von Kárman Equation for flat plates ( dp e / dx ≠ 0)

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

von Kárman Equation for flat plates (dpe/dx≠0)

For laminar or turbulent flows: in the turbulent case we take time-average velocity and pressure.

Procedure: Mass and momentum balance of the following control volume:

dx

dxdx

d

x x+dx


dx

xmdxxm

m

dxudydx

dudym

x

dxx

00

Mass balance: dxxxVC mmmMdt

d

Steady Flow

o Flow rate: xmx

x udym

0

o Flow rate : dxxm

dxudydx

d

0

von Kárman Equation forflat plates (dpe/dx≠0)


dx

xmdxxm

m

dxudydx

dUmUq

xqm

0

o x Momentum flow rate through y=δ:

dxudydx

dm

0

Mass balance :



dxxqmxqxqmxq

qmxq

VCxx

VC

xdxqmxqmxqmxx Fqqq

dt

dK

o Difference : xqmxdxxqmx qq

dxdyudx

d

0

2

x momentum balance:

Steady flow



xqmxqdxxqmxq

qmxq

qmxqmxdxqmxx qqqF

xxVC

x momentum balance:

dxdyudx

d

0

2 dxudydx

dU

0



dxddppddpppFVCx 02

1

Forces along x:

p+dpp

p+1/2dp

τ0

dxdx

dpF

VCx

0

dx

dUU

dx

dpe



00

0 dyuUdx

dUudyuU

x

Final result:

Using the definition of d and δm:

dx

dUUU

dx

ddm 2

0



When dpe/dx=0 (dU/dx=0):

dx

dU m 2

0 dx

d

Uc m

f

2

21 2

0


When dpe/dx=0 (dU/dx=0) we have m=a

(a takes different values in laminar and turbulent flow):

dx

dac f

2 Boundary layer grows faster

when Cf is higher


Approximate solutions for laminar boundary layer for dpe/dx=0

Blasius solution shows that fU

u

xx

yRewith

andxx Re

5

yx

x

y5

5

y

fU

u

a – constant along the BD

1

0

1

yd

U

u

U

um



von Kárman Equation:dx

dU m 2

0

but

0

0

y

y

u

β - constant

dx

daU

2

0

yyd

UudU

U

dx

daU

U

2xax Re

12 00

Integrating



Remark: a and β depend on the velocity profile, however δ/x, cf and CD do not vary much with profile shape

xf

ac

Re

2

xax Re

12

U0

We have

and

LD

aC

Re

22

UUc f

2

220


Approximate profiles for Laminar BD

a β c f

Linear 0,167 1

Parabolic 0,133 2

Sinusoidal 0,137

Blasius

0dxdp e

y

U

u

2

2

yy

U

u

y

U

u

2sin

xRe

789,4

xRe

484,5

xRe

461,3

xRe

5

2

xRe

578,0

xRe

729,0

xRe

656,0

xRe

664,0

x


ma 0 yydUud


Contents:– von Kármàn Equation;

– Simplification for ;

– Approximate solutions for laminar Boundary Layers with

zero pressure gradient .

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

0dx

dpe

0dx

dpe


Recommended Study Elements:– Sabersky – Fluid Flow: 8.6, 8.7

– White – Fluid Mechanics: 7.3, 7.4

Von Kárman Equation for a flat plate


Problem on the Von Kármàn Equation

von Kárman Equation for flat plates ( dp e / dx ≠ 0)

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