Page 128 - Introduction to Computational Fluid Dynamics

P1: IWV
May 20, 2005
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5.1 INTRODUCTION
ρ and
were not functions of scalar s then the flow equations for = u 1 , u 2 12:28 107
will be independent of the scalar transport equations. This is the first reason for
distinguishing the flow-field equations from other scalar transport equations. To
appreciate the second reason, we first set out the equations governing the flow field
(the Navier–Stokes equations):
∂(ρ) 1 ∂ 1 ∂
+ {r ρ u f1 } + {r ρ u f2 } = 0, (5.3)
∂t r ∂x 1 r ∂x 2
∂(ρ u 1 ) 1 ∂ 1 ∂
+ {r ρ u f1 u 1 } + {r ρ u f2 u 1 }
∂t r ∂x 1 r ∂x 2

∂p 1 ∂ ∂u 1 1 ∂ ∂u 1
=− + r µ eff + r µ eff + S u1 , (5.4)
∂x 1 r ∂x 1 ∂x 1 r ∂x 2 ∂x 2
∂(ρ u 2 ) 1 ∂ 1 ∂
+ {r ρ u f1 u 2 } + {r ρ u f2 u 2 }
∂t r ∂x 1 r ∂x 2
∂p 1 ∂ ∂u 2 1 ∂ ∂u 2
=− + r µ eff + r µ eff + S u2 . (5.5)
∂x 2 r ∂x 1 ∂x 1 r ∂x 2 ∂x 2
A few comments having a bearing on the solution strategy are now in order.

1. In Equations 5.3–5.5, there are three unknowns (u 1 , u 2 , and p). Therefore, the
equation set is solvable.
2. In boundary layer flows, the pressure gradient is specified (external flows) or is
evaluated via the overall duct mass flow rate balance (internal flows). In elliptic
flows, however, ∂p/∂x 1 and ∂p/∂x 2 are not a priori known.
3. Thus, if we regard Equation 5.4 as the determinant of u 1 field and Equation 5.6
as the determinant of u 2 field, then the pressure field can be established only via
the mass conservation equation (5.3). The situation is somewhat similar to the
case of internal boundary layer flows but is not as straightforward.
4. The suffix f is attached to velocities satisfying the mass conservation equa-
tion. The velocity field without suffix f may or may not satisfy mass conserva-
tion directly although, in a continuum, it is expected that the u i and u fi fields
are identically overlapping and, therefore, the former must also satisfy mass
conservation.
5. The reader may find this distinction between the u i and u fi fields somewhat
unfamiliar. This is because most textbooks a priori assume a fluid continuum.
Numerical solutions are, however, developed in a discretised space and the
distinction mentioned here becomes relevant. This will become clear in a later
section.

These points reveal the fact that there is no explicit differential equation for
determination of the pressure field with p (or its variant) as the dependent variable.