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Authors

Yury S. Kachanov, Khristianovich Institute of Theoretical and Applied Mechanics SB RAS, Novosibirsk, 630090
Vladimir I. Borodulin, Khristianovich Institute of Theoretical and Applied Mechanics SB RAS, Novosibirsk, 630090
Andrey V. Ivanov, Khristianovich Institute of Theoretical and Applied Mechanics SB RAS, Novosibirsk, 630090

Abstract

Investigations of mechanisms of transformation of vortical freestream disturbances into cross-flow (CF) instability vortices and waves are very important for subsequent development of modern methods of prediction of the laminar-turbulent transition location in three-dimensional boundary layers (on swept wings, blades of turbomachines, etc.). At the same time, the amount of quantitative experimental studies, devoted to these mechanisms, is still very restricted. The present experimental investigation is a continuation and subsequent development of studies [1-3]. Problems of the localized vortical receptivity of a swept-wing boundary layer were examined previously for cases of scattering of freestream vortices on surface roughness [1] and surface vibrations [2]. The distributed vortical receptivity at excitation of the CF-waves by streamwise freestream vortices on a smooth surface was investigated in [3]. In contrast to these experiments, the surface was not smooth in the present study; it had several kinds of controlled two-dimensional (i.e. spanwise-uniform) waviness.

The experiments were carried out in a low-turbulence wind tunnel on a model of a swept wing with sweep angle of 25 degree at freestream speed of about 12 m/s. The measurements were performed at fully controlled disturbance conditions when both the surface non-uniformities and the freestream vortices were excited by specially developed techniques. To simulate the surface non-uniformities, special films of a variable thickness (wavy in the chordwise direction and uniform in the spanwise direction) were glued onto the surface. The streamwise freestream vortices were generated by a vibrating wire, which was placed upstream from the experimental model, was parallel to the leading edge and was equipped with a special small-scale nonuniformity.

Based on the developed technique of approximation of experimental distributions by analytical solutions of evolutionary equations for distributed excitation of CF-modes, values of the coefficients of the distributed vortex-roughness receptivity are obtained experimentally for the first time. Dependences of these coefficients on the spanwise wavenumber (and the angle of propagation of CF-waves) and on the disturbance frequency are investigated.

This study is supported by the Boeing Operations International, Inc. and by the Russian Academy of Sciences. 

Borodulin V.I., Ivanov A.V., Kachanov Y.S., Roschektayev A.P. J. Fluid Mech., 716:487–527, 2013. 

Borodulin V.I., Ivanov A.V., Kachanov Y.S., Roschektayev A.P. J. Fluid Mech., 793:162–208, 2016.

Borodulin V.I., Ivanov A.V., Kachanov Y.S. In: XV Intl. Conf. on Methods of Aerophysical Research. Proc / Ed. V.M. Fomin – Novosibirsk: ITAM SB RAS 2010, 10 pp.

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Authors

Anatoly I. Ruban, Imperial College London
Marina A. Kravtsova, Imperial College London
Sharad Keshari, Imperial College London

Abstract

In compressible flows, the free-stream turbulence is composed of the vorticity and
entropy waves. In fact, one can show, based on the Euler equations, that any small amplitude
perturbation to the uniform flow may be represented as a superposition of three
modes: acoustic waves, vorticity waves and entropy waves. The receptivity of the
boundary layer to acoustic noise and to vorticity waves has been studied extensively
by various authors. The entropy waves did not attract such attention. In this
presentation, we will show that in high speed subsonic and transonic flows, the
entropy waves are as important for the receptivity as the vorticity waves

The entropy waves produce oscillations of the gas temperature and density, but
the velocity and the pressure remain unperturbed. This precludes the entropy
waves from penetrating the boundary layer, as this happens, for example, with
acoustic waves. However, when the boundary layer encounters a wall roughness,
the flow near the roughness appears to be perturbed. The steady flow perturbations
induced by the roughness are not confined to the boundary layer but also extend
to the inviscid flow region outside the boundary layer. The latter come into
interaction with the density perturbations in the entropy wave. As a result, a
localised `forcing' is created that is periodic in time.

When analysing the response of the boundary layer to this forcing, we relied on
the so-called `double resonance' principle which states that an efficient generation
of the Tollmien--Schlichting waves is observed when, first, the frequency of the
entropy wave is close to the frequency of the generated Tollmien--Schlichting
wave, and, second, the longitudinal size of the wall roughness is of the same order
as the wave length of the Tollmien--Schlichting wave. Under these two
conditions, the triple-deck theory is applicable to describe the receptivity process.
For linear receptivity, the triple-deck equations can be solved analytically both for
subsonic and transonic flow regimes. When dealing with the non-linear receptivity,
we had to solve these equations numerically. We found that the non-linearity
enhances the receptivity significantly, especially when a local separation region
forms near the roughness.

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Authors

Simon Schmidt, Technische Universität Berlin
Kilian Oberleithner, Technische Universität Berlin

Abstract

The destabilization and breakup of liquid jets is relevant to a broad range of industrial applications, including ink-jet printers or fuel injection in e.g. diesel engines or gas turbines. In such cases a high velocity liquid stream is injected into a still gaseous environment through a high-pressure nozzle. The resulting two-fluid system is subjected to extrinsic and intrinsic mechanisms of destabilization. Extrinsic factors for instance, are given by pressure fluctuations in the supplied liquid stream or forced oscillatory movement of the jet within the nozzle. Forced destabilization can be achieved with the use of a fluidic oscillator, to create a spatially or temporarily oscillating jet.
In this talk we analyze the instability growth and development of coherent structures in a weakly forced laminar liquid jet in a still gaseous environment by means of spectral decomposition of the underlying two-fluid velocity field. Its full unsteady representation is obtained via two-dimensional fully nonlinear numerical simulation of a planar liquid jet, forced harmonically at the domain inlet.
Depending on the forcing frequency and amplitude the destabilization of the jet is dominated by the growth of the fundamental wave or higher harmonic waves during downstream convection of the disturbance. The influence of higher harmonic growth manifests in an increasingly complex topology of the liquid surface, which can be explained with the increase of baroclinic torque along the liquid/gas interface. The subsequent shear-layer separation in the gaseous phase leads to vortex roll-up and the development of complex vortical structures, which potentially feed turbulent fluctuations in the late stage of destabilization.
The stability of the investigated jets is further assessed using a mean flow approach to local linear stability analysis. Both phases are discretized separately and are coupled along the interface. The spreading of the mean flow raises necessity to introduce a downstream correction of the mean interface position. For low amplitude forcing where the flow is dominated by growth of the fundamental wave the linear stability model gives credible results and the onset of saturation is successfully predicted using the mean flow. For larger forcing amplitudes, where the influence of higher harmonic waves is pronounced, visible deviations between the linear model and the fully nonlinear flow representation are exhibited.

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Authors

Marco Placidi, University of Surrey
Michael Gaster, City, University of London
Chris Atkin, City, University of London

Abstract

Maintaining natural or hybrid laminar flow over an aircraft wing offers the tantalising prospect of minimising fuel consumption and pollutant emissions thanks to to the well-known potential reduction in the skin friction drag. However, there are many processes that affect the laminar/turbulent transition that are not yet fully understood. In this presentation, we focus on the receptivity of two-dimensional boundary layers to acoustic fields and two-dimensional roughness. In the presence of a freestream oscillatory component (usually due to an acoustic field), the flow develops a Stokes Layer (SL), which is superimposed on the Tollmien-Schlichting (T-S) wave excited by the receptivity process. The challenge, in experimental work, is to be able to measure the disturbance component in isolation and hence characterise the flow instability. 
We performed wind tunnel experiments on the receptivity of T-S instability to acoustic disturbances and roughness in our low-turbulence wind tunnel on a flat plate model. To recreate the conditions described above, the freestream was subjected to a plain acoustic wave so to generate a Stokes Layer, which is superimposed on the amplitude of the existing T-S disturbances. 
In the course of this presentation, we intend to propose a novel approach to measure the T-S component amplitude; this is based on a vanishing 2D roughness technique, which is controlled during the data acquisition. Our technique allows for the exploration of both protuberances and cavities in the depth range −750μm < h < 750μm. These depths are designed to cover both the predicted linear and non-linear amplification regimes of the T-S instability. Some of these imperfections were experimentally never explored before due to limitations of previous work. 
It will be shown that we are able to successfully measure within the linear regime of the instability for both positive and negative roughness heights. We will also highlight the link between the severity of the departure from the linear regime and the geometry of the surface imperfection, attempting to link these findings with the flow physics in the near field of the roughness.

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Authors

Henrique Raposo, Airbus, Imperial College London
Shahid Mughal, Imperial College London
Richard Ashworth, Airbus

Abstract

The generation of the first-mode instability through scattering of an acoustic wave by localised surface roughness, suction or heating is studied with the time-harmonic adjoint linearised compressible Navier-Stokes equations. Often in the literature for subsonic Mach numbers (i.e. M<1), acoustic receptivity studies are restricted to downstream travelling waves, together with the assumption that the acoustic wavelength remains large compared to the boundary layer thickness. Under such conditions, the usual boundary layer assumptions can be introduced to model the forced response of the boundary layer to plane acoustic waves leading to a parabolic set of governing equations, the so-called linearised unsteady boundary layer equations (LUBLE).

In this work we remove the assumption that the acoustic wave travels in a single direction and investigate the validity of the LUBLE across the Mach number and angle of incidence parameter space. For near upstream travelling waves at high subsonic Mach numbers the comparatively short wavelengths of the acoustic waves cause sizeable pressure oscillations across the boundary layer, thus invalidating the LUBLE. In such cases, the LUBLE should be replaced by a more accurate model of the acoustic signature within the boundary layer. We explore the use of the inviscid compressible linear stability equations (ICLSE) as an alternative model. Firstly we show how the so-called reflection coefficient can be determined independently of the full flow field profiles. The sum of the incident and reflected acoustic waves then drives the unsteady Stokes motion within the boundary layer, for which a double-layer high-Strouhal asymptotic solution is obtained. The inner-layer solution is introduced to satisfy the no-slip condition, violated by the ICLSE.

Using these two distinct models of the acoustic field, we proceed to investigate the influence of the acoustic wave angle of incidence on the T-S wave amplitude triggered by a localised roughness element. When the Reynolds number at the location of the roughness element is sufficiently high, very good agreement is obtained between the LUBLE and ICLSE-based receptivity models for moderate Mach numbers and downstream propagating waves. At the other end of the parameter space, i.e. for near upstream propagating waves and high subsonic Mach numbers, clear differences between the two arise. The ICLSE-based model is in close qualitative agreement with published results based on the full forced linear stability equations.

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Authors

Samuele Viaro, University of Sheffield
Pierre Ricco, University of Sheffield

Abstract

The evolution of unsteady Gortler vortices generated by free-stream disturbances
in a compressible boundary layer over a concave wall is investigated theoretically
and numerically.
The effect of the curvature, Mach number, frequency, and wavelengths are presented
through a rigorous initial boundary value framework derived from the full compressible
Navier-Stokes equations.
This receptivity analysis accounts for the forcing of the oncoming free-stream vortical
disturbances, which is synthesized through appropriate initial and boundary conditions
derived from matched asymptotic expansions. An eigenvalue approach and a further
asymptotic analysis in the limit of large Gortler number reveal the different stages
of the downstream evolution of the flow.
The vortices tend to move toward the core of the boundary layer when the flow
is more stable, i.e., as the frequency and the Mach number increase, or curvature decreases.
The more unstable the flow, the lower the streamwise wavelength of the Gortler vortices
with respect to the free-stream streamwise wavelength.
The growth rates obtained through the receptivity and the eigenvalue approaches agree
sufficiently downstream where the free-stream vortical disturbances have fully decayed,
but the amplitude at any distance and the correct initial growth can only be computed
via the receptivity approach.
The neutral curves of the Gortler-vortex flow, i.e., curves that distinguish the perturbation
flow conditions of growth and decay, are also computed for different Gortler numbers,
wavelengths, low frequencies, and Mach numbers. The perturbations are defined as Klebanoff
modes, low-frequency oblique Tollmien-Schlichting (TS) waves, or Gortler vortices depending
on their growth rate. The critical Gortler number below which the inviscid instability
due to the curvature never occurs is obtained and the conditions for which only Klebanoff modes
or TS waves exist are thus revealed.
A triple-deck analysis proves the negligible influence of mild curvature on the TS-waves
growth.
Finally, the theoretical results explain the physics behind direct numerical simulation findings
in the hypersonic regime and are compared with the available experimental data in the
supersonic regime.

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Authors

Hans Peter Barth, DLR, Institute of Aerodynamics and Flow Technology
Stefan Hein, DLR, Institute of Aerodynamics and Flow Technology

Abstract

Research into laminar-turbulent transition delay in three-dimensional boundary layers is motivated by the possible drag reduction of aircraft with swept wings. Saric et al. discovered a nonlinear transition-delaying mechanism which is now known as “Upstream Flow Deformation“ (UFD). Because the effect can be triggered by artificially exciting an exponentially amplified instability mode, it has the potential to be used as an energetically very efficient method for boundary-layer stabilization compared to approaches such as distributed suction and other actuation concepts aimed at reducing the crossflow component of the laminar boundary layer.
Saric et al. used discrete roughness elements to artificially excite a stationary instability mode as control mode. Naturally, the stabilizing nonlinear effects strongly depend on initial conditions which motivates the research into actuation concepts better suited for controlling them. Therefore, different actuation methods with a controllable excitation amplitude have already been investigated, such as discrete pneumatic and plasma actuation. In the case of pneumatic actuation, the UFD effect has been successfully employed for transition delay, while no such successful attempt with a plasma actuator is known to the authors.
In the present work, an alternative actuation concept based on spanwise periodic surface heating and its effect on the boundary-layer flow field were investigated using the redesigned DLR swept-flat plate experiment at a Reynolds number of Re=1.6*10⁶ in the 1MG wind tunnel in Göttingen.
An actuator has been developed which can create a spanwise periodic wall-temperature distribution at 8 different chordwise positions to introduce an initial disturbance into the boundary-layer flow field. Hot-wire anemometry in the boundary layer downstream has shown the successful excitation of the control mode and quantified its amplitude depending on the actuation position and the amplitude of the surface-temperature periodicity.
For a selected combination of actuation position and amplitude, the impact on the boundary layer downstream of the actuator was investigated in detail. The unsteady boundary-layer flow field was sampled at 14 chordwise, 60 spanwise and 30 wallnormally distributed positions for a duration of 4 seconds each with a sampling rate of 32 kHz. This allows for the quantitative characterization of the relevant stationary and non-stationary phenomena across the complete transition scenario, including the onset of secondary instabilities and subsequent transition itself. Particular attention is paid in the presentation to the development of travelling crossflow instabilities because their inadvertent elevation by other actuation methods has been known to cause the UFD method to fail.

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Authors

Jonathan Morrison, Imperial College
Hari Vemuri, Bristol University
Richard Bosworth, Oliver Wyman
Eric Kerrigan, Imperial College

Abstract

This experiment focusses on transition delay where a TS wave packet is generated through surface receptivity to free-stream vorticity. Although deterministic approaches to attenuate TS waves with open-loop control have been shown to provide good attenuation, they lack the essential robustness required to adapt to inherent uncertainty. Therefore, only a stable, real-time, closed-loop system can provide a realistic and practical control scheme. In recent years, there have been some significant advances in adapting modern robust control to fluid flows and especially to the attenuation of TS waves using adaptive and robust control techniques. In this work, we attenuate the growth of 2D TS waves using a single-input and single-output (SISO) feedback system, using an oscillating ribbon placed above the Blasius boundary layer. The aim is to investigate the performance of the SISO control system in attenuating single-frequency, two-dimensional disturbances, as well as wave packets generated by these configuations. We obtain the necessary plant models using system identification, then design controllers based on the obtained models and implement them in real-time to test their performance. Measurements are performed in a tunnel with free-stream turbulence intensity of less than 0:05%. The flat plate has an asymmetric modified super-elliptic leading edge, designed to minimise any disturbance generated by a discontinuity in surface curvature at the termination of the leading-edge. A zero pressure gradient along the plate is ensured by the use of a trailing-edge control flap. A surface-mounted hot wire downstream of the actuator provides the difference of streamwise velocity on which the controller is based. The Orr-Sommerfeld equation is fourth-order in space, so that there is no direct way in which a numerical model may be obtained. The controller design is based on an experimentally dertermined frequency response. The near-field is avoided because the initial disturbance that gives rise to the TS wave packet decays with distance: this is not modelled. The near field is minimised by placing two actuator slots side by side such that the upstream wall-mounted hot wire is immediately downstream of the intervening gap. The actuator operates approximately 300 mm downstream of the source, the feedback loop operating between the actuator and the wall-mounted hot wire a further 15 mm downsteam. A Nyquist stability plot is used to optimise the controller gain in closed loop while still ensuring stability. A second-order controller is used for maximum attentuation: the rms streamwise velocity fluctuation is reduced by 35% over a significant domain.

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Authors

Onofrio Semeraro, Laboratoire d'informatique pour la mécanique et les sciences de l'ingénieur (LIMSI) - CNRS
Michele Alessandro Bucci, Laboratoire d'informatique pour la mécanique et les sciences de l'ingénieur (LIMSI) - CNRS
Alexandre Allauzen, Laboratoire d'informatique pour la mécanique et les sciences de l'ingénieur (LIMSI) - CNRS
Guillaume Wisniewski, Laboratoire d'informatique pour la mécanique et les sciences de l'ingénieur (LIMSI) - CNRS
Laurent Cordier, Institut Pprime, CNRS, Universite’ de Poitiers, ENSMA
Lionel Mathelin, Laboratoire d'informatique pour la mécanique et les sciences de l'ingénieur (LIMSI) - CNRS

Abstract

In the last decades, numerous research efforts have been devoted to the application of control theory in fluid flows. A rather common approach consists in obtaining reduced order representations of the original system, most of the time within the limit of linear dynamics, for the design of the controllers. Once the reduced-order model is available, several strategies have been tested, ranging from the Linear Quadratic Gaussian (LQG) controllers, to the class of adaptive and Model Predictive Controllers (MPC). Many of these strategies are based on minimization of an objective function and are characterized by rather different levels of robustness. Despite some successes, these control techniques usually exhibit limitations due to the off-line design as well as the neglected nonlinearities.

We here propose a shift of paradigm based on the application of Deep Reinforcement Learning (DRL) for the control of fluid flows. This strategy is quickly spreading in the machine learning community as a viable strategy for applying nonlinear optimal control. In fact, the origin of DRL can be traced back to the generalization of the optimal control problem within the nonlinear framework, leading – in the continuous formulation – to the Hamilton-Jacobi-Bellman (HJB) equation. Within this framework, the only a-priori requirement is the definition of an instantaneous reward as measure of the relevance of an action when the system is in a given state. The objective function to be maximized is then defined as the expectation of the cumulative rewards in the limit of an infinite future time horizon, the so-called action-value. This function, associating to each state of the system the value of the objective function, is here approximated with neural networks, of which recent applications have determined the success of the DRL.

We are successfully testing DRL algorithms for the control of the Kuramoto-Sivashinsky (KS) equation in one-dimension, by means of localized actuation, partial knowledge of the state relying on sensor measurements and model-free DRL controllers. We consider a chaotic regime, in combination with a setup that closely resembles realistic settings (limited number of sensors and actuators). We are also analyzing the physical action of this control strategy in the state space of the full nonlinear system under consideration. In the final contribution, we will show how this strategy can be used with fluid flow configurations such as the flow over an open cavity or around a NACA profile.

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Authors

George Papadakis, Imperial College
Dandan Xiao, University of Nottingham

Abstract

The work considers the nonlinear optimal control of transition in a boundary layer flow subjected to a pair of free-stream vortical perturbations using a receding horizon approach. The optimal control problem is solved using the Lagrange variational technique that results in a set of linearised adjoint equations, which are used to obtain the optimal wall actuation (blowing and suction from a control slot located in the transition region). The receding horizon approach enables the application of control action over a long time period, and this allows the extraction of time-averaged statistics as well as investigation of the control effect downstream of the control slot. The results show that the controlled flow energy is initially reduced in the streamwise direction and then increased because transition still occurs. The distribution of the optimal control velocity responds to the flow activity above and upstream of the control slot. The control effect propagates downstream of the slot and the flow energy is reduced up to the exit of the computational domain. The mean drag reduction is 55% and 10% in the control region and downstream of the slot, respectively. The control mechanism is investigated by examining the second order statistics and the two-point correlations. It is found that in the upstream (left) side of the slot, the controller counteracts the near wall high speed streaks and reduces the turbulent shear stress; this is akin to opposition control in channel flow, and because the time-average control velocity is positive, it is more similar to blowing-only opposition control. In the downstream (right) side of the slot, the controller reacts to the impingement of turbulent spots that have been produced upstream and inside the boundary layer (top-bottom mechanism). The control velocity is positive and increases in the streamwise direction, and the flow behavior is similar to that of uniform blowing. 

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Authors

Louis Cattafesta, Florida State University
Ross Richardson, Florida State University
Adam Edstrand, Florida State University, Sandia National Laboratory
Yiyang Sun, Florida State University, University of Minnesota
Kunihiko Taira, Florida State University, UCLA
Peter Schmid, Imperial College

Abstract

We present an investigation of physics-based attenuation of a trailing vortex.  Trailing vortices have many adverse effects in both aeronautical and maritime applications, including increased drag, wake-hazard, vortex detection and noise. Unfortunately, designing an effective control scheme for such complex three-dimensional flow fields proves to be non-trivial.  Intuitive control strategies often lead to suboptimal control. In this research, we utilize parabolized stability analysis to guide the design of an effective control scheme for a trailing vortex flow field aft of a NACA0012 half-wing at an angle of attack of 5 degrees and a chord-based Reynolds number Re = 1000.  The stability results show that the unstable mode with the smallest growth rate (fifth wake mode) provides a pathway to excite a vortex instability, whereas the principal unstable mode does not. Inspired by this finding, we perform direct numerical simulations that excite each mode with body forces matching the shape function from the stability analysis. Relative to the uncontrolled case, the controlled flows show increased attenuation of circulation and peak streamwise vorticity, with the fifth-mode-based control set-up outperforming the principal-mode-based set-up. Finally, we discuss progress on the implementation of this physics-based active control system to excite the wake instabilities along the trailing edge of the wing surface in wind tunnel experiments over a range of flow conditions using a variety of flow control actuation schemes.

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Authors

Pierluigi Morra, KTH Royal Institute of Technology
Kenzo Sasaki, Instituto Tecnologico de Aeronautica
Ardeshir Hanifi, KTH Royal Institute of Technology
André V. G. Cavalieri, Instituto Tecnologico de Aeronautica
Dan S. Henningson, KTH Royal Institute of Technology

Abstract

Laminar-turbulent transition delay is of great interest because the laminar flow state is characterized by lower friction drag than the turbulent one, which implies less energy consumption for many applications, such as transportation means like trains and aircrafts. The transition to turbulence of laminar boundary-layer (BL) flows is usually due to growth and breakdown of initially small perturbations. Thus, aiming for damping their growth can be an efficient method to avoid or delay transition to turbulence. The current work presents a realizable way to control streaky disturbances in BL flows and delay transition to turbulence by means of active flow control. Numerical simulations of the nonlinear transitional regime in a Blasius BL are performed. Streaky structures are excited in the BL by means of a high level of free-stream turbulence (FST), and the generated disturbances are then sensed by means of localized wall-shear-stress sensors and controlled using near-wall actuators that model plasma actuators. Each actuator is powered by a time-varying signal whose amplitude is computed by processing signals from the sensors. The actuation signal is computed by means of the Linear Quadratic Gaussian regulator (LQG) and the Inverse Feed-Forward Control technique. The use of the LQG requires a state-space representation of the system dynamics, so the flow is described by means of a linear time-invariant operator that captures only the most relevant information of the dynamics and results in a Reduced Order Model (ROM). The ROM is computed by means of the Eigensystem Realization Algorithm, which is based on the impulse responses of the real system. Collecting such impulse responses may be unfeasible when the source of disturbances is FST because of the high number of states needed for a precise description of the phenomenon. In order to determine the relevant system dynamics and generate the needed impulse responses a new method is proposed. The method is based on system identification of FST-induced disturbances in the upstream BL. The effectiveness of the approach in delaying bypass transition is shown. The work presents a systematic way to deal with high dimensional disturbances in order to build ROMs for a feasible application of control theory techniques. Moreover, it is shown that even when considering practical constraints, such as the type and size of actuators and sensors, it is possible to achieve at least as large transition delay as that obtained in more idealized cases found in literature.

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Authors

Tariq Saeed, Imperial College, Arup
Jonathan Morrison, Imperial College

Abstract

A crossflow experiment, utilising boundary-layer hot-wire anemometry, has been performed for a 40° swept-back wing in a wind tunnel with turbulence intensity of approximately 0.13%, at chord-Re=10^6. The effects of a three-dimensional roughness array and a two-dimensional strip on the initiation and development of crossflow instabilities were studied, by varying: the heights of the three-dimensional roughness, and both the height and chordwise location of the roughness strip. The results include roughness receptivity coefficients, disturbance amplitude evolution, contours of mean and fluctuating velocity across a spanwise region, power spectral density plots of fluctuating mean and unsteady velocities, and two-wire cross-correlation measurements.

Receptivity to three-dimensional roughness of stationary crossflow disturbances was seen to be nearly linear, in contrast to that of travelling-wave crossflow disturbances which were nonlinear. The downstream amplification of both disturbance types remained linear with no significant alteration to growth rate. It is suspected that travelling-wave dominated crossflow transition is brought about through an increased receptivity of travelling-wave disturbances to free-stream turbulence at the roughness site, and not by modulation of stationary crossflow disturbance by unsteadiness as the boundary-layer evolves downstream.

Various phenomena were observed of the interaction of the forced crossflow boundary layer with a two-dimensional strip. The strip, located near the neutral point, resulted in amplification of stationary crossflow disturbances. Further increases in height up to around 80% of the boundary layer thickness impeded the evolution of the stationary crossflow disturbance; however, it resulted in an unsteady signal, rich in high-frequency fluctuations with peaks at: 430 Hz, 860 Hz and 1290 Hz. Moving the strip downstream resulted in a lower critical roughness height, with broadband excitation of fluctuations in the frequency range: 300 Hz - 2000 Hz.

The response of crossflow disturbances to a two-dimensional strip located near the neutral point, in the absence of a forcing 3D-roughness array, was seen to excite both stationary and travelling-wave disturbances, albeit increasing disturbance levels by a small amount. It is concluded that amplification of crossflow disturbances brought about by placing the roughness strip downstream of a 3D-roughness array is due to a complex interaction mechanism between the established crossflow instabilities with the strip, rather than increased receptivity of these disturbances to free-stream turbulence because of the presence of the strip itself. Furthermore, we conclude that the generation of longitudinal streamwise vorticity by 3D-roughness arrays has a greater impact on roughness receptivity than a two-dimensional roughness strip.

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Authors

Isabella Fumarola, City University London

Abstract

One of the receptivity mechanisms for laminar to turbulence transition is due to the freestream turbulence interacting with the boundary layer. In particular, the effect of freestream turbulence on the cross-flow instability has been largely reported in literature, but there is still a lack of understanding in the physical process of interaction between freestream turbulence and boundary layer. This work focuses on the freestream turbulence and boundary layer interaction in the region upstream the attachment-line of swept circular cylinders.The stagnation point region was, firstly, investigated in 1928 by Piercy and Richardson, who observed that it is a region of high level of disturbances. The phenomenon has been the matter of many discussions and experiments. In 1960s Sutera et al. proposed the vorticity amplification theory, which demonstrated that the vorticity, if properly oriented, may amplify in the stagnation region. The problem was later widely studied on two-dimensional bodies, but not deeply investigated on swept bodies.This work shows some experiments carried out in the T2 wind tunnel at City, University of London with a freestream turbulence level around 1%. The cylinder has been mounted in five sweep configurations from 0° to 30°. For each sweep angle, the flow approaching the attachment-line has been measured from a distance of four times the radius of the cylinder up to the wall, focusing on the boundary layer. The measurements technique adopted was multi-component LDA, which has allowed us to gather information of all the velocity component at the same time. The mean flow at the attachment-line was found to be laminar, and compared well with the theory, in all the configurations. The fluctuations showed an increment in the RMS level with a maximum at the edge of the boundary layer, except at the highest sweep angle. The influence of the angle over the amplification of the fluctuation is presented and discussed analysing the data in the frequency domain.

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Authors

Kentaro Kato, Linné Flow Centre / KTH Mechanics, Royal Institute of Technology
Antonio Segalini, Linné Flow Centre/KTH Mechanics, Royal Institute of Technology
P. Henrik Alfredsson, Linné Flow Centre/KTH Mechanics, Royal Institute of Technology
R. J. Lingwood, Department of Mechanical and Aerospace Engineering/Brunel University London; Linné Flow Centre/KTH Mechanics, Royal Institute of Technology

Abstract

The instability and transition to turbulence of the boundary layer on a cone rotating in a still fluid have been studied. The flow geometry is defined by the cone-apex angle, and earlier research has shown that on a wide cone (including the half angle of 90 degrees, i.e. the disk) cross-flow instabilities control the flow, whereas on sharper cones, with half angles below about 40 degrees, centrifugal instability becomes important. Our long-term aim is to shed light on complex mechanisms involving these different instabilities and how they affect laminar-turbulent transition; in the present work, however, we limit the study to a cone with a half angle of 60 degrees. The cone has a base diameter of 474 mm and is rotating in still air.

In our experiments the boundary-layer thickness is of the order of 1 mm making velocity measurements a challenging task. Here a laboratory-fixed single hot-wire probe was used to measure the azimuthal velocity. The experiments were made both on a clean cone and with 24 uniformly distributed roughness elements (several different roughness heights were explored).

The measurements captured the following development stages of the spiral vortices, which are initially stationary with respect to the rotating-cone surface: linear then nonlinear development stages; saturation stage; and collapse stage. In the early stage, the vortices develop along the cone surface with a particular, shallow inclination (around 10 degrees with respect to the azimuthal direction). This observation agrees well with local linear stability analysis but at a threshold finite amplitude the nonlinearity becomes significant. Then, the growth saturates and the vortex location becomes unsteady, which is observed as a vortex meandering motion. Also, high-frequency oscillations are observed, which may be related to a secondary instability, and finally the vortices collapse, leading to transition.

In addition to the stationary vortex disturbances, our results also provide evidence of non-stationary growing disturbances, which seem at first to develop independently from the stationary structures.

In the case with non-uniform roughness heights, the vortices merge and split, which affects the collapse process of the vortices. Large structures are formed at the collapse stage, similar to structures that have been observed in previous flow visualizations.

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Authors

Shumpei Hara, KTH-Royal Institute of Technology
Santhosh Babu Mamidala, KTH-Royal Institute of Technology
Jens Henrik Mikael Fransson, KTH-Royal Institute of Technology

Abstract

A deeper understanding of boundary-layer transition in different aerodynamic conditions would lead to improved control methods for skin-friction drag reduction. Transition is sensitive to both free-stream turbulence and surface roughness disturbances. The former disturbance can be characterized by its intensity and its integral length scale, which together sets the spanwise scale of the streaky structures consisting of alternating high- and low-speed regions inside the boundary layer. The latter disturbance is determined by its geometrical shape. Simplified, three-dimensional discrete roughness and its effect on transition is often characterized by its roughness Reynolds number, which is defined by the roughness height and the corresponding local velocity. Depending on the roughness Reynolds number and the geometrical setting of the discrete roughness it has been shown that transition can either be promoted or delayed. However, to our knowledge, no systematic experimental study has focused on the physical interaction between the aforementioned parameters. The objective of the present study is to gain further insight into this physical interaction by controlling the integral length scale of the free-stream turbulence while keeping the turbulence intensity constant under variable roughness height. The experiment is being performed in flat-plate boundary layers developing under zero-pressure-gradient conditions in a low-turbulence level wind tunnel at KTH. The variation of free-stream turbulence characteristics is obtained by means of varying the free-stream turbulence generating grid. Different grid mesh widths and bar diameters are used and the relative position of the grid with respect to the leading edge is changed. The localised roughness is cylindrical with variable diameter and with adjustable height of micro-meter precision. The measurements of the streamwise velocity are performed using hot-wire anemometry in which the hot-wire probe moves in three dimensions in the wake region behind the localised roughness and the fixed one is implemented to calculate the spatial correlation function as well. Finally, we summarize the effect of the free-stream turbulence and discrete roughness on flow structures and the transition mechanism characterised by the transition Reynolds number considered from the intermittency function.

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Authors

Masaharu Matsubara, Shinshu University
Yu Imanishi, Shinshu University
Yuya Tanada, Shinshu University
Sattaya Yimprasert, Shinshu University
Yutaro Endo, Shinshu University
Tatsuya Tsumura, Shinshu University

Abstract

In the last two decades it has been revealed that the large-scale and very-large-scale motions exist in wall-bounded shear flows. In high-Reynolds-number turbulent pipe and channel and flows, Monty et al. [J. Fluid Mech 589, 147-156] observed a very large features that have the streamwise length size ten times larger than the pipe diameter or channel width. In the other hand, Seki et al. [Physics of Fluids 24, 124102] found very-low-frequency peak of the streamwise velocity spectra in low Reynolds number channel flow. Though the low frequency corresponds to about ten channel widths that is the same as the size of turbulent patches appearing in the transitional channel flow, the peak continues even in the turbulent flow. In this study, the hot-wire measurement and flow visualization have been performed with gradually increasing Reynolds number of the channel flow. The intermittency factor, or turbulent fraction, was determined by means of the linear slope fitting for the probability density function of the second derivative of the streamwise velocity fluctuation. The intermittency factor curve shows that the flow for 1400 ≤ Re ≤ 2600 is transitional state and turbulent patches appear in the flow visualization, where Re is the Reynolds number based on the mean velocity and the channel width d. The power spectra of the velocity fluctuation indicates the low-frequency peak in the transitional range and it continues even in turbulent flow for Re ≤ 3000. Furthermore the low-frequency plateau remains up to Re = 4000. In the flow visualization over Re = 2600 the turbulent patches are not distinguished from the flow totally filled of turbulence. This means the low-frequency peak is not due to turbulent patch appearance. Careful observation of the movies of the flow visualization indicates that there exist clusters of the longitudinal streaks. The streamwise size of the cluster is ten times larger than the channel width dthat corresponds to the low-frequency peak. The two-point cross spectra of the velocity fluctuation was obtained with two hot wire sensors separated in the spanwise direction. The cross spectra map indicates drastic change at Re = 4000 that the spanwise size of the very-large feature suddenly decreases from 8 d to 1 d.  This demonstrates that the very-large features in the channel flow alternates between a patch-like structure and an elongate streak strongly dependent on Reynolds number.

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Authors

Jens Henrik Mikael Fransson, KTH-Royal Institute of Technology

Abstract

Free-stream turbulence (FST) and its effect on boundary-layer transition is an intricate problem. Elongated streamwise streaks of low- and high speed are created inside the boundary layer and their amplitude and spanwise wavelength are believed to be important for the onset of transition. The streaks are unsteady, and their amplitude grows with the square root of the downstream distance until the appearance of the first turbulent spots. The spanwise wavelength is often said to adopt to the boundary-layer thickness, giving streaky structures of aspect ratio one after some adaptation length. Furthermore, the transitional Reynolds number is often simply correlated with the turbulence intensity (Tu) and the characteristic length scales of the FST are often considered to have a small influence on the transition location. Here, we present new results from a large experimental hot-wire measurement campaign performed at KTH-Royal Institute of Technology, where both the Tu and the characteristic length scales of the FST are varied. Our results show that the integral length scale (Lx) of the FST affects the transition location differently depending on Tu in a given boundary layer. On the one hand, for small Tu an increase in Lx advances transition, in agreement with established results. On the other hand, for large Tu an increase in length scale postpones transition. In the present experimental setup both trends have been observed and a hypothesis for the trend change is formulated and tested. In addition, we believe that one has overlooked the significance of Lx in the transition process. For low Tu levels a 12% increase of Lx can advance the transition point by as much as 35%, and for high Tu levels an 18% increase of Lx can delay transition by 22%. These results contradict previous results that the streaks adopt to the boundary layer thickness independent of the FST characteristics, i.e. if the streamwise streaks are important for the transition onset, they are also likely to be affected by the FST. Our measurements show that the aspect ratio of the streaky structures correlates with an FST Reynolds number and that the aspect ratio can change by a factor of two at the location of transition. Based on these results a new transition prediction model has successfully been developed and will be presented at the symposium.

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Authors

Jeanne Cam-Tu Methel, ONERA, ISAE-SUPAERO
Maxime Forte, ONERA
Olivier Vermeersch, ONERA
Grégoire Casalis, ISAE-SUPAERO

Abstract

As laminar flow control by boundary layer suction is becoming an increasingly viable commercial application, airframe integration issues still need to be addressed. The current configuration to implement wall suction on an aerodynamic surface involves the juxtaposition of a perforated suction panel next to a solid-wall panel. Surface imperfections are expected to occur at this junction. Just as wall suction is generally known to stabilize the boundary layer, surface defects commonly tend to destabilize it. The competition between these two effects needs to be investigated to understand if current tolerances in manufacturing and assembly are acceptable to justify the use of wall suction.

The focus of this experimental study is to investigate the effect of surface defects on the laminar-turbulent transition of a boundary layer undergoing suction. The objective is to provide some critical height or depth criteria below which suction could be effective. A flat plate with a suction region comprised of nine suction chambers was placed in a two-dimensional Blasius flow. Reference transition positions were first determined for four suction configurations, without any defect. Surface defects were then introduced on the flat plate in the form of wires, forward-facing steps and gaps.

For the wires, the parameter of interest is the relative height (diameter) with respect to the boundary layer displacement thickness. Transition positions were determined for a range of relative heights and suction configurations. Below a critical relative height of 0.4, transition is still the result of viscosity-driven instabilities (Tollmein-Schlichting instabilities) and suction can effectively delay transition with respect to the case without suction.

Past the critical relative height value, transition occurs immediately downstream of the wire, regardless of whether or not suction is applied. In these critical cases, boundary layer profiles immediately downstream of the wire have a much more well-defined inflection point than the boundary layer profiles downstream of a subcritical wire. Above the critical relative height, transition mechanisms change and are the result of inflection-type instabilities. Wall suction is therefore ineffective because of the change in transition mechanism.

Experiments with the forward-facing steps and gaps are ongoing, and results will be available by the time of the conference.

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Authors

Viola Wartemann, DLR
Giannino Ponchio Camillo, DLR
Alexander Wagner, DLR

Abstract

In the present study the influence of transpiration cooling on hypersonic boundary layer transition is investigated. The study is carried out on a 7° half angle, blunted cone with a porous carbon-carbon insert through which gas is injected into the boundary layer. The analyses are focused on test cases with low cooling gas mass fluxes which do not induce shocks upstream of the porous insert.

At the considered hypersonic free stream conditions second mode waves are the dominant boundary layer instabilities, which are consequently the focus of this study. The stability analyses are performed by means of the stability code NOLOT, NOnLOcal Transition analysis, of the German Aerospace Center (DLR).

It was found, that the strongest effect of mass injection on the second mode is in the region of the effusion element and quickly decreases in downstream direction. It was observed that the growth rate increases, caused by the change of the local Reynolds number, with increasing mass flux. Due to the increase of the boundary layer thickness because of the mass injection, the most amplified frequencies shift to lower values.

The numerical predictions are compared with experimental results. The experimental data referred to in the present study were obtained in the DLR High Enthalpy Shock Tunnel Göttingen (HEG).

A previous numerical / experimental analysis of a first HEG test case demonstrated, that it is possible to use LST (Linear Stability Theory) for test cases with low mass injection rates, to study the second modes. A good agreement between the measured and predicted characteristics (looking at frequency range as well as the increase of the second modes) were demonstrated for the first maxima. LST predictions of second mode stabilization shortly downstream of effusion were confirmed experimentally. In the experiment as well as in the prediction second maxima appeared, which vanished in downstream direction. Thus, a region near the effusion element was found, where the predictions matched quite well. However, the transition location, further downstream, based on N-factor distribution was inaccurate, indicating an underprediction of the instability growth rate. The assumed reason for this behaviour are non-linear effects, which are not detectable with LST. Hence, in this paper in addition to LST calculations PSE (Parabolized Stability Equations) will be performed. The comparison to the experimental data will be extended to various experimental test cases with different mass fluxes.

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Authors

Christian Stemmer, Technical University of Munich
Antonio Di Giovanni, Technical University of Munich

Abstract

At hypersonic speeds, the wall heat transfer in the case of a turbulent boundary layer is considerably larger than in the laminar case. Therefore, the prediction of the transition location is an important issue in the design of re-entry vehicles. For blunt re-entry capsules with a smooth surface, the accelerated boundary layer is modally stable. However, roughness is always present on the capsule wall due to thermal ablation and manufacturing uncertainties. Previous investigations on hemispherical geometries with a rough wall under cold wind-tunnel conditions (M=5.9) showed that skewed roughness geometries can generate crossflow-type vortices similar to the ones known for subsonic three-dimensional boundary layers [1]. These roughness-induced crossflow-type vortices can support the presence of highly unstable modes. Moreover, in the altitudes where laminar-turbulent transition occurs, chemical reactions and non-equilibrium significantly affect the state of the boundary layer and can destabilize the flow in the roughness wake.
In this study, we numerically investigate the laminar-turbulent transition of the boundary layer on a capsule-like geometry with distributed roughness in a typical re-entry scenario (M=20). By means of Direct Numerical Simulations, unsteady disturbances are forced in the roughness wake, revealing the presence of different instability modes in correspondence with roughness-induced crossflow-type vortices. Both the linear and non-linear growth of the unstable modes are considered, including laminar-turbulent breakdown.
Chemical and thermal non-equilibrium significantly change the temperature profiles of the boundary layer and have a non-negligible influence on the stability properties of the roughness wake. The various non-equilibrium effects on the steady base flow and on the (non-)linear growth of the unsteady disturbances are compared based on different simulations conducted with gas models of increasing complexity (equilibrium air, chemical non-equilibrium and thermochemical non-equilibrium). Furthermore, the location of the laminar-turbulent transition is identified by a sudden augmentation of the wall heat flux observed in the unsteady flow compared to the steady flow.  The influence of the gas modeling on the transition location is analyzed, showing the importance of considering the effects of non-equilibrium to correctly predict location and values of the heat augmentation induced by the laminar-turbulent breakdown in the capsule configuration.

[1] Di Giovanni, A., Stemmer, C., “Crossflow-type breakdown induced by distributed roughness in the boundary layer of a hypersonic capsule configuration”, JFM 856, 2018.

[2] Hein, S., Theiss, A., Di Giovanni, A., Stemmer, C., Schilden, T., Schröder, W., Paredes, P., Choudhari, M. M., Li, F., Reshotko, E., “Numerical Investigation of Roughness Effects on Transition on Spherical Capsules”, JSR 2019.

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Authors

Iván Padilla Montero, von Karman Institute for Fluid Dynamics
Fernando Miró Miró, von Karman Institute for Fluid Dynamics
Fabio Pinna, von Karman Institute for Fluid Dynamics

Abstract

Natural boundary-layer transition at hypersonic speeds is known to be strongly influenced by the presence of isolated roughness elements on the surface. Examples include damaged heat-shield tiles, rivets or gap fillers, among many others. Such three-dimensional elements with heights comparable to the local boundary-layer thickness generally induce counter-rotating streamwise vortices in the wake flow, giving rise to low-velocity streaks that are surrounded by regions of high-detached shear. The perturbations generated by these elements can accelerate the growth of incoming disturbances and introduce additional instability mechanisms in the flow field, eventually leading to a premature occurrence of transition.

In recent years, many researchers have studied the stability characteristics of the wake induced by isolated roughness elements, both experimentally and numerically. Their analyses have revealed that the roughness wake supports the growth of sinuous and varicose instability modes that develop in the high-shear regions introduced by the counter-rotating vortex pair, and that these modes can undergo a substantial growth during the linear stages of the transition process. The aforementioned numerical studies have assumed air to behave like a calorically perfect gas. However, linear stability analyses have shown that this assumption must be abandoned when modeling high-enthalpy environments. Instead, flow assumptions accounting for the excitation of the internal energy modes and the dissociation of air molecules are necessary. In practical applications, roughness-induced transition often occurs in high-temperature environments, requiring the use of the mentioned more physically-inclusive flow assumptions.

The present work investigates the influence of high-temperature effects on the instability of the wake induced by a cuboidal roughness element mounted on top of a flat plate at freestream conditions that are representative of the Space Shuttle reentry trajectory. Three different flow assumptions are considered, namely, a calorically perfect gas (CPG), a chemically-frozen thermally perfect gas (TPG) and a gas in chemical nonequilibrium (CNE). For each case, two-dimensional linear stability theory (also known as BiGlobal stability theory) is employed on spanwise planes together with base flows obtained from steady laminar Navier-Stokes simulations. For each of the aforementioned flow assumptions, the most unstable perturbations growing in the wake flow field are analyzed and compared, identifying the frequency range that provides the highest growth rate through their evolution along the wake. The resulting N-factor envelope is compared to that without the roughness element, quantifying the upstream movement of the predicted transition onset location due to the roughness element with the various flow assumptions.

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Authors

Shahram Karami, Monash Universuty
Vassilis Theofilis, University of Liverpool
Julio Soria, Monash Universuty

Abstract

A turbulent under-expanded jet impinging normally on a flat plate generates very intense pressure (acoustic) waves, the presence of shocks also promotes the generation of entropy and vortical linear flow instabilities. The interactions of acoustic waves with flow in a confined geometry result in self-sustained oscillations. Full understanding of this problem is far from being achieved presently, due to the multi-faceted nature of the linear mechanisms involved and the ensuing nonlinear dynamics. In order to address the linear instability generation mechanisms in the problem at hand, global instability analysis of under-expanded supersonic impinging jets is performed to reveal the dominant structures and interactions of coherent structures with the shocks and the impinging plate.

In this turbulent configuration instability analysis is performed around the ensemble-averaged mean flow, using a matrix-forming methodology as implemented in a massively parallel in-house code. Expanding upon earlier work, appropriate treatment of the shock location is incorporated in the analysis. The mean flow is obtained using an in-house parallel large eddy simulation code, which has also generated a large database of time-accurate flow fields against which instability analysis results are compared. 

Two cases are studied at a Reynolds number of 50,000 (where the Re is defined based on jet diameter (D) and inlet gas velocity, having a nozzle pressure ratio of 3.4 and “short” and “long” nozzle-to-wall distances of h=2D and 5D, respectively. The instantaneous flow fields reveal a complex interaction of the vortical structures and first oblique shock which is stronger in the short-distance case. The coherent structures formed in the shear layer are found to grow more rapidly in the h=2D case. The mean flow fields show the formation of a recirculation bubble at h=2D, the bubble being absent at h=5D. To characterise the flow further, spatial cross-correlation analysis of the near field pressure and velocity is performed. The correlation coefficient is found to have large values at the Mach disk and the oblique shocks in the h=2D case, while the high correlation region turns towards the shear layer at h=5D. In both cases, the high correlation of the pressure at the nozzle lip and streamwise velocity in the shear layer suggests that flow receptivity is key to active flow control. Results on manipulation of the pressure field near to the nozzle lip to either enhance or diminish the formation and evolution of the coherent structures (depending on desired output) will be presented at the conference. 

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Authors

Jianxin Liu, Tianjin University
Xuesong Wu, Tianjin University, Imperial College London

Abstract

The streaky structures of the boundary layers may be excited by the turbulence in the freestream or the roughness elements on the wall. They have a strong effect on the stability of the boundary layers which is able to delay or advance the transition. The impact on the spanwise periodic streaky structures in supersonic boundary layers is investigated at present. It have been know that these streaky structures are able to stabilize or destabilize the boundary layers. It is concerned that how these structures affect the lower branch of first modes which are dominated by the triple-deck theory and the second modes which are dominated by the supersonic relative Mach number in the compressible boundary layer in this work. By extending the triple-deck theory for the three-dimensional incompressible boundary layers, we derived the compressible one for investigating the lower-branch of the first modes. Both the artificial streaks and the realistic roughness elements inducing streaks were investigated for these type modes. It is found that there are the even modes and odd modes in the flow and the first modes at some frequencies can be stabilized by the three-dimensional effects but the others can be destabilized by the streaks. In addition, the even modes of the streaky boundary layer are easy to radiate spontaneously at a lower frequency. The effect on the Mack mode waves in the boundary layers are also investigated by the inviscid Bi-Global instability equations. The streaky boundary layers are generated by the turbulence in the freestream. The instability is analyzed at different amplitude of the streak. The stability analysis suggests that the Mack mode can be destabilized significantly but the least stable mode in the streaky boundary layers is still dominated by the shear of the streak. The spontaneous radiated instability modes can also be found in the flow.

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Authors

Christoph Hader, University of Arizona
Hermann Fasel, University of Arizona

Abstract

In hypersonic transition wind-tunnel experiments, transition is caused “naturally” by free-stream disturbances even when so-called quiet tunnels are employed such as the Boeing/AFOSR Mach 6 Quiet Tunnel (BAM6QT) at Purdue University. The nature and composition of the free-stream disturbance environment in high-speed transition experiments is difficult to assess and therefore largely unknown. Therefore, for our investigations, a random (broadband) forcing approach was implemented into a high-order accurate finite-difference Navier-Stokes code to model “natural” transition in hypersonic boundary layers. The approach was validated for a flared cone geometry for the quiet flow conditions of the experiments at the BAM6QT facility (Hader & Fasel, 2018 JFM, Vol. 847). The validation results revealed that with this approach many of the flow characteristics observed in the experiments are captured more accurately compared to “controlled” fundamental breakdown simulations (Hader & Fasel, AIAA 2016-3344). The random forcing approach was found to be a promising strategy to investigate nonlinear breakdown in hypersonic boundary layers (when detailed information regarding the free-stream disturbance environment is not available) without introducing any bias towards any nonlinear mechanism, and/or the selection of specific frequencies or wave numbers, that are required in “controlled” breakdown simulations. Currently, simulations are carried out employing this broadband forcing approach to investigate boundary-layer transition for flared and straight cones at Mach 6 for the “noisy” conditions of the BAM6QT. The results will be compared to experimental measurements and results for the quiet “natural” transition simulations. In addition, DNS are performed for an increased free-stream stagnation enthalpy to allow the investigation of the “natural” transition for more realistic free-flight conditions (more realistic wall-to-edge-temperature ratios, Tw/Te). A comprehensive comparison between all simulation cases will be presented. Comparisons to experimental measurements, where available, will be provided to highlight the differences and similarities of the transition mechanisms, with special emphasis on the nonlinear stages, that are relevant for quiet, noisy ground-based test conditions as well as for free-flight conditions. The ultimate goal of our investigations is to understand the “natural” transition process for various flow conditions and model geometries. This understanding is essential in order to shed light on the implications with respect to extrapolating findings from wind-tunnel experiments to free flight. Furthermore, the understanding of the relevant nonlinear mechanisms is crucial for developing effective and efficient control strategies for delaying transition in hypersonic boundary layers. Towards this end, results from simulations of promising flow control strategies, both active and passive, will also be presented and discussed.

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Authors

Ivan Egorov, Central aerohydrodynamic Institute, Moscow Institute of Physics and Technology
Alexander Fedorov, Moscow Institute of Physics and Technology
Andrey Novikov, Central aerohydrodynamic Institute, Moscow Institute of Physics and Technology

Abstract

The problem of laminar-turbulent transition (LTT) in hypersonic boundary-layer flows is one of the main still unsolved problems of high-speed aerodynamics. To predict and control LTT it is needed to clarify basic physical mechanisms driving the transition process. In the case of low free-stream disturbances typical for flight conditions, LTT includes the three main stages [1]: receptivity to external disturbances, growth of unstable modes (such as first and second Mack modes, cross flow instability and Görtler vortices) and the nonlinear breakdown of disturbances leading to a fully turbulent flow regime. A holistic computation of LTT including the all stages is possible only using direct numerical simulations (DNS), where the full unsteady three-dimensional (3D) Navier–Stokes equations are solved without any restriction on the mean (unperturbed laminar) flow and disturbance amplitudes. In addition, as opposed to physical experiments, DNS gives full information about 3D disturbance field, which enables to identify and study in detail different LTT mechanisms. The modern methods of parallel computations and rapid developments of multi-processor supercomputers make it feasible to conduct such numerical experiments for hypersonic boundary layers for simple configurations such as a flat plate and a cone at zero angle of attack [2]. Further progress in computational hardware will allow us to handle more and more complicated and practical configurations. However, there is a stumbling block associated with sensitivity of transition to the initial and boundary conditions, which are related to receptivity problem associated with excitation of unstable modes by freestream and body-induced disturbances. This talk will demonstrate that differences in ambient disturbance fields can result in significant changes in the LTT location and even in its physics. In particular, we discuss our DNS of a transitional flow over a compression corner at the freestream Mach number 5.4 and high unit Reynolds numbers. The flow parameters are relevant to the Hyper-X model tested in the NASA LaRC 20-Inch Mach 6 Air Tunnel [3]. It is shown that depending on the location and frequency of external forcing, LTT can evolved in qualitatively different ways. Similar issues related to receptivity of high-speed boundary layer to solid particulates [4], thermal fluctuations [5,6] and free-stream turbulence will be addressed also.
1. Fedorov A. Rev. Fluid Mech., 43, 2011.
2. Zhong X., and Wang X. Rev. Fluid Mech., 44, 2012.
3. Berry S.A., DiFulvio M., and Kowalkowski M.K. NASA TM-2000-210316, Aug. 2000.
4. Chuvakhov, P.V., Fedorov, A.V., and Obraz, A.O. Fluid Mech., 859, 2019.
5. Fedorov A.V. Procedia IUTAM , 14, 2015.
6. Fedorov A., and Tumin A. AIAA J., 55(7), 2017.

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Authors

Miguel Beneitez, KTH Royal Institute of Technology
Yohann Duguet, LIMSI-CNRS, Campus Universitaire d’Orsay, Université Paris-Saclay
Philipp Schlatter, KTH Royal Institute of Technology
Dan S. Henningson, KTH Royal Institute of Technology

Abstract

Some recent advances in the subcritical transition to turbulence are due to new concepts borrowed from the theory of dynamical systems. Among these ideas, the computation of special trajectories lying on the laminar-turbulent separatrix has proven useful to map the state space and identifying the relevant basins of attraction [1]. Extending these concepts to spatially developing boundary layer flows is not trivial and it is expensive [2,3]. We show here new computations of such edge trajectories over long time horizons in a Blasius boundary layer. A novel technique featuring a moving time-dependent computational domain is used in order to track the corresponding perturbations further in space and time. The associated flow field contains a solitary pair of streaks of finite length with simplified bursting dynamics. This state is linearly unstable by construction, and its stable manifold divides the state space into two parts corresponding respectively to bypass transition and to transition by Tollmien-Schlichting waves. The finite-time linear stability properties of this time-dependent coherent structure is addressed by the computation of the so-called OptimallyTime-Dependent (OTD) modes [4]. The potential of this approach as a toolbox to understand and quantify the nucleation of incipient spots, central in bypasstransition, will be discussed.

[1] Eckhardt, B., Schneider, T. M., Hof, B. & Westerweel, J. (2007) ”Turbulence transition in pipe flow.” Annu. Rev. Fluid Mech. 39, 447–468

[2] Cherubini, S., Palma, P. De, Robinet, J-Ch. & Bottaro, A. (2011) ”Edge states in a boundary layer.” Physics Of Fluids 23(5), 051705.

[3] Duguet, Y., Schlatter, P., Henningson, D. S. & Eckhardt, B. (2012) ”Self-sustained localized structures in a boundary-layer flow.” Physical review letters108(4), 044501.

[4] Babaee, H. & Sapsis, T. P. (2016) ”A minimization principle for the description of modes associated with finite-time instabilities.” Proc. R. Soc.A472(2186), 20150779

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Authors

Georgios Rigas, Caltech
Denis Sipp, ONERA
Tim Colonius, Caltech

Abstract

In a linear frequency-domain framework, the most amplified instabilities are typically described by considering singular vectors of the resolvent operator of the linearized Navier-Stokes equations. In this study, we extend the methodology to take into account nonlinear triadic interactions by considering a finite number of harmonics in the frequency domain using a Harmonic Balance Method. Two approaches will be compared for solving the nonlinear Harmonic-Balanced Navier-Stokes equations: a global approach considering the full Jacobian of the problem and an efficient spatial marching technique based on the One-Way Navier-Stokes (OWNS) equations, which substantially reduces the computational cost.  

We demonstrate the framework on a zero-pressure flat-plate boundary layer by considering three-dimensional spanwise-periodic perturbations triggered by a few optimal forcing modes of finite amplitude. Optimal nonlinear forcing mechanics that lead to transition and maximize the skin-friction coefficient are identified using direct-adjoint looping. Two cases are considered, corresponding to fundamental and subharmonic forcing. The proposed frequency-domain framework identifies in a computationally efficient and accurate manner worst-case disturbances for wall-bounded transition.

*G.R. and T.C. acknowledge support from ONR (grant N00014-16-1-2445) and The Boeing Company (CT-BA-GTA-1)

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Authors

M. J. Philipp Hack, Center for Turbulence Research, Stanford University
Zhu Huang, Center for Turbulence Research, Stanford University
Tim Flint, Center for Turbulence Research, Stanford University

Abstract

The non-orthogonal nature of the linearized Navier-Stokes operators allows significant transient amplification of disturbances and thus opens pathways to turbulence that are independent from the exponential stability of a flow. Energetically optimal disturbances computed in linear transient-growth analysis qualitatively explain the amplification of perturbations that lead for instance to the formation of highly energetic streaks which can play an important role in advancing the flow towards turbulence. Considered as a whole, the process of laminar-turbulent transition is nonetheless a strongly nonlinear phenomenon, which has motivated the study of transient perturbation growth using the full nonlinear governing equations. Nonlinear optimal disturbances have been widely investigated and so-called minimal seeds which most effectively trigger transition to turbulence have been identified in canonical configurations of incompressible flows. Transition to turbulence is nonetheless of particular practical relevance in various flows with non-negligible compressibility effects. Within this work, we develop a variational framework for the analysis of nonlinear optimal perturbations in compressible flows. The flow solver is based on the compressible Navier-Stokes equations in conserved variables and supports complex geometries via a curvilinear formulation which is spatially discretized using high-order schemes. The objective of finding the specific perturbations which undergo the highest nonlinear amplification is solved via a variational formulation. The outcome is an iterative scheme based on the successive time integration of the forward and adjoint nonlinear compressible Navier-Stokes equations which allows the first identification and study of general nonlinear optimal disturbances in compressible flows. The algorithm is applied in the analysis of nonlinear optimal disturbances in compressible pipe flow. The computed perturbations originate from spatially localized initial conditions and undergo several stages of energy amplification. During their amplification, the perturbations extract energy from the mean flow through the Orr mechanism. Nonlinear interactions subsequently redistribute energy between the scales of the perturbations before highly energetic streaks are generated by means of the lift-up mechanism. In the limit of low Mach numbers, the evolution of the perturbations agrees with those identified in earlier studies of incompressible channel flow.

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Authors

Rishi Kumar, Imperial College London
Andrew Walton, Imperial College London

Abstract

The nonlinear stability of plane Poiseuille-Couette flow to three-dimensional disturbances is investigated asymptotically at large values of the Reynolds number R based on channel half-width and the maximum velocity of the Poiseuille component. The asymptotic theory, aimed at a detailed understanding of the physical mechanisms governing the amplitude-dependent stability properties of the flow, shows that the phase shifts induced across the critical layer and a near-wall shear layer are comparable when the disturbance size Delta=O(R^{-4/9}). In addition, it emerges that at this crucial size both streamwise and spanwise wavelengths of the travelling wave disturbance are comparable with the channel width, with an associated phasespeed of O(1). Neutral solutions are found to exist in the range 0<V<2 with c_0 = V to leading order, where c_0 and V are non-dimensional quantities representing the dominant phasespeed of the nonlinear travelling waves and the wall sliding speed respectively. Moreover, these instability modes exist at sliding speeds well in excess of the linear instability cut-off. The amplitude equation governing these modes is derived analytically and we further find that this asymptotic structure breaks down in the limit V→2 when the disturbance streamwise wavelength decreases to O(R^{-1/3}) and the maximum of the basic flow becomes located at the upper wall. Further examination of the strongly nonlinear structure in the limit V→2 is carried out and shows that as the sliding speed is increased, the critical layer moves ever closer to the O(R^{-1/3}) shear layer adjacent to the upper wall. This new short-scale structure possesses comparable-sized derivatives in the streamwise and wall-normal directions. In addition the amplitude of the travelling waves can be shown to rise to O(R^{-1/3}) while the phasespeed remains O(1). The governing balances in this new nonlinear near-wall region are the full 2D Navier-Stokes equations but with unit Reynolds number and are subject to matching conditions to the O(1) Rayleigh region and a link to the Stokes region (now of thickness O(R^{-2/3})) near the lower wall. Even in the case of 3D disturbances, the leading-order equations apparently remain 2D due to the fact that the streamwise and wall-normal scales are shorter than that imposed in the spanwise direction. These equations are reminiscent of those governing the `production layer' that forms an integral part of a self-sustaining process in boundary layer instability.

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Authors

philip hall, Monash Uni.

Abstract

It is now widely believed that vortex-wave interaction theory (VWI) or its finite Reynolds number manifestation self-sustained process theory (SSP) is deeply implicated with the formation and support of near-wall streaks in turbulent shear flows. However, the turbulent structure away from the wall in for example the log layer cannot be supported by a single interaction and so multiple waves of different frequencies would be needed in a SSP description of such structures. That would prove a challenge beyond present computer capabilities but we show how the high Reynolds number VWI approach can be used to describe arrays of interactions. The large-scale properties are shown to be governed by two slow evolution equations and lead naturally to a region where the flow is completely self-similar and taking on a form entirely consistent with the Townsend attached eddy hypothesis. Moreover, within that layer we show how many of the classic properties of turbulent shear flows, including a logarithmic profile, naturally arise within that layer. The structure found in this layer is shown to be generic to all shear flows independent of geometry. We show how the breakdown of self-similarity leads to the formation of transition layers where the nonlinear Ray theory approach needed to describe distributed VWI. The two transition layers are shown to take on many similarities with the buffer layer and wake regions of wall-bounded turbulent shear flows.

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Authors

Masato Nagata, Tianjin University, Kyoto University
Baofang Song, Tianjin University
Darren P. Wall, Kyushu University

Abstract

The problem of transition from laminar flow to turbulence in plane Couette flow subject to a span-wise system rotation is considered, where the fluid motion is governed by two non-dimensional parameters, R (shear-induced Reynolds number) and Ω (system rotation rate). This system has long been serving as a testing ground for comparison between experiments and theories for understanding mechanisms of transition in fluid dynamics, and it is believed that the basic laminar state first loses stability to stream-wise independent disturbances with increasing R and/or Ω, resulting in the onset of secondary flow with a stream-wise independent roll-cell structure, followed by successive supercritical bifurcations originating from three-dimensional instabilities of the secondary flow.
In the present paper, we re-examine this seemingly well-known instability property of the secondary flow, especially for small R, with a surprising result. We shall demonstrate that the solution branch for a three-dimensional tertiary flow with small stream-wise wave numbers resulting from the afore-mentioned secondary instability connects with the so-called Ribbon, which is composed of two tilted vortex flows, elongated in the stream-wise direction with the same but opposite tilted angles. Ribbons bifurcate from the basic laminar state as well as in the case of the stream-wise independent two-dimensional flows, and, moreover, in the limit of small stream-wise wavenumbers, the bifurcation point of the Ribbons coincides with that of the stream-wise independent flows. Thus, the stream-wise independent secondary flow becomes unstable to disturbances with small stream-wise wavenumbers as soon as it bifurcates from the basic state, so that there exist no stable stream-wise independent roll-cell structures. 
It should be noted that one of the sides is bounded by a free surface and the other by a solid surface in experiments, whereas the periodic boundary condition in the span-wise direction has been applied in our theoretical analysis. Nevertheless, we conclude that the roll-cells, which are observed in experiments as the secondary flow, might be actually three-dimensional tilted vortex flow with long stream-wise wave lengths, free from the different types of boundary condition at the sides.

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Authors

Michael Karp, Center for Turbulence Research, Stanford University
M. J. Philipp Hack, Center for Turbulence Research, Stanford University

Abstract

Boundary layers are prone to separation when subjected to adverse pressure gradients. Flow separation often reduces the performance of aerodynamic and hydrodynamic vehicles. Vortex generators have been utilized successfully as passive flow control devices; however they often trigger turbulence, which increases drag and heat transfer. Moreover, finding the optimal parameters for their design is oftentimes based on trial and error. In this study we seek by means of nonlinear optimization the velocity disturbances at a given upstream position, which delay laminar separation as far downstream as possible. Our approach leverages insights gained from the non-modal growth of disturbances in boundary layers and their role in generating a mean flow distortion which augments the shear at the wall. An informed guess based on linear analysis of transient perturbation growth leads to significant delay of separation and serves as a starting point for the nonlinear optimization algorithm. Nonlinear optimal perturbations, found by means of the conjugate gradient algorithm, enable further delay of the separation location, with moderate improvements relative to the linear analysis. The mechanism of separation delay is the generation of a mean flow distortion by nonlinear interactions during perturbation growth. The mean flow distortion enhances the momentum close to the wall, and thus counteracts the deceleration of the flow in that region. Similar results are obtained for two optimization objectives – maximal mean separation location and minimal mean bubble length. The influence of the disturbance spanwise wavenumber on separation delay is investigated. It is found that the optimal spanwise wavenumber leading to maximal separation delay is significantly smaller than the one associated with maximal linear energy growth. Analysis of the secondary instability of the optimized base flow is conducted to identify potential absolute instabilities. Both varicose and sinuous secondary disturbances are found to be absolutely unstable, with the varicose instabilities occurring for relatively low spanwise wavenumbers and sinuous instabilities observed for a limited region of higher spanwise wavenumbers.

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Authors

Daiane I. Dolci, University of São Paulo
Bruno S. Carmo, University of São Paulo

Abstract

Instabilities in the laminar flow around a bluff body has attracted great interest of the fluid mechanics community as it presents diverse physical mechanisms that are coupled together. There is boundary layer separation due to adverse pressure gradient, wakes, time-periodic flows, interaction between vortices, to name a few of the most important ones. The dynamics are even richer when the body is free to move or vibrate. Recently, there have been papers studying the normal stability of the flow around cylinders that are allowed to move. However, to the best of our knowledge, no work has yet addressed non-normal stability analysis of the flow around an elastically mounted circular cylinder. Here we present the adjoint operator of the fluid-structure coupled problem, in which the cylinder is modelled as a mass-spring-damper system, allowed to move in the cross-stream direction, and the flow is viscous and incompressible, governed by the Navier-Stokes equations. We then adapt well known algorithms to obtain the optimal perturbations of the flow for a series of time horizons, i.e., we find the perturbations that lead to the largest linear growth for a given time interval. We analyse the importance of the non-normal growth of the disturbance in the first instability of this flow, in which the steady flow suffers a Hopf bifurcation and becomes time periodic.

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Authors

Nathaniel Hildebrand, University of Minnesota
Anubhav Dwivedi, University of Minnesota
Sidharth GS, University of Minnesota
Joseph Nichols, University of Minnesota
Mihailo Jovanovic, University of Southern California
Graham Candler, University of Minnesota

Abstract

We study the evolution of three-dimensional small perturbations in nominally two-dimensional (2D) laminar shock boundary layer interactions. Our study is aimed to uncover mechanisms that can cause large perturbation growth and lead to the breakdown of the laminar flow. The analysis utilizes the global linear system associated with the 2D interaction as the base flow and investigates global stability, transient response and response of stable interactions to exogenous disturbances. We consider interactions in two different configurations: a compression ramp, and an impinging oblique shock. Differences and similarities in the two configurations are noted.

Global stability analysis reveals that the interaction becomes unstable to three-dimensional perturbations past a critical freestream Reynolds number Re. Alternately, the interactions at a fixed Re can also destabilize if the inviscid pressure rise associated with the interaction increases (achieved by increasing the ramp angle in the compression ramp case, or the impingement angle in the oblique shock case). The global mode associated with this mode is found to grow monotonically at zero frequency. The mode is present in the entirety of the bubble and its spanwise wavelength scales with the length of the separation zone.  The instability manifests as the emergence of three-dimensionality in the nominally two-dimensional interaction. As the mode grows, the topology of the separation bubble morphs into spanwise recirculation cells. The reattachment line develops undulations and streamwise streaks in velocity and temperature persist downstream in the reattached flow. As the perturbations grow larger,  stable spiral foci appear in the separation zone wall streamline pattern. Wavemaker analysis reveals that the spatial origin of the mode is located close to the core of the separation bubble.

While modal mechanisms dominate the long-time dynamics of unstable configurations, stable shock boundary layer interactions can support large growth of three-dimensional perturbations due to the non-normality of the underlying linearized dynamics. It is shown that the centrifugal effects due to the concave curvature of the reattaching flow play an important role. Transient growth in time and input-output analyses point to selective amplification of low-frequency upstream vortical perturbations by the interaction. The spanwise length scale of the vortices experiencing largest growth scales with the reattaching boundary layer thickness.

Understanding the modal and non-modal mechanisms of perturbation growth will help contribute towards predictive transition tools in the flows with shock boundary layer interactions.

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Authors

Tim Gebler, Chair of Fluid Dynamics, Technische Universität Darmstadt
Judith Kahle, Chair of Fluid Dynamics, Technische Universität Darmstadt
Dominik Plümacher, Chair of Fluid Dynamics, Technische Universität Darmstadt
Martin Oberlack, Chair of Fluid Dynamics, Technische Universität Darmstadt; 2) Graduate School of Excellence Computational Engineering, Technische Universität Darmstadt

Abstract

The non-modal stability theory developed in the 1990s profoundly changed our understanding of transition. It turned out, that certain flows admit a short-time (transient) growth of disturbances, even if a modal analysis predicts stability for long times for all eigenvalues. Here, we present a novel approach to non-modal stability based on the symmetries of the linearized stability equation.
It has been shown by the present group, that the normal-mode ansatz has its basis in three classical symmetries, i.e. translation in time and space (homogeneous directions) and scaling. Most importantly, many canonical shear flows admit additional symmetries, which in turn lead to eigenfunctions with algebraic growth in time.
We demonstrate our new approach for a two-dimensional disk-type flow, where the laminar azimuthal velocity profile is given by a potential vortex, though viscosity is retained in the entire analysis. The flow is defined on an open domain with a thin rotating wire at the centre as well as vanishing disturbances at the centre of the flow and at infinity. The solution of the stability problem using the normal mode ansatz shows, that all eigenvalues imply stability.
As opposed to this result, the new symmetry induced ansatz enables a representation of unstable modes with a slow algebraic growth rate, though with a fractional exponent. These modes are not represented by the normal mode ansatz. The corresponding stability problem is solved analytically. Interestingly enough, this instability leads to vortex spirals in the disk-type flow.
The present approach has important implications for both theory and applications of stability problems. The new ansatz is deeply rooted in the mathematics of the stability equations and can be applied to other shear flows. An extension of the non-modal stability method to a larger class of rotational shear flows is currently under investigation. The investigated disk-type flow is a limiting case for the wide gap Taylor Couette flow between two concentric cylinders with infinitesimal inner cylinder and infinite outer cylinder radius. In wide gap Taylor Couette experiments with an inner rotating cylinder, columnar vortices are observed. We believe that our theory gives new insights into a limiting case of TC for an infinite radius ratio of outer to inner cylinder radius. In particular, we are able to show, that two-dimensional vortex instabilities exist for a particular choice of boundary conditions.

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Authors

Yongyun Hwang, Department of Aeronautics, Imperial College London
Lloyd Fung, Department of Aeronautics, Imperial College London

Abstract

It is well known that stratification can stabilise shear flow. In a vertical shear flow, the Miles-Howard's criterion firmly indicates that flow should be stable if the local Richardson number is greater than one fourth. However, a recent inviscid stability analysis (J. Candelier, S. Le Dizès, and C. Millet, 2011, J. Fluid Mech., 685:191) showed that if shear is tilted with a non-zero angle from vertical, an instability can arise even under very strong stratification, and such an instability was recently observed in a titled wake flow at low Reynolds number (Meunier, 2012, J. Fluid Mech., 699:174). In the present study, a linear stability analysis is performed on a tilted parallel wake in a strongly stratied fluid at low Reynolds numbers. A particular emphasis of the present study is given to the understanding of the low-Froude-number mode observed by the recent experiment by Meunier (2012). It is shown that  in the limit of low Froude number, the linearised equations of motion can be reduced to the Orr-Sommerfeld equation on the horizontal plane, except the viscous term that contains vertical dissipation. It is subsequently shown that the low-Froude-number mode is a horizontal inflectional instability and that the resulting most unstable mode should remain two dimensional as long as the tilting angle is set to be small enough. To support this theoretical claim, the asymptotic regime where this equation would be strictly valid is also discussed in relation to previous arguments on the proper vertical length scale. Finally, an absolute and convective instability analysis of parallel wake is performed, revealing qualitatively good agreement with the experimental result. The low-Froude-number mode is stabilised on increasing Froude number like in the experiment, and it was found that this is due to the the emergence of small vertical velocity at finite Froude number, which modifies the horizontal inflectional instability and leads to pradoxically stabilising buoyancy force on increasing Froude number.

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Authors

Jean Helder Marques Ribeiro, University of California, Los Angeles
Chi-An Yeh, University of California, Los Angeles
Kunihiko Taira, University of California, Los Angeles; Florida State University

Abstract

Resolvent analysis has provided a systematic approach to examine the input-output relationship of sustained perturbations with respect to a given base flow (Trefethen et al 1993).  While the original formulation considered the use of stable laminar flow as a base state, recent studies have allowed for the use of turbulent mean flows as well as unstable base states (McKeon and Sharma 2010).  The resolvent analysis requires the base flow and the discrete linearized Navier-Stokes operator to construct the resolvent operator (transfer function).  The size of this operator ($n \times n$) is proportional to the size of the spatial grid ($n$) required to resolve the perturbations about the base flow of interest, which can become extremely large for turbulent flows and triglobal analysis. The major bottlenecks in performing the analysis are the computational cost of singular value decomposition (SVD) $\mathcal{O}(n^3)$ and the memory requirement during the SVD of the resolvent operator.  For this reason, the applications of resolvent analysis have been limited to simple flows, unless high-performance computers are available.   

To address these issues, we incorporate randomized numerical linear algebra (Halko et al. 2011) into the resolvent analysis.  We first constructthe sketch matrix of the full resolvent operator by passing a random test matrix.  This sketch matrix can be used to determine the low-dimensional basis to construct the low-dimensional representation of the resolvent operator, on which the SVD can be performed.  The findings from the SVD analysis on the low-dimensional subspace can be projected back to the physical space to retrieve the dominant response and forcing modes as well as the gain.  We find that the computation through the present formulation is reduced to order $\mathcal{O}(n^2)$ and the memory requirement is significantly lowered.  

To demonstrate the capability of the current formulation, we compare the findings from the randomized resolvent analysis with that of the full analysis for a high-dimensional turbulent flow. The randomized analysis can find the resolvent and forcing modes with remarkable fidelity while achieving speed up. We also report the influence of the size of the random test matrix and the dominant gain spacing on the randomized resolvent analysis.  With the present randomized formulation, we can perform resolvent analysis on a standard workstation without any high-performance resources, which empowers the resolvent analysis to tackle high-dimensional complex flows. 

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Authors

Chi-An Yeh, University of California, Los Angeles
Muralikrishnan Gopalakrishnan Meena, University of California, Los Angeles
Kunihiko Taira, University of California, Los Angeles; Florida State University

Abstract

We present a network-based formulation to examine the input-output relationship for perturbation over a time-varying base flow. The foundation of this technique builds upon the concept of a time-evolving network (Grindrod et al. 2011) to identify the key dynamical paths along which perturbations amplify. To demonstrate the strength of this approach, we consider a canonical two-dimensional isotropic turbulent flow. This flow is described as a network of vortical elements, where the time-varying edge-strength is characterize by the induced velocity between each vortical element according to the Biot-Savart's law. Within this vortical-interaction network, we extract important structures that influence the future evolution of the flow by taking the right singular vector of the resolvent of the associated adjacency matrix. This singular vector, or broadcast mode, gives insights into the vortical elements at which information can be effectively spread over the entire network when it evolves in time. The present time-evolving network analysis complements the resolvent analysis for fluid flows (Trefethen et al. 1993, McKeon \& Sharma 2010). Furthermore, a memory parameter in the time-evolving network formulation (Grindrod \& Higham 2014) can be introduced to investigate the effective broadcasting structures over a short-time window as the flow evolves in time. According to the broadcasting structures, vortices are shown to be the key players for flow evolution. This finding was also observed by Jim\'enez (2018) using machining learning through exhaustive simulations. With the present finite-time network approach, we find that the locations where strong vortices pass by are also effective choices to broadcast information and change the flow evolution.

The insights given by the broadcasting structure are leveraged to design distributive forcing to the flow, aiming to modify its time evolution. The forcing are introduced in the form of either an offset in the initial condition or a right-hand-side forcing in the simulations. We observe that the forcing designed through the time-evolving network model is able to perturb the flow in an effective manner. Moreover, kinetic energy dissipation is enhanced with the use of forcing suggested by the time-evolving model. The findings from the current approach provide stimulating use of network analysis to guide flow control efforts.

This work was supported by Army Research Office (W911NF-14-1-0224), Air Force Office of Scientific Research (FA9550-18-1-0040) and Office of Naval Research (N00014-15-R-FO13).

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Authors

Kouta Watanabe, SOKA University
Hideki Shiiba, SOKA University
Yoshio Ishii, SOKA University

Abstract

It is known that the laminar flow is maintained if the flow velocity is small and it becomes turbulent in the middle of the flow if the flow velocity is high by the Reynolds experiment (1988). Especially in the intermediate region, it has also been found that laminar flow and turbulence may occur intermittently, Existence of the region in which this turbulent and laminar flow coexist locally continued to be difficult to understand until recently. This is because the laminar flow is stable even at an infinitely fast flow rate from the linear stabilization analysis, and experimentally, the laminar flow is sometimes maintained up to a relatively large flow velocity in some cases. In particular, it is also known that, when artificially disturbance of a nonlinear perturbation in the flow is given, there is a boundary of returning to laminar flow or turbulence. At that boundary, it appears an interesting localized interior structure in turbulence which are coherent structure or puff structure. When this puff disappears, it returns to laminar flow, but recent studies have predicted that transition from laminar to turbulent flow is similar to Directed Percolation (DP) transition.In this study, based on these results, we have attempted to design a discrete DP model, and considered the mechanism on the transition to turbulent flow using discrete DP model by numerical calculation and analysis. Especially, discrete mathematical model is ultra discretized (tropical discretization), its behavior is investigated. Moreover, inverse ultra discretization is performed to reconstruct discrete mathematical model and its properties are examined.

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Authors

Anton Pershin, University of Leeds
Cedric Beaume, University of Leeds
Steven Tobias, University of Leeds

Abstract

Transition to turbulence in shear flows is often analyzed from the viewpoint of the spatiotemporal dynamics of localized perturbations. In numerical studies, the initial spots used for time-integration are usually generated from random perturbations or turbulence at supercritical Reynolds numbers. In contrast, we consider how exact localized states of plane Couette flow can be used to study dynamics in transitional flows. These are found on two branches of equilibria and travelling waves intertwined in a bifurcation pattern known as homoclinic snaking. We will first discuss homoclinic snaking and its role as a generator of localized initial conditions. Then, we will present a schematic of parameter space which summarizes the various regimes localized initial conditions exhibit involving oscillatory dynamics, spot splitting and front growth. We will explain some of these phenomena in terms of connections between the localized states and other exact solutions in plane Couette flow and, if time permits, explore relaminarization control perspectives.

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Authors

Marcello A. F. Medeiros, University of Sao Paulo
Fernando H. T. Himeno, University of Sao Paulo
Marlon S. Mathias, University of Sao Paulo
Andrés G. Martinez, University of Sao Paulo

Abstract

On airplanes, transition often initiates from small irregular disturbances introduced into the boundary layer which produce TS waves modulated in streamwise and spanwise directions. Such waves have not been studied to the same level as monochromatic ones and there remains open questions about their nonlinear regime.

Modulated waves are well represented by wavepackets, whose most salient nonlinear feature is the growth of oblique low frequency waves. Their frequency is substantially higher than the subharmonic frequency which is generally attributed to detunned subharmonic resonance. However detunned resonance must involve two modes symmetric with respect to the subharmonic frequency, a feature not observed in the packet. This question, among others, motivated this investigation.

The work presents DNS results aimed at reproducing two experimental studies of the literature. The subharmonic stage of the experimental packets was accuratelly reproduced. The numerics were cleaner and the evolution of each mode composing the packet was traced in detail. Analysis showed the evolution of the packet modes agreed almost exactly with the corresponding monochromatic wave case. The above mentioned frequency mismatch was explained as follows. The amplitude of each mode involved is below 0.1% of the free-stream velocity. Hence the resonance does not affect the primary waves, which remain linear. Resonance initiates when primary waves reach a threshold amplitude, and fades away when they reach the same threshold amplitude on the decaying stage after the second branch of the instability loop. Hence when subharmonic waves reach their peak and start to decay, primary waves have already decayed substantially. Since in the boundary layer the frequency of the most unstable wave decays with distance from the leading edge, the combination of these factors produce the mentioned frequency mismatch.

The resonance explained the observation that the nonlinear activity in the packet is concentrated at its trailing edge. A nonlinear activity prior to subharmonic resonance was also identified and shown to be essential to explain the paket evolution. The experiments show that prior to transition there is a rapid nonlinear growth of fundamental oblique modes, a feature that has not been reproduced by previous numerical simulations of wavepackets. The current study reproduced some features of these modes which resemble fundamental instability. However their amplitude was not accurately reproduced. The results suggest the amplitude mismatch may be associated with minute roughnesses on the plate, and this possibility is under investigation with simulations of other experiments. Results should be available for the meeting.

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Authors

Zhongyu Zhang, Tianjin University
Xuesong Wu, Tianjin University, Imperial College London

Abstract

     The nonlinear evolution of multiple helical waves in a subsonic large-radius circular jet, which contains a generalised inflection point, is studied. Helical waves in large-radius circular jets are similar to near-planar oblique waves in planar shear layers as discussed in Wu (1993), which can be characterised by integer azimuthal wavenumbers $m=\pm1,\pm2,\pm3,...\,$. Their amplitudes vary slowly in the circumferential direction, the same magnitude as the streamwise direction as well as time, due to the large nozzle radius naturally. Distinguishing from planar disturbances, or the axisymmetric mode which should be described by a strongly nonlinear theory, helical modes are driven by a weakly nonlinear one when the linear growth rate decreases to $O(\epsilon^{2/5})$ instead of $O(\epsilon^{1/2})$ for the axisymmetric mode; $\eps\ll 1$ is the initial amplitude of the disturbance. A significant feature of the modulation equation is that the derivation with respect to circumferential variable $\theta$ appears in the history-dependent nonlinear terms and displays the three-dimensionality if the large radius $r_c$ is assigned by $r_c^{-1}=O(\epsilon^{1/5})$. Mean-flow distortions of velocity as well as temperature are produced by nonlinear interactions and further drive the evolution. Besides, through the critical-layer analysis, the non-parallelism is also covered by not only the base flow, the streamwise varying and the transverse velocity included, but also a translating-critical-layer effect (see also in Cowley 1985; Wu \& Zhuang 2016). Some proper limits of the spatial-temporal modulation equation are also studied. Finally, it is noted that this is a general theory for inviscid instability on circular shear layers and a necessary step towards finding the acoustic source and radiation of the jet noise. 

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Authors

André V. G. Cavalieri, Instituto Tecnológico de Aeronáutica
Kenzo Sasaki, Instituto Tecnológico de Aeronáutica
Pierluigi Morra, Royal Institute of Technology
Ardeshir Hanifi, Royal Institute of Technology
Dan S. Henningson, Royal Institute of Technology

Abstract

Up to 50% of drag and therefore fuel consumptions of modern aircraft is to overcome the skin friction due to turbulent boundary layers. Any reduction on the skin friction, such as that resulting from the laminar-turbulent transtion delay, would imply in significant savings for the operational costs of such aircraft. With this in mind, this work deals with the control of bypass transition, which is related to strong transient amplifications of upstream perturbations taking the form of streamwise elongated "streaky" structures in a zero-pressure gradient boundary layer. Our approach consists of numerical simulations of a transitional boundary layer, where free-stream turbulence, modelled via a superposition of 200 modes taken from the continous branch of the Orr-Sommerfeld operator induces the streaks. Two levels of FST are considered, corresponding to 3.0% and 3.5%, both of which represent challenging scenarios, where the transition to turbulence occurs via the algebraic growth of the resulting streaks. The set-up consists of localized actuators and sensors which measure wall-shear stress. We develop reduced-order models (ROMs) for such flow by means of empirical transfer functions. Such ROMs allow us to design linear quadratic gaussian regulators for the problem, without having direct access to the impulse responses of the perturbations. Three different actuators are developed, with changes in their corresponding spatial support; a baseline shape, with a vertical component forcing only and two shapes obtained via optimization procedures. A computationally efficient method is derived to obtain an actuator which aims to induce the exact structures which are inside the boundary layer, given in terms of their first spectral proper orthogonal decomposition (SPOD) mode, and an actuator that maximizes the energy of the induced downstream structures. All three actuators lead to significant delays in transition and were shown to be robust to mild variations in the FST levels. Differences are understood in terms of the SPOD of actuation and FST-induced fields along with the causality of the control scheme for a cancellation of the disturbances. Accordingly, the actuator optimized to generate the leading SPOD mode, representing the streaks in the open-loop flow, leads to the highest transition delay. Finally, it is shown that even with the actuator located downstream of the input measurement, it may become impossible to cancel incoming disturbances in a causal way, depending on the wall-normal position of the output and on the actuator considered, which limits sensor/actuator placement capable of good closed-loop performance.

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Authors

Caihong Su, Tianjin University
BingBing Wan, China Aerodynamics Research and Development Center

Abstract

Hypersonic boundary-layer receptivity to free-stream disturbance is investigated for Mach 6 flow over a 5 degree half-angle blunt cone with nose radius of 5.1mm. The governing equations are full Navier-Stokes equations, and 5th order WENO scheme and 6th central difference scheme are used for space discretization and 3rd Runge-Kutta scheme for time integration. A strong entropy layer is formed due to the large transverse entropy gradient behind the shock, which supports a group of instability at low frequencies with an inviscid nature. The base flow filed therefore can be divided into three regions in turn starting from the wall: the boundary-layer region, the entropy-layer region and the outer region. Two different frequencies, i.e. 33kHz and 150kHz of slow acoustic wave with zero incidence angle are considered, which correspond those of the first mode and the second mode, respectively. The free-stream disturbance is superposed on the outer boundary and DNS is performed until the unstable mode is excited in the boundary layer. This case serves as a baseline. In order to figure out the specific route the disturbance takes to trigger the instability mode, a smaller computational domain is taken out from the former one along with the already obtained base flow. Several different disturbances including the boundary-layer disturbances, the entropy-layer disturbances and the perturbations in the outer region are introduced at the inlet and/or the upper boundary of the new computational domain. Numerical simulations show that, for the first mode, the entropy-layer mode is able to excite the first mode near the neutral point, but its efficiency is very low. The transmitted slow acoustic wave behind the shock plays a leading role. For the second mode, the fast mode is first generated in the boundary layer, while downstream the entropy-layer disturbances manifest themselves through forcing mechanism and eventually generate the second mode.

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Authors

Ming Dong, Department of Mechanics, Tianjin University

Abstract

Receptivity of the instability modes in boundary-layer flows is a fundamental topic that determines the initial amplitude of the perturbations in the early stage of laminar-turbulent transition. Unlike the subsonic/incompressible regime, in which the receptivity was described by the triple-deck theory successfully (see for instance Ruban(1984) and Goldstein (1985)), the receptivity of the supersonic inviscid modes is less understood. Although there have already been quite a few numerical investigations on the supersonic receptivity, in this paper, I will employ the large-Reynolds-number asymptotic approach and demonstrate the physical mechanism more clearly. The receptivity to be concerned in this paper is induced by scattering of the freestream acoustic waves by a local roughness element mounted on a flat plate in a supersonic stream. Depending on the wavelength of the instability mode, two mechanisms are taken into account, i.e. the ordinary inviscid mode (the wavelength is comparable with the local boundary-layer thickness) and the long-wavelength first mode (the wavelength is much longer than the local boundary-layer thickness). The width of the roughness is assumed to be comparable with the wavelength of the instability mode. In the large Reynolds number limit, the roughness-induced mean-flow distortion of the former mechanism exhibits a double-deck structure (see Choudhari & Duck (1996)), while that of the latter exhibits a triple-deck structure (see Smith (1973)). The acoustic signature in the main deck is described by the continuous mode of the Rayleigh equation. The interaction of the acoustic wave and the mean-flow distortion in the main deck gives birth to the instability modes, whose initial amplitude is determined by using the Residual theorem. In order to show the numerical solution, I choose three typical cases, in which the oncoming Mach numbers are 3.0, 4.5 and 5.92, respectively. For the first-mode disturbance, the receptivity to a slow acoustic wave is stronger than that to a fast one, while a reversed trend is observed for the second mode. Increase of the roughness height or decrease of the difference between the roughness width and the perturbation wavelength strengthens the receptivity efficiency. The most profound receptivity appears at a certain incident angle of the acoustic wave, and this angle varies from 30 degree to 60 degree depending on the parameters. Interesting enough, for a hypersonic case when the second mode synchronizes with the first mode at the unstable branch, the receptivity coefficient undergoes a sharp increase in the vicinity of the synchronization frequency.

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Authors

Neil D. Sandham, University of Southampton
Nicola De Tullio, University of Southampton

Abstract

Direct numerical simulations are run for a NACA0012 airfoil at Mach 0.4 and Reynolds number 50,000. The flow fields are characterised by transitional separation bubbles that occupy 50-60% of the suction surface and are followed by a region of attached turbulent flow. With increasing sweep angle the transitional flow structures arising in the separation bubble change orientation in a manner that is not explained by any changes in leading-edge normal Reynolds number. At 20 degrees sweep the dominant structures are oblique relative to the leading edge, while at 40 degrees they are perpendicular to the leading edge. At both these angles the flow structures are more coherent than those that arise at zero sweep angle from an oblique mode mechanism, involving a pair of equal and opposite oblique modes. The change at 20 degrees sweep is associated with symmetry breaking, leading to a preference for a single oblique mode. The further change at 40 degrees sweep is associated with the emergence in the DNS of a broadband peak in the energy spectrum with a maximum at Strouhal numbers of 3.5, which are lower than those seen at lower sweep angles.

The changes in the flow are further studied by a global stability analysis based on the time and span-averaged flowfields. At the largest sweep angle a strongly unstable global mode emerges, peaking at St=3 and with a range 1 < St <5 that is in good agreement with the DNS spectra. From the global eigenmode it is possible to discover the mechanism responsible for the mode. Strong spatial amplification is observed in the detached shear layer, consistent with the inflectional nature of the velocity profiles. However the peak growth rate of the global mode does not correspond to the peak amplification factor along the shear layer, which occurs at high frequencies than the global mode. The reason for this is the other components of the mechanism, namely the sound radiation from the trailing edge and the receptivity of the laminar boundary layer upstream, which are enhances at lower frequencies. Thus the dominant global mode amplification arises from a combination of factors. Having identified the important role played by acoustic waves and receptivity, further insight is obtained by considering the dispersion relation of upstream-propagating acoustic waves. This quantifies the aforementioned symmetry-breaking and also places limits on the combinations of spanwise wavenumbers and frequencies that are possible.

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Authors

Vladimir I. Borodulin, Khristianovich Institute of Theoretical and Applied Mechanics SB RAS, Novosibirsk, 630090
Andrey V. Ivanov, Khristianovich Institute of Theoretical and Applied Mechanics SB RAS, Novosibirsk, 630090
Yury S. Kachanov, Khristianovich Institute of Theoretical and Applied Mechanics SB RAS, Novosibirsk, 630090

Abstract

The mechanisms of transformation of free-stream turbulence into the cross-flow (CF) instability modes in 3D boundary layers are of great fundamental and practical interest because they can provide a basis for designing modern advanced methods of the transition prediction. However, such mechanisms are rather poorly studied at present. Most of known investigations in this field are theoretical ones and devoted to purely 2D problems. The goal of the present study was to narrow the gap mentioned above and to obtain quantitative description of one of 3D receptivity mechanisms.
   The experiments were carried out in a low-turbulence wind tunnel on a model of a 25° swept wing at low subsonic freestream speed. The model was equipped with two disturbance sources. One of them was a tiny blowing-suction port, which was used for the excitation of CF wave-trains and obtaining their stability characteristics. Another source was a thin vibrating wire installed upstream the model and equipped with a small bulge. The wire itself generated 2D vortex street, while the bulge produced streamwise oriented vortices propagating near the external border of the boundary layer. This source was shown to excite CF-instability modes by a distributed receptivity mechanism under study. The measurements were carried out by a hot-wire anemometer both inside and outside the boundary layer. Both downstream evolution of the CF-modes and the vortex input were quantified properly.
   The most important results of the study include: (i) a definition and a procedure of determination of the distributed vortex-receptivity coefficients, based on approximation of the experimental data by analytical solutions of an evolutionary equation, (ii) two data sets containing information on characteristics of the base-flow instability to the CF-modes and on its distributed receptivity, (iii) the distributed vortex-receptivity coefficients determined in Fourier space as functions of the spanwise wavenumber and frequency.
   It is found that for all frequencies the receptivity to quasi-2D free-stream vortices very much weaker compared to the essentially 3D vortices. Excitation of very large amplitudes of the CF-waves is found in a range of negative spanwise wavenumbers. This phenomenon is not connected with the spanwise-wavenumber dependence of the distributed receptivity coefficients but is explained by presence of (or closeness to) the streamwise-wavenumber resonance, possible in the 3D subsonic boundary layers but impossible in 2D ones.
   This study is supported by the Boeing Operations International, Inc. and by the Russian Academy of Sciences.

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Authors

Barry Crowley, City University London
Chris Atkin, City University London

Abstract

A set of experiments were performed to investigate the physical mechanisms that cause discrete suction perforations to destabilise boundary layers when the suction rate is very large. As is well established in the literature, at low suction rates or when the suction is uniformly distributed: suction has a stabilising effect on boundary layers and thus suppresses transition to turbulence. At high suction rates the localised flow distortions induced by discrete suction perforations, are known to adversely affect the transition process. In this study, two different transition mechanisms were found; the dominant mechanism, depended on how far the transition front was located from the suction perforations, before suction was applied.

It was found that if the transition front was close to the suction perforations: low frequency disturbances amplified by the suction perforations appeared to interact with low frequency oblique modes that dominate the weakly non-linear stages of an N/H-type transition process. This interaction would increase the amplitude of the oblique modes, causing the transition front to move forward. In the absence of a transition front, or if the transition front was sufficiently far from the suction perforations, it was found that stationary modes induced by the suction perforations would dominate the transition process. It was found that these stationary modes were spatially amplified across the suction array, and would produce inflectional stream-wise velocity profiles (along the wall-normal) if sufficiently large, after this a high frequency secondary wave would be strongly amplified and would dominate in the non-linear stages of the transition process.

The low frequency disturbances are suspected to be amplified by inflectional span-wise velocity profiles. In the absence of a non-linearly unstable flow with which to interact, these low frequency structures dissipate down-stream of the suction array. In this case the growth of the stationary modes is sufficient to cause high frequency secondary waves to grow which amplify more rapidly than the low frequency, and thus dominate the transition process.

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Authors

Helen Reed, Texas A&M University,

Abstract

This presentation will begin with an overview of the theoretical and computational boundary-layer receptivity, stability, and control research conducted by the speaker over the past four decades in collaboration with her colleagues and students. Her overarching approach has been one of verification and validation, including working closely with experimentalists (both with ground facilities and flight platforms) to achieve closure between theory and experiment. Her team has worked to identify instability mechanisms in complex 3-D configurations and develop physics-based models/tools, guidelines, and correlations (surface features). She will present recent results in subsonic and hypersonic flow, and then close with a discussion of current opportunities and challenges.

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Authors

Jeffrey Crouch, Boeing Company

Abstract

Approaches to transition prediction are considered within the context of applications to aircraft, where surface-induced distortions are an integral part of the boundary-layer flow. Different routes to transition are discussed as a function of the type and level of surface irregularity perturbing the flow. Amplitude methods offer the best potential for predicting the gradual movement of transition with increasing levels of flow distortion. Global stability analysis offers the potential for capturing abrupt transition near the source of the flow distortion (typically characterized as bypass transition). A semi-empirical amplitude method, based on a combination of stability theory and experiments, is considered for a-priori predictions to capture gradual changes to transition for realistic surface irregularities. This predictive framework is then used to characterize the different mechanisms and flow regimes responsible for boundary-layer transition in the presence of surface steps and gaps.