DS s Lecture Boundary Layer4

8/12/2019 DS s Lecture Boundary Layer4

1/73

BOUNDARY LAYER

FRICTION FORCE)


2/73

Implementations of momentumtransfer, heat transfer and masstransfer principles

Momentumtransfer

Heat transferoperations(conduction,convection,radiation)

Mass transfer operations(distillation,absorption,extraction,humidification)

Fluid mechanics (momentumbalance, flow inconduits, pumpsand compressors,boundary layer,particulate flow(fixed bed, fluidisedbed, entrainedbed))


3/73

Applications in ChemicalEngineering


4/73

Higher flow induces higher velocity gradient close to the wall .Higher du/dy higher shear stress higher pump power incase of flow in pipe. ; Higher du/dy higher shear stress

higher convective heat transfer (h) in case of HT from inside tooutside the pipe


5/73

Boundary layer inside the pipe

Development of fully developed laminar and turbulentflow in circular pipe (a) laminar flow (b) turbulent flow(after Douglas et. al, 2001)


6/73

Effect of boundary layer on the friction coefficient inside the pipe


7/73

Q hot Q cold

Th Ti,wall

To,wall

Tc

Region I : Hot Liquid-

Solid ConvectionNEWTONS LAW OFCCOLING

. . x h h iwdq h T T dA

Region II : Conduction Across Copper Wall

FOURIERS LAW

. x

dT dq k

dr

Region III: Solid Cold LiquidConvection

NEWTONS LAWOF CCOLING

. . x c ow cdq h T T dA

Thermal boundary layerEnergy moves from hotfluid to a surface byconvection, throughthe wall by conduction,and then by convectionfrom the surface to thecold fluid.


8/73

7.1. Reynolds-Number andGeometry Effects

The technique of boundary-layer (BL) analysis canbe used to compute viscous effects near solid walls and to patch (to join) these onto the outer inviscid

motion.In Fig. 7.1 a uniform stream U moves parallel to asharp flat plate of length L. If the Reynolds numberUL/ is low (Fig. 7.1 a ), the viscous region is very

broad and extends far ahead and to the sides of theplate due to retardation of the oncoming streamgreatly by the plate.


9/73

Fig. 7.1 Comparison of flow past a sharp flat plate at low and highReynolds numbers: ( a ) laminar, low-Re flow; ( b) high-Re flow


10/73

At a high-Reynolds-number flow (Fig. 7.1b) theviscous layers , either laminar or turbulent, are verythin , thinner even than the drawing shows.We define the boundary layer thickness as thelocus of points where the velocity u parallel to theplate reaches 99 % of the external velocity U .

As we shall see in Sec. 7.4, the accepted formulasfor flat-plate flow are


11/73

where Re x = Ux / is called the local Reynolds number of the flow along the plate surface. The turbulent-flowformula applies for Re x > approximately 10 6.

The blanks indicate that the formula is not applicable.In all cases these boundary layers are so thin thattheir displacement effect on the outer inviscid layer isnegligible.Thus the pressure distribution along the plate can becomputed from inviscid theory as if the boundary layer

were not even there.


12/73

This external pressure field then drives theboundary-layer flow, acting as a forcing function inthe momentum equation along the surface.

For slender bodies , such as plates and airfoilsparallel to the oncoming stream, the assumption ofnegligible interaction between the boundary layerand the outer pressure is an excellent approximation

because pressure along plate is constant .For a blunt-body (bluff body) flow , however , there is apressure distribution over the surface body, so thepressure must be taken into account.


13/73

Figure 7.2 shows two sketches of flow past a two-or three-dimensional blunt body.In the idealized sketch (7.2a), there is a thin film ofboundary layer about the body and a narrow sheetof viscous wake in the rear.In a actual flow (Fig. 7.2 b), the boundary layer is

thin on the front , or windward, side of the body,where the pressure decreases along the surface (favorable pressure gradient ).


14/73

.

Fig. 7.2 Illustration of the strong interaction betweenviscous and inviscid regions in the rear of blunt bodyflow: ( a ) idealized and definitely false picture ofblunt- body flow (according to Bernoullis law); ( b)actual picture of blunt body flow.


15/73

But in the rear the boundary layer encountersincreasing pressure (adverse pressure gradient )and breaks off, or separates, into a broad, pulsating

wake.The mainstream is deflected by this wake , so thatthe external flow is quite different from theprediction from inviscid theory with the addition of a

thin boundary layer


16/73


17/73


18/73

7 2 von Karmans Momentum -Integral Estimates

A boundary layer of unknown thickness growsalong the sharp flat plate in Fig. 7.3.The no-slip wall condition retards the flow, makingit into a rounded profile u(y ), which merges into theexternal velocity U constant at a thickness y ( x ).By utilizing the control volume of Fig. 7.3, we found

(without making any assumptions about laminarversus turbulent flow) that the drag force on theplate is given by the momentum integral across theexit plane due to the change of velocity from U to

u(x,y)


19/73

Fig. 7.3 Growth of a boundary layer on a flat plate.

Drag force = V x mass rate = ((U u ) x b (width) x u x y).This represents how much the momentum is lost due to friction


20/73

where b is the plate width into the paper and theintegration is carried out along a vertical plane x constant (von Krmn, 1921).

Imagine that the flow remains at uniform velocityU , but the surface of the plate is moved upwards to reduce momentum flux = momentum flux lossboundary layer actually does. Then, the drag force

Momentum thickness is defined as the loss ofmomentum per unit width divided by U 2 due to the

presence of the growing boundary layer.


21/73

By comparing this with Eq. (7.4) Krmn arrived atwhat is now called the momentum integral relation forflat-plate boundary-layer flow ( w )

It is valid for either laminar or turbulent flat-plate flow .

(von Krmn, 1921).

Vertical variable

horisontal variable


22/73

Variables inmomentumintegral relation(7.5)

Local, vertical-variable

Local, horisontalvariable


23/73

Eq. 7.9 is the desired thickness estimate . It is allapproximate, of course, part of Krmns momentum-integral theory [7], but it is startlinglyaccurate, being only 10 percent higher than theknown exact solution for laminar flat-plate flow,which we gave as Eq. (7.1 a ).By combining Eqs. (7.9) and (7.7) we also obtain a

shear-stress estimate along the plate


24/73

Again this estimate , in spite of the crudeness of theprofile assumption (7.6) is only 10 % > the knownexact laminar-plate-flow solution c f = 0.664/ Re x 1/2 ,

treated in Sec. 7.4.The dimensionless quantity c f , called the skin-frictioncoefficient , is analogous to the friction factor f inducts.

A boundary layer can be judged as thin if, say, theratio / x < about 0.1 . / x = 0.1 = 5.0/ Re x 1/2 or Re x >2500.


25/73

For Re x < 2500 we can estimate that boundary-layer theory fails because the thick layer has asignificant effect on the outer inviscid flow

(thickness creates pressure distribution across theboundary layer ).The upper limit on Re x for laminar flow is about 3 x10 6, where measurements on a smooth flat plate

[8] show that the flow undergoes transition to aturbulent boundary layer.From 3 x 10 6 upward the turbulent Reynoldsnumber may be arbitrarily large, and a practical

limit at present is 5 x 109

for oil supertankers.


26/73

Displacement Thickness

Another interesting effect of a boundary layer is itssmall but finite displacement of the outer streamlines.

As shown in Fig. 7.4, outer streamlines must deflectoutward a distance *( x ) to satisfy conservation ofmass between the inlet and outlet as a result offluid entrainment from fluid flow to boundary layer.

The quantity * is called the displacement thickness ofthe boundary layer.To relate it to u(y ), cancel and b from Eq. (7.11),evaluate the left integral, and add and subtract U from

the right integrand:


27/73

Bernoullieq. applies

Navier Stokes

eq. applies

Fig. 7.4 Displacement effect of a boundarylayer. Fluid entrainment occurs from free fluidflow to the boundary layer, so mass rate at 0 =mass rate at 1

Hypothetical layer attributes to fluidentrainment

1

Fluidentrainment


28/73

Imagine that the flow remains atuniform velocity U , but the surface ofthe plate (wall) is moved upwards(displaced) * to reduce mass flux =mass flux loss of the main flow the BLgenerates.

0

*U w U u dy w

Local, horisontalvariable


29/73

Introducing von Karmans profile approximation (7.6)into (7.12), we obtain by integration the approximateresult

These estimates are only 6% away from the exactsolutions for laminar flat-plate flow given in Sec. 7.4:

* = 0.344 = 1.721 x /Re x 1/2

. Since *


30/73

The usefulness of this interpretation of displacementthickness becomes obvious if we consider uniform flowentering a channel bounded by two parallel walls. As theboundary layers grow on the upper and lower walls , theirrotational core flow must accelerate to satisfyconservation of mass .From the point of view of the core flow between theboundary layers, the boundary layers cause the channelwalls to appear to converge - the apparent distancebetween the walls decreases as x increases.


31/73

In this area, we can assumethat the flow is inviscid.

This layer is viscous

If we assume that massdisplacement is small for certainaxial distance in the inviscid area,then, mass balance and Bernoulli

equation may apply


32/73

EXAMPLE 7.2 Are low-speed, small-scale air and water boundarylayers really thin? Consider flow at U = 1 ft/s past aflat plate 1 ft long . Compute the boundary-layerthickness at the trailing edge for (a ) air and (b)water at 20C.


33/73

(Re L > 2500, then BL is thin)


34/73

7.3 Prandtls Boundary-LayerEquations based on Navier-

Stokes equations)Derivation for Two-Dimensional Flow

We consider only steady two-dimensionalincompressible viscous flow with the x directionalong the wall and y normal to the wall, as in Fig.7.3. We neglect gravity, which is important only inboundary layers where fluid buoyancy is dominant.From Chap. 4, the complete equations of motionconsist of continuity and the x- and y -momentumrelations


35/73

These should be solved for u, v, and p subject to

typical no-slip, inlet, and exit boundary conditions,but in fact they are too difficult to handle for mostexternal flows.

v


36/73

In 1904 Prandtl deduced that a shear layer mustbe very thin if the Reynolds number is large , sothat the following approximations apply:

Applying these approximations to Eq. (7.14 c)results in a powerful simplification .


37/73

In other words, the y-momentum equation can beneglected entirely , and the pressure varies onlyalong the boundary layer , not through it (not in y).The pressure-gradient term in Eq. (7.14b) isassumed to be known from Bernoullis equationapplied to the outer inviscid flow (general case)

Meanwhile, one term in Eq. (7.14 b) is negligibledue to Eqs. (7.15).


38/73

However, neither term in the continuity relation(7.14 a) can be neglected - another warning thatcontinuity is always a vital part of any fluid-flowanalysis.

The net result is that the three full equations ofmotion (7.14) are reduced to Prandtls two boundary-layer equations


39/73

These are to be solved for u(x, y) and v(x, y), withU(x) assumed to be a known function from theouter inviscid-flow (Bernoulli) analysis .There are two boundary conditions on u and oneon v :


40/73

Unlike the Navier-Stokes equations (7.14), whichare mathematically elliptic and must be solvedsimultaneously over the entire flow field, theboundary-layer equations (7.19) aremathematically parabolic and are solved bybeginning at the leading edge and goingdownstream as far as you like, stopping at theseparation point or earlier if you prefer.


41/73

7.4 The Flat-Plate BoundaryLayer: Blasiuss Laminar flow

The classic and most often used solution ofboundary-layer theory is for flat-plate flow, as inFig. 7.3, which can represent either laminar orturbulent flow.For laminar flow past the plate , the boundary-layer equations (7.19) can be solved exactly for uand v, assuming that the free-stream velocity U isconstant (d U/dx = 0).


42/73

The solution was given by Prandtls student

Blasius, in 1908.With a coordinate transformation, Blasius showedexperimentally (exactly) that the dimensionlessvelocity profile u/U is a function only of the single

composite dimensionless variable (y)[U/( x)] 1/2

:

The boundary conditions (7.20) become

This is the Blasius equation.


43/73


44/73

Some tabulated values of the velocity-profileshape f( )= u/U are given in Table 7.1.


45/73

Since u/U 1.0 only as y infinity , it iscustomary to select the boundary layer thicknessat that point where u/U = 0.99. From the table, thisoccurs at 5.0:

With the profile known, Blasius, of course, couldalso compute the wall shear and displacementthickness. The slope at y = 0 is

0 332

/

.

d u U

d


46/73

Notice how close these are to Karmans integralestimates , Eqs. (7.9), (7.10), and (7.13). When c f is converted to dimensional form, we have

The wall shear drops off with x 1/2

because ofboundary-layer growth and varies as velocity U tothe 1.5 power. This is in contrast to laminar pipeflow, where w is proportional to U and is

independent of x.


47/73

If w (x) is substituted into Eq. (7.4), we computethe total drag force

The drag increases only as x 1/2 . The nondimensional drag coefficient is defined as

.


48/73

Thus, for laminar plate flow, C D = 2 x the value ofthe skin-friction coefficient at the trailing edge. Thisis the drag on one side of the plate.

Krmn pointed out that the drag could also becomputed from the momentum relation (7.2). Indimensionless form, Eq. (7.2) becomes

This can be rewritten in terms of the momentumthickness at the trailing edge (at x = L )


49/73


50/73

If we plot the Blasius velocity profile from Table 7.1in the form of u/U vs y/ , we can see why thesimple integral-theory guess from von Karman, Eq.(7.6), was such a great success. This is done inFig. 7.5.

The simple parabolic approximation is not far fromtrue Blasiuss profile (based on experiments) ;hence its momentum thickness is within 10 percentof the true value.


51/73


52/73

EXAMPLE 7.3 A sharp flat plate with L =1 m and b = 3 m isimmersed parallel to a stream of velocity 2 m/s .Find the drag on one side of the plate, and at thetrailing edge find the thicknesses , *, and for(a) air, =1.23 kg/m 3 and =1.46x10 -5 m 2/s , and(b) water, =1000 kg/m 3 and =1.02 x 10 -6 m 2/s.


53/73

Part a.


54/73

Part b.


55/73

The drag is 215 x more for water in spite of thehigher Reynolds number and lower drag coefficientbecause water is 57 x more viscous and 813 xdenser than air.From Eq. (7.26), in laminar flow, it should have(57) 1/2 (813) 1/2 = 7.53(28.5) = 215 x more drag.


56/73

The water layer is 3.8 times thinner than the airlayer, which reflects the square root of the 14.3ratio of air to water kinematic viscosity. ( 14.3 = 3.8 )

7 4 The Flat Plate Boundary


57/73

7.4 The Flat-Plate BoundaryLayer: Turbulent flow Karman

+Prandtl) We begin with Eq. (7.5), which is valid for laminaror turbulent flow.

From the definition of c f , Eq. (7.10), this can berewritten as, which is valid for laminar or turbulent

flow.

Going back to Fig. 6.9, we see that flat-plate flowis very nearly logarithmic , with a slight outer profile

and a thin viscous sublayer.


58/73


59/73

Therefore, just as in turbulent pipe flow, weassume that the logarithmic law (Eq. 6.21) holds allacross the boundary layer

with, as usual, = 0.41 and B = 5.0. At the outer

edge of the boundary layer , y = and u = U, andEq. (7.34) becomes


60/73

By relating c f and w according to Eq 7.10 andsubstituting definition of u* , then

Substituting Eq 7.36 to Eq. (7.35) is results in forturbulent flat-plate flow

It is a complicated law , but we can at least solvefor a few values and list them:


61/73

Following a suggestion of Prandtl , convert complexlog friction law (7.37) to a simple power-lawapproximation

This we shall use as the left-hand side of Eq. (7.33).For the right-hand side of (7.33), we need anestimate for ( x) in terms of (x).


62/73

Following another suggestion of Prandtl,approximate turbulent profiles in Fig. 7.5 to a 1/7-power law

This is shown as a dashed line in Fig. 7.5.With this simple approximation, the momentumthickness (7.28) can easily be evaluated:

In principle, by knowing u/U = f(y/ ), = f( ) canbe found using momentum integral relation .


63/73

Substitute Eqs. (7.38) and (7.40) into Krmns momentum law (7.33)

Separate the variables and integrate, assuming = 0 at x = 0:

Thus the thickness of a turbulent boundary layerincreases as x 6/7 , >> the laminar increase as x 1/2 .


64/73

Equation (7.42) is the solution to the problem ,because all other parameters are now available. Forexample, combining Eqs. (7.42) and (7.38), we

obtain the friction variation

Writing this out in dimensional form, we have

Turbulent plate friction drops slowly with x, increases

nearly as and U 2

, and is rather insensitive to .


65/73

We can evaluate the drag coefficient from Eq.(7.29)

Then C D is only 16 % > the trailing-edge skinfriction (at x=L) , c f (L) [in laminar boundary layer C D is 100% > c f (L) , see Eq. (7.27)].

The displacement thickness can be estimated fromthe one-seventh-power law and Eq. (7.12):


66/73

The turbulent flat-plate shape factor isapproximately

Figure 7.6 shows flat-plate drag coefficients for bothlaminar-and turbulent-flow conditions . The smooth-wall relations (7.27) and (7.45) are shown, alongwith the effect of wall roughness, which is quitestrong.The proper roughness parameter here is x/ or L/ ,by analogy with the pipe parameter /d.


67/73


68/73

In the fully rough regime , C D is independent of theReynolds number , so that the drag varies exactlyas U 2 and is independent of .

Schlichting [1] gives a curve fit for skin friction anddrag in the fully rough regime :


69/73

Equation (7.48 b) is plotted to the right of thedashed line in Fig. 7.6.The figure also shows the behavior of the drag

coefficient in the transition region 5x105

< Re L < 8x10 7, where the laminar drag at the leading edge isan appreciable fraction of the total drag.Schlichting [1] suggests the following curve fits for

these transition drag curves depending upon theReynolds number Re trans where transition begins:


70/73

EXAMPLE 7.4 A hydrofoil 1.2 ft long and 6 ft wide is placed in awater flow of 40 ft/s, with = 1.99 slugs/ft 3 and =0.000011 ft 2/s.(a) Estimate the boundary-layer thickness at theend of the plate.Estimate the friction drag for (b) turbulent smooth-

wall flow from the leading edge, (c) laminarturbulent flow with Re trans = 5 x 10 5, and (d)turbulent rough-wall flow with = 0.0004 ft.


71/73


72/73


73/73

Documents

DS s Lecture Boundary Layer4