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CHAPTER 9: EXTERNAL* INCOMPRESSIBLE VISCOUS FLOWS *(unbounded)

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Page 1: CHAPTER 9: EXTERNAL* INCOMPRESSIBLE VISCOUS FLOWS *(unbounded)

CHAPTER 9: EXTERNAL* INCOMPRESSIBLE VISCOUS FLOWS

*(unbounded)

Page 2: CHAPTER 9: EXTERNAL* INCOMPRESSIBLE VISCOUS FLOWS *(unbounded)

CHAPTER 9: EXTERNAL INCOMPRESSIBLE VISCOUS FLOWS

adverse pressure gradient

leads to separationdifficult to use theory

thicker

•Re = Ux/; Re = Uc/; …•laminar and turbulent boundary layers• displaced inviscid outer flow• adverse pressure gradient and separation

= separated bdy layer

Page 3: CHAPTER 9: EXTERNAL* INCOMPRESSIBLE VISCOUS FLOWS *(unbounded)

Boundary Layer Provides Missing Link Between Theory and Practice

Boundary layer, , where viscous stresses (i.e. velocity gradient) are important we’ll define as where u(x,y) = 0 to 0.99 U above boundary.

Page 4: CHAPTER 9: EXTERNAL* INCOMPRESSIBLE VISCOUS FLOWS *(unbounded)

In August of 1904 Ludwig Prandtl, a 29-year old professor presenteda remarkable paper at the 3rd International Mathematical Congress in Heidelberg. Although initially largely ignored, by the 1920s and 1930s the powerful ideas of that paper helped create modern fluid dynamics out of ancient hydraulics and 19th-century hydrodynamics.

(only 8 pages long, but arguably one of the most important fluid-dynamics papers ever written)

Page 5: CHAPTER 9: EXTERNAL* INCOMPRESSIBLE VISCOUS FLOWS *(unbounded)

• Prandtl assumed no slip condition• Prandtl assumed thin boundary layer region where shear forces are

important because of large velocity gradient• Prandtl assumed inviscid external flow• Prandtl assumed boundary so thin that within it p/y 0; v << u

and /x << /y • Prandtl outer flow drives boundary layer boundary layer can greatly

effect outer outer “inviscid” flow if separates

Page 6: CHAPTER 9: EXTERNAL* INCOMPRESSIBLE VISCOUS FLOWS *(unbounded)

Theodore von Karman

- 1904 PrandtlFluid Motion with Very Small Friction2-D boundary layer equations

- 1908 BlasiusThe Boundary Layers in Fluids with Little FrictionSolution for laminar, 0-pressure gradient flow

- 1921 von Karman Integral form of boundary layer equations

- 1924 Sir Horace LambHydrodynamics ~ one paragraph on bdy layers

- 1932 Sir Horace LambHydrodynamics ~ entire section on bdy layers

BOUNDARY LAYER HISTORY

Page 7: CHAPTER 9: EXTERNAL* INCOMPRESSIBLE VISCOUS FLOWS *(unbounded)

INTERNAL EXTERNAL

FULLY DEVELOPED?

CAN BE NEVER

WAKE? NEVER USUALLY - PLATE IS EXCEPTION

THEORY

LAMINAR

PIPES, DUCTS,.. FLAT PLATE & ZERO PRESSURE GRADIENT

GROWING BOUNDARY

LAYER?

NOT WHEN

FULLY

DEVELOPED

ALWAYS

ADVERSE PRESSURE

GRADIENT

PIPE/DUCT=N0

DIFFUSER=YES

PLATE=MAYBE

BODIES=USUALLY

TURBULENT

EXPERIMENT

PIPE (EXAMPLE)

u(r)/Uc/l = (y/R)1/n

PLATE (EXAMPLE)

u(y)/Uo = (y/)1/n

Page 8: CHAPTER 9: EXTERNAL* INCOMPRESSIBLE VISCOUS FLOWS *(unbounded)

Note – throughout figures theboundary layer thickness*,,

is greatly exaggerated!(disturbance layer*)

Airline industry had to develop flat face rivets.

Page 9: CHAPTER 9: EXTERNAL* INCOMPRESSIBLE VISCOUS FLOWS *(unbounded)

Re = 20,000

Angle of attack = 6o

Symmetric Airfoil

16% thick

Page 10: CHAPTER 9: EXTERNAL* INCOMPRESSIBLE VISCOUS FLOWS *(unbounded)

Flat Plate (no pressure gradient)~ what is velocity profile?~ wall shear stress/drag?~ displacement of free stream?~ laminar vs turbulent flow?

Immersed Bodies~ wall shear stress/drag?~ lift?~ minimize wake

Page 11: CHAPTER 9: EXTERNAL* INCOMPRESSIBLE VISCOUS FLOWS *(unbounded)

breath

Page 12: CHAPTER 9: EXTERNAL* INCOMPRESSIBLE VISCOUS FLOWS *(unbounded)

FLAT PLATE – ZERO PRESSURE GRADIENT

Page 13: CHAPTER 9: EXTERNAL* INCOMPRESSIBLE VISCOUS FLOWS *(unbounded)

Laminar Flow/x ~ 5.0/Rex

1/2

THEORY

Turbulent Flow Rextransition > 500,000u(y)/U = (y/)1/7

/x ~ 0.382/Rex1/5

EXPERIMENTAL

Page 14: CHAPTER 9: EXTERNAL* INCOMPRESSIBLE VISCOUS FLOWS *(unbounded)

“At these Rex numbersbdy layers so thin that displacement effect onouter inviscid layer is small”

No simple theory for Re < 1000;(can’t assume

is thin)

Page 15: CHAPTER 9: EXTERNAL* INCOMPRESSIBLE VISCOUS FLOWS *(unbounded)

ReL = 10,000 Visualization is by air bubbles see that boundary+ layer, , is thin and that outer free stream is displaced, *, very little.

FLAT PLATE – ZERO PRESSURE GRADIENT

+ Disturbance Thickness, (x) (pg 412); boundary layer thickness, (x) (pg 415)

outside (x), U is constant so P is constant

u(x,y) is not constant, (x) is thin so assume P inside (x) is impressed from the outside

Page 16: CHAPTER 9: EXTERNAL* INCOMPRESSIBLE VISCOUS FLOWS *(unbounded)

ReL = 10,000 Visualization is by air bubbles see that boundary+ layer, , is thin and that outer free stream is displaced, *, very little.

FLAT PLATE – ZERO PRESSURE GRADIENT

+ Disturbance Thickness, (x) (pg 412); boundary layer thickness, (x) (pg 415)

Rex = Ux/Assume Rextransition ~ 500,000

x

ReL = Ux/

L

Page 17: CHAPTER 9: EXTERNAL* INCOMPRESSIBLE VISCOUS FLOWS *(unbounded)

SIMPLIFYING ASSUMPTIONS OFTEN MADE FOR ENGINERING ANALYSIS OF BOUNDARY LAYER FLOWS

Page 18: CHAPTER 9: EXTERNAL* INCOMPRESSIBLE VISCOUS FLOWS *(unbounded)

Development of laminar boundary layer (0.01% salt water, free stream velocity 0.6 cm/s, thickness

of the plate 0.5 mm, hydrogen bubble method).

** * * *

Rex 1000

Page 19: CHAPTER 9: EXTERNAL* INCOMPRESSIBLE VISCOUS FLOWS *(unbounded)

breath

Page 20: CHAPTER 9: EXTERNAL* INCOMPRESSIBLE VISCOUS FLOWS *(unbounded)

FLAT PLATE – ZERO PRESSURE GRADIENT: (x)

BOUNDARY OR DISTURBANCE LAYER

Page 21: CHAPTER 9: EXTERNAL* INCOMPRESSIBLE VISCOUS FLOWS *(unbounded)

(x) *

BOUNDARY OR DISTURBANCE LAYER

Page 22: CHAPTER 9: EXTERNAL* INCOMPRESSIBLE VISCOUS FLOWS *(unbounded)

Definition: u(x,) = 0.99 of U=U=Ue

(within 1 % of U)

Boundary+ Layer Thickness (x)

+Disturbance

Page 23: CHAPTER 9: EXTERNAL* INCOMPRESSIBLE VISCOUS FLOWS *(unbounded)

is at y location where u(x,y) = 0.99 U

Because the change in u in the boundary layer takes place asymptotically, there is some indefiniteness in determining exactly.

Page 24: CHAPTER 9: EXTERNAL* INCOMPRESSIBLE VISCOUS FLOWS *(unbounded)

NOTE: boundary layer is much thicker in turbulent flow.

Blasius showed theoretically for laminar flow that /x = 5/(Rex)1/2

(Rex = Ux/) x1/2

Experimentally found*for turbulent flow that

x4/5

Page 25: CHAPTER 9: EXTERNAL* INCOMPRESSIBLE VISCOUS FLOWS *(unbounded)

NOTE: velocity gradient at wall (w = du/dy) is significantly greater.

At same x: U/L > U /T At same x: wL < wT

Page 26: CHAPTER 9: EXTERNAL* INCOMPRESSIBLE VISCOUS FLOWS *(unbounded)

Note, boundary layer is not a streamline!

From theory (Blasius 1908, student of Prandtl):

= 5x/(Rex1/2) = 5x/(U/[x])1/2 = 51/2x1/2/U1/2

d/dx = 5 (/U)1/2 (½) x-1/2 = 2.5/(Rex)1/2

V/U = dy/dxstreamline = 0.84/(Rex1/2)

dy/dxstreamline d/dx so not streamline

Page 27: CHAPTER 9: EXTERNAL* INCOMPRESSIBLE VISCOUS FLOWS *(unbounded)

Behavior of a fluid particle traveling along a streamline through a boundary layer along a flat plate.

U = U = Ue

Page 28: CHAPTER 9: EXTERNAL* INCOMPRESSIBLE VISCOUS FLOWS *(unbounded)

ASIDE ~ LAMINAR TO TURBULENT TRANSITION

Page 29: CHAPTER 9: EXTERNAL* INCOMPRESSIBLE VISCOUS FLOWS *(unbounded)

LAMINAR TO TURBULENT TRANSITION

Page 30: CHAPTER 9: EXTERNAL* INCOMPRESSIBLE VISCOUS FLOWS *(unbounded)

Emmons spot ~ Rex = 200,000Spots grow approximately linearly downstream at downstream

speed that is a fraction of the free stream velocity.

NOTE: Turbulence is not initiated at Retr all along the width of the plate

Page 31: CHAPTER 9: EXTERNAL* INCOMPRESSIBLE VISCOUS FLOWS *(unbounded)

Emmons spot, Rex = 400,000 smoke in wind tunnel

Page 32: CHAPTER 9: EXTERNAL* INCOMPRESSIBLE VISCOUS FLOWS *(unbounded)

Transition not fixed but usually around Rex ~ 500,000 (2x105-3x106, MYO)

For air at standard conditions and U = 30 m/s, xtr ~ 0.24 m

x=0

Turbulent boundary layer is thicker and grows faster.

Page 33: CHAPTER 9: EXTERNAL* INCOMPRESSIBLE VISCOUS FLOWS *(unbounded)

Nevertheless: Treat transition as it happens all along Rex = 500,000

Experimentally transition occurs around Rex ~ 5 x 105

Water moving around 4 m/s past a ship, transitions after about 0.14 m from the bow,

representing only about 0.1 % of total length of 143 m long ship.

Page 34: CHAPTER 9: EXTERNAL* INCOMPRESSIBLE VISCOUS FLOWS *(unbounded)

breath

Page 35: CHAPTER 9: EXTERNAL* INCOMPRESSIBLE VISCOUS FLOWS *(unbounded)

FLAT PLATE – ZERO PRESSURE GRADIENT: *(x)

DISPLACEMENT THICKNESS

Page 36: CHAPTER 9: EXTERNAL* INCOMPRESSIBLE VISCOUS FLOWS *(unbounded)

*(x)

DISPLACEMENT THICKNESS

Page 37: CHAPTER 9: EXTERNAL* INCOMPRESSIBLE VISCOUS FLOWS *(unbounded)

Definition: * = 0 (1 – u/U)dy

Displacement Thickness *(x)

* is displacement of outer streamlines due to boundary layer

Page 38: CHAPTER 9: EXTERNAL* INCOMPRESSIBLE VISCOUS FLOWS *(unbounded)

By definition, no flow passes through streamline, so mass through 0 to h at x = 0is the same as through 0 to h + * at x = L.

x = L

Displacement thickness *

Page 39: CHAPTER 9: EXTERNAL* INCOMPRESSIBLE VISCOUS FLOWS *(unbounded)

Uh = 0h+* udy = 0h+* (U + u - U)dy

Uh = 0h+* Udy + 0h+* (u - U)dy

Uh = U(h + *) + 0h+*(u - U)dy

-U* = 0h+*(u -U)dy

* = 0h+*(– u/U + 1)dy 0 (1 – u/U)dy

Page 40: CHAPTER 9: EXTERNAL* INCOMPRESSIBLE VISCOUS FLOWS *(unbounded)

* 0 (1 – u/U)dy 0

(1 – u/U)dy

displacement of outerstreamlines due to (x)

function of x!

Page 41: CHAPTER 9: EXTERNAL* INCOMPRESSIBLE VISCOUS FLOWS *(unbounded)

breath

Page 42: CHAPTER 9: EXTERNAL* INCOMPRESSIBLE VISCOUS FLOWS *(unbounded)

Definition: * = 0 (1 – u/U)dy

U*w = 0 (U – u)dyw 0 (U – u)dyw

the deficit in mass flux through area w due to the presence of the boundary layer.

=mass flux passing through an area [*w] in the absence

of a boundary layer

Displacement Thickness *

Page 43: CHAPTER 9: EXTERNAL* INCOMPRESSIBLE VISCOUS FLOWS *(unbounded)

Definition: * = 0 (1 – u/U)dy

U* = 0 (U – u)dy 0 (U – u[y])dy

Displacement Thickness *

Page 44: CHAPTER 9: EXTERNAL* INCOMPRESSIBLE VISCOUS FLOWS *(unbounded)

* problem

Page 45: CHAPTER 9: EXTERNAL* INCOMPRESSIBLE VISCOUS FLOWS *(unbounded)

Displacement Thickness, *, Problem

Suppose given velocity profile:u = 0 from y = 0 to u = Ue for y > .

Show that * =

=

Page 46: CHAPTER 9: EXTERNAL* INCOMPRESSIBLE VISCOUS FLOWS *(unbounded)

Displacement Thickness * Suppose u = 0 from y = 0 to and u = Ue for y > .

Show that * = * = 0 (1 – u/Ue)dy = 0 (1 – 0/Ue)dy + (1 – Ue/Ue)dy

* = 0 dy =

=

Page 47: CHAPTER 9: EXTERNAL* INCOMPRESSIBLE VISCOUS FLOWS *(unbounded)

* = Distance that an equivalent inviscid flow is displaced from a solid boundary as a consequence of

slow moving fluid in the boundary layer.

*(x) ~ 1/3 (x)

Blasius* = 1.721x/(Rex)1/2

Laminar flow on flat plate in uniform free stream

Blasius = 5x/(Rex1/2)

Page 48: CHAPTER 9: EXTERNAL* INCOMPRESSIBLE VISCOUS FLOWS *(unbounded)

Displacement Thickness *

Definition: * = 0 (1 – u/Ue)dy From the point of view of the flow outside the boundary layer, * can be interpreted as the distance that the presence of the boundary layer appears to “displace” the flow outward (hence its name). To the external flow, this streamline displacement also looks like a slight thickening of the body shape.

and *are greatly magnified

*(x)(x)

Page 49: CHAPTER 9: EXTERNAL* INCOMPRESSIBLE VISCOUS FLOWS *(unbounded)

breath

Page 50: CHAPTER 9: EXTERNAL* INCOMPRESSIBLE VISCOUS FLOWS *(unbounded)

*(x) EXAMPLE

Page 51: CHAPTER 9: EXTERNAL* INCOMPRESSIBLE VISCOUS FLOWS *(unbounded)

2-D DUCT

PROBLEM: Find Ue as a function of U, D and *

Ue(x)

Ue(x) = ?

Page 52: CHAPTER 9: EXTERNAL* INCOMPRESSIBLE VISCOUS FLOWS *(unbounded)

Continuity equation (per W):

{UD = -D2D/2 udy + -D2D/2 Uedy - -D2D/2 Uedy}

UD = -D2D/2 Uedy – {-D2D/2 (Ue-u)dy}

u(y)U f(y) Ue(x)

Page 53: CHAPTER 9: EXTERNAL* INCOMPRESSIBLE VISCOUS FLOWS *(unbounded)

Continuity equation (per W):

UD = -D2D/2 Uedy – {-D2D/2 (Ue-u)dy}

UD =

-D2D/2 Uedy–{(-D/2) (-D/2+) (Ue-u)dy+(D/2- ) D/2 (Ue-u)dy}

(used approximation that outside boundary layer u =Ue)(note that Ue = function of x but not y)

INLET OF DUCT

Page 54: CHAPTER 9: EXTERNAL* INCOMPRESSIBLE VISCOUS FLOWS *(unbounded)

Continuity equation (per W):

UD = -D2D/2 Uedy – {(-D/2) (-D/2+) (Ue-u)dy + (D/2- ) D/2 (Ue-u)dy}

UD = -D2D/2 Uedy – Ue{(-D/2) (-D/2+) (1-u/Ue)dy}

+ Ue{(D/2- ) D/2 (1-u/Ue)dy}

INLET OF DUCT*

*

* = 0 (1 –u/Ue)dy 0 (1 – u/Ue)dy

Page 55: CHAPTER 9: EXTERNAL* INCOMPRESSIBLE VISCOUS FLOWS *(unbounded)

INLET OF DUCT*

*

Continuity equation (per W):

UD = -D2D/2 Uedy–Ue{(-D/2) (-D/2+) (1-u/Ue)dy+(D/2- ) D/2 (1-u/Ue)dy}

-{ * + *}

UD= UeD – 2Ue* = Ue[D – 2*]

Ue(x) = UD/[D – 2*(x)] QED

Page 56: CHAPTER 9: EXTERNAL* INCOMPRESSIBLE VISCOUS FLOWS *(unbounded)

from Continuity Equation

UD = Ue [D – 2*]

We see that the free stream velocity in the duct is given by the effective decrease in the cross

sectional area due to the growth of the boundary layers, and this decrease in area is measured by the displacement thickness!!!

Page 57: CHAPTER 9: EXTERNAL* INCOMPRESSIBLE VISCOUS FLOWS *(unbounded)

FLAT PLATE – ZERO PRESSURE GRADIENT: (x)

MOMENTUM THICKNESS

Page 58: CHAPTER 9: EXTERNAL* INCOMPRESSIBLE VISCOUS FLOWS *(unbounded)

* (x)Momentum Thickness

MOMENTUM THICKNESS

Page 59: CHAPTER 9: EXTERNAL* INCOMPRESSIBLE VISCOUS FLOWS *(unbounded)

Definition:

= 0 u/U(1 – u/U)dy

Momentum Thickness

(x)

How define drag due to boundary layer?

Page 60: CHAPTER 9: EXTERNAL* INCOMPRESSIBLE VISCOUS FLOWS *(unbounded)

Momentum Thickness

Definition: = 0 u/Ue(1 – u/Ue)dy

Ue2

w = 0 u(Ue – u)dyw

The momentum thickness, , is defined as the thickness of a layer of fluid, with velocity Ue, for which the momentum flux is equal to the

deficit of momentum flux through the boundary layer. (Mass flux through = 0 udyw)

Page 61: CHAPTER 9: EXTERNAL* INCOMPRESSIBLE VISCOUS FLOWS *(unbounded)

u

U[(u)([U-u])]xdy

[Flux of momentum deficit]

UU w

Page 62: CHAPTER 9: EXTERNAL* INCOMPRESSIBLE VISCOUS FLOWS *(unbounded)

= 0.99U (within 1 %) * = 0(1 – u/Ue)dy

0(1 – u/Ue)dy =0 u/Ue(1 – u/Ue)dy

0 u/Ue(1 – u/Ue)dy* & Easier to calculate form data

and more physical significancebut “can be” dependent on

Summary:

Page 63: CHAPTER 9: EXTERNAL* INCOMPRESSIBLE VISCOUS FLOWS *(unbounded)

breath

Relate to drag

Page 64: CHAPTER 9: EXTERNAL* INCOMPRESSIBLE VISCOUS FLOWS *(unbounded)

Control volume analysis of drag force on a flat plate due to boundary shear

Relate to Drag

Page 65: CHAPTER 9: EXTERNAL* INCOMPRESSIBLE VISCOUS FLOWS *(unbounded)

Relate to Drag

PRESSURE IS UNIFORM - NO NET PRESSURE FORCE!- NO DU/DX!

Page 66: CHAPTER 9: EXTERNAL* INCOMPRESSIBLE VISCOUS FLOWS *(unbounded)

1. From (0,0) to (0,h): V•n = -Uo = -Ue; V = Ue

2. From (0,h) to (L,): streamline, V•n = 0 (no shear)3. From (L, ) to (L,0): V•n = u(x=L,y) 4. From (L, 0) to (0,0): V•n = 0 (shear)

Fx = -Don CV = (d/dt) udVol + u(V•n)dA

D

0(NO BODY FORCES) (STEADY)

Page 67: CHAPTER 9: EXTERNAL* INCOMPRESSIBLE VISCOUS FLOWS *(unbounded)

Fx = -D = u(V•n)dA

u(V•n)dy(w) = 0

h Uo(-Uo)wdy + 0 u(L,y) u(L,y)wdy

1. From (0,0) to (0,h): V•n = -Uo = -Ue; V = Ue

2. From (0,h) to (L,): no shear, V•n = 03. From (L, ) to (L,0): V•n = u(x=L,y); V = u(x,y) 4. From (L, 0) to (0,0): V•n = 0

Page 68: CHAPTER 9: EXTERNAL* INCOMPRESSIBLE VISCOUS FLOWS *(unbounded)

-D = - Uo2wh + 0 u(L,y)2wdy

Not so useful because don’t know h as a function of (x)

Page 69: CHAPTER 9: EXTERNAL* INCOMPRESSIBLE VISCOUS FLOWS *(unbounded)

-D = - Uo2wh + 0 u(x,y)2wdy

Get rid of h by using conservation of mass.

(V•n)dA= 0 0Uowdy + 0u(x,y)wdy = 0

Uoh = 0u(x,y)dy

-D = - Uow 0u(x,y)dy + 0 u(x,y)2wdy

h

Page 70: CHAPTER 9: EXTERNAL* INCOMPRESSIBLE VISCOUS FLOWS *(unbounded)

-D = - Uow 0(x)u(y)dy + 0

(x) u(x,y)2wdy

D = w [0(x)Uou(x,y)dy - u(x,y)2dy]

D = w [0(x) u(x,y) [Uo - u(x,y)]dy

D w [0 u(x,y) [Uo - u(x,y)]dy

Uo2w [0

[u(x,y)/Uo][1o – [u(x,y)/Uo]]dy

D(x) = Uo2w (x)

(First derived by Von Karman in 1921)

Page 71: CHAPTER 9: EXTERNAL* INCOMPRESSIBLE VISCOUS FLOWS *(unbounded)

breath

Relate to wall shear stress

Page 72: CHAPTER 9: EXTERNAL* INCOMPRESSIBLE VISCOUS FLOWS *(unbounded)

D(x) = Uo2w (x)

D(x) = 0x wwdx = Uo

2 w (x)

w(x) = d/dx {Uo2 (x)}

No pressure gradient; dU/dx = 0

w(x)/ = Uo2 d/dx {(x)}

The shear stress at the wall is proportional to the rate of change of momentum thickness with distance.

(Cf =w(x)/[(1/2) Uo2)])

Page 73: CHAPTER 9: EXTERNAL* INCOMPRESSIBLE VISCOUS FLOWS *(unbounded)

Book ~ derivation includes pressure gradient casew(x)/ = d/dx { Uo(x)

2 (x)} *Uo(x)dUo(x)/dx

NOTE

Class ~ derivation for constant pressure casew(x)/ = Uo

2 d/dx {(x)}

Page 74: CHAPTER 9: EXTERNAL* INCOMPRESSIBLE VISCOUS FLOWS *(unbounded)

w(x)/ = Uo2 d/dx {(x)}

w(x)/ = d/dx { Uo(x) 2 (x)} *Uo(x)dUo(x)/dx

book

me

Uo = constant; dP/dx = 0

Uo constantdP/dx 0

Page 75: CHAPTER 9: EXTERNAL* INCOMPRESSIBLE VISCOUS FLOWS *(unbounded)

breath

Estimating (x), *(x), (x), Drag

For laminar / turbulent / and pressure gradient flows

Page 76: CHAPTER 9: EXTERNAL* INCOMPRESSIBLE VISCOUS FLOWS *(unbounded)

w(x)/ = Uo2 d/dx {(x)}

w(x)/ = d/dx { Ue(x) 2 (x)} *Ue(x)dUe(x)/dx

book

me

Page 77: CHAPTER 9: EXTERNAL* INCOMPRESSIBLE VISCOUS FLOWS *(unbounded)

w(x)/ = Uo2 d/dx {(x)}

No pressure gradient; dU/dx = 0

w(x)/ = Uo2 d/dx {0 u/Ue(1 – u/Ue)dy

u(x)/U is assumed to be similar for all x and is normally specified as a function of y/(x)

Defining = y/

w(x)/=Uo2d/dx{ 01 u/Ue(1– u/Ue)d}

w(x)/=Uo2d/dx{01 u/Ue(1– u/Ue)d}

Page 78: CHAPTER 9: EXTERNAL* INCOMPRESSIBLE VISCOUS FLOWS *(unbounded)

Dimensionless velocity profile for a laminar boundary layer:comparison with experiments by Liepmann, NACA Rept. 890,1943. Adapted from F.M. White, Viscous Flow, McGraw-Hill, 1991

y/

Page 79: CHAPTER 9: EXTERNAL* INCOMPRESSIBLE VISCOUS FLOWS *(unbounded)

turbulent bdy layer velocity profileU = 10 m/s; x1=10cm

00.020.040.060.080.10.120.140.16

600 650 700 750 800 850 900 950 1000 1050

u (cm/sec)

y (cm

)

turbulent bdy layer velocity profileU = 10 m/s; x1=100cm

0

0.2

0.4

0.6

0.8

1

600 650 700 750 800 850 900 950 1000 1050

u (cm/sec)

y (cm

) 0

0.2

0.4

0.6

0.8

1

1.2

0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 1.05

Turbulent bdy layer* ~ u/Ue = (y/)1/7; Ue = 1000 cm/s

y/(x)

u(x)/U

* du/dy|wall =

Page 80: CHAPTER 9: EXTERNAL* INCOMPRESSIBLE VISCOUS FLOWS *(unbounded)

Blasius developed an exact solution (but numerical integration was necessary) for laminar flow with no pressure variation.

Blasius could theoretically predict boundary layer thickness (x), velocity profile u(x,y)/U vs y/, and wall shear stress w(x).

Von Karman and Poulhausen derived momentum integral equation (approximation) which can be

used for both laminar (with and without pressure gradient) and

turbulent flow

Page 81: CHAPTER 9: EXTERNAL* INCOMPRESSIBLE VISCOUS FLOWS *(unbounded)

Von Karman and Polhausen method (MOMENTUM INTEGRAL EQ. Section 9-4)

devised a simplified method by satisfying only the boundary

conditions of the boundary layer flow rather than satisfying Prandtl’s

differential equations for each andevery particle within the boundary layer.

Page 82: CHAPTER 9: EXTERNAL* INCOMPRESSIBLE VISCOUS FLOWS *(unbounded)
Page 83: CHAPTER 9: EXTERNAL* INCOMPRESSIBLE VISCOUS FLOWS *(unbounded)

breath

Page 84: CHAPTER 9: EXTERNAL* INCOMPRESSIBLE VISCOUS FLOWS *(unbounded)

QUESTIONS

Page 85: CHAPTER 9: EXTERNAL* INCOMPRESSIBLE VISCOUS FLOWS *(unbounded)

? In laminar flow along a plate, (x), (x), *(x) andw(x):(a) Continually decreases(b) Continually increases(c) Stays the same

? In turbulent flow along a plate, (x), (x), *(x) andw(x):(a) Continually decreases(b) Continually increases(c) Stays the same

?At transition from laminar to turbulent flow, (x), (x), *(x) andw(x):(a) Abruptly decreases(b) Abruptly increases(c) Stays the same

Page 86: CHAPTER 9: EXTERNAL* INCOMPRESSIBLE VISCOUS FLOWS *(unbounded)

forced

natural

6:1 ellipsoid

wall

Turbulent

Laminar

TurbulentLaminar

Page 87: CHAPTER 9: EXTERNAL* INCOMPRESSIBLE VISCOUS FLOWS *(unbounded)

What is wrong with this figure?

Page 88: CHAPTER 9: EXTERNAL* INCOMPRESSIBLE VISCOUS FLOWS *(unbounded)

What is wrong with this figure?

Page 89: CHAPTER 9: EXTERNAL* INCOMPRESSIBLE VISCOUS FLOWS *(unbounded)

Are Antarctic Icebergs Towable Arctic News Record – Summer 1984; 36

Cf = 0.074/Re1/5 (Fox:Cf = 0.0594/Re1/5); Area = 1 km long x 0.5 km wide

sea water = 1030 kg/m3; sea water = 1.5 x 10-6 m2/secPower available = 10 kW; Maximum speed = ?

Page 90: CHAPTER 9: EXTERNAL* INCOMPRESSIBLE VISCOUS FLOWS *(unbounded)

P = UD104 = U Cf½ U2A = U [0.074 1/5/(U1/5L1/5)]( ½ U2A)

104 = U14/5 [0.074(1.5x10-6)1/5/10001/5] x [½ (1030)(1000)(500)]

U14/5 = 0.03054

U = 0.288 m/s

Rex ~ 3x108

Page 91: CHAPTER 9: EXTERNAL* INCOMPRESSIBLE VISCOUS FLOWS *(unbounded)

80% of fresh water found in world – Antarctic ice

Biggest problem not meltingbut is = ?

Page 92: CHAPTER 9: EXTERNAL* INCOMPRESSIBLE VISCOUS FLOWS *(unbounded)

THE END

Page 93: CHAPTER 9: EXTERNAL* INCOMPRESSIBLE VISCOUS FLOWS *(unbounded)

extra “stuff”

Page 94: CHAPTER 9: EXTERNAL* INCOMPRESSIBLE VISCOUS FLOWS *(unbounded)

breath

Derivation

Page 95: CHAPTER 9: EXTERNAL* INCOMPRESSIBLE VISCOUS FLOWS *(unbounded)

Blasius developed an exact solution (but numerical integration was necessary) for laminar flow with no pressure variation.

Blasius could theoretically predict boundary layer thickness (x), velocity profile u(x,y)/U vs y/, and wall shear stress w(x).

Von Karman and Poulhausen derived momentum integral equation (approximation) which can be

used for both laminar (with and without pressure gradient) and

turbulent flow

Page 96: CHAPTER 9: EXTERNAL* INCOMPRESSIBLE VISCOUS FLOWS *(unbounded)

MOMENTUM INTEGRAL EQUATIONdP/dx is not a constant!

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Deriving: MOMENTUM INTEGRAL EQ

so can calculate (x), w.

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u(x,y)

Surface Mass Flux Through Side ab

w

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Surface Mass Flux Through Side cd

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Surface Mass Flux Through Side bc

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Surface Mass Flux Through Side bc

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Assumption : (1) steady (3) no body forces

Apply x-component of momentum eq. to differential control volume abcd

u

Page 103: CHAPTER 9: EXTERNAL* INCOMPRESSIBLE VISCOUS FLOWS *(unbounded)

mf represents x-component of momentum fluxFsx will be composed of shear force on boundaryand pressure forces on other sides of c.v.

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X-momentumFlux =

cvuVdA

Surface Momentum Flux Through Side ab

uw

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X-momentumFlux =

cvuVdA

Surface Momentum Flux Through Side cd

u

Page 106: CHAPTER 9: EXTERNAL* INCOMPRESSIBLE VISCOUS FLOWS *(unbounded)

U=Ue=U

X-momentumFlux =

cvuVdA

Surface Momentum Flux Through Side bc

u

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a-b c-d

c-d b-c

X-Momentum Flux Through Control Surface

Page 108: CHAPTER 9: EXTERNAL* INCOMPRESSIBLE VISCOUS FLOWS *(unbounded)

IN SUMMARYX-Momentum Equation

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X-Force on Control Surface

w is unit width into pagep(x)

Surface x-Force On Side bc

Note that p f(y)inside bdy layer w

Page 110: CHAPTER 9: EXTERNAL* INCOMPRESSIBLE VISCOUS FLOWS *(unbounded)

w is unit width into pagep(x+dx)

Surface x-Force On Side cd

w

Page 111: CHAPTER 9: EXTERNAL* INCOMPRESSIBLE VISCOUS FLOWS *(unbounded)

Note that since the velocity gradient goes to zero at the top of the boundary layer, then viscous shears go to zero there (bc).

p + ½ (dp/dx)dx

Surface x-Force On Side bc

Page 112: CHAPTER 9: EXTERNAL* INCOMPRESSIBLE VISCOUS FLOWS *(unbounded)

Why w and not w (bc)?

pressure

Page 113: CHAPTER 9: EXTERNAL* INCOMPRESSIBLE VISCOUS FLOWS *(unbounded)

Surface x-Force On Side bc

p + ½ dp/dx along bc

In x-direction: [p + ½ (dp/dx)] wd

Page 114: CHAPTER 9: EXTERNAL* INCOMPRESSIBLE VISCOUS FLOWS *(unbounded)

-(w + ½ dw/dx]xdx)wdx

Surface x-Force On Side ad

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Fx = Fab + Fcd + Fbc + Fad

Fx = pw -(p + [dp/dx]x dx) w( + d) + (p + ½ [dp/dx]xdx)wd - (w + ½ (dw/dx)xdx)wdx

Fx = pw -(p w + p wd + [dp/dx]x dx) w + [dp/dx]x dx w d) + (p wd + ½ [dp/dx]xdxwd) - (w + ½ (dw/dx)xdx)wdx

d <<

=

dw << w

p(x)

** ++

#

#

Page 116: CHAPTER 9: EXTERNAL* INCOMPRESSIBLE VISCOUS FLOWS *(unbounded)

=

Page 117: CHAPTER 9: EXTERNAL* INCOMPRESSIBLE VISCOUS FLOWS *(unbounded)

ad = 0

ab -cd bc

U

Page 118: CHAPTER 9: EXTERNAL* INCOMPRESSIBLE VISCOUS FLOWS *(unbounded)

Divide by wdx

dp/dx = -UdU/dx for inviscid flow outside bdy layer

= from 0 to of dy

-(dp/dx) = -(0dy) (-UdU/dx) = (dU/dx)(0

Udy)

Page 119: CHAPTER 9: EXTERNAL* INCOMPRESSIBLE VISCOUS FLOWS *(unbounded)

dp/dx = -UdU/dx for inviscid flow outside bdy layer

= from 0 to of dy

p + ½ U2 + gz = constant outside boundary layer (“inviscid”)

dp/dx + UdU/dx = 0 outside bdy layer (dz = 0)

Page 120: CHAPTER 9: EXTERNAL* INCOMPRESSIBLE VISCOUS FLOWS *(unbounded)

Integration by parts

Multiply by U2/U2 Multiply by U/U

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QED

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FLAT PLATE – ZERO PRESSURE GRADIENTLAMINAR FLOW

Blasius Theoretical Solution

for u(x,y), (x), *(x), (x)Background

Page 123: CHAPTER 9: EXTERNAL* INCOMPRESSIBLE VISCOUS FLOWS *(unbounded)

Blasius* (1908) developed an exact solution for laminar flow over a flat plate with no pressure variation. Blasius could

theoretically predict: (x), *(x), (x),

velocity profile u(x,y)/U vs y/,and wall shear stress w(x).

*(first graduate student of Prandtl)

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Blasius* (1908) developed an exact solution but required numerical integration,

not good if there is a pressure gradient and not good for turbulent flow

hence we develop the momentum integral equation.

Page 125: CHAPTER 9: EXTERNAL* INCOMPRESSIBLE VISCOUS FLOWS *(unbounded)

Newton’s Law: F = mau(u/x) + v(u/y) = -(1/) p/x + (2u/x2 + 2u/y2)u(v/x) + v(v/y) = -(1/) p/y + (2v/x2 + 2v/y2)

Cons. of Massu/x + v/y = 0

Boundary Conditions: u = v = 0 at y = 0

Ue outside boundary layer

2-D Governing Equations

Blasius Solution for Laminar, dP/dx=dUe/dx=0 Flow

Page 126: CHAPTER 9: EXTERNAL* INCOMPRESSIBLE VISCOUS FLOWS *(unbounded)

v << u; d/dx << d/dy Conservation of Mass

u/x + v/y = 0

Prandtl made some simplifying assumptions, based on the premise that the boundary layer is very thin (v and small compared to U and L) so 2-D equations over flat

plate more tractable:

Blasius Solution for Laminar, dP/dx=dUe/dx=0 Flow

Page 127: CHAPTER 9: EXTERNAL* INCOMPRESSIBLE VISCOUS FLOWS *(unbounded)

v << u; d/dx << d/dyConservation of Y-Momentum

u(v/x) + v(v/y) = -(1/) p/y + (2v/x2 + 2v/y2)

p/y 0

Prandtl made some simplifying assumptions, based on the premise that the boundary layer is very thin (v and small compared to U and L) so 2-D equations over flat

plate more tractable:

Pressure can vary only along boundary layer not through it!

Blasius Solution for Laminar, dP/dx=dUe/dx=0 Flow

Page 128: CHAPTER 9: EXTERNAL* INCOMPRESSIBLE VISCOUS FLOWS *(unbounded)

v << u; d/dx << d/dyConservation of X-Momentum

u(u/x) + v(u/y) = -(1/) p/x + (2u/x2 + 2u/y2)p/y 0

u(u/x) + v(u/y) = -(1/) dp/dx + (2u/x2 + 2u/y2)

u(u/x) + v(u/y) UdU/dx + (1/)(/y)

Prandtl made some simplifying assumptions, based on the premise that the boundary layer is very thin (v and small compared to U and L) so 2-D equations over flat

plate more tractable:

B.Eq.

Blasius Solution for Laminar, dP/dx=dUe/dx=0 Flow

Page 129: CHAPTER 9: EXTERNAL* INCOMPRESSIBLE VISCOUS FLOWS *(unbounded)

u(u/x) + v(u/y) = (2u/y2)u/x + v/y = 0u = v = 0 at y = 0

u = Ue at y = (outside boundary layer)

With Prandtl’s simplifying assumptions plus nopressure gradient for laminar flow, left the 2-D mass and momentum

equations over flat plate “ripe” for Blasius:

BOUNDARY LAYER EQUATIONS

Blasius Solution for Laminar, dP/dx=dUe/dx=0 Flow

Page 130: CHAPTER 9: EXTERNAL* INCOMPRESSIBLE VISCOUS FLOWS *(unbounded)

By an ingenious coordinate transformation of the boundary layer equation Blasius showed that the dimensionless velocity distribution, u(x,y)/Ue,

was only a function of one composite dimensionless variable, [y/x][Uex/]1/2.

u(x,y)/Ue = f([y/x][Uex/]1/2) = f() = y/

Dimensional analysis = u(x,y)/Ue = f (Uex/, y/x)

Page 131: CHAPTER 9: EXTERNAL* INCOMPRESSIBLE VISCOUS FLOWS *(unbounded)

Dimensionless velocity profile for a laminar boundary layer:comparison with experiments by Liepmann, NACA Rept. 890,1943. Adapted from F.M. White, Viscous Flow, McGraw-Hill, 1991

y/(x)

Page 132: CHAPTER 9: EXTERNAL* INCOMPRESSIBLE VISCOUS FLOWS *(unbounded)

From Blasius solution it is found that:

= 5 (x/U)1/2 or /x = 5(Rex)-1/2

* = (x/U)1/2 or */x = 1.72(Rex)-1/2

= (x/U)1/2 or /x = 0.664(Rex)-1/2

w = 0.332 U3/2 (/x)

u(x,y)/Ue =1 at y(Ue/[x])1/2 = u(x,y)/Ue = 0.99 at y(Ue/[x])1/2 5.0

Page 133: CHAPTER 9: EXTERNAL* INCOMPRESSIBLE VISCOUS FLOWS *(unbounded)

u(x,y)