principios de flujo turbulento

8/18/2019 Principios de Flujo Turbulento

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SECTION 9.5 TURBULENTBOUNDARYLAYER

finally, beyond that region is the velocity defect region. Each of these velocity zones will bediscussed separa tely.

Viscous Sublayer The zone immediately adjace nt to the wall is a layer of fluid that is essentially laminar because the presence of the wall dampens the cross-stream mixing and turbulentfluctuations. This very thin layer is called the viscous sublayer . 1his thin layer behaves as aCo uette flow introduced in Sect ion 9.1. In the viscous sub layer , T is v irtuall y constant and

equal to the shea r stress at the wall, T0. Thus du dy = Toff..L which on integration yieldsToY

u f L

Dividing the numerator and denominator by p gives

Tu/Pu = y

f L/ p

u YT P- yvTJP v

(9.19)

(9.20)

1hc combination of variabl es v TJP has the dimensions of velocity and recurs again and againin derivations involving boundary-layer theory. t has been given the special name s hearvelocity. Ihe shear velocity (which is also sometimes called frict ion velocity) is symbolized as u .Thus, by definition,

f= p (9.21)Now substituting u. for VTJP in Eq. (9.20), yields the nondimensional velocity distributionin the viscous sublayer:

u yu. v u.

(9.22)

Expe riment al results show that the limit of visco us sublayer occurs when yu /v is approximately 5. Consequen tly, the thickness of the viscous sub ayer, identified by 8' , is given as

8, = vu.

(9.23)

The thickness of the viscous sublayer is very small ( typicall y less than one-tenth the thicknessof a dime). The thickness of the visco us sublayer increases as the wall shear stress decreases inthe downstream direction.

The Logarithmic Velocity Distribution The flow zone outside the viscous sublayer is turbulent; therefore, a complete ly different type of flow is involved. The mixing action of turbulen ce

ca uses small fluid masses to be swept back and forth in a direction transverse to the mean flowdirec tion. A small mass of fluid swept from a low-velocity zone next to the viscous sublayerinto a higher-velocity zone farther out in the stream has a retarding effect on the higher-velo citystream. Similarly, a small mass of fluid that originates farther out in the boundary layer in ahig h v elocity flow zone and is swep t into a region of low velocity has the effec t of acceleratingthe lower -velocity fluid. Altho ugh the process just described is primaril y a momentum exchange phenomenon , it has the same effect as applying a shea r stress to the fluid; thu s in tu rbu lent flow these stresses are termed apparent shear stresses, or Reynolds stresses after theBritish scientis t-engineer who first did extensive research in turbulent flow in the late 1800s.

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8 CHAPTER 9 PREDICTING SHEAR FORCE. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ·········· ······

FIGURE 9 8

Velocity fluctuationsin turbulent flow.

The mixing action of turbulence causes the velocities at a given po int in a flow to fluctuwith time. If one places a velocity-sensing device, such as a hot-wire anemometer , in a turlent flow, one can measure a fluctuating velocity, as illustrated in Fig. 9.8. It is convenienthink of the velocity as composed of two parts: a mean value, u plus a fluctuating part, u'.fluctuating part of the velocity is responsible for the mixing act ion and the momentuexchange, which manifests itself as an apparent shear stress as noted previously. In fact,apparent shear stress is related to the fluctuating part of the velocity by

z:·g r r rT IVIn : r TT- r. . ; :ij>J

3c

]

Ta pp = u v 9

Time

where u and v' refe r to the x andy components of the velocity fluctuations , respectiveand the bar over these terms denotes the product of u' v ' ave raged over a perioof time.* The express ion for apparent shear st ress is n ot ve ry useful in this form,Prandtl developed a theory to relate the apparent shear stress to the tem poral mean velocdis tri bution .

The theory developed by Prandt l is analogous to the idea of mo lecular transport creatinshear stress presented in Chapter 2 ln the turbulent boundar y layer, the principal flow is palel to the boundary. However, because of turbulent eddies, there are fluctuating compon etransverse to the principal flow di rect io n . These fluctuating velocity components are assoated with small masses of fluid, as s h own in Fig. 9.8, that move across the boundary layer.the mass moves from the lower-velocity region to th e higher -velocity region, it tends to retits or iginal velocity. The difference in velocity between the surrounding fluid and the tranported mass is identified as the fluctuating velocity componen t u , For the mass shown ifig. 9.8, u ' would be negative and approximated byt

u = edudy

where du dy is the mean velocity gradient and e s the distance the small fluid mass travels inthe t ransverse direction. Prandtl identified this distance as the mixing length: ' PrandLIsu med that the magnitude of the transverse fluctuating velocity component is proportionalthe mag nitude of th e fluctuating component in the principal flow direction: I I Iwhich seems to be a reasonable assumption because both compo nents arise from the same s

Equation (9.24) c an be derived b y considering the momenrum ex change that results when the transverse componeof urbulent flow passe5 through an area p ~ r a U e tto the x z plane. Or, by including the fluctuating ve locity componenin the Navier-Stokes equations, one can obtain the apparent shear stress terms, one of which is Eq. (9.24). De taiLhese d e r i v t i o n ~appear in Chapter 18 of Schlichting (4).1 for convenience, the bar used to denote time-averaged velocity is deleted.


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SECTION 9 5 TURBULENT BO UNDARY L Y R

of eddies. Also, it shou ld be noted that a positive v' will be associate d with a negat ive u', so theproduct u v will be negative. Thus the apparent shear stress can be expressed as

( du)2

Tapp = - pu v = pez y 9 .25 )

A more general form ofEq. (9 .25) is

_ 2 1dulduTapp - p y y

which ensures tha t the sign for the apparent shear stress is correct.The theory leading to Eq. (9.25) is called Prandtl s mixing-length theory and is used

extensively in analyses involving turbulent flow.* Pran dtl also made the important andclever assumption that the mixing length is proportional to the distance from the wall( = Ky for the region close to the wall. If one considers the velocity distribution in aboundary layer where du dy is positive, as is shown in Fig. 9.9, and s ubstitutes KYfore , thenEq. (9.25) reduces to

J r L .

·· 1 - - - - - - - - - r - - - - - + 1Fluid mas

r - i r u - · b -r u ~_• t - • o _ n ~= ~ ~ ~ x

-u

For the zone of flow near the boundary, it is assumed that the shear stress is uniform andapproximate ly equal to the shear stress at the wall. Thus the foregoing equation becomes

Taking the square root of each side of Eq. (9.26) and rearranging yields

VTJP yu =

y

Integrating the above equation and substitu ting u. for VTJP gives

u 1- = - lny + Cu. K

9.26 )

9 .27 )

*Prandtl published an account of his mixing -length concept in 1925. G. I. Taylor (5) published a similar con cept in1915, but the idea has been traditionall y attributed to Prandtl.

• 0 . . . . .

FIGUR 9 9

o nce pl o f mixing len t h


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34 CHAPTER9 PREDICTING SHEAR FORCE

FIGURE 9 1 0

Velocity distribu tton in aturbulent boundary layer.

Experiments on smooth boundaries indicate that the constant of integration C can be giveterms of u. v and a pure number as

1 vC = 5 . 5 6 - n

K •

When thi s exp ression for Cis substituted into Eq. 9.27), the result is

u 1 yu .- = - In - 5.56U • K V

9

1n Eq. 9.28), K has som etim es been called the univer sal turbul ence constant, or Karman s con sExperiments show that this constant is approximately 0.41 3) for the turbulent wne next toviscous sublayer. Introducing this value forK into Eq. 9.28) gives the logarithmic velocitydistribu

u yu.- = 2.44ln - + 5.56u v 9

Obviously the region where this model is valid is limited because the mixing length cancontinuousl y increase to the boundary layer edge. 1his distr ibution is valid for values of yranging from approximately 30 to 500.

The region between the viscous sublayer and the logarithmic velocity distribut ion is

buffer zone. There isno

equation for the velocity distri bution in thi s zone, although varioempirical expression s have been develop ed 6). However , it is co mmon practice to extrapothe velocity profile for the viscous sub ayer to larger value s o f yu / v and the logar ithmic veloprofile to smaller values of yu./v until the velocity profile s inter sect as shown in Fig. 9.10.

~""8E0

< ::

8c"li-5

";:;

il:

1000

100

11.8410

5

Range o f experimental data

.. .. = 2.44 In - • + 5.56• v

uu

Loganthmicvelocity

distributi on

~uffer

zone

+isco ussublayer

Velocitydefectlawappl ies

Law of

the wall

~ ~ ~0 10 20 30

fJ; rclanvc ve locity)


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SECTION 9 5 TURBULENT BOUNDARY LAY R 34

The intersection occu rs at yu . v = 11.84 and is regarded as the demar cation between the viscoussublayer and the logarithmic profile. The nominal thickness of the viscous sub ayer is

v8 V = 11.84 -

u.9.30)

The combination of the viscous and logarithmic velocity profile for the range of yu ./ vfrom 0 to approximately 500 is called the law of the wall .

Making a semilogarithmic plot of the velocity distribution in a turbulent bound ry layer,as shown in Fig. 9.10, makes it straightforward to identify the velocity distribution in the viscoussublayer and in the region where the logarithmic equation applies . However, the logarithmicnature of this plot accentuates the nondimensional distance yu . /v near the wall. A better perspective of the rela t ive extent of the regions is obtained by plotting the graph on a linear scale,as shown in Fig. 9.11. From this plot one notes that the laminar sublayer nd buffer zone are avery small part of the thickness of the turbulent boundary layer.

700

600

500

400

yu .I '

300 uu. Logarithmicvelocity

0.15 nd yu . / v > 500 the velocity profile correspondingto the law of the wall become s increasingly inadequate to match experimental data, so athird zone , called the velocity defect region, is identified. The velocity in this region is represented by the velocity defect law, which for a flat plate with zero pressure gradien t is simply

expressed as

~ =u. 8

9.31)

and the correlation with experimental data is plotted in Fig. 9.12 . At the edge of the boundarylayer y = 8 nd 0 - u u. = 0 sou = 0 or the free-stream velocity. Thi s law applies torough as well as smooth surface s . However , the functional relationship has to be modified forflows with free-stream pressure gradients .

FIGURE 9 11

Ve locily distribution in oturbulent boundory oy r-lineor sc o le s.


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342 CHAPTER 9 PREDICTIN SHEAR FOR CE

FIGURE 9 12

Velocity defect la wfor boundary layerson flat p late zeropress ure grad ient).[Aher Rouse 6 ) ]

y

0

1.0 - - .

0.8

0.6

Ran ge o f experi me ntal data - -0.4

0.2

20

As shown in Fig. 9.9, the demarcation between the law of the wall and the velocity defect regions i s somewhat arbitrar y so there is considerable overlap b etween the two r egions. The thzones of the turbul ent boundary layer and their range of appl icability are summarized in Table

TABLE 9 2 Zones for Turbulent Boundary layer on Flat Plate

Zone Velocity Distribution

Viscous Sub ayer u yu.u. v

Logarithmic Velocity Distribution u yu .- = 2.44ln - + 5.56u. v

Velocity Defect Law U0 - u = ( ~)I u. BI

Range

yu.0 < - < 11. 84

v

yu.11.84 :S - < 5 0v

500 :S yu. > 0 .15v s

Power Law Formula for Velocity Distribution Anal yses have shown that f or a w ide rangReynolds numbers (10 5 < Re < 10\ the velocity profile in the turbulent bounda ry layer oflat plate i s approximated rea sonabl y by th e pow er l w equ ation

~ = ~) 7 9

Compari sons with experim ental r esult s sh ow that thi s fo rmul a conf orms to tho se re sver y closely o ve r about 90 of the boundar y layer (0.1 < y/8 < 1 . O bviously it is not valithe surface bec ause du) ( dy iy o c o ,which implies infinite surface shear str e ss. For inner 10 of the boundar y layer , one mu st resort to equations for the law of the wall (Fig . 9.10) to obtain a more precise pr ediction of velocity . Beca use Eq. (9.32 ) is valid othe ma jor portion of the boundary la yer , it is used to advant age in deriving the overall thicknesof the boundar y layer as well as other relation s for the turbulent boundar y layer .

principios de flujo turbulento

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