Analysis of Physical Intuition …
P M V SubbaraoProfessor
Mechanical Engineering Department
I I T Delhi
Two-dimensional Boundary Layer Flows
Division of Flow at Higher Reynolds Numbers
Prandtls Large Reynolds Number 2-D Incompressible Flow
The free-stream velocity will accelerate for non-zero values of β:
m
edge L
xUxU
where L is a characteristic length and m is a dimensionless constant that depends on β:
1
2
m
m
The condition m = 0 gives zero flow acceleration corresponding to the Blausius solution for flat-plate flow.
The Measure of Wedge Angle
The boundary layer is seen to grow in thickness as x moves from 0 to L.
Two-dimensional Boundary Layer Flows
In dimensionless variables the steady Incompressible Navier-Stokes equations in two dimensions may be written:
0Re
1 2
ux
puv
0Re
1 2
vy
pvv
0
y
v
x
u
The boundary layer is seen to grow in thickness as x moves from 0 to L.
The Art of Asymptotic Thinking
This suggests that the term in x-momentum equation can be properly estimated as of order U2/L
uv .
In the dimensionless formulation, should be taken as O(1) at large Re.
uv .
If this term is to balance the viscous stress term, then the natural choiceis to assume that the y-derivatives of u are so large that the balance is with .
2
2
Re
1
y
u
This is due to the fact that the boundary layer on the plate is observed to be so thin.Thus it makes sense to define
A stretched variable yy Re
Local Reynolds Number
xURe
Shape of Boundary Layer In Stretched Coordinates
The stretched N-S Equations
2-D incompressible continuity equations
0
y
v
x
u
yy Re
0
Re
yv
x
u
In order to keep this essential equation intact and as of order unity: x
u
The stretched variable must be compensated by a stretched form of the y-velocity component:
y
yv Re
Stretched coordinate:
0
y
v
x
u
2-D incompressible continuity equations in stretched coordinates:
Prandtls Intuition
Prandtl would have been comfortably guessed this definition.
The boundary layer on the plate was so thin that there could have been only a small velocity component normal to its surface.
Thus the continuity equation will survive our limit Re .
0
y
v
x
u
X - Momentum Equation in Stretched Coordinates
0Re
12
2
2
2
y
u
x
u
x
p
y
uv
x
uu
Returning now to consideration of x-momentum equation, retain the pressure term as O(1).
x
p
0Re
12
2
2
2
y
u
x
u
x
puv
x-momentum equation in stretched coordinates:
In the limit Re , with stretched variables, this amounts to dropping the term
2
2
Re
1
x
u
02
2
y
u
x
p
y
uv
x
uu
y-Momentum Equation in 2-D Boundary Layer Flows
Use these stretched variables in y-momentum equation
0y
p
0Re
1
Re
1Re
ReRe 2
2
2
2
23
y
v
x
v
y
p
y
vv
x
vu
2
2
22
2
Re
1
Re
1
x
v
y
v
y
vv
x
vu
y
p
0Re
1 2
vy
pvv
Thus in the limit Re the vertical momentum equation reduces to
The Conclusions from Intuitive Mathematics
• The pressure does not change as we move vertically through the thin boundary layer.
• That is, the pressure throughout the boundary layer at a station x must be the pressure outside the layer.
• At this point a crucial contact is made with inviscid fluid theory.
• The “pressure outside the boundary layer” should be determined by the inviscid theory.
• Since the boundary layer is thin and will presumably not disturb the inviscid flow very much.
• In particular for a flat plate the Euler flow is the uniform stream- the plate has no effect and so the pressure has its constant free-stream value.