Numerical calculation of developing laminar flow in pipes of arbitrary curvature radius

Elliptic forms of the Navier‐Stokes equations in rigorously formulated toroidal coordinates are solved numerically for laminar, incompressible, isothermal flow in curved pipes of small curvature radius and 90° deflection angle. As a special case, the flow through curved annuli is also considered in which the presence of the inner curved wall causes the appearance of two (rather than one) pair of cross‐stream vortices. Recirculation in the main flow direction is observed at the outer curvature wall of a pipe with Rc/d = 1 and Re = 1800 over a deflection angle corresponding to 9° ≤ θ ≤ 21°. Flow reversal is attributed to the large and unfavorable longitudinal pressure gradient prevailing at the outer curvature wall over the first half of the pipe. Calculations show that the cross‐steam flow in a strongly curved pipe can be as large as 80% of the bulk average velocity at θ = 45° but is reduced in magnitude between θ = 45° and 90°. The numerical procedure used for the present calculations is a special case of a more general method which will solve flows in arbitrary curvilinear orthogonal coordinates provided the appropriate scale factors and extra source terms are included. A detailed derivation of the difference equations in curvilinear orthogonal coordinates is, therefore, presented here.