A nonlinear, asymptotic investigation of the stationary modes of instability of the three-dimensional boundary layer on a rotating disc

1987 ◽  
Vol 413 (1845) ◽  
pp. 497-513 ◽  
Keyword(s):  
Boundary Layer ◽  
Three Dimensional ◽  
Nonlinear Effects ◽  
Finite Amplitude ◽  
Neutral Stability ◽  
Lower Branch ◽  
Rotating Disc ◽  
Dimensional Boundary ◽  
High Reynolds ◽  
Set Up

There exist two types of stationary instability of the flow over a rotating disc corresponding to the upper, inviscid mode and the lower-branch mode, which has a triple-deck structure, of the neutral stability curve. The linear problem has been investigated by P. Hall ( Proc. R. Soc. Lond. A 406, 93-106 (1986)) and the asymptotic structure of the wavenumber and orientation of these modes has been obtained. Here, a nonlinear investigation of high Reynolds number, stationary instabilities in the three-dimensional boundary layer on a rotating disc is given for the lower branch mode. By considering nonlinear effects and following the framework set up by Hall, asymptotic solutions are obtained that enable the finite amplitude growth of a disturbance close to the neutral location to be described.

scholarly journals The effects of suction on the nonlinear stability of the three-dimensional boundary layer above a rotating disc

1992 ◽  
Vol 436 (1897) ◽  
pp. 405-415 ◽  
Keyword(s):  
Linear Problem ◽  
Three Dimensional ◽  
Threshold Value ◽  
Neutral Stability ◽  
Lower Branch ◽  
Rotating Disc ◽  
Weakly Nonlinear ◽  
Nonlinear Properties ◽  
Dimensional Boundary

There exist two types of stationary instability of the flow over a rotating disc corresponding to the upper-branch, inviscid mode and the lower-branch mode, which has a triple deck structure, of the neutral stability curve. A theoretical investigation of the linear problem and an account of the weakly nonlinear properties of the lower branch-modes have been undertaken by P. Hall and S. MacKerrell respectively. Motivated by recent reports of experimental sightings of the lower-branch mode and an examination of the role of suction on the linear stability properties of the flow here we investigate the effects of suction on the nonlinear disturbance described by S. MacKerrell. The additional analysis required to incorporate suction is relatively straightforward and enables us to derive an amplitude equation which describes the evolution of the mode. For each value of the suction a threshold value of the disturbance amplitude is obtained; modes of size greater than this threshold grow without limit as they develop away from the point of neutral stability.

Non-stationary cross-flow vortices in three-dimensional boundary-layer flows

1988 ◽  
Vol 417 (1852) ◽  
pp. 179-212 ◽  
Keyword(s):  
Boundary Layer ◽  
Three Dimensional ◽  
Cross Flow ◽  
Neutral Stability ◽  
Two Dimensional ◽  
Boundary Layer Flows ◽  
Rotating Disc ◽  
Dimensional Boundary ◽  
Flow Vortex ◽  
The Waves

The upper-branch neutral stability of three-dimensional disturbances imposed on a three-dimensional boundary-layer profile is considered and in particular we investigate non-stationary cross-flow vortices. The wave speed is taken to be small initially and with a disturbance structure analogous to that occurring in two-dimensional boundary-layer stability the linear and nonlinear eigenrelations are derived for profiles with more than one critical layer. For the flow due to a rotating disc we show that linear viscous neutral modes exist for all wave angles between 10.6° and 39.6°. As the extremes of this range are approached the flow structure evolves to another, either the viscous mode of Hall ( Proc. R. Soc. Lond . A 406, 93(1986)) or the non-stationary inviscid modes considered by Stuart in Gregory et al . ( Phil. Trans. R. Soc. Lond . A 248, 155 (1955)). In the former case, corresponding to wave angles of 39.6°, the waves become almost stationary and in the latter case, with wave angles of 10.6° the waves are travelling much faster with a disturbance structure based on the Rayleigh scalings. The analysis is extended to include O (1) wavespeeds and we show that as the wavespeed of the cross-flow vortex approaches the tree-stream value, the corresponding disturbance amplitude increases, the growth here being slower than that for two-dimensional boundary layers.

On the instability of a three-dimensional attachment-line boundary layer: weakly nonlinear theory and a numerical approach

1986 ◽  
Vol 163 ◽  
pp. 257-282 ◽  
Author(s):  
Philip Hall ◽  
Mujeeb R. Malik
Keyword(s):  
Boundary Layer ◽  
Nonlinear Theory ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Numerical Approach ◽  
Finite Amplitude ◽  
Lower Branch ◽  
Weakly Nonlinear ◽  
Weakly Nonlinear Theory ◽  
Attachment Line

The instability of a three-dimensional attachment-line boundary layer is considered in the nonlinear regime. Using weakly nonlinear theory, it is found that, apart from a small interval near the (linear) critical Reynolds number, finite-amplitude solutions bifurcate subcritically from the upper branch of the neutral curve. The time-dependent Navier–Stokes equations for the attachment-line flow have been solved using a Fourier–Chebyshev spectral method and the subcritical instability is found at wavenumbers that correspond to the upper branch. Both the theory and the numerical calculations show the existence of supercritical finite-amplitude (equilibrium) states near the lower branch which explains why the observed flow exhibits a preference for the lower branch modes. The effect of blowing and suction on nonlinear stability of the attachment-line boundary layer is also investigated.

Secondary instability of crossflow vortices and swept-wing boundary-layer transition

1999 ◽  
Vol 399 ◽  
pp. 85-115 ◽  
Author(s):  
MUJEEB R. MALIK ◽  
FEI LI ◽  
MEELAN M. CHOUDHARI ◽  
CHAU-LYAN CHANG
Keyword(s):  
Experimental Data ◽  
Boundary Layer ◽  
Three Dimensional ◽  
Finite Amplitude ◽  
Secondary Instability ◽  
Boundary Layer Transition ◽  
Swept Wing ◽  
Layer Transition ◽  
Transition Prediction ◽  
Dimensional Boundary

Crossflow instability of a three-dimensional boundary layer is a common cause of transition in swept-wing flows. The boundary-layer flow modified by the presence of finite-amplitude crossflow modes is susceptible to high-frequency secondary instabilities, which are believed to harbinger the onset of transition. The role of secondary instability in transition prediction is theoretically examined for the recent swept-wing experimental data by Reibert et al. (1996). Exploiting the experimental observation that the underlying three-dimensional boundary layer is convectively unstable, non-linear parabolized stability equations are used to compute a new basic state for the secondary instability analysis based on a two-dimensional eigenvalue approach. The predicted evolution of stationary crossflow vortices is in close agreement with the experimental data. The suppression of naturally dominant crossflow modes by artificial roughness distribution at a subcritical spacing is also confirmed. The analysis reveals a number of secondary instability modes belonging to two basic families which, in some sense, are akin to the ‘horseshoe’ and ‘sinuous’ modes of the Görtler vortex problem. The frequency range of the secondary instability is consistent with that measured in earlier experiments by Kohama et al. (1991), as is the overall growth of the secondary instability mode prior to the onset of transition (e.g. Kohama et al. 1996). Results indicate that the N-factor correlation based on secondary instability growth rates may yield a more robust criterion for transition onset prediction in comparison with an absolute amplitude criterion that is based on primary instability alone.

Streamwise absolute instability of a three-dimensional boundary layer at high Reynolds numbers

1998 ◽  
Vol 373 ◽  
pp. 111-153 ◽  
Author(s):  
OLEG S. RYZHOV ◽  
EUGENE D. TERENT'EV
Keyword(s):  
Boundary Layer ◽  
Wind Tunnel ◽  
Wave Packets ◽  
Three Dimensional ◽  
Reynolds Numbers ◽  
Absolute Instability ◽  
Streamwise Direction ◽  
High Reynolds Numbers ◽  
Dimensional Boundary ◽  
High Reynolds

The simplest receptivity problem of linear disturbances artificially excited in a three-dimensional boundary layer adjacent to a solid surface is studied in the framework of the generalized triple-deck theory. In order to provide a mathematical model to be compared with experimental data from wind-tunnel tests we consider the base flow over a swept flat plate. Then crossflow in the near-wall region originates owing to an almost constant pressure gradient induced from outside with a displacement body on top. A pulsed or vibrating ribbon installed on the solid surface serves as an external agency provoking initially weak pulsations. A periodic dependence of the ribbon shape on a coordinate normal to the streamwise direction makes the receptivity problem effectively two-dimensional, thereby allowing a rigorous analysis to be carried out without additional assumptions.The most striking result from the asymptotic theory is the discovery of streamwise absolute instability intrinsic to a three-dimensional boundary layer at high Reynolds numbers. However, due to limitations imposed on the receptivity problem no definite conclusions can be made with regard to possible continued convection of disturbances in the crossflow direction. An investigation of the dispersion-relation roots points to the fact that wave packets of different kinds can be generated by an external source operating in the pulse mode. Rapidly growing wave packets sweep downstream, weaker wave packets move against the oncoming stream. Insofar as the amplitude of all of the modulated signals increases exponentially in time and space, the excitation process gives rise to absolutely unstable disturbances in the streamwise direction. The computation confirms the theoretical prediction about the existence of upstream-advancing wave packets. They can be prevented from being persistently amplified only in a region ahead of the ribbon where nearly critical values of the Reynolds number are attained.The results achieved are shown to be broadly consistent with wind-tunnel measurements. Hence a conjecture is made that the onset of transition is probably associated, under some environmental conditions, with the mechanism of streamwise absolute instability in the supercritical range of the Reynolds numbers.

Circulation control in magnetohydrodynamic rotating flows

2017 ◽  
Vol 829 ◽  
pp. 328-344 ◽  
Author(s):  
V. D. Borisevich ◽  
E. P. Potanin ◽  
J. Whichello
Keyword(s):  
Magnetic Field ◽  
Boundary Layer ◽  
Exact Analytical Solution ◽  
External Rotation ◽  
Three Dimensional ◽  
Flow Characteristics ◽  
Rotating Cylinder ◽  
Rotating Disc ◽  
Angular Velocities ◽  
Energy Equations

A model of a laminar viscous conducting flow, near a dielectric disc in a uniform magnetic field and in the presence of external rotation, is considered, where there is a uniform suction and an axial temperature gradient between the flow and the disc’s surface. It is assumed that the parameters of the suction or the magnetohydrodynamic (MHD) interaction are such that the nonlinear inertial terms, related to the circulation flow, are negligible in the differential equations of the MHD boundary layer on a rotating disc. Analysis of the motion and energy equations, taking the dependence of density on temperature into account, is carried out using the Dorodnitsyn transformation. The exact analytical solution for the boundary layer and heat transfer equations is obtained and analysed, neglecting the viscous and Joule dissipation. The dependence of the flow characteristics in the boundary layer on the rate of suction and the magnetic field induction is studied. It is shown that the direction of the radial flow in the boundary layer on a disc can be changed, not only by variation of the ratio between the angular velocities in the external flow and the boundary layer, but also by changing the ratio of the temperatures in these two flows, as well as by varying the hydrodynamic Prandtl number. The approximate calculation of a three-dimensional flow in a rotating cylinder with a braking disc (or lid) is carried out, demonstrating that a magnetic field slows the circulation velocity in a rotating cylinder.

An Assessment of Three-Dimensional Turbulent Boundary Layer Prediction Methods

1973 ◽  
Vol 95 (3) ◽  
pp. 415-421 ◽  
Author(s):  
A. J. Wheeler ◽  
J. P. Johnston
Keyword(s):  
Boundary Layer ◽  
Eddy Viscosity ◽  
Three Dimensional ◽  
High Sensitivity ◽  
Boundary Layer Flows ◽  
Mean Velocity ◽  
Eddy Viscosity Model ◽  
Separating Flow ◽  
Dimensional Boundary ◽  
Stress Models

Predictions have been made for a variety of experimental three-dimensional boundary layer flows with a single finite difference method which was used with three different turbulent stress models: (i) an eddy viscosity model, (ii) the “Nash” model, and (iii) the “Bradshaw” model. For many purposes, even the simplest stress model (eddy viscosity) was adequate to predict the mean velocity field. On the other hand, the profile of shear stress direction was not correctly predicted in one case by any model tested. The high sensitivity of the predicted results to free stream pressure gradient in separating flow cases is demonstrated.

Experimental and Theoretical Investigation of the Supersonic Boundary Layer Stability on the Swept Wing

2008 ◽  
Vol 3 (3) ◽  
pp. 34-38
Author(s):  
Sergey A. Gaponov ◽  
Yuri G. Yermolaev ◽  
Aleksandr D. Kosinov ◽  
Nikolay V. Semionov ◽  
Boris V. Smorodsky
Keyword(s):  
Boundary Layer ◽  
Three Dimensional ◽  
Leading Edge ◽  
Supersonic Boundary Layer ◽  
Mean Flow ◽  
Swept Wing ◽  
Wing Model ◽  
Dimensional Boundary ◽  
The Stability ◽  
Good Agreement

Theoretical and an experimental research results of the disturbances development in a swept wing boundary layer are presented at Mach number М = 2. In experiments development of natural and small amplitude controllable disturbances downstream was studied. Experiments were carried out on a swept wing model with a lenticular profile at a zero attack angle. The swept angle of a leading edge was 40°. Wave parameters of moving disturbances were determined. In frames of the linear theory and an approach of the local self-similar mean flow the stability of a compressible three-dimensional boundary layer is studied. Good agreement of the theory with experimental results for transversal scales of unstable vertices of the secondary flow was obtained. However the calculated amplification rates differ from measured values considerably. This disagreement is explained by the nonlinear processes observed in experiment

Sign in / Sign up

Export Citation Format

Share Document