Page 85 - Computational Fluid Dynamics for Engineers

70 2. Conservation Equations

2.7 Boundary Conditions

The nature of the equations and their domains of dependence and zones of
influence have implications for boundary conditions. Thus, for example, the
steady, two-dimensional incompressible form of the momentum equation for
laminar flow,
2
2
du du _ I dp (d u d u\ -
dx dy Q dx I dx 2 dy 2 J
is elliptic, and boundary conditions are required on all sides of the solution do-
main (see Section 4.5). The equation is always solved together with the equation
of continuity and the normal momentum equation so that u, v and p or their
gradients must be specified for a problem on its boundaries.
In the case of steady, two-dimensional boundary-layer flows, the equation
2
du du 1 dp d u
2
dx dy g dx dy
is parabolic so that, with the continuity equation and with known pressure,
u and v or their gradients are required on three sides of the solution domain.
The forms of these equations appropriate to turbulent flows are presumed to
have the same requirements. The boundary conditions vary considerably from
problem to problem and there are two main types.
In external flows, the shear layers flowing in the x-direction adjoin an effec-
tively "inviscid" freestream extending to y = oo. On the lower side there may
be either a solid surface, usually taken as y = 0 as in Fig. 2.5a, in which case the
viscous region is called a "wall shear layer," or there may be another inviscid