J Rafael Pacheco Assistant Professor Departamento de Ingenieria Meca nica Instituto Tecnolo gico de Monterrey Monterrey NJ 64849 Me xico Mem ASME e mail rpacheco asu edu Arturo Pacheco Vega Visiting Assistant Professor Department of Aerospace and Mechanical Engineering University of Notre Dame Notre Dame IN 46556 Mem ASME Analysis of Thin Film Flows Using a Flux Vector Splitting1 We propose a flux vector splitting FVS for the solution of film flows radially spreading on a flat surface created by an impinging jet using the shallow water approximation The governing equations along with the boundary conditions are transformed from the physical to the computational domain and solved in a rectangular grid A first order upwind finite difference scheme is used at the point of the shock while a second order upwind differentiation is applied elsewhere Higher order spatial accuracy is achieved by introducing a MUSCL approach Three thin film flow problems 1 one dimensional dam break problem 2 radial flow without jump and 3 radial flow with jump are investigated with emphasis in the prediction of hydraulic jumps Results demonstrate that the method is useful and accurate in solving the shallow water equations for several flow conditions DOI 10 1115 1 1538626 Introduction A phenomenon of interest in free surface film flows under certain conditions is the formation of a sudden discontinuity in the depth of a flowing liquid i e a hydraulic jump as the flow in the radial direction makes a transition from supercritical to a subcritical regime Significant back flow may be present at the jump location with a corresponding loss of mechanical energy due to the abrupt change in depth In general the region of increased film thickness corresponds to a zone of substantially amplified turbulence From a thermal standpoint the location of the hydraulic jump is critical because of the degraded transport characteristics that exist there This region is unsteady and depends heavily on the drainage configuration downstream Thin film flows can be characterized as shallow water flows in which the vertical dimension is much smaller than any typical horizontal length scale Important applications of the shallowwater equations include the prediction of weather simulation of tidal flows storm surges river flows and dam break waves For engineering applications numerical solution of the governing equations is a useful tool for aiding in the prediction of local flow properties of the fluid From a numerical standpoint a major difficulty in analyzing the hydraulic jump stems from the discontinuity of the fluid depth across the jump which must be accurately captured by the numerical method Due to the similarity between the depth averaged equations and the two dimensional compressible Navier Stokes equations the methods developed to solve the compressible flow may also be used to study the shallow water equations and therefore to analyze hydraulic jumps in thin flows Several finite difference schemes have been developed for the simulation of gas dynamics equations and are available in the literature The main difference among them is the way they address the problem of shock waves formation which is of fundamental importance in the overall accuracy of the calculations These techniques are based either on central difference formulations 1 2 or on upwind difference discretizations Though the earlier schemes give fairly accurate results in the case of smooth and weak shocks the precision given by the latter methods even for problems containing strong shock waves is superior 3 Representatives of upwind schemes are those by Roe 4 and Yee and Harten 5 based on flux difference splitting FDS techniques and those by Steger and Warming 6 van Leer 7 and Walters et al 8 structured on the flux vector splitting FVS approach Since the shallow water equations have the property of being nonhomogeneous functions of the primitive variables the application of FVS schemes to the solution of these equations has been limited mainly due to the difficulty in finding the proper split of the flux vectors However a number of numerical studies for the shallow water equations using schemes based on the FDS method have been carried out Examples include the study of flow in a channel of infinite width 9 the simulation of open channel flows coupled with flux limiters 10 and the analysis of twodimensional free surface flow equations employing a high order Godunov type scheme based on monotone upstream centered scheme for conservation laws MUSCL variable extrapolation and slope limiters 11 among others In the current work we are interested in the analysis of freesurface thin film flows for which a hydraulic jump may exist using the shallow water equations as mathematical model To this end we use a finite difference scheme based on a new split of the flux vector to solve the governing equations A first order upwind differentiation is enforced at the location of the shock whereas a higher order upwind differentiation is employed elsewhere A straight forward formulation of the governing equations is presented followed by a discussion of the split of the flux vector that accounts for the inhomogeneity of the shallow water model The effects of the FVS on the accuracy and consistency of the spatial discretization are studied next Finally three thin film problems 1 one dimensional dam break problem 2 radial flow without jump and 3 radial flow with jump are investigated with emphasis in the capability of the technique to capture discontinuities in the flow field Mathematical Model Governing Equations We present the governing NavierStokes and continuity equations for incompressible fluid with constant properties 1 Address correspondence to ASU P O Box 6106 Tempe AZ 85287 6106 Contributed by the Fluids Engineering Division for publication in the JOURNAL OF FLUIDS ENGINEERING Manuscript received by the Fluids Engineering Division December 15 1999 revised manuscript received October 11 2002 Associate Editor Y Matsumoto Journal of Fluids Engineering u 0 1 u u u p 2 u gez t 2 where ez is the unit vector in the vertical direction u u v w Copyright 2003 by ASME MARCH 2003 Vol 125 365 represents the Cartesian velocity vector components p is the pressure is the kinematic viscosity is the density and g is the gravitational acceleration In the following the physical space will be denoted by x y z and the computational space by Assuming that vertical acceleration wind stresses and