As sensors and flow control actuators become smaller, cheaper, and more pervasive, the use of feedback control to manipulate the details of fluid flows becomes increasingly attractive. One of the challenges is to develop mathematical models that describe the fluid physics relevant to the task at hand, while neglecting irrelevant details of the flow in order to remain computationally tractable. A number of techniques are presently used to develop such reduced-order models, such as proper orthogonal decomposition (POD), and approximate snapshot-based balanced truncation, also known as balanced POD. Each method has its strengths and weaknesses: for instance, POD models can behave unpredictably and perform poorly, but they can be computed directly from experimental data; approximate balanced truncation often produces vastly superior models to POD, but requires data from adjoint simulations, and thus cannot be applied to experimental data.In this paper, we show that using the Eigensystem Realization Algorithm (ERA) [15], one can theoretically obtain exactly the same reduced order models as by balanced POD. Moreover, the models can be obtained directly from experimental data, without the use of adjoint information. The algorithm can also substantially improve computational efficiency when forming reduced-order models from simulation data. If adjoint information is available, then balanced POD has some advantages over ERA: for instance, it produces modes that are useful for multiple purposes, and the method has been generalized to unstable systems. We also present a modified ERA procedure that produces modes without adjoint information, but for this procedure, the resulting models are not balanced, and do not perform as well in examples. We present a detailed comparison of the methods, and illustrate them on an example of the flow past an inclined flat plate at a low Reynolds number.

PurposeTo study the effects of velocity correction schemes for a temperature transforming model (TTM) for convection controlled solid‐liquid phase‐change problem.Design/methodology/approachThe effects of three different solid velocity correction schemes, the ramped switch‐off method (RSOM), the ramped source term method (RSTM) and the variable viscosity method (VVM), on a TTM for numerical simulation of convection controlled solid‐liquid phase‐change problems are investigated in this paper. The comparison is accomplished by analyzing numerical simulation and experimental results of a convection/diffusion phase‐change problem in a rectangular cavity. Model consistency of the discretized TTM is also examined in this paper. The simulation results using RSOM, RSTM and VVM in TTM are compared with experimental results.FindingsIn order to efficiently use the discretized TTM model and obtain convergent and reasonable results, a grid size must be chosen with a suitable time step (which should not be too small). Applications of RSOM and RSTM‐TTM yield identical results which are more accurate than VVM.Originality/valueThis paper provides generalized guidelines about the solid velocity correction scheme and criteria for selection of time step/grid size for the convection controlled phase change problem.

This paper addresses a feed-forward optimal control problem for one rigid body to approach to and align with another arbitrarily rotating rigid body, with an application to the satellite rendezvous problem. In particular, we focus on the satellite rendezvous strategy of finding an optimal trajectory, and the required thrust force profiles, which will guide the chasing spacecraft to approach the tumbling satellite such that the two vehicles will eventually have no relative rotation and thus a subsequent docking or capture operation can be safely performed with a normal docking or capture mechanism. Our approach is to model the system using rigid-body dynamics and apply Pontryagin's Maximum Principle for the optimal control. A planar problem is presented as a case study, in which together with the Maximum Principle, the Lie algebras associated with the system are used to examine the existence of singular extremals for the time-optimal control problem. Also, optimal trajectories and the corresponding set of control force/torque profiles are numerically generated for the time/fuel-consumption optimal control problem.

SUMMARYIn this paper we present a systematic and general method for developing variational integrators for LiePoisson Hamiltonian systems living in a finite-dimensional space g * , the dual of Lie algebra associated with a Lie group G. These integrators are essentially different discretized versions of the Lie-Poisson variational principle, or a modified Lie-Poisson variational principle proposed in this paper. We present three different integrators, including symplectic, variational Lie-Poisson integrators on G ×g * and on g×g * , as well as an integrator on g * that is symplectic under certain conditions on the Hamiltonian. Examples of applications include simulations of free rigid body rotation and the dynamics of N point vortices on a sphere. Simulation results verify that some of these variational Lie-Poisson integrators are good candidates for geometric simulation of those two Lie-Poisson Hamiltonian systems.

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