Incompressible, Laminar Couette Flow - ULisboa · Aerodynamics Masters of Mechanical Engineering...

Aerodynamics

Masters of Mechanical Engineering

• Steady-flow,

• Flow independent of z direction,(two-dimensional)

• Fully developed flow,

h

U

x

0=∂

∂

t

0=∂

∂

z

0=∂

∂

x

vr

y

Incompressible, Laminar Couette Flow

Aerodynamics


y

h

U

• Boundary conditions

- Impermeability of the walls:

- No-slip condition:

x

000 =⇒==⇒= vhyvy

Uuhyuy ˆ00 =⇒==⇒=


Aerodynamics


• Continuity equation

• Boundary condition

0=v

.0 constvy

v=⇔=

∂

∂


Aerodynamics


• Momentum balance,


• Pressure is only a function of

must be indepedent of

y

2

21

0y

u

x

p

∂

∂+

∂

∂−= ν

ρ

y

p

∂

∂−=

ρ

10

x

x

=

∂

∂0

x

vx

r

dx

dp


Aerodynamics



• Boundary conditions

y

u

ydx

dp

y

u

yx

yx

∂

∂=

∂

∂==

∂

∂

µτ

τ

ρρν

112

2

x

Uuhy

uy

ˆ

00

=⇒=

=⇒=


Aerodynamics


• Solution

• Reference values for length and velocity

( )

−+=

−−=

2

ˆ

2

1ˆ

hy

dx

dp

h

U

yhydx

dpU

h

yu

yx µτ

µ

UU

hL

ref

ref

ˆ=

=


Aerodynamics


• Solution in dimensionless variables

• Non-dimensional numbers

−Λ−=

−Λ−=

h

y

U

h

y

h

y

U

u

yx

2

121

Re

2

ˆ21

11ˆ

2ρ

τ

dx

dp

U

h

hURe

µ

ν

2

ˆ

2

=Λ

= Reynolds number

Pressure gradient parameter


Aerodynamics


• Non-dimensional numbers

Reynolds number

Pressure gradient parameter

2

2

2

ˆ

ˆ

ˆ

h

U

dx

dp

h

U

h

U

Re

µ

µ

ρ

=Λ

∝


Aerodynamics


-0.25 0 0.25 0.5 0.75 1 1.250

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Λ=-2

Λ=-1

Λ=0

Λ=1

Λ=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

Λ=-2

Λ=-1

Λ=0

Λ=1

Λ=2

h

y

h

y

UU ˆ 2UR yxe ρτ


Aerodynamics


∂

∂

∂

∂+

∂

∂+

∂

∂

∂

∂+

∂

∂−=

∂

∂+

∂

∂

∂

∂+

∂

∂

∂

∂+

∂

∂

∂

∂+

∂

∂−=

∂

∂+

∂

∂

=∂

∂+

∂

∂

y

v

yx

v

y

u

xy

p

y

vv

x

vu

x

v

y

u

yx

u

xx

p

y

uv

x

uu

y

v

x

u

ννρ

ννρ

21

21

0

Two-dimensional, IncompressibleSteady Flow

Aerodynamics


Constant viscosity, ν=constant

∂

∂+

∂

∂+

∂

∂−=

∂

∂+

∂

∂

∂

∂+

∂

∂+

∂

∂−=

∂

∂+

∂

∂

=∂

∂+

∂

∂

2

2

2

2

2

2

2

2

1

1

0

y

v

x

v

y

p

y

vv

x

vu

y

u

x

u

x

p

y

uv

x

uu

y

v

x

u

νρ

νρ


Aerodynamics



Making the equations dimensionless

Reference values

Velocity

LengthPressure *22

**

**

,

,

pUpU

LyyLxxL

vUvuUuU

ee

eee

ρρ =→

==→

==→

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

y

v

x

u

e

ee

eee

e

e

µ

ρ

νµ

ρ

Aerodynamics


y

u

∂

∂= µτ (uni-dimensional shear-stress)

Ar µ � 1,8×10-5kgm-1s-1 ν �1,1×10-5m2s-1

Água µ � 1,0×10-3kgm-1s-1 ν �1,0×10-6m2s-1


• Practical applications are usually flows athigh Reynolds numbers, 5

10>eR

Aerodynamics



• Effects of shear-stresses are restricted to small regions that exhibit large velocity variations in small distances

• Thin shear layers- Thickness of the shear layer, δ, is much smaller than the reference length L, δ/L≪ 1

Aerodynamics


Boundary-layer Wake

Mixing layerJet


Aerodynamics


Thick shear layers (Bluff bodies)


Aerodynamics


Boundary-Layer Approximations

Prandtl simplifications (1904)

Analysis of the order of magnitude of the terms included in the continuity and momentum balance equations

Starting hypothesis: Re≫1. (δ/L≪1)ν

xUR e

e =

Aerodynamics



Prandtl Simplifications (1904)

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

Known orders of magnitude

O[x]→ L

O[y] → δ O[u]→ Ue

Aerodynamics


[ ]

[ ]L

Uv

v

L

U

y

v

x

u

e

e

δ

δ

=

=+

=∂

∂+

∂

∂

0

0

O

O


Continuity equation

Aerodynamics


L

U

L

p

dx

dp

dx

dUU

dx

dp

constUp

ee

ee

e

e

2

2

11

0

.2

1

==

=+

=+

ρρ

ρ

ρ

O


Bernoulli’s equation applied to the outer flow(ideal fluid)

Aerodynamics


++=+

++=+

++=+

∂

∂+

∂

∂+

∂

∂−=

∂

∂+

∂

∂

2

22222

22

222

2

2

2

2

11

111

1

1

δ

δ

ν

δν

νρ

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 x direction

Aerodynamics


+

2

11

δ

L

Re

111

01

2

2

2

2

2

ee

e

RL

L

Rx

u

Rx

u

=⇒

=

∂

∂

≅=

∂

∂

δ

δν

ν

O

O

≪


Momentum balance in the x direction

Analysis of diffusion

Aerodynamics


O

O

++

∂

∂−=+

++

∂

∂−=+

∂

∂+

∂

∂+

∂

∂−=

∂

∂+

∂

∂

2

2

2

2

2

2

2

2

2

32

2

2

2

2

2

2

2

1

1

1

δ

δδν

ρ

δδ

δ

δν

ρ

δδ

νρ

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 y direction

Aerodynamics


2

2

2

2

1

111

11

L

U

y

p

Ry

p

U

L

e

ee

δ

ρ

ρδ

=

∂

∂−

++

∂

∂−=+

eRL

=

2

δ

O

O



Using we obtain

Aerodynamics


0

2

11 222

20

≅∂

∂

=

=

∂

∂∫

y

p

UUR

UL

dyy

pee

e

e ρρρδδ

O ≪



Across the boundary-layer

Therefore,

Aerodynamics


∂

∂+−=

∂

∂+

∂

∂

=∂

∂+

∂

∂

2

21

0

y

u

dx

dp

y

uv

x

uu

y

v

x

u

νρ

• The selected coordinate system must respect thefollowing conditions:

1. The x coordinate must be aligned with the outer flow

2. The y coordinate is normal to the surface


Aerodynamics


∂

∂+−=

∂

∂+

∂

∂

=∂

∂+

∂

∂

2

21

0

y

u

dx

dp

y

uv

x

uu

y

v

x

u

νρ

• 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


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 x direction (initial value problem).

∂

∂+−=

∂

∂+

∂

∂

=∂

∂+

∂

∂

2

21

0

y

u

dx

dp

y

uv

x

uu

y

v

x

u

νρ


Aerodynamics


0≅∂

∂

y

p

02

2

≅

∂

∂

x

uν

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,

Aerodynamics


x

p

x

p e

∂

∂≅

∂

∂

02

2

≅

∂

∂

x

uν


• Parabolized Navier-Stokes equations

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


Aerodynamics


02

2

≅

∂

∂

x

uν


• Reduced Navier-Stokes equations


― Pressure determination makes the problem elliptic

Incompressible, Laminar Couette Flow - ULisboa · Aerodynamics Masters of Mechanical Engineering...

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