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.

Issue Section:
Technical Papers
Keywords:
film flow, computational fluid dynamics, finite difference methods, boundary-value problems, water waves
Topics:
Boundary-value problems, Dams, Flow (Dynamics), Radial flow, Shallow water equations, Thin films, Water, Hydraulic jump, Eigenvalues, Shock (Mechanics), Film flow, Jacobian matrices
1.
MacCormack, R. W., 1969, “The Effect of Viscosity in Hypervelocity Impact Craterings,” AIAA Paper No. 69-354.
2.
Beam
,
R. M.
, and
Warming
,
R. F.
,
1976
, “
An Implicit Finite-Difference Algorithm for Hyperbolic Systems in Conservation Law Form
,”
Comput. Phys.
,
22
, pp.
87
–
110
.
3.
Yee, H. C., 1989, “A Class of High-Resolution Explicit and Implicit Shock-Capturing Methods,” technical report, von Karman Institute for Fluid Dynamics,
4.
Roe
,
P. L.
,
1981
, “
Approximate Riemann Solvers, Parameter Vectors, and Difference Schemes
,”
Comput. Phys.
,
43
, pp.
357
–
372
.
5.
Yee
,
H. C.
, and
Harten
,
A.
,
1987
, “
Implicit TVD Schemes for Hyperbolic Conservation-Laws in Curvilinear Coordinates
,”
AIAA J.
,
25
, pp.
266
–
274
.
6.
Steger, J. L., and Warming, R. F., 1979, “Flux Vector Splitting of the Inviscid Gasdynamic Equation With Application to Finite Difference Methods,” technical report, NASA TM-78605.
7.
Van Leer, B., 1982, “Flux-Vector Splitting for the Euler Equations,” Krause E., editor, Lecture, Notes in Physics, E. Krause, ed., Springer, Berlin, 170, pp. 507–512.
8.
Walters, R. W., Thomas, J. L., and Van Leer, B., 1986, “An Implicit Flux-Split Algorithm for the Compressible Euler and Navier-Stokes Equations,” Lecture Notes in Physics, F. G. Zhuang and Y. L. Zhu, eds., Springer, Berlin, 264, pp. 628–635.
9.
Glaister
,
P.
,
1988
, “
Approximate Riemann Solutions of the Shallow Water Equations
,”
J. Hydraulic Res.
,
26
, pp.
293
–
306
.
10.
Glaister
,
P.
,
1993
, “
Flux Difference Splitting for Open-Channel Flows
,”
Int. J. Numer. Methods Fluids
,
16
, pp.
629
–
654
.
11.
Alcrudo
,
F.
, and
Garcia-Navarro
,
P.
,
1993
, “
A High-Resolution Godunov-Type Scheme in Finite Volumes for the 2D Shallow-Water Equations
,”
Int. J. Numer. Methods Fluids
,
16
, pp.
489
–
505
.
12.
Chapman
,
R. S.
, and
Kuo
,
C. Y.
,
1985
, “
Application of the Two-Equation κ-ε Turbulence Model to a Two-Dimensional, Steady, Free Surface Flow Problem With Separation
,”
Int. J. Numer. Methods Fluids
,
5
, pp.
257
–
268
.
13.
Vreugdenhill, C. B., 1993, Numerical Methods for Shallow-Water Flows, Kluwer, Dordrecht, Netherlands.
14.
Cunge, J., 1975, “Rapidly Varying Flow in Power and Pumping Canals,” Unsteady flow in open channels, K. Mahmood and V. Yevjevich, eds., Water Resources Publications, Littleton, CO, pp. 539–586.
15.
Thomas, J. L., and Walters, R. W., 1985, “Upwind Relaxation Algorithms for the Navier-Stokes Equations,” AIAA Paper No. 85–1501.
16.
Hirsch, C., 1990, Numerical Computation of Internal and External Flows, John Wiley and Sons, New York.
17.
Stoker, J. J., 1957, Water Waves, Wiley Interscience, New York,
18.
Sadler, C. D., and Higgins, M. S., 1963, “Radial Free Surfaces Flows,” master’s thesis, Massachussetts Institute of Technology, Cambridge, MA.
19.
Younus
,
M.
, and
Chaudhry
,
M. H.
,
1994
, “
A Depth-Averaged κ-ε Turbulence Model for the Computation of Free Surface Flow
,”
J. Hydraul Res.
,
32
(
3
), pp.
415
–
444
.
20.
Hunt, J. C. R., Pacheco, J. R., Mahalov, A., and Fernando, H. J. S., 2003, “Effects of Rotation and Sloping Terrain on Fronts of Density Currents,” J. Fluid Mech., (submitted).
21.
Ahmad, D., 1967, “Circular Hydraulic Jump,” master’s thesis, Colorado State University, Fort Collings, CO.
Copyright © 2003
 by ASME
You do not currently have access to this content.