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Adimensionalizacin de las ecuaciones de Navier-Stokes (Libro de Munson: pp 429 a 432

Fluido Newtoniano incompresible

Flujo Bi-dimensionalEcuacin de Continuidad

Ecuaciones de Navier-Stokes

El eje y es verticalCondiciones de Borde

oEjemplos:Componentes del vector velocidad

conocidas para ciertas posicionesPresin conocida en parte de los

lmites del campo de flujoEn caso de estado no estacionario se

deben conocer los valores iniciales detodas las variables

Las ecuaciones que gobiernan el caso estudiadoson las ecuaciones de continuidad y las de

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2

Navier-Stokes, ms las ecuaciones queespecifican las condiciones iniciales y de borde.

Prximo paso: Definir un conjunto de variablesadimensionales.Se selecciona un valor de referencia para

cada variable (tambin llamado valorcaracterstico)

Variables en este problema:

Se requerirn:oVelocidad de referencia =oPresin de referencia =oLongitud de referencia =oTiempo de referencia =

Los valores de referencia son parmetrosdel problema.

Variables adimensionales:

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3

Las ecuaciones que gobiernan el caso deestudio se re-escriben en trminos de lasvariables adimensionales. Por ejemplo:

Otros trminos que aparecen en las ecuacionespueden expresarse en una forma similar. Lasecuaciones re-escritas en trminos de lasvariables adimensionales quedan como sigue:

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Los trminos entre corchetes contienen las

cantidades de referencia (valores caracteristicos) ypueden interpretarse como indices de las distintasfuerzas (por unidad de volumen) involucradas.

= Fuerzas inerciales locales (por u. de vol.)

= Fuerzas inerciales convectivas (p./u.de vol.)

= Fuerza de presin (por u. de vol.)

= Fuerza de gravedad (por u. de vol.)

= Fuerzas viscosas (por unidad de volumen)

El paso final consiste en hacer que los factores en lasecuaciones diferenciales sean adimensionales: se

dividen las ecuaciones (7.32) y (7.33) por uno de losfactores entre corchetes.

Se puede seleccionar cualquiera de los factores.

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5

Lo convencional es dividir por el factor .La forma final adimensional es la siguiente:

Los factores entre corchetes son grupos (o nmeros)adimensionales estndar (o sus recprocas) quesurgen de otro enfoque llamado anlisisdimensional.

= Nmero de Strouhal= Nmero de Euler

= Recproca del cuadrado del nmero deFroude

= Recproca del nmero de Reynolds

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6

Es claro que cada nmero adimensional puedeinterpretarse como una relacin entre dos fuerzas.

Las formas adimensionales vistas para lasecuaciones que gobiernan el fenmeno considerado(ecs 7.31, 7.34 y 7.35) pueden utilizarse paraestablecer los llamados requerimientos de similitud.

Si dos sistemas distintos estn gobernados por lasmismas ecuaciones adimensionales, entonces lassoluciones en trminos de

ysern las mismas si los cuatro parmetros

yson iguales para los dos sistemas. La igualdad detales parmetros es lo que se denomina similitud

dinmica

Las condiciones iniciales y de borde expresadas enforma adimensional tambin tienen que ser igualespara los dos sistemas. Esto requerir que hayacompleta similitud geomtrica.

Si el flujo es estacionario el grupo adimensionaldesaparece.

El nivel de presin es importante si existe laposibilidad de cavitacin.

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De otra manera se puede imponer igual ao , y el nmero de Euler se puede

eliminar de los requerimientos de similitud.

Si hay posibilidades de cavitacin la presincaracterstica se define como relativa a la

presin de vapor: donde esuna presin de referencia en el campo de flujo.

De esta manera, el parmetro de similaridadpasa a ser .

oEste parmetro suele escribirse como: a esta forma se la

llama nmero de cavitacin

Conclusin: si la cavitacin no es unapreocupacin, no se necesita un parmetro de

similitud que involucre a . Pero si se requieremodelar la cavitacin, el nmero de

cavitacin pasa a ser un importante parmetrode similitud.

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Nmero de Froude:Surge por la inclusin del efecto de la

gravedad.

Es importante en problemas donde hayauna superficie libre (ej.: resistencia almovimiento de barcos).

oLa forma de las superficie libre estinfluenciada por la gravedadelnmero de Froude se vuelve un

importante parmetro de similitud.

oSi no hay superficies libres, se puededefinir una nueva presin:

-> con tal cambio elnmero de Froude no aparecer en lasecuaciones adimensionales.

Conclusin: En el caso de flujo en estadoestacionario, de un fluido incompresible,sin superficies libres, la similitud dinmicay cinemtica se obtendrn (para sistemasgeomtricamente similares) si existesimilitud para el nmero de Reynolds.

oSi hay superficies libres, debe existir similitudpara el nmero de Froude.

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oSi se incluyen los efectos de la tensin superficial,los mismos aparecern en la condicin de bordede la superficie libre y el nmero de

Weber aparecer como un parmetro desimilitud adicional.

oEnfatizamos que las ecuaciones que describen elmovimiento de los fluidos, expresadas en trminosde variables adimensionales, conducen a laaparicin de grupos adimensionales (o nmeros

adimensionales), que permiten obtener leyes desimilitud.

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Similitud Geomtric

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In the preceding sections of this chapter, dimensional analysis has been used to obtain sim-

ilarity laws. This is a simple, straightforward approach to modeling, which is widely used.

The use of dimensional analysis requires only a knowledge of the variables that influencethe phenomenon of interest. Although the simplicity of this approach is attractive, it must be

recognized that omission of one or more important variables may lead to serious errors in

the model design. An alternative approach is available if the equations 1usually differentialequations2governing the phenomenon are known. In this situation similarity laws can bedeveloped from the governing equations, even though it may not be possible to obtain ana-

lytic solutions to the equations.

To illustrate the procedure, consider the flow of an incompressible Newtonian fluid.

For simplicity we will restrict our attention to two-dimensional flow, although the results are

applicable to the general three-dimensional case. From Chapter 6 we know that the gov-

erning equations are the continuity equation

(7.28)

and the NavierStokes equations

(7.29)

(7.30)

where the y axis is vertical, so that the gravitational body force, only appears in the y

equation. To continue the mathematical description of the problem, boundary conditions are

required. For example, velocities on all boundaries may be specified; that is, and

at all boundary points and In some types of problems it may be nec-

essary to specify the pressure over some part of the boundary. For time-dependent problems,

initial conditions would also have to be provided, which means that the values of all depen-

dent variables would be given at some time 1usually taken at 2.Once the governing equations, including boundary and initial conditions, are known,

we are ready to proceed to develop similarity requirements. The next step is to define a new

set of variables that are dimensionless. To do this we select a reference quantity for each type

of variable. In this problem the variables are and tso we will need a reference

velocity, V, a reference pressure, a reference length, and a reference time, These

reference quantities should be parameters that appear in the problem. For example, may

be a characteristic length of a body immersed in a fluid or the width of a channel through

which a fluid is flowing. The velocity, V, may be the free-stream velocity or the inlet veloc-

ity. The new dimensionless 1starred2variables can be expressed as

The governing equations can now be rewritten in terms of these new variables. For

example,

0u

0x

0Vu*

0x*

0x*

0x

V

/

0u*

0x*

x* x

/ y* y

/ t* t

t

u*

u

V v

*

v

V

p*

p

p0

/

t./,p0,

u, v,p,x,y,

t 0

y yB.xxBv vB

u uB

rg,

ra 0v0t

u0v

0x v

0v

0yb 0p

0y rg ma 02v

0x2

02v

0y2b

ra 0u0t u

0u

0x v

0u

0yb 0p

0x ma 02u

0x2

02u

02y

b

0u

0x

0v

0y

0

7.10 Similitude Based on Governing Differential Equations 429

7.10 Similitude Based on Governing Differential Equations

Similarity laws can

be directly devel-

oped from the equa-

tions governing the

phenomenon of

interest.

MUNSON

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and

The other terms that appear in the equations can be expressed in a similar fashion. Thus, interms of the new variables the governing equations become

(7.31)

and

(7.32)

(7.33)

The terms appearing in brackets contain the reference quantities and can be interpreted as

indices of the various forces 1per unit volume2that are involved. Thus, as is indicated inEq. 7.33, inertia 1local2force, inertia 1convective2force, pressure force,

gravitational force, and viscous force. As the final step in the nondimensional-

ization process, we will divide each term in Eqs. 7.32 and 7.33 by one of the bracketed quan-

tities. Although any one of these quantities could be used, it is conventional to divide by the

bracketed quantity which is the index of the convective inertia force. The final nondi-

mensional form then becomes

We see that bracketed terms are the standard dimensionless groups 1or their recipro-cals2which were developed from dimensional analysis; that is, is a form of Strouhalnumber, the Euler number, the reciprocal of the square of the Froude number,

and the reciprocal of the Reynolds number. From this analysis it is now clear howeach of the dimensionless groups can be interpreted as the ratio of two forces, and how these

groups arise naturally from the governing equations.

Although we really have not helped ourselves with regard to obtaining an analytical

solution to these equations 1they are still complicated and not amenable to an analyticalsolution2, the dimensionless forms of the equations, Eqs. 7.31, 7.34, and 7.35, can be usedto establish similarity requirements. From these equations it follows that if two systems are

governed by these equations, then the solutions will be

the same if the four parameters and are equal for the two sys-rV/m/tV,p0rV2, V2g/,

1in terms ofu*, v*,p*,x*,y*, and t*2

mrV

/

g/V2p0rV

2

/tV

rV2/

FVFG

Fp FIc FI/

FVFGFP

cp0/

d 0p*0y*

3rg 4 cmV/

2d a 02v*

0x*2

02v*

0y*2b

FIcFI/

c rVt

d 0v*0t*

c rV2/

d au* 0v*0x*

v*0v*

0y*b

c rVt

d 0u*0t*

c rV2/

d au* 0u*0x*

v*0u*

0y*b cp0

/d 0p*

0x* c mV

/2

d a 02u*0x*2

0

2u*

0y*2b

0u*

0x*

0v*

0y* 0

02u

0x2

V

/

0

0x*a 0u*

0x*b0x*

0x

V

/2

02u*

0x*2

430 Chapter 7/ Similitude, Dimensional Analysis, and Modeling

Governing equa-

tions expressed in

terms of dimension-

less variables lead

to the appropriate

dimensionless

groups. (7.35) c p0rV2

d 0p*0y*

c g/V2

d c mrV/

d a 02v*0x*2

0

2v*

0y*2b

c /tV

d 0v*0t*

u*0v*

0x* v*

0v*

0y*

(7.34)c /tV

d 0u*0t*

u*0u*

0x* v*

0u*

0y* c p0

rV2d 0p*0x*

c mrV/

d a 02u*0x*2

0

2u*

0y*2b

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tems. The two systems will be dynamically similar. Of course, boundary and initial condi-

tions expressed in dimensionless form must also be equal for the two systems, and this will

require complete geometric similarity. These are the same similarity requirements that would

be determined by a dimensional analysis if the same variables were considered. However,

the advantage of working with the governing equations is that the variables appear naturallyin the equations, and we do not have to worry about omitting an important one, provided the

governing equations are correctly specified. We can thus use this method to deduce the con-

ditions under which two solutions will be similar even though one of the solutions will most

likely be obtained experimentally.

In the foregoing analysis we have considered a general case in which the flow may be

unsteady, and both the actual pressure level, and the effect of gravity are important. A

reduction in the number of similarity requirements can be achieved if one or more of these

conditions is removed. For example, if the flow is steady the dimensionless group, can

be eliminated.

The actual pressure level will only be of importance if we are concerned with cavita-

tion. If not, the flow patterns and the pressure differences will not depend on the pressure

level. In this case, can be taken as and the Euler number can be eliminated

as a similarity requirement. However, if we are concerned about cavitation 1which will occurin the flow field if the pressure at certain points reaches the vapor pressure, 2, then theactual pressure level is important. Usually, in this case, the characteristic pressure, is

defined relative to the vapor pressure such that where is some reference pres-

sure within the flow field. With defined in this manner, the similarity parameter

becomes This parameter is frequently written as and in this

form, as was noted previously in Section 7.6, is called the cavitation number. Thus we can

conclude that if cavitation is not of concern we do not need a similarity parameter involving

, but if cavitation is to be modeled, then the cavitation number becomes an important

similarity parameter.

The Froude number, which arises because of the inclusion of gravity, is important for

problems in which there is a free surface. Examples of these types of problems include the

study of rivers, flow through hydraulic structures such as spillways, and the drag on ships.

In these situations the shape of the free surface is influenced by gravity, and therefore the

Froude number becomes an important similarity parameter. However, if there are no free sur-

faces, the only effect of gravity is to superimpose a hydrostatic pressure distribution on the

pressure distribution created by the fluid motion. The hydrostatic distribution can be elimi-

nated from the governing equation 1Eq. 7.302by defining a new pressure,and with this change the Froude number does not appear in the nondimensional governing

equations.

We conclude from this discussion that for the steady flow of an incompressible fluid

without free surfaces, dynamic and kinematic similarity will be achieved if1for geometri-cally similar systems2Reynolds number similarity exists. If free surfaces are involved, Froudenumber similarity must also be maintained. For free-surface flows we have tacitly assumed

that surface tension is not important. We would find, however, that if surface tension is in-

cluded, its effect would appear in the free-surface boundary condition, and the Weber num-ber, would become an additional similarity parameter. In addition, if the governing

equations for compressible fluids are considered, the Mach number, would appear as an

additional similarity parameter.

It is clear that all the common dimensionless groups that we previously developed by

using dimensional analysis appear in the governing equations that describe fluid motion when

these equations are expressed in terms of dimensionless variables. Thus, the use of the gov-

erning equations to obtain similarity laws provides an alternative to dimensional analysis.

This approach has the advantage that the variables are known and the assumptions involved

Vc,

rV2/s,

p p rgy,

p0

1 prpv212rV2,1 prpv2rV2.p0rV

2p0

prp0 prpv

p0,

pv

rV2

1or

12rV

2

2,p0

/tV,

p0,

7.10 Similitude Based on Governing Differential Equations 431

The use of govern-

ing equations to ob-

tain similarity laws

provides an alterna-

tive to conventional

dimensional analy-

sis.

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are clearly identified. In addition, a physical interpretation of the various dimensionless groups

can often be obtained.

432 Chapter 7/ Similitude, Dimensional Analysis, and Modeling

1. Bridgman, P. W., Dimensional Analysis, Yale UniversityPress, New Haven, Conn., 1922.

2. Murphy, G., Similitude in Engineering, Ronald Press, NewYork, 1950.

3. Langhaar, H. L.,Dimensional Analysis and Theory of Mod-els, Wiley, New York, 1951.

4. Huntley, H. E.,Dimensional Analysis, Macdonald, London,1952.

5. Duncan, W. J., Physical Similarity and Dimensional Analy-sis: An Elementary Treatise, Edward Arnold, London, 1953.

6. Sedov, K. I., Similarity and Dimensional Methods in Me-

chanics, Academic Press, New York, 1959.7. Ipsen, D. C., Units,Dimensions, and Dimensionless Num-

bers, McGraw-Hill, New York, 1960.

8. Kline, S. J., Similitude and Approximation Theory, Mc-Graw-Hill, New York, 1965.

9. Skoglund, V. J., SimilitudeTheory and Applications, In-ternational Textbook, Scranton, Pa., 1967.

10. Baker, W. E., Westline, P. S., and Dodge, F. T., SimilarityMethods in Engineering DynamicsTheory and Practice of

Scale Modeling, Hayden 1Spartan Books2, Rochelle Park, N.J.,1973.

11. Taylor, E. S.,Dimensional Analysis for Engineers, Claren-don Press, Oxford, 1974.

12. Isaacson, E. de St. Q., and Isaacson, M. de St. Q.,Dimen-sional Methods in Engineering and Physics, Wiley, New York,1975.

13. Schuring, D. J., Scale Models in Engineering, PergamonPress, New York, 1977.

14. Yalin, M. S., Theory of Hydraulic Models, Macmillan, Lon-don, 1971.

15. Sharp, J. J., Hydraulic Modeling, Butterworth, London,1981.

16. Schlichting, H., Boundary-Layer Theory, 7th Ed., Mc-Graw-Hill, New York, 1979.

17. Knapp, R. T., Daily, J. W., and Hammitt, F. G., Cavitation,McGraw-Hill, New York, 1970.

References

Key Words and TopicsIn the E-book, click on any key word

or topic to go to that subject.

Basic dimensions

Buckingham pi theorem

Cauchy number

Correlation of experimental data

Dimensional analysis

Dimensionless group

Dimensionless product

Distorted model

Euler number

Flow around immersed bodies

Flow through closed conduits

Flow with a free surface

Froude number

Mach number

Model

Model design conditions

Modeling laws

Pi term

Prediction equation

Reference dimensions

Repeating variables method

Reynolds number

Scales of models

Similarity requirements

Similitude

Similitude based on differential

equations

Strouhal number

Theory of models

True model

Weber number

Review Problems

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Problems

Note: Unless otherwise indicated, use the values of fluid prop-erties found in the tables on the inside of the front cover. Prob-lems designated with an 1*2are intended to be solved with theaid of a programmable calculator or a computer. Problems des-ignated with a 12are open-ended problems and require crit-ical thinking in that to work them one must make various as-sumptions and provide the necessary data. There is not a uniqueanswer to these problems.

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