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TRANSCRIPT
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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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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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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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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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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
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