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.

Direct simulation of evolution and control of three-dimensional instabilities in attachment-line boundary layers

1995 ◽  
Vol 291 ◽  
pp. 369-392 ◽  
Author(s):  
Ronald D. Joslin
Keyword(s):  
Boundary Layer ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Plate Boundary ◽  
Navier Stokes ◽  
Two Dimensional ◽  
Computationally Efficient ◽  
Simulation Results ◽  
Dimensional Simulation ◽  
Attachment Line

The spatial evolution of three-dimensional disturbances in an attachment-line boundary layer is computed by direct numerical simulation of the unsteady, incompressible Navier–Stokes equations. Disturbances are introduced into the boundary layer by harmonic sources that involve unsteady suction and blowing through the wall. Various harmonic-source generators are implemented on or near the attachment line, and the disturbance evolutions are compared. Previous two-dimensional simulation results and nonparallel theory are compared with the present results. The three-dimensional simulation results for disturbances with quasi-two-dimensional features indicate growth rates of only a few percent larger than pure two-dimensional results; however, the results are close enough to enable the use of the more computationally efficient, two-dimensional approach. However, true three-dimensional disturbances are more likely in practice and are more stable than two-dimensional disturbances. Disturbances generated off (but near) the attachment line spread both away from and toward the attachment line as they evolve. The evolution pattern is comparable to wave packets in flat-plate boundary-layer flows. Suction stabilizes the quasi-two-dimensional attachment-line instabilities, and blowing destabilizes these instabilities; these results qualitatively agree with the theory. Furthermore, suction stabilizes the disturbances that develop off the attachment line. Clearly, disturbances that are generated near the attachment line can supply energy to attachment-line instabilities, but suction can be used to stabilize these instabilities.

Stability and Three-Wave Interaction of Disturbances in a Supersonic Boundary Layer With Cooling

2010 ◽  
Vol 5 (3) ◽  
pp. 52-62
Author(s):  
Sergey A. Gaponov ◽  
Natalya M. Terekhova
Keyword(s):  
Boundary Layer ◽  
Three Dimensional ◽  
Wave Interaction ◽  
Considerable Change ◽  
Supersonic Boundary Layer ◽  
Surface Cooling ◽  
Weakly Nonlinear ◽  
Linear Evolution ◽  
Weakly Nonlinear Theory ◽  
Compressed Gas

In linear and nonlinear approach (weakly nonlinear theory of stability) interaction of disturbances on a boundary layer of compressed gas is considered at surface cooling. The regimes of moderate (Max number М = 2) and high (М = 5.35) are considered at supersonic speeds. It is established that the surface cooling leads to considerable change of linear evolution of disturbances: the vortical disturbances of the first mode are stabilised, and the acoustic disturbances of the second mode are destabilised, the change degree is defined by the degree of change of the temperature factor. The nonlinear interaction in three-wave systems on high (М = 5.35) supersonic regimes on a boundary layer of compressed gas is carried out between waves of the different nature (acoustic and vortical) in a regime of a parametrical resonance. As a rating wave the flat acoustic wave which raises three-dimensional subharmonic components of the vortical modes. However, the similar interactions for vortical waves at М = 2 considerably weaken. It is possible to expect that surface cooling will lead to delay of a laminar regime at М = 2 and to accelerate of turbulization at М = 5.35

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

1990 ◽  
Vol 217 ◽  
pp. 367-390 ◽  
Author(s):  
Philip Hall ◽  
Sharon O. Seddougui
Keyword(s):  
Boundary Layer ◽  
Reynolds Number ◽  
Stokes Equations ◽  
Hydrodynamic Instability ◽  
Three Dimensional ◽  
Low Frequency ◽  
Leading Edge ◽  
Two Dimensional ◽  
Oblique Wave ◽  
Attachment Line

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 attachment-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. This has been done in the linear and weakly nonlinear regimes by Hall, Malik & Poll (1984) and Hall & Malik (1986). Here the corresponding problem is studied for oblique waves and their interaction with two-dimensional waves is investigated. In fact oblique modes cannot be described exactly at finite Reynolds number so it is necessary to make a high-Reynolds-number approximation and use triple-deck theory. 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. First a low-frequency mode closely related to the viscous stationary crossflow mode discussed by Hall (1986) and MacKerrell (1987) is a possible cause of breakdown. Secondly a class of oblique wave with frequency comparable with that of the two-dimensional mode is another cause of breakdown. It is shown that the relative importance of the modes depends on the distance from the attachment line.

scholarly journals Can weakly nonlinear theory explain Faraday wave patterns near onset?

2015 ◽  
Vol 777 ◽  
pp. 604-632 ◽  
Author(s):  
A. C. Skeldon ◽  
A. M. Rucklidge
Keyword(s):  
Nonlinear Theory ◽  
Stokes Equations ◽  
Single Mode ◽  
Theoretical Work ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Weakly Nonlinear ◽  
Weakly Nonlinear Theory ◽  
Temporal Modes ◽  
Pattern Forming

The Faraday problem is an important pattern-forming system that provides some middle ground between systems where the initial instability involves just a single mode, and in which complexity then results from mode interactions or secondary bifurcations, and cases where a system is highly turbulent and many spatial and temporal modes are excited. It has been a rich source of novel patterns and of theoretical work aimed at understanding how and why such patterns occur. Yet it is particularly challenging to tie theory to experiment: the experiments are difficult to perform; the parameter regime of interest (large box, moderate viscosity) along with the technical difficulties of solving the free-boundary Navier–Stokes equations make numerical solution of the problem hard; and the fact that the instabilities result in an entire circle of unstable wavevectors presents considerable theoretical difficulties. In principle, weakly nonlinear theory should be able to predict which patterns are stable near pattern onset. In this paper we present the first quantitative comparison between weakly nonlinear theory of the full Navier–Stokes equations and (previously published) experimental results for the Faraday problem with multiple-frequency forcing. We confirm that three-wave interactions sit at the heart of why complex patterns are stabilised, but also highlight some discrepancies between theory and experiment. These suggest the need for further experimental and theoretical work to fully investigate the issues of pattern bistability and the role of bicritical/tricritical points in determining bifurcation structure.

scholarly journals A Numerically Supported Investigation of the 3D Flow in Centrifugal Impellers: Part I — The Validation Base

1996 ◽  
Author(s):  
Ch. Hirsch ◽  
S. Kang ◽  
G. Pointel
Keyword(s):  
Boundary Layer ◽  
Three Dimensional ◽  
Boundary Layer Separation ◽  
Numerical Approach ◽  
Secondary Flows ◽  
Pump Impeller ◽  
High Loss ◽  
Leakage Flows ◽  
Fine Mesh ◽  
Radial Pump

The three-dimensional flow in centrifugal impellers is investigated on the basis of a detailed analysis of the results of numerical simulations. In order to gain confidence in this process, an in-depth validation is performed, based on computations of Krain’s centrifugal compressor and of a radial pump impeller, both with vaneless diffusers. Detailed comparisons with available experimental data provide high confidence in the numerical tools and results. The appearance of a high loss ‘wake’ region results from the transport of boundary layer material from the blade surfaces to the shroud region and its location depends on the balance between secondary and tip leakage flows and is not necessarily connected to 3D boundary layer separation. Although the low momentum spots near the shroud can interfere with 3D separated regions, the main outcome of the present analysis is that these are two distinct phenomena. Part I of this paper focuses on the validation base of the numerical approach, based on fine mesh simulations, while Part II presents an analysis of the different contributions to the secondary flows and attempts to estimate their effect on the overall flow pattern.

Effects of Inlet Distortion on the Flow Field in a Transonic Compressor Rotor

1996 ◽  
Author(s):  
Chunill Hah ◽  
Douglas C. Rabe ◽  
Thomas J. Sullivan ◽  
Aspi R. Wadia
Keyword(s):  
Boundary Layer ◽  
Flow Field ◽  
Viscous Flow ◽  
Total Pressure ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Aerodynamic Loss ◽  
Transonic Compressor ◽  
Inlet Distortion ◽  
Compressor Rotor

The effects of circumferential distortions in inlet total pressure on the flow field in a low-aspect-ratio, high-speed, high-pressure-ratio, transonic compressor rotor are investigated in this paper. The flow field was studied experimentally and numerically with and without inlet total pressure distortion. Total pressure distortion was created by screens mounted upstream from the rotor inlet. Circumferential distortions of 8 periods per revolution were investigated at two different rotor speeds. The unsteady blade surface pressures were measured with miniature pressure transducers mounted in the blade. The flow fields with and without inlet total pressure distortion were analyzed numerically by solving steady and unsteady forms of the Reynolds-averaged Navier-Stokes equations. Steady three-dimensional viscous flow calculations were performed for the flow without inlet distortion while unsteady three-dimensional viscous flow calculations were used for the flow with inlet distortion. For the time-accurate calculation, circumferential and radial variations of the inlet total pressure were used as a time-dependent inflow boundary condition. A second-order implicit scheme was used for the time integration. The experimental measurements and the numerical analysis are highly complementary for this study because of the extreme complexity of the flow field. The current investigation shows that inlet flow distortions travel through the rotor blade passage and are convected into the following stator. At a high rotor speed where the flow is transonic, the passage shock was found to oscillate by as much as 20% of the blade chord, and very strong interactions between the unsteady passage shock and the blade boundary layer were observed. This interaction increases the effective blockage of the passage, resulting in an increased aerodynamic loss and a reduced stall margin. The strong interaction between the passage shock and the blade boundary layer increases the peak aerodynamic loss by about one percent.

scholarly journals Finite-amplitude steady waves in plane viscous shear flows

1985 ◽  
Vol 160 ◽  
pp. 281-295 ◽  
Author(s):  
F. A. Milinazzo ◽  
P. G. Saffman
Keyword(s):  
Boundary Layer ◽  
Reynolds Number ◽  
Stokes Equations ◽  
Reynolds Numbers ◽  
Finite Amplitude ◽  
Linear Instability ◽  
Navier Stokes ◽  
Unidirectional Flow ◽  
Viscous Shear ◽  
Finite Amplitude Waves

Computations of two-dimensional solutions of the Navier–Stokes equations are carried out for finite-amplitude waves on steady unidirectional flow. Several cases are considered. The numerical method employs pseudospectral techniques in the streamwise direction and finite differences on a stretched grid in the transverse direction, with matching to asymptotic solutions when unbounded. Earlier results for Poiseuille flow in a channel are re-obtained, except that attention is drawn to the dependence of the minimum Reynolds number on the physical constraint of constant flux or constant pressure gradient. Attempts to calculate waves in Couette flow by continuation in the velocity of a channel wall fail. The asymptotic suction boundary layer is shown to possess finite-amplitude waves at Reynolds numbers orders of magnitude less than the critical Reynolds number for linear instability. Waves in the Blasius boundary layer and unsteady Rayleigh profile are calculated by employing the artifice of adding a body force to cancel the spatial or temporal growth. The results are verified by comparison with perturbation analysis in the vicinity of the linear-instability critical Reynolds numbers.

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