongated surface obstacles on a concave surface is examined in detail, and also contrasted with the flow pattern due to roughness elements with aspect ratio of order unity on flat surfaces. Finally, the applicability of the Briggs-Bers criterion to unstable physical systems of this type is questioned by providing a counterexample in the form of the inviscid limit of interactive Goertler vortices.

Choudhari, Meelan↗

NASA NTRS1992

Fully nonlinear Goertler vortices in constricted channel flows and their effect on the onset of separation

The development of fully nonlinear Goertler vortices in high Reynolds number flow in a symmetrically constricted channel is investigated. Attention is restricted to the case of 'strongly' constricted channels considered by Smith and Daniels (1981) for which the scaled constriction height is asymptotically large. Such flows are known to develop a Goldstein singularity and subsequently become separated at some downstream station past the point of maximum channel constriction. It is shown that these flows can support fully nonlinear Goertler vortices, of the form elucidated by Hall and Lakin (1988), for constrictions which have an appreciable region of local concave curvature upstream of the position at which separation occurs. The effect on the onset of separation due to the nonlinear Goertler modes is discussed. A brief discussion of other possible nonlinear states which may also have a dramatic effect in delaying (or promoting) separation is given.

Denier, James P.↗

NASA NTRS1992

Streamwise vortices in heated boundary layers

The nonlinear instability of the boundary layer on a heated flat plate placed in an oncoming flow is investigated. Such flows are unstable to stationary vortex instabilities and inviscid traveling wave disturbances governed by the Taylor-Goldstein equation. For small temperature differences the Taylor-Goldstein equation reduces to Rayleigh's equation. When the temperature difference between the wall and free stream is small the preferred mode of instability is a streamwise vortex. It is shown in this case that the vortex, assumed to be of small wavelength, restructures the underlying mean flow to produce a profile which can be massively unstable to inviscid traveling waves. The mean state is shown to be destabilized or stabilized to inviscid waves depending on whether the Prandtl number is less or greater than unity.

Hall, Philip↗

NASA NTRS1992

Crossflow effects on the growth rate of inviscid Goertler vortices in a hypersonic boundary layer

The effects of crossflow on the growth rate of inviscid Goertler vortices in a hypersonic boundary layer with pressure gradient are studied. Attention is focused on the inviscid mode trapped in the temperature adjustment layer; this mode has greater growth rate than any other mode. The eigenvalue problem which governs the relationship between the growth rate, the crossflow amplitude, and the wavenumber is solved numerically, and the results are then used to clarify the effects of crossflow on the growth rate of inviscid Goertler vortices. It is shown that crossflow effects on Goertler vortices are fundamentally different for incompressible and hypersonic flows. The neutral mode eigenvalue problem is found to have an exact solution, and as a by-product, we have also found the exact solution to a neutral mode eigenvalue problem which was formulated, but unsolved before, by Bassom and Hall (1991).

Fu, Yibin↗

NASA NTRS1992

Wave interaction theory and LFC

Attention is drawn to the influence of preexisting finite-amplitude instabilities on the growth of other disturbances; current design tools for LFC take no notice of this kind of interaction. When a rational accounting is accomplished for the evolution of incoming disturbances in finite-amplitude solutions of the equations of motion, future transition-prediction methods will need to take these wave interactions into account. Attention is given here to interactions in the presence of crossflow vortices and interactions involving Goertler vortices.

Hall, Philip↗

NASA NTRS1991

On the nonlinear development of the most unstable Goertler vortex mode

The nonlinear development of the most unstable Gortler vortex mode in boundary layer flows over curved walls is investigated. The most unstable Gortler mode is confined to a viscous wall layer of thickness O(G -1/5) and has spanwise wavelength O(G 11/5); it is, of course, most relevant to flow situations where the Gortler number G is much greater than 1. The nonlinear equations covering the evolution of this mode over an O(G -3/5) streamwise lengthscale are derived and are found to be of a fully nonparallel nature. The solution of these equations is achieved by making use of the numerical scheme used by Hall (1988) for the numerical solution of the nonlinear Gortler equations valid for O(1) Gortler numbers. Thus, the spanwise dependence of the flow is described by a Fourier expansion, whereas the streamwise and normal variations of the flow are dealt with by employing a suitable finite difference discretization of the governing equations. Our calculations demonstrate that, given a suitable initial disturbance, after a brief interval of decay, the energy in all the higher harmonics grows until a singularity is encountered at some downstream position. The structure of the flowfield as this singularity is approached suggests that the singularity is responsible for the vortices, which are initially confined to the thin viscous wall layer, moving away from the wall and into the core of the boundary layer.

Denier, James P.↗

NASA NTRS1991

Nonlinear development and secondary instability of Gortler vortices in hypersonic flows

In a hypersonic boundary layer over a wall of variable curvature, the region most susceptible to Goertler vortices is the temperature adjustment layer over which the basic state temperature decreases monotonically to its free stream value. Except for a special wall curvature distribution, the evolution of Goertler vortices trapped in the temperature adjustment layer will in general be strongly affected by the boundary layer growth through the O(M sup 3/2) curvature of the basic state, where M is the free stream Mach number. Only when the local wavenumber becomes as large as of order M sup 3/8, do nonparallel effects become negligible in the determination of stability properties. In the latter case, Goertler vortices will be trapped in a thin layer of O(epsilon sup 1/2) thickness which is embedded in the temperature adjustment layer; here epsilon is the inverse of the local wavenumber. A weakly nonlinear theory is presented in which the initial nonlinear development of Goertler vortices in the neighborhood of the neutral position is studied and two coupled evolution equations are derived. From these, it can be determined whether the vortices are decaying or growing depending on the sign of a constant which is related to wall curvature and the basic state temperature.

Fu, Yibin B.↗

NASA NTRS1990

The effect of wall compliance on the Goertler vortex instability

The stability of the flow of a viscous incompressible fluid over a curved compliant wall to longitudinal Goertler vortices is investigated. The compliant wall is modeled by a particularly simple equation relating the induced wall displacement to the pressure in the overlying fluid. Attention is restricted to the large Goertler number regime; this regime being appropriate to the most unstable Goertler mode. The effect of wall compliance on this most unstable mode is investigated.

Denier, J. P.↗

NASA NTRS1990

On a class of unsteady three-dimensional Navier Stokes solutions relevant to rotating disc flows: Threshold amplitudes and finite time singularities

A class of exact steady and unsteady solutions of the Navier Stokes equations in cylindrical polar coordinates is given. The flows correspond to the motion induced by an infinite disc rotating with constant angular velocity about the z-axis in a fluid occupying a semi-infinite region which, at large distances from the disc, has velocity field proportional to (x,-y,O) with respect to a Cartesian coordinate system. It is shown that when the rate of rotation is large, Karman's exact solution for a disc rotating in an otherwise motionless fluid is recovered. In the limit of zero rotation rate a particular form of Howarth's exact solution for three-dimensional stagnation point flow is obtained. The unsteady form of the partial differential system describing this class of flow may be generalized to time-periodic equilibrium flows. In addition the unsteady equations are shown to describe a strongly nonlinear instability of Karman's rotating disc flow. It is shown that sufficiently large perturbations lead to a finite time breakdown of that flow whilst smaller disturbances decay to zero. If the stagnation point flow at infinity is sufficiently strong, the steady basic states become linearly unstable. In fact there is then a continuous spectrum of unstable eigenvalues of the stability equations but, if the initial value problem is considered, it is found that, at large values of time, the continuous spectrum leads to a velocity field growing exponentially in time with an amplitude decaying algebraically in time.

Hall, Philip↗

NASA NTRS1990

On the Goertler instability in hypersonic flows: Sutherland law fluids and real gas effects

The Goertler vortex instability mechanism in a hypersonic boundary layer on a curved wall is investigated. The precise roles of the effects of boundary layer growth, wall cooling, and gas dissociation is clarified in the determination of stability properties. It is first assumed that the fluid is an ideal gas with viscosity given by Sutherland's law. It is shown that when the free stream Mach number M is large, the boundary layer divides into two sublayers: a wall layer of O(M sup 3/2) thickness over which the basic state temperature is O(M squared) and a temperature adjustment layer of O(1) thickness over which the basic state temperature decreases monotonically to its free stream value. Goertler vortices which have wavelengths comparable with the boundary layer thickness are referred to as wall modes. It is shown that their downstream evolution is governed by a set of parabolic partial differential equations and that they have the usual features of Goertler vortices in incompressible boundary layers. As the local wavenumber increases, the neutral Goertler number decreases and the center of vortex activity moves towards the temperature adjustment layer. Goertler vortices with wavenumbers of order one or larger must necessarily be trapped in the temperature adjustment layer and it is this mode which is most dangerous. For this mode, it was found that the leading order term in the Goertler number expansion is independent of the wavenumber and is due to the curvature of the basic state. This term is also the asymptotic limit of the neutral Goertler numbers of the wall mode. To determine the higher order corrections terms in the Goertler number expansion, two wall curvature cases are distinguished. Real gas effects were investigated by assuming that the fluid is an ideal dissociating gas. It was found that both gas dissociation and wall cooling are destabilizing for the mode trapped in the temperature adjustment layer, but for the wall mode trapped near the wall the effect of gas dissociation can be either destabilizing or stabilizing.

Fu, Yibin B.↗

NASA NTRS1990

The inviscid secondary instability of fully nonlinear longitudinal vortex structures in growing boundary layers

The inviscid instability of a longitudinal vortex structure within a steady boundary layer is investigated. The instability has a wavelength comparable with the boundary layer thickness so that a quasi-parallel approach to the instability problem can be justified. The generalization of the Rayleigh equation to such a flow is obtained and solved for the case when the vortex structure is induced by curvature. Two distinct modes of instability are found; these modes correspond with experimental observations on the breakdown process for Goertler vortices.

Hall, Philip↗

NASA NTRS1990

Vortex instabilities in 3D boundary layers: The relationship between Goertler and crossflow vortices

The inviscid and viscous stability problems are addressed for a boundary layer which can support both Goertler and crossflow vortices. The change in structure of Goertler vortices is found when the parameter representing the degree of three-dimensionality of the basic boundary layer flow under consideration is increased. It is shown that crossflow vortices emerge naturally as this parameter is increased and ultimately become the only possible vortex instability of the flow. It is shown conclusively that at sufficiently large values of the crossflow there are no unstable Goertler vortices present in a boundary layer which, in the zero crossflow case, is centrifugally unstable. The results suggest that in many practical applications Goertler vortices cannot be a cause of transition because they are destroyed by the 3-D nature of the basic state. In swept wing flows the Goertler mechanism is probably not present for typical angles of sweep of about 20 degrees. Some discussion of the receptivity problem for vortex instabilities in weakly 3-D boundary layers is given; it is shown that inviscid modes have a coupling coefficient marginally smaller than those of the fastest growing viscous modes discussed recently by Denier, Hall, and Seddougui (1990). However the fact that the growth rates of the inviscid modes are the largest in most situations means that they are probably the most likely source of transition.

Bassom, Andrew↗

NASA NTRS1990

Wave interactions in a three-dimensional attachment-line boundary layer

The three-dimensional boundary layer on a swept wing can support different types of hydrodynamic instability. Here attention is focused on the so-called 'spanwise instability' problem which occurs when the attachment-line boundary layer on the leading edge becomes unstable to Tollmien-Schlichting waves. In order to gain insight into the interactions that are important in that problem a simplified basic state is considered. This simplified flow corresponds to the swept attachemnt-line boundary layer on an infinite flat plate. The basic flow here is an exact solution of the Navier-Stokes equations and its stability to two-dimensional waves propagating along the attachment line can be considered exactly at finite Reynolds number. The corresponding problem is studied for oblique waves, and their interaction with two-dimensional waves is investigated. It is shown that there are two types of oblique wave which, if excited, cause the destabilization of the two-dimensional mode and the breakdown of the disturbed flow at a finite distance from the leading edge.

Hall, Philip↗

NASA NTRS1990

On the interaction of stationary crossflow vortices and Tollmien-Schlichting waves in the boundary layer on a rotating disc

There are many fluid flows where the onset of transition can be caused by different instability mechanisms which compete among themselves. The interaction is considered of two types of instability mode (at an asymptotically large Reynolds number) which can occur in the flow above a rotating disc. In particular, the interaction is examined between lower branch Tollmien-Schlichting (TS) waves and the upper branch, stationary, inviscid crossflow vortex whose asymptotic structure has been described by Hall (1986). This problem is studied in the context of investigating the effect of the vortex on the stability characteristics of a small TS wave. Essentially, it is found that the primary effect is felt through the modification to the mean flow induced by the presence of the vortex. Initially, the TS wave is taken to be linear in character and it is shown (for the cases of both a linear and a nonlinear stationary vortex) that the vortex can exhibit both stabilizing and destabilizing effects on the TS wave and the nature of this influence is wholly dependent upon the orientation of this latter instability. Further, the problem is examined with a larger TS wave, whose size is chosen so as to ensure that this mode is nonlinear in its own right. An amplitude equation for the evolution of the TS wave is derived which admits solutions corresponding to finite amplitude, stable, traveling waves.

Bassom, Andrew P.↗

NASA NTRS1990

Concerning the interaction of non-stationary cross-flow vortices in a three-dimensional boundary layer

Recently there has been much work devoted to considering some of the many and varied interaction mechanisms which may be operative in three-dimensional boundary layer flows. This paper is concerned with resonant triads of crossflow vortices. The effects of interactions upon resonant triads is examined where each member of the triad has the property of being linearly neutrally stable so that the importance of the interplay between modes can be relatively easily assessed. Modes within the boundary layer flow above a rotating disc are investigated because of the similarity between this disc flow and many important practical flows and, secondly, because the selected flow is an exact solution of the Navier-Stokes equations which makes its theoretical analysis especially attractive. It is demonstrated that the desired triads of linearly neutrally stable modes can exist within the chosen boundary layer flow. Evolution equations are obtained to describe the development of the amplitudes of these modes once the interaction mechanism is accounted for. It is found that the coefficients of the interaction terms within the evolution equations are, in general, given by quite intricate expressions although some elementary numerical work shows that the evaluation of these coefficients is practicable. The basis of the work lends itself to generalization to more complicated boundary layers, and effects of detuning or non-parallelism could be provided for within the asymptotic framework.

Bassom, Andrew P.↗

NASA NTRS1990

On the instability of hypersonic flow past a flat plate

The instability of hypersonic boundary-layer flows over flat plates is considered. The viscosity of the fluid is taken to be governed by Sutherland's law, which gives a much more accurate representation of the temperature dependence of fluid viscosity at hypersonic speeds than Chapman's approximate linear law; although at lower speeds the temperature variation of the mean state is less pronounced so that the Chapman law can be used with some confidence. Attention is focussed on the so-called (vorticity) mode of instability of the viscous hypersonic boundary layer. This is thought to be the fastest growing inviscid disturbance at hypersonic speeds; it is also believed to have an asymptotically larger growth rate than any viscous or centrifugal instability. As a starting point the instability of the hypersonic boundary layer which exists far downstream from the leading edge of the plate is investigated. In this regime the shock that is attached to the leading edge of the plate plays no role, so that the basic boundary layer is non-interactive. It is shown that the vorticity mode of instability of this flow operates on a significantly different lengthscale than that obtained if a Chapman viscosity law is assumed. In particular, it is found that the growth rate predicted by a linear viscosity law overestimates the size of the growth rate by O(M(exp 2). Next, the development of the vorticity mode as the wavenumber decreases is described, and it is shown that acoustic modes emerge when the wavenumber has decreased from it's O(1) initial value to O(M (exp -3/2). Finally, the inviscid instability of the boundary layer near the leading edge in the interaction zone is discussed and particular attention is focussed on the strong interaction region which occurs sufficiently close to the leading edge. It is found that the vorticity mode in this regime is again unstable, and that it is concentrated in the transition layer at the edge of the boundary layer where the temperature adjusts from its large, O(M(exp 2), value in the viscous boundary layer, to its O(1) free stream value. The existence of the shock indirectly, but significantly, influences the instability problem by modifying the basic flow structure in this layer.

Blackaby, Nicholas↗

NASA NTRS1990

On the receptivity problem for Goertler vortices: Vortex motions induced by wall roughness

The receptivity problem for Goertler vortices induced by wall roughness is investigated. The roughness is modelled by small amplitude perturbations to the curved wall over which the flow takes place. The amplitude of these perturbations is taken to be sufficiently small for the induced Goertler vortices to be described by linear theory. The roughness is assumed to vary in the spanwise direction on the boundary layer lengthscale, while in the flow direction the corresponding variation is on the lengthscale over which the wall curvature varies. In fact the latter condition can be relaxed to allow for a faster streamwise roughness variation so long as the variation does not become as fast as that in the spanwise direction. The function which describes the roughness is assumed to be such that its spanwise and streamwise dependences can be separated; this enables progress by taking Fourier or Laplace transforms where appropriate. The cases of isolated and distributed roughness elements are investigated and the coupling coefficient which relates the amplitude of the forcing and the induced vortex amplitude is found asymptotically in the small wavelength limit. It is shown that this coefficient is exponentially small in the latter limit so that it is unlikely that this mode can be stimulated directly by wall roughness. The situation at 0(1) wavelengths is quite different and this is investigated numerically for different forcing functions. It is found that an isolated roughness element induces a vortex field which grows within a wedge at a finite distance downstream of the element. However, immediately downstream of the obstacle the disturbed flow produced by the element decays in amplitude. The receptivity problem at larger Goertler numbers appropriate to relatively large wall curvature is discussed in detail.

Denier, James P.↗

NASA NTRS1990

On the instability of hypersonic flow past a wedge

The instability of a compressible flow past a wedge is investigated in the hypersonic limit. Particular attention is paid to Tollmien-Schlichting waves governed by triple-deck theory though some discussion of inviscid modes is given. It is shown that the attached shock can have a significant effect on the growth rates of Tollmien-Schlichting waves. Moreover, the presence of the shock allows for more than one unstable Tollmien-Schlichting wave. Indeed an infinite discrete spectrum of unstable waves is induced by the shock, but these modes are unstable over relatively small but high frequency ranges. The shock is shown to have little effect on the inviscid modes considered by previous authors and an asymptotic description of inviscid modes in the hypersonic limit is given.

Cowley, Stephen↗