Instabilities and Structures in Free Surface Flows

Vortex wakes and thrust generation of flapping foils

Teis Schnipper has in his Ph.D. project made an experimental study of vortex shedding and wakes of a foil with periodic pitching oscillations in a two-dimensional free stream created by a flowing soap film. The flapping foil is a simple realization of a model fish and the setup makes it possible to study generic features of vortex shedding with relevance for swimming and flying.
The setup allowed the visualization of both the vortex formation and the wake evolution and made it possible to make a systematic wake map in the frequency and amplitude plane and to determine the transition from von Kármán vortex streets, usually associated with drag, to inverted von Kármán vortex streets, usually associated with thrust.
The figure below shows a time-series of the vortex formation leading to a wake in which two vortex pairs are shed in each flapping period. Vortices are shed both at the round leading edge and at the sharp trailing edge of the foil. Such wake structures are common behind swimming fish. 

Additionally, we have jointly with Jens H. Walther (DTU Mechanical Engineering) explored the connection between wake structure and thrust generation by combining simulations and experiments.

T. Schnipper, A. Andersen, and T. Bohr, Vortex wakes of a flapping foil, Journal of Fluid Mechanics 633, 411-423 (2009) (pdf) 

A. Andersen, T. Bohr, and T. Schnipper, Separation vortices and pattern formation, Theoretical and Computational Fluid Dynamics 24, 329-334 (2010) (pdf)

A. Andersen, T. Bohr, T. Schnipper, and J. H. Walther, Wake structure and thrust generation of a flapping foil in two-dimensional flow, Journal of Fluid Mechanics 812, R4 (2017) (pdf)

Walking droplets versus quantum mechanics

In a series of thought-provoking studies initiated by Yves Couder and Emmanuel Fort in Paris it has been shown that liquid droplets walking on a vertically vibrated bath can display a type of macroscopic wave-particle duality. But how closely does this new macroscopic wave-particle duality resemble the mysterious wave-particle duality at the microscopic scale of atoms? This fascinating question has so far been left unanswered. We argue that the double-slit experiment with walking droplets can possibly lead to spectacular interference patterns (Fig. 1), but that these will be fundamentally different from the interference in the renowned double-slit experiment with single electrons that is so central in quantum mechanics (Fig. 2). Our conclusions are based on experiments on walking droplets and general theoretical arguments. By presenting a modified double-slit experiment with a central splitter plate we theoretically pinpoint the fundamental difference between the two systems, namely that the droplet singles out a particular path whereby destroying the quantum mechanical democracy of paths experienced by an electron.

FIG. 1. Double-slit experiment with a walking droplet. (a), (b) The droplet is propelled by its pilot-wave towards the double-slit. (c) The droplet follows a well-defined trajectory through one of the slits, whereas the pilot-wave is free to pass through both slits. (d) – (f) After passage the droplet moves along a deflected path. The red dot shows the position of the droplet, and the double-slit is accentuated in white.
FIG. 2. Simulation of the quantum mechanical double-slit experiment. A single wave packet passes the double-slit and evolves according to the Schrödinger equation into a characteristic interference pattern.

A. Andersen, J. Madsen, C. Reichelt, S. Rosenlund Ahl, B. Lautrup, C. Ellegaard, M.T. Levinsen and T. Bohr, Double-slit experiment with single wave-driven particles and its relation to quantum mechanics, Physical Review E 92, 013006 (2015) (pdf) 

T. Bohr, A. Andersen and B. Lautrup, Bouncing droplets, pilot-waves, and quantum mechanics, Recent Advances in Fluid Dynamics with Environmental Applications, 335-349, Springer International Publishing (2016) (pdf) 

T. Bohr, Quantum physics dropwise, Nature Physics, News and Views, 1-2 (2017) (pdf) 

Vortex flows with a free surface

The bathtub vortex forms when water drains out of a container through a small drain-hole at the bottom and a dip in the water surface is formed which makes the vortex beautifully visible. This everyday phenomenon is the prototype fluid vortex and it has features in common with many flows in nature and technology.

We have worked on theory of the flow-structure and the surface shape of the bathtub vortex, and we have performed experiments, including the following surface photographs and flow-visualisations.
The free surface of a time-independent bathtub vortex in a rotating cylindrical container with a drain-hole at the bottom. The rotation rate of the container is 6 rpm (left), 12 rpm (center), and 18 rpm (right). The water in the central region is spinning fast and the depth of the surface dip increases when the rotation rate is increased.
An overlay of two flow visualisations made at 12 rpm by using a vertical light sheet through the axis of symmetry and adding fluorescent dye at the surface (red) and at the bottom inlet (green), respectively. The red dye is flowing in a thin surface layer to the central vortex core where it is diverted down-wards to the drain-hole. The free surface dip is seen above the central red region (the red pattern above the surface is due to reflections). The green dye spirals inward in the thin Ekman boundary layer at the bottom, then up-wards around the central drain-pipe forming the layered structure, and finally down the drain.

The bathtub vortex was discussed in a radio interview with BBC 4 for the science magazine program Leading Edge in October 2003, and our work was published in Physical Review Letters and Journal of Fluid Mechanics. 

A. Andersen, T. Bohr, B. Stenum, J. Juul Rasmussen, and B. Lautrup, Anatomy of a Bathtub Vortex, Physical Review Letters 91, 104502 (2003) (pdf)

A. Andersen, B. Lautrup, and T. Bohr, An averaging method for nonlinear laminar Ekman layers, Journal of Fluid Mechanics 487, 81-90 (2003) (pdf)

A. Andersen, T. Bohr, B. Stenum, J. Juul Rasmussen, and B. Lautrup, The bathtub vortex in a rotating container, Journal of Fluid Mechanics 556, 121-146 (2006) (pdf)

Bubble pinch-off in a rotating flow

The tip of the bathtub vortex is unstable at high rotation rates and air bubbles are released and carried down by the sink-flow. We have shown that a considerable air flow takes place from the tip and upwards through the collapsing neck and that the associated pressure reduction drives the pinch-off and makes the intense rotational motion irrelevant in the final stages of the collapse. 

Flow with intense rotation and bubble pinch-off at the tip of a bathtub vortex. Stable tip at low rotation rate in (a) and bubble formation at high rotation rate in (b) and (c). The frequency of bubble formation increases and the bubble size decreases with increasing rotation rate as shown in (b) and (c).
Time-evolution of the collapsing air-filled neck near pinch-off. The images were obtained using high-speed digital video, and the time to pinch-off in milliseconds is shown above each frame.

R. Bergmann, A. Andersen, D. van der Meer, and T. Bohr, Bubble Pinch-Off in a Rotating Flow, Physical Review Letters 102, 204501 (2009) (pdf)

Hydraulic jumps

Hydraulic jumps are fascinating examples of free surface flows with strongly deformed free surfaces. We have studied circular hydraulic jumps in boundary layers since the 90ies, as an interesting case of non-linear pattern formation. In the early works we looked at the size and shape of such jumps, and shown that even though there is separation in the jump region, averaging techniques can be quite useful. We have shown that hydraulic jumps can bifurcate between two forms, a “Type 1” form where there is separation only on the bottom of the plate and a “Type 2” form, where there is separation at the fluid surface. The Type II states show a surprising spontanuous symmetry breaking transition, where the circular shape become deformed into a polygonal structure.

In a recent paper (link to separate story here) (Martens et al. Phys. Rev E, 2012) we discuss the mechanism for the formation of the polygons.

We have also looked at hydraulic jumps in flows predominantly in one direction, created either by confining the flow to a narrow channel with parallel walls or by providing an inflow in the form of a narrow sheet (Bonn et al. J. Fluid Mech. 2009) (link). In the channel flow, we found a linear height profile upstream of the jump as expected for a supercritical one-dimensional boundary layer flow, but we found that the surface slope was up to an order of magnitude larger than expected and independent of flow rate. We explained this as an effect of turbulent fluctuations creating an enhanced eddy viscosity, and we modeled the results in terms of Prandtl's mixing-length theory with a mixing length that was proportional to the height of the fluid layer. In the sheet flow, we found that the jump had the shape of a rhombus with sharply defined oblique shocks. The experiment showed that the variation of the opening angle of the rhombus with flow rate was determined by the condition that the normal velocity at the jump was constant. The work was published in the Journal of Fluid Mechanics and was featured on the cover showing the rhombic hydraulic jump in the sheet experiment.

D. Bonn, A. Andersen, and T. Bohr, Hydraulic jumps in a channel, Journal of Fluid Mechanics 618, 71-87 (2009) (pdf)

Asymptotic Properties of the Circular Hydraulic Jump
T. Bohr, P. Dimon and V. Putkaradze
J. Fluid Mech. 254, 635 (1993)

Hydraulic Jumps, Flow Separation and Wave Breaking: an Experimental Study
C. Ellegaard, A. Espe Hansen and A. Haaning
Physica B, 228, 1 (1996)

Averaging theory for the structure of hydraulic jumps and separation in laminar free surface flows
V. Putkaradze and S. Watanabe
Phys. Rev. Lett. 79 1038 (1997).

Creating corners in kitchen sinks
C. Ellegaard, A. Espe Hansen, A. Haaning, K. Hansen, A. Marcussen, J. Lundbek Hansen, and S. Watanabe
Nature 392, 767 (1998)

Cover illustration: Polygonal hydraulic jumps
C. Ellegaard, A. Espe Hansen, A. Haaning, K. Hansen, A. Marcussen, J. Lundbek Hansen, and S. Watanabe
Nonlinearity 12, 1-7 (1999)

Integral methods for shallow free-surface flows with separation
S. Watanabe, V. Putkaradze, and T. Bohr
Journal of fluid mechanics 480, 233-265 (2003)

Hydraulic jumps in a channel
D. Bonn, A. Andersen, and T. Bohr
Journal of Fluid Mechanics 618, 71-87 (2009)

Separation vortices and pattern formation
A. Andersen, T. Bohr, and T. Schnipper
Theoretical and Computational Fluid Dynamics 24, 329-334 (2010) 

Rotating Polygons

The free surface of a fluid in a cylindrical container with a rotating bottom plate can undergo a surprising instability by which the surface shape spontaneously breaks rotational symmetry and turns into a “rotating polygon” as shown in the movie here. Isaac Newton considered an important rotating flow: a cylindrical container with a free surface, the so-called “Newton’s bucket”. The fact that the fluid at rest in the rotating system has a parabolic surface was for Newton proof that inertial systems are special, being free of “fictitious forces”. In Jansson et al. 2006, based on the bachelor project by Thomas Jansson, Martin Haspang and Kaare H. Jensen, we made a slight modification of Newton’s bucket, where we allowed only the bottom plate of the container to rotate. Although there now can be no coordinate frame where the fluid is at rest, one would still expect the surface to be curved in an axially symmetric way. What we found instead was that a class of new stable states exist, where the surface loses axial symmetry and deforms into the shape of a uniformly rotating polygon. This is even more surprizing due to the fact that the flow becomes strongly turbulent. Typical shapes (triangle and pentagon) are shown below. After the publication of our paper, it turned out that these polygonal states had been found earlier by Vatistas and coworkers (Vatistas 1990).

In our recent work: Rotating Polygon Instability of a Swirling Free Surface Flow (L. Tophøj, J. Mougel, T. Bohr, and D. Fabre, Phys. Rev. Lett. 110, 194502),a collaboration between DTU and IMFT in Toulouse) we explain the rotating polygon instability in terms of resonant interactions between gravity waves on the outer part of the surface and centrifugal waves on the inner part. Our model is based on potential flow theory, linearized around a potential vortex flow with a free surface for which we show that unstable resonant states appear. Limiting our attention to the lowest order mode of each type of wave and their interaction, we obtain an analytically soluble model, which, together with estimates of the circulation based on angular momentum balance, reproduces the main features of the experimental phase diagram. The generality of our arguments implies that the instability should not be limited to flows with a rotating bottom (implying singular behavior near the corners), and indeed we show that we can obtain the polygons transiently by violently stirring liquid nitrogen in a hot container, as shown in the movie here.

T.R.N. Jansson, K.H. Jensen, M.P. Haspang, P. Hersen, and T. Bohr
Physical Review Letters 96, 174502 (2006) (pdf) [Erratum 98, 049901 (2007)]

Polygon formation and surface flow on a rotating fluid surface
R. Bergmann, L. Tophøj, T.A.M. Homan, P. Hersen, A. Andersen, and T. Bohr
Journal of Fluid Mechanics 679, 415-431 (2011) (pdf)[Erratum 691, 605-606 (2012)]

Stationary ideal flow on a free surface of a given shape
L. Tophøj and T. Bohr
Journal of Fluid Mechanics 721, 28-45 (2013) (pdf)

The Rotating Polygon Instability of a Swirling Free Surface  Flow
L. Tophøj, J. Mougel, T. Bohr and D. Fabre
Phys. Rev. Lett. 110, 194502 (2013) (pdf)

From Newton's bucket to rotating polygons: experiments on surface instabilities in swirling flows
B. Bach, E.C. Linnartz, M.H. Vested, A. Andersen, and T. Bohr
Submitted to Journal of Fluid Mechanics (2014) (pdf)

On the instabilities of a potential vortex with a free surface
J. Mougel, D. Fabre, L. Lacaze and T. Bohr
Journal of Fluid Mechanics 824, 230-264 (2017) (pdf)

Japanese Fan Flow

The picture shows the downward oriented "breeze" due to a pitching fan: The sharp trailing edge shed vortices that are advected downstream while being deformed by the flow. The colorful pattern is due to thickness variations that are visualized with thin-film interferometry.

Japanese fan flow, T. Schnipper, L. Tophøj, A. Andersen, and T. Bohr, Physics of Fluids 22, 091102 (2010) (pdf)

Unsteady aerodynamics of fluttering and tumbling cards

Leaves, tree seeds, and paper cards falling in air are spectacular examples of time-dependent fluid mechanics at intermediate Reynolds numbers at which both inertial and viscous effects are important. The trajectory of a falling card typically appears to be very complex with the card either oscillating from side to side (fluttering) or rotating and drifting sideways (tumbling). Despite that fluttering and tumbling are common everyday phenomena that have interested scientists for centuries, it turns out that little is known theoretically about this type of aerodynamics and the nature of the transitions between periodic fluttering, periodic tumbling, and steady descent. 

Anders Andersen has - in collaboration with Umberto Pesavento and Z. Jane Wang - made a combined experimental, numerical, and theoretical study of this problem. In a table top experiment we have explored the two-dimensional dynamics of rigid metal plates falling in a water tank and measured the trajectories using high speed digital video. Examples of the observed trajectories include periodic fluttering (blue) and periodic tumbling (red). The measured trajectories give us the translational and the rotational plate acceleration and since the plate dynamics is governed by gravity and fluid forces alone this permits us to extract the fluid forces directly.

A. Andersen, U. Pesavento, and Z. J. Wang, Unsteady aerodynamics of fluttering and tumbling plates, Journal of Fluid Mechanics 541, 65-90 (2005) (pdf) 

A. Andersen, U. Pesavento, and Z. J. Wang, Analysis of transitions between fluttering, tumbling, and steady descent of falling cards, Journal of Fluid Mechanics 541, 91-104 (2005) (pdf)

Cavitation in fluids

Cavitation arises when at pressure reduction the tensile strength of a liquid is exceeded. Then vapour bubbles grow, and the single-phase liquid becomes a two-phase system. The performance of ship propellers, turbine blades and other hydraulic components is often strongly affected by cavitation, and the collapse of cavitation bubbles may cause serious erosion of adjacent solid surfaces. Theoretically, pure water has a tensile strength of approximately 1400 bar, a value actually measured with extremely pure water, but tap water has a tensile strength of at most a few bar, unless special precautions are made. The actual tensile strength is of great technical importance, and it makes studies of the physics of cavitation an important issue. At DTU Physics, research in cavitation erosion and cavity dynamics has been undertaken during 60 years. At present the cavitation nuclei determining the tensile strength of a liquid are in focus. We study these, and calculate their tensile strength from high-speed video recordings of the initial growth of bubbles generated by a tensile pulse as illustrated below. Based on our experiments a theory of cavitation nuclei in hydraulic systems was recently presented, and we aim at obtaining a coherent understanding of bubble nucleation in liquids generally.

In the above video sequence a cavitation nucleus of tensile strength 0.53 bar became critical ~360 µs after the arrival of the tensile pulse, causing growth of a vapour bubble, indicated by “1” at t = 417 µs. A new bubble, indicated by “2” at t = 577 µs, of tensile strength 0.19 bar only, started growing at ~490 µs. This teaches us that during exposure to tensile stress cavitation nuclei change their tensile strength - they are time dependent. The length of the scale bar is 2 mm.

A. Andersen and K.A. Mørch, Cavitation nuclei in water exposed to transient pressures, Journal of Fluid Mechanics 771, 424-448 (2015) (pdf)

K.A. Mørch, Cavitation inception from bubble nuclei, Interface Focus 5, 20150006 (2015) (pdf)