Abstract
Although one-dimensional analysis is a classical, accepted method used extensively in the literature and has been commonly used for the comparison with approximation methods for the solution of free surface flows, the requirement to specify the skin friction coefficient value has always been a constraint with which previous investigators have had trouble. The alternative method of solving the full Navier-Stokes equations requires substantial computer time because of the iteration necessary to resolve the location of the free surface boundary. A relatively simple parabolic numerical method for the thin-layer equations in two dimensions to solve the free surface flows without the need of assuming the skin friction coefficient and the need of iterating the free surface boundary is the purpose of this investigation. An order-of-magnitude analysis is used to reduce the elliptic governing Navier-Stokes equations to a parabolized set for high Reynolds numbers. The resulting equations are discretized and solved numerically for different sub and supercritical flows, various Pr, Fr, and Re numbers and for many different thermal boundary conditions using an implicit marching method employing the tridiagonal algorithm.
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