  We present numerical analyses of 2-D electrohydrodynamic (EHD) flows of a
dielectric liquid between a wire electrode and two plate electrodes with a
Poiseuille flow, using direct numerical simulation and global stability
analysis. Both conduction and injection mechanisms for charge generation are
considered. In this work, we focused on the intensity of the cross-flow and
studied the EHD flows without a cross-flow, with a weak cross-flow and with a
strong cross-flow. (1)In the case without a cross-flow, we investigated its
nonlinear flow structures and linear dynamics. We found that the flow in the
conduction regime is steady, whereas the flow in the injection regime is
oscillatory, which can be explained by a global stability analysis. (2)The EHD
flow with a weak cross-flow is closely related to the flow phenomena in
electrostatic precipitator (ESP). Our analyses indicate that increasing the
cross-flow intensity or the electric Reynolds number leads to a less stable
flow. Based on these results, we infer that one should adopt a relatively low
voltage and weak cross-flow in the wire-plate EHD flow to avoid flow
instability, which may hold practical implications for ESP. (3)The case of
strong cross-flow is examined to study the EHD effect on the wake flow. By
comparing the conventional cylindrical wake with the EHD wake in linear and
nonlinear regimes, we found that the EHD effect brings forward the vortex
shedding in wake flows. Besides, the EHD effect reduces the drag coefficient
when the cross-flow is weak, but increases it when it is strong.
