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2004 Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST Boundary layer Equations Contents: Boundary Layer Equations; Boundary Layer Separation; Effect of londitudinal pressure gradient on boundary layer evolution Blasius Solution Integral parameters: Displacement thickness and momentum thickness

Boundary layer Equations

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Boundary layer Equations. Contents : Boundary Layer Equations; Boundary Layer Separation; Effect of londitudinal pressure gradient on boundary layer evolution Blasius Solution Integral parameters: Displacement thickness and momentum thickness. - PowerPoint PPT Presentation

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Page 1: Boundary layer Equations

2004 Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST

Boundary layer Equations Contents:

– Boundary Layer Equations;

– Boundary Layer Separation;

– Effect of londitudinal pressure gradient on boundary layer evolution

– Blasius Solution

– Integral parameters: Displacement thickness and momentum thickness

Page 2: Boundary layer Equations

2004 Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST

Laminar Thin Boundary Layer Equations (<<x) over flat plate

Steady flow, constant and . Streamlines slightly divergent0 yp

dxdpxp e

2

2

2

21

y

u

x

u

x

p

y

uv

x

uu

2D Navier-Stokes Equations along x direction:

Compared with 2

2

y

u

dxdpe

Page 3: Boundary layer Equations

2004 Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST

Laminar Thin Boundary Layer Equations (<<x) over flat plate

2

21

y

u

dx

dp

y

uv

x

uu e

Laminar thin boundary layer equations (<<x) for flat plates

pe external pressure, can be calculated with Bernoulli’s Equation as there are no viscous effects outside the Boundary Layer

Note 1. The plate is considered flat if is lower then the local curvature radius

Note 2. At the separation point, the BD grows a lot and is no longer thin

Page 4: Boundary layer Equations

2004 Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST

z

wu

y

vu

x

uu

y

u

dx

dp

y

uv

x

uu e

2

21

Turbulent Thin Boundary Layer Equations (<<x) over flat plate

2D Thin Turbulent Boundary Layer Equation (<<x) to flat plates:

Resulting from Reynolds Tensions (note the w term)

0 0

Page 5: Boundary layer Equations

2004 Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST

Boundary Layer Separation Boundary Layer Separation: reversal of the flow by

the action of an adverse pressure gradient (pressure increases in flow’s direction) + viscous effects

mfm: BL / Separation / Flow over edges and blunt bodies

Page 6: Boundary layer Equations

2004 Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST

Boundary Layer Separation Boundary layer separation: reversal of the flow by the

action of an adverse pressure gradient (pressure increases in flow’s direction) + viscous effects

Page 7: Boundary layer Equations

2004 Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST

Boundary Layer Separation Bidimensional (2D) Thin Boundary Layer (<<x)

Equations to flat plates:

2

21

y

u

dx

dp

y

uv

x

uu e

Close to the wall (y=0) u=v=0 :

dx

dp

y

u e

y1

0

2

2

Similar results to turbulent boundary layer - close to the wall there is laminar/linear sub-layer region.

Page 8: Boundary layer Equations

2004 Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST

Boundary Layer Separation Outside Boundary layer: 0

2

2

y

u

The external pressure gradient can be:o dpe/dx=0 <–> U0 constant (Paralell outer streamlines):

o dpe/dx>0 <–> U0 decreases (Divergent outer streamlines):

o dpe/dx<0 <–> U0 increases (Convergent outer streamlines):

Close to the wall (y=0) u=v=0 :

dx

dp

y

u e

y1

0

2

2

Same sign

Page 9: Boundary layer Equations

2004 Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST

Zero pressure gradient:dpe/dx=0 <–> U0 constant (Paralell outer streamlines):

y

u

Inflection point at the wall

No separation of boundary layer

02

2

yy

u

00

2

2

yy

u

Boundary Layer Separation

Curvature of velocity profile is constant

Page 10: Boundary layer Equations

2004 Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST

Favourable pressure gradient:dpe/dx<0 <–> U0 increases (Convergent outer streamlines):

02

2

yy

u y

00

2

2

yy

u

Curvature of velocity profile remains constant

No boundary layer separation

Boundary Layer Separation

Page 11: Boundary layer Equations

2004 Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST

Adverse pressure gradient:dpe/dx>0 <–> U0 decreases (Divergent outer streamlines):

02

2

yy

u

00

2

2

yy

u Curvature of velocity profile can change

Boundary layer Separation can occur

y

P.I.

Boundary Layer Separation

Separated Boundary Layer

Page 12: Boundary layer Equations

2004 Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST

Sum of viscous forces:2

2

y

u

Become zero with velocity

Can not cause by itself the fluid stagnation (and the separation of Boundary Layer)

Boundary Layer Separation

Page 13: Boundary layer Equations

2004 Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST

Effect of longitudinal pressure gradient:

0dx

dpe (Convergent outer streamlines)

0dx

dpe (Divergent outer streamlines)

Viscous effects retarded Viscous effects reinforced

Fuller velocity profiles

Less full velocity profiles

...11

dx

dp

ux

u e

Decreases BL growth Increases BL growths

Boundary Layer Separation

Page 14: Boundary layer Equations

2004 Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST

Effect of longitudinal pressure gradient:

Fuller velocity profiles

Less full velocity profiles

...11

dx

dp

ux

u e

Decreases BL growth Increases BL growths

Fuller velocity profiles – more resistant to adverse pressure gradients

Turbulent flows (fuller profiles)- more resistant to adverse pressure gradients

Boundary Layer Separation

Page 15: Boundary layer Equations

2004 Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST

Boundary Layer Sepaation

Longitudinal and intense adverse pressure gradient does not cause separation

=> there’s not viscous forces

Page 16: Boundary layer Equations

2004 Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST

Blasius Solution to Laminar Boundary Layer Equation over a flat plate with dpe/dx=0

Bidimensional (2D) Thin Boundary Layer (<<x) Equations to flat plates:

2

2

y

u

y

uv

x

uu

0

y

v

x

u

Boundary Condition: y=0 u=v=0y=∞ u=U

Page 17: Boundary layer Equations

2004 Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST

Blasius hypothesis: with fU

u

The introdution of η corresponds to recognize that the nondimension velocity profile is stabilized.

nx

Ay

A and n are unknowns

Blasius Solution to Laminar Boundary Layer Equation over a flat plate with dpe/dx=0

Remark: eyx

A

y n

x

ny

x

nA

x n

1

Page 18: Boundary layer Equations

2004 Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST

Blasius Solution to Laminar Boundary Layer Equation over a flat plate with dpe/dx=0

Procedure:

oUsing current function:y

u

x

v

o Remark:

yyd

d

A

xUf

A

xu

nn

F

dfA

xU

n

o Replace u/U=f(η) e at the boundary layer equation, choose n such that the resulting equation does not depend on x and A in order to simplify the equation.

.

xv

Page 19: Boundary layer Equations

2004 Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST

Blasius Solution to Laminar Boundary Layer Equation over a flat plate with dpe/dx=0

o FUy

u

results:

Fx

nU

x

u

o

o Fx

UA

y

un

o Fx

UA

y

un

2

2

2

2

o FxnFnxA

U

xv nn

11

FA

xU

n

From:

Page 20: Boundary layer Equations

2004 Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST

Blasius Solution to Laminar Boundary Layer Equation over a flat plate with dpe/dx=0

We will obtain:

02

12

FFA

UnxF

n

2

2

y

u

y

uv

x

uu

00, xu

00, xv

Uxu ,

Boundary Conditions:

o Making n=1/2 and the equation comes:UA

02 FFFx

Uy

with

00 FU 00 F

00 FF 00 F

UFU 1F

Page 21: Boundary layer Equations

2004 Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST

Graphical Solution:

Blasius Solution to Laminar Boundary Layer Equation over a flat plate with dpe/dx=0

0 0 0,33211 0,3298 0,3232 0,6298 0,26683 0,8461 0,16144 0,9555 0,06425 0,9916 0,00596 0,999 0,00247 0,999 0,00028 1 0,0001

x

Uy

FU

u F

0

0,4

0,8

1,2

0 2 4 6 8 10

x

Uy

FU

u

F

Page 22: Boundary layer Equations

2004 Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST

Solution:

Blasius Solution to Laminar Boundary Layer Equation over a flat plate with dpe/dx=0

0 0 0,33211 0,3298 0,3232 0,6298 0,26683 0,8461 0,16144 0,9555 0,06425 0,9916 0,00596 0,999 0,00247 0,999 0,00028 1 0,0001

x

Uy

FU

u F

0Fx

UU

x

FUx Re

664,00

2

0

0

y

y

u

oShear stress at the wall

2

0

2

1U

fc

o Friction coefficent

Page 23: Boundary layer Equations

2004 Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST

Solution:

Blasius Solution to Laminar Boundary Layer Equation over a flat plate with dpe/dx=0

0 0 0,33211 0,3298 0,3232 0,6298 0,26683 0,8461 0,16144 0,9555 0,06425 0,9916 0,00596 0,999 0,00247 0,999 0,00028 1 0,0001

x

Uy

FU

u F

dxDL

o 0

o Drag

L

D

LU

DC

Re

328,1

21 2

o Drag Coefficent

UL

L Re

02

1F

L

UU

Page 24: Boundary layer Equations

2004 Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST

Solution :

Blasius Solution to Laminar Boundary Layer Equation over a flat plate with dpe/dx=0

Uyu 99,0

o Boundary layer thickness

0 0 0,33211 0,3298 0,3232 0,6298 0,26683 0,8461 0,16144 0,9555 0,06425 0,9916 0,00596 0,999 0,00247 0,999 0,00028 1 0,0001

x

Uy

FU

u F

xUxx Re

55

η=5

%8,10

5

0

F

F

o Shear stress at y=

Page 25: Boundary layer Equations

2004 Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST

Displacement thickness:

Blasius Solution to Laminar Boundary Layer Equation over a flat plate with dpe/dx=0

0

* 1dyuU

Ud

0

1dyuU

Ud

0

udyUU d

U

0

dyuU

Ideal Fluid flow rate

Real Flow rate

Déficit of flow rate due to velocity reduction at BD

Page 26: Boundary layer Equations

2004 Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST

Displacement thickness :

Blasius Solution to Laminar Boundary Layer Equation over a flat plate with dpe/dx=0

0

1dyuU

Ud

0

1dyuU

Ud

0

udyUU d

Ideal Fluid flow rate

Real Flow rate

Déficit of flow rate due to velocity reduction at BD

dUq

Page 27: Boundary layer Equations

2004 Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST

Blasius Solution to Laminar Boundary Layer Equation over a flat plate with dpe/dx=0

Displacement thickness :

0

1dyuU

Ud

0

1dyuU

Ud

0

1udy

Ud

Initial deviation of BD

δ

Deviation of outer streamlines

Section where the streamline become part of boundary layer

δdq/U LC

Page 28: Boundary layer Equations

2004 Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST

Blasius Solution to Laminar Boundary Layer Equation over a flat plate with dpe/dx=0

Blasius Solution for displacement thickness:

x

d

x Re

72,1

δ dq/U LC

Ux

x Recom

344,0dou

Page 29: Boundary layer Equations

2004 Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST

Blasius Solution to Laminar Boundary Layer Equation over a flat plate with dpe/dx=0

Momentum thickness:

0

2

1udyuU

Um

0

2

1udyuU

Um

dU mdUdyu

2

0

2

mUudyUdyu

2

00

2

0

2

0

2 dyuudyUU m

Page 30: Boundary layer Equations

2004 Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST

Blasius Solution to Laminar Boundary Layer Equation over a flat plate with dpe/dx=0

Momentum flow rate through a section of BD:

mdqm UUUdyuqx

222

0

2

Momentum flow rate of uniform

profile

UU

Reduction due to deficit of flow

rate

dUU

Reduction due to deficit

momentum flow rate at BD

mUU

Page 31: Boundary layer Equations

2004 Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST

Blasius Solution to Laminar Boundary Layer Equation over a flat plate with dpe/dx=0

Longitudinal momentum balance between the leading edge and a cross section at x:

xxqmxqm xx

qqD

0

dU 2 mU d 2

mU 2

δ d-dLC

x

Page 32: Boundary layer Equations

2004 Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST

Blasius Solution to Laminar Boundary Layer Equation over a flat plate with dpe/dx=0

Blasius Solution to momentum thickness:

x

m

x Re

664,0

Ux

x Rewith

133.0mor

Page 33: Boundary layer Equations

2004 Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST

Laminar Boundary Layer Equations Contents:

– Thin Boundary Layer Equations with Zero Pressure Gradient;

– Boundary Layer Separation;

– Effect of longitudinal pressure gradient on the evolution of Boundary Layer

– Blasius Solution

– Local Reynolds Number and Global Reynolds Number

– Integral Parameters: displacement thickness and momentum thickness

Page 34: Boundary layer Equations

2004 Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST

Recommended study elements:– Sabersky – Fluid Flow: 8.3, 8.4

– White – Fluid Mechanics: 7.4 (sem método de Thwaites)

Blasius Solution to Laminar Boundary Layer Equation over a flat plate with dpe/dx=0

Page 35: Boundary layer Equations

2004 Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST

=1,2 kg/m3, =1,810-5 Pa.s(Rex)c =106.

Exercise

L=2m

U=2m/s 0dxdpe

Large plate with neglectable thickness, lenght L=2m. Parallel and non-disturbed air flow. (=1,2 kg/m3, =1,810-5 Pa.s) with U=2 m/s. Zero pressure gradient over the flat plate. Transition to turbulent at Rex=106.

Page 36: Boundary layer Equations

2004 Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST

a) Find boundary layer thickness at sections S1 and S2, at distance x1=0,75 m and x2=1,5 m of the leading edge

Exercise

mU

x cxc 5,72

2,1108,110Re

56

Find xc:

xx Re

5

Laminar Boundary layer at x1 and x2 – We can apply Blasius

Solution

=1,2 kg/m3, =1,810-5 Pa.s(Rex)c =106.

L=2m

U=2m/s 0dxdpe

Page 37: Boundary layer Equations

2004 Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST

Exercise

xx Re

5

Laminar Boundary layer at x1 and x2 – We can apply Blasius

Solution

mx 75,01 5

51 10105,1

75,02Re

x m0119,010

575,051

mx 5,12 52 102Re x

m0168,02

a) Find boundary layer thickness at sections S1 and S2, at distance x1=0,75 m and x2=1,5 m of the leading edge

Page 38: Boundary layer Equations

2004 Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST

Exercise

b) Check that it is a thin boundary layer.

=1,2 kg/m3, =1,810-5 Pa.s(Rex)c =106.

L=2m

U=2m/s 0dxdpe

A: Thin Blayer if /x<<1: 0159,075,0

0119,0

1

x

0112,05,1

0168,0

2

x

Why/x at 2 is lower than /x at 1?

y=(x)

Page 39: Boundary layer Equations

2004 Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST

Exercise

d) Find the value of y1 at x1 of the streamline passing through the coordinates x2=1,5 and y2=.

=1,2 kg/m3, =1,810-5 Pa.s(Rex)c =106.

L=2m

U=2m/s 0dxdpe

Streamline

x2=1,5mx1=0,75m

y1=?

A: We have the same flow rate between the streamline and the plate at both cross sections

Flow rate through a cross section of BD: dUq

Flow rate through section 2: 22 dUq

Flow rate through section 1: 11 yU 1dU 1

1

11

00

1

yy

udyudyudyq

1221 ddy

Page 40: Boundary layer Equations

2004 Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST

Exercise

d) Find the value of y1 at x1 of the streamline passing through the coordinates x2=1,5 and y2=.

=1,2 kg/m3, =1,810-5 Pa.s(Rex)c =106.

L=2m

U=2m/s 0dxdpe

Linha de corrente

x2=1,5mx1=0,75m

y1=?

A: We have the same flow rate between the streamline and the plate at both cross sections

1221 ddy Laminar BD: 344,0d

0,0168m 0,0058m 0,0041m y1=0,0151m

Page 41: Boundary layer Equations

2004 Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST

Exercise

e) Find the force per unit leght between sections S1 and S2.

L=2m

U=2m/s 0dxdpe=1,2 kg/m3, =1,810-5 Pa.s(Rex)c =106.

xmx UD 2,0

A: There are no other forces applied except that imposed by the resistance (Drag) of plate:

The applied force between the leading edge and the cross section at x is:

Laminar BD: 133,0m

Drag force to section 2: D0,2=0,0107N/mmm 00223,0133,0 22 mm 00158,0133,0 11 Drag force to section 1: D0,1=0,0076N/m

Drag force between 1 and 2: D1,2=D0,2-D0,1=0,0031N/m

Page 42: Boundary layer Equations

2004 Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST

Exercise

f) True or False?: ”Under the conditions of the problem, if the plate was sufficiently long (L ), the boundary layer would eventually separate?

L=2m

U=2m/s 0dxdpe=1,2 kg/m3, =1,810-5 Pa.s(Rex)c =106.

False: The BD will separate only with adverse pressure gradient. The drag forces will decrease with the velocity

over the plate. The drga forces are not able to stop the fluid flow.