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 .

Euler Solution of Wedge Flow

m

edge L

xUxU

xUpxp 212

1

121

2

1m

L

xUpxp

1

2

m

m

12

2

m

L

xm

U

dx

dp

Velocity Outside Boundary Layer for Cylinder

4

4

2

22 2cos21

r

R

r

RUxU

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.

The ejection of vorticity into the free stream

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.