scholarly journals The minimal seed of turbulent transition in the boundary layer

2011 ◽  
Vol 689 ◽  
pp. 221-253 ◽  
Author(s):  
S. Cherubini ◽  
P. De Palma ◽  
J.-C. Robinet ◽  
A. Bottaro
Keyword(s):  
Boundary Layer ◽  
Stokes Equations ◽  
Base Flow ◽  
Flow Structures ◽  
Time Interval ◽  
Short Time Interval ◽  
Optimization Approach ◽  
Computational Domain ◽  
Energy Growth ◽  
Optimal Disturbances

AbstractThis paper describes a scenario of transition from laminar to turbulent flow in a spatially developing boundary layer over a flat plate. The base flow is the Blasius non-parallel flow solution; it is perturbed by optimal disturbances yielding the largest energy growth over a short time interval. Such perturbations are computed by a nonlinear global optimization approach based on a Lagrange multiplier technique. The results show that nonlinear optimal perturbations are characterized by a localized basic building block, called the minimal seed, defined as the smallest flow structure which maximizes the energy growth over short times. It is formed by vortices inclined in the streamwise direction surrounding a region of intense streamwise disturbance velocity. Such a basic structure appears to be a robust feature of the base flow since it is practically invariant with respect to the initial energy of the perturbation, the target time, the Reynolds number and the dimensions of the computational domain. The minimal seed grows very rapidly in time while spreading, and it triggers nonlinear effects which bring the flow to turbulence in a very efficient manner, through the formation of a turbulence spot. This evolution of the initial optimal disturbance has been studied in detail by direct numerical simulations. Using a perturbative formulation of the Navier–Stokes equations, each linear and nonlinear convective term of the equations has been analysed. The results show the fundamental role of the streamwise inclination of the vortices in the process. The nonlinear coupling of the finite amplitude disturbances is crucial to sustain such streamwise inclination, as well as to generate dislocations within the flow structures, and local inflectional velocity distributions. The analysis provides a picture of the transition process characterized by a sequence of structures appearing successively in the flow, namely, $ \mrm{\Lambda} $ vortices, hairpin vortices and streamwise streaks. Finally, a disturbance regeneration cycle is conceived, initiated by the fast nonlinear amplification of the minimal seed, providing a possible scenario for the continuous regeneration of the same fundamental flow structures at smaller space and time scales.

scholarly journals Weakly nonlinear optimal perturbations

2015 ◽  
Vol 785 ◽  
pp. 135-151 ◽  
Author(s):  
Jan O. Pralits ◽  
Alessandro Bottaro ◽  
Stefania Cherubini
Keyword(s):  
Stokes Equations ◽  
Initial Conditions ◽  
Navier Stokes ◽  
Simple Shear Flow ◽  
Time Interval ◽  
Short Time Interval ◽  
Weakly Nonlinear ◽  
Spatially Extended ◽  
Initial Excitation ◽  
Optimal Disturbances

A simple approach is described for computing spatially extended, weakly nonlinear optimal disturbances, suitable for maintaining a disturbance-regeneration cycle in a simple shear flow. Weakly nonlinear optimals, computed over a short time interval for the expansion used to remain tenable, are oblique waves which display a shorter streamwise and a longer spanwise wavelength than their linear counterparts. Threshold values of the initial excitation energy, separating the region of damped waves from that where disturbances grow without bounds, are found. Weakly nonlinear optimal solutions of varying initial amplitudes are then fed as initial conditions into direct numerical simulations of the Navier–Stokes equations and it is shown that the weakly nonlinear model permits the identification of flow states which cause rapid breakdown to turbulence.

Global three-dimensional optimal disturbances in the Blasius boundary-layer flow using time-steppers

2010 ◽  
Vol 650 ◽  
pp. 181-214 ◽  
Author(s):  
ANTONIOS MONOKROUSOS ◽  
ESPEN ÅKERVIK ◽  
LUCA BRANDT ◽  
DAN S. HENNINGSON
Keyword(s):  
Boundary Layer ◽  
Boundary Layer Flow ◽  
Wave Packets ◽  
Three Dimensional ◽  
Computational Domain ◽  
Initial Condition ◽  
Oblique Wave ◽  
Layer Flow ◽  
Optimal Disturbances ◽  
Matrix Free

The global linear stability of the flat-plate boundary-layer flow to three-dimensional disturbances is studied by means of an optimization technique. We consider both the optimal initial condition leading to the largest growth at finite times and the optimal time-periodic forcing leading to the largest asymptotic response. Both optimization problems are solved using a Lagrange multiplier technique, where the objective function is the kinetic energy of the flow perturbations and the constraints involve the linearized Navier–Stokes equations. The approach proposed here is particularly suited to examine convectively unstable flows, where single global eigenmodes of the system do not capture the downstream growth of the disturbances. In addition, the use of matrix-free methods enables us to extend the present framework to any geometrical configuration. The optimal initial condition for spanwise wavelengths of the order of the boundary-layer thickness are finite-length streamwise vortices exploiting the lift-up mechanism to create streaks. For long spanwise wavelengths, it is the Orr mechanism combined with the amplification of oblique wave packets that is responsible for the disturbance growth. This mechanism is dominant for the long computational domain and thus for the relatively high Reynolds number considered here. Three-dimensional localized optimal initial conditions are also computed and the corresponding wave packets examined. For short optimization times, the optimal disturbances consist of streaky structures propagating and elongating in the downstream direction without significant spreading in the lateral direction. For long optimization times, we find the optimal disturbances with the largest energy amplification. These are wave packets of Tollmien–Schlichting waves with low streamwise propagation speed and faster spreading in the spanwise direction. The pseudo-spectrum of the system for real frequencies is also computed with matrix-free methods. The spatial structure of the optimal forcing is similar to that of the optimal initial condition, and the largest response to forcing is also associated with the Orr/oblique wave mechanism, however less so than in the case of the optimal initial condition. The lift-up mechanism is most efficient at zero frequency and degrades slowly for increasing frequencies. The response to localized upstream forcing is also discussed.

scholarly journals Various approaches to determine active regions in an unstable global mode: application to transonic buffet

2019 ◽  
Vol 881 ◽  
pp. 617-647 ◽  
Author(s):  
Edoardo Paladini ◽  
Olivier Marquet ◽  
Denis Sipp ◽  
Jean-Christophe Robinet ◽  
Julien Dandois
Keyword(s):  
Boundary Layer ◽  
Flow Field ◽  
Stokes Equations ◽  
Active Regions ◽  
Physical Models ◽  
Computational Domain ◽  
Global Mode ◽  
Pass Filter ◽  
Low Pass Filter ◽  
Transonic Buffet

The transonic flow field around a supercritical airfoil is investigated. The objective of the present paper is to enhance the understanding of the physical mechanics behind two-dimensional transonic buffet. The paper is composed of two parts. In the first part, a global stability analysis based on the linearized Reynolds-averaged Navier–Stokes equations is performed. A recently developed technique, based on the direct and adjoint unstable global modes, is used to compute the local contribution of the flow to the growth rate and angular frequency of the unstable global mode. The results allow us to identify which zones are directly responsible for the existence of the instability. The technique is firstly used for the vortex-shedding cylinder mode, as a validating case. In the second part, in order to confirm the results of the first part, a selective frequency damping method is locally used in some regions of the flow field. This method consists of applying a low-pass filter on selected zones of the computational domain in order to damp the fluctuations. It allows us to identify which zones are necessary for the persistence of the instability. The two different approaches give the same results: the shock foot is identified as the core of the instability; the shock and the boundary layer downstream of the shock are also necessary zones while damping the fluctuations on the lower surface of the airfoil; and outside the boundary layer between the shock and the trailing edge or above the supersonic zone does not suppress the shock oscillation. A discussion on the several physical models, proposed until now for the buffet phenomenon, and a new model are finally offered in the last section.

Direct numerical simulation of a separated turbulent boundary layer

1998 ◽  
Vol 374 ◽  
pp. 379-405 ◽  
Author(s):  
Y. NA ◽  
P. MOIN
Keyword(s):  
Numerical Simulation ◽  
Boundary Layer ◽  
Shear Stress ◽  
Turbulent Boundary Layer ◽  
Direct Numerical Simulation ◽  
Shear Layer ◽  
Stokes Equations ◽  
Pressure Fluctuations ◽  
Shear Stresses ◽  
Computational Domain

A separated turbulent boundary layer over a flat plate was investigated by direct numerical simulation of the incompressible Navier–Stokes equations. A suction-blowing velocity distribution was prescribed along the upper boundary of the computational domain to create an adverse-to-favourable pressure gradient that produces a closed separation bubble. The Reynolds number based on inlet free-stream velocity and momentum thickness is 300. Neither instantaneous detachment nor reattachment points are fixed in space but fluctuate significantly. The mean detachment and reattachment locations determined by three different definitions, i.e. (i) location of 50% forward flow fraction, (ii) mean dividing streamline (ψ=0), (iii) location of zero wall-shear stress (τw=0), are in good agreement. Instantaneous vorticity contours show that the turbulent structures emanating upstream of separation move upwards into the shear layer in the detachment region and then turn around the bubble. The locations of the maximum turbulence intensities as well as Reynolds shear stress occur in the middle of the shear layer. In the detached flow region, Reynolds shear stresses and their gradients are large away from the wall and thus the largest pressure fluctuations are in the middle of the shear layer. Iso-surfaces of negative pressure fluctuations which correspond to the core region of the vortices show that large-scale structures grow in the shear layer and agglomerate. They then impinge on the wall and subsequently convect downstream. The characteristic Strouhal number St=fδ*in/U0 associated with this motion ranges from 0.0025 to 0.01. The kinetic energy budget in the detachment region is very similar to that of a plane mixing layer.

Recent developments in hydrodynamic stability of two-phase flows

2012 ◽  
Vol 9 (1) ◽  
pp. 125-130
Author(s):  
A.N. Osiptsov ◽  
S.A. Boronin
Keyword(s):  
Boundary Layer ◽  
Particle Concentration ◽  
Volume Fraction ◽  
Time Interval ◽  
Dusty Gas ◽  
Two Phase Flows ◽  
Two Phase ◽  
Layer Width ◽  
Dispersed Flows ◽  
Optimal Disturbances

In the framework of two-continuum model, the stability of plane-parallel dispersed flows is analyzed. Several flow configurations are considered and several new factors are analyzed. The factors include: particle velocity slip and particle concentration non-uniformity in the main flow, non-Stokesian components of the interphase force and finite volume fraction of the dispersed phase. It is found that the new factors modify significantly the parameters of the fastest growing mode and change the critical Reynolds number of two-phase flows. A method for studying algebraic (non-modal) instability and optimal disturbances to dispersed flows is proposed. While studying the non-modal instability of the dusty-gas boundary-layer flow with a non-uniform particle concentration, we found that the disturbances with the maximum energy gain at a limited time interval are streamwise-elongated structures (streaks). As compared to the flow of a particle-free fluid, optimal disturbances to the dusty-gas flow gain much larger kinetic energy even at the boundary layer width-averaged mass concentration of ten percent, which leads to significant amplification of non-modal instability mechanism due to the presence of suspended particles.

scholarly journals On self-sustained oscillations in two-dimensional compressible flow over rectangular cavities

2002 ◽  
Vol 455 ◽  
pp. 315-346 ◽  
Author(s):  
CLARENCE W. ROWLEY ◽  
TIM COLONIUS ◽  
AMIT J. BASU
Keyword(s):  
Boundary Layer ◽  
Shear Layer ◽  
Stokes Equations ◽  
Boundary Layer Transition ◽  
Computational Domain ◽  
Two Dimensional ◽  
Mean Flow ◽  
Recirculating Flow ◽  
Layer Mode ◽  
Mach Numbers

Numerical simulations are used to investigate the resonant instabilities in two-dimensional flow past an open cavity. The compressible Navier–Stokes equations are solved directly (no turbulence model) for cavities with laminar boundary layers upstream. The computational domain is large enough to directly resolve a portion of the radiated acoustic field, which is shown to be in good visual agreement with schlieren photographs from experiments at several different Mach numbers. The results show a transition from a shear-layer mode, primarily for shorter cavities and lower Mach numbers, to a wake mode for longer cavities and higher Mach numbers. The shear-layer mode is characterized well by the acoustic feedback process described by Rossiter (1964), and disturbances in the shear layer compare well with predictions based on linear stability analysis of the Kelvin–Helmholtz mode. The wake mode is characterized instead by a large-scale vortex shedding with Strouhal number independent of Mach number. The wake mode oscillation is similar in many ways to that reported by Gharib & Roshko (1987) for incompressible flow with a laminar upstream boundary layer. Transition to wake mode occurs as the length and/or depth of the cavity becomes large compared to the upstream boundary-layer thickness, or as the Mach and/or Reynolds numbers are raised. Under these conditions, it is shown that the Kelvin–Helmholtz instability grows to sufficient strength that a strong recirculating flow is induced in the cavity. The resulting mean flow is similar to wake profiles that are absolutely unstable, and absolute instability may provide an explanation of the hydrodynamic feedback mechanism that leads to wake mode. Predictive criteria for the onset of shear-layer oscillations (from steady flow) and for the transition to wake mode are developed based on linear theory for amplification rates in the shear layer, and a simple model for the acoustic efficiency of edge scattering.

On the breakdown of boundary layer streaks

2001 ◽  
Vol 428 ◽  
pp. 29-60 ◽  
Author(s):  
PAUL ANDERSSON ◽  
LUCA BRANDT ◽  
ALESSANDRO BOTTARO ◽  
DAN S. HENNINGSON
Keyword(s):  
Boundary Layer ◽  
Flat Plate ◽  
Travelling Waves ◽  
Base Flow ◽  
Transition To Turbulence ◽  
Mean Flow ◽  
Critical Amplitude ◽  
Flow Modification ◽  
Streamwise Streaks ◽  
Optimal Disturbances

A scenario of transition to turbulence likely to occur during the development of natural disturbances in a flat-plate boundary layer is studied. The perturbations at the leading edge of the flat plate that show the highest potential for transient energy amplification consist of streamwise aligned vortices. Due to the lift-up mechanism these optimal disturbances lead to elongated streamwise streaks downstream, with significant spanwise modulation. Direct numerical simulations are used to follow the nonlinear evolution of these streaks and to verify secondary instability calculations. The theory is based on a linear Floquet expansion and focuses on the temporal, inviscid instability of these flow structures. The procedure requires integration in the complex plane, in the coordinate direction normal to the wall, to properly identify neutral modes belonging to the discrete spectrum. The streak critical amplitude, beyond which streamwise travelling waves are excited, is about 26% of the free-stream velocity. The sinuous instability mode (either the fundamental or the subharmonic, depending on the streak amplitude) represents the most dangerous disturbance. Varicose waves are more stable, and are characterized by a critical amplitude of about 37%. Stability calculations of streamwise streaks employing the shape assumption, carried out in a parallel investigation, are compared to the results obtained here using the nonlinearly modified mean fields; the need to consider a base flow which includes mean flow modification and harmonics of the fundamental streak is clearly demonstrated.

scholarly journals Transient growth in the flow past a three-dimensional smooth roughness element

2013 ◽  
Vol 724 ◽  
pp. 642-670 ◽  
Author(s):  
S. Cherubini ◽  
M. D. De Tullio ◽  
P. De Palma ◽  
G. Pascazio
Keyword(s):  
Boundary Layer ◽  
Boundary Layer Flow ◽  
Roughness Element ◽  
Three Dimensional ◽  
Reynolds Numbers ◽  
Base Flow ◽  
Energy Gain ◽  
Roughness Elements ◽  
Layer Flow ◽  
Optimal Disturbances

AbstractThis work provides a global optimization analysis, looking for perturbations inducing the largest energy growth at a finite time in a boundary-layer flow in the presence of smooth three-dimensional roughness elements. Amplification mechanisms are described which can bypass the asymptotical growth of Tollmien–Schlichting waves. Smooth axisymmetric roughness elements of different height have been studied, at different Reynolds numbers. The results show that even very small roughness elements, inducing only a weak deformation of the base flow, can localize the optimal disturbance characterizing the Blasius boundary-layer flow. Moreover, for large enough bump heights and Reynolds numbers, a strong amplification mechanism has been recovered, inducing an increase of several orders of magnitude of the energy gain with respect to the Blasius case. In particular, the highest value of the energy gain is obtained for an initial varicose perturbation, differently to what found for a streaky parallel flow. Optimal varicose perturbations grow very rapidly by transporting the strong wall-normal shear of the base flow, which is localized in the wake of the bump. Such optimal disturbances are found to lead to transition for initial energies and amplitudes considerably smaller than sinuous optimal ones, inducing hairpin vortices downstream of the roughness element.

The behaviour of Tollmien–Schlichting waves undergoing small-scale localised distortions

2016 ◽  
Vol 792 ◽  
pp. 499-525 ◽  
Author(s):  
Hui Xu ◽  
Spencer J. Sherwin ◽  
Philip Hall ◽  
Xuesong Wu
Keyword(s):  
Boundary Layer ◽  
Flat Plate ◽  
Transmission Coefficient ◽  
Stokes Equations ◽  
Base Flow ◽  
Navier Stokes ◽  
Small Scale ◽  
Two Dimensional ◽  
Navier Stokes Equations ◽  
Tollmien Schlichting

This paper is concerned with the behaviour of Tollmien–Schlichting (TS) waves experiencing small localised distortions within an incompressible boundary layer developing over a flat plate. In particular, the distortion is produced by an isolated roughness element located at $\mathit{Re}_{x_{c}}=440\,000$. We considered the amplification of an incoming TS wave governed by the two-dimensional linearised Navier–Stokes equations, where the base flow is obtained from the two-dimensional nonlinear Navier–Stokes equations. We compare these solutions with asymptotic analyses which assume a linearised triple-deck theory for the base flow and determine the validity of this theory in terms of the height of the small-scale humps/indentations taken into account. The height of the humps/indentations is denoted by $h$, which is considered to be less than or equal to $x_{c}\mathit{Re}_{x_{c}}^{-5/8}$ (corresponding to $h/{\it\delta}_{99}<6\,\%$ for our choice of $\mathit{Re}_{x_{c}}$). The rescaled width $\hat{d}~(\equiv d/(x_{c}\mathit{Re}_{x_{c}}^{-3/8}))$ of the distortion is of order $\mathit{O}(1)$ and the width $d$ is shorter than the TS wavelength (${\it\lambda}_{TS}=11.3{\it\delta}_{99}$). We observe that, for distortions which are smaller than 0.1 of the inner deck height ($h/{\it\delta}_{99}<0.4\,\%$), the numerical simulations confirm the asymptotic theory in the vicinity of the distortion. For larger distortions which are still within the inner deck ($0.4\,\%<h/{\it\delta}_{99}<5.5\,\%$) and where the flow is still attached, the numerical solutions show that both humps and indentations are destabilising and deviate from the linear theory even in the vicinity of the distortion. We numerically determine the transmission coefficient which provides the relative amplification of the TS wave over the distortion as compared to the flat plate. We observe that for small distortions, $h/{\it\delta}_{99}<5.5\,\%$, where the width of the distortion is of the order of the boundary layer, a maximum amplification of only 2 % is achieved. This amplification can however be increased as the width of the distortion is increased or if multiple distortions are present. Increasing the height of the distortion so that the flow separates ($7.2\,\%<h/{\it\delta}_{99}<12.8\,\%$) leads to a substantial increase in the transmission coefficient of the hump up to 350 %.

Free-stream coherent structures in growing boundary layers: a link to near-wall streaks

2015 ◽  
Vol 778 ◽  
pp. 451-484 ◽  
Author(s):  
Kengo Deguchi ◽  
Philip Hall
Keyword(s):  
Boundary Layer ◽  
Reynolds Number ◽  
Turbulent Boundary Layer ◽  
Coherent Structures ◽  
Free Stream ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Base Flow ◽  
Interaction Problem ◽  
Near Wall

In a recent paper, Deguchi & Hall (J. Fluid Mech., vol. 752, 2014a, pp. 602–625) described a new kind of exact coherent structure which sits at the edge of an asymptotic suction boundary layer at high values of the Reynolds number $Re$. At a distance $\ln Re$ from the wall, the structure is driven by the fully nonlinear interaction of tiny rolls, waves and streaks convected downstream at almost the free-stream speed. The interaction problem satisfies the unit-Reynolds-number three-dimensional Navier–Stokes equations and is localized in a layer of the same depth as the unperturbed boundary layer. Here, we show that the interaction problem is generic to any boundary layer that approaches its free-stream form through an exponentially small correction. It is shown that away from the layer where it is generated the induced roll–streak flow is dominated by non-parallel effects which now play a major role in the streamwise evolution of the structure. The similarity with the parallel boundary layer case is restricted only to the layer where it is generated. It is shown that non-parallel effects cause the structure to persist only over intervals of finite length in any growing boundary layer and lead to a flow structure reminiscent of turbulent boundary layer simulations. The results found shed light on a possible mechanism to couple near-wall streaks with coherent structures located towards the edge of a turbulent boundary layer. Some discussion of how the mechanism adapts to a three-dimensional base flow is given.

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