Laminar and Turbulent Flow in Water

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Laminar and turbulent flow inwaterH G Riveros1 and D Riveros-Rosas2

1 Institute of Physics, Universidad Nacional Autonoma de Mexico, Ciudad Universitaria,Mexico, D.F., Mexico2 Departamento de Ingeniera Qumica, Universidad de Sonora, Hermosillo, Sonora, Mexico

E-mail: [email protected]

AbstractThere are many ways to visualize flow, either for laminar or turbulent flows.A very convincing way to show laminar and turbulent flows is by theperturbations on the surface of a beam of water coming out of a cylindricaltube. Photographs, taken with a flash, show the nature of the flow of water inpipes. They clearly show the difference between turbulent and laminar flow,and let, in an accessible way, data be taken to analyse the conditions underwhich both flows are present. We found research articles about turbulencemeasurements, using sophisticated equipment, but they do not use theperturbation of the free surface of the flowing liquid to show or measure theturbulence.

IntroductionThe problem of transporting water through pipesexists since this is the way it is distributed tocommunities; if the water is moving withoutfriction (viscosity) it could be distributed with verylittle energy dissipation. The viscosity opposes themotion of a layer of water over another, and actsas a friction force, transferring part of the energyof flow into thermal energy.

The differential equations for flow havetwo solutions: either time-independent or time-dependent. In the first case, the fluid velocityis constant in time at each point, and thecorresponding flow is called laminar. If thevelocity changes over time, the flow is turbulent;in this case, the solutions correspond either tothe stationary or transient state. There are manytechniques for flow visualization [1], most of thembased on the dispersion of light produced by tracerparticles or gas bubbles. We use the shape of thesurface of a cylindrical water beam, smooth forlaminar flow or rough for turbulent flow [2]. The

roughness grows with the turbulence as measuredby the Reynolds number.

The volume flow rate through a surface S iscalculated knowing the velocity of the fluid at eachpoint of the surface to calculate the integral:

=

v dS. (1)

This equation can be easily integrated when thevelocity is constant over the surface S: =vS. However, to achieve a constant velocity ona surface is not easy. Instead, we can measurethe flow in the tube and then divide by its areato obtain the average velocity of the water. Tomove a cylindrical segment of water inside a hose,one needs to apply a pressure difference betweenits ends to compensate for the effects of viscositythat oppose its motion. For laminar flows, thewater velocity next to the tube wall is zero andmaximum at its centre, a result that is obtainedby considering a cylinder of water interacting andpressure applied to its ends. Thus, a steady state

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Laminar and turbulent flow in water

is reached some time after applying pressure. Themaximum speed and the speed at each point areproportional to the applied pressure difference andflow is proportional to pressure. If the pressure isincreased substantially, the speeds of the differentregions of the water will change continually inmagnitude and direction, resulting in a turbulentflow in which each point is changing its velocitywith time. Turbulence increases with increasingpressure. If the flow is laminar, the pressuredifference between the ends is proportional towater flow; and if the flow is turbulent, the pressuredifference is proportional to the square of the flow.The Reynolds number (Re = vD/) helps us toidentify the type of flow we have: if its value isless than 2300 the flow is laminar and if greaterthan 4000 the flow is turbulent (in cylindricalpipes). Between 2300 and 4000 it is considereda transition regime. If we express the density ,velocity v, the diameter D and viscosity in thesame system of units, the Re Reynolds number hasno units. The velocity v is the velocity averagedover the area of the tube and is easily measuredby measuring the volume flowing in a certaintime; the flow/area gives us the average speed.For laminar flow, equation (1) can be integratedand we obtain the Poiseuille law, stating that thepressure difference between the ends of the tube isproportional to flow:

P = 8L/r 4 (2)where is viscosity, L is the length of the tube, is the flow, and r is the inner radius of the tubethrough which fluid flows. For a turbulent flow, adifferent expression for a horizontal tube is found:

P = Lv2/4r = L2/42r 5 (3)where is the coefficient of friction of DArcyWeisbach [3], is the density, v is the velocityand is the flow. We see that the pressuredifference is proportional to the square of theflow , if we neglect the change in with flowrate. Equation (3) applies to steady-state flow ofan incompressible fluid. The friction coefficient depends on whether the flow is laminar,transitional or turbulent, and the roughness of thetube. Usually, to observe laminar flow, ink or airbubbles have been used; turbulent flow is detectedby the light scattered by small particles suspendedin the fluid. We present a new way to observe the

Figure 1. Photographs taken with a 2 megapixel digitalcamera with flash. The flows are laminar.

turbulent flows coming out of tubes into the air, asperturbations in the surface exposed to air in thefluid. Turbulence implies chaotic movements insmall volumes within the fluid. When those smallvolumes reach the surface, the surface tension ofthe liquid returns them to the beam, but the surfacemust be deformed for this to be achieved; the effectcan be seen in high speed photographs taken withflash to freeze motion.

ExperimentsThe photos in figures 13 display the flow for avariety of Reynolds numbers calculated for theflow at the mouth of the tube. As the diameterof the water beam changes, so does the Reynoldsnumber.

Laminar flows do not disturb the surface inpoints throughout said flow; the changes observedare due to the attempts of the superficial tensionforce to reduce the area by forming smallerdiameter drops. However, on increasing therate of the flow to reach turbulent regimes, thephotographs clearly show the disturbances at thefree surface of the water beam.

In the photographs in figure 2 there isan increment in the turbulence, shown asdeformations in the surface of the water beam. Thebeam surface is deformed by changes in speed of

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H G Riveros and D Riveros-Rosas

Figure 2. On increasing the flow rate, the water beamchanged to a turbulent flow. Note that the surfacetension prevents the turbulence breaking the beam intodrops.

the internal portions of the water beam, which arearrested by the surface tension of the water.

In the case in which the initial flow ishorizontal, there are similar figures that changewith turbulence, and we can observe the samegeneral characteristics as for vertical flow. First wecan observe changes in output speed, resulting inchanges in the path of the water beam, and finallywe can see turbulent deformations observed on thesurface of the water beam. This is indicative of theturbulence in the internal portions. In this case, theflow can be measured by taking the time it takes tofill a container or by the equation of the parabolacorresponding to the water path [4].

With a horizontal tube, measuring the heightY of the tube (above the table) and the horizontaldistance X travelled by the water beam (on thetable), we can calculate the horizontal speed Vat the mouth of the tube. The falling time T isobtained from Y = gT 2/2 and V = X/T ; thenthe water flow is calculated as V A, where A isthe internal area of the tube. Using this procedurewe obtain a flow of (6.46 0.12) l min1 andmeasuring the volume collected in a given time theflow was (6.35 0.14) l min1. Both values are

Figure 3. Photographs of the turbulence in a horizontalwater beam; they show how the Reynolds numberincreases with water speed.

in agreement, within the respective experimentaluncertainty.

From figures 13, we can conclude that theflow of water in the pipes that we use is generallyturbulent and therefore the pressure drop betweenthe ends is not proportional to water flow.

Turbulence implies that different regionswithin the water beam have different magnitudesand directions of velocity; with changes in thepressure or violent changes in direction suchas those introduced by a change of 90 in thedirection of the velocity. Figure 4 shows two flowswith the same water speed, but flowing throughthe straight portion of a T-junction, or bending 90using the other arm of the T-junction.

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Laminar and turbulent flow in water

a

bFigure 4. (a) A water beam passing through the straightsection of a T-junction. (b) A water beam at the samerate of flow, after being deflected 90 in the T-junction.

Driving the water with a piston pump,the pressure changes between maximum andminimum values, both of which diminish as thewater travels through the hose. Figure 5 showsthe change in the water path at the outlet of thehose. Note that the maximum pressure producesa high velocity jet at the exit, which producesthe parabola joining the maxima in the jet. Theminimum follows a parabola below. The waterbeam is held together by cohesion and by thesurface tension force.

ConclusionsPhotographs provide a very graphical way to showthe turbulence in a fluid flowing in a beam inthe open air. The deformation at a point in thesurface of the beam shows the chaotic behaviourof an internal portion of the fluid. Also shown

Water driven for a piston pump

Figure 5. Photograph of a water beam powered by apiston pump. We can see clearly noticeable changes inthe output speed. The maximum pressure produces theupper parabolic path and the minimum produces thelower.

with extraordinary clarity is the relation betweenthe Reynolds number and the kind of flow regimeobserved at the water beam and the transition froma laminar to a turbulent flow.

Acknowledgment

We thank the reviewer who improved our articlewith many useful suggestions.

Received 18 January 2010, in final form 16 February 2010doi:10.1088/0031-9120/45/3/010

References[1] Clayton B R and Massey B S 1967 J. Sci. Instrum.

44 211[2] Sears F W, Zemansky M W and Young H D 1985

College Physics (Reading, MA:Addison-Wesley) p 275

[3] www.engineeringtoolbox.com/darcy-weisbach-equation-d 646.html

[4] Castro P M, Delfino A, Vieira E and Faria V A2000 Pin-hole water flow from cylindricalbottles Phys. Educ. 35 1109

H G Riveros has been a researcher at theInstituto de Fsica de la UniversidadNacional Autonoma de Mexico (UNAM)since 1963. He has written several booksand designed many experiments anddemonstrations. He writes the columnLos Placeres del Pensamiento in theBulletin of the Mexican Physical Society.

D Riveros-Rosas is a physicist from theUniversidad Nacional Autonoma deMexico (UNAM) and obtained his PhDin 2008 in the energy area. He has writtenseveral books in physics education and isa researcher at the Geophysics Institute atthe UNAM.

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Laminar and Turbulent Flow in Water