Investigation on large responses of the streamwise velocity with the effect of spanwise Lorentz force

Abstract

The flow of a weakly conductive fluid (i.e. seawater) can be controlled by Lorentz forces generated by the suitably chosen magnetic and electric fields, which has significant effects for applications in the drag reduction and oscillatory suppression. However, the control efficiency is very low due to the application of large amplitude for Lorentz force. Therefore, the large response, induced by a small Lorentz force, is the key to enhance the flow control efficiency. In this paper, the amplification mechanism of the velocity response with the effect of Lorentz force in weakly conductive fluids is investigated, where the Lorentz force is applied on the lower wall in the channel. The analytic solutions of the velocity responses in linear stage are derived with linear stability theory, when the amplitude of Lorentz force is far less than 1. From the discussions of the analytic solutions, the mechanism is revealed on large responses which is induced by the small Lorentz force in the flow field. The results show that the flow along the spanwise direction is induced by Lorentz force, which leads to the momentum exchange of fluids in the wall-normal direction. Therefore, the large responses of velocity are generated due to the high-speed fluids transferring to the wall. Moreover, the responses depend on the parameters of Lorentz force, i.e. wave number Kz , effective penetration Δ, including proportional to amplitude A and square of Reynolds number Re2. With the increase of Kz , the response decreases monotonously. However, with the increase of Δ, the response increases first (Δ<0.4), reaches the maximum at Δ=0.4, and then decreases (Δ>0.4). Finally, the maximum response is obtained and the corresponding amplification is 480, when the two parameters are optimized, i.e. Kz=2, Δ=0.4.