Computational Fluid Structure Interaction - Computational Hemodynamics: Theory, Modelling and Applications - page 122

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Fig. 5.26 Conservation principle applied to a single control volume

Fig. 5.27 Control volume
schematic for the finite vol-
ume approach

where a nb is the neighbouring node coefficient, n φ is the flow variable at the
neighbouring node, and b is a constant. There are a number of discretisation schemes
available but for it to produce physically realistic solution, it must satisfy rules of:
conservativeness, boundedness, and transportiveness.
Conservativeness refers to flux consistency at the control volume faces so that
the flux leaving a control volume is equal to the flux entering the adjacent control
volume through the same face. Boundedness requires the discretized equations to
have coefficients satisfying

∑

a
aS

nb

(5.46)

−

p p

where S p comes from source terms. This constraint can be satisfied if a p is as
large as possible, and S p is negative. All coefficients should be of the same sign,
preferably positive so that an increase in ϕ at one node should result in an increase
in ϕ at neighbouring nodes. For transportiveness, the Peclet number is used which
we saw earlier is the relative strengths of convection and diffusion

convection
diffusion

u
v L

(5.47)

Pe = =

where u is the fluid velocity, v is the kinematic velocity, and L is a characteristic
length. The Peclet number, Pe is related with the directionality of influence and
this is determined by the discretisation scheme. This is presented schematically in
Fig. 5.28 where a value of Pe = 0 implies pure diffusion, and the contour of influ-
ence are concentric circles for constant values of ф since diffusion tends to spread
out evenly in all directions. As Pe increases, the influence curve becomes more