scholarly journals An optimal path to transition in a duct

2008 ◽  
Vol 367 (1888) ◽  
pp. 529-544 ◽  
Author(s):  
Damien Biau ◽  
Alessandro Bottaro
Keyword(s):  
Travelling Waves ◽  
Optimal Path ◽  
Optimal Solution ◽  
Optimization Procedure ◽  
Travelling Wave ◽  
Base Flow ◽  
Mean Flow ◽  
Flow Distortion ◽  
Optimal Perturbation ◽  
Optimal Disturbances

This paper is concerned with the transition of the laminar flow in a duct of square cross section. As in the similar case of pipe flow, the motion is linearly stable for all Reynolds numbers, rendering this flow a suitable candidate for a study of the ‘bypass’ path to turbulence. It has already been shown that the classical linear optimal perturbation problem, yielding optimal disturbances in the form of longitudinal vortices, fails to provide an ‘optimal’ path to turbulence, i.e. optimal perturbations do not elicit a significant nonlinear response from the flow. Previous simulations have also indicated that a pair of travelling waves generates immediately, by nonlinear quadratic interactions, an unstable mean flow distortion, responsible for rapid breakdown. By the use of functions quantifying the sensitivity of the motion to deviations in the base flow, the optimal travelling wave associated with its specific defect is found by a variational approach. This optimal solution is then integrated in time and shown to display a qualitative similarity to the so-called ‘minimal defect’, for the same parameters. Finally, numerical simulations of an ‘edge state’ are conducted, to identify an unstable solution that mediates laminar–turbulent transition and relate it to results of the optimization procedure.

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 Stability of zero-pressure-gradient boundary layer distorted by unsteady Klebanoff streaks

2011 ◽  
Vol 681 ◽  
pp. 116-153 ◽  
Author(s):  
NICHOLAS J. VAUGHAN ◽  
TAMER A. ZAKI
Keyword(s):  
Boundary Layer ◽  
Pressure Gradient ◽  
Low Frequency ◽  
Finite Amplitude ◽  
Base Flow ◽  
Streamwise Vorticity ◽  
Mean Flow ◽  
S Wave ◽  
Flow Distortion ◽  
Tollmien Schlichting

The secondary instability of a zero-pressure-gradient boundary layer, distorted by unsteady Klebanoff streaks, is investigated. The base profiles for the analysis are computed using direct numerical simulation (DNS) of the boundary-layer response to forcing by individual free-stream modes, which are low frequency and dominated by streamwise vorticity. Therefore, the base profiles take into account the nonlinear development of the streaks and mean flow distortion, upstream of the location chosen for the stability analyses. The two most unstable modes were classified as an inner and an outer instability, with reference to the position of their respective critical layers inside the boundary layer. Their growth rates were reported for a range of frequencies and amplitudes of the base streaks. The inner mode has a connection to the Tollmien–Schlichting (T–S) wave in the limit of vanishing streak amplitude. It is stabilized by the mean flow distortion, but its growth rate is enhanced with increasing amplitude and frequency of the base streaks. The outer mode only exists in the presence of finite amplitude streaks. The analysis of the outer instability extends the results of Andersson et al. (J. Fluid Mech. vol. 428, 2001, p. 29) to unsteady base streaks. It is shown that base-flow unsteadiness promotes instability and, as a result, leads to a lower critical streak amplitude. The results of linear theory are complemented by DNS of the evolution of the inner and outer instabilities in a zero-pressure-gradient boundary layer. Both instabilities lead to breakdown to turbulence and, in the case of the inner mode, transition proceeds via the formation of wave packets with similar structure and wave speeds to those reported by Nagarajan, Lele & Ferziger (J. Fluid Mech., vol. 572, 2007, p. 471).

scholarly journals Low-frequency sound radiated by a nonlinearly modulated wavepacket of helical modes on a subsonic circular jet

2009 ◽  
Vol 637 ◽  
pp. 173-211 ◽  
Author(s):  
XUESONG WU ◽  
PATRICK HUERRE
Keyword(s):  
Acoustic Field ◽  
Low Frequency ◽  
Base Flow ◽  
Matched Asymptotic Expansion ◽  
Sound Waves ◽  
Mean Flow ◽  
Flow Distortion ◽  
Experimental Conditions ◽  
Modulation Equation ◽  
Instability Modes

A possible fundamental physical mechanism by which instability modes generate sound waves in subsonic jets is presented in the present paper. It involves a wavepacket of a pair of helical instability modes with nearly the same frequencies but opposite azimuthal wavenumbers. As the wavepacket undergoes simultaneous spatial–temporal development in a circular jet, the mutual interaction between the helical modes generates a strong three-dimensional, slowly modulating ‘mean-flow distortion’. It is demonstrated that this ‘mean field’ radiates sound waves to the far field. The emitted sound is of very low frequency, with characteristic time and length scales being comparable with those of the envelope of the wavepacket, which acts as a non-compact source. A matched-asymptotic-expansion approach is used to determine, in a self-consistent manner, the acoustic field in terms of the envelope of the wavepacket and a transfer factor characterizing the refraction effect of the background base flow. For realistic jet spreading rates, the nonlinear development of the wavepacket is found to be influenced simultaneously by non-parallelism and non-equilibrium effects, and so a composite modulation equation including both effects is constructed in order to describe the entire growth–attenuation–decay cycle. Parametric studies pertaining to relevant experimental conditions indicate that the acoustic field is characterized by a single-lobed directivity pattern beamed at an angle about 45°–60° to the jet axis and a broadband spectrum centred at a Strouhal number St ≈ 0.07–0.2. As the nonlinear effect increases, the radiation becomes more efficient and the noise spectrum broadens, but the gross features of the acoustic field remain robust, and are broadly in agreement with experimental observations.

Mach number effects on the global mixing modes induced by ramp injectors in supersonic flows

2014 ◽  
Vol 757 ◽  
pp. 403-431 ◽  
Author(s):  
Luca Massa
Keyword(s):  
Mach Number ◽  
Free Stream ◽  
Mixing Layer ◽  
Three Dimensional ◽  
Base Flow ◽  
Recirculation Region ◽  
Mean Flow ◽  
Mixing Rate ◽  
Flow Distortion ◽  
The Mean

AbstractModern injectors for supersonic combustors (hypermixers) augment the fuel–air mixing rate by energizing the perturbation in the mixing layer. From an instability point of view, the increased perturbation growth is linked to the increased complexity of the equilibrium base flow when compared to the axisymmetric mixing layer. Common added features are streamwise vortex streaks, oblique recompression shocks and Prandtl–Meyer expansions. One of the main effects of such distortions of the mean flow is to transform the instability responsible for the creation of fine scales from a local amplified mode to a global self-sustained fluctuation. The focus of the present research is on the flow distortion induced by flushed ramps for free-stream Mach numbers in the range 2.5–3.5. The principal mean flow features are the recirculation region due to the recompression of the flow after the ramp, the shear layer over the recirculation region and the vortex streaks propagating from the ramp corners. A global three-dimensional stability analysis and three-dimensional direct numerical simulations of small perturbations of the mean flow are performed. The growth and energy distribution of the dominant and subdominant fluctuations supported by the three-dimensional steady laminar base flow are computed. The main results are the growth rates of the self-sustained varicose and sinuous modes and their correlation to the variation in the free-stream Mach number. The complex three-dimensional wavemaker is investigated by evaluating the three-dimensional eigenfunctions of the direct and adjoint modes, and the effects of the axial vorticity generated by the ramp corners are discussed.

scholarly journals Streamwise-travelling waves of spanwise wall velocity for turbulent drag reduction

2009 ◽  
Vol 627 ◽  
pp. 161-178 ◽  
Author(s):  
MAURIZIO QUADRIO ◽  
PIERRE RICCO ◽  
CLAUDIO VIOTTI
Keyword(s):  
Drag Reduction ◽  
Travelling Waves ◽  
Phase Speed ◽  
Travelling Wave ◽  
Wall Turbulence ◽  
Energetic Efficiency ◽  
Friction Drag ◽  
Mean Flow ◽  
The Waves ◽  
Spanwise Velocity

Waves of spanwise velocity imposed at the walls of a plane turbulent channel flow are studied by direct numerical simulations. We consider sinusoidal waves of spanwise velocity which vary in time and are modulated in space along the streamwise direction. The phase speed may be null, positive or negative, so that the waves may be either stationary or travelling forward or backward in the direction of the mean flow. Such a forcing includes as particular cases two known techniques for reducing friction drag: the oscillating wall technique (a travelling wave with infinite phase speed) and the recently proposed steady distribution of spanwise velocity (a wave with zero phase speed). The travelling waves alter the friction drag significantly. Waves which slowly travel forward produce a large reduction of drag that can relaminarize the flow at low values of the Reynolds number. Faster waves yield a totally different outcome, i.e. drag increase (DI). Even faster waves produce a drag reduction (DR) effect again. Backward-travelling waves instead lead to DR at any speed. The travelling waves, when they reduce drag, operate in similar fashion to the oscillating wall, with an improved energetic efficiency. DI is observed when the waves travel at a speed comparable with that of the convecting near-wall turbulence structures. A diagram illustrating the different flow behaviours is presented.

scholarly journals Streamwise vortices in heated boundary layers

1993 ◽  
Vol 252 ◽  
pp. 301-324 ◽  
Author(s):  
Philip Hall
Keyword(s):  
Free Stream ◽  
Travelling Waves ◽  
Streamwise Vortex ◽  
Travelling Wave ◽  
Streamwise Vortices ◽  
Mean Flow ◽  
Small Temperature ◽  
Wave Disturbances ◽  
Stationary Vortex ◽  
The Mean

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 travelling 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 travelling waves. The mean state is shown to be destabilized if the Prandtl number is less than unity.

Travelling Wave Control of Rotating Discs and Analysis of its Spillover Effect

1988 ◽  
Vol 202 (2) ◽  
pp. 119-127 ◽  
Author(s):  
C-S Kim ◽  
C-W Lee
Keyword(s):  
Travelling Waves ◽  
Spillover Effect ◽  
Polynomial Equation ◽  
Travelling Wave ◽  
Rotating Disc ◽  
System A ◽  
Finite Dimensional ◽  
Control Scheme ◽  
Actuator System ◽  
Wave Control

A modal control scheme for rotating disc systems is developed based upon the finite-dimensional sub-system model including a few lower backward travelling waves important to the disc response. For the single discrete sensor and actuator system, a polynomial equation, which determines the closed-loop system poles, is derived and the spillover effect is analysed, providing a sufficient condition for stability. Finally, simulation studies are performed to show the effectiveness of the travelling wave control scheme proposed.

TRAVELLING WAVES AND OSCILLATIONS IN SAL’NIKOV’S COMBUSTION REACTION IN A COMPRESSIBLE GAS

2015 ◽  
Vol 56 (3) ◽  
pp. 233-247 ◽  
Author(s):  
RHYS A. PAUL ◽  
LAWRENCE K. FORBES
Keyword(s):  
Asymptotic Solution ◽  
Multiple Scales ◽  
Travelling Waves ◽  
Method Of Lines ◽  
Travelling Wave ◽  
Reaction Scheme ◽  
Compressible Viscous Gas ◽  
The Method Of Lines ◽  
Temperature Waves ◽  
Parameter Values

We consider a two-step Sal’nikov reaction scheme occurring within a compressible viscous gas. The first step of the reaction may be either endothermic or exothermic, while the second step is strictly exothermic. Energy may also be lost from the system due to Newtonian cooling. An asymptotic solution for temperature perturbations of small amplitude is presented using the methods of strained coordinates and multiple scales, and a travelling wave solution with a sech-squared profile is derived. The method of lines is then used to approximate the full system with a set of ordinary differential equations, which are integrated numerically to track accurately the evolution of the reaction front. This numerical method is used to verify the asymptotic solution and investigate behaviours under different conditions. Using this method, temperature waves progressing as pulsatile fronts are detected at appropriate parameter values.

scholarly journals Influence of the Atmospheric Surface Layer on a Turbulent Flow Downstream of a Ship Superstructure

2013 ◽  
Vol 30 (8) ◽  
pp. 1803-1819 ◽  
Author(s):  
Luksa Luznik ◽  
Cody J. Brownell ◽  
Murray R. Snyder ◽  
Hyung Suk Kang
Keyword(s):  
Surface Layer ◽  
Atmospheric Surface Layer ◽  
The United States ◽  
Naval Academy ◽  
Mean Flow ◽  
Flow Distortion ◽  
Turbulent Statistics ◽  
Sonic Anemometers ◽  
Flight Deck ◽  
Inflow Conditions

Abstract This paper describes a set of turbulence measurements at sea in the area of high flow distortion in the near-wake and recirculation zone behind a ship's superstructure that is similar in geometry to a helicopter hangar/flight deck arrangement found on many modern U.S. Navy ships. The instrumented ship is a 32-m-long training vessel operated by the United States Naval Academy that has been modified by adding a representative flight deck and hangar structure. The flight deck is instrumented with up to seven sonic anemometers/thermometers that are used to obtain simultaneous velocity measurements at various spatial locations on the flight deck, and one sonic anemometer at bow mast is used to characterize inflow atmospheric boundary conditions. Data characterizing wind over the deck at an incoming angle of 0° (head winds) and wind speeds from 2 to 10 m s−1 obtained in the Chesapeake Bay are presented and discussed. Turbulent statistics of inflow conditions are analyzed using the Kaimal universal turbulence spectral model for the atmospheric surface layer and show that for the present dataset this approach eliminates the need to account for platform motion in computing variances and covariances. Conditional sampling of mean flow and turbulence statistics at the flight deck indicate no statistically significant variations between unstable, stable, and neutral atmospheric inflow conditions, and the results agree with the published data for flows over the backward-facing step geometries.

Nonlinear travelling internal waves with piecewise-linear shear profiles

2018 ◽  
Vol 856 ◽  
pp. 984-1013 ◽  
Author(s):  
K. L. Oliveras ◽  
C. W. Curtis
Keyword(s):  
Piecewise Linear ◽  
Travelling Waves ◽  
Tangential Velocity ◽  
Travelling Wave ◽  
Numerical Continuation ◽  
Travelling Wave Solutions ◽  
Water Wave Problem ◽  
Two Fluids ◽  
Wave Solutions ◽  
Linear Shear

In this work, we study the nonlinear travelling waves in density stratified fluids with piecewise-linear shear currents. Beginning with the formulation of the water-wave problem due to Ablowitz et al. (J. Fluid Mech., vol. 562, 2006, pp. 313–343), we extend the work of Ashton & Fokas (J. Fluid Mech., vol. 689, 2011, pp. 129–148) and Haut & Ablowitz (J. Fluid Mech., vol. 631, 2009, pp. 375–396) to examine the interface between two fluids of differing densities and varying linear shear. We derive a systems of equations depending only on variables at the interface, and numerically solve for periodic travelling wave solutions using numerical continuation. Here, we consider only branches which bifurcate from solutions where there is no slip in the tangential velocity at the interface for the trivial flow. The spectral stability of these solutions is then determined using a numerical Fourier–Floquet technique. We find that the strength of the linear shear in each fluid impacts the stability of the corresponding travelling wave solutions. Specifically, opposing shears may amplify or suppress instabilities.

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