Page 191 - Basic Structured Grid Generation

7

Moving grids and time-dependent

co-ordinate systems

7.1 Time-dependent co-ordinate transformations

The numerical solution of time-dependent (otherwise known as transient, evolution,
or unsteady) transport equations may require time-dependent ‘moving’ grids in phys-
ical space. For example, transonic flow problems and similar applications involving
the propagation of shocks demand moving grids, with a continuing re-distribution of
grid-nodes, to capture a shock. Some problems will involve fixed boundaries in phys-
ical space, with internal grid-nodes moving in response to the flow developed. Other
problems involve moving boundaries, for example the internal combustion chamber.
In either case a time variable t enters the hosted equations and the grid generation
equations as an independent parameter. Any boundary-conforming grid at any time
i
t with curvilinear co-ordinates x ,i = 1, 2, 3 (in three dimensions), however, may
i
still be mapped to a fixed uniform rectangular grid in x -computational space. The
transformation involved will now be time-dependent and of the form
3
2
1
y i = y i (x ,x ,x ,t), i = 1, 2, 3, (7.1)
with inverse
i
i
x = x (y 1 ,y 2 ,y 3 ,t), i = 1, 2, 3, (7.2)
where y i are the cartesian co-ordinates with respect to some rectangular cartesian
reference system of a point in physical space which at time t has curvilinear co-
i
ordinates x . We assume that the Jacobian
j
J = det(∂y i /∂x )

of the transformation remains non-zero at all times.
An arbitrary function f(r,t) of space (the physical domain) and time has differential
(small increment to first order)

∂f ∂f
df = dy i + dt,
∂y i ∂t
t y i