incompressible, laminar couette flow

8/12/2019 Incompressible, laminar couette flow

1/32

Aerodynamics

Masters of Mechanical Engineering

Steady-flow,

Flow independent of zdirection,(two-dimensional)

Fully developed flow,

h

U

x

0=

t

0=

z

0=

x

vr

y

Incompressible, Laminar Couette Flow


2/32

Aerodynamics


y

h

U

Boundary conditions

- Impermeability of the walls:

- No-slip condition:

x

000 ==== vhyvy

Uuhyuy 00 ====



3/32

Aerodynamics


Continuity equation

Boundary condition

0=v

.0 constvy

v==



4/32

Aerodynamics


Momentum balance,

Momentum balance,

Pressure is only a function of

must be indepedent of

y

2

21

0y

u

x

p

+

=

y

p

=

10

x

x

=

0

x

vx

r

dx

dp



5/32

Aerodynamics


Momentum balance,

Boundary conditions

yu

ydx

dp

y

u

yx

yx

=

==

112

2

x

Uuhy

uy

00

==

==



6/32

Aerodynamics


Solution

Reference values for length and velocity

( )

+=

=

2

2

1

hy

dx

dp

h

U

yhydx

dpUh

yu

yx

UU

hL

ref

ref

=

=



7/32

Aerodynamics


Solution in dimensionless variables

Non-dimensional numbers

=

=

h

y

U

h

y

h

y

U

u

yx

2

121

Re

2

21

11

2

dx

dp

U

h

hURe

2

2

=

= Reynolds number

Pressure gradient parameter



8/32

Aerodynamics


Non-dimensional numbers

Reynolds number

Pressure gradient parameter

2

2

2

hU

dx

dp

h

U

h

U

Re

=



9/32

Aerodynamics


-0.25 0 0.25 0.5 0.75 1 1.250

0.1

0.2

0.3

0.4

0.50.6

0.7

0.8

0.9

1

=-2

=-1

=0

=1

=2

-3 -2 -1 0 1 2 30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

=-2

=-1

=0

=1

=2

h

y

h

y

UU 2

UR yxe



10/32

Aerodynamics


+

+

+

=

+

+

+

+

=

+

=

+

y

v

yx

v

y

u

xy

p

y

v

vx

v

u

x

v

y

u

yx

u

xx

p

y

uvx

uu

y

v

x

u

2

1

21

0

Two-dimensional, IncompressibleSteady Flow


11/32

Aerodynamics


Constant viscosity, =constant

+

+

=

+

+

+

=

+

=

+

2

2

2

2

2

2

2

2

1

1

0

y

v

x

v

y

p

y

vvx

vu

y

u

x

u

x

p

y

uv

x

uu

y

v

x

u



12/32

Aerodynamics



Making the equations dimensionless

Reference values

Velocity

Length

Pressure *22

**

**

,

,

pUpU

LyyLxxL

vUvuUuU

ee

eee

=

==

==


13/32

Aerodynamics


convectiondiffusionO[ ]


( ) ( )

( ) ( )

====

+

+

=

+

+

+

=

+

=+

2

2*

*2

2*

*2

*

*

*

*

*

*

*

*

2

*

*2

2

*

*2

*

*

*

*

*

*

*

*

*

*

*

*

1

1

0

L

UL

UU

LULUR

y

v

x

v

Ry

p

y

vv

x

vu

y

u

x

u

Rx

p

y

uv

x

uu

yv

xu

e

ee

eee

e

e


14/32

Aerodynamics


yu

= (uni-dimensional shear-stress)

Ar 1,810-5kgm-1s-1 1,110-5m2s-1

gua 1,010-3kgm-1s-1 1,010-6m2s-1


Practical applications are usually flows at

high Reynolds numbers,510>eR


15/32

Aerodynamics



Effects of shear-stresses are restricted tosmall regions that exhibit large velocityvariations in small distances

Thin shear layers- Thickness of the shear layer, , is much

smaller than the reference length L, /L 1


16/32

Aerodynamics


Boundary-layer Wake

Mixing layerJet



17/32

Aerodynamics


Thick shear layers (Bluff bodies)



18/32

Aerodynamics


Boundary-Layer Approximations

Prandtl simplifications (1904)

Analysis of the order of magnitude of the termsincluded in the continuity and momentumbalance equations

Starting hypothesis: Re1. (/L1)

xUR ee=


19/32

Aerodynamics



Prandtl Simplifications (1904)

Order of magnitude of variable , O[], is given bythe upper limit of the variation

Known orders of magnitude

O[x] L

O[y] O[u] Ue


20/32

Aerodynamics


[ ]

[ ]L

Uv

v

L

U

y

v

x

u

e

e

=

=+

=

+

0

0

O

O


Continuity equation


21/32

Aerodynamics


LU

Lp

dxdp

dx

dUU

dx

dp

constUp

ee

ee

e

e

2

2

11

0

.

2

1

==

=+

=+

O


Bernoullis equation applied to the outer flow(ideal fluid)


22/32

Aerodynamics


++=+

++=+

++=+

+

+

=

+

2

22222

22

222

2

2

2

2

11

111

1

1

L

R

L

LUL

U

L

U

L

U

L

U

U

L

U

L

U

L

U

L

U

y

u

x

u

x

p

y

uv

x

uu

e

e

eeee

eeeee


Momentum balance in the xdirection


23/32

Aerodynamics


+

2

11

L

Re

1

11

01

2

2

2

2

2

ee

e

RL

L

Rx

u

Rx

u

=

=

=

O

O


Momentum balance in the xdirection

Analysis of diffusion


24/32

Aerodynamics


O

O

++

=+

++

=+

+

+

=

+

2

2

2

2

2

2

2

2

2

32

2

2

2

2

2

2

2

1

1

1

L

L

U

L

U

LUy

p

L

U

L

U

L

U

L

U

y

p

L

U

L

U

y

v

x

v

y

p

y

vv

x

vu

ee

e

ee

eeee


Momentum balance in the ydirection


25/32

Aerodynamics


2

2

2

2

1

111

11

L

U

y

p

Ry

p

U

L

e

ee

=

++

=+

eRL =

2

O

O



Using we obtain


26/32

Aerodynamics


0

2

11 2222

0

=

=

yp

UU

R

U

L

dy

y

pee

e

e

O



Across the boundary-layer

Therefore,


27/32

Aerodynamics


+=

+

=

+

2

21

0

y

u

dx

dp

y

uv

x

uu

y

v

x

u

The selected coordinate system must respect thefollowing conditions:

1. The xcoordinate must be aligned with theouter flow

2. The ycoordinate is normal to the surface



28/32

Aerodynamics


+=

+

=

+

2

21

0

y

u

dx

dp

y

uv

x

uu

y

v

x

u

Static pressure is independent of the coordinate y.Pressure change with x(dp/dx) may be obtainedfrom the outer flow, p(x)pe(x). Therefore, the

pressure does not belong to the unknowns.The pressure is part of the input

of a boundary-layer problem



29/32

Aerodynamics


The equations are no longer elliptic in the xdirection. For a given value of x, the solution dependsonly on the upstream conditions. Therefore, it is

possible to solve the problem using a marchingprocedure in the xdirection (initial value problem).

+=

+

=

+

2

21

0

y

u

dx

dp

y

uv

x

uu

y

v

x

u



30/32

Aerodynamics


0

y

p

02

2

xu

Simplified Forms of the Navier-Stokes Equations

Boundary layer, thin shear layer equations

Pressure determined by the outer flow,

Diffusion in the main direction of the flowneglected,


31/32

Aerodynamics


x

p

x

p e

02

2

x

u


Parabolized Navier-Stokes equations

Pressure derivative in the main direction of theflow determined by the outer flow,

Diffusion in the main direction of the flow

neglected,


32/32

Aerodynamics


02

2

x

u


Reduced Navier-Stokes equations

Diffusion in the main direction of the flowneglected,

Pressure determination makes the problemelliptic

incompressible, laminar couette flow

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