Abstract
A two-dimensional pulsating flow of a viscous fluid in a plane channel whose wall has rectangular microcavities partially or completely filled with a compressible gas is investigated. This problem formulation can clarify the friction reduction mechanism in a laminar sublayer of a turbulent viscous boundary layer flow over a textured stripped superhydrophobic surface containing periodically arranged rectangular micro-cavities filled with gas. It is assumed that the dimensions of the cavities are much smaller than the channel thickness. On the macroscale, the problem of one-dimensional unsteady viscous flow in a plane channel with no-slip conditions on the walls and a harmonic variation of the pressure difference is solved. The solution obtained in this way is used for formulating non-stationary in time and periodic in space boundary conditions for the flow on the scale of a chosen cavity (microscale), with the instantaneous volume of the gas bubble in the cavity depending on the instantaneous pressure over the cavity. The flow on the microscale near a cavity with a gas bubble occurs in the Stokes regime. The numerical solution is obtained using an original version of the boundary element method. A parametric numerical study of the flow field in a pulsating shear flow over a cavity with a compressible gas bubble is performed. The averaged parameters characterizing the effective ‘velocity slip’ of viscous fluid and the friction reduction in a pulsating flow over a stripped superhydrophobic surface are calculated.
Highlights
As the external flow over a cavity with a gas bubble we will consider a pulsating flow of a viscous fluid in a plane channel with width H* in Cartesian coordinates x*, y*, where the x*-coordinate is directed along the lower channel wall
Θ is the angle between the horizontal and the bubble surface. These streamline patterns show the main specific features of the flow restructuring which may occur during one fluctuation period of the bubble surface, such as: the formation of one large eddy occupying the entire space in the cavity over the bubble; the formation of local eddies near the edge points of the meniscus; and a shear flow of the fluid along the phase interface when the cavity is completely occupied by the gas bubble
On the basis of parametric calculations of a pulsating viscous-fluid flow over a two-dimensional microcavity with a gas bubble, the estimates of variation of the average parameters of the f low over the cavity are obtained for different modes of imposed pressure and velocity fluctuations, and different initial positions of the bubble surface in the cavity
Summary
As the external flow over a cavity with a gas bubble we will consider a pulsating flow of a viscous fluid in a plane channel with width H* in Cartesian coordinates x*, y*, where the x*-coordinate is directed along the lower channel wall. From the linearity of Eq (1.2) it follows that u1(t, y) can be sought as the real part of the solution of an equation for a complex-valued function V(t, Y) This equation with the corresponding boundary conditions read:. The obtained expressions for the instantaneous pressure in the f luid p(t, x) and the velocity profile u(t, y) = u0(y) + u1(t, y) will be used below in specifying the boundary conditions for the flow over a small chosen rectangular cavity located on the lower channel wall, with a compressible gas bubble inside. According to the solution of the Navier-Stokes equations, simultaneous fluctuations of the pressure and the longitudinal velocity near the wall may result in both the co-directed and counter-directed actions on the average friction in the flow over a microcavity with a gas bubble
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