Computational Fluid Structure Interaction - Computational Hemodynamics: Theory, Modelling and Applications - page 128

Fig. 5.33 Transport of the scalar variable ϕ by pure convection. Line a-a' is a horizontal line half
way up the domain. a The variable is convected through the domain aligned with the computa-
tional grid lines. b The variable is convected through the domain at an angle of 45°to the compu-
tational grid lines. c false diffusion of temperature in computational domain

assuming u e = u w . The convective fluxes are defined at the control volume faces as
ϕ e = ϕ P and ϕ w = ϕ W . Expanding ϕ W through a Taylor series we get

22

d φ ∆ xd φ

φφ = −∆ x + −…

WP

2

dx 2!

dx

and rearranging

2

φφ φ

PW d

− ∆

=− +∆

xd φ

2

(5.58)

Ox ( )

2

∆ x dx 2

dx

If we substitute Eq. (5.58) into (5.57) we get

d φ ∆

−

u x d d φ

2

u +∆= O( x ) 0 (5.59)

dx 2 dx dx

Fig. 5.34 Solutions for
the convection transport
of ϕ. The upper differenc-
ing scheme using a 10 × 10
control volume mesh shows
that the profile is closer to the
exact solution