Effective transition of steady flow over a square leading-edge plate

2012 ◽  
Vol 698 ◽  
pp. 335-357 ◽  
Author(s):  
Mark C. Thompson
Keyword(s):  
Reynolds Number ◽  
Shear Layer ◽  
Three Dimensional ◽  
Leading Edge ◽  
Maximum Energy ◽  
Transient Growth ◽  
Two Dimensional ◽  
Optimal Perturbation ◽  
Perturbation Mode ◽  
Optimal Mode

AbstractPrevious experimental studies have shown that the steady recirculation bubble that forms as the flow separates at the leading-edge corner of a long plate, becomes unsteady at relatively low Reynolds numbers of only a few hundreds. The reattaching shear layer irregularly releases two-dimensional vortices, which quickly undergo three-dimensional transition. Similar to the flow over a backward-facing step, this flow is globally stable at such Reynolds numbers, with transition to a steady three-dimensional flow as the first global instability to occur as the Reynolds number is increased to 393. Hence, it appears that the observed flow behaviour is governed by transient growth of optimal two-dimensional transiently growing perturbations (constructed from damped global modes) rather than a single three-dimensional unstable global mode. This paper quantifies the details of the transient growth of two- and three-dimensional optimal perturbations, and compares the predictions to other related cases examined recently. The optimal perturbation modes are shown to be highly concentrated in amplitude in the vicinity of the leading-edge corners and evolve to take the local shape of a Kelvin–Helmholtz shear-layer instability further downstream. However, the dominant mode reaches a maximum amplitude downstream of the position of the reattachment point of the shear layer. The maximum energy growth increases at 2.5 decades for each increment in Reynolds number of 100. Maximum energy growth of the optimal perturbation mode at a Reynolds number of 350 is greater than $1{0}^{4} $, which is typically an upper limit of the Reynolds number range over which it is possible to observe steady flow experimentally. While transient growth analysis concentrates on the evolution of wavepackets rather than continuous forcing, this appears consistent with longitudinal turbulence levels of up to 1 % for some water tunnels, and the fact that the optimal mode is highly concentrated close to the leading-edge corner so that an instantaneous projection of a perturbation field from a noisy inflow onto the optimal mode can be significant. Indeed, direct simulations with inflow noise reveal that a root-mean-square noise level of just 0.1 % is sufficient to trigger some unsteadiness at $\mathit{Re}= 350$, while a 0.5 % level results in sustained shedding. Three-dimensional optimal perturbation mode analysis was also performed showing that at $\mathit{Re}= 350$, the optimal mode has a spanwise wavelength of 11.7 plate thicknesses and is amplified 20 % more than the two-dimensional optimal disturbance. The evolved three-dimensional mode shows strong streamwise vortical structures aligned at a shallow angle to the plate top surface.

Three-dimensional instabilities in the boundary-layer flow over a long rectangular plate

2011 ◽  
Vol 681 ◽  
pp. 411-433 ◽  
Author(s):  
HEMANT K. CHAURASIA ◽  
MARK C. THOMPSON
Keyword(s):  
Reynolds Number ◽  
Shear Layer ◽  
Three Dimensional ◽  
Reynolds Numbers ◽  
Vortical Flow ◽  
Forced Flow ◽  
Recirculation Region ◽  
Two Dimensional ◽  
Optimal Perturbation ◽  
Dimensional Instability

A detailed numerical study of the separating and reattaching flow over a square leading-edge plate is presented, examining the instability modes governing transition from two- to three-dimensional flow. Under the influence of background noise, experiments show that the transition scenario typically is incompletely described by either global stability analysis or the transient growth of dominant optimal perturbation modes. Instead two-dimensional transition effectively can be triggered by the convective Kelvin–Helmholtz (KH) shear-layer instability; although it may be possible that this could be described alternatively in terms of higher-order optimal perturbation modes. At least in some experiments, observed transition occurs by either: (i) KH vortices shedding downstream directly and then almost immediately undergoing three-dimensional transition or (ii) at higher Reynolds numbers, larger vortical structures are shed that are also three-dimensionally unstable. These two paths lead to distinctly different three-dimensional arrangements of vortical flow structures. This paper focuses on the mechanisms underlying these three-dimensional transitions. Floquet analysis of weakly periodically forced flow, mimicking the observed two-dimensional quasi-periodic base flow, indicates that the two-dimensional vortex rollers shed from the recirculation region become globally three-dimensionally unstable at a Reynolds number of approximately 380. This transition Reynolds number and the predicted wavelength and flow symmetries match well with those of the experiments. The instability appears to be elliptical in nature with the perturbation field mainly restricted to the cores of the shed rollers and showing the spatial vorticity distribution expected for that instability type. Indeed an estimate of the theoretical predicted wavelength is also a good match to the prediction from Floquet analysis and theoretical estimates indicate the growth rate is positive. Fully three-dimensional simulations are also undertaken to explore the nonlinear development of the three-dimensional instability. These show the development of the characteristic upright hairpins observed in the experimental dye visualisations. The three-dimensional instability that manifests at lower Reynolds numbers is shown to be consistent with an elliptic instability of the KH shear-layer vortices in both symmetry and spanwise wavelength.

Transient growth in strongly stratified shear layers

2014 ◽  
Vol 758 ◽  
Author(s):  
A. K. Kaminski ◽  
C. P. Caulfield ◽  
J. R. Taylor
Keyword(s):  
Shear Layer ◽  
Normal Mode ◽  
Richardson Number ◽  
Mixing Layer ◽  
Three Dimensional ◽  
Buoyancy Frequency ◽  
Transient Growth ◽  
Time Intervals ◽  
Optimal Perturbation ◽  
Target Time

AbstractWe investigate numerically transient linear growth of three-dimensional perturbations in a stratified shear layer to determine which perturbations optimize the growth of the total kinetic and potential energy over a range of finite target time intervals. The stratified shear layer has an initial parallel hyperbolic tangent velocity distribution with Reynolds number $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\mathit{Re}=U_0 h/\nu =1000$ and Prandtl number $\nu /\kappa =1$, where $\nu $ is the kinematic viscosity of the fluid and $\kappa $ is the diffusivity of the density. The initial stable buoyancy distribution has constant buoyancy frequency $N_0$, and we consider a range of flows with different bulk Richardson number ${\mathit{Ri}}_b=N_0^2h^2/U_0^2$, which also corresponds to the minimum gradient Richardson number ${\mathit{Ri}}_g(z)=N_0^2/(\mathrm{d}U/\mathrm{d} z)^2$ at the midpoint of the shear layer. For short target times, the optimal perturbations are inherently three-dimensional, while for sufficiently long target times and small ${\mathit{Ri}}_b$ the optimal perturbations are closely related to the normal-mode ‘Kelvin–Helmholtz’ (KH) instability, consistent with analogous calculations in an unstratified mixing layer recently reported by Arratia et al. (J. Fluid Mech., vol. 717, 2013, pp. 90–133). However, we demonstrate that non-trivial transient growth occurs even when the Richardson number is sufficiently high to stabilize all normal-mode instabilities, with the optimal perturbation exciting internal waves at some distance from the midpoint of the shear layer.

Large-eddy simulation of boundary-layer separation and transition at a change of surface curvature

2001 ◽  
Vol 439 ◽  
pp. 305-333 ◽  
Author(s):  
ZHIYIN YANG ◽  
PETER R. VOKE
Keyword(s):  
Boundary Layer ◽  
Shear Layer ◽  
Three Dimensional ◽  
Leading Edge ◽  
Boundary Layer Transition ◽  
Two Dimensional ◽  
Layer Transition ◽  
Transition Mechanism ◽  
The Mean ◽  
Dimensional Instability

Transition arising from a separated region of flow is quite common and plays an important role in engineering. It is difficult to predict using conventional models and the transition mechanism is still not fully understood. We report the results of a numerical simulation to study the physics of separated boundary-layer transition induced by a change of curvature of the surface. The geometry is a flat plate with a semicircular leading edge. The Reynolds number based on the uniform inlet velocity and the leading-edge diameter is 3450. The simulated mean and turbulence quantities compare well with the available experimental data.The numerical data have been comprehensively analysed to elucidate the entire transition process leading to breakdown to turbulence. It is evident from the simulation that the primary two-dimensional instability originates from the free shear in the bubble as the free shear layer is inviscidly unstable via the Kelvin–Helmholtz mechanism. These initial two-dimensional instability waves grow downstream with a amplification rate usually larger than that of Tollmien–Schlichting waves. Three-dimensional motions start to develop slowly under any small spanwise disturbance via a secondary instability mechanism associated with distortion of two-dimensional spanwise vortices and the formation of a spanwise peak–valley wave structure. Further downstream the distorted spanwise two-dimensional vortices roll up, leading to streamwise vorticity formation. Significant growth of three-dimensional motions occurs at about half the mean bubble length with hairpin vortices appearing at this stage, leading eventually to full breakdown to turbulence around the mean reattachment point. Vortex shedding from the separated shear layer is also observed and the ‘instantaneous reattachment’ position moves over a distance up to 50% of the mean reattachment length. Following reattachment, a turbulent boundary layer is established very quickly, but it is different from an equilibrium boundary layer.

Hydromagnetic instability of a free shear layer at small magnetic Reynolds numbers

1971 ◽  
Vol 49 (1) ◽  
pp. 21-31 ◽  
Author(s):  
Kanefusa Gotoh
Keyword(s):  
Magnetic Field ◽  
Reynolds Number ◽  
Shear Layer ◽  
Critical Reynolds Number ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Free Shear Layer ◽  
Electrically Conducting ◽  
The Stability ◽  
The Magnetic Field

The effect of a uniform and parallel magnetic field upon the stability of a free shear layer of an electrically conducting fluid is investigated. The equations of the velocity and the magnetic disturbances are solved numerically and it is shown that the flow is stabilized with increasing magnetic field. When the magnetic field is expressed in terms of the parameter N (= M2/R2), where M is the Hartmann number and R is the Reynolds number, the lowest critical Reynolds number is caused by the two-dimensional disturbances. So long as 0 [les ] N [les ] 0·0092 the flow is unstable at all R. For 0·0092 < N [les ] 0·0233 the flow is unstable at 0 < R < Ruc where Ruc decreases as N increases. For 0·0233 < N < 0·0295 the flow is unstable at Rlc < R < Ruc where Rlc increases with N. Lastly for N > 0·0295 the flow is stable at all R. When the magnetic field is measured by M, the lowest critical Reynolds number is still due to the two-dimensional disturbances provided 0 [les ] M [les ] 0·52, and Rc is given by the corresponding Rlc. For M > 0·52, Rc is expressed as Rc = 5·8M, and the responsible disturbance is the three-dimensional one which propagates at angle cos−1(0·52/M) to the direction of the basic flow.

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 From three-dimensional to quasi-two-dimensional: transient growth in magnetohydrodynamic duct flows

2018 ◽  
Vol 861 ◽  
pp. 382-406 ◽  
Author(s):  
Oliver G. W. Cassells ◽  
Tony Vo ◽  
Alban Pothérat ◽  
Gregory J. Sheard
Keyword(s):  
Magnetic Field ◽  
Reynolds Number ◽  
Layer Thickness ◽  
Hartmann Number ◽  
Three Dimensional ◽  
Transient Growth ◽  
Two Dimensional ◽  
Growth Mechanisms ◽  
Electrically Conducting ◽  
Magnetohydrodynamic Models

This study seeks to elucidate the linear transient growth mechanisms in a uniform duct with square cross-section applicable to flows of electrically conducting fluids under the influence of an external magnetic field. A particular focus is given to the question of whether at high magnetic fields purely two-dimensional mechanisms exist, and whether these can be described by a computationally inexpensive quasi-two-dimensional model. Two Reynolds numbers of $5000$ and $15\,000$ and an extensive range of Hartmann numbers $0\leqslant \mathit{Ha}\leqslant 800$ were investigated. Three broad regimes are identified in which optimal mode topology and non-modal growth mechanisms are distinct. These regimes, corresponding to low, moderate and high magnetic field strengths, are found to be governed by the independent parameters; Hartmann number, Reynolds number based on the Hartmann layer thickness $R_{H}$ and Reynolds number built upon the Shercliff layer thickness $R_{S}$, respectively. Transition between regimes respectively occurs at $\mathit{Ha}\approx 2$ and no lower than $R_{H}\approx 33.\dot{3}$. Notably for the high Hartmann number regime, quasi-two-dimensional magnetohydrodynamic models are shown to be excellent predictors of not only transient growth magnitudes, but also the fundamental growth mechanisms of linear disturbances. This paves the way for a precise analysis of transition to quasi-two-dimensional turbulence at much higher Hartmann numbers than is currently achievable.

Experimental Investigation of Separated Shear Layer Over a Flat Plate for Various Angles of Attack and Tail Flap Deflections

2014 ◽  
Author(s):  
K. Anand ◽  
S. Sarkar
Keyword(s):  
Reynolds Number ◽  
Shear Layer ◽  
Flat Plate ◽  
Three Dimensional ◽  
Leading Edge ◽  
Stream Velocity ◽  
Pressure Gradients ◽  
Bubble Bursting ◽  
Separated Shear Layer ◽  
Bubble Length

Shear layer development over a thick flat plate with a semi-circular leading edge is investigated for a range of angles of attack under different imposed pressure gradients for a Reynolds number of 2.44×105 (based on chord and free-stream velocity). The features of the separated shear layer are very well documented through a combination of surface pressure measurement and flow visualization by particle image velocimetry (PIV). The instability of the separated layer occurs because of enhanced receptivity of perturbations leading to the development of significant unsteadiness and three-dimensional motions in the second-half of the bubble. The onset of separation, transition and the point of reattachment are identified for varying angles of attack and imposed pressure gradients. The reattachment point shifts from 12.5% to 53% of chord resulting in enhancement of bubble length from 5% to 47%, while onset of transition shifts upstream from 14% to 7.5% as α increases. The Reynolds number based on the length of laminar shear layer is found to be in the range of 0.7×104 to 2.0×104. The separated shear layer fails to reattach attributing to bubble bursting at α = 12° for β = −45°, while, it bursts at α = 5° for β = +45°. The bubble falls in the category of short bubble for α < 3°, whereas, it becomes long for α ≥ 3°. The data concerning laminar portion and reattachment points agree well with the literature.

scholarly journals Non-Modal Three-Dimensional Optimal Perturbation Growth in Thermally Stratified Mixing Layers

Fluids ◽  
2021 ◽  
Vol 6 (1) ◽  
pp. 37
Author(s):  
Helena Vitoshkin ◽  
Alexander Gelfgat
Keyword(s):  
Mixing Layer ◽  
Three Dimensional ◽  
Stable Stratification ◽  
Base Flow ◽  
Transient Growth ◽  
Two Dimensional ◽  
Optimal Perturbation ◽  
Dimensional Growth ◽  
Layer Flow ◽  
Perturbation Growth

A non-modal transient disturbances growth in a stably stratified mixing layer flow is studied numerically. The model accounts for a density gradient within a shear region, implying a heavier layer at the bottom. Numerical analysis of non-modal stability is followed by a full three-dimensional direct numerical simulation (DNS) with the optimally perturbed base flow. It is found that the transient growth of two-dimensional disturbances diminishes with the strengthening of stratification, while three-dimensional disturbances cause significant non-modal growth, even for a strong, stable stratification. This non-modal growth is governed mainly by the Holmboe modes and does not necessarily weaken with the increase of the Richardson number. The optimal perturbation consists of two waves traveling in opposite directions. Compared to the two-dimensional transient growth, the three-dimensional growth is found to be larger, taking place at shorter times. The non-modal growth is observed in linearly stable regimes and, in slightly linearly supercritical regimes, is steeper than that defined by the most unstable eigenmode. The DNS analysis confirms the presence of the structures determined by the transient growth analysis.

Bifurcation Characteristics of Flows in Rectangular Sudden Expansion Channels

2005 ◽  
Author(s):  
Francine Battaglia ◽  
George Papadopoulos
Keyword(s):  
Reynolds Number ◽  
Laminar Flow ◽  
Critical Reynolds Number ◽  
Three Dimensional ◽  
Reynolds Numbers ◽  
Expansion Ratio ◽  
Sudden Expansion ◽  
Two Dimensional ◽  
Effective Expansion ◽  
Three Dimensional Simulations

The effect of three-dimensionality on low Reynolds number flows past a symmetric sudden expansion in a channel was investigated. The geometric expansion ratio of in the current study was 2:1 and the aspect ratio was 6:1. Both experimental velocity measurements and two- and three-dimensional simulations for the flow along the centerplane of the rectangular duct are presented for Reynolds numbers in the range of 150 to 600. Comparison of the two-dimensional simulations with the experiments revealed that the simulations fail to capture completely the total expansion effect on the flow, which couples both geometric and hydrodynamic effects. To properly do so requires the definition of an effective expansion ratio, which is the ratio of the downstream and upstream hydraulic diameters and is therefore a function of both the expansion and aspect ratios. When the two-dimensional geometry was consistent with the effective expansion ratio, the new results agreed well with the three-dimensional simulations and the experiments. Furthermore, in the range of Reynolds numbers investigated, the laminar flow through the expansion underwent a symmetry-breaking bifurcation. The critical Reynolds number evaluated from the experiments and the simulations was compared to other values reported in the literature. Overall, side-wall proximity was found to enhance flow stability, helping to sustain laminar flow symmetry to higher Reynolds numbers in comparison to nominally two-dimensional double-expansion geometries. Lastly, and most importantly, when the logarithm of the critical Reynolds number from all these studies was plotted against the reciprocal of the effective expansion ratio, a linear trend emerged that uniquely captured the bifurcation dynamics of all symmetric double-sided planar expansions.

Two- and Three-Dimensional Simulations of the Flow Around a Cylinder Fitted With Strakes

2012 ◽  
Author(s):  
Bruno S. Carmo ◽  
Rafael S. Gioria ◽  
Ivan Korkischko ◽  
Cesar M. Freire ◽  
Julio R. Meneghini
Keyword(s):  
Reynolds Number ◽  
Vortex Shedding ◽  
Free Stream ◽  
Three Dimensional ◽  
The Body ◽  
Vortex Wake ◽  
Two Dimensional ◽  
Flow Around A Cylinder ◽  
Three Dimensional Simulations ◽  
Angle Of Rotation

Two- and three-dimensional simulations of the flow around straked cylinders are presented. For the two-dimensional simulations we used the Spectral/hp Element Method, and carried out simulations for five different angles of rotation of the cylinder with respect to the free stream. Fixed and elastically-mounted cylinders were tested, and the Reynolds number was kept constant and equal to 150. The results were compared to those obtained from the simulation of the flow around a bare cylinder under the same conditions. We observed that the two-dimensional strakes are not effective in suppressing the vibration of the cylinders, but also noticed that the responses were completely different even with a slight change in the angle of rotation of the body. The three-dimensional results showed that there are two mechanisms of suppression: the main one is the decrease in the vortex shedding correlation along the span, whilst a secondary one is the vortex wake formation farther downstream.

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