7/23/2019 Fluids EBook_ Similitude

http://slidepdf.com/reader/full/fluids-ebook-similitude 1/2

07/12/2015 Fluids eBook: Similitude

https://ecourses.ou.edu/cgi-bin/ebook.cgi?doc=&topic=fl&chap_sec=06.2&page=theory 1/2

Ch 6. Modeling and Similitude   Multimedia Engineering Fluids

Dimensional Analysis

  Similitude  Dimensionless

Groups 

Fluids Similitude   Case Intro   Theory   Case Solution   Simulation

 Chapter  1. Basics 2. Fluid Statics 3. Kinematics 4. Laws (Integral) 5. Laws (Diff.) 6. Modeling/Similitude

 7. Inviscid 8. Viscous 9. External Flow 10. Open-Channel

 Appendix

 Basic Math Units Basic Equations Water/Air Tables Sections

 Search

 eBooks

 Dynamics Fluids Math Mechanics Statics Thermodynamics

 Author(s):

Chean Chin NgoKurt Gramoll

  ©Kurt Gramoll

FLUID MECHANICS - THEORY

 

In the Dimensional Analysis section, methods toidentify dimensionless parameters governing aparticular model were presented. The discussion in thissection is focused on how to relate a model to anactual prototype (i.e., the real full scale structure). Inother words, it is desired to find out under whatconditions will testing or experiments done on a modelaccurately represent or predict the actual phenomena.

 

Similitude

Wind Tunnel Testing

 

In the study of fluid mechanics, models are frequentlyused for testing and development purposes inlaboratories before a full scale prototype is built. Themodel can be either smaller than the prototype (e.g.,

design of dam, airplane and automobiles) or larger thanthe prototype (e.g., study of interaction between redblood cells and the vessel wall).

Take the aircraft industry for example. It is a commonpractice to design and develop an aircraft by testingthe model in a wind tunnel. This way, the expenses of product development will be reduced considerablycompared to conducting the design and developmentprocess on a full scale aircraft.

The next question would be under what conditions willthe model accurately portray the actual full scale

prototype. In order to yield useful information from themodel being tested, the model and prototype shouldfulfill the requirement of geometric, kinematic anddynamic similitude, as discussed in the subsequentparagraphs.

Geometric SimilitudeThe first requirement is for the model and prototype tobe geometrically similar. The model and prototypeshould have the same shape, and their dimensionsshould be scaled proportionally. In addition, their orientation and surface roughness should also have thesame scale. 

Kinematic SimilitudeTo obtain useful information from the model testing,the second requirement is for the model and prototypeto be kinematically similar. In order to satisfy thiscondition, the flow conditions should be the same. Inother words, the velocities and accelerations atcorresponding points should have the same directionsand scaled magnitudes. Model and prototype that arekinematic similar are also geometrically similar.

Dynamic SimilitudeIn order to achieve dynamic similitude, all forces (e.g.,

pressure, shear, viscous, and surface tension forces)at corresponding points of the model and prototypeshould have the same directions and scaledmagnitudes. Model and prototype that are dynamicallysimilar are both geometrically and kinematicallysimilar.

7/23/2019 Fluids EBook_ Similitude

http://slidepdf.com/reader/full/fluids-ebook-similitude 2/2

07/12/2015 Fluids eBook: Similitude

https://ecourses.ou.edu/cgi-bin/ebook.cgi?doc=&topic=fl&chap_sec=06.2&page=theory 2/2

 

An Example: Flow Past a Sphere

Flow Past a Sphere: Geometric andKinematic Similitude

Click to view movie (29k)

Flow Past a Sphere: Dynamic SimilitudeClick to view movie (16k)

 

Take flow past a sphere for example. A prototype andmodel satisfying geometric similitude are shown in thefigure. The diameters of the spheres are proportional.Based on dimensional analysis, it can be determinedthat the drag force (D) can be modeled as,

  D/(d

2

ρV

2

) = function (Re)

where Re is the Reynolds number, and is given by

  Re = ρVd/μ

Both the model and prototype should follow the samerelationship. In order to satisfy the dynamic similituderequirement, the Reynolds number   and dimensionlessdrag for the model and prototype should be the same:

  Rem  = Rep  and

Dm/(dm

2

 ρmVm

2

) = Dp/(dp

2ρpVp

2

)

where the subscripts m and p refer to the model andprototype, respectively.

From this example, it is shown that for the model andprototype to fulfill the requirement of similitude, all thecorresponding pi terms  should be equal. In other applications, the Froude number  will be the same for both the model and prototype when free surface flow isinvolved. The Weber number  will be the same for model and prototype when surface tension force isimportant.