Numerical experiment of measurement-integrated simulation to reproduce turbulent flows with feedback loop to dynamically compensate the solution using real flow information

Abstract

Reproduction of the exact structure of real turbulent flows is crucial in many applications. Four Dimensional variation (4D-VAR) is widely used in numerical weather forecasting, but it requires huge computational power to repeatedly solve flow dynamics and its adjoint, and, therefore, is not suitable to apply to problems of real-time flow reproduction such as feedback flow control. Kalman filter and observer, in which numerical solution converges to the real state asymptotically by means of the feedback signal proportional to the difference between the calculated state and the real state, requiring much less computational load than the variational method, are potential candidates to solve the problem. By comparing Kalman filter and observer, the latter has simpler structure retaining essential part of the state estimation. This study deals with a special type of observer, or measurement-integrated simulation (MI simulation), in which a SIMPLER-based flow solver is used as the mathematical model of the system in place of approximate small dimensional linear differential equations usually used in observers. Reproduction of the exact structure of a turbulent flow was investigated by a MI simulation. A numerical experiment was performed for a fully developed turbulent flow in a pipe with a square cross section. The MI simulation was performed with the feedback from the standard solution in the flow domain for the cases using: (1) all velocity components at all grid points, (2) partial velocity components at all grid points, or (3) all velocity components at partial grid points. Convergence of the MI simulation to the standard solution was investigated using the steady error norm for the convergent state and the time constant for the transient state. The result of the MI simulation using all the velocity information exponentially converges to the standard solution with a steady state error reduced from that of the ordinary simulation in a range of the feedback gain. Decreasing the feedback gain reduces the effect of feedback, and a feedback gain which is too large destabilizes the closed loop system, resulting in large error. The time constant decreases almost inversely proportional to the feedback gain as long as the feedback system is stable. For the MI simulation with the feedback using limited information, feedback using two velocity components by omitting one transverse velocity component showed a good result, although the other results were not satisfactory. For the MI simulation with the feedback using limited grid points, the result of the MI simulation applying the feedback at the grid points on every 20th plane in the x 1 direction was almost the same as that using all grid points at some feedback gain, while the result with the feedback on the planes skipped in the x 2 direction requires 10 times more planes to achieve the same reduction rate.