The theory of characteristics is used to determine the boundary
conditions at the inlet and exit of the computational domain. For
subsonic inlet flow four quantities are specified and one is
extrapolated from the interior of the computational domain. In
particular, the total pressure, v and w velocity components, and
the downstream running Riemann invariant, 
(or the total temperature 
), can be specified as a
function of the radius. The upstream running Riemann invariant, 
, is extrapolated from the interior of
the computational domain. For simulations containing inlet hot
streaks, the boundary conditions within the hot streak must be
modified. Within the hot streak the inlet flow variables used to
define the specified boundary conditions can be written as
where 
is the temperature within the hot streak and
is the temperature of the undisturbed inlet flow. The
static and total pressure within the hot streak are assumed to be equal
to that of the undisturbed inlet flow. In the current investigation,
the temperature profile within the hot streak is based on a
hyperbolic-tangent distribution with the center located at 40% of
span. This distribution is consistent with the experimental geometry
of Butler et al. (1989).
For subsonic outflow one flow quantity is specified and four are
extrapolated from the interior of the computational domain. The v
and w velocity components, entropy, and the downstream running
Riemann invariant are extrapolated from the interior of the
computational domain. The pressure ratio, 
, is
specified at mid-span of the computational exit and the pressure at
all other radial locations at the exit is obtained by integrating the
equation for radial equilibrium. Periodicity is enforced along the
outer boundaries of the H-grids in the circumferential ( 
)
direction.
For viscous simulations, no-slip boundary conditions are enforced along the surfaces of the stator and rotor airfoils. Absolute no-slip boundary conditions are enforced at the hub and tip end walls of the stator regions, along the surface of the vane, and along the outer casing (tip end wall) of the rotor blade. Relative no-slip boundary conditions are imposed at the hub and along the surface of the rotor blade. It is assumed that the normal derivative of the pressure is zero at solid wall surfaces. In addition, a specified temperature distribution is held constant in time along the solid surfaces.
The flow variables of Q at zonal boundaries are explicitly updated after each time step by interpolating values from the adjacent grid. The zonal boundary conditions are non-conservative, but for subsonic flow this should not affect the accuracy of the final flow solution.