Laser Doppler Velocimetry (LDV) is a method for measuring the speed of small particles. When particles are suspended in a fluid in the presence of a known electric field, a velocity measurement is a measure of the mobility of the particles. Small particles suspended in fluid are illuminated by a laser beam and the light scattered to various angles is compared to light in a reference beam to determine the doppler shift of the scattered light. The doppler shift of the light depends on the speed of the particles and the angle of measurement. In this experiment LDV will be used to determine two profiles of laminar flow and one profile of turbulent flow [1].
2.Theory
2.1.a Hydrodynamics
In this experiment the flow profile in a straight cylindrical tube is to be measured. In such a tube two types of flow profiles can exist: a laminar flow and a turbulent flow. Which of the two will be encountered depends on the Reynolds number of the flow. The Reynolds number is given by [2,3]:
(1)
(2)
with the density of the fluid( / ), R is radius of the tube( ), 2R the inner diameter of the tube, the average velocity in the axial direction of the tube ( ) and the absolute viscosity of the fluid ( ) and is kinematics viscosity ( ). Reynolds number is a dimensionless quantity that characterises the flow. At low values of the Reynolds number ( ) the flow will be laminar, at higher values ( ) the flow will be turbulent.
Laminar Flow
Laminar flow is characterised by a situation in which the fluid particles flow in regular layers of different velocity. The flow is quite orderly. In a long cylindrical tube the flow velocity at distance r from the axis of the tube can be represented by [2,3]
(3)
Figure1. Laminar flow profile [4].
Figure 1 shows that the velocity is zero at the solid wall of the tube and increases parabolically with flow, reaching its maximum at the centre of the tube in axis. At the axis of the tube the velocity of the fluid is given by . This is a useful criteria to see whether the flow is laminar or not.
Figure2. Laminar flow profile in 3D perspective [5].
Turbulent Flow
At higher fluid velocities the order in the flow decreases; the fluid particles do not longer move straight lines through the tube, but there is momentum transfer in directions perpendicular to the axis of the tube. This results in a chaotic flow with
vertices that do not damp out. Turbulent flow is extremely complex. A complete theoretical description is still to be presented. However a description that is often used for the average flow profile is given by Nikoradze [2,6].
(4)
With the maximum velocity of the flow at the axis of the tube and the constant die represents the degree of turbulence [2].
Figure 3. Turbulent flow profile [4].
Figure 3 shows at the turbulent flow velocity is zero at the solid wall of the tube, but the face velocity is straighter and squared up. As the velocity of the fluid continues to increase the face velocity will continue to straighten up until all particles are moving at the same velocity ( except at the solid wall of the tube where the flow remains at zero) [4].
Experimental values of n are 6 ( at ), 7 ( at ), or even n=10 at extremely high values of [6].
Figure 4. Turbulent flow profile according to the formula of Nikoradze.
For the description of turbulent flow one may also use the following empirical formula:
(5)
This profile is shown in Figure 5. Close to the wall of the tube the formula 4 and 5 are equivalent.
Figure 5. Continuous turbulent profile according to formula 5.
Transient effects
At transitions in the diameter of the tube the flow profile will generally be disturbed over some distance: the flow needs a certain distance to stabilise again.
This effect can also
1. http://www.dipmec.unian.it/misure/strumenti/LDA/LDA en html
2. Practicum Module,Laser Doppler Velocity Measurement, Natuurkunde Rug
3. F.W.Sears, M.W.Zemansky, University Physics, Part 1,(Addison Wesley,1963), blz. 254, [Practicumbibliotheek AN41].
4. http://www.coleparmer.com/techinfo/techinfo.asp?htmlfile=V Pdeviations.htm
5. http://hyperphysics.phy-astr.gsu.edu/hbase/pfric.html#vel
6. H.Schlichting, Boundary-Layer Theory 6th edition, (McGraw-Hill, 1968), Hoofdstuk V en XX.[Bibliotheek Nat./Scheik. 113 A 35]
7. R.S. Brodkey, The Phenomena of Fluid Motions, (Addison-Wesley,1967), Hoofdstuk 9-5, 14-1 en 14-3D. [Bibliotheek Nat./Scheik. 113 A 48].
8. F.A. Jenkins H.E.White,Fundamentals of Optics 4th edition,( McGraw-Hill, 1981). [Practicumbibliotheek OP6]
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