Post on 03-Jun-2018
8/12/2019 Incompressible, laminar couette flow
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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
8/12/2019 Incompressible, laminar couette flow
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Aerodynamics
Masters of Mechanical Engineering
y
h
U
Boundary conditions
- Impermeability of the walls:
- No-slip condition:
x
000 ==== vhyvy
Uuhyuy 00 ====
Incompressible, Laminar Couette Flow
8/12/2019 Incompressible, laminar couette flow
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Aerodynamics
Masters of Mechanical Engineering
Continuity equation
Boundary condition
0=v
.0 constvy
v==
Incompressible, Laminar Couette Flow
8/12/2019 Incompressible, laminar couette flow
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Aerodynamics
Masters of Mechanical Engineering
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
Incompressible, Laminar Couette Flow
8/12/2019 Incompressible, laminar couette flow
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Aerodynamics
Masters of Mechanical Engineering
Momentum balance,
Boundary conditions
yu
ydx
dp
y
u
yx
yx
=
==
112
2
x
Uuhy
uy
00
==
==
Incompressible, Laminar Couette Flow
8/12/2019 Incompressible, laminar couette flow
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Aerodynamics
Masters of Mechanical Engineering
Solution
Reference values for length and velocity
( )
+=
=
2
2
1
hy
dx
dp
h
U
yhydx
dpUh
yu
yx
UU
hL
ref
ref
=
=
Incompressible, Laminar Couette Flow
8/12/2019 Incompressible, laminar couette flow
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Aerodynamics
Masters of Mechanical Engineering
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
Incompressible, Laminar Couette Flow
8/12/2019 Incompressible, laminar couette flow
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Aerodynamics
Masters of Mechanical Engineering
Non-dimensional numbers
Reynolds number
Pressure gradient parameter
2
2
2
hU
dx
dp
h
U
h
U
Re
=
Incompressible, Laminar Couette Flow
8/12/2019 Incompressible, laminar couette flow
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Aerodynamics
Masters of Mechanical Engineering
-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
Incompressible, Laminar Couette Flow
8/12/2019 Incompressible, laminar couette flow
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Aerodynamics
Masters of Mechanical Engineering
+
+
+
=
+
+
+
+
=
+
=
+
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
8/12/2019 Incompressible, laminar couette flow
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Aerodynamics
Masters of Mechanical Engineering
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
Two-dimensional, IncompressibleSteady Flow
8/12/2019 Incompressible, laminar couette flow
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Aerodynamics
Masters of Mechanical Engineering
Two-dimensional, IncompressibleSteady Flow
Making the equations dimensionless
Reference values
Velocity
Length
Pressure *22
**
**
,
,
pUpU
LyyLxxL
vUvuUuU
ee
eee
=
==
==
8/12/2019 Incompressible, laminar couette flow
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Aerodynamics
Masters of Mechanical Engineering
convectiondiffusionO[ ]
Two-dimensional, IncompressibleSteady Flow
( ) ( )
( ) ( )
====
+
+
=
+
+
+
=
+
=+
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
8/12/2019 Incompressible, laminar couette flow
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Aerodynamics
Masters of Mechanical Engineering
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
Two-dimensional, IncompressibleSteady Flow
Practical applications are usually flows at
high Reynolds numbers,510>eR
8/12/2019 Incompressible, laminar couette flow
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Aerodynamics
Masters of Mechanical Engineering
Two-dimensional, IncompressibleSteady Flow
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
8/12/2019 Incompressible, laminar couette flow
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Aerodynamics
Masters of Mechanical Engineering
Boundary-layer Wake
Mixing layerJet
Two-dimensional, IncompressibleSteady Flow
8/12/2019 Incompressible, laminar couette flow
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Aerodynamics
Masters of Mechanical Engineering
Thick shear layers (Bluff bodies)
Two-dimensional, IncompressibleSteady Flow
8/12/2019 Incompressible, laminar couette flow
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Aerodynamics
Masters of Mechanical Engineering
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=
8/12/2019 Incompressible, laminar couette flow
19/32
Aerodynamics
Masters of Mechanical Engineering
Boundary-Layer Approximations
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
8/12/2019 Incompressible, laminar couette flow
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Aerodynamics
Masters of Mechanical Engineering
[ ]
[ ]L
Uv
v
L
U
y
v
x
u
e
e
=
=+
=
+
0
0
O
O
Boundary-Layer Approximations
Continuity equation
8/12/2019 Incompressible, laminar couette flow
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Aerodynamics
Masters of Mechanical Engineering
LU
Lp
dxdp
dx
dUU
dx
dp
constUp
ee
ee
e
e
2
2
11
0
.
2
1
==
=+
=+
O
Boundary-Layer Approximations
Bernoullis equation applied to the outer flow(ideal fluid)
8/12/2019 Incompressible, laminar couette flow
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Aerodynamics
Masters of Mechanical Engineering
++=+
++=+
++=+
+
+
=
+
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
Boundary-Layer Approximations
Momentum balance in the xdirection
8/12/2019 Incompressible, laminar couette flow
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Aerodynamics
Masters of Mechanical Engineering
+
2
11
L
Re
1
11
01
2
2
2
2
2
ee
e
RL
L
Rx
u
Rx
u
=
=
=
O
O
Boundary-Layer Approximations
Momentum balance in the xdirection
Analysis of diffusion
8/12/2019 Incompressible, laminar couette flow
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Aerodynamics
Masters of Mechanical Engineering
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
Boundary-Layer Approximations
Momentum balance in the ydirection
8/12/2019 Incompressible, laminar couette flow
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Aerodynamics
Masters of Mechanical Engineering
2
2
2
2
1
111
11
L
U
y
p
Ry
p
U
L
e
ee
=
++
=+
eRL =
2
O
O
Boundary-Layer Approximations
Momentum balance in the ydirection
Using we obtain
8/12/2019 Incompressible, laminar couette flow
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Aerodynamics
Masters of Mechanical Engineering
0
2
11 2222
0
=
=
yp
UU
R
U
L
dy
y
pee
e
e
O
Boundary-Layer Approximations
Momentum balance in the ydirection
Across the boundary-layer
Therefore,
8/12/2019 Incompressible, laminar couette flow
27/32
Aerodynamics
Masters of Mechanical Engineering
+=
+
=
+
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
Boundary-Layer Approximations
8/12/2019 Incompressible, laminar couette flow
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Aerodynamics
Masters of Mechanical Engineering
+=
+
=
+
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
Boundary-Layer Approximations
8/12/2019 Incompressible, laminar couette flow
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Aerodynamics
Masters of Mechanical Engineering
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
Boundary-Layer Approximations
8/12/2019 Incompressible, laminar couette flow
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Aerodynamics
Masters of Mechanical Engineering
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,
8/12/2019 Incompressible, laminar couette flow
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Aerodynamics
Masters of Mechanical Engineering
x
p
x
p e
02
2
x
u
Simplified Forms of the Navier-Stokes Equations
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,
8/12/2019 Incompressible, laminar couette flow
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Aerodynamics
Masters of Mechanical Engineering
02
2
x
u
Simplified Forms of the Navier-Stokes Equations
Reduced Navier-Stokes equations
Diffusion in the main direction of the flowneglected,
Pressure determination makes the problemelliptic