A flow with periodic variations (fluid dyamics)
In fluid dynamics, a flow with periodic variations is known as pulsatile flow, or as Womersley flow. The flow profiles was first derived by John R. Womersley (1907–1958) in his work with blood flow in arteries. The cardiovascular system of chordate animals is a very good example where pulsatile flow is found, but pulsatile flow is also observed in engines and hydraulic systems, as a result of rotating mechanisms pumping the fluid.
Equation.
The pulsatile flow profile is given in a straight pipe by
formula_1
where:
Properties.
Womersley number.
The pulsatile flow profile changes its shape depending on the Womersley number
formula_2
For formula_3, viscous forces dominate the flow, and the pulse is considered quasi-static with a parabolic profile.
For formula_4, the inertial forces are dominant in the central core, whereas viscous forces dominate near the boundary layer. Thus, the velocity profile gets flattened, and phase between the pressure and velocity waves gets shifted towards the core.
Function limits.
Lower limit.
The Bessel function at its lower limit becomes
formula_5
which converges to the Hagen-Poiseuille flow profile for steady flow for
formula_6
or to a quasi-static pulse with parabolic profile when
formula_7
In this case, the function is real, because the pressure and velocity waves are in phase.
Upper limit.
The Bessel function at its upper limit becomes
formula_8
which converges to
formula_9
This is highly reminiscent of the Stokes layer on an oscillating flat plate, or the skin-depth penetration of an alternating magnetic field into an electrical conductor.
On the surface formula_10, but the exponential term becomes negligible once formula_11 becomes large, the velocity profile becomes almost constant and independent of the viscosity. Thus, the flow simply oscillates as a plug profile in time according to the pressure gradient,
formula_12
However, close to the walls, in a layer of thickness formula_13, the velocity adjusts rapidly to zero. Furthermore, the phase of the time oscillation varies quickly with position across the layer. The exponential decay of the higher frequencies is faster.
Derivation.
For deriving the analytical solution of this non-stationary flow velocity profile, the following assumptions are taken:
Thus, the Navier-Stokes equation and the continuity equation are simplified as
formula_14
and
formula_15
respectively. The pressure gradient driving the pulsatile flow is decomposed in Fourier series,
formula_16
where formula_17 is the imaginary number, formula_18 is the angular frequency of the first harmonic (i.e., formula_19), and formula_20 are the amplitudes of each harmonic formula_21. Note that, formula_22 (standing for formula_23) is the steady-state pressure gradient, whose sign is opposed to the steady-state velocity (i.e., a negative pressure gradient yields positive flow). Similarly, the velocity profile is also decomposed in Fourier series in phase with the pressure gradient, because the fluid is incompressible,
formula_24
where formula_25 are the amplitudes of each harmonic of the periodic function, and the steady component (formula_23) is simply Poiseuille flow
formula_27
Thus, the Navier-Stokes equation for each harmonic reads as
formula_28
With the boundary conditions satisfied, the general solution of this ordinary differential equation for the oscillatory part (formula_29) is
formula_30
where formula_31 is the Bessel function of first kind and order zero, formula_32 is the Bessel function of second kind and order zero, formula_33 and formula_34 are arbitrary constants, and formula_35 is the dimensionless Womersley number. The axisymmetric boundary condition (formula_36) is applied to show that formula_37 for the derivative of above equation to be valid, as the derivatives formula_38 and formula_39 approach infinity. Next, the wall non-slip boundary condition (formula_40) yields formula_41. Hence, the amplitudes of the velocity profile of the harmonic formula_21 becomes
formula_43
where formula_44 is used for simplification.
The velocity profile itself is obtained by taking the real part of the complex function resulted from the summation of all harmonics of the pulse,
formula_45
Flow rate.
Flow rate is obtained by integrating the velocity field on the cross-section. Since,
formula_46
then
formula_47
Velocity profile.
To compare the shape of the velocity profile, it can be assumed that
formula_48
where
formula_49
is the shape function.
It is important to notice that this formulation ignores the inertial effects. The velocity profile approximates a parabolic profile or a plug profile, for low or high Womersley numbers, respectively.
Wall shear stress.
For straight pipes, wall shear stress is
formula_50
The derivative of a Bessel function is
formula_51
Hence,
formula_52
Centre line velocity.
If the pressure gradient formula_20 is not measured, it can still be obtained by measuring the velocity at the centre line. The measured velocity has only the real part of the full expression in the form of
formula_54
Noting that formula_55, the full physical expression becomes
formula_56
at the centre line. The measured velocity is compared with the full expression by applying some properties of complex number. For any product of complex numbers (formula_57), the amplitude and phase have the relations formula_58 and formula_59, respectively. Hence,
formula_60
and
formula_61
which finally yield
formula_62
References.
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