Hivel2D users manual
Natural boundary conditions are applied to the sidewall boundaries through the weak
form. The side wall boundaries are "no-flux" boundaries; that is, there is no net flux of
mass or momentum through these boundaries. These boundary conditions are enforced
through the line integral in the weak form.
Petrov-Galerkin Test Function
The Petrov-Galerkin test function yi is defined (Berger 1993) as:
ϕ i = φi I + ϕi
(10)
φi
φi
ϕi
= β ∆x
B
A + ∆y
(11)
y
x
where β is a dissipation coefficient varying in value from 0 to 0.5, the ∆ terms are the
linear basis functions, and ∆ x and ∆ y are the grid intervals. A detailed explanation of
^
^
this test function, in particular A and B, is given in Berger (1993).
Shock Capturing
The coefficient β scales the dissipation needed for numerical stability. More
dissipation is needed in the vicinity of shocks such as hydraulic jumps than in smooth
regions of the flow field. Because a lower value of β ( β = β SM ) is more precise, a
large value of β ( β = βSH = 0.5 where β SM and βSH are the Petrov-Galerkin parameters
for smooth flow and for shocks, respectively) is applied only in regions in which it is
needed. HIVEL2D employs a mechanism that detects shocks and increases β automati-
cally. Therefore, βSH is implemented only when needed as determined by evaluation of
the element energy deviation. In a similar manner, the eddy viscosity coefficient C varies
from CSM to CSH, the effect being that eddy viscosity is increased only in areas of greatest
element energy deviation.
Temporal Derivatives
A finite difference expression is used for the temporal derivatives. The general
expression for the temporal derivative of a variable Qj is:
6
Chapter 2 HIVEL 2D overview