Hovering of a rigid pyramid in an oscillatory airflow
Annie Weathers, Brendan Folie, Bin Liu, Stephen Childress and Jun Zhang
Journal of Fluid Mechanics 650, 415-425 (2010).
Abstract: We investigate the dynamics of rigid bodies (hollow 'pyramids') placed within a background airflow, oscillating with zero mean. The asymmetry of the body introduces a net upward force. We find that when the amplitude of the airflow is above a threshold, the net lift exceeds the weight and the object starts to hover. Our results show that the objects hover at far smaller air amplitudes than would be required by a quasi-steady theory, although this theory accounts qualitatively for the behaviour of the system as the body mass becomes small.
Shape Optimization of Peristaltic Pumping
S. Walker and M. Shelley
Journal of Computational Physics 229, 1260-1291 (2010).
Abstract: Transport is a fundamental aspect of biology and peristaltic pumping is a fundamental mechanism to accomplish this; it is also important to many industrial processes. We present a variational method for optimizing the wave shape of a peristaltic pump. Specifically, we optimize the wave profile of a two dimensional channel containing a Navier-Stokes fluid with no assumption on the wave profile other than it is a traveling wave (e.g. we do not assume it is the graph of a function). Hence, this is an infinite-dimensional optimization problem. The optimization criteria consists of minimizing the input fluid power (due to the peristaltic wave) subject to constraints on the average flux of fluid and area of the channel. Sensitivities of the cost and constraints are computed variationally via shape differential calculus and we use a sequential quadratic programming (SQP) method to find a solution of the first order KKT conditions. We also use a merit-function based line search in order to balance between decreasing the cost and keeping the constraints satisfied when updating the channel shape. Our numerical implementation uses a finite element method for computing a solution of the Navier-Stokes equations, adjoint equations, as well as for the SQP method when computing perturbations of the channel shape. The walls of the channel are deformed by an explicit front-tracking approach. In computing functional sensitivities with respect to shape, we use L2-type projections for computing boundary stresses and for geometric quantities such as the tangent field on the channel walls and the curvature; we show error estimates for the boundary stress and tangent field approximations. As a result, we find optimized shapes that are not obvious and have not been previously reported in the peristaltic pumping literature. Specifically, we see highly asymmetric wave shapes that are far from being sine waves. Many examples are shown for a range of fluxes and Reynolds numbers up to Re = 500 which illustrate the capabilities of our method.
Each frame of the movies show the current shape with the steady-state flow field illustrated by streamlines. Everything is plotted with respect to the wave frame of the traveling wave. Periodic boundary conditions are imposed on the left and right ends of the channel. The Reynolds number is Re = 500.
- Asymmetry Allowed
In these movies, both top and bottom walls are independent but each moves with the same wave speed.
Asymmetric: Medium Flux Constraint (MPG)
Asymmetric: Large Flux Constraint (MPG)
- Symmetry Enforced
In these movies, the bottom wall corresponds to a line of symmetry. Only the top-half of the domain is shown.
Symmetric: Medium Flux Constraint (MPG)
Symmetric: Large Flux Constraint (MPG)
Modeling simple locomotors in Stokes flow
A. Kanevsky, M. Shelley and A.-K. Tornberg
Journal of Computational Physics 229, 958-977 (2010).
Abstract: Motivated by the locomotion of flagellated micro-organisms and by recent experiments of chemically driven nanomachines, we study the dynamics of bodies of simple geometric shape that are propelled by specified tangential surface stresses. We develop a mathematical description of the body dynamics based on a mixed-type boundary integral formulation. We also derive analytic axisymmetric solutions for the case of a single locomoting sphere and ellipsoid based on spherical and ellipsoidal harmonics, and compare our numerical results to these. The hydrodynamic interactions between two spherical and ellipsoidal swimmers in an infinite fluid are then simulated using second-order accurate spatial and temporal discretizations. We find that the near-field interactions result in complex and interesting changes in the locomotors' orientations and trajectories. Stable as well as unstable pairwise swimming motions are observed, similar to the recent findings of Pooley et al.
Hydrodynamic mobility of Chiral colloidal aggregates
Eric E. Keaveny and Michael J. Shelley
Physical Review E79, 051405 (2009).
Abstract: A recent advance in colloidal technology [Zerrouki et al., Nature 455, 380 (2008)] uses magnetic aggregation to enable the formation of micron-scale particle clusters with helical symmetry. The basic building blocks of these aggregates are doublets composed of two micron-scale beads of different radii bonded together by a magnetic cement. Such self-assembled structures offer potential for controllable transport and separation in a low Reynolds number environment using externally applied magnetic or electric fields. Establishing the hydrodynamic properties of the aggregates, in particular the coupling between rotation and translation afforded by the cluster geometry, is an essential initial step toward the design of microfluidic devices employing these aggregates. To quantify this coupling, we first determine parameterized expressions that describe the positions of the beads in an aggregrate as a function of size ratio of the two beads composing the doublets. With the geometry of the structure known, we perform hydrodynamic calculations to ascertain entries of the mobility matrix for the aggregate and establish the relationship between the applied torque about the helical axis and translations parallel to this direction. We find that for larger values of the particle radius ratio the coupling between rotations and translations changes sign as the number of doublets in the aggregate increases. This feature indicates that the clusters possess a more complex superhelical structure.
Transition to mixing and oscillations in a Stokesian viscoelastic flow
Becca Thomases and Michael Shelley
Physical Review Letters 103, 094501 (2009).
Abstract: In seeking to understand experiments on low-Reynolds-number mixing and flow transitions in viscoelastic fluids, we simulate the dynamics of the Oldroyd-B model, with a simple background force driving the flow. We find that at small Weissenberg number, flows are "slaved" to the extensional geometry imposed by forcing. For large Weissenberg number, such solutions become unstable and transit to a structurally dissimilar state dominated by a single large vortex. This new state can show persistent oscillatory behavior with the production and destruction of smaller-scale vortices that drive mixing.
These movies show the dynamics of the flow and stress tensor for the Weissenberg 10 simulations and run for t=0 to 2000. They correspond with Figures 1 and 2 in the paper.
TS2009-Vort.mov: Movie of the dynamics of the vorticity. Note that there is a change in the scale of the axes at t=500.
TS2009-StrmLine.mov: Movie of the dynamics of the streamlines of the flow. At each time step, streamlines are calculated and these are then animated over time.
TS2009-TrS.mov: Movie of the dynamics of the trace of the stress tensor.
TS2009-S12.mov: Movie of the dynamics of the shear stress S12.
TS2009-Vort_2.mov: Movie of the dynamics of the vorticity in the case referred to at the end of the paper where the vortex does not relax to a single quadrant, but instead switches dynamically and persistently from quadrant to quadrant.
The mechanics of slithering locomotion
David L. Hu, Jasmine Nirody, Terri Scott, and Michael J. Shelley
Proceedings of the National Academy of Science (USA) doi:10.1073/pnas.0812533106 (2009).
Abstract: In this experimental and theoretical study, we investigate the slithering of snakes on flat surfaces. Previous studies of slithering have rested on the assumption that snakes slither by pushing laterally against rocks and branches. In this study, we develop a theoretical model for slithering locomotion by observing snake motion kinematics and experimentally measuring the friction coefficients of snakeskin. Our predictions of body speed show good agreement with observations, demonstrating that snake propulsion on flat ground, and possibly in general, relies critically on the frictional anisotropy of their scales. We have also highlighted the importance of weight distribution in lateral undulation, previously difficult to visualize and hence assumed uniform. The ability to redistribute weight, clearly of importance when appendages are airborne in limbed locomotion, has a much broader generality, as shown by its role in improving limbless locomotion.
Click here for images and videos of Limbless Locomotion
Snakes use scales to slither: Mathematical model suggests 'sideways' friction is key, Roberta Kwok, NatureNews, 8 June 2009
Snakes' Locomotion Appears a Matter of Scales, Henry Fountain, Observatory, The New York Times, June 06, 2009
Shape-changing bodies in fluid: Hovering, ratcheting, and bursting
Saverio E. Spagnolie and Michael J. Shelley
Physics of Fluids 21, 013103 (2009).
Abstract: Motivated by recent experiments on the hovering of passive bodies, we demonstrate how a simple shape-changing body can hover or ascend in an oscillating background flow. We study this ratcheting effect through numerical simulations of the two-dimensional Navier-Stokes equations at intermediate Reynolds number. This effect could describe a viable means of locomotion or transport in such environments as a tidal pool with wave-driven sloshing. We also consider the velocity burst achieved by a body through a rapid increase in its aspect ratio, which may contribute to the escape dynamics of such organisms as jellyfish.
Click here for video of "Ratcheting Upward Against Gravity"
Instabilities, pattern formation and mixing in active suspensions
D. Saintillan and M. Shelley
Physics of Fluids 20, 123304 (2008).
Abstract: Suspensions of self-propelled particles, such as swimming microorganisms, are known to undergo complex dynamics as a result of hydrodynamic interactions. To elucidate these dynamics, a kinetic theory is developed and applied to study the linear stability and the non-linear pattern formation in these systems. The evolution of a suspension of self-propelled particles is modeled using a conservation equation for the particle configurations, coupled to a mean-field description of the flow arising from the stress exerted by the particles on the fluid. Based on this model, we first investigate the stability of both aligned and isotropic suspensions. In aligned suspensions, an instability is shown to always occur at finite wavelengths, a result that extends previous predictions by Simha and Ramaswamy [Hydrodynamic fluctuations and instabilities in ordered suspensions of self-propelled particles," Phys. Rev. Lett. 89, 058101 (2002)]. In isotropic suspensions, we demonstrate the existence of an instability for the active particle stress, in which shear stresses are eigenmodes and grow exponentially at long scales. Non-linear effects are also investigated using numerical simulations in two dimensions. These simulations confirm the results of the stability analysis, and the long-time non-linear behavior is shown to be characterized by the formation of strong density fluctuations, which merge and break up in time in a quasi-periodic fashion. These complex motions result in very efficient fluid mixing, which we quantify by means of a multiscale mixing norm.
Click here to see a movie of swimmer-driven mixing
Anomalous Hydrodynamic Drafting of Interacting Flapping Flags
Leif Ristroph and Jun Zhang
Physical Review Letters 101, 194502 (2008).
Abstract: In aggregates of objects moving through a fluid, bodies downstream of a leader generally experience reduced drag force. This conventional drafting holds for objects of fixed shape, but interactions of deformable bodies in a flow are poorly understood, as in schools of fish. In our experiments on "schooling" flapping flags we find that it is the leader of a group who enjoys a significant drag reduction (of up to 50 %), while the downstream flag suffers a drag increase. This counterintuitive inverted drag relationship is rationalized by dissecting the mutual influence of shape and flow in determining drag. Inverted drafting has never been observed with rigid bodies, apparently due to the inability to deform in response to the altered flow field of neighbors.
Wired Science, The Weird and Beautiful World of Fluid Dynamics: Inverted Drafting, Jane J. Lee, June 22, 2011
NewsDay.com, Health, Kathy Wollard, How Come?: Why do flags flap in the wind?, March 29, 2009
Nature Physics, Research Highlights, Follow the Leader, Dec. 2008
Physics Today, back scatter Flapping flags in tandem, p. 108, November, 2008
Nature, Research Highlights - physics Flags and drag, v456, p284, 2008
The Economist, Science & Technology, Aerodynamics Blowin' in the wind, Flapping flags may shed light on how fish school and birds flock, Nov. 27th, 2008
Peristaltic pumping and irreversibility of a Stokesian viscoelastic fluid
J. Teran, L. Fauci, and M. Shelley
Physics of Fluids 20, 073101 (2008).
Abstract: Peristaltic pumping by wavelike contractions is a fundamental biomechanical mechanism for fluid and material transport and is used in the esophagus, intestine, oviduct, and ureter. While peristaltic pumping of a Newtonian fluid is well understood, in many important settings, as in the fluid dynamics of reproduction, the fluids have non-Newtonian responses. Here, we present a numerical method for simulating an Oldroyd-B fluid coupled to contractile, moving walls. A marker and cell grid-based projection method is used for the fluid equations and an immersed boundary method is used for coupling to a Lagrangian representation of the deforming walls. We examine numerically the peristaltic transport of a highly viscous Oldroyd-B fluid over a range of Weissenberg numbers and peristalsis wavelengths and amplitudes.
Click here to see a movie demonstrating irreversibility of Stokesian viscoelastic flows
or Click here for a quicktime movie of the above (for Linux, MacOSX)
Self-Induced Cyclic Reorganization of Free Bodies through Thermal Convection
Bin Liu and Jun Zhang
Physical Review Letters 100, 244501 (2008).
Abstract: We investigate the dynamics of a thermally convecting fluid as it interacts with freely moving solid objects. This is a previously unexplored paradigm of interactions between many free bodies mediated by thermal convection, which gives rise to surprising robust oscillations between different large-scale circulations. Once begun, this process repeats cyclically, with the collection of objects (solid spheres) entrained and packed from one side of the convection cell to the other. The cyclic frequency is highest when the spheres occupy about half of the cell bottom and their size coincides with the thickness of the thermal boundary layer. Our work shows that a deformable mass stimulates a thermally convecting fluid into oscillation, a collective behavior that may be found in nature.
Click here for New Scientist Video of the Experiment -- Earth-in-a-box may explain continental drift (on youTube)
New Scientist Environment -- Why continents split up and get back together, by Devin Powell, 02 July 2008
Physical Review Focus -- Desktop Continental Drift, 12 June 2008 (contains videos from the experiment)
physicsworld.com Table-top experiment could explain why continents drift, June 24, 2008 - (one must sign-in to read article).
Instabilities and pattern formation in active particle suspensions: Kinetic theory and continuum simulations
David Saintillan and Michael J. Shelley
Physical Review Letters 100, 178103 (2008).
Abstract: We use kinetic theory and non-linear continuum simulations to study the collective dynamics in suspensions of self-propelled particles. The stability of aligned suspensions is first analyzed, and we demonstrate that such suspensions are always unstable to fluctuations, a result that generalizes previous predictions by Simha and Ramaswamy (2002). Isotropic suspensions are also considered, and it is shown that an instability for the particle stress occurs in that case. Using simulations, non-linear effects are investigated, and the long-time behavior of the suspensions is observed to be characterized by the formation of strong density fluctuations, resulting in efficient fluid mixing.
An experimental investigation and a simple model of a valveless pump
Thomas T. Bringley, Stephen Childress, Nicolas Vandenberghe, and Jun Zhang
Physics of Fluids 20, 033602 (2008).
Abstract: We construct a valveless pump consisting of a section of elastic tube and a section of rigid tube connected in a closed loop and filled with water. By periodically squeezing the elastic tube at an asymmetric location, a persistent flow around the tubes is created. This effect, called the Liebau phenomenon or valveless pumping, has been known for some time but is still not completely understood. We study the flow rates for various squeezing locations, frequencies, and elastic tube rigidities. To understand valveless pumping, we formulate a simple model that can be described by ordinary differential equations. The time series of flow velocities generated by the model are qualitatively and quantitatively similar to those seen in the experiment. The model provides a physical explanation of valveless pumping, and it allows us to identify the essential pumping mechanisms.
Growth of anti-parallel vorticity in Euler flows
Physica D 237, 1921-1925 (2008).
Abstract: In incompressible Euler flows, vorticity is intensified by line stretching, a process that can come either from the action of shear, or from advection with curvature. Focusing on the latter process, we derive some estimates on the maximal growth of vorticity in axisymmetric flow without swirl, given that vorticity support volume or kinetic energy is fixed. This leads to consideration of locally 2D anti-parallel vortex structures in three dimensions. We exhibit a class of line motions which lead to infinite vorticity in a finite time, with only a finite total line stretching. If the line is replaced by a locally 2D Euler flow, we obtain a class of models of vorticity growth which are similar to the paired vortex structures studied by Pumir and Siggia. We speculate on the mechanisms which can suppress the nonlinear effects necessary for the finite-time singularity exhibited by the moving line problem.
Flapping states of a flag in an inviscid fluid: bistability and the transition to chaos
Silas Alben and Michael J. Shelley
Physical Review Letters 100, 074301 (2008).
Abstract: We investigate the "flapping flag" instability through a model for an inextensible flexible sheet in an inviscid 2D flow with a free vortex sheet. We solve the fully-nonlinear dynamics numerically and find a transition from a power spectrum dominated by discrete frequencies to an apparently continuous spectrum of frequencies. We compute the linear stability domain which agrees with previous approximate models in scaling but differs by large multiplicative factors. We also find hysteresis, in agreement with previous experiments.
Erratum: correction of parameters and Fig. 2
First Periodic State
Second Periodic State
Third Periodic State
Validation of a simple method for representing spheres and slender bodies in an immersed boundary method for Stokes flow on an unbounded domain
Thomas T. Bringley, Charles S. Peskin
Journal of Computational Physics 227, 5397-5425 (2008).
Abstract: We test the efficacy of using a single Lagrangian point to represent a sphere, and a one-dimensional array of such points to represent a slender body, in a new immersed boundary method for Stokes flow. A numerical parameter, the spacing of the Eulerian grid, is used to determine the effective radius of the immersed sphere or slender body. Such representations are much less expensive computationally than those with two or three-dimensional meshes of Lagrangian points. To perform this test, we develop a numerical method to solve the discretized Stokes equations on an unbounded Eulerian grid which contains an arbitrary configuration of Lagrangian points that apply force to the fluid and that move with the fluid. We compare results computed with this new immersed boundary method to known results for spheres and rigid cylinders in Stokes flow in R3. We find that, for certain choices of parameters, the interactions with the fluid of a single Lagrangian point accurately replicate those of a sphere of some particular radius, independent of the location of the point with respect to the Eulerian grid. The interactions of a linear array of Lagrangian points, for certain choices of parameters, accurately replicate those of a cylinder of some particular radius, independent of the position and orientation of the array with respect to the Eulerian grid. The effective radius of the sphere and the effective radius of the cylinder turn out to be related in a simple and natural way. Our results suggest recipes for choosing parameters that should be useful to practitioners. One surprising result is that one must not use too many Lagrangian points in an array. Another is that the approximate delta functions traditionally used in the immersed boundary method perform much better than higher order delta functions with the same support.
Rotational dynamics of a superhelix towed in a Stokes fluid
Sunghwan Jung, Kathleen Mareck, Lisa Fauci, and Michael J. Shelley
Physics of Fluids 19, 103105 (2007).
Abstract: Motivated by the intriguing motility of spirochetes of helically shaped bacteria that screw through viscous fluids due to the action of internal periplasmic flagella, we examine the fundamental fluid dynamics of superhelices translating and rotating in a Stokes fluid. A superhelical structure may be thought of as a helix whose axial centerline is not straight, but also a helix. We examine the particular case in which these two superimposed helices have different handedness, and employ a combination of experimental, analytic, and computational methods to determine the rotational velocity of superhelical bodies being towed through a very viscous fluid. We find that the direction and rate of the rotation of the body is a result of competition between the two superimposed helices; for small axial helix amplitude, the body dynamics is controlled by the short-pitched helix, while there is a crossover at larger amplitude to control by the axial helix.We find far better, and excellent, agreement of our experimental results with numerical computations based upon the method of Regularized Stokeslets than upon the predictions of classical resistive force theory.
Liquid crystal droplet production in a microfluidic device
B. Hamlington, B. Steinhaus, J. Feng, D. Link, A.-Q. Shen, and M. Shelley
Liquid Crystals 34, 861-870 (2007).
Abstract: Liquid crystal drops dispersed in a continuous phase of silicone oil are generated with a narrow distribution in droplet size in microfluidic devices both above and below the nematic-to-isotropic transition temperature. Our experiments show that the surface properties of the channels can be critical for droplet formation. We observe different dynamics in liquid crystal droplet generation and coalescence, and distinct droplet morphology on altering the microchannel surface energy. This is explained by the thermodynamic description of the wetting dynamics of the system. The effect of the nematic-to-isotropic transition on the formation of liquid crystal droplets is also observed and related to the capillary number. We also investigate how the nematic droplet size varies with the flow rate ratio and compare this behaviour with a Newtonian reference system. The effect of the defect structures of the nematic liquid crystal can lead to distinctly different scaling of droplet size in comparison with the Newtonian system. When the nematic liquid crystal phase is stretched into a thin filament before entering the orifice, different defect structures and numbers of defect lines can introduce scatter in the drop size. Capillary instabilities in thin nematic liquid crystal filament have an additional contribution from anisotropic effects such as surface gradients of bending stress, which can provide extra instability modes compared with that of isotropic fluids.
Emergence of singular structures in Oldroyd-B fluids
Becca Thomases and Michael Shelley
Physics Of Fluids 19, 103103 (2007).
Abstract: Numerical simulations reveal the formation of singular structures in the polymer stress field of a viscoelastic fluid modeled by the Oldroyd-B equations driven by a simple body force. These singularities emerge exponentially in time at hyperbolic stagnation points in the flow and their algebraic structure depends critically on the Weissenberg number. Beyond a first critical Weissenberg number the stress field approaches a cusp singularity, and beyond a second critical Weissenberg number the stress becomes unbounded exponentially in time. A local approximation to the solution at the hyperbolic point is derived from a simple ansatz, and there is excellent agreement between the local solution and the simulations. Although the stress field becomes unbounded for a sufficiently large Weissenberg number, the resultant forces of stress grow subexponentially. Enforcing finite polymer chain lengths via a FENE-P penalization appears to keep the stress bounded, but a cusp singularity is still approached exponentially in time.
Orientational order and instabilities in suspensions of self-locomoting rods
David Saintillan and Michael Shelley
Physical Review Letters 99, 058102 (2007).
Abstract: The orientational order and dynamics in suspensions of self-locomoting slender rods are investigated numerically. In agreement with previous theoretical predictions, nematic suspensions of swimming particles are found to be unstable at long wavelengths as a result of hydrodynamic fluctuations. Nevertheless, a local nematic ordering is shown to persist over short length scales and to have a significant impact on the mean swimming speed. Consequences of the large-scale orientational disorder for particle dispersion are also discussed.
Click here to watch the dynamics of an active particle suspension
Stretch-Coil Transition and Transport of Fibers in Cellular Flows
Yuan-Nan Young and Michael Shelley
Physical Review Letters 99, 058303 (2007).
Abstract: It is shown that a slender elastic fiber moving in a Stokesian fluid can be susceptible to a buckling instability -- termed the "stretch-coil" instability -- when moving in the neighborhood of a hyperbolic stagnation point of the flow. When the stagnation point is embedded in an extended cellular flow, it is found that immersed fibers can move as random walkers across time-independent closed-streamline flows. It is also found that the flow is segregated into transport regions around hyperbolic stagnation points and their manifolds, and closed entrapment regions around elliptic points.
Click here to see a simulation of buckling-driven transport
Modeling the dynamics of a free boundary on turbulent thermal convection
Jin-Qiang Zhong and Jun Zhang
Physical Review E 76, 016307 (2007).
Abstract: Based on our previous experimental study, we present a one-dimensional, phenomenological model of a thermal blanket floating on the upper surface of a thermally convecting fluid. The model captures the most important interactions between the floating solid and the fluid underneath. By the thermal blanketing effect, the presence of the solid plate modifies the flow structure below; in turn, the flow exerts a viscous drag that causes the floating boundary to move. An oscillatory state and a trapped state are found in this model, which is in excellent agreement with experimental observations. The model also offers details on the transition between the states, and gives useful insights on this coupled system without the need for full-scale simulations.
Dynamical states of a mobile heat blanket on a thermally convecting fluid
Jin-Qiang Zhong and Jun Zhang
Physical Review E 75, 055301(R) (2007).
Abstract: We experimentally study the dynamical states of a freely moving, floating heat blanket that is coupled with a thermally convecting fluid. This floating boundary modifies the large-scale flow pattern in the bulk and destabilizes the coupled system, leading to spontaneous oscillations. As the moving boundary exceeds a critical size, the system makes a transition from an oscillatory state to a weakly confined state, in which the moving boundary executes only small excursions in response to random bypassing thermal plumes. To explain the observed states and transition, we provide a low-dimensional model that appears to capture the underlying mechanism of this coupled system.
Surface waves on a semitoroidal water ring
Sungwhan Jung, Erica Kim, Michael Shelley and Jun Zhang
Physics of Fluids 19, 058105 (2007).
Abstract: We study the dynamics of surface waves on a semitoroidal ring of water that is excited by vertical vibration. We create this specific fluid volume by patterning a glass plate with a hydrophobic coating, which confines the fluid to a precise geometric region. To excite the system, the supporting plate is vibrated up and down, thus accelerating and decelerating the fluid ring along its toroidal axis. When the driving acceleration is sufficiently high, the surface develops a standing wave, and at yet larger accelerations, a traveling wave emerges. We also explore frequency dependencies and other geometric shapes of confinement.
Click here for a movie from the experiment
Instructions to view this movie on Windows, MacOSX, and Linux platforms
Hovering of a passive body in an oscillating airflow
Stephen Childress, Nicolas Vandenberghe and Jun Zhang
Physics of Fluids 18, 117103 (2006).
Abstract: Small flexible bodies are observed to hover in an oscillating air column. The air is driven by a large speaker at frequencies in the range 10-65 Hz at amplitudes 1-5 cm. The bodies are made of stiffened tissue paper, bent to form an array of four wings, symmetric about a vertical axis. The flapping of the wings, driven by the oscillating flow, leads to stable hovering. The hovering position of the body is unstable under free fall in the absence of the airflow. Measurements of the minimum flow amplitude as a function of flow frequency were performed for a range of self-similar bodies of the same material. The optimal frequency for hovering is found to vary inversely with the size. We suggest, on the basis of flow visualization, that hovering of such bodies in an oscillating flow depends upon a process of vortex shedding closely analogous to that of an active flapper in otherwise still air. A simple inviscid model is developed illustrating some of the observed properties of flexible passive hoverers at high Reynolds number.
Click here for a movie from experiment 1 - divX avi movie
Click here for a movie from experiment 2 - wmv3 movie
The video frame rate here is close to the flapping frequency so the bug does not seem to be flapping, but it is.
Instructions to view these movies on Windows, MacOSX, and Linux platforms
Dynamics of a Deformable Body in a Fast Flowing Soap Film
Sungwhan Jung, Kathleen Mareck, Michael Shelley, and Jun Zhang
Physical Review Letters 97, 134502 (2006).
Abstract: We study the behavior of an elastic loop embedded in a flowing soap film. This deformable loop is wetted into the film and is held fixed at a single point against the oncoming flow. We interpret this system as a two-dimensional flexible body interacting in a two-dimensional flow. This coupled fluid-structure system shows bistability, with both stationary and oscillatory states. In its stationary state, the loop remains essentially motionless and its wake is a von Kármán vortex street. In its oscillatory state, the loop sheds two vortex dipoles, or more complicated vortical structures, within each oscillation period. We find that the oscillation frequency of the loop is linearly proportional to the flow velocity, and that the measured Strouhal numbers can be separated based on wake structure.
Click here for a movie from the experiment
Periodic sedimentation in a Stokesian fluid
Sungwhan Jung, Saverio Spagnolie, Karishma Parikh, Michael Shelley, and Anna-Karin Tornberg
Physical Review E 74, 035302(R) (2006)
Abstract: We study the sedimentation of two identical but nonspherical particles sedimenting in a Stokesian fluid. Experiments and numerical simulations reveal periodic orbits wherein the bodies mutually induce an in-phase rotational motion accompanied by periodic modulations of sedimentation speed and separation distance. We term these “tumbling orbits” and find that they appear over a broad range of body shapes.
Movies of the Experiments and Simulations:
Sedimenting disks (experiment)
Periodic Tumbling: Three disks
Instability of three (initially perturbed) sedimenting bodies
Two meandering disks
APS/DFD 2005, Video Presentation - Periodic Parachutes
On Unidirectional Flight of a Free Flapping Wing
Nicolas Vandenberghe, Stephen Childress and Jun Zhang
Physics of Fluids 18, 014102 (2006)
Abstract: We study the dynamics of a rigid, symmetric wing that is flapped vertically in a fluid. The motion of the wing in the horizontal direction is not constrained. Above a critical flapping frequency, forward flight arises as the wing accelerates to a terminal state of constant speed. We describe a number of measurements which supplement our previous work. These include (a) a study of the initial transition to forward flight near the onset of the instability, (b) the separate effects of flapping amplitude and frequency, (c) the effect of wing thickness, (d) the effect of asymmetry of the wing planform, and (e) the response of the wing to an added resistance. Our results emphasize the robustness of the mechanisms determining the forward flight speed as observed in our previous study.
Coherent Locomotion as an Attracting State for a Free Flapping Body
S. Alben and M. Shelley
Proceedings of the National Academy of Science (USA) 102, 11163-11166 (2005)
Abstract: A common strategy for locomotion through a fluid uses appendages, such as wings or fins, flapping perpendicularly to the direction of travel. This is in marked difference to strategies using propellers or screws, ciliary waves, or rowing with limbs or oars which explicitly move fluid in the direction opposite to travel. Flapping locomotion is also never observed for microorganisms moving at low Reynolds number. To understand the nature of flapping locomotion we study numerically the dynamics of a simple body, flapped up and down within a viscous fluid and free to move horizontally. We show here that, at sufficiently large frequency Reynolds number, unidirectional locomotion emerges as an attracting state for an initially nonlocomoting body. Locomotion is generated in two stages: first, the fluid field loses symmetry by the classical von Karman instability; and second, precipitous interactions with vortical structures shed in previous flapping cycles push the body into locomotion. Body mass and slenderness play central and unexpected roles in each stage. Conceptually, this work demonstrates how locomotion can be transduced from the simple oscillations of a body through an interaction with its fluid environment.
Click here for movies referenced in the publication
Thermal convection with a freely moving top boundary
Jin-qiang Zhong and Jun Zhang
Physics of Fluids 17, 115105 (2005).
Abstract: In thermal convection, coherent flow structures emerge at high Rayleigh numbers as a result of intrinsic hydrodynamic instability and self-organization. They range from small-scale thermal plumes that are produced near both the top and the bottom boundaries to large-scale circulations across the entire convective volume. These flow structures exert viscous forces upon any boundary. Such forces will affect a boundary which is free to deform or change position. In our experiment, we study the dynamics of a free boundary that floats on the upper surface of a convective fluid. This seemingly passive boundary is subjected solely to viscous stress underneath. However, the boundary thermally insulates the fluid, modifying the bulk flow. As a consequence, the interaction between the free boundary and the convective flows results in a regular oscillation. We report here some aspects of the fluid dynamics and discuss possible links between our experiment and continental drift.
Heavy flags undergo spontaneous oscillations in flowing water
M. Shelley, N. Vandenberghe, and J. Zhang
Physical Review Letters 94, 094302 (2005).
Abstract: By immersing a compliant yet self-supporting sheet into flowing water, we study a heavy, stream-lined and elastic body interacting with a fluid. We find that above a critical flow velocity a sheet aligned with the flow begins to flap with a Strouhal frequency consistent with animal locomotion. This transition is subcritical. Our results agree qualitatively with a simple fluid dynamical model that predicts linear instability at a critical flow speed. Both experiment and theory emphasize the importance of body inertia in overcoming the stabilizing effects of finite rigidity and fluid drag.
Click here for a movie from the experiment
Moore's Law and the Saffman-Taylor Instability
Petri Fast and Michael J. Shelley
Journal of Computational Physics 212, 1-5 (2005).
Abstract: Ten years ago Hou, Lowengrub and Shelley published a state-of-the-art boundary integral simulation of a classical viscous fingering problem, the Saffman-Taylor instability. In terms of complexity and level of detail, those computations are still among the most ramified and accurately computed interfacial instability patterns that have appeared in the literature. Since 1994, the computational power of a standard workstation has increased a hundredfold as predicted by Moore's law. The purpose of this Note is to consider Moore's law and its consequences in computational science, and in particular, its impact on studying the Saffman-Taylor instability. We illustrate Moore's law and fast algorithms in action by presenting the worlds largest viscous fingering simulation to date.
Click here for an animation from the data
Instructions to view this movie on Windows, MacOSX, and Linux platforms
Marvin A. Jones and Michael J. Shelley
Journal of Fluid Mechanics 540, 393-425 (2005).
Abstract: In this study we consider the unsteady separated flow of an inviscid fluid around a falling flat plate of small thickness and high aspect ratio. The motion of the plate, which is initially released from rest, is unknown in advance and is determined as part of the solution. The flow solution is assumed two-dimensional and to consist of a bound vortex sheet coincident with the plate and two free vortex sheets that emanate from each of the plate's two sharp edges. Throughout its motion, the plate continually sheds vorticity from each of its two sharp edges and the unsteady Kutta condition, which states the fluid velocity must be bounded everywhere, is applied at each edge. The coupled equations of motion for the plate and its trailing vortex wake are derived and are shown to depend only on a modified Froude number.
A computational fluid dynamics of 'clap and fling' in the smallest insects
Laura A. Miller and Charles S. Peskin
The Journal of Experimental Biology 208, 195-212 (2005).
Abstract: In this paper, we have used the immersed boundary method to solve the two-dimensional Navier-Stokes equations for two immersed wings performing an idealized 'clap and fling' stroke and a 'fling' half-stroke. We calculated lift coefficients as functions of time per wing for a range of Reynolds numbers (Re) between 8 and 128. We also calculated the instantaneous streamlines around each wing throughout the stroke cycle and related the changes in lift to the relative strength and position of the leading and trailing edge vortices. Our results show that lift generation per wing during the 'clap and fling' of two wings when compared to the average lift produced by one wing with the same motion falls into two distinct patterns. For Re=64 and higher, lift is initially enhanced during the rotation of two wings when lift coefficients are compared to the case of one wing. Lift coefficients after fling and during the translational part of the stroke oscillate as the leading and trailing edge vortices are alternately shed. In addition, the lift coefficients are not substantially greater in the two-winged case than in the one-winged case. This differs from three-dimensional insect flight where the leading edge vortices remain attached to the wing throughout each half-stroke. For Re=32 and lower, lift coefficients per wing are also enhanced during wing rotation when compared to the case of one wing rotating with the same motion. Remarkably, lift coefficients following two-winged fling during the translational phase are also enhanced when compared to the one-winged case. Indeed, they begin about 70% higher than the one-winged case during pure translation. When averaged over the entire translational part of the stroke, lift coefficients per wing are 35% higher for the two-winged case during a 4.5 chord translation following fling. In addition, lift enhancement increases with decreasing Re. This result suggests that the Weis-Fogh mechanism of lift generation has greater benefit to insects flying at lower Re. Drag coefficients produced during fling are also substantially higher for the two-winged case than the one-winged case, particularly at lower Re.
Click here for articles and simulations of Tiny Insect Flight.
Symmetry Breaking Leads to Forward Flapping Flight
Nicolas Vandenberghe, Jun Zhang, and Stephen Childress
Journal of Fluid Mechanics 506, 147-155 (2004).
Abstract: Flapping flight is ubiquitous in Nature, yet cilia and flagella, not wings, prevail in the world of micro-organisms. This paper addresses this dichotomy. We investigate experimentally the dynamics of a wing, flapped up and down and free to move horizontally. The wing begins to move forward spontaneously as a critical frequency is exceeded, indicating that 'flapping flight' occurs as a symmetry-breaking bifurcation from a pure flapping state with no horizontal motion. A dimensionless parameter, the Reynolds number based on the flapping frequency, characterizes the point of bifurcation. Above this bifurcation, we observe that the forward speed increases linearly with the flapping frequency. Visualization of the flow field around the heaving and plunging foil shows a symmetric pattern below transition. Above threshold, an inverted von Kármán vortex street is observed in the wake of the wing. The results of our model experiment, namely the critical Reynolds number and the behaviour above threshold, are consistent with observations of the flapping-based locomotion of swimming and flying animals.
Click here to see a video of coherent locomotion resulting from a symmetry-breaking instability
How flexibility induces streamlining in a two-dimensional flow.
Silas Alben, Michael Shelley, and Jun Zhang
Physics of Fluids 16, 1694-1713 (2004).
Abstract: Recent work in bio-fluid dynamics has studied the relation of fluid drag to flow speed for flexible organic structures, such as tree leaves, seaweed, and coral beds, and found a reduction in drag growth due to body reconfiguration with increasing flow speed. Our theoretical and experimental work isolates the role of elastic bending in this process. Using a flexible glass fibre wetted into a vertical soap-film tunnel, we identify a transition in flow speed beyond which fluid forces dominate the elastic response, and yield large deformation of the fibre that greatly reduce drag. We construct free-streamline models that couple fluid and elastic forces and solve them in an efficient numerical scheme. Shape self-similarity emerges, with a scaling set by the balance of forces in a small "tip region" about the flow's stagnation point. The result is a transition from the classical U2 drag scaling of rigid bodies to a U4/3 drag law. The drag scaling is derived from an asymptotic expansion in the length scale of similarity, and it is found that the tip region induces the far-field behavior. The drag law persists, with a simple modification, under variations of the model suggested by the experiment, such as the addition of flow tunnel walls, and a back pressure in the wake.
Transition from ciliary to flapping mode in a swimming mollusc: flapping flight as a bifurcation in Reω
Stephen Childress and Robert Dudley
Journal of Fluid Mechanics 498, 257-288 (2004).
Abstract: From observations of swimming of the shell-less pteropod mollusc Clione antarctica we compare swimming velocities achieved by the organism using ciliated surfaces alone with velocities achieved by the same organism using a pair of flapping wings. Flapping dominates locomotion above a swimming Reynolds number Re in the range 5-20. We test the hypothesis that Re ≈ 5-20 marks the onset of 'flapping flight' in these organisms. We consider the proposition that forward, reciprocal flapping flight is impossible for locomoting organisms whose motion is fully determined by a body length L and a frequency ω below some finite critical value of the Reynolds number Reω = ω L2/ν. For a self-similar family of body shapes, the critical Reynolds number should depend only upon the geometry of the body and the cyclic movement used to locomote. We give evidence of such a critical Reynolds number in our data, and study the bifurcation in several simplified theoretical models. We argue further that this bifurcation marks the departure of natural locomotion from the low Reynolds number or Stokesian realm and its entry into the high Reynolds number or Eulerian realm. This occurs because the equilibrium swimming or flying speed Uf obtained at the instability is determined by the mechanics of a viscous fluid at a value of Ref=Uf L/ν that is not small.
Click here for videos of the Antarctic Pteropods
Research at McMurdo Station, Antarctica. - Steve Childress' journal, photo gallery, research notes
Simulations of the whirling instability by the immersed boundary method
Sookkyung Lim and Charles S. Peskin
Siam J. Sci. Comput. 25, (6), 2066-2083 (2004).
Abstract: When an elastic filament spins in a viscous incompressible fluid it may undergo a whirling instability, as studied asymptotically by Wolgemuth, Powers, and Goldstein [Phys. Rev. Lett., 84 (2000), pp. 16-23]. We use the immersed boundary (IB) method to study the interaction between the elastic filament and the surrounding viscous fluid as governed by the incompressible Navier-Stokes equations. This allows the study of the whirling motion when the shape of the filament is very different from the unperturbed straight state.
Click here for animations of twirling and overwhirling motions.
Simulating the dynamics and interactions of flexible fibers in Stokes flows
Anna-Karin Tornberg and Michael J. Shelley
Journal of Computational Physics 196 8-40 (2004).
Abstract: The dynamics of slender filaments or fibers suspended in Stokesian fluids are fundamental to understanding many flows arising in physics, biology and engineering. Such filaments can have aspect ratios of length to radius ranging from a few tens to several thousands. Full discretizations of such 3D flows are very costly. Instead, we employ a non-local slender body theory that yields an integral equation, along the filament centerline, relating the force exerted on the body to the filament velocity. This hydrodynamical description takes into account the effect of the filament on the fluid, and is extended to capture the interaction of multiple filaments as mediated by the intervening fluid. We consider filaments that are inextensible and elastic. Replacing the force in the slender body integral equation by an explicit expression that uses Euler-Bernoulli theory to model bending and tensile forces yields an integral expression for the velocity of the filament centerlines, coupled to auxiliary integro-differential equations for the filament tensions. Based on a regularized version of these slender body equations that is asymptotically equivalent to the original formulation, we construct a numerical method which uses a combination of finite differences, implicit time-stepping to avoid severe stability constraints, and special quadrature methods for nearly singular integrals. We present simulations of single flexible filaments, as well as multiple interacting filaments, evolving in a background shear flow. These simulations show shear induced buckling and relaxation of the filaments, leading to the storage and release of elastic energy. These dynamics are responsible for the development of positive first normal stress differences, commonly associated with fiasco-elastic fluids that are suspensions of microscopic elastic fibers.
Click here for animations of the simulation
Instability of Semiflexible Filaments in Shear Flow Yields First Normal Stress Difference by Leif Becker and Michael Shelley, Physical Review Letters 87, 198301 (2001)
The Growth and Buckling of Smectic Liquid Crystal Filaments, Michael Shelley and Tetsuji Ueda
A Moving Overset Grid Method for Interface Dynamics applied to Non-Newtonian Hele-Shaw Flow
Petri Fast and Michael J. Shelley
Journal of Computational Physics 195, 117 (2004).
Abstract: We present a novel moving overset grid scheme for the accurate and efficient long-time simulation of an air bubble displacing a non-Newtonian fluid in the prototypical thin film device, the Helehaw cell. We use a two-dimensional generalization of Darcy's law that accounts for shear thinning of a non-Newtonian fluid. In the limit of weak shear thinning, the pressure is found from a ladder of two linear elliptic boundary value problems, each to be solved in the whole fluid domain. A moving body fitted grid is used to resolve the flow near the interface, while most of the fluid domain is covered with a fixed Cartesian grid. Our use of body-conforming grids reduces grid anisotropy effects and allows the accurate modeling of boundary conditions.
Chaotic mixing in a torus map
Jean-Luc Thiffeault and Stephen Childress
Chaos 13, (2), 502-507 (2003).
Abstract: The advection and diffusion of a passive scalar is investigated for a map of the 2-torus. The map is chaotic, and the limit of almost-uniform stretching is considered. This allows an analytic understanding of the transition from a phase of constant scalar variance (for short times) to exponential decay (for long times). This transition is embodied in a short superexponential phase of decay. The asymptotic state in the exponential phase is an eigenfunction of the advection-diffusion operator, in which most of the scalar variance is concentrated at small scales, even though a large-scale mode sets the decay rate. The duration of the superexponential phase is proportional to the logarithm of the exponential decay rate; if the decay is slow enough then there is no superexponential phase at all.
Drag reduction through self-similar bending of a flexible body.
Silas Alben, Michael Shelley, and Jun Zhang
Nature 420, 479-481 (2002).
Abstract: The classical theory of high-speed flow predicts that a moving rigid object experiences a drag proportional to the square of its speed. However, this reasoning does not apply if the object in the flow is flexible, because its shape then becomes a function of its speed -- for example, the rolling up of broad tree leaves in a stiff wind. The reconfiguration of bodies by fluid forces is common in nature, and can result in a substantial drag reduction that is beneficial for many organisms. Experimental studies of such flow structure interactions generally lack a theoretical interpretation that unifies the body and flow mechanics. Here we use a flexible fibre immersed in a flowing soap film to measure the drag reduction that arises from bending of the fibre by the flow. Using a model that couples hydrodynamics to bending, we predict a reduced drag growth compared to the classical theory. The fibre undergoes a bending transition, producing shapes that are self-similar; for such configurations, the drag scales with the length of self-similarity, rather than the fibre profile width. These predictions are supported by our experimental data.
Nature News and Views -- Bend and Survive, by Victor Steinberg
Nature's Secret to Building for Strength: Flexibility, Kenneth Chang, The New York Times, December 17,2002
Stark durch Nachgeben, Andrea Naica-Loebell, Telepolis, 11.12.2002
Dynamic Patterns and Self-Knotting of a Driven Hanging Chain
Andrew Belmonte, Shaden Eldakar, Michael Shelley, and Chris Wiggins
Physical Review Letters 87, (11), 114301 (2001).
Abstract: When shaken vertically, a hanging chain displays a startling variety of distinct behaviors. We find experimentally that instabilities occur in tonguelike bands of parameter space, to swinging or rotating pendular motion, or to chaotic states. Mathematically, the dynamics are described by a nonlinear wave equation. A linear stability analysis predicts instabilities within the well-known resonance tongues; their boundaries agree very well with experiment. Full simulations of the 3D dynamics reproduce and elucidate many aspects of the experiment. The chain is also observed to tie knots in itself, some quite complex. This is beyond the reach of the current analysis and simulations.
Click here for movies of the experiment
Click here for movies of the simulation
Maths helps magicians knot, NatureNews, 11 Sept. 2001
Heart simulation by an immersed boundary method with formal second-order accuracy and reduced numerical viscosity
David M. McQueen and Charles S. Peskin
Mechanics for a New Millennium, Proceedings of the International Conference on Theoretical and Applied Mechanics (ICTAM) 2000, (H. Aref and J.W. Phillips, eds.) Kluwer Academic Publishers, (2001).
Abstract: This paper describes a formally second-order accurate version of the immersed boundary method and its application to the computer simulation of blood flow in a three-dimensional model of the human heart.
Click here for heart animations computed by the immersed boundary method
Two-dimensional simulations of valveless pumping using the immersed boundary method
Eunok Jung and Charles S. Peskin
SIAM J. Sci. Comput. 23, 19-45 (2001).
Abstract: Flow driven by pumping without valves is examined, motivated by biomedical applications: cardiopulmonary resuscitation (CPR) and the human fetus before the development of the heart valves. The direction of flow inside a loop of tubingwhic h consists of (almost) rigid and flexible parts is investigated when the boundary of one end of the flexible segment is forced periodically in time. Despite the absence of valves, net flow around the loop may appear in these simulations. The magnitude and even the direction of this flow depend on the driving frequency of the periodic forcing.
An experiment on valveless pumping - AML experiment
Flexible filaments in a flowing soap film as a model for one-dimensional flags in a two-dimensional wind.
Jun Zhang, Stephen Childress, Albert Libchaber, and Michael Shelley
Nature 408, 835-839 (2000).
Abstract: The dynamics of swimming fish and flapping flags involves a complicated interaction of their deformable shapes with the surrounding fluid flow. Even in the passive case of a flag, the flag exerts forces on the fluid through its own inertia, elastic responses, and is likewise acted on by hydrodynamic pressure and drag. But such couplings are not well understood. Here we study these interactions experimentally, using an analogous system of flexible filaments in flowing soap films. We find that, for a single filament (or 'flag') held at its upstream end and otherwise unconstrained, there are two distinct, stable dynamical states. The first is a stretched-straight state: the filament is immobile and aligned in the flow direction. The existence of this state seems to refute the common belief that a flag is always unstable and will flap. The second is the flapping state: the filament executes a sinuous motion in a manner akin to the flapping of a flag in the wind. We study further the hydrodynamically coupled interaction between two such filaments, and demonstrate the existence of four different dynamical states.
Click here for ABC News video, March 2003. The late Mr. Peter Jennings reported about the work in the AML on flapping flags. - video
Online articles related to this research
Eulerian mean flow from an instability of convective plumes.
Chaos 10, 1054 (2000).
Abstract: The origin of large-scale flows in systems driven by concentrated Archimedean forces is considered. A two-dimensional model of plumes, such as those observed in thermal convection at large Rayleigh and Prandtl numbers, is introduced. From this model, we deduce the onset of mean flow as an instability of a convective state consisting of parallel vertical flow supported by buoyancy forces. The form of the linear equation governing the instability is derived and two modes of instability are discussed, one of which leads to the onset of steady Eulerian mean flow in the system. We are thus able to link the origin of the mean flow precisely to the profiles of the unperturbed plumes. The form of the nonlinear partial differential equation governing the Eulerian mean flow , including nonlinear effects, is derived in one special case. The extension to three dimensions is outlined.
Periodic boundary motion in thermal turbulence
Jun Zhang, Albert Libchaber
Phys. Rev. Lett. 84, 4361 (2000).
Abstract: A free-floating plate is introduced in a Bénard convection cell with an open surface. It covers partially the cell and distorts the local heat flux, inducing a coherent flow that in turn moves the plate. Remarkably, the plate can be driven to a periodic motion even under the action of a turbulent fluid. The period of the oscillation depends on the coverage ratio, and on the Rayleigh number of the convective system. The plate oscillatory behavior observed in this experiment, may be related to a geological model, in which continents drift in a quasi-periodic fashion.
Non-Boussinesq effect: asymmetric velocity profiles in thermal convection
Jun Zhang, S. Childress, and A. Libchaber
Physics of Fluids 10, 1534 (1998).
Abstract: In thermal convection at high Rayleigh numbers, in the hard turbulent regime, a large scale flow is present. When the viscosity of the fluid strongly depends on temperature, the top-bottom symmetry is broken. In addition to the asymmetric temperature profile across the convection cell, the velocity profiles near the plate boundaries show dramatic differences from the symmetric case. We report here that the second derivative of the velocity profiles are of opposite signs in the thermal sub-layers, through measurements derived from the power spectrum of temperature time-series. As a result, the stress rate applied at the plates is maintained constant within a factor of 3, while the viscosity changes by a factor of 53, in qualitative agreement with previous theory.
Non-Boussinesq effect: thermal convection with broken symmetry
Jun Zhang, S. Childress, and A. Libchaber
Physics of Fluids 9, (4)a, 1034 (1997).
Abstract: We investigate large Rayleigh number (106-109) and large Prandtl number (102-103) thermal convection in glycerol in an aspect ratio one cubic cell. The kinematic viscosity of the fluid strongly depends upon the temperature. The symmetry between the top and bottom boundary layers is thus broken, the so-called non-Boussinesq regime. In a previous paper Wu and Libchaber have proposed that in such a state the two thermal boundary layers adjust their length scales so that the mean hot and cold temperature fluctuations are equal in the center of the cell. We confirm this equality. A simplified two-dimensional model for the mean center temperature based on an equation for the thermal boundary layer is presented and compared with the experimental results. The conclusion is that the central temperature adjusts itself so that the heat fluxes from the boundary layers are equal, temperature fluctuations at the center symmetrical, at a cost of very different temperature drops and Rayleigh number for each boundary.