Visualizing the geometry of state space in plane Couette flow

Motivated by recent experimental and numerical studies of coherent structures in wall-bounded shear flows, we initiate a systematic exploration of the hierarchy of unstable invariant solutions of the Navier-Stokes equations. We construct a dynamical, 10^5-dimensional state-space representation of plane Couette flow at Re = 400 in a small, periodic cell and offer a new method of visualizing invariant manifolds embedded in such high dimensions. We compute a new equilibrium solution of plane Couette flow and the leading eigenvalues and eigenfunctions of known equilibria at this Reynolds number and cell size. What emerges from global continuations of their unstable manifolds is a surprisingly elegant dynamical-systems visualization of moderate-Reynolds turbulence. The invariant manifolds tessellate the region of state space explored by transiently turbulent dynamics with a rigid web of continuous and discrete symmetry-induced heteroclinic connections.Comment: 32 pages, 13 figures submitted to Journal of Fluid Mechanic

Time-Dependent 2-D Vector Field Topology: An Approach Inspired by Lagrangian Coherent Structures

This paper presents an approach to a time-dependent variant of the concept of vector field topology for 2-D vector fields. Vector field topology is defined for steady vector fields and aims at discriminating the domain of a vector field into regions of qualitatively different behaviour. The presented approach represents a generalization for saddle-type critical points and their separatrices to unsteady vector fields based on generalized streak lines, with the classical vector field topology as its special case for steady vector fields. The concept is closely related to that of Lagrangian coherent structures obtained as ridges in the finite-time Lyapunov exponent field. The proposed approach is evaluated on both 2-D time-dependent synthetic and vector fields from computational fluid dynamics

Cluster-based feedback control of turbulent post-stall separated flows

We propose a novel model-free self-learning cluster-based control strategy for general nonlinear feedback flow control technique, benchmarked for high-fidelity simulations of post-stall separated flows over an airfoil. The present approach partitions the flow trajectories (force measurements) into clusters, which correspond to characteristic coarse-grained phases in a low-dimensional feature space. A feedback control law is then sought for each cluster state through iterative evaluation and downhill simplex search to minimize power consumption in flight. Unsupervised clustering of the flow trajectories for in-situ learning and optimization of coarse-grained control laws are implemented in an automated manner as key enablers. Re-routing the flow trajectories, the optimized control laws shift the cluster populations to the aerodynamically favorable states. Utilizing limited number of sensor measurements for both clustering and optimization, these feedback laws were determined in only $O(10)$ iterations. The objective of the present work is not necessarily to suppress flow separation but to minimize the desired cost function to achieve enhanced aerodynamic performance. The present control approach is applied to the control of two and three-dimensional separated flows over a NACA 0012 airfoil with large-eddy simulations at an angle of attack of $9^\circ$, Reynolds number $Re = 23,000$ and free-stream Mach number $M_\infty = 0.3$. The optimized control laws effectively minimize the flight power consumption enabling the flows to reach a low-drag state. The present work aims to address the challenges associated with adaptive feedback control design for turbulent separated flows at moderate Reynolds number.Comment: 32 pages, 18 figure

Geometry of the ergodic quotient reveals coherent structures in flows

Dynamical systems that exhibit diverse behaviors can rarely be completely understood using a single approach. However, by identifying coherent structures in their state spaces, i.e., regions of uniform and simpler behavior, we could hope to study each of the structures separately and then form the understanding of the system as a whole. The method we present in this paper uses trajectory averages of scalar functions on the state space to: (a) identify invariant sets in the state space, (b) form coherent structures by aggregating invariant sets that are similar across multiple spatial scales. First, we construct the ergodic quotient, the object obtained by mapping trajectories to the space of trajectory averages of a function basis on the state space. Second, we endow the ergodic quotient with a metric structure that successfully captures how similar the invariant sets are in the state space. Finally, we parametrize the ergodic quotient using intrinsic diffusion modes on it. By segmenting the ergodic quotient based on the diffusion modes, we extract coherent features in the state space of the dynamical system. The algorithm is validated by analyzing the Arnold-Beltrami-Childress flow, which was the test-bed for alternative approaches: the Ulam's approximation of the transfer operator and the computation of Lagrangian Coherent Structures. Furthermore, we explain how the method extends the Poincar\'e map analysis for periodic flows. As a demonstration, we apply the method to a periodically-driven three-dimensional Hill's vortex flow, discovering unknown coherent structures in its state space. In the end, we discuss differences between the ergodic quotient and alternatives, propose a generalization to analysis of (quasi-)periodic structures, and lay out future research directions.Comment: Submitted to Elsevier Physica D: Nonlinear Phenomen

Mixing in convective thermal fluxes in unsteady nonhomogeneous flows generating complex three dimensional vorticity patterns

Diffusion and scaling of the velocity and vorticity in a thermoelectric driven heating and cooling experimental device is presented in order to map the different patterns and transitions between two and three dimensional convection in an enclosure with complex driven flows. The size of the water tank is of 0.2 x 0.2 x 0.1 m and the heat sources or sinks can be regulated both in power and sign [1-3]. The thermal convective driven flows are generated by means of Peltier effects in 4 wall extended positions of 0.05 x 0.05 cm each. The parameter range of convective cell array varies strongly with the Topology of the boundary conditions. Side heat and momentum fluxes are a function of Rayleigh, Peclet and Nusselt numbers, [4-6] Visualizations are performed by PIV, Particle tracking and shadowgraph. The structure of the flow is shown by setting up a convective flow generated by buoyant heat fluxes. The experiments described here investigate high Prandtl number mixing using brine and fresh water in order to form a density interface and low Prandtl number mixing with temperature gradients. The evolution of the mixing fronts are compared and the topological characteristics of the merging of the convective structures are examined for different configurations. Based on two dimensional Vorticity spectral analysis, new techniques can be very useful to determine the evolution of scales considering the multi-fractal structure of the convective flows.Peer ReviewedPostprint (published version