Computational Fluid Structure Interaction - Computational Hemodynamics: Theory, Modelling and Applications - page 123

Fig. 5.28 Influencing regions
from the flow field on nodal
points in the discretised
domain

elliptic and the value of ф at the E node is influenced more by the upstream node.
When Pe →∞ the flow is in pure convection, and the influence contour is stretched
where the value of ϕ at E is affected only by upstream conditions.

5.4.3 One-Dimensional Steady State Convection-Diffusion

in Finite Volume

The convection terms move a scalar variable, ϕ in the flow direction by the velocity
component while diffusion distributes the variable in all directions by its diffusion
coefficient, Γ. If we ignore source terms and focus on steady convection and diffu-
sion of the variable ϕ in one dimension, its equation becomes

(5.48)

du () φ φ

=Γ

d d

dx dx dx

This convection diffusion equation, is representative of the momentum equation if
ϕ = u and Γ= . The 1D mesh for nodal point P , and its neighbouring nodes is given

in Fig. 5.29 .
Integration of Eq. (5.48) over the control volume gives

ν

d φ d φ

φ () ()

e

u φ − =Γ −Γ u

w e w

dx dx

e w

convective flux

diffusive flux

where the two diffusive fluxes are approximated with a linear interpolation as

d φ φφ − d φ φφ −

EP PW

Γ =Γ = D ( φφ − ); Γ =Γ = D ( φφ − )

e e eE P w w wP W

dx ∆ x dx ∆ x

PE WP

e w

where D =Γ ∆ / x and D x

wwWP

=Γ ∆ /

e e PE