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Lecture 16

Boundary Layer Flows (1)

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Lecture 16 Boundary Layer Flows (1)

Contents

16.1 Fluid Flow Past Solid Boundaries

16.2 Boundary Layer Concept

Objectives

- Study flow phenomena near a solid boundary

- Understand the concept of boundary layers

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Navier-Stokes Eq.

16.1 Fluid Flow Past Solid Boundaries

2

2

2

2

2

21

z

u

y

u

x

u

x

p

z

uw

y

uv

x

uu

t

u

2

2

2

2

2

21

z

v

y

v

x

v

y

p

z

vw

y

vv

x

vu

t

v

gz

w

y

w

x

w

z

p

z

ww

y

wv

x

wu

t

w

2

2

2

2

2

21

0

z

w

y

v

x

u

- Continuity Eq.

Viscous force

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Solution of Navier-Stokes Eq.

1) Navier-Stokes 방정식에서점성항무시 → Euler 방정식 → Bernoulli 공식

2) Potential flow 로해석

• d’Alembert’s paradox:

- 점성력의영향을무시해서는안되는영역이있음

- 고체경계면부근에서는점성계수의값이작더라도속도경사의크기가매우크기때

문에전단력이커짐.

16.1 Fluid Flow Past Solid Boundaries

du

dy

- 경계층에서는점성력의영향이크며에너지의손실에직접영향을미침

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Boundary layer theory (Prandtl, 1904)

- 평판이일정한유속 U 로움직이는유체속에있을때이평판이유체흐름에미치는

영향해석

- 경계층은시작점에서작으나뒤로갈수록커짐

- U 가커질수록경계층의두께는얇아짐

- 경계층은매우얇으며따라서경계층내의압력은밖의압력과같음 → 미지수감소

- 층류경계층에이어하류구간에난류경계층이형성됨

16.1 Fluid Flow Past Solid Boundaries

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16.1 Fluid Flow Past Solid Boundaries

• Flow phenomena near a solid boundary is of great value in engineering

problems.

- external flows: flow around an object immersed in the fluid

(over a wing or a flat plate, fluid force on vehicles)

- internal flows: flow between solid boundaries

(flow in pipes and channels)

16.1.1 Flow around an object immersed in the fluid

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16.1 Fluid Flow Past Solid Boundaries

- Flow over a wing

- Fluid force (drag and lift) on vehicles

- Design of building

- Vortex around the cylinder

Engineering problems of external flows

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16.1 Fluid Flow Past Solid Boundaries

1. Laminar flow over smooth or rough boundaries

- possesses essentially the same properties

- the velocity is being zero at the boundary surface → no slip

- the shear stress throughout the flow is given as

- surface roughness has no effect on the flow

as long as the roughness are small relative to

the flow cross section size.

- viscous effects dominates the whole flow

du

dy (16.1)

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16.1 Fluid Flow Past Solid Boundaries

2. Turbulent flow

- Roughness of the boundary will affect the fluid motion

1) Flow over a smooth boundary

- Flow is always separated from the boundary by a sublayer of viscosity-

dominated flow (laminar sublayer).

[Re] Laminar sublayer

In a region very close to the boundary, the available

mixing length is reduced to zero (i.e., the turbulence

is completely extinguished).

→ A film of viscous flow over the boundary results

in.

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16.1 Fluid Flow Past Solid Boundaries

dv

dy

2

2 dvl

dy

- Shear stress:

Inside the viscous sublayer:

Outside the viscous sublayer:

- Between the turbulent region and the viscous sublayer lies a transition

zone in which shear stress results from a complex combination of both

turbulent and viscous action.

- The thickness of the viscous sublayer varies with time because the

sublayer flow is unsteady.

(16.2)

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External flow:

potential flow

Boundary layer

flow: turbulent

rotational flow

Intermittent

nature

16.1 Fluid Flow Past Solid Boundaries

Laminar sub

layer

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16.1 Fluid Flow Past Solid Boundaries

ev

- Roughness of the boundary surface affects the physical properties

(velocity, shear, friction) of the fluid motion.

- The effect of the roughness is dependent on the relative size of

roughness ( ) and viscous sublayer ( ).

2) Flow over a rough boundary

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16.1 Fluid Flow Past Solid Boundaries

Classification of boundary surfaces

0.3v

e

i) Smooth surface:

- Roughness projections are completely submerged in viscous sublayer.

→ They have no effect on the turbulence.

- Classification of surfaces based on ratio of absolute roughness to viscous

sublayer thickness.

0.3 10v

e

10v

e

ii) Transition:

iii) Rough surface:

(16.3a)

(16.3b)

(16.3c)

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16.1 Fluid Flow Past Solid Boundaries

- However, the thickness of the viscous sublayer depends on certain

properties of the flow.

→ The same boundary surface behave as a smooth one or a rough one

depending on the size of the Reynolds number and of the viscous sublayer.

1v

e

fR

) roughsurfacee v

i V R

) smoothsurfacee v

ii V R

(16.4)

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16.1 Fluid Flow Past Solid Boundaries

At the entrance to a pipe, viscous effects begin their influence to lead a

growth of the boundary layer.

• Unestablished flow zone:

~ is dominated by the growth of boundary layers accompanied by diminishing

core of irrotational fluid at the center of the pipe.

Irrotational flow Rotational flow

16.1.2 Flow between solid boundaries

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16.1 Fluid Flow Past Solid Boundaries

• Established flow zone:

~ Influence of wall friction is felt throughout the flow field.

~ There is no further changes in the velocity profiles.

~ Flow is everywhere rotational.

2100e

VdR

- Flow in a boundary layer may be laminar if , or

turbulent if Re ≥ 4,000.

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16.1 Fluid Flow Past Solid Boundaries

1) Laminar boundary layer

x

2,100100

20 20

ex R

d

length of unestablished flow zone

Thus, x < 100 d → unestablished flow

(16.5)

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16.1 Fluid Flow Past Solid Boundaries

1) Turbulent boundary layer

- The boundary layer may be laminar at the upstream end of the

unestablished flow zone, followed by turbulent boundary layers in the

downstream portion of this region.

- In the turbulent boundary layer zone, there exists the viscous sublayer near

the solid boundary.

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Potentia

l core Boundary

layer

- Laminar flow in pipes → parabolic profile (Poiseuille flow)

- Turbulent flow in pipes → logarithmic profile

16.1 Fluid Flow Past Solid Boundaries

• Velocity profile at the established flow zone

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• Prandtl (1904) suggested the boundary layer concept to explain the

resistance of streamlined bodies, and flat plates parallel to the flow.

- The frictional effects of the flow are confined to the boundary layer.

- Outside boundary layer, the flow is frictionless (irrotational, potential).

16.2.1 Introduction

16.2 Boundary Layer Concept

Outer zone:

potential flow

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Laminar

boundary layer

flow

Separation

occurs at the

crest.

Turbulent

boundary layer

flow

16.2 Boundary Layer Concept

External flow:

potential flow

Boundary layer

flow: turbulent

rotational flow

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• Mechanism of boundary layer growth on a smooth flat plate

- Boundary layer flow: laminar → transition → turbulent

- Velocity of the fluid particle at the body wall is zero.

- Velocity gradient in the vicinity of the boundary is very high.

- Large frictional (shear) stresses in the boundary layer slow down

successive fluid elements.

- Boundary layers steadily thicken downstream along the body.

16.2 Boundary Layer Concept

u

y

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Division of boundary layers

0x

V xR

0VR

500,000x

R

3,900R

(16.6)

1) Laminar boundary layer

(16.7)

16.2 Boundary Layer Concept

2) Turbulent boundary layer

500,000x

R

3,900R

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Growth of the boundary layer

- The boundary layer starts with zero thickness at the leading edge of the

plate where the viscous action begins.

- The boundary layer steadily increases in thickness as increased viscous

action extends into the flow and slows down more and more fluid.

- At the end of the laminar boundary layer, instabilities in the boundary layer

produce breakdown of the laminar structure of the flow and cause

turbulence to begin.

- At the onset of turbulence, the transition boundary layer thickens rapidly,

developing into a turbulent boundary layer.

- A thin laminar sublayer exists between the solid surface and the turbulent

zone in the boundary layer.

16.2 Boundary Layer Concept

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16.2.2 Boundary layer theory

16.2 Boundary Layer Concept

- For low viscosity and high acceleration of the fluid motion, significant

viscous shear occurs only within a relatively thin layer next to the boundary.

→ Boundary layer flow (inner flow)

- Outside the layer (outer flow), the flow will suffer only a minor influence of

the viscous forces. → Ideal fluid theory

- At the outer flow zone, flow will be determined primarily by the relation

among inertia, pressure gradient, and body forces. → potential flow

(irrotational flow)

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16.2 Boundary Layer Concept

1) Solve first the outer problem of ideal fluid motion ignoring viscous effects

entirely

2) Using outer solution values of velocity and pressure for the upper edge of

the boundary layer, the inner viscous flow problem is solved.

- Outer solution gives the pressure distribution, lift force of immersed body

- Inner solution gives an estimate of the friction force or drag on the shape

Engineering approach to solution of boundary layer flow

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16.2 Boundary Layer Concept

- In the boundary layer, because of the relative thinness, u is much greater

than v.

, ,u u v v

y x x y

(16.8)

- As a consequence of the velocity behavior in the boundary layer, the

pressure variation in the y direction is negligible.

0,p p dp

y x dx

(16.9)