We consider the evolution of finite uniform-vorticity regions in an unbounded in viscid fluid. We perform a perturbation analysis based on the assumption that the regions are remote from each other and nearly circular. Thereby we obtain a self-consistent infinite system of ordinary differential equations governing the physical-space moments of the individual regions. Truncation yields an Nth-order moment model. Special attention is given to the second-order model where each region is assumed elliptical. The equations of motion conserve local area, global centroid, total angular impulse and global excess energy, and the system can be written in canonical Hamiltonian form. Computational comparison with solutions to the contour-dynamical representation of the Euler equations shows that the model is useful and accurate. Because of the internal degrees of freedom, namely aspect ratio and orientation, two like-signed vorticity regions collapse if they are near each other. Although the model becomes invalid during a collapse, we find a striking similarity with the merger process of the Euler equations.

In this paper we develop the Optimal Control Approach to the rotary control of the cylinder wake. We minimize the functional which represents the sum of the work needed to resist the drag force and the work needed to control the flow, where the rotation rate φ̇(t) is the control variable. Sensitivity of the functional to control is determined using the adjoint equations. We solve them in the “vorticity” form, which is a novel approach and leads to computational advantages. Simulations performed at Re=75 and Re=150 reveal systematic decrease of the total power and drag achieved using a very small amount of control effort. We investigate the effect of the optimization horizon on the performance of the algorithm and the correlation of the optimal controls with the changes of the flow pattern. The algorithm was also applied to the control of the subcritical flow at Re=40, however, no drag reduction was achieved in this case. Based on this, limits of the performance of the algorithm are discussed.

A new self-consistent model of the incompressible Euler equations in two dimensions is presented. The vorticity is assumed to be distributed in well separated disjoint piecewiseconstant elliptical finite-area vortex regions (FAVORs) D k with area A k , The evolution equations for four variables that describe each FAVOR are derived by truncating a physicalspace moment description by omitting terms O ((A k /R ka ) 2 ). (R ka is the inter-FAVOR centroid distance.) The model is validated by comparing steady-state configurations and dynamical evolutions with contour dynamical results.PACS numbers: 47.10. + g Our new self-consistent low-order model of the two-dimensional (2D) incompressible Euler equations (EE) is a decisive improvement over the well-known singular (point) or invariant-core vortex models. 1 The model, derived by perturbation methods, is validated by comparing solutions with corresponding contour dynamical (CD) results. 2 We show that in first order, the elliptical shape is physically reasonable and mathematically cogent for the well-separated, disjoint finite-area vortex regions (FAVORs) which create the flow. The elliptical shape is used to approximate the vortical structures observed experimentally in a strain field 3 or after a "merger" or "pairing" event. 4,5 The mathematical justification is given below. Thus, each elliptical FAVOR is described by four dynamical variables: the centroid, 3c k = (x k ,y k )\ the aspect ratio, \ k \ and the orientation of the major axis, k .The \ k and k introduce sufficient "internal" degrees of freedom per FAVOR so that a pair of isolated regions can attract and "collapse" (i.e., centroid coalescence in a finite time)--a process analogous initially to merger or pairing. Note that when a collapse event is in progress, the low-order model begins to breakdown for it violates the assumption of the perturbation expansion. However, critical collapse distances are close to the corresponding critical merger distances for the full EE. 2 We derive the model from the EE in vorticity/stream-function form,where the velocity field u = (u, v) = (\ff y , -i// x ). We consider the flow created by constant vorticity (o k in Ndisjoint domains, D k (t), with constant area A k , where the diameter of each (ocA k^2 ) is assumed to be much less than the smallest inter-FAVOR distance, R ka , as shown in Fig.