Chapter 4 T URBULENCE
Description
When flow in a straight cylindrical pipe is relatively low, fluid particles move smoothly in concentric layers. This type of flow is called laminar flow. The relation between the pressure gradient and flow is linear and described by Poiseuille’s law (box figure, left). As flow becomes increasingly larger, the smooth parallel fluid motion becomes wavy, leading to vortices propagated downstream, subsequently the number of vortices increases and finally fluid motion becomes irregular [2]. This irregular and seemingly random fluid particle motion is called turbulence. Turbulent flow is energetically more costly than laminar flow, because part of the mechanical energy used to maintain flow (i.e., pressure gradient) is lost in the erratic motion between the fluid particles. The resistance to flow is thus higher, which is reflected by the change in slope in the relation between pressure drop and flow (box figure).
To judge whether a fluid flow is laminar or turbulent, the Reynolds
number, Re, is often used. Re is defined as with the fluid density, v the mean fluid velocity, D the tube inner diameter and fluid
viscosity. The Reynolds number is the ratio of inertia and viscous effects. For low Reynolds numbers the viscous effects are dominant and laminar flow exists. Thus, it is not only the fluid velocity that determines whether or not the flow is laminar, but tube size, viscosity and blood density also play a role.
There exists a transitional zone around critical Reynolds number of 2200
where flow is neither strictly laminar nor strictly turbulent. Also when flow is
16 Basics of Hemodynamics slowly increased turbulence may start at Reynolds numbers somewhat higher than 2200 and, inversely, when flow is decreased from a turbulent case it may remain turbulent for Reynolds numbers smaller than 2200. In some hemodynamic texts the radius is used instead of the diameter; the critical Reynolds number is then 1100.
Physiological and clinical relevance At normal resting conditions arterial flows are laminar. For instance, in the human aorta at rest with a Cardiac Output, CO, of 6 1/min the Reynolds number can be calculated as follows.
Mean velocity and with an it equals about The Reynolds number is then, assuming
blood density to be and
blood viscosity to be 3.5 cP:
This Reynolds number is far below the critical number of 2200 and thus flow is
laminar. With heavy exercise, where flow can increase by a factor of 5 or so, the Reynolds number increases to values above 2200 and turbulence occurs.
The criterion for transition to turbulence, i.e., Re > 2200 applies to steady flow in straight tubes. Because arterial flows are highly pulsatile, this criterion does not strictly apply. For pulsatile flow laminar flow persists longer and transition to turbulence takes place at higher Reynolds numbers.
Turbulence is delayed when the fluid is accelerating whereas transition to turbulence occurs faster in decelerating flows. Loss of pressure due to turbulence is an effective means to decelerate flow fast. A classical example is turbulence distal to a stenosis. Fluid particles, which have been accelerated through the converging part of the stenosis need to decelerate fast in the distal expanding part, flow separates and turbulence develops. Turbulence in severe stenoses can be initiated for Reynolds numbers as low as 50.
Turbulence may affect endothelial function and play an important role in certain pathologies. For example, it has been suggested that turbulence distal to stenoses contributes to the phenomenon of post-stenotic dilatation. Aortic dilatation in valvular stenosis is also known to exist. Also, turbulence occurring at the venous anastomoses of vascular access grafts used in hemodialysis patients has been correlated with the local development of intima hyperplasia, which ultimately leads to a stenosis and graft failure.
TURBULENCE evidenced as rapid fluctuation in the aortic velocity signal measured in (A) a patient with normal aortic valve and a normal cardiac output of 5.3 l/min, and (B) a patient with normal aortic valve but with an elevated cardiac output of 12.9 l/min. Turbulence is much more present and intense in the case of high aortic flow. Adapted from [1], used by permission.
References
1.2.
Nichols WW, O’Rourke MF. McDonald’s blood flow in arteries. 1990, London, Edward Arnold, 3rd edn.
Munson BR, Young DF, Okiishi TH. Fundamentals of Fluid mechanics. 1994, New York, John Wiley & Sons.
