Mean potential flows through groups of fixed bodies

4 Mean potential flows through groups of fixed bodies 

Far upstream of the bodies, the flow is a uniform stream of speed U. Due to the linearity of Laplace s equation, the flow generated by N bodies, labelled as 1,. .., k,. .., N, of volume Vsk, can be decomposed as the sum of the contributions from a uniform flow and from the individual bodies, 

The interaction between each body is implicitly included through the added-mass coefficient (or dipole strength) which increases with a - this increase was explicitly calculated for groups of cylinders by Dalton Helfinstine [140], In an unbounded flow, the outside flow uf. is characterised by a total dipole strength which is equal to the sum of the individual contributions from all the bodies, so that for x x,  

The Eulerian mean velocity is computed by averaging over the whole fluid region between the bodies, as in (7.31). It can be expressed, using Gauss s Theorem, as 

To illustrate the sensitivity of the mean flow within a groups of bodies and their impact on the ambient external flow, we consider the influence of the shape of bodies on the mean flow estimates. For simplicity, in the ensuing calculations, the bodies are assumed to be identical (with volume Vb and added-mass coefficient Cm) and the volume fraction, a, is small. 

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