@article{463, abstract = {We investigate transient behaviors induced by magnetic fields on the dynamics of the flow of a ferrofluid in the gap between two concentric, independently rotating cylinders. Without applying any magnetic fields, we uncover emergence of flow states constituted by a combination of a localized spiral state (SPIl) in the top and bottom of the annulus and different multi-cell flow states (SPI2v, SPI3v) with toroidally closed vortices in the interior of the bulk (SPIl+2v = SPIl + SPI2v and SPIl+3v = SPIl + SPI3v). However, when a magnetic field is presented, we observe the transient behaviors between multi-cell states passing through two critical thresholds in a strength of an axial (transverse) magnetic field. Before the first critical threshold of a magnetic field strength, multi-stable states with different number of cells could be observed. After the first critical threshold, we find the transient behavior between the three- and two-cell flow states. For more strength of magnetic field or after the second critical threshold, we discover that multi-cell states are disappeared and a localized spiral state remains to be stimulated. The studied transient behavior could be understood by the investigation of various quantities including a modal kinetic energy, a mode amplitude of the radial velocity, wavenumber, angular momentum, and torque. In addition, the emergence of new flow states and the transient behavior between their states in ferrofluidic flows indicate that richer and potentially controllable dynamics through magnetic fields could be possible in ferrofluic flow.}, author = {Altmeyer, Sebastian and Do, Younghae and Ryu, Soorok}, issn = {1054-1500}, journal = {Chaos}, number = {11}, publisher = {AIP Publishing}, title = {{Transient behavior between multi-cell flow states in ferrofluidic Taylor-Couette flow}}, doi = {10.1063/1.5002771}, volume = {27}, year = {2017}, } @article{513, abstract = {We present an experimental setup that creates a shear flow with zero mean advection velocity achieved by counterbalancing the nonzero streamwise pressure gradient by moving boundaries, which generates plane Couette-Poiseuille flow. We obtain experimental results in the transitional regime for this flow. Using flow visualization, we characterize the subcritical transition to turbulence in Couette-Poiseuille flow and show the existence of turbulent spots generated by a permanent perturbation. Due to the zero mean advection velocity of the base profile, these turbulent structures are nearly stationary. We distinguish two regions of the turbulent spot: the active turbulent core, which is characterized by waviness of the streaks similar to traveling waves, and the surrounding region, which includes in addition the weak undisturbed streaks and oblique waves at the laminar-turbulent interface. We also study the dependence of the size of these two regions on Reynolds number. Finally, we show that the traveling waves move in the downstream (Poiseuille) direction.}, author = {Klotz, Lukasz and Lemoult, Grégoire M and Frontczak, Idalia and Tuckerman, Laurette and Wesfreid, José}, journal = {Physical Review Fluids}, number = {4}, publisher = {American Physical Society}, title = {{Couette-Poiseuille flow experiment with zero mean advection velocity: Subcritical transition to turbulence}}, doi = {10.1103/PhysRevFluids.2.043904}, volume = {2}, year = {2017}, } @article{1021, abstract = {Most flows in nature and engineering are turbulent because of their large velocities and spatial scales. Laboratory experiments on rotating quasi-Keplerian flows, for which the angular velocity decreases radially but the angular momentum increases, are however laminar at Reynolds numbers exceeding one million. This is in apparent contradiction to direct numerical simulations showing that in these experiments turbulence transition is triggered by the axial boundaries. We here show numerically that as the Reynolds number increases, turbulence becomes progressively confined to the boundary layers and the flow in the bulk fully relaminarizes. Our findings support that turbulence is unlikely to occur in isothermal constant-density quasi-Keplerian flows.}, author = {Lopez Alonso, Jose M and Avila, Marc}, issn = {0022-1120}, journal = {Journal of Fluid Mechanics}, pages = {21 -- 34}, publisher = {Cambridge University Press}, title = {{Boundary layer turbulence in experiments on quasi Keplerian flows}}, doi = {10.1017/jfm.2017.109}, volume = {817}, year = {2017}, } @article{792, abstract = {The chaotic dynamics of low-dimensional systems, such as Lorenz or Rössler flows, is guided by the infinity of periodic orbits embedded in their strange attractors. Whether this is also the case for the infinite-dimensional dynamics of Navier–Stokes equations has long been speculated, and is a topic of ongoing study. Periodic and relative periodic solutions have been shown to be involved in transitions to turbulence. Their relevance to turbulent dynamics – specifically, whether periodic orbits play the same role in high-dimensional nonlinear systems like the Navier–Stokes equations as they do in lower-dimensional systems – is the focus of the present investigation. We perform here a detailed study of pipe flow relative periodic orbits with energies and mean dissipations close to turbulent values. We outline several approaches to reduction of the translational symmetry of the system. We study pipe flow in a minimal computational cell at Re=2500, and report a library of invariant solutions found with the aid of the method of slices. Detailed study of the unstable manifolds of a sample of these solutions is consistent with the picture that relative periodic orbits are embedded in the chaotic saddle and that they guide the turbulent dynamics.}, author = {Budanur, Nazmi B and Short, Kimberly and Farazmand, Mohammad and Willis, Ashley and Cvitanović, Predrag}, issn = {0022-1120}, journal = {Journal of Fluid Mechanics}, pages = {274 -- 301}, publisher = {Cambridge University Press}, title = {{Relative periodic orbits form the backbone of turbulent pipe flow}}, doi = {10.1017/jfm.2017.699}, volume = {833}, year = {2017}, } @article{824, abstract = {In shear flows at transitional Reynolds numbers, localized patches of turbulence, known as puffs, coexist with the laminar flow. Recently, Avila et al. (Phys. Rev. Lett., vol. 110, 2013, 224502) discovered two spatially localized relative periodic solutions for pipe flow, which appeared in a saddle-node bifurcation at low Reynolds number. Combining slicing methods for continuous symmetry reduction with Poincaré sections for the first time in a shear flow setting, we compute and visualize the unstable manifold of the lower-branch solution and show that it extends towards the neighbourhood of the upper-branch solution. Surprisingly, this connection even persists far above the bifurcation point and appears to mediate the first stage of the puff generation: amplification of streamwise localized fluctuations. When the state-space trajectories on the unstable manifold reach the vicinity of the upper branch, corresponding fluctuations expand in space and eventually take the usual shape of a puff.}, author = {Budanur, Nazmi B and Hof, Björn}, issn = {0022-1120}, journal = {Journal of Fluid Mechanics}, publisher = {Cambridge University Press}, title = {{Heteroclinic path to spatially localized chaos in pipe flow}}, doi = {10.1017/jfm.2017.516}, volume = {827}, year = {2017}, } @article{651, abstract = {Superhydrophobic surfaces reduce the frictional drag between water and solid materials, but this effect is often temporary. The realization of sustained drag reduction has applications for water vehicles and pipeline flows. }, author = {Hof, Björn}, issn = {0028-0836}, journal = {Nature}, number = {7636}, pages = {161 -- 162}, publisher = {Nature Publishing Group}, title = {{Fluid dynamics: Water flows out of touch}}, doi = {10.1038/541161a}, volume = {541}, year = {2017}, } @article{662, abstract = {We report a direct-numerical-simulation study of the Taylor-Couette flow in the quasi-Keplerian regime at shear Reynolds numbers up to (105). Quasi-Keplerian rotating flow has been investigated for decades as a simplified model system to study the origin of turbulence in accretion disks that is not fully understood. The flow in this study is axially periodic and thus the experimental end-wall effects on the stability of the flow are avoided. Using optimal linear perturbations as initial conditions, our simulations find no sustained turbulence: the strong initial perturbations distort the velocity profile and trigger turbulence that eventually decays.}, author = {Shi, Liang and Hof, Björn and Rampp, Markus and Avila, Marc}, issn = {1070-6631}, journal = {Physics of Fluids}, number = {4}, publisher = {American Institute of Physics}, title = {{Hydrodynamic turbulence in quasi Keplerian rotating flows}}, doi = {10.1063/1.4981525}, volume = {29}, year = {2017}, } @article{673, abstract = {We present a numerical study of wavy supercritical cylindrical Couette flow between counter-rotating cylinders in which the wavy pattern propagates either prograde with the inner cylinder or retrograde opposite the rotation of the inner cylinder. The wave propagation reversals from prograde to retrograde and vice versa occur at distinct values of the inner cylinder Reynolds number when the associated frequency of the wavy instability vanishes. The reversal occurs for both twofold and threefold symmetric wavy vortices. Moreover, the wave propagation reversal only occurs for sufficiently strong counter-rotation. The flow pattern reversal appears to be intrinsic in the system as either periodic boundary conditions or fixed end wall boundary conditions for different system sizes always result in the wave propagation reversal. We present a detailed bifurcation sequence and parameter space diagram with respect to retrograde behavior of wavy flows. The retrograde propagation of the instability occurs when the inner Reynolds number is about two times the outer Reynolds number. The mechanism for the retrograde propagation is associated with the inviscidly unstable region near the inner cylinder and the direction of the global average azimuthal velocity. Flow dynamics, spatio-temporal behavior, global mean angular velocity, and torque of the flow with the wavy pattern are explored.}, author = {Altmeyer, Sebastian and Lueptow, Richard}, issn = {2470-0045}, journal = {Physical Review E}, number = {5}, publisher = {American Physical Society}, title = {{Wave propagation reversal for wavy vortices in wide gap counter rotating cylindrical Couette flow}}, doi = {10.1103/PhysRevE.95.053103}, volume = {95}, year = {2017}, } @article{745, abstract = {Fluid flows in nature and applications are frequently subject to periodic velocity modulations. Surprisingly, even for the generic case of flow through a straight pipe, there is little consensus regarding the influence of pulsation on the transition threshold to turbulence: while most studies predict a monotonically increasing threshold with pulsation frequency (i.e. Womersley number, ), others observe a decreasing threshold for identical parameters and only observe an increasing threshold at low . In the present study we apply recent advances in the understanding of transition in steady shear flows to pulsating pipe flow. For moderate pulsation amplitudes we find that the first instability encountered is subcritical (i.e. requiring finite amplitude disturbances) and gives rise to localized patches of turbulence ('puffs') analogous to steady pipe flow. By monitoring the impact of pulsation on the lifetime of turbulence we map the onset of turbulence in parameter space. Transition in pulsatile flow can be separated into three regimes. At small Womersley numbers the dynamics is dominated by the decay turbulence suffers during the slower part of the cycle and hence transition is delayed significantly. As shown in this regime thresholds closely agree with estimates based on a quasi-steady flow assumption only taking puff decay rates into account. The transition point predicted in the zero limit equals to the critical point for steady pipe flow offset by the oscillation Reynolds number (i.e. the dimensionless oscillation amplitude). In the high frequency limit on the other hand, puff lifetimes are identical to those in steady pipe flow and hence the transition threshold appears to be unaffected by flow pulsation. In the intermediate frequency regime the transition threshold sharply drops (with increasing ) from the decay dominated (quasi-steady) threshold to the steady pipe flow level.}, author = {Xu, Duo and Warnecke, Sascha and Song, Baofang and Ma, Xingyu and Hof, Björn}, issn = {0022-1120}, journal = {Journal of Fluid Mechanics}, pages = {418 -- 432}, publisher = {Cambridge University Press}, title = {{Transition to turbulence in pulsating pipe flow}}, doi = {10.1017/jfm.2017.620}, volume = {831}, year = {2017}, } @article{661, abstract = {During embryonic development, mechanical forces are essential for cellular rearrangements driving tissue morphogenesis. Here, we show that in the early zebrafish embryo, friction forces are generated at the interface between anterior axial mesoderm (prechordal plate, ppl) progenitors migrating towards the animal pole and neurectoderm progenitors moving in the opposite direction towards the vegetal pole of the embryo. These friction forces lead to global rearrangement of cells within the neurectoderm and determine the position of the neural anlage. Using a combination of experiments and simulations, we show that this process depends on hydrodynamic coupling between neurectoderm and ppl as a result of E-cadherin-mediated adhesion between those tissues. Our data thus establish the emergence of friction forces at the interface between moving tissues as a critical force-generating process shaping the embryo.}, author = {Smutny, Michael and Ákos, Zsuzsa and Grigolon, Silvia and Shamipour, Shayan and Ruprecht, Verena and Capek, Daniel and Behrndt, Martin and Papusheva, Ekaterina and Tada, Masazumi and Hof, Björn and Vicsek, Tamás and Salbreux, Guillaume and Heisenberg, Carl-Philipp J}, issn = {1465-7392}, journal = {Nature Cell Biology}, pages = {306 -- 317}, publisher = {Nature Publishing Group}, title = {{Friction forces position the neural anlage}}, doi = {10.1038/ncb3492}, volume = {19}, year = {2017}, } @article{1494, abstract = {Turbulence is one of the most frequently encountered non-equilibrium phenomena in nature, yet characterizing the transition that gives rise to turbulence in basic shear flows has remained an elusive task. Although, in recent studies, critical points marking the onset of sustained turbulence have been determined for several such flows, the physical nature of the transition could not be fully explained. In extensive experimental and computational studies we show for the example of Couette flow that the onset of turbulence is a second-order phase transition and falls into the directed percolation universality class. Consequently, the complex laminar–turbulent patterns distinctive for the onset of turbulence in shear flows result from short-range interactions of turbulent domains and are characterized by universal critical exponents. More generally, our study demonstrates that even high-dimensional systems far from equilibrium such as turbulence exhibit universality at onset and that here the collective dynamics obeys simple rules.}, author = {Lemoult, Grégoire M and Shi, Liang and Avila, Kerstin and Jalikop, Shreyas V and Avila, Marc and Hof, Björn}, journal = {Nature Physics}, number = {3}, pages = {254 -- 258}, publisher = {Nature Publishing Group}, title = {{Directed percolation phase transition to sustained turbulence in Couette flow}}, doi = {10.1038/nphys3675}, volume = {12}, year = {2016}, } @article{1588, abstract = {We investigate the Taylor-Couette system where the radius ratio is close to unity. Systematically increasing the Reynolds number, we observe a number of previously known transitions, such as one from the classical Taylor vortex flow (TVF) to wavy vortex flow (WVF) and the transition to fully developed turbulence. Prior to the onset of turbulence, we observe intermittent bursting patterns of localized turbulent patches, confirming the experimentally observed pattern of very short wavelength bursts (VSWBs). A striking finding is that, for a Reynolds number larger than that for the onset of VSWBs, a new type of intermittently bursting behavior emerges: patterns of azimuthally closed rings of various orders. We call them ring-bursting patterns, which surround the cylinder completely but remain localized and separated in the axial direction through nonturbulent wavy structures. We employ a number of quantitative measures including the cross-flow energy to characterize the ring-bursting patterns and to distinguish them from the background flow. These patterns are interesting because they do not occur in the wide-gap Taylor-Couette flow systems. The narrow-gap regime is less studied but certainly deserves further attention to gain deeper insights into complex flow dynamics in fluids.}, author = {Altmeyer, Sebastian and Do, Younghae and Lai, Ying}, journal = {Physical Review E}, number = {5}, publisher = {American Physical Society}, title = {{Ring-bursting behavior en route to turbulence in narrow-gap Taylor-Couette flows}}, doi = {10.1103/PhysRevE.92.053018}, volume = {92}, year = {2015}, } @article{1589, abstract = {We investigate the dynamics of ferrofluidic wavy vortex flows in the counter-rotating Taylor-Couette system, with a focus on wavy flows with a mixture of the dominant azimuthal modes. Without external magnetic field flows are stable and pro-grade with respect to the rotation of the inner cylinder. More complex behaviors can arise when an axial or a transverse magnetic field is applied. Depending on the direction and strength of the field, multi-stable wavy states and bifurcations can occur. We uncover the phenomenon of flow pattern reversal as the strength of the magnetic field is increased through a critical value. In between the regimes of pro-grade and retrograde flow rotations, standing waves with zero angular velocities can emerge. A striking finding is that, under a transverse magnetic field, a second reversal in the flow pattern direction can occur, where the flow pattern evolves into pro-grade rotation again from a retrograde state. Flow reversal is relevant to intriguing phenomena in nature such as geomagnetic reversal. Our results suggest that, in ferrofluids, flow pattern reversal can be induced by varying a magnetic field in a controlled manner, which can be realized in laboratory experiments with potential applications in the development of modern fluid devices.}, author = {Altmeyer, Sebastian and Do, Younghae and Lai, Ying}, journal = {Scientific Reports}, publisher = {Nature Publishing Group}, title = {{Magnetic field induced flow pattern reversal in a ferrofluidic Taylor-Couette system}}, doi = {10.1038/srep18589}, volume = {5}, year = {2015}, } @article{1664, abstract = {Over a century of research into the origin of turbulence in wall-bounded shear flows has resulted in a puzzling picture in which turbulence appears in a variety of different states competing with laminar background flow. At moderate flow speeds, turbulence is confined to localized patches; it is only at higher speeds that the entire flow becomes turbulent. The origin of the different states encountered during this transition, the front dynamics of the turbulent regions and the transformation to full turbulence have yet to be explained. By combining experiments, theory and computer simulations, here we uncover a bifurcation scenario that explains the transformation to fully turbulent pipe flow and describe the front dynamics of the different states encountered in the process. Key to resolving this problem is the interpretation of the flow as a bistable system with nonlinear propagation (advection) of turbulent fronts. These findings bridge the gap between our understanding of the onset of turbulence and fully turbulent flows.}, author = {Barkley, Dwight and Song, Baofang and Vasudevan, Mukund and Lemoult, Grégoire M and Avila, Marc and Hof, Björn}, journal = {Nature}, number = {7574}, pages = {550 -- 553}, publisher = {Nature Publishing Group}, title = {{The rise of fully turbulent flow}}, doi = {10.1038/nature15701}, volume = {526}, year = {2015}, } @article{1679, author = {Lemoult, Grégoire M and Maier, Philipp and Hof, Björn}, journal = {Physics of Fluids}, number = {9}, publisher = {American Institute of Physics}, title = {{Taylor's Forest}}, doi = {10.1063/1.4930850}, volume = {27}, year = {2015}, } @article{1868, abstract = {We investigate high-dimensional nonlinear dynamical systems exhibiting multiple resonances under adiabatic parameter variations. Our motivations come from experimental considerations where time-dependent sweeping of parameters is a practical approach to probing and characterizing the bifurcations of the system. The question is whether bifurcations so detected are faithful representations of the bifurcations intrinsic to the original stationary system. Utilizing a harmonically forced, closed fluid flow system that possesses multiple resonances and solving the Navier-Stokes equation under proper boundary conditions, we uncover the phenomenon of the early effect. Specifically, as a control parameter, e.g., the driving frequency, is adiabatically increased from an initial value, resonances emerge at frequency values that are lower than those in the corresponding stationary system. The phenomenon is established by numerical characterization of physical quantities through the resonances, which include the kinetic energy and the vorticity field, and a heuristic analysis based on the concept of instantaneous frequency. A simple formula is obtained which relates the resonance points in the time-dependent and time-independent systems. Our findings suggest that, in general, any true bifurcation of a nonlinear dynamical system can be unequivocally uncovered through adiabatic parameter sweeping, in spite of a shift in the bifurcation point, which is of value to experimental studies of nonlinear dynamical systems.}, author = {Park, Youngyong and Do, Younghae and Altmeyer, Sebastian and Lai, Yingcheng and Lee, Gyuwon}, issn = {1539-3755}, journal = {Physical Review E}, number = {2}, publisher = {American Physical Society}, title = {{Early effect in time-dependent, high-dimensional nonlinear dynamical systems with multiple resonances}}, doi = {10.1103/PhysRevE.91.022906}, volume = {91}, year = {2015}, } @article{2030, abstract = {A hybrid-parallel direct-numerical-simulation method with application to turbulent Taylor-Couette flow is presented. The Navier-Stokes equations are discretized in cylindrical coordinates with the spectral Fourier-Galerkin method in the axial and azimuthal directions, and high-order finite differences in the radial direction. Time is advanced by a second-order, semi-implicit projection scheme, which requires the solution of five Helmholtz/Poisson equations, avoids staggered grids and renders very small slip velocities. Nonlinear terms are evaluated with the pseudospectral method. The code is parallelized using a hybrid MPI-OpenMP strategy, which, compared with a flat MPI parallelization, is simpler to implement, allows to reduce inter-node communications and MPI overhead that become relevant at high processor-core counts, and helps to contain the memory footprint. A strong scaling study shows that the hybrid code maintains scalability up to more than 20,000 processor cores and thus allows to perform simulations at higher resolutions than previously feasible. In particular, it opens up the possibility to simulate turbulent Taylor-Couette flows at Reynolds numbers up to O(105). This enables to probe hydrodynamic turbulence in Keplerian flows in experimentally relevant regimes.}, author = {Shi, Liang and Rampp, Markus and Hof, Björn and Avila, Marc}, journal = {Computers and Fluids}, number = {1}, pages = {1 -- 11}, publisher = {Elsevier}, title = {{A hybrid MPI-OpenMP parallel implementation for pseudospectral simulations with application to Taylor-Couette flow}}, doi = {10.1016/j.compfluid.2014.09.021}, volume = {106}, year = {2015}, } @article{1804, abstract = {It is known that in classical fluids turbulence typically occurs at high Reynolds numbers. But can turbulence occur at low Reynolds numbers? Here we investigate the transition to turbulence in the classic Taylor-Couette system in which the rotating fluids are manufactured ferrofluids with magnetized nanoparticles embedded in liquid carriers. We find that, in the presence of a magnetic field transverse to the symmetry axis of the system, turbulence can occur at Reynolds numbers that are at least one order of magnitude smaller than those in conventional fluids. This is established by extensive computational ferrohydrodynamics through a detailed investigation of transitions in the flow structure, and characterization of behaviors of physical quantities such as the energy, the wave number, and the angular momentum through the bifurcations. A finding is that, as the magnetic field is increased, onset of turbulence can be determined accurately and reliably. Our results imply that experimental investigation of turbulence may be feasible by using ferrofluids. Our study of transition to and evolution of turbulence in the Taylor-Couette ferrofluidic flow system provides insights into the challenging problem of turbulence control.}, author = {Altmeyer, Sebastian and Do, Younghae and Lai, Ying}, journal = {Scientific Reports}, publisher = {Nature Publishing Group}, title = {{Transition to turbulence in Taylor-Couette ferrofluidic flow}}, doi = {10.1038/srep10781}, volume = {5}, year = {2015}, } @article{1837, abstract = {Transition to turbulence in straight pipes occurs in spite of the linear stability of the laminar Hagen-Poiseuille flow if both the amplitude of flow perturbations and the Reynolds number Re exceed a minimum threshold (subcritical transition). As the pipe curvature increases, centrifugal effects become important, modifying the basic flow as well as the most unstable linear modes. If the curvature (tube-to-coiling diameter d/D) is sufficiently large, a Hopf bifurcation (supercritical instability) is encountered before turbulence can be excited (subcritical instability). We trace the instability thresholds in the Re - d/D parameter space in the range 0.01 ≤ d/D\ ≤ 0.1 by means of laser-Doppler velocimetry and determine the point where the subcritical and supercritical instabilities meet. Two different experimental set-ups are used: a closed system where the pipe forms an axisymmetric torus and an open system employing a helical pipe. Implications for the measurement of friction factors in curved pipes are discussed.}, author = {Kühnen, Jakob and Braunshier, P and Schwegel, M and Kuhlmann, Hendrik and Hof, Björn}, journal = {Journal of Fluid Mechanics}, number = {5}, publisher = {Cambridge University Press}, title = {{Subcritical versus supercritical transition to turbulence in curved pipes}}, doi = {10.1017/jfm.2015.184}, volume = {770}, year = {2015}, } @article{2224, abstract = {This work investigates the transition between different traveling helical waves (spirals, SPIs) in the setup of differentially independent rotating cylinders. We use direct numerical simulations to consider an infinite long and periodic Taylor-Couette apparatus with fixed axial periodicity length. We find so-called mixed-cross-spirals (MCSs), that can be seen as nonlinear superpositions of SPIs, to establish stable footbridges connecting SPI states. While bridging the bifurcation branches of SPIs, the corresponding contributions within the MCS vary continuously with the control parameters. Here discussed MCSs presenting footbridge solutions start and end in different SPI branches. Therefore they differ significantly from the already known MCSs that present bypass solutions (Altmeyer and Hoffmann 2010 New J. Phys. 12 113035). The latter start and end in the same SPI branch, while they always bifurcate out of those SPI branches with the larger mode amplitude. Meanwhile, these only appear within the coexisting region of both SPIs. In contrast, the footbridge solutions can also bifurcate out of the minor SPI contribution. We also find they exist in regions where only one of the SPIs contributions exists. In addition, MCS as footbridge solution can appear either stable or unstable. The latter detected transient solutions offer similar spatio-temporal characteristics to the flow establishing stable footbridges. Such transition processes are interesting for pattern-forming systems in general because they accomplish transitions between traveling waves of different azimuthal wave numbers and have not been described in the literature yet.}, author = {Altmeyer, Sebastian}, issn = {0169-5983}, journal = {Fluid Dynamics Research}, number = {2}, publisher = {IOP Publishing}, title = {{On secondary instabilities generating footbridges between spiral vortex flow}}, doi = {10.1088/0169-5983/46/2/025503}, volume = {46}, year = {2014}, }