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  • Calculate Reynolds number.
  • Use the Reynolds number for a system to determine whether it is laminar or turbulent.

Sometimes we can predict if flow will be laminar or turbulent. We know that flow in a very smooth tube or around a smooth, streamlined object will be laminar at low velocity. We also know that at high velocity, even flow in a smooth tube or around a smooth object will experience turbulence. In between, it is more difficult to predict. In fact, at intermediate velocities, flow may oscillate back and forth indefinitely between laminar and turbulent.

An occlusion, or narrowing, of an artery, such as shown in [link] , is likely to cause turbulence because of the irregularity of the blockage, as well as the complexity of blood as a fluid. Turbulence in the circulatory system is noisy and can sometimes be detected with a stethoscope, such as when measuring diastolic pressure in the upper arm’s partially collapsed brachial artery. These turbulent sounds, at the onset of blood flow when the cuff pressure becomes sufficiently small, are called Korotkoff sounds . Aneurysms, or ballooning of arteries, create significant turbulence and can sometimes be detected with a stethoscope. Heart murmurs, consistent with their name, are sounds produced by turbulent flow around damaged and insufficiently closed heart valves. Ultrasound can also be used to detect turbulence as a medical indicator in a process analogous to Doppler-shift radar used to detect storms.

Figure shows a rectangular section of a blood vessel. The blood flow is shown toward right. The blood vessel is shown to be broader at one end and narrow toward the opposite end. The flow is shown to be laminar as shown by horizontal parallel lines. The velocity is v one in the broader section of blood vessel. The junction where the tube narrows the velocity is v two. The lines of flow are shown to bend. The regions where the blood vessels are narrow, the flow is shown to be turbulent as shown to by curling arrows. The velocity is given by v three toward right. The length of the arrows depicting the velocities represent that v three is greater than v two greater than v one.
Flow is laminar in the large part of this blood vessel and turbulent in the part narrowed by plaque, where velocity is high. In the transition region, the flow can oscillate chaotically between laminar and turbulent flow.

An indicator called the Reynolds number     N R size 12{N rSub { size 8{R} } } {} can reveal whether flow is laminar or turbulent. For flow in a tube of uniform diameter, the Reynolds number is defined as

N R = 2 ρ vr η (flow in tube), size 12{N rSub { size 8{R} } = { {2ρ ital "vr"} over {η} } } {}

where ρ size 12{ρ} {} is the fluid density, v size 12{v} {} its speed, η size 12{η} {} its viscosity, and r size 12{r} {} the tube radius. The Reynolds number is a unitless quantity. Experiments have revealed that N R size 12{N rSub { size 8{R} } } {} is related to the onset of turbulence. For N R size 12{N rSub { size 8{R} } } {} below about 2000, flow is laminar. For N R size 12{N rSub { size 8{R} } } {} above about 3000, flow is turbulent. For values of N R size 12{N rSub { size 8{R} } } {} between about 2000 and 3000, flow is unstable—that is, it can be laminar, but small obstructions and surface roughness can make it turbulent, and it may oscillate randomly between being laminar and turbulent. The blood flow through most of the body is a quiet, laminar flow. The exception is in the aorta, where the speed of the blood flow rises above a critical value of 35 m/s and becomes turbulent.

Is this flow laminar or turbulent?

Calculate the Reynolds number for flow in the needle considered in Example 12.8 to verify the assumption that the flow is laminar. Assume that the density of the saline solution is 1025 kg/ m 3 .

Strategy

We have all of the information needed, except the fluid speed v size 12{v} {} , which can be calculated from v ¯ = Q / A = 1.70 m/s size 12{ {overline {v}} =Q/A=1 "." "70"" m/s"} {} (verification of this is in this chapter’s Problems and Exercises).

Solution

Entering the known values into N R = 2 ρ vr η size 12{N rSub { size 8{R} } = { {2ρ ital "vr"} over {η} } } {} gives

N R = 2 ρ vr η = 2 ( 1025 kg/m 3 ) ( 1.70 m/s ) ( 0.150 × 10 3 m ) 1 . 00 × 10 3 N s/m 2 = 523 .

Discussion

Since N R size 12{N rSub { size 8{R} } } {} is well below 2000, the flow should indeed be laminar.

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Source:  OpenStax, Introduction to physics for vanguard high school (derived from college physics). OpenStax CNX. Oct 15, 2014 Download for free at http://legacy.cnx.org/content/col11715/1.1
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