Computational Fluid Structure Interaction - Computational Hemodynamics: Theory, Modelling and Applications - page 103

and we see that total acceleration (i.e. sum of the local acceleration and convection )
of the fluid is driven by the pressure gradient in the x -direction. The negative sign
denotes that a positive pressure produces a decrease in acceleration, and vice-versa.
For a steady flow, we get

∂ u ∂ P

=−

ρ u

(5.17)

∂ x ∂ x

convection pressure

and when this is integrated along the streamline, produces Bernoulli's equation
which we have seen from Chap. 4. For example,

2

2

∂ u ∂

=− ⇒ +

1 P Px () u x ()

2

PU

∞ ∞

u = +

∂ x ρ ∂ x ρ ρ 2

where P ∞ is the upstream pressure, and U ∞ is the freestream velocity. If we set the
upstream velocity to U ∞ = 1 m/s and a radius of R = 1 m the above equation becomes

2

ρ ρ 1

2 2

Px () −= − = −− P ( U u x ()) 1 1

∞ ∞

2

2 2

x

Along the streamline, the velocity profile u ( x ) and the acceleration is plotted in
Fig. 5.9 . The velocity drops very rapidly as the fluid approaches the bifurcation.
At the bifurcation midpoint, the velocity is zero (stagnation point) and the surface
pressure is a maximum. The pressure difference p ( x ) - p atm demonstrates that the
pressure increases as the fluid approaches the stagnation point. With the density ρ
set to 1 kg/m 3 it reaches a maximum value of 0.5, i.e. p stag - p atm =

2
(1 / 2 ) ρ ∞ as

u ( x ) → 0 near the stagnation point.
The total acceleration profile depicted in Fig. 5.10 also shows strong decelera-
tion of the fluid as it approaches the cylinder. The maximum deceleration occurs at
x = − 1.29 m with a magnitude of − 0.372 m/s 2 .

Fig. 5.9 a Velocity profile u ( x ) along the stagnation streamline. b Pressure difference profile
P ( x ) - P ∞ along the stagnation streamline