Speaker: Herman Haustein, Ph.D. School of Mechanical Engineering Tel Aviv University, Israel

A NOVEL SOLUTION TO THE AGE OLD PROBLEM OF FLOW OVER A SPHERE

February 16, 2018 | 12:00 PM | DeWalt Seminar Room, 2164 Martin Hall

Abstract:

The flow over a sphere has been a challenging problem ever since d’Alembert’s paradox (1752). Despite incentives to resolve this fundamental case with relevance to so many fields (particle flows, ballistics, etc.), this intensely curved bluff body with its three-dimensional flow, is mathematically insurmountable. While some existing approximate methods contend with the boundary layer (BL) up to separation (Blasius series, Karman-Pohlhausen), they remain unresolved without accounting for the free-stream alteration by the finite BL, flow separation and the resulting wake. Instead, individual aspects of the flow have been empirically correlated: the separation angle, the drag coefficient and wake characteristics (Clift, 1978). Here a new approach is taken to obtain the free-stream velocity distribution as dependent on Re, addressing primarily the modifications to the potential flow solution.

For this mathematically convenient inviscid solution to represent realistic viscous conditions a novel concept is first introduced, whereby the boundary layer’s displacement effect is emulated by a moving boundary. Examining the inviscid flow around a steadily expanding sphere in transformed (stretching) coordinates generates a constant Rankine-body, resembling a sphere with a wake. Considering the viscous flow separation angle, a translational function is found relating the inviscid expansion rate to Re. Thereby, the closed-form separation angle model, derived by a similarity problem, predicts slightly better than the best existing (two-part) empirical correlation. The Rankine-body is seen to represent the normalized displacement thickness by comparison to simulations over Re=60-1500, with its predicted separation value comparing well to the other methods mentioned.

Starting with an asymptotic BL growth description (Re→¥, Blasius series), the minimal modification of the free stream is obtained for low Re. While at high Re the BL is negligible and the primary contribution is from free stream deflection by flow separation. Superimposing these effects results in a closed-form free-stream prediction dependent only on Re, up to the point of separation. Together with already predicted BL parameters and using approximate methods mentioned, leads to an overall BL thickness prediction. Furthermore, this gives the dimensional velocity distribution and wall-shear within the laminar BL up to separation. Using Euler’s equation to convert the free-stream to pressure and incorporating its decay to the ambient in the wake, gives the pressure distribution over the entire sphere, to good agreement with present simulations and previous numerical and experimental data (150<Re<160,000). Finally, the pressure and shear are integrated over the sphere to give the total drag, shown to reproduce the well-known experimental C_{d} curve, including the minimum around Re=5,000.