Theory of viscous flow in curved shallow channels

SUMMARY

The axisymmetrical, viscous flow in curved channels is considered in the case where the hydraulic radius of the cross-section is small with respect to the average of curvature of the bend.

F t Ananyan1

s theory on this subject is reconsidered, using a regular per-turbat method. The results are applied to a specific channel with a shallow rectangular cross-section. This yields solutions for the tangential velocity-component, the secondary circulation and the influence of the latter on the former.

Second the case of a shallow, rectangular cross-section is treated, using the method of matched asymptotic expansions .. This yields solutions of the velocity-components three separa~e regions of the cross-section (viz. near each side-wall and near the central axis of the cross-section), which turn out to be sums

(constant, depending on )

~the

and~

(function of the

x

~ized

(co-ordinates

normal- )

vertical~

)

Within the limitations of the theory, the functions, which have been tabulated here, are valid for all channel~ with a shallow, rectangular cross-section. Once these functions being known, it is possible to determine the velocity-components without solving the differential equations.

Finally, in order to find continuous velocity-profiles, the solutions in the three separate regions are composed to solutions valid in the entire cross-section. These composite solutions turn out to agree well with the results of Ananyan's theory.