Abstract:
The flow of the weak electrolyte solution can be controlled by electromagnetic forces generated by the suitably chosen magnetic and electric fields, which has significant effects for applications in the drag reduction, lift amplification, and oscillatory suppression. However, the control efficiency is very low due to the application of a large amplitude for the electromagnetic force. Therefore, the large response, induced by a small electromagnetic force, is the key to enhance the flow control efficiency. In this paper, based on the laminar flow of a weakly conductive fluid in a channel, the flow responses are induced by the electromagnetic force, which is applied on the lower wall of the channel and is cosine distribution along the spanwise direction. The analytic solutions of the velocity responses in linear stage are deduced with linear stability theory, and the numerical solutions of these responses in nonlinear stage are calculated with direct numerical simulation (DNS). For the periodic characters of the channel flow in the streamwise and spanwise directions, the dealiased Fourier method is used in these two directions, while the Chebyshev-tau method is used in the wall-normal direction. Moreover, the usual no-slip and no-penetration conditions are used on the walls. The time advancement is performed with third-order accuracy by using a semi-implicit back-differentiation formula method. To eliminate residual divergence, the pressure term and the linear term are solved with a Chebyshev-tau influence matrix method. For the non-linear term, a spectral truncation method is used to remove aliasing errors. Combing the analytic and numerical solutions, the amplification mechanisms are revealed, and the influences of the parameters of electromagnetic forces and flow field are discussed. The results show that the amplification of velocity response along the streamwise direction is proportional to
Re2 for small amplitude, i.e., the responses in the linear stage. Moreover, the amplification of response increases firstly and then decreases with the increase of the effective penetration
Δ, while the amplification monotonically decreases with the increase of wave number along the spanwise direction. However, with the increase of amplitude, the response amplification approaches the nonlinear stage. The amplification monotonically decreases, while the amplitude value of velocity response increases firstly and then decreases. Moreover, the amplitude value reaches the maximum which is more than 0.2, and the corresponding amplification is the order of 10
2. Therefore, the flow induced by the electromagnetic force is amplified with the effect of flow field, which is the key to enhance the efficiency of flow control.