On the numerical solution of film and jet flow problems
Abstract
The analysis of many fluid flows of practical interest such as
liquid films and jets are complicated by the presence of free surface
boundaries.
Exact solutions are scare
and increasingly, numerical methods are being applied to predict local flow properties
and hydrodynamic structure.
Like the unknown pressure and velocities, the shape and the position of the boundary must
be determined as part of the solution. This information is needed for the design
of cooling schemes in high-temperature applications, to optimize
heat treatment of metals, and to improve material production processes.
The objective of this research is to develop an accurate and
efficient numerical method that can be applied to the simulation of
free surface flow problems, i.e. horizontal jets, impinging jets and films
with and without rotation.
The governing equations are written in terms of primitive variables
and solved by the non-staggered grid fractional step method when
hydraulic jumps are absent in the flow.
The physical
domain is transformed to a rectangle for two-dimensional problems or
a parallelepiped for three-dimensional problems by means of a numerical
mapping technique. The pressure Poisson equation is formulated in
the same manner as on a staggered grid and solved with a SOR method.
The location of the phase boundary is accomplished by applying
both, the normal-stress condition or the kinematic boundary condition
depending on the physical force that regulates the behavior of the
flow.
The method was applied to solve plane Newtonian jet flows. The
numerical predictions are in good agreement with the results based
on the finite and spectral element methods as well as the
finite difference streamfunction vorticity formulation.
The boundary conditions at the free surface are more accurately
satisfied when compared with available data.
In the presence of hydraulic jumps, the problem is modeled using
the shallow-water approximation and the governing equations are
solved using shock capturing schemes.
The governing equations were discretized using both the
lambda and flux vector splitting methods. The finite difference
technique incorporates a numerical mapping so that the flow
regime is transformed to a regular domain for numerical integration.
These methods were applied for the simulation of a thin film flowing
radially outward on a stationary disk. In this formulation, a first-order
forward difference approximation for the time derivative was used.
The results showed the location of the hydraulic jump could be
predicted.
Among the advantages of the non-staggered grid fractional step
method are: the accuracy is second order in
space and time, it can handle three-dimensional problems in complex
geometries such as flows that turn 90 degrees, it is possible to
perform large eddy simulations and to implement turbulent models, and
because of local orthogonality at the surface, melting problems can
be studied with this method. The flux vector splitting technique
can be used to analyze thin films with singularities present in the
flow field.
