analysis of obviously boundary layer… p m v subbarao professor mechanical engineering department i...
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Analysis of Obviously Boundary Layer…
P M V SubbaraoProfessor
Mechanical Engineering Department
I I T Delhi
Flat Plate Boundary Layer Flows
Prandtls Viscous Flow Past a General Body• Prandtl’s striking insight is clearer when we consider flow past a
general smooth body.
• The boundary layer is taken as thin in the neighborhood of the body.
• Curvilinear coordinates can be introduced.
• x the arc length along curves paralleling the body surface and y the coordinate normal to these curves.
• In the stretched variables, and in the limit for large Re, it turns out that we again get only 0
y
p
• must be interpreted to mean that the pressure is what would be computed from the inviscid flow past the body.
0y
p
Bernoulli’s Pressure Prevails in Prandtls Boundary Layer
• If p and U are the free stream values of p and u, then Bernoulli’s theorem for steady flow yields along the body surface
xpxUppeuler 21
2
1
It is this p(x) which now applies in the boundary layer.
02
2
y
u
x
p
y
uv
x
uu
x
p
Thus the inviscid flow past the body determines the pressure variation which is then imposed on the boundary layer through the now known function in .
Prandtl’s Boundary Layer Equations
• We note that the system of equations given below are usually called the Prandtl boundary-layer equations.
02
2
y
u
x
p
y
uv
x
uu
0y
p
0
y
v
x
u
Closing Remarks on Prandtl’s Idea
• We have discussed here the essence of Prandtl’s idea without any indication of possible problems in implementing it for an arbitrary body.
• The main problem which will arise is that of boundary layer separation.
• It turns out that the function p(x), which is determined by the inviscid flow, may lead to a boundary layer which cannot be continued indefinitely along the surface of the body.
• What can and does occur is a breaking away of the boundary layer from the surface, the ejection of vorticity into the free stream, and the creation of free separation streamline.
• Separation is part of the stalling of an airfoil at high angles of attack.
Blasius’ solution for a semi-infinite flat plate
Prandtl’s boundary layer equations for flat plate:
02
2
y
u
y
uv
x
uu0
y
v
x
u
Subject to the conditions:
00at 0 &xyvu 0 &xyasUu
To find an exact solution for the velocity distribution, Blasius introduced the dimensionless coordinates as:
y
L
x &
10
NOMINAL BOUNDARY LAYER THICKNESS
xy
U
U
u
plate
Until now we have not given a precise definition for boundary layer thickness. Here we use to denote nominal boundary thickness, which is defined to be the value of y at which u = 0.99 U, i.e.
Uyxuy
99.0),(
x
y
u
U
u = 0.99 U
The choice 0.99 is arbitrary; we could have chosen 0.98 or 0.995 or whatever we find reasonable.
Streamwise Variation of Boundary Layer Thickness
Consider a plate of length L. Based on the Boundary Layer Approximations, maximum value of is estimated as
UL
L ReRe ,)(~ 2/1
2/12/1
~
U
LCor
U
L
2/12/1
~
U
xCor
U
x
xy
U
U
u
plateL
Similarly the local value of is estimated as
SIMILARITYOne triangle is similar to another triangle if it can be mapped onto the other triangle by means of a uniform stretching.
The red triangles are similar to the blue triangle.
The red triangles are not similar to the blue triangle.
Perhaps the same idea can be applied to the solution of our problem:
0,2
2
y
v
x
u
y
u
y
uv
x
uu
Uuvuyyy
,0,000
13
Similarity SolutionSuppose the solution has the property that when u/U is plotted against y/ (where (x) is the previously-defined nominal boundary layer thickness) a universal function is obtained, with no further dependence on x. Such a solution is called a similarity solution.
plate
xy
U
U
uu
U
x1 x2
00
1
1u/U
y/profiles at x1 and x2
00
1
1u/U
y/profile at x1
profile at x2
Similarity satisfied
Similarity not satisfied
Similarity is satisfied if a plot of u/U versus y/ defines exactly the same function regardless of
the value of x.
Discovery of SIMILARITY Variable
For a solution obeying similarity in the velocity profile we must have
)(1 x
yf
U
u
2/12/1
~
U
xCor
U
x
x
Uy
Ux
yf
U
u
2/1,)(
where f1 is a universal function, independent of x (position along the plate).
Since we have reason to believe that
We can rewrite any such similarity form as
Note that is a dimensionless variable.