External Flows. Boundary Layer concepts

Henryk Kudela

Contents

1 Introduction

External flows past objects encompass an extremely wide variety of fluid mechanics phenom-ena. Clearly the character of the flow field is a function of theshape of the body. For a given-shaped object,the characteristics of the flow depend very strongly on various parameters suchas size,orientation,speed,and fluid properties. According to dimensional analysis arguments,thecharacter of the flow should depend on the various dimensionless parameters involved. For typicalexternal flows the most important of these parameters are theReynolds numberRe = UL/ν , whereL– is characteristic dimension of the body. For many high-Reynolds-number flows the flow fieldmay be divided into two region

1. a viscous boundary layer adjacent to the surface of the vehicle

2. the essentially inviscid flow outside the boundary layer

W know that fluids adhere the solid walls and they take the solid wall velocity. When the walldoes not move also the velocity of fluid on the wall is zero. In region near the wall the velocityof fluid particles increases from a value of zero at the wall tothe value that corresponds to theexternal ”frictionless” flow outside the boundary layer (see figure).

2 Boundary layer concepts

The concept of the boundary layer was developed by Prandtl in1904. It provides an importantlink between ideal fluid flow and real-fluid flow.

Fluids having relatively small viscosity , the effect of internal friction in a fluid isappreciable only in a narrow region surrounding the fluid boundaries.

Since the fluid at the boundaries has zero velocity, there is asteep velocity gradient from theboundary into the flow. This velocity gradient in a real fluid sets up shear forces near the boundary

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Figure 1: Visualization of the flow around the car. It is visible the thin layer along the bodycause by viscosity of the fluid. The flow outside the narrow regin near the solid boundary can beconsidered as ideal (ivicied).

that reduce the flow speed to that of the boundary.That fluid layer which has had its velocityaffected by the boundary shear is calledthe boundary layer.For smooth upstream boundaries the boundary layer starts out as alaminar boundary layer inwhich the fluid particles move in smooth layers. As the laminar boundary layer increases inthickness, it becomes unstable and finally transforms into aturbulent boundary layer in whichthe fluid particles move in haphazard paths. When the boundary layer has become turbulent, thereis still a very thon layer next to the boundary layer that has laminar motion. It is called thelaminarsublayer.Various definitions of boundary–layer thicknessδ have been suggested. The most basic definition

Figure 2: The development of the boundary layer for flow over aflat plate, and the different flowregimes. The vertical scale has been greatly exaggerated and horizontal scale has been shortened.

refers to the displacement of the main flow due to slowing downod particles in the boundary zone.

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This thicknessδ ∗1,calledthe displacement thickness is expressed by

Uδ1∗ =

∫ δ

0(U −u)dy (1)

Figure 3: Definition of boundary layer thickness:(a) standard boundary layer(u = 99%U ),(b)boundary layer displacement thickness .

The boundary layer thickness is defined also as that distancefrom the plate at which the fluidvelocity is within some arbitrary value of the upstream velocity. Typically, as indicated in figure(??a),δ = y whereu = 0.99U .Another boundary layer characterstic, called asthe boundary layer momentum thickness, Θ as

Θ =∫ δ

0

uU

(1− uU

)dy (2)

All three boundary layer thickness definitionδ , δ ∗1, Θ are use in boundary layer analysis.

3 Scaling analysis

Prandtl obtained the simplified equation of fluid motion inside the boundary layer by scaling anal-ysis called a rrelative order of magnitude analysis. Let as recall the steady equation of motion forlongitudinal component of velocity

u∂u∂x

+ u∂v∂y

= −∂ p∂x

+ ν(

∂ 2u∂x2 +

∂ 2u∂y2

)

(3)

The left terms of the eq. (??) is called as advective term of acceleration. The term proportionalto the viscosity represent viscous forces. At first Prandtl’s boundary layer theory is a applicableif δ ≪ L , it is the thickness of the boundary layer is much smaller that the then streamwise

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(longitudinal) length of body.Let a characteristic magnitude ofu in the flow field beU . Let L be the streamwise distance overwhich u changes appreciably (from 0 toU ). A measure of∂u

∂x is thereforeUL , so that the advective

termu partialu∂x may be estimated

u∂u∂x

∼ U2

L(4)

where∼ is to be interpreted as ”of order”. We can regarded the termU2

L as a measure of the inertialforces. A measure of the viscus term in eq. (??) is

ν∂ 2u∂y2 ∼ νU

δ 2 (5)

The term(∂ 2u∂x2 ∼ U

L2 is much smaller than term(∂ 2u∂y2 ∼ U

L2 may be drops from the equations. Prandtlassumed that within the boundary layer, the viscous forces and inertial forces are the same order.It means that

νU

δ 2 :U2

L=

νLU

(

Lδ

)2

=∼ 1 (6)

Recognizing thatUL/ν = ReL,we see immediately that

δ ∼ L√ReL

(7)

The coefficient of the proportionality, that correspond to the thickness of boundary layer according

Figure 4: An order–of–magnitude analysis of the laminar boundary layer equations along a flatplate revels thatδ grows like

√x

to the definition ofδ asu = 0.99U in eq. (??) is commonly taken as equal to 5. The thickness ofthe boundary layer along the flat plate depend on x and can be calculated form

δ (x) =5x√Rex

, Rex =Uxν

(8)

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The set of equations of the motion for a steady, incompressible laminar boundary layer inxy–plane without significant gravitational effects are

u∂u∂x

+ u∂v∂y

= −∂ p∂x

+ ν∂ 2u∂y2 (9)

0 = −∂ p∂y

(10)

∂u∂x

+∂v∂y

= 0 (11)

Due to fact that in boundary layerv ≪ u y–components equation of momentum reduced to the(??). Equation (??) says that the pressure is approximately uniform (constant) across the boundarylayer. The pressure at the surface is therefore equal to thatat the edge of the boundary layer wecan apply the Bernoulli equation to the outer flow region. Differentiating with respect to x theBernoulli equation we obtain

pρ

+12

U2 = const → 1ρ

∂ p∂x

= −UdUdx

The equations (??) and (??) are used to determineu andv in boundary layer. The boundarylayer conditions are

u(x,0) = 0 (12)

v(x,0) = 0 (13)

u(x0,y) = uin(y) (14)

4 Momentum Equation Applied to the Boundary Layer

In order to calculate boundary layers approximately, we often use methods where the equationsof motion are not satisfied everywhere in the field but only in integral means across the thicknessof the boundary layer. The starting point for these integralmethods is usually the momentumequation which can be derived by applying the continuity equation and the balance of momentumin its integral form. Let us applied the balance of momentum to the flow over the flat plate. Thecontrol volume was chosen as it is shown in figure (??). Pay attention that due to action of theboundary layer the streamline 2 is displaced from the solid wall. There is no mass flow throughthe streamline 2. The equation of the linear momentum is

∫

Sρu(u ·n)dS =

∫

StdS = −FD (15)

The pressure is assumed uniform, and so it has no net force on the plate. Evaluating the integralin on left side of the eq. (??) on obtain

−FD = ρ∫

1u(u ·ndS+ρ

∫

3u(u ·ndS = ρ

∫ h

0U0(−U0)b dy+ρ

∫ δ

0uub dy =−ρU2

0 bh+ρb∫ δ

0dy

(16)

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Figure 5: Analysis of the drag force on a flat plate due to boundary share by applying the linearmomentum principle.

or

FD = ρU20 bh−ρb

∫ δ

0dy (17)

whereb is a width of the plate. Now using the integral form of continuity equation (conservationof mass = the flow rate through section (1) must equal that through section (2)) we obtain

ρb∫ h

0vektu ·n dy = bUρh = ρ

∫ δ

0u dy (18)

Introduce the value ofh to (??) we obtain

FD = ρb∫ δ

0u(U −u) dy|x=L (19)

The development of Eq.(??and its use was first derived by Theodore von Karman in 1921. Bycomparing Eqs. (??) and (??) we see that the drag can be written in terms of the momentumthicknessΘ, as

FD = ρbU2Θ (20)

Momentum thicknessΘ measure of total plate drag. But the integral wall shear stress along theplate gives also the drag force

FD(x) = b∫ x

0τw(x) dx, and

dFD

dx= bτw (21)

Meanwhile, the derivative of eq. (??), with U=contstant, is

dFD

dx= ρbU2 dΘ

dx= bτw (22)

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Equation

ρU2dΘdx

= τ (23)

ix called themomentum-integral relation for the flat–plate boundary layer flow.The equation (?? one can also write in non– dimensional form

dΘdx

=τ

ρU2 =12

C f (x) (24)

whereτw/ρU2/2 is defined as a skin friction coefficient,C f . The point is to use eqn. (??) todetermineu/U when we do not have an exact solution. To do this, we guess the solution in theform u/U = f (y/δ ). This guess is made in such a way that it will fit the following four thingsthat are true of the velocity profile (boundary values on the solid wall, y = 0, and on the edge ofboudary layer,y = δ ):

u = 0 for y = 0 (on the solid boundary) (25)

u = U at y = δ (on the edge of boundary layer) (26)

dudy

= 0 at y = δ (27)

d2udy2 = 0 at y = 0 (28)

If f (y/δ ) is written as a polynomial with four constantsa, b ,c andd

uU

= a+ byδ

+ c( y

δ

)2+ d

( yδ

)3(29)

the four bove condtion about teh velocity porfile give

• 0 = a, which eliminatea immiediatly

• 1 = b+ c+ d

• 0 = b+2c+3d

• 0 = 2c

Solving the middke two above equations forb andd we obaind = −1/2 andb = 3/2, so

uU

=32

yδ− 1

2

( yδ

)3(30)

This makes possible to estimate both momentum thicness and wall shear

Θ =∫ δ

0

(

32

yδ− 1

2

( yδ

)3)(

1− 32

yδ

+12

( yδ

)3)

dy = δ39280

(31)

τw = µ∂u∂y

|y=0 =32

µUδ

(32)

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By substituting (??) into (??) and rearranging we obtain

δdδ =14013

νU

dx (33)

whereν = µ/ρ . We can integrate from 0 intox, assuming thatδ = 0 atx = 0

δ 2 =28013

νxU

(34)

orδx

=4.64√

Rex(35)

This is the desired thickness estimate. It is accurate, being 5.6% smaller than the known exactsolution for laminar flat–plate flow (δ/x = 5/

√Rex)

we can also obtain a shear–stress, skin–friction coefficient

C f =2τ2

ρU=

34.64

1√Rex

=0.6466√

Rex, Rex =

Uxν

(36)

Exact laminar–plate–solution isC f = 0.664/Re1/2.

During this course I will be used the following books:

References

[1] F. M. White, 1999.Fluid Mechanics, McGraw-Hill.

[2] B. R. Munson, D.F Young and T. H. Okiisshi, 1998.Fundamentals of Fluid Mechanics, JohnWiley and Sons, Inc. .

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