Eddy (fluid dynamics)

In fluid dynamics, an eddy is the swirling of a fluid and the reverse current created when the fluid is in a turbulent flow regime.[2] The moving fluid creates a space devoid of downstream-flowing fluid on the downstream side of the object. Fluid behind the obstacle flows into the void creating a swirl of fluid on each edge of the obstacle, followed by a short reverse flow of fluid behind the obstacle flowing upstream, toward the back of the obstacle. This phenomenon is naturally observed behind large emergent rocks in swift-flowing rivers.

Swirl and eddies in engineering

The propensity of a fluid to swirl is used to promote good fuel/air mixing in internal combustion engines.

In fluid mechanics and transport phenomena, an eddy is not a property of the fluid, but a violent swirling motion caused by the position and direction of turbulent flow.[3]

Reynolds number and turbulence

In 1883, scientist Osborne Reynolds conducted a fluid dynamics experiment involving water and dye, where he adjusted the velocities of the fluids and observed the transition from laminar to turbulent flow, characterized by the formation of eddies and vortices.[4] Turbulent flow is defined as the flow in which the system's inertial forces are dominant over the viscous forces. This phenomenon is described by Reynolds number, a unit-less number used to determine when turbulent flow will occur. Conceptually, the Reynolds number is the ratio between inertial forces and viscous forces.[5]

The general form for the Reynolds number flowing through a tube of radius r (or diameter d):

${\displaystyle \mathrm {Re} ={2v\rho r \over \mu }={\rho vd \over \mu }}$

where v is the velocity of the fluid, ρ is its density, r is the radius of the tube, and μ is the viscosity of the fluid.

The transition from laminar to turbulent flow in a fluid is defined by the critical Reynolds number, for a closed pipe this works out to approximately

${\displaystyle \mathrm {Re} _{\text{c}}\approx 2000.}$

In terms of the critical Reynolds number, the critical velocity is represented as

${\displaystyle v_{\text{c}}={\frac {\mathrm {Re} _{\text{c}}\mu }{\rho d}}.}$

Research and development

Hemodynamics

Hemodynamics is the study of blood flow in the circulatory system. Blood flow in straight sections of the arterial tree are typically laminar (high, directed wall stress), but branches and curvatures in the system cause turbulent flow.[2] Turbulent flow in the arterial tree can cause a number of concerning effects, including atherosclerotic lesions, postsurgical neointimal hyperplasia, in-stent restenosis, vein bypass graft failure, transplant vasculopathy, and aortic valve calcification.

Industrial processes

Lift and drag properties of golf balls are customized by the manipulation of dimples along the surface of the ball, allowing for the golf ball to travel further and faster in the air.[6][7]

The data from turbulent-flow phenomena has been used to model different transitions in fluid flow regimes, which are used to thoroughly mix fluids and increase reaction rates within industrial processes.[8]

Fluid currents and pollution control

Oceanic and atmospheric currents transfer particles, debris, and organisms all across the globe. While the transport of organisms, such as phytoplankton, are essential for the preservation of ecosystems, oil and other pollutants are also mixed in the current flow and can carry pollution far from its origin.[9][10] Eddy formations circulate trash and other pollutants into concentrated areas which researchers are tracking to improve clean-up and pollution prevention.

Mesoscale ocean eddies play crucial roles in transferring heat poleward, as well as maintaining heat gradients at different depths.[11]

Computational fluid dynamics

These are turbulence models in which the Reynolds stresses, as obtained from a Reynolds averaging of the Navier–Stokes equations, are modelled by a linear constitutive relationship with the mean flow straining field, as:

${\displaystyle -\rho \langle u_{i}u_{j}\rangle =2\mu _{t}S_{i,j}-{2 \over 3}\rho \kappa \delta _{i,j}}$

where

• ${\displaystyle \mu _{t}}$ is the coefficient termed turbulence "viscosity" (also called the eddy viscosity)
• ${\displaystyle \kappa ={\tfrac {1}{2}}(\langle u_{1}u_{1}\rangle +\langle u_{2}u_{2}\rangle +\langle u_{3}u_{3}\rangle )}$ is the mean turbulent kinetic energy
• ${\displaystyle S_{i,j}}$ is the mean strain rate
Note that that inclusion of ${\displaystyle {\tfrac {2}{3}}\rho \kappa \delta _{i,j}}$ in the linear constitutive relation is required by tensorial algebra purposes when solving for two-equation turbulence models (or any other turbulence model that solves a transport equation for ${\displaystyle \kappa }$ .[12]

Mesoscale ocean eddies

Eddies are common in the ocean, and range in diameter from centimeters to hundreds of kilometers. The smallest scale eddies may last for a matter of seconds, while the larger features may persist for months to years.

Eddies that are between about 10 and 500 km (6.2 and 310.7 miles) in diameter and persist for periods of days to months are known in oceanography as mesoscale eddies.[13]

Mesoscale eddies can be split into two categories: static eddies, caused by flow around an obstacle (see animation), and transient eddies, caused by baroclinic instability.

When the ocean contains a sea surface height gradient this creates a jet or current, such as the Antarctic Circumpolar Current. This current as part of a baroclinically unstable system meanders and creates eddies (in much the same way as a meandering river forms an ox-bow lake). These types of mesoscale eddies have been observed in many of major ocean currents, including the Gulf Stream, the Agulhas Current, the Kuroshio Current, and the Antarctic Circumpolar Current, amongst others.

Mesoscale ocean eddies are characterized by currents that flow in a roughly circular motion around the center of the eddy. The sense of rotation of these currents may either be cyclonic or anticyclonic (such as Haida Eddies). Oceanic eddies are also usually made of water masses that are different from those outside the eddy. That is, the water within an eddy usually has different temperature and salinity characteristics to the water outside the eddy. There is a direct link between the water mass properties of an eddy and its rotation. Warm eddies rotate anti-cyclonically, while cold eddies rotate cyclonically.

Because eddies may have a vigorous circulation associated with them, they are of concern to naval and commercial operations at sea. Further, because eddies transport anomalously warm or cold water as they move, they have an important influence on heat transport in certain parts of the ocean.

References

1. Tansley, Claire E.; Marshall, David P. (2001). "Flow past a Cylinder on a Plane, with Application to Gulf Stream Separation and the Antarctic Circumpolar Current" (PDF). Journal of Physical Oceanography. 31 (11): 3274–3283. Bibcode:2001JPO....31.3274T. doi:10.1175/1520-0485(2001)031<3274:FPACOA>2.0.CO;2.
2. Chiu, Jeng-Jiann; Chien, Shu (2011-01-01). "Effects of Disturbed Flow on Vascular Endothelium: Pathophysiological Basis and Clinical Perspectives". Physiological Reviews. 91 (1): 327–387. doi:10.1152/physrev.00047.2009. ISSN 0031-9333. PMC 3844671. PMID 21248169.
3. Lightfoot, R. Byron Bird ; Warren E. Stewart ; Edwin N. (2002). Transport phenomena (2. ed.). New York, NY [u.a.]: Wiley. ISBN 0-471-41077-2.
4. Kambe, Tsutomu (2007). Elementary Fluid Mechanics. World Scientific Publishing Co. Pte. Ltd. p. 240. ISBN 978-981-256-416-0.
5. "Pressure". hyperphysics.phy-astr.gsu.edu. Retrieved 2017-02-12.
6. Arnold, Douglas. "The Flight of a Golf Ball" (PDF).
7. "Why are Golf Balls Dimpled?". math.ucr.edu. Retrieved 2017-02-12.
8. Dimotakis, Paul. "The Mixing Transition in Turbulent Flows" (PDF). California Institute of Technology Information Tech Services.
9. "Ocean currents push phytoplankton, and pollution, around the globe faster than thought". Science Daily. 16 April 2016. Retrieved 2017-02-12.
10. "Ocean Pollution". National Oceanic and Atmospheric Administration.
11. "Ocean Mesoscale Eddies – Geophysical Fluid Dynamics Laboratory". www.gfdl.noaa.gov. Retrieved 2017-02-12.
12. "Linear eddy viscosity models -- CFD-Wiki, the free CFD reference". www.cfd-online.com. Retrieved 2017-02-12.
13. Tansley, Claire E.; Marshall, David P. (2001). "Flow past a Cylinder on a β Plane, with Application to Gulf Stream Separation and the Antarctic Circumpolar Current". Journal of Physical Oceanography. 31 (11): 3274–3283. Bibcode:2001JPO....31.3274T. doi:10.1175/1520-0485(2001)031<3274:FPACOA>2.0.CO;2. ISSN 1520-0485.