On the determination of solenoidal or compressible velocity fields from measurements of passive or reactive scalars

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Abstract
Several techniques have been proposed for determining two‐ or three‐dimensional velocity fields from measurements of one passive scalar. It is shown that measurements of one scalar and knowledge of the equation governing its transport determine a velocity field, only up to an additive vector field locally perpendicular to the gradient of the scalar field but otherwise arbitrary. Three previously proposed procedures for selecting a unique velocity field from among the uncountable infinity consistent with the scalar transport data and equation are then discussed, and it is shown that a recent ‘‘iterative inversion’’ procedure for ‘‘solution’’ of a singular linear equation system (obtained using only measurements of one scalar and the equation governing its transport) cannot converge as claimed. A method for determining the correct n‐dimensional (n=2 or 3) divergence‐free velocity field from measurements of n−1 passive or reactive scalars is then developed. Finally, it is shown how the velocity field in an n‐dimensional compressible flow can be determined from measurements of density and n−1 passive or reactive scalars.