Page 102 - Biomedical Engineering and Design Handbook Volume 1, Fundamentals

PHYSICAL AND FLOW PROPERTIES OF BLOOD 79

where c = wave speed
w = 2pHR/60 = angular frequency
HR = heart rate
A = input pressure amplitude
1
J and J = Bessel functions of order 0 and 1 of the first kind
0
1
r and r = blood and wall densities
F T
R = undisturbed radius of the tube
0
Excellent recent summaries on pulsatile blood flow may be found in Nichols and O’Rourke (1998)
and Zamir (2000).
3.3.5 Turbulence

Turbulence has been shown to exist in large arteries of a living system. It is especially pronounced
when the flow rate increases in exercise conditions (Yamaguchi and Parker, 1983). Turbulence is
characterized by the appearance of random fluctuations in the flow. The transition to turbulence is a
very complex procedure, which schematically can be described by a hierarchy of motions: growth
of two-dimensional infinitesimal disturbances to final amplitudes, three-dimensionality of the flow,
and a maze of complex nonlinear interactions among the large-amplitude, three-dimensional modes
resulting in a final, usually stochastically steady but instantaneously random motion called turbulent
flow (Akhavan et al., 1991; Einav and Sokolov, 1993).
In a turbulent flow field, all the dynamic properties (e.g., velocity, pressure, vorticity) are random
functions of position and time. One thus looks at the statistical aspects of the flow characteristics
(e.g., mean velocity, rms turbulent intensity). These quantities are meaningful if the flow is stochas-
tically random (i.e., its statistics are independent of time) (Nerem and Rumberger, 1976). The time
average of any random quantity is given by
1 ∞
)
f ≡ lim T→∞ ∫ ft dt (3.29)
(
T 0
One can thus decompose the instantaneous variables u and v as follows:

ux t) = U x() + ′ (, (3.30)
u x t)
(,
vy t) = V y() + ′ (, (3.31)
v y t)
(,
Px t) = P x() + ′ (, (3.32)
p x t)
(,
We assume that u′ is a velocity fluctuation in the x direction only and v′ in the y direction only. The
overbar denotes time average, so that by definition, the averages of u′, v′, and p′ fluctuations are zero
(stochastically random), and the partial derivatives in time of the mean quantities UV P,, are zeros
(the Reynolds turbulence decomposition approach, according to which velocities and pressures can
be decomposed to time-dependent and time-independent components).
By replacing u with U + u′ etc. in the Navier-Stokes equation and taking time average, it can be
shown that for the turbulent case the two-dimensional Navier-Stokes equation in cartesian coordi-
nates becomes
∂ U ∂ U ∂ P ∂ ⎡ ∂ U ⎤
ρU + ρV =− + ⎢ μ − ρuv ′′ ⎥
x ∂ y ∂ x ∂ y ∂ ⎣ y ∂ ⎦
(3.33)
∂ U
=→ U = U y() V = 0
0
∂x x