The effect of a uniform through-surface flow on a cylinder and sphere

Abstract

The effect of a uniform through-surface flow (velocity U b ) on a rigid and stationary cylinder and sphere (radius a) fixed in a free stream (velocity U ∞ ) is analysed analytically and numerically. The flow is characterised by a dimensionless blow velocity Λ (= U b /U ∞ ) and Reynolds number Re (= 2aU ∞ /ν, where ν is the kinematic viscosity). High resolution numerical calculations are compared against theoretical predictions over the range −3 Λ 3 and Re = 1, 10, 100 for planar flow past a cylinder and axisymmetric flow past a sphere. For −Λ 1, the flow is viscously dominated in a thin boundary layer of thickness ν/|U b | adjacent to the rigid surface which develops in a time ν/U 2 b ; the surface vorticity scales as Re|Λ|U ∞ /a for a cylinder and sphere. A boundary layer analysis is developed to analyse the unsteady viscous forces. Numerical results show that the surface pressure and vorticity distribution within the boundary layer agrees with a steady state analysis. The flow downstream of the body is irrotational so the wake volume flux, Q w , is zero and the drag force is F D = −ρU ∞ Q b , where ρ is the density of the fluid and Q b is the normal flux through the body surface. The drag coefficient is therefore −2πΛ or −8Λ for a cylinder or sphere, respectively. A dissipation argument is applied to analyse the drag force; the rate of working of the drag force is balanced by viscous dissipation, flux of stagnation pressure and rate of work by viscous stresses due to sucking. At large Re|Λ|, the drag force is largely determined by viscous dissipation for a cylinder, with a weak contribution by the normal viscous stresses, while for a sphere, only 3/4 of the drag force is determined by viscous dissipation with the remaining 1/4 due to the flux of stagnation pressure through the sphere surface. When Λ 1, the boundary layer thickness initially grows linearly with time as vorticity is blown away from the rigid surface. The vorticity in the boundary layer is weakly dependent on viscous effects and scales as U ∞ /aΛ or U ∞ /aΛ 3/2 for a cylinder and sphere, respectively. For large blow velocity, the vorticity is swept into two well-separated shear layers and the maximum vorticity decreases due to diffusion. The drag force is related to the vorticity distribution on the body surface and an approximate expression can be derived by considering the first term of a Fourier expansion in the surface vorticity. It is found that the drag coefficient C D for a cylinder (corrected for flow boundedness) is weakly dependent on Λ while for a sphere, C D decreases with Λ.

The effect of inertia and vertical confinement on the flow past a circular cylinder in a Hele-Shaw configuration

“…The corresponding drag coefficient is defined as which can be split into the pressure and the viscous components, where is the unit vector in the streamwise direction (Klettner et al. 2016). The drag coefficient is plotted in figure 15( b ) with the numerical simulations compared with Lee & Fung's (1969) Stokes solution and also a direct calculation with (4.1) using Thompson's analysis.…”

“…where x is the unit vector in the streamwise direction (Klettner et al 2016). The drag coefficient is plotted in figure 15 Fung 's (1969) Stokes solution and also a direct calculation with (4.1) using Thompson's analysis.…”

A comprehensive mathematical model for nanofibre formation in centrifugal spinning methods

“…where n is the unit normal vector to the interface pointing outwards from the jet surface and H denotes the mean local curvature of the interface. Moreover, Σ = −pI + Π andΣ = −pI +Π are the stresses exerted by the liquid jet and air on the interface, respectively; the latter is responsible for the aerodynamic drag force (see Batchelor 1967;Klettner et al 2016). It is worth mentioning that the first and second terms on the left-hand side of (3.41) represent the forces per unit area exerted by air and the jet on the interface, respectively; the term on the right-hand side stands for the normal curvature force related to the local curvature of the air-jet interface.…”

Flow and passive transport in planar multipolar flows

“…Between 300 000 and 2 × 10 6 nodes were used in the computations. The system of equations (2.5 -2.7) was expressed in a finite element formulation and solved using an in-house code 'ACEsim' with the characteristic based split scheme (Nicolle & Eames 2011;Klettner et al 2016). The defining equations were solved in two steps: involving first the calculation of the velocity field until the flow field ran to steady-state.…”