scholarly journals Nonlinear dynamics of surfactant-laden two-fluid Couette flows in the presence of inertia

2016 ◽  
Vol 802 ◽  
pp. 5-36 ◽  
Author(s):  
A. Kalogirou ◽  
D. T. Papageorgiou
Keyword(s):  
Stokes Equations ◽  
Travelling Waves ◽  
Type Theorem ◽  
Three Dimensional ◽  
Two Dimensional ◽  
One Dimensional ◽  
Wide Range ◽  
Time Periodic ◽  
Two Fluid ◽  
Permanent Form

The nonlinear stability of immiscible two-fluid Couette flows in the presence of inertia is considered. The interface between the two viscous fluids can support insoluble surfactants and the interplay between the underlying hydrodynamic instabilities and Marangoni effects is explored analytically and computationally in both two and three dimensions. Asymptotic analysis when one of the layers is thin relative to the other yields a coupled system of nonlinear equations describing the spatio-temporal evolution of the interface and its local surfactant concentration. The system is non-local and arises by appropriately matching solutions of the linearised Navier–Stokes equations in the thicker layer to the solution in the thin layer. The scaled models are used to study different physical mechanisms by varying the Reynolds number, the viscosity ratio between the two layers, the total amount of surfactant present initially and a scaled Péclet number measuring diffusion of surfactant along the interface. The linear stability of the underlying flow to two- and three-dimensional disturbances is investigated and a Squire’s type theorem is found to hold when inertia is absent. When inertia is present, three-dimensional disturbances can be more unstable than two-dimensional ones and so Squire’s theorem does not hold. The linear instabilities are followed into the nonlinear regime by solving the evolution equations numerically; this is achieved by implementing highly accurate linearly implicit schemes in time with spectral discretisations in space. Numerical experiments for finite Reynolds numbers indicate that for two-dimensional flows the solutions are mostly nonlinear travelling waves of permanent form, even though these can lose stability via Hopf bifurcations to time-periodic travelling waves. As the length of the system (that is the wavelength of periodic waves) increases, the dynamics becomes more complex and includes time-periodic, quasi-periodic as well as chaotic fluctuations. It is also found that one-dimensional interfacial travelling waves of permanent form can become unstable to spanwise perturbations for a wide range of parameters, producing three-dimensional flows with interfacial profiles that are two-dimensional and travel in the direction of the underlying shear. Nonlinear flows are also computed for parameters which predict linear instability to three-dimensional disturbances but not two-dimensional ones. These are found to have a one-dimensional interface in a rotated frame with respect to the direction of the underlying shear and travel obliquely without changing form.

Spatio-temporal dynamics of a periodically driven cavity flow

2003 ◽  
Vol 478 ◽  
pp. 197-226 ◽  
Author(s):  
M. J. VOGEL ◽  
A. H. HIRSA ◽  
J. M. LOPEZ
Keyword(s):  
Free Surface ◽  
Stokes Equations ◽  
Temporal Dynamics ◽  
Three Dimensional ◽  
Spanwise Direction ◽  
Two Dimensional ◽  
Measured Velocity ◽  
Time Symmetry ◽  
Time Flow ◽  
Time Periodic

The flow in a rectangular cavity driven by the sinusoidal motion of the floor in its own plane has been studied both experimentally and computationally over a broad range of parameters. The stability limits of the time-periodic two-dimensional base state are of primary interest in the present study, as it is within these limits that the flow can be used as a viable surface viscometer (as outlined theoretically in Lopez & Hirsa 2001). Three flow regimes have been found experimentally in the parameter space considered: an essentially two-dimensional time-periodic flow, a time-periodic three-dimensional flow with a cellular structure in the spanwise direction, and a three-dimensional irregular (in both space and time) flow. The system poses a space–time symmetry that consists of a reflection about the vertical mid-plane together with a half-period translation in time (RT symmetry); the two-dimensional base state is invariant to this symmetry. Computations of the two-dimensional Navier–Stokes equations agree with experimentally measured velocity and vorticity to within experimental uncertainty in parameter regimes where the flow is essentially uniform in the spanwise direction, indicating that in this cavity with large spanwise aspect ratio, endwall effects are small and localized for these cases. Two classes of flows have been investigated, one with a rigid no-slip top and the other with a free surface. The basic states of these two cases are quite similar, but the free-surface case breaks RT symmetry at lower forcing amplitudes, and the structure of the three-dimensional states also differs significantly between the two classes.

scholarly journals Turbulent 2.5-dimensional dynamos

2016 ◽  
Vol 799 ◽  
pp. 246-264 ◽  
Author(s):  
K. Seshasayanan ◽  
A. Alexakis
Keyword(s):  
Turbulent Flows ◽  
Large Scale ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Reynolds Numbers ◽  
Navier Stokes ◽  
Two Dimensional ◽  
Navier Stokes Equations ◽  
Wide Range ◽  
Two Dimensional Flow

We study the linear stage of the dynamo instability of a turbulent two-dimensional flow with three components $(u(x,y,t),v(x,y,t),w(x,y,t))$ that is sometimes referred to as a 2.5-dimensional (2.5-D) flow. The flow evolves based on the two-dimensional Navier–Stokes equations in the presence of a large-scale drag force that leads to the steady state of a turbulent inverse cascade. These flows provide an approximation to very fast rotating flows often observed in nature. The low dimensionality of the system allows for the realization of a large number of numerical simulations and thus the investigation of a wide range of fluid Reynolds numbers $Re$, magnetic Reynolds numbers $Rm$ and forcing length scales. This allows for the examination of dynamo properties at different limits that cannot be achieved with three-dimensional simulations. We examine dynamos for both large and small magnetic Prandtl-number turbulent flows $Pm=Rm/Re$, close to and away from the dynamo onset, as well as dynamos in the presence of scale separation. In particular, we determine the properties of the dynamo onset as a function of $Re$ and the asymptotic behaviour in the large $Rm$ limit. We are thus able to give a complete description of the dynamo properties of these turbulent 2.5-D flows.

Topology of three-dimensional steady cellular flow in a two-sided anti-parallel lid-driven cavity

2017 ◽  
Vol 826 ◽  
pp. 302-334 ◽  
Author(s):  
Francesco Romanò ◽  
Stefan Albensoeder ◽  
Hendrik C. Kuhlmann
Keyword(s):  
Reynolds Number ◽  
Limit Cycle ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Convection Cell ◽  
Two Dimensional ◽  
Three Dimensional Flow ◽  
Driven Cavity ◽  
Dimensional Flow ◽  
Wide Range

The structure of the incompressible steady three-dimensional flow in a two-sided anti-symmetrically lid-driven cavity is investigated for an aspect ratio $\unicode[STIX]{x1D6E4}=1.7$ and spanwise-periodic boundary conditions. Flow fields are computed by solving the Navier–Stokes equations with a fully spectral method on $128^{3}$ grid points utilizing second-order asymptotic solutions near the singular corners. The supercritical flow arises in the form of steady rectangular convection cells within which the flow is point symmetric with respect to the cell centre. Global streamline chaos occupying the whole domain is found immediately above the threshold to three-dimensional flow. Beyond a certain Reynolds number the chaotic sea recedes from the interior, giving way to regular islands. The regular Kolmogorov–Arnold–Moser tori grow with increasing Reynolds number before they shrink again to eventually vanish completely. The global chaos at onset is traced back to the existence of one hyperbolic and two elliptic periodic lines in the basic flow. The singular points of the three-dimensional flow which emerge from the periodic lines quickly change such that, for a wide range of supercritical Reynolds number, each periodic convection cell houses a double spiralling-in saddle focus in its centre, a spiralling-out saddle focus on each of the two cell boundaries and two types of saddle limit cycle on the walls. A representative analysis for $\mathit{Re}=500$ shows chaotic streamlines to be due to chaotic tangling of the two-dimensional stable manifold of the central spiralling-in saddle focus and the two-dimensional unstable manifold of the central wall limit cycle. Embedded Kolmogorov–Arnold–Moser tori and the associated closed streamlines are computed for several supercritical Reynolds numbers owing to their importance for particle transport.

A numerical simulation method for bionic fish self-propelled swimming under control based on deep reinforcement learning

2020 ◽  
Vol 234 (17) ◽  
pp. 3397-3415
Author(s):  
Lang Yan ◽  
Xinghua Chang ◽  
Runyu Tian ◽  
Nianhua Wang ◽  
Laiping Zhang ◽  
...  
Keyword(s):  
Reinforcement Learning ◽  
Motion Control ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Coupling Method ◽  
Computational Domain ◽  
Two Dimensional ◽  
Fish Swimming ◽  
Fish Model ◽  
Wide Range

In order to simulate the under control self-propelled swimming of bionic fishes, a coupling method of hydrodynamics/kinematics/motion-control is presented in this paper. The Navier-Stokes equations in the arbitrary Lagrangian-Eulerian framework are solved in parallel based on the computational domain decomposition to simulate the unsteady flow field efficiently. The flow dynamics is coupled with the fish dynamics in an implicit way by a dual-time stepping approach. In order to discretize the computational domain during a wide range maneuver, an overset grid approach with a parallel implicit hole-cutting technique is adopted and coupled with morphing hybrid grids around the undulation body. The motion control of the fish swimming is realized by a deep reinforcement learning algorithm, which makes the fish model choose proper undulation manner according to a specific purpose. By adding random disturbances in the training process of fish swimming along a straight line, a simplified two-dimensional fish model obtains the ability to swim along a specific trajectory. Then in subsequent tests, the two-dimensional fish model is able to swim along more complex curves with obstacles. Finally, the starting process of a three-dimensional tuna-like model is simulated preliminarily to validate the ability of the coupling method for three-dimensional complex configurations. The numerical results demonstrate that this study could be used to explore the swimming mechanism of fishes in complex environments and to guide how robotic fishes can be controlled to accomplish their tasks.

Finite-amplitude equilibrium states in plane Couette flow

1997 ◽  
Vol 342 ◽  
pp. 159-177 ◽  
Author(s):  
A. CHERHABILI ◽  
U. EHRENSTEIN
Keyword(s):  
Couette Flow ◽  
Stokes Equations ◽  
Travelling Waves ◽  
Three Dimensional ◽  
Reynolds Numbers ◽  
Finite Amplitude ◽  
Equilibrium States ◽  
Navier Stokes ◽  
Two Dimensional ◽  
Plane Couette Flow

A numerical bifurcation study in plane Couette flow is performed by computing successive finite-amplitude equilibrium states, solutions of the Navier–Stokes equations. Plane Couette flow being linearly stable for all Reynolds numbers, first two-dimensional equilibrium states are computed by extending nonlinear travelling waves in plane Poiseuille flow through the Poiseuille–Couette flow family to the plane Couette flow limit. The resulting nonlinear states are stationary with a spatially localized structure; they are subject to two-dimensional and three-dimensional secondary disturbances. Three-dimensional disturbances dominate the dynamics and three-dimensional stationary equilibrium states bifurcating at criticality on the two-dimensional equilibrium surface are computed. These nonlinear states, periodic in the spanwise direction and spatially localized in the streamwise direction, are computed for Reynolds numbers (based on half the velocity difference between the walls and channel half-width) close to 1000. While a possible relationship between the computed solutions and experimentally observed coherent structures in turbulent plane Couette flow has to be assessed, the present findings reinforce the idea that subcritical transition may be related to the existence of finite-amplitude states which are (unstable) fixed points in a dynamical systems formulation of the Navier–Stokes system.

scholarly journals A Framework for Approximation of the Stokes Equations in an Axisymmetric Domain

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Niklas Ericsson
Keyword(s):  
Fourier Expansion ◽  
Stokes Equations ◽  
Stokes Problem ◽  
Three Dimensional ◽  
Countable Family ◽  
Angular Variable ◽  
Two Dimensional ◽  
Sobolev Norms ◽  
Well Posed ◽  
Variational Formulations

Abstract We develop a framework for solving the stationary, incompressible Stokes equations in an axisymmetric domain. By means of Fourier expansion with respect to the angular variable, the three-dimensional Stokes problem is reduced to an equivalent, countable family of decoupled two-dimensional problems. By using decomposition of three-dimensional Sobolev norms, we derive natural variational spaces for the two-dimensional problems, and show that the variational formulations are well-posed. We analyze the error due to Fourier truncation and conclude that, for data that are sufficiently regular, it suffices to solve a small number of two-dimensional problems.

Transformations of two-dimensional layered zinc phosphates to three-dimensional and one-dimensional structures

2002 ◽  
Vol 12 (4) ◽  
pp. 1044-1052 ◽  
Author(s):  
Amitava Choudhury ◽  
S. Neeraj ◽  
Srinivasan Natarajan ◽  
C. N. R. Rao
Keyword(s):  
Three Dimensional ◽  
Two Dimensional ◽  
One Dimensional ◽  
Zinc Phosphates

Mine Impact Burial Prediction From One to Three Dimensions

2008 ◽  
Vol 62 (1) ◽  
Author(s):  
Peter C. Chu
Keyword(s):  
Three Dimensional ◽  
Time History ◽  
Three Dimensions ◽  
Vertical Position ◽  
Burial Depth ◽  
Prediction Errors ◽  
Two Dimensional ◽  
Dimensional Model ◽  
One Dimensional ◽  
Dimensional Models

The Navy’s mine impact burial prediction model creates a time history of a cylindrical or a noncylindrical mine as it falls through air, water, and sediment. The output of the model is the predicted mine trajectory in air and water columns, burial depth/orientation in sediment, as well as height, area, and volume protruding. Model inputs consist of parameters of environment, mine characteristics, and initial release. This paper reviews near three decades’ effort on model development from one to three dimensions: (1) one-dimensional models predict the vertical position of the mine’s center of mass (COM) with the assumption of constant falling angle, (2) two-dimensional models predict the COM position in the (x,z) plane and the rotation around the y-axis, and (3) three-dimensional models predict the COM position in the (x,y,z) space and the rotation around the x-, y-, and z-axes. These models are verified using the data collected from mine impact burial experiments. The one-dimensional model only solves one momentum equation (in the z-direction). It cannot predict the mine trajectory and burial depth well. The two-dimensional model restricts the mine motion in the (x,z) plane (which requires motionless for the environmental fluids) and uses incorrect drag coefficients and inaccurate sediment dynamics. The prediction errors are large in the mine trajectory and burial depth prediction (six to ten times larger than the observed depth in sand bottom of the Monterey Bay). The three-dimensional model predicts the trajectory and burial depth relatively well for cylindrical, near-cylindrical mines, and operational mines such as Manta and Rockan mines.

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