Some relations between drag and flow pattern of viscous flow past a sphere and a cylinder at low and intermediate Reynolds numbers

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Abstract
The results of a numerical evaluation of the Navier-Stokes equations of motion for the case of a viscous fluid streaming past a sphere are presented in terms of the length of the standing eddy behind the sphere and in terms of the angle of flow separation at the sphere. Emphasis was placed on calculating these quantities at Reynolds numbers between 20 and 40 where no reliable theoretical or experimental values are available. In support of these calculations, it is shown that the values for the drag on a sphere previously calculated by us from the Navier-Stokes equations of motion by the same numerical technique as that used for calculating the eddy length and angle of flow separation agree well with our recent, extensive drag measurements for a wide Reynolds number interval. Our results are used to make a comparison between drag and flow field as predicted by analytical solutions and numerical solutions to the Navier-Stokes equations of motion. Some limitations of the analytical solutions to predict correct values for the drag, and to describe the correct nature of the flow field, are pointed out. It is shown further that a plot of [(D/Ds) – 1] versus log NRe, where D is the actual drag on a sphere, Ds is the Stokes drag, and NRe is the Reynolds number, reveals that the variation of the drag on a sphere with Reynolds number follows well defined régimes, which correlate well with the régimes of the flow field around a sphere. A similar relationship between ‘drag-régime’ and flow field pattern is discussed for the case of viscous flow past a cylinder.

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