Session D29: Nonlinear Dynamics: Coherent Structures II

Asymptotic descriptions of self-sustaining processes in shear and Langmuir turbulence: A comparative study

It has long been observed that stress-driven turbulence either in the presence or the absence of surface waves is characterized by streamwise-oriented roll vortices and streamwise streaks associated with spanwise variations in the streamwise flow. To elucidate the fundamental differences between wave-free (``shear") and wave-catalyzed (``Langmuir") turbulence, two separate asymptotic theories are developed in parallel. First, a large Reynolds number analysis of the Navier--Stokes equations that describes a self-sustaining process (SSP) operative in linearly stable wall-bounded shear flows is recounted. This theory is contrasted with that emerging from an asymptotic reduction in the strong wave-forcing limit of the Craik--Leibovich (CL) equations governing Langmuir turbulence. The comparative analysis reveals important structural and dynamical differences between the SSPs in shear flows with and without surface waves and lends further support to the view that Langmuir turbulence in the upper ocean is a distinct turbulence regime.   

Experimental Observation of Exact Coherent Structures in a Weakly Turbulent Quasi-Two-Dimensional Flow

The dynamical systems approach to fluid turbulence relies on understanding the role of {\it unstable}, non-chaotic solutions -- such as equilibria, traveling waves, and periodic orbits -- of the Navier-Stokes equations. These solutions, called Exact Coherent Structures, exist in the same parameter regime as turbulence, but being unstable, are observed in experiments only as short transients. In this talk, we present experimental evidence for the existence and dynamical relevance of unstable equilibria in a weakly turbulent quasi-two-dimensional (Q2D) Kolmogorov flow. In the experiment, this Q2D flow is generated in an electromagnetically driven {\it shallow} layer of electrolyte. The numerical simulations, however, use a strictly 2D model which incorporates the effects of the finite thickness of the fluid layer in the experiment. During its evolution, there are instances when the dynamics of a weakly turbulent flow slow down, rather dramatically. Using experimental flow fields from such instances, and by means of a Newton-Solver, we numerically compute several unstable equilibria. Additionally, using numerical simulations, we show that the dynamics of a turbulent flow in the neighbourhood of an equilibrium are accurately described by the unstable manifold of the equilibrium.   

Symmetry Broken Exact Coherent Structures in Plane Couette Flow

Invariant solutions of the fully resolved Navier-Stokes equation, known as exact coherent structures (ECS) are an exciting and potentially revolutionary method for understanding turbulent dynamics. The geometry of plane Couette flow leads to the existence of ECS with a high degree of symmetry. However, turbulent flows do not display a high degree of symmetry, so it is unclear whether these symmetric ECS can truly capture the turbulent dynamics. We report the discovery of four new periodic orbits -- P85 and P60 which are fully symmetric, and P32 and P8, which have partially broken symmetry. Projections of these periodic orbits in the dissipation-energy input plane reveal that P32, P60 and P85 lie in the turbulent region of the state space, whereas P8 lies very far away from this region. Parametric continuation in the spanwise periodic cell length $L_z$ suggests that P8 undergoes two bifurcations, which are verified by analysis of various properties of P8 in the dissipation-energy input plane, and by observations of changes in the stability of eigenvectors that are consistent with bifurcations.   

Wavebreaking of Interfacial Stokes Flows

Viscous fluid conduits provide a versatile system for the study of dissipationless, dispersive hydrodynamics. A dense, viscous fluid serves as the background media through which a less dense, less viscous fluid buoyantly rises. If fluid is continuously injected into the exterior fluid, an interface forms that behaves like a deformable pipe. Conservation of mass implies that the interfacial dynamics are conservative, i.e., they behave like a superfluid. Through buoyancy, high viscosity contrast, and a long wave assumption, conduit interfacial dynamics can be modeled by a scalar, nonlinear, dispersive wave equation with no assumption on amplitude. Experiments involving solitons, wavebreaking leading to dispersive shock waves (DSWs), and their interactions will be presented. The results include the refraction and absorption of a soliton by a DSW and the refraction of a DSW by a second DSW, resulting in two-phase behavior. Excellent agreement between nonlinear wave (Whitham) averaging, numerics, and laboratory experiments will be presented. The nonlinear wave dynamics observed in this model system have implications for a broad range of other conservative dispersive hydrodynamic systems.   

Frozen Fronts selection in flow against self-sustained chemical waves

Laboratoire Fluides Automatique et Syst\`{e}mes Thermiques, Universit\'{e} Paris Sud, C.N.R.S. (UMR7608), B\^{a}timent 502, Campus Universitaire, 91405 Orsay Cedex, France Autocatalytic reaction fronts between two reacting species in the absence of fluid flow propagate as solitary waves. The coupling between autocatalytic reaction front and simple forced hydrodynamic flows may lead to stationary fronts whose velocity and shape depend on the underlying flow field. We focus on the issue of the chemo-hydrodynamic coupling between forced advection opposed to self-sustained chemical waves which can lead to Frozen Fronts, i.e. static stationary fronts, S. Saha et al, EPL 101, 38003 (2013). We perform experiments and numerical simulations with the well characterized autocatalytic Iodate Arsenous Acid reaction (IAA) over a wide range of flow velocities around a solid disk. We delineate the range over which we do observe these Frozen Fronts. We compare the shape of the observed Frozen Fronts to the computed ones in the so-called eikonal, thin front limit. In this limit, we are able to provide a scenario for the selection of the observed frozen states.   

Experimental studies of reaction front barriers in a three-dimensional nested vortex flow

We present experiments that study the behavior of reaction fronts propagating in three-dimensional, laminar fluid flows. The primary flow is a chain of nested horizontal and vertical vortices, a flow that has been shown to produce chaotic mixing even if time-independent\footnote{M.A. Fogleman, M.J. Fawcett and T.H. Solomon, Phys. Rev. E {\bf 63}, 020101(R) (2001).} The fronts are produced by the excitable, Ruthenium-catalyzed Belousov-Zhabotinsky chemical reaction. When illuminated with a near-UV laser beam, the Ru indicator fluoresces everywhere except where there is a reaction front. By scanning the laser beam and imaging from above, we are able to do a full 3D-visualization of the reaction front propagating through the flow. The fronts are observed to encounter tube- and sheet-like barriers, whose properties we measure experimentally. We interpret the results by generalizing a recent theory of ``burning invariant manifolds'' \footnote{J. Mahoney, D. Bargteil, M. Kingsbury, K. Mitchell and T. Solomon, Europhys. Lett. {\bf 98}, 44005 (2012).} which have been shown previously to act as one-way barriers for reaction fronts propagating in two-dimensional fluid flows.   

Experimental studies of pinned and unpinned reaction fronts in two-dimensional, vortex-dominated flows

We present experiments that study the propagation and pinning of reaction fronts in laminar, two-dimensional fluid flows. The flows are forced using magnetohydrodynamic techniques and are composed of vortex chains and arrays with or without an imposed wind. The reaction fronts are produced by the excitable, ferroin-catalyzed Belousov-Zhabotinsky chemical reaction. We consider how the addition of time-periodic oscillations of the flow can affect the pinning of reaction fronts.\footnote{P.W. Megson, M.L. Najarian, K.E. Lilienthal and T.H. Solomon, Phys. Fluids {\bf 27}, 023601 (2015).} Furthermore, we measure the speed at which reaction fronts propagate in the flow, looking for scaling of the measured front propagation speed with the non-dimensional reaction-diffusion (no flow) speed. We analyze all of these results by considering the role of one-way barriers produced by ``burning invariant manifolds.''\footnote{J. Mahoney, D. Bargteil, M. Kingsbury, K. Mitchell and T. Solomon, Europhys. Lett. {\bf 98}, 44005 (2012).}   

Vertically localized equilibrium solutions in the large eddy simulation of homogeneous shear flow

Equilibrium solutions in a large eddy simulation (LES) of statistically-stationary homogeneous shear flow with zero molecular viscosity are numerically obtained by a Newton-Krylov-hookstep method. The energy input is done by the mean shear at scales comparable to the spanwise width $L_z$ of the computational domain, while energy dissipation is represented by the eddy viscosity term at the small scale of the order of the Smagorinsky length $C_S \Delta$ ($C_S$ is the static Smagorinsky constant and $\Delta$ is the grid scale). It is shown that these solutions appear by a saddle-node bifurcation as $C_S \Delta/L_z$ decreases, and have the sinuous symmetry of Nagata's equilibrium solution in Couette flow (JFM 217, 519-527 (1990)). Both lower- and upper-branch solutions are vertically localized. The upper-branch solution is characterized by tall structures, while the lower-branch forms in the critical layer as in the asymptotic theory of shear flows at high-Reynolds numbers (K. Deguchi \& P. Hall, Phil. Trans. R. Soc A, 372:20130352 (2014)).   

Exact localized free-stream coherent structures in a parallel boundary layer

The dynamical systems description of transitional turbulence is based on exact invariant solutions of the 3D Navier-Stokes equations. We present a new family of exact traveling wave solutions in the asymptotic suction boundary layer. The solutions are localized in wall-normal and spanwise direction. Instead of being attached to the wall, the solutions are dominated by vortical structures reaching far into the free-stream region. These invariant solutions thus suggest that dynamical systems concepts, so far mostly studied in confined geometries, can carry over to open boundary layers and are relevant for turbulence far from confining walls.   

Exact laminar solutions for flows in channels with sinusoidal walls

We compute exact solutions for steady, incompressible, laminar flows in sinusoidal channels using Newton's method, employing domain transformation with spectral resolution in all spatial directions. Aligning the walls to be in phase has made computations considerably cheap (runtime/case $\sim$ 10 minutes on 1 core); Newton's method has allowed tracing solutions into high Reynolds number ranges, where solutions are temporally unstable. We identify four parameters: the amplitude, maximum slope, and streamwise inclination of the grooves/furrows in the surfaces, as well as the mean pressure gradient that drives the flow. Results are presented for amplitudes ranging from 0.1\% to 10\% of channel half-height, and maximum slopes ranging from 0.3 to 3.0, for a set of inclinations and Reynolds numbers. We look at the onset and sizes of steady recirculation zones, their effect on the volume flux, and relative contributions of pressure and wall-shear to total drag. The strengths of shear layers and the wall-normal gradients of circulation are considered as indicators for Kelvin-Helmholtz and centrifugal instabilities respectively. Future work will focus on computing other classes of exact solutions and understanding their significance to transition and turbulence.