Research
Papers
Archetypes in mechanics
Hoepffner
Mon Habilitation à diriger les recherches. Soutenance le mardi 15 septembre 2015 à 10h. Jussieu: barre 5565, troisième étage, salle 311. La soutenance sera en français.
[pdf]
Image:
Front matter.

Abstract:
 There too, I have a comparison, because I like to work with comparisons. I would say that this is a little bit the problem of painting at the time of Chassériaux or Ingres, and of impressionist painting. There was a time when we got lost in the perfect detail and we did not see that what was important was to get a large contrast, the fact that this shadow was blue and not black, things like that...
 Yes, but this was partly induced by photography.
 You are absolutely right! Photography killed Ingres, in some way, and I would say that simulations play a similar role to photography when it comes to our subject; this is the same thing.
And the reports from the three reviewers: Médéric Argentina from Nice, Tomas Bohr from Copenhagen and Hamid Kellay from Bordeaux:
rapports_HDR.pdf

Instability and breakup of the capillary bridge with throughflow: the "capillary Venturi"
Paré & Hoepffner
Journal of Fluid Mechanics, submitted.
[pdf]
Image:
Sketch of the flow configuration: a capillary bridge with throughflow.

Abstract:
The capillary bridge is an axisymmetric body of liquid held between two discs. Its bifurcations is rich, as it can breakup for two different reasons: if its length is larger than its perimeter, it is unstable to the RayleighPlateau instability, and if its volume is decreased for a given length and radius, there is a fold bifurcation that leads to the disappearance of the steady solution. To this classical system we add a dynamic effect: we consider the behavior of the bridge when there is a flow through it.
By analogy with the instability of a bandsaw, we show that the RayleighPlateau instability is a particular case of the dynamic instability of the capillary Venturi. We also describe a model that explains why the throughflow decreases the range of possible volumes of the capillary bridge. We compare these analyses to the nonlinear bifurcation diagram of a 1D model and numerical simulations of the NavierStokes equations.

A model for the global structure of selfsimilar vortex sheet rollup
Hoepffner & Fontelos
Physics of fluids, REJECTED.
[pdf]
Image:
Numerical simulation of the Euler equation for a vortex sheet with a local defect, showing the selfsimilar growth of a doublevortex solution.

Abstract:
Most vortices are born from the rollup of a shear layer. The rollup is traditionally modeled as the selfsimilar winding of an infinite spiral connected to a thin shear layer: a vortex sheet. We demonstrate a composite vortex sheetpoint vortex model to quantify the global structure of two archetypal cases of selfsimilar rollup. These cases are Kaden's single spiral solution of the wingtip vortex and Pullin's double spiral solution of the nonlinear impulse response of the KelvinHelmholtz instability. The model consists in replacing the spiral with a point vortex of equal vortical intensity and accounting for the mutual interaction of the point vortex and the untouched sheet. We show that the model itself has an attractive selfsimilar solution which compares well with numerical experiments.

Vortices catapult droplets in atomization
Jerome, Marty, Matas, Zaleski & Hoepffner
Physics of Fluids, vol. 25, 112109.
[pdf]
http://dx.doi.org/10.1063/1.4831796
Image:
A sketch of the catapult mechanism by which the vortex shedding sequence behind a liquid wave throws droplets very high.

Abstract:
A droplet ejection mechanism in planar twophase mixing layers is examined. Any disturbance on the gasliquid interface grows into a KelvinHelmholtz wave and the wave crest forms a thin liquid ﬁlm that ﬂaps as the wave grows downstream. Increasing the gas speed, it is observed that the ﬁlm breaksup into droplets which are eventually thrown into the gas stream at large angles. In a ﬂow where most of the momentum is in the horizontal direction, it is surprising to observe these large ejection angles. Our experiments and simulations show that a recirculation region grows downstream of the wave and leads to vortex shedding similar to the wake of a backwardfacing step. The ejection mechanism results from the interaction between the liquid ﬁlm and the vortex shedding sequence: a recirculation zone appears in the wake of the wave and a liquid ﬁlm emerges from the wave crest; the recirculation region detaches into a vortex and the gas ﬂow over the wave momentarily reattaches due to the departure of the vortex; this reattached ﬂow pushes the liquid ﬁlm down; by now, a new recirculation vortex is being created in the wake of the wave—just where the liquid ﬁlm is now located; the liquid ﬁlm is blownup from below by the newly formed recirculation vortex in a manner similar to a bagbreakup event; the resulting droplets are catapulted by the recirculation vortex.
The movie showing the experiments, included as supplementary material can be seen here: catapult_light.mov[11Mo] or here: catapult_heavy.mov[24Mo].

Instability regimes in the primary breakup region of planar coflowing sheets
Fuster, Matas, Marty, Popinet, Hoepffner, Cartellier & Zaleski
Journal of Fluid mechanics, vol. 736, pages 150176.
[pdf]
http://dx.doi.org/10.1017/jfm.2013.536
Image:
The distribution of vorticity during One cycle of the instability.

Abstract:
This article investigates the appearance of instabilities in two planar coﬂowing ﬂuid sheets with diﬀerent densities and viscosities via experiments, numerical simulation and linear stability analysis. At low dynamic pressure ratios a convective instability is shown to appear for which the frequency of the waves in the primary atomization region is inﬂuenced by both liquid and gas velocities. For large dynamic pressure ratios an asymptotic regime is obtained in which frequency is solely control led by gas velocity and the instability becomes absolute. We show that in this regime the splitter plate thickness can also aﬀect the nature of the instability if it is larger than the gas vorticity thickness. Computational and experimental results are in agreement with the predictions of a spatiotemporal stability analysis.

Recoil of a liquid filament: escape from pinchoff through creation of a vortex ring
Hoepffner & Paré
Journal of Fluid Mechanics, vol 734 (2013) pages 183197.
[pdf]
http://dx.doi.org/10.1017/jfm.2013.472
Image:
Display of our experiment: water freefalling from a vertical straw. It seems that the tip is going to pinch, but no... the pinching is escaped. Why? 
Abstract:
A liquid ﬁlament recoils because of its surface tension. It may recoil to one sphere: the geometrical shape with lowest surface, or otherwise segment to several pieces which individually will recoil to spheres. This experiment is classical and its exploration is fundamental to the understanding of how liquid volumes relax. In this paper, we uncover a mechanism involving the creation of a vortex ring which plays a central role in escaping segmentation. The retracting blob is connected to the untouched ﬁlament by a neck. The radius of the neck decreases in time such that we may expect pinchoﬀ. There is a ﬂow through the neck because of the retraction. This ﬂow may detach into a jet downstream of the neck when ﬂuid viscosity exceeds a threshold. This sudden detachment creates a vortex ring which strongly modiﬁes the ﬂow pressure: ﬂuid is expelled back into the neck which in turn reopens.
The movie showing the experiments, included as supplementary material can be seen here: pinching.mov[8Mo]
The parameter file for the numerical simulations with gerris and explanations is here: parameterfile.pdf

Elasticity in the physics of sports
Hoepffner
Conference "The physics of sports", April 2012, http://www.ladhyx.polytechnique.fr/home/physicsofsports.
[pdf]
Image:
Understanding the role played by the elasticity
of muscle/tendons by taking the example of the bounce of
the kangaroo. 
Abstract:
Elasticity plays a role in many instances of sports. The aim
of this introductory paper is to sketch a general viewpoint
that would help building links between the different contributions to the conference involving elasticity.

Models for an alternative pole vault
Hoepffner
Conference "The physics of sports", April 2012, http://www.ladhyx.polytechnique.fr/home/physicsofsports.
[pdf]
Image:
The two free parameters for the pole are its length and its stiffness. This diagram shows how the window of good properties gets thinner for longer poles. 
Abstract:
Pole vault is the athletic discipline of using a soft pole to jump as high as possible. The vaulter should run fast, and must carefully fit the length and stiffness of his pole to his size, strength and gymnastics ability. In this paper, we propose a springmass model for the vault as a tool to explore the possibilities in trajectories and performance.

Le monde: sciences appliquées
Citation de mon travail dans l'article du quotidien, le jours de la médaille d'or olympique de Renaud Lavillénie 5m97, et un dossier de "l'équipe", avec une interview intéressante du champion:
lemonde.pdf

The evolution of a localized nonlinear wave of the KelvinHelmholtz instability with gravity
Orazzo and Hoepffner.
Physics of fluids, Vol 24, issue 11 (2012)
[pdf]
http://link.aip.org/link/?PHF/24/112106
Image:
In gray the selfsimilar wave, and in black the same thing but with gravity: the wave reaches a plateau at a height such that the hydrostatic pressure gradient in the liquid is proportional to the wind stagnation pressure.

Abstract:
At the interface between a liquid and a gas in the presence of gravity, there are well known surface waves which can propagate for long distance with little attenuation, as it is for instance the case at the surface of the sea. If wind is present, these wave can progressively accumulate energy as they propagate and grow to large sizes during storms. On the other hand, for a given wind strength, there is potential for the spontaneous creation of a localized wave: this wave can reach a size such that the hydrostatic pressure drop from top to bottom equals the stagnation pressure of the wind. This process for the disruption of the flat interface is localized and nonlinear. We study the properties of this localized and nonlinear wave using numerical simulations of the NavierStokes equations.

Oblique waves lift the flapping flag
Hoepffner & Naka.
Physical Review Letters, Novembre 2011, Vol 107, Issue 19
[gallery] 
[pdf]
http://link.aps.org/doi/10.1103/PhysRevLett.106.104502
Parallel simulation of multiphase flows using octree adaptivity and the volumeoffluid method
Agbaglah, Delaux, Fuster, Hoepffner, Josserand, Popinet, Ray, Scardovelli and Zaleski.
Comptes rendus de l'académie des sciences vol 339 (2011), pages 194207
[pdf]
Image:
This is a simulation of the impact of a falling liquid drop (in blue) into a bath of the same liquid (in green), like for instance a raindrop falling into a pool. The splash happens very quickly, much faster than the eye could see, and brings about a growing corrola which itself destabilizes into a crown of smaller drops. This simulation is obtained by solving the NavierStokes equations.

Abstract:
We describe computations performed using the Gerris code, an opensource software implementing ﬁnite volume solvers on an octree adaptive grid together with a piecewise linear volume of ﬂuid interface tracking method. The parallelisation of Gerris is achieved by domain decomposition. We show examples of the capabilities of Gerris on several types of problems. The impact of a droplet on a layer of the same liquid results in the formation of a thin air layer trapped between the droplet and the liquid layer that the adaptive reﬁnement allows to capture. It is followed by the jetting of a thin corolla emerging from below the impacting droplet. The jet atomisation problem is another extremely challenging computational problem, in which a large number of small scales are generated. Finally we show an example of a turbulent jet computation in an equivalent resolution of 6 × 1024^3 cells. The jet simulation is based on the conﬁguration of the Deepwater Horizon oil leak.

Selfsimilar wave produced by local perturbation of the KelvinHelmholtz shearlayer instability
Hoepffner, Blumenthal & Zaleski.
Physical Review Letters, March 2011, Vol 106, No 10
[gallery] 
[pdf]
http://link.aps.org/doi/10.1103/PhysRevLett.106.104502
Image: We show the difference between a shear layer excited by a wavelike perturbation (right, the Kelvin case), and excited by a dirac (left, the Helmholtz case). We see clearly the growth of a selfsimilar structure.

Abstract:
We show that the KelvinHelmholtz instability excited by a localized perturbation yields a selfsimilar wave. The instability of the mixing layer was first conceived by Helmholtz as the inevitable growth of any localized irregularity into a spiral, but the search and uncovering of the resulting selfsimilar evolution was hindered by the technical success of Kelvin's wavelike perturbation theory. The identification of a selfsimilar solution is useful since its specific structure is witness of a subtle nonlinear equilibrium among the forces involved. By simulating numerically the NavierStokes equations, we analyze the properties of the wave: growth rate, propagation speed and the dependency of its shape upon the density ratio of the two phases of the mixing layer. 
Realizing turbulent statistics
Hoepffner, Naka & Fukagata.
Journal of Fluid Mechanics, vol. 676, 2011, pages 5480.
[gallery] 
[pdf]
http://journals.cambridge.org/abstract_S0022112011000322
Image: A turbulent shear layer is simulated (top), and the statistics at x=10, 20, 30 are artificially reproduced using the filter technique at the inlet of three new simulations. The images show snapshots of the vorticity field.

Abstract:
The twopoints twotimes covariance precisely describes the fluctuations of a turbulent
flow. We present a technique to produce a random field with a given twopoints two
times covariance. It is obtained as the output of a linear filter fed with white noise. We
use this method to generate fully turbulent inflow conditions in numerical simulation of
fluid flows. The method is illustrated on the simulation of a turbulent free shear layer.
The filter coefficients are obtained from the resolution of the YuleWalker equation, and
the computation can be performed efficiently using a recursive solution procedure. The
method should also be useful in the study of flow receptivity, when the processes of
transition to turbulence are sensitive to the perturbation environment.

The phase relationship in laminar channel flow controlled by wavelike blowing/suction
Mamori & Fukagata & Hoepffner.
Physical review E, april 2010, volume 81, number 4.
[pdf]
http://link.aps.org/doi/10.1103/PhysRevE.81.046304
Image:
Blowing and suction is applied at the walls of a channel, in the form of a traveling wave. Here we define the critical layer: this is the position in the channel where the fluid velocity is equal to the speed of propagation of the wave imposed at the walls.

Abstract:
The phase relationship between the streamwise and the wallnormal velocity disturbances induced by a traveling wavelike blowing/suction control [T. Min et al., J. Fluid Mech. 558, 309 (2006)] in a twodimensional laminar Poiseuille ﬂow is investigated. The investigation is done by solving the linearized NavierStokes equation and by using the identity equation between the skinfriction drag and the Reynolds shear stress [K. Fukagata et al., Phys. Fluids 14, L73 (2002)]. It has been known that a traveling wave creates a nonquadrature between the velocity disturbances and generates the positive phase shift of the streamwise velocity disturbance in the case of a skinfriction drag reduction. The present analysis further reveals that this nonquadrature consists of an inviscid base phase relationship and a nearwall phase shift induced by the viscosity. The analogy between the present control and Stokes’ second problem is discussed. The thickness of the nearwall region in which the viscous phase shift takes place is found to be scaled similarly to the Stokes’ second problem.

Mechanisms of nonmodal energy amplification in channel flow between compliant walls
Hoepffner, Bottaro & Favier.
Journal of Fluid Mechanics, vol. 642, january 2010, pages 489507.
[gallery] 
[preprint (1M)]
http://journals.cambridge.org/action/displayAbstract?aid=6902140
Image:
The waves propagating along the elastic membrane are associated with fluid movement in the channel: here sinuous waves for three different wavelengthes. We show how these waves can be related to transient growth of energy, when they form standing patterns.

Abstract: The mechanisms leading to large transient growth of disturbances for the
flow in a channel with compliant walls are investigated. The walls are modeled as
thin springbacked plates, and the flow dynamics is modeled using the
NavierStokes equations linearised about the parabolic Poiseuille profile.
Analysis for streamwiseinvariant perturbations show that this fluidstructure
system can sustain oscillatory energy evolution of large amplitude, in the form
of spanwise standing waves. Such waves are related to the travelling
waves which free wall can support, modified to account for an 'added mass' effect.
Simple scaling arguments are found to provide results in excellent agreement with
computations of optimal disturbances, for low to moderate values of the
stiffness parameter characterising the compliant surface. Movies are available with the online version of the paper.

Pumping or drag reduction?
Hoepffner & Fukagata.
Journal of Fluid Mechanics, vol. 635, 2009, pages 171187.
[gallery] 
[preprint (1M)]
http://journals.cambridge.org/action/displayAbstract?aid=6166000
Image: In both cases, the flow is subject to a wave going from left to right. On the top, there is a wave of wall deformation (peristalsis) and the pumping is to the right; on the bottom, there is a wave of blowing and suction through the wall, and the pumping is to the left.

Abstract: Two types of wall actuation in channel flow are considered: traveling waves of
wall deformation (peristalsis) and traveling waves of blowing and suction. The
flow response and its mechanisms are analyzed using nonlinear and weakly
nonlinear computations. We show that both actuations induce a flux in the
channel in absence of imposed pressure gradient and can thus be characterized as
pumping. In the context of flow control, pumping and drag reduction are strongly
connected, and we seek to define them properly based on these two actuation
examples. Movies showing the flow motion for the two types of actuations are
available with the online version of this paper.

Inputoutput analysis and control design applied to a linear model of spatially developing flows
Bagheri, Hoepffner, Schmid & Henningson.
Applied mechanics review, 2009, vol. 62, Issue 2.
[gallery] 
[preprint (2M)]
Image:
We analyse and control a one dimentional model of fluid flow: the complex GinzburgLandau equation. Top: position of perturbation input, sensor and actuator for feedback control, and bottom: two views of the response of the system when the control is turned on or off. These techniques can be used to control the instabilities in fluid flows.

Abstract: This review presents a framework for the inputoutput analysis, model reduction and control design for fluid dynamical systems using examples applied to the linear complex GinzburgLandau equation. Major advances in hydrodynamics stability, such as global modes in spatially inhomogeneous systems and transient growth of nonnormal systems is reviewed. Inputoutput analysis generalizes hydrodynamic stability analysis by considering a finitetime horizon over which energy amplification, driven by a special input (disturbances/actuator) and measured at a specified output (sensor), is observed. In the control design the loop is closed between the output and the input through a feedback gain. Model reduction approximates the system with a loworder model, making modern control design computationally tractable for systems of large dimensions. Methods from control theory are reviewed and applied to the GinzburgLandau equation in a manner that is readily generalized to fluid mechanics problems, thus giving a fluid mechanics audience an accessible introduction to the subject.

Stochastic approach to the receptivity problem applied to bypass transition in boundary layers
Hoepffner & Brandt.
Physics of Fluids, 20, 024108, 2008.
[gallery] 
[preprint (1.7M)]
http://link.aip.org/link/doi/10.1063/1.2841621
Image: comparison of the most growing flow perturbations (left) to the structures observed in a simulation of a turbulent boundary layer (right). We show that using tools of stochastic analysis can be usefull to understand the transition to turbulence.

Abstract: To study the flow behaviour in the presence of external disturbances of chaotic nature, a stochastic approach is pursued. In particular, transition to turbulence in boundary layers exposed to high levels of freestream turbulence is considered. The late stages of this transition scenario, characterised by the growth and breakdown of streamwiseelongated streaks, are examined by considering the linear evolution of perturbations to a base flow consisting of the Blasius profile and the streaks. A stochastic initial condition is considered where the freestream perturbations are described by
the correlations of isotropic homogeneous turbulence. The spatial correlation of the excited flow at later times can be computed by the numerical solution of a Lyapunov equation. It is shown that freestream turbulence has the necessary features to excite secondary energy growth, thus playing a central role in the transition to turbulence. The method proposed here can be used to examine the receptivity of other flows to external noise whose statistical properties are known or can be modelled.

Optimal growth, model reduction and control in a separated boundarylayer flow using global modes
Åkervik, Hoepffner, Ehrenstein & Henningson.
Journal of Fluid Mechanics, vol. 579, 2007, pages 305314.
[preprint (800k)]
http://journals.cambridge.org/action/displayAbstract?aid=1006684
Image:
Several eigenmodes (oscillating solutions around the steady state), for the recirculating flow in a boundary layer. Several of these modes are unstable.
We show what is the instability mechanisms, based on the pressure,we show that these eigenmodes can be combined to obtain a large transient growth of energy,and we use these modes to perform feedback control,using one sensor and one actuator.

Abstract:
Twodimensional global eigenmodes are used as a projection basis both for analysing the
dynamics and building a reduced model for control in a prototype separated boundary layer
flow. In the present configuration, a high aspect ratio smooth cavitylike geometry
confines the separation bubble. Optimal growth analysis using the reduced basis shows
that the sum of the highly nonnormal global eigenmodes are able to describe a localized
disturbance. Subject to this worstcase initial condition, a large transient growth
associated with the development of a wavepacket along the shear layer followed by a
global cycle related to the two unstable global eigenmodes is found. The
flow simulation
procedure is coupled to a measurement feedback controller, which senses the wall
shear stress at the downstream lip of the cavity and actuates at the upstream lip. A reduced
model for the control optimization is obtained by a projection on the least stable
global eigenmodes, and the resulting linearquadraticgaussian controller is applied to
the NavierStokes time integration. It is shown that the controller is able to damp out
the global oscillations.

Steady solutions of the NavierStokes equations by selective frequency damping
Åkervik, Brandt, Henningson, Hoepffner, Marxen & Schlatter.
Physics of Fluids, 18, 068102, 2006.
[preprint (500k)]
http://link.aip.org/link/doi/10.1063/1.2211705
Image:
The unstable steady state of a recirculating boundary layer can be computed using this technique.

Abstract:
A new method, enabling the computation of steady solutions of the NavierStokes
equations in globally unstable configurations, is presented. We show that it is
possible to reach a steady state by damping the unstable frequencies. This is
achieved by adding a dissipative relaxation term proportional to the
highfrequency content of the velocity fluctuations. Results are presented for
boundarylayer cavitydriven separation and a separation bubble induced by an
external pressure gradient.

Linear feedback control and estimation
applied to instabilities in spatially
developing boundary layers
Chevalier, Hoepffner, Åkervik & Henningson.
Journal of Fluid Mechanics, vol. 588, 2007, pages 163187.
[preprint (1M)]
http://journals.cambridge.org/action/displayAbstract?aid=1345888
Image:
A boundary layer controlled at the wall by feedback control, viewed from above. We see the unstable waves growing from left to right. Top: without control, bottom: with control.

Abstract:
This paper presents the application of feedback control to spatially
developing boundary layers.
It is the natural followup of Högberg and Henningson (2001), where exact
knowledge of the entire flow state was assumed for control. We apply recent
developments stochastic models for the external sources of disturbances that
allow the efficient use of several wall measurement for estimation of the flow
evolution: the two components of the skinfriction and the pressure fluctuation
at the wall.
Perturbations to base flow profiles of the family of FalknerSkanCooke
boundary layers are estimated by use of wall measurements. The estimated state
is in turn fed back for control in order to reduce the kinetic energy of the
perturbations. The control actuation is achieved by means of unsteady blowing
and suction at the wall.
Flow perturbations are generated at the upstream region in the
computational box and are propagating in the boundary
layer. Measurement are extracted downstream over a thin strip,
followed by a second thin strip where the actuation is performed.
It is shown that flow disturbances can be efficiently estimated and
controlled in spatially evolving boundary layers for a wide range of
base flows and disturbances.

Transient growth on boundary layer streaks
Hoepffner, brandt & Henningson.
Journal of Fluid Mechanics, vol. 537, 2005, pages 91100.
[preprint (500k)]
http://journals.cambridge.org/action/displayAbstract?aid=325458
Image:
Visualisation of a structure that may grow to a large amplitude on the top of a boundary layer streaks. Top: initial condition, bottom: at the time of maximum growth. The flow is from left to right.

Abstract:
The linear perturbations evolving on streamwise boundary layer
streaks which yield maximum energy growth are computed. The steady and spanwise
periodic streaks arising from the nonlinear saturation of optimally growing
streamwise vortices are considered as base flow. It is shown that significant
transient growth may occur for both sinuous antisymmetric perturbations and for
varicose symmetric modes. The energy growth is observed at amplitudes
significantly below the threshold beyond which the streaks become linearly
unstable and is largest for sinuous perturbations, to which the base flow
considered first become unstable. The optimal initial condition consists of
velocity perturbations localised in the regions of highest shear of the streak
base flow, tilted upstream from the wall. The optimal response is still
localised in the areas of largest shear but it is tilted in the flow
direction. The most amplified perturbations closely resemble the unstable
eigenfunctions obtained for streaks of higher amplitudes. The results suggest
the possibility of a transition scenario characterised by the nonmodal growth
of a primary perturbations, the streaks, followed by the secondary transient
growth of higher frequency perturbations. Implication for turbulent flow is also
discussed.

State estimation in wallbounded flow systems:
Part 2. Turbulent flows
Chevalier, Hoepffner, Bewley & Henningson.
Journal of Fluid Mechanics, vol. 552, 2006, pages 167187
[preprint (1.7M)]
http://journals.cambridge.org/action/displayAbstract?aid=420478
Image: We use the statistics of the turbulent flow to optimize for the best estimator. Here are shown the covariance matrices for two wavenumbers.

Abstract:
This work extends the estimator developed in Part 1 of this
study (Hoepffner et al., J. Fluid Mech., to appear) to the problem of
estimating a turbulent channel flow at Re tau=100 based on a history of noisy
measurements on the wall. The key advancement enabling this work is the
development and implementation of an efficient technique to extract, from direct
numerical simulations, the relevant statistics of an appropriatelydefined
``external forcing'' term on the NavierStokes equation linearized about the
mean turbulent flow profile. This forcing term is designed to account for the
unmodelled (nonlinear) terms during the computation of the (linear) Kalman
filter feedback gains It is demonstrated that by applying the optimal feedback
gains, satisfactory correlation between the actual and estimated flow is
obtained in the nearwall regions. Both Kalman and extended Kalman filters are
evaluated and naturally the extended filter is giving better correlations
between the actual and estimated flow, however the Kalman filter gives good
performance in the nearwall regions.

State estimation in wallbounded flow systems:
Part 1. Perturbed laminar flows
Hoepffner, Chevalier, Bewley & Henningson.
Journal of Fluid Mechanics, vol. 534, 2005, pages 263294.
[preprint (1.7M)]
http://journals.cambridge.org/action/displayAbstract?aid=314562
Image:
We measure the pressure and stress at the wall in the boundary layer. From this information, we can reconstruct (right) the full flow field of the evolving wavepacket (left).

Abstract:
In applications involving the modelbased control of transitional
wallbounded flow systems, one often desires to estimate the interior
flow state based on a history of noisy measurements from an array of
flushmounted skinfriction and pressure sensors on the wall. This
paper considers this estimation problem, using a Kalman filter based
on the linearised NavierStokes equations and appropriate stochastic
models for the relevant statistics of the initial conditions, sensor
noise, and external disturbances acting on the system. We show that a
physically relevant parameterisation of these statistics is key to
obtaining well resolved feedback kernels with appropriate spatial
extent for all three types of flow measurements available on the wall.
The effectiveness of the resulting Kalman and extended Kalman filters
that implement this feedback is verified for both infinitesimal and
finiteamplitude disturbances in direct numerical simulations of a
perturbed laminar channel flow. The consideration of timevarying
feedback kernels is shown to be particularly advantageous to
accelerate the convergence of the estimator from unknown initial
conditions. A companion paper (Part 2) considers the extension of
such estimators to the case of fullydeveloped turbulence.

Conference proceedings
Une vague autosemblable pour l'atomisation
Hoepffner, Blumenthal and Zaleski.
Rencontre du non linéaire 2010
[preprint]
Pumping effects by surface traveling waves (in japanese)
Hoepffner and Fukagata.
Proc. JSME Fluid Engineering Conference 2009, Nagoya, November 78.
[preprint]
Friction drag reduction by traveling wavelike surface heating and cooling (in japanese)
Mamori, Hoepffner and Fukagata.
Proc. JSME Fluid Engineering Conference 2009, Nagoya, November 78.
[preprint]
Une instabilité d'origine visqueuse pour l'atomisation
Anne Bagué, Jérôme Hoepffner, Stéphane Zaleski.
Congrès Français de mécanique 2009, Marseille.
[preprint]
Pumping or drag reduction?
Hoepffner, Fukagata.
Proc. 2008 meeting of the ICTAM, Adelaide.
[preprint]
Linear analysis of drag deduction by traveling wavelike body force in channel flow (in Japanese)
Mamori, Fukagata, Hoepffner & Obi.
Proc. 2008 Annual Meeting of Japan Society of Fluid Mechanics, Kobe.
[Abstract] 
[preprint (300K)]
Cavity flow control
Congrès Français de mécanique 2007, Grenoble.
[preprint]
Stochastic excitation on boundary layer streaks
Congrès Français de mécanique 2007, Grenoble.
[preprint]
Identification of sources of sound in low mach
number flows by the use of flow field eigenmodes
Proceedings of 13th international congress on sound and vibrations, Vienna, July 2006.
[Abstract] 
[preprint (300K)]
Control of cavitydriven separated boundary layer
Proceedings of the Conference on active flow control, Berlin, September 2006.
[Abstract] 
[preprint (1.5M)]
Modeling flow statistics using convex optimization
proceedings of the 44th IEEE Conference on Decision
and Control (CDC) and European Control Conference (ECC) 2005.
[Abstract] 
[preprint (200k)]
Control of instabilities in a cavitydriven separated boundarylayer flow
Proceedings of the 3rd Symposium on Global Flow instability and control, Crete.
Linear feedback control of transition in shear flows
Proceedings of the 6th IUTAM symposium on LaminarTurbulent transition, Bangalore 2004..
[Abstract] 
[preprint (300k)]
Theses
Stability and control of shear flows subject to stochastic excitations
Doctoral thesis (I can mail printout version on demand)
[Abstract]
[preprint (5.4M)] 
[preprint (12M)]
Control and estimation of wall bounded flow systems
Licentiate thesis (I can mail printout version on demand)
[Abstract] 
[preprint (1.2M)]
talks
Follow the link on the title if available
for the slides of the presentation, and the link on the location for
related web pages.
2013

October 
Recoil of a liquid filament: escape from pinchoff through creation of a vortex rin
The work on the retraction of the capillary ligament, together with my PhD student Gounséti Paré. Invited seminar at the universtity of bristol.
School of mathematics
Bristol, United Kingdoms.

September 
Selfsimilar KelvinHelmholtz wave
The work on the stability of the mixing layer. Conference organized by the GDR Phenix at université Paris Diderot.
Nonlinear hydrodynamic waves: wave interactions and wave turbulence
Paris, France.

July 
Selfsimilar evolution of the KelvinHelmholtz instability
The work on the stability of the mixing layer. Conference organized at Ecole Polytechnique.
Trends in Open Shear Flow Instability
Paris, France.

June 
Segmentation capillaire
Présentation du travail avec Gounséti Paré sur la segmentation capillaire et le tourbillon à l'intérieur du filament liquide. Journée Nicoise de mécanique des fluides.
Laboratoire LPMC
Nice, France.

April 
Segmentation capillaire
Présentation du travail avec Gounséti Paré sur la segmentation capillaire et le tourbillon à l'intérieur du filament liquide.
ESPCI, laboratoire PMMH
Paris, France.

January 
Segmentation capillaire
Présentation du travail avec Gounséti Paré sur la segmentation capillaire et le tourbillon à l'intérieur du filament liquide.
MSC,groupe dynamique des systèmes hors équilibre
Paris, France.

2012

August 
Ondes obliques sur le drapeau
Journées de la matière condensée, organisées par la société Française de Physique. Mini colloque "Elasticité et géométrie des objets de basse dimensionnalité"organisé par José Bico et Benoit Roman.
http://www.icgm.fr/spip.php?article1001,
Montpellier, France.


Oblique waves on the flapping flag
The waves on the flag, for a talk at the ICTAM conference in Beijin. (ICTAM=International Conference on Theoretical and Applied Mechanics).
ICTAM 2012,
Beigin, China.

May 
Capillary segmentation
During my visit at LEGI in Grenoble to work with JeanPhilippe Matas on the atomisation problem, I present for the first time our results with Gounséti Paré of the vortex ring inside the liquid filament.
LEGI,
Grenoble, France.

April 
Models for an alternative pole vault
Presentation of my work on pole vaulting at the conference "Physics of sports" organized by Christophe Clanet at Ecole Polytechnique.
Physics of sports,
Paris, France.

2011

December 
Ondes obliques sur le drapeau
L’instabilité du drapeau est un cas d’école d’instabilité fluide/structure, ainsi qu’une piste pour l’étude de la nage des poissons. Cette instabilité est également un phénomène que nous pouvons observer quotidiennement. Nous montrons que contrairement à l’imaginaire collectif qui les veut verticales, les ondes sur le drapeau sont obliques. Nous construisons un modèle et évaluons ses implications, notamment, les ondes obliques contribuent au soutient du drapeau par un effet de portance.
Laboratoire Matière et Systèmes Complexes,
Paris, France.

September 
Instabilité de KelvinHelmholtz localisée, non linéaire et autosemblable
Plutôt que d'aborder la fameuse instabilité comme Kelvin avec une onde de
faible amplitude, nous l'excitons comme Helmholtz l'avait d'abord pensée,
avec une perturbation localisée violente. On met ainsi en évidence une
vague autosemblable modèle dont nous chercherons les traces dans les
processus d'atomisation et à la surface des océans en tempêtes.
LadHyx,
Paris, France.

2010

November 
Pumping or drag reduction?
The work with Koji Fukagata, presented at the Division of fluid mechanics conference of the Americal Physical Society.
APS DFD 2010,
Long Beach, California, USA.


A selfsimilar wave for spray atomization
The work with Ralf Blumenthal and Stephane Zaleski on atomization presented during a visit to the lab of François Gallaire.
Laboratoire de mécanique de fluides et instabilités, LFMI, Ecole polytechnique fédérale de Lausanne, EPFL
Lausanne, Suisse.

October 
A selfsimilar wave for spray atomization
The work with Ralf Blumenthal and Stephane Zaleski on atomization presented at the new "groupement de recherche" on liquid films.
Premières journées du GDR films liquides,
Paris, France.

September 
A selfsimilar wave for atomization
The work with Ralf Blumenthal and Stephane Zaleski on atomization.
European fluid mechanics conference 8,
Bad Reichenhall, Germany.

June 
Roues souples
Travaux en commun avec Sébastien Neukirch sur le roulement de structures souples.
Laboratoire FAST,
Paris, France.

May 
Une vague pour l'atomisation
Groupe de travail "Instabilité hydrodynamiques et ondes non linéaires".
Village de Peyresq,
France.

March 
Une vague autosemblable pour l'atomisation [2M]
Présentation de nos travaux sur l'atomisation à la communauté française du non linéaire.
Rencontre du non linéaire 2010,
Paris, France.

2009

November 
Anatomy of a wave [2M]
We look at a twophase mixing layer, and describe the growth of one single wave of large amplitude: selfsimilar growth in time.
Conference APSdivision of fluid dynamics 2009,
Minneapolis, Minnesota.

June 
Energy growth in the compliant channel [2M]
Pour les animations, voir la "gallerie".
Conference Fluid and Elasticity 2009,
Carry le Rouet, France.

April 
Pumping from the walls: "la distinction des mécanismes" [17M],
[5M]
Pour les animations, voir la "gallerie".
PMMH, Ecole supérieure de physique et de chimie industrielles (ESPCI),
Paris.

February 
Pumping from the walls: "la distinction des mécanismes" [17M],
[5M]
Pour les animations, voir la "gallerie".
LMFA, école centrale de Lyon,
Lyon.

January 
Pumping from the walls: "la distinction des mécanismes" [17M],
[5M]
Présentation d'un thème de recherche du laboratoire aux étudiants du L3/M1 de l'ENS. Pour les animations, voir la "gallerie".
Ecole normale supérieure,
Paris.

2008

October 
Réalisation de statistiques turbulentes
Institut D'Alembert,
Paris.

August 
CONF: Pumping or drag reduction?
ICTAM 2008, Adelaide,
Australia.

June 
Feedback control of fluid flow and stochastic methods
An overview of these topics and applications.
Kawahara laboratory,
Osaka University, Japan.

March 
Pumping from the walls
Different ways to pump in a pipe/Channel: peristalsis and more. See animations in the
gallery.
Institut Jean le Rond d'Alembert,
Paris, France.
Energy growth in the compliant channel
Response of the compliant channel to external excitations. See animations in the
gallery.
ONERA Val Fleury,
Paris, France.

2007

December 
Transition to turbulence and flow control
A brief handdrawn introduction to these topics to the members of my new lab.
Fukagata/Obi lab., KEIO University,
Tokyo, Japan.

August 
CONF: Cavity control/Stochastic excitations on streaks
Congrès Francais de Mécanique 2007,
Grenoble, France.

June 
CONF: How to build a quantitatively accurate simple model
SCAT Vortex summer School,
Porquerolles, France.

May 
Méthodes matricielles pour la stabilité des écoulements nonparallèlles
GDR turbulence/DYCOEC, IRPHÉ,
Marseille, France.

April 
Outils pour le contrôle des écoulements
Séminaire invité au MSNMGP,
Marseille, France.

January 
Contrôle des écoulements: la sténose
Séminaire à l'IRPHÉ.
Marseille, France.

2006

June 
CONF: Stochastic excitation on boundary layer streaks European Fluid Mechanics Conference
(EFMC06), MiniSymposium on hydrodynamic stability,
Stockholm, Sweden
CONF: Control of shear flows subject to stochastic excitations ERCOFTAC workshop
"LaminarTurbulent Transition Mechanisms,
Prediction and Control"
in the archipelago of Stockhom.
Nässlingen, Sweden

May 
Stability and control of shear flows subject to stochastic excitations
PhD defence, KTH.
KTH, Sweden

2005

December 
CONF: Modeling flow statistics using convex optimization
Conference CDCECC 2005 (Conference on Decision and Control/European Control Conference) in Seville.
CDC/ECC 2005, Seville, Spain

November 
CONF: Control of instabilities in a cavitydriven separated boundary layer flow
Presentation at the APS conference, division of fluid mechanics.
APS 2005, Chicago, USA

October 
Résonance et contrôle en cavité ouverte
Presentation at the IRPHÉ of
the latest results for cavity control and resonances.
IRPHÉ, Marseille, France

September 
CONF: Control of instabilities in a cavitydriven separated boundarylayer flow
3rd Symposium on Global Flow Instability and Control, Crete

Mayjune 
Contrôle et estimation en écoulements cisaillés
Visite des labos: PMMH, LIMSI, LMM, LML, IMFT en France.
Paris, Lille, Toulouse

February 
Transient growth on boundary layer streaks
Presentation to the department of mechanics of the results obtained with
Luca Brandt on secondary instability of streaks.
KTH Mechanics, Sweden

2004

December 
CONF: Linear feedback control of transition in shear flows
Presentation to the 2004 IUTAM conference in India. Compensation results,
and some examples on the flexibility of the quadratic objective function.
IUTAM 2005, Bangalore, India

June 
Control and estimation of wallbounded flow systems
Presentation of my licentiate thesis to the department; "mid.PhD" exam and thesis.
KTH mechanics, Sweden

April

CONF: Coupling sensors and actuators in flow control
Abisko workshop, in the north of Sweden. Description of a potential
transfer function formulation for flow control.
Abisko meeting, Sweden

March 
Turbulent channel flow estimation
Model for the perturbation to the linear dynamics in transitionnal and turbulent
channel flow. Use of those statistics for optimal estimation.
(Slides by Mattias Chevalier for the 2003
APS conference in New Jersey.)
THTLAB, Tokyo university

2003

November 
State estimation in wallbounded flow systems
Talk at the APS conference. Presentation of the paper "Estimation of wallbounded flow systems" written together with Mattias Chevalier and Thomas Bewley.
APS conference 2003, New Jersey, USA

September 
Estimation of wal bounded flow systems
Presentation of my research activity to the department of
Mechanics. Part of the "seminar course"
KTH mechanics, Sweden

August 
Estimation of wallbounded shear flows
At the conference "Svenska mekanik dagar" (The swedish mechanics days)
SMD 2003, Gothenburg, Sweden

May 
Linear feedback control and estimation in wall bounded shear flows
Presentation to the LadHyX group of our recent results on time varying estimation in the channel flow.
LadHyX, Ecole Polytechnique, Paris, France.

March 
Estimation of stochastically excited flows
Presentation of the stochastic approach to the estimation problem. Definitions
and procedures for infinite dimentional systems.
Åre meeting, Sweden

January 
Optimal Flow Estimation Presentation of
our results from San Diego at the department of mechanics at KTH. This
was one of the "flow control seminars".
KTH mechanics, Sweden

2002

December 
Optimal Flow
Estimation Presentation to the "dynamic and system control
group" at the department of MAE of the work done with Mattias
Chevalier and Thomas bewley at UCSD during the fall. Dept of
MAE, UCSD, USA

September 
Model reduction in boundary layer compensation
Presentation to the "flow control lab." of the work done during the past year in Sweden with Dan Henningson
Flow control Lab, UCSD, USA

March 
Boundary layer stability, control and estimation, and model reduction
A review of the basics leading to the model reduction problem.
Åre meeting, Sweden

Abstracts
Realizing turbulent statistics
Jérôme Hœpffner
Yoshitsugu Naka
Koji Fukagata
Shinnosuke Obi
Abstract:
Linear analysis of drag deduction by traveling wavelike body force in channel flow
Hiroya Mamori
Koji Fukagata
Jérôme Hœpffner
Shinnosuke Obi
Abstract:
A linear analysis is made for the skinfriction drag reduction in channel ﬂow by traveling wavelike surface heat
ing/cooling. The analysis reveals that downstream traveling waves can reduce the skinfriction drag. The buoyancy
force induced by the surface traveling wave creates the disturbance ﬁelds of velocities, pressure and temperature. Due
to a subtle effect of ﬂuid viscosity, a phase lead of streamwise velocity disturbance (u′ ) is induced. Therefore, the
phases of u′ and v′ (wallnormal velocity disturbance) depart from quadrature, inducing a nonzero Reynolds shear
stress near the walls. resulting drag reduction effect.
Pumping or drag reduction?
Jérôme Hœpffner
Koji Fukagata
Abstract:
Two types of wall actuation in channel flow are considered: traveling waves of
wall deformation (peristalsis) and traveling waves of blowing and suction. The
flow response and its mechanisms are analyzed using nonlinear and weakly
nonlinear computations. We show that both actuations induce a flux in the
channel in absence of imposed pressure gradient and can thus be characterized as
pumping. In the context of flow control, pumping and drag reduction are strongly
connected, and we seek to define them properly based on these two actuation
examples. Movies showing the flow motion for the two types of actuations are
available with the online version of this paper.
Mechanisms of nonmodal energy amplification in channel flow between compliant walls
Jérôme Hœpffner
Alessandro Bottaro
Juien Favier
Abstract:
The mechanisms leading to large transient growth of disturbances for the
flow in a channel with compliant walls are investigated. The walls are modeled as
thin springbacked plates, and the flow dynamics is modeled using the
NavierStokes equations linearised about the parabolic Poiseuille profile.
Analysis for streamwiseinvariant perturbations show that this fluidstructure
system can sustain oscillatory energy evolution of large amplitude, in the form
of spanwise standing waves. Such waves are related to the travelling
waves which free wall can support, modified to account for an 'added mass' effect.
Simple scaling arguments are found to provide results in excellent agreement with
computations of optimal disturbances, for low to moderate values of the
stiffness parameter characterising the compliant surface. Movies are available with the online version of the paper.
Inputoutput analysis and control design applied to a linear model of spatially developing flows
Shervin Bagheri
Jérôme Hœpffner
Peter Schmid
Dan Henningson
Abstract:
This review presents a framework for the inputoutput analysis, model reduction and control design for fluid dynamical systems using examples applied to the linear complex GinzburgLandau equation. Major advances in hydrodynamics stability, such as global modes in spatially inhomogeneous systems and transient growth of nonnormal systems is reviewed. Inputoutput analysis generalizes hydrodynamic stability analysis by considering a finitetime horizon over which energy amplifation, driven by a speciþ®¡Ä input (disturbances/actuator) and measured at a specified output (sensor), is observed. In the control design the loop is closed between the output and the input through a feedback gain. Model reduction approximates the system with a loworder model, making modern control design computationally tractable for systems of large dimensions. Methods from control theory are reviewed and applied to the GinzburgLandau equation in a manner that is readily generalized to fluid mechanics problems, thus giving a fluid mechanics audience an accessible introduction to the subject.
Stochastic approach to the receptivity problem applied to bypass transition in boundary layers
Jérôme Hœpffner
Luca Brandt
Abstract:
To study the flow behaviour in the presence of external disturbances of chaotic nature, a stochastic approach is pursued. In particular, transition to turbulence in boundary layers exposed to high levels of freestream turbulence is considered. The late stages of this transition scenario, characterised by the growth and breakdown of streamwiseelongated streaks, are examined by considering the linear evolution of perturbations to a base flow consisting of the Blasius profile and the streaks. A stochastic initial condition is considered where the freestream perturbations are described by
the correlations of isotropic homogeneous turbulence. The spatial correlation of the excited flow at later times can be computed by the numerical solution of a Lyapunov equation. It is shown that freestream turbulence has the necessary features to excite secondary energy growth, thus playing a central role in the transition to turbulence. The method proposed here can be used to examine the receptivity of other flows to external noise whose statistical properties are known or can be modelled.
Optimal growth, model reduction and control in a separated boundarylayer flow using global modes
Espen Åkervik
Jérôme Hœpffner
Uwe Ehrenstein
Dan Henningson
Abstract:
Twodimensional global eigenmodes are used as a projection basis both for analysing the
dynamics and building a reduced model for control in a prototype separated boundary layer
flow. In the present configuration, a high aspect ratio smooth cavitylike geometry
confines the separation bubble. Optimal growth analysis using the reduced basis shows
that the sum of the highly nonnormal global eigenmodes are able to describe a localized
disturbance. Subject to this worstcase initial condition, a large transient growth
associated with the development of a wavepacket along the shear layer followed by a
global cycle related to the two unstable global eigenmodes is found. The
flow simulation
procedure is coupled to a measurement feedback controller, which senses the wall
shear stress at the downstream lip of the cavity and actuates at the upstream lip. A reduced
model for the control optimization is obtained by a projection on the least stable
global eigenmodes, and the resulting linearquadraticgaussian controller is applied to
the NavierStokes time integration. It is shown that the controller is able to damp out
the global oscillations.
Steady solutions of the NavierStokes equations by selective frequency damping
Espen Åkervik
Luca Brandt
Dan Henningson
Jérôme Hœpffner
Olaf Marxen
Philipp Schlatter
Abstract:
A new method, enabling the computation of steady solutions of the NavierStokes
equations in globally unstable configurations, is presented. We show that it is
possible to reach a steady state by damping the unstable frequencies. This is
achieved by adding a dissipative relaxation term proportional to the
highfrequency content of the velocity fluctuations. Results are presented for
boundarylayer cavitydriven separation and a separation bubble induced by an
external pressure gradient.
Control of cavitydriven separated boundary layer
Jérôme Hœpffner
Espen Åkervik
Uwe Ehrenstein
Dan Henningson
Abstract:
The aim of this paper is to use the eigenmodes of the 2D flow,
first to analyse its dynamics, and second to build a reduced model
for control design. The cavity driven separated boundary layer presents
similarities with the flat plate boundary layer separation bubble,
but with fixed separation and reatachment points. The flow dynamics
is dominated by the shear layer instability, and the pressure is found
to play a coupling role between separation and reatachment,
potentially leading to global instability. The structure of the computed eigenmodes
is discussed in details. The large dimensionality of the
discretized flow system is a challenge for control design. A reduced
dynamic model is constructed by projection on a basis of eigenmodes, and the
controller optimized for this reduced model is found to perform well on the full system.
Linear feedback control and estimation applied to instabilities in spatially developing boundary layers
Mattias Chevalier
Jérôme Hœpffner
Espen Åkervik
Dan Henningson
Abstract:
This paper presents the application of feedback control to spatially
developing boundary layers.
It is the natural followup of Högberg and Henningson (2001), where exact
knowledge of the entire flow state was assumed for control. We apply recent
developments stochastic models for the external sources of disturbances that
allow the efficient use of several wall measurement for estimation of the flow
evolution: the two components of the skinfriction and the pressure fluctuation
at the wall.
Perturbations to base flow profiles of the family of FalknerSkanCooke
boundary layers are estimated by use of wall measurements. The estimated state
is in turn fed back for control in order to reduce the kinetic energy of the
perturbations. The control actuation is achieved by means of unsteady blowing
and suction at the wall.
Flow perturbations are generated at the upstream region in the
computational box and are propagating in the boundary
layer. Measurement are extracted downstream over a thin strip,
followed by a second thin strip where the actuation is performed.
It is shown that flow disturbances can be efficiently estimated and
controlled in spatially evolving boundary layers for a wide range of
base flows and disturbances.
Transient growth on boundary layer streaks
Jérôme Hœpffner
Luca Brandt
Dan Henningson
Abstract:
The linear perturbations evolving on streamwise boundary layer
streaks which yield maximum energy growth are computed. The steady and spanwise
periodic streaks arising from the nonlinear saturation of optimally growing
streamwise vortices are considered as base flow. It is shown that significant
transient growth may occur for both sinuous antisymmetric perturbations and for
varicose symmetric modes. The energy growth is observed at amplitudes
significantly below the threshold beyond which the streaks become linearly
unstable and is largest for sinuous perturbations, to which the base flow
considered first become unstable. The optimal initial condition consists of
velocity perturbations localised in the regions of highest shear of the streak
base flow, tilted upstream from the wall. The optimal response is still
localised in the areas of largest shear but it is tilted in the flow
direction. The most amplified perturbations closely resemble the unstable
eigenfunctions obtained for streaks of higher amplitudes. The results suggest
the possibility of a transition scenario characterised by the nonmodal growth
of a primary perturbations, the streaks, followed by the secondary transient
growth of higher frequency perturbations. Implication for turbulent flow is also
discussed.
State estimation in wallbounded flow systems. Part 2. Turbulent flows
Mattias Chevalier
Jérôme Hœpffner
Thomas Bewley
Dan Henningson
Abstract: This work extends the estimator developed in Part 1 of this
study (Hoepffner et al., J. Fluid Mech., to appear) to the problem of
estimating a turbulent channel flow at Re tau=100 based on a history of noisy
measurements on the wall. The key advancement enabling this work is the
development and implementation of an efficient technique to extract, from direct
numerical simulations, the relevant statistics of an appropriatelydefined
``external forcing'' term on the NavierStokes equation linearized about the
mean turbulent flow profile. This forcing term is designed to account for the
unmodelled (nonlinear) terms during the computation of the (linear) Kalman
filter feedback gains It is demonstrated that by applying the optimal feedback
gains, satisfactory correlation between the actual and estimated flow is
obtained in the nearwall regions. Both Kalman and extended Kalman filters are
evaluated and naturally the extended filter is giving better correlations
between the actual and estimated flow, however the Kalman filter gives good
performance in the nearwall regions.
State estimation in wallbounded flow systems. Part 1. Perturbed laminar flows
Jérôme Hœpffner
Mattias Chevalier
Thomas Bewley
Dan Henningson
Abstract:
In applications involving the modelbased control of transitional
wallbounded flow systems, one often desires to estimate the interior
flow state based on a history of noisy measurements from an array of
flushmounted skinfriction and pressure sensors on the wall. This
paper considers this estimation problem, using a Kalman filter based
on the linearised NavierStokes equations and appropriate stochastic
models for the relevant statistics of the initial conditions, sensor
noise, and external disturbances acting on the system. We show that a
physically relevant parameterisation of these statistics is key to
obtaining well resolved feedback kernels with appropriate spatial
extent for all three types of flow measurements available on the wall.
The effectiveness of the resulting Kalman and extended Kalman filters
that implement this feedback is verified for both infinitesimal and
finiteamplitude disturbances in direct numerical simulations of a
perturbed laminar channel flow. The consideration of timevarying
feedback kernels is shown to be particularly advantageous to
accelerate the convergence of the estimator from unknown initial
conditions. A companion paper (Part 2) considers the extension of
such estimators to the case of fullydeveloped turbulence.
Modeling flow statistics using convex optimization
Jérôme Hœpffner
Abstract:
A method is proposed to estimate the covariance of
disturbances to a stable linear system when its state covariance is known and a
dynamic model is available. This is an issue of fundamental interest for
estimation and control of fluid mechanical systems whose dynamics is described
by the linearized Navier Stokes equations. The problem is formulated in terms of
a matrix norm minimisation with linear matrix inequality constraint, and solved
numerically by means of alternating convex projection. The method is tested on
covariance estimation in a low Reynolds number channel flow.
Linear feedback control of transition in shear flows
Jérôme Hœpffner
Mattias Chevalier
Thomas Bewley
Dan Henningson
Abstract:
This work focuses on the application of linear feedback control to
transition to turbulence in shear flows. The controller uses
wallmounted sensor information to estimate the flow disturbances and
uses wall actuators to prevent transition to turbulence. The flow
disturbances are induced by external sources of perturbations described
by means of a stochastic volume forcing. We
show that improved performance can be achieved if the proper
destabilisation mechanisms are targeted.
Control and estimation of wall bounded flow systems
Jérôme Hœpffner
Licentiate thesis, KTH Mechanics, Stockholm, Sweden.
Abstract:
This thesis focuses on the application of linear feedback control and
estimation to channel flow. Both the initial stage of the transition
and the low Reynolds number turbulent cases are studied. From sensors
at the wall, the state of the flow is estimated, using a stochastic
description of the flow disturbances. The estimated state is in turn
fed back to the flow system in order to achieve a control objective.
This model based scheme uses the linearised NavierStokes equations
as a dynamic model for the flow evolution. The emphasis is here put on
the estimation procedure, that was so far the limiting factor for the
overall control performance. We show that the estimation performance
rely on a correct description of the flow disturbances. We apply
model reduction on the controller, and show that we can maintain the
control performance even with a highly truncated system. We then
introduce a representation of the feedback by means of transfer
functions, and discuss the implication of the transfer function for
the interpretation of the feedback, and for possible implementation of
the control loop.
Download:
[preprint (1.2M)]
Stability and control of shear flows subject to stochastic disturbances
Jérôme Hœpffner
Doctoral thesis, KTH Mechanics, Stockholm, Sweden.
Abstract:
In this thesis, we adapt and apply methods from linear control theory to shear
flows. The challenge of this task is to build a linear dynamic system that models
the evolution of the flow, using the NavierStokes equations, then to define
sensors and actuators, that can sense the flow state and affect its evolution.
We consider flows exposed to stochastic excitations. This framework allows to
account for complex sources of excitations, often present in engineering
applications. Once the system is built, including dynamic model, sensors,
actuators, and sources of excitations, we can use standard optimization
techniques to derive a feedback law. We have used feedback control to stabilize
unstable flows, and to reduce the energy level of sensitive flows subject to
external excitations.
The files of the Matlab codes included in the thesis can be downloaded in [computer codes].
Download:
[preprint (5.4M)]
[preprint (12M)]
Identification of sources of sound in low mach
number flows by the use of flow field eigenmodes
Axel Kierkegaard
Gunilla Ephraimson
Jérôme Hœpffner
Espen Åkervik
Dan Henningson
Mats Åbom
Proceedings of 13th international congress on sound and vibrations, Vienna, July 2006
Abstract:
We present a method to study sound generation processes in low Mach Number flows. Instead
of the full flow field obtained from e.g. a DNS, we consider a base flow together with
a timedependent perturbation, where the perturbation satisfy the NavierStokes equations
linearized around the baseflow. In a reduced model the perturbation is approximated by a
linear combination of the eigenmodes of a corresponding eigenvalue problem. The behavior
in time is determined by the corresponding eigenvalues. Curle's equation is used to calculate
the acoustic field. By studying the source terms in Curle's equation, it is possible to identify
mechanisms for sources of sound. This makes it possible to study how the different sources
of sound depend on different structures of the flow field.We apply the methodology on a twodimensional
flow over a cavity with smoothed corners. Results of acoustic pressure in the far
field and source strengths for different superpositions of eigenmodes are presented.
Download:
[preprint (300K)]
