l8 ch4 boundary layers

8/8/2019 L8 Ch4 Boundary Layers

1/12

Differential Analysis Laminar BL

Start with N-S Equations: 2D, steady flow

0!x

x

x

x

y

v

x

u

2

2

2

2

2

2

2

2

yv

xv

yp

yvv

xvu

y

u

x

u

x

p

y

uv

x

uu

xx

xx

xx!

xx

xx

x

x

x

x

x

x!

x

x

x

x

QQVV

QQVV

Simplify by doing order-of-magnitude analysis:

u>>v xp/xx >> xp/xy(recall pres.~const. normal to surface)

2

2

2

2

y

u

x

u

x

u

y

u

x

x""

x

x

x

x""

x

x


2/12

y-mom eliminated and x-mom simplifies to:

Differential Analysis (cont.)

2

2

y

u

x

p

y

uv

x

uu

x

x

x

x!

x

x

x

xQVV

For a flat, horizontal surface, dp/dx = 0

For a curved or flat surface angled relative to the flow, dp/dx nonzero.

From Eulers equation (inviscid flow outside of b.l. used to calculatepressure gradient):

dx

duu

dx

dp

x

p ee

e V!!x

x


3/12

Zero pressure gradient case: Blasius Equation

2

2

y

u

y

u

vx

u

u x

x

!x

x

x

x

QVV

Want to find u(x,y) such that at y=0, u=v=0 and at large y, u=ue, v=0

0!x

x

x

x

y

v

x

u

Convert PDEs above into an ODE using a coordinate transformation:

and a stream function (]) transformation: ),('x

vy

uff

u

u

e x

x!

x

x!!

x

x!

]]

L

x

uy e

RL

2!

Substitute these transformed variables into above PDEs:

x

u

x

u

x

x

x

x!

x

x

where ''fuu

e!x

x

y

u

y

u

x

x

x

x!

x

x

(basic idea is that at any x location, the shapeof the velocity profile u(y) will be similar, i.e.same function, and will always approach ue)


4/12

Blasius Solution

After some manipulations, the following ODE is obtained for f(L):

f f + f = 0

The solution to this may be found in Table 4.3.

Recall the b.l. thickness H was defined as location y where u/ue=0.99In the transformed variables, this means f=0.99

From Table 4.3, this corresponds to when L=3.5 (edge of b.l.)

Converting back to cartesian coordinates:x

x Re

0.5!

H


5/12

Additional Results for Flat Plate B.L.

From definitions of displacement and momentum thickness andskin friction:

xx Re

72.1*!

H

f

x

Cx

!!Re

664.0U

x

ue3

332.0Q

X !

L

fL

C

Re

328.12!!

U

Total skin-friction coefficient over plate of length L:


6/12

Flow with a Pressure Gradient

So far we neglected the pressure variation along the flow in aboundary layer

This is not valid for boundary layer over curved surface like airfoil

Owing to objects shape the free stream velocity just outside theboundary layer varies along the length of the surface.

As per Bernoullis equation, the static pressure on the surface of theobject, therefore, varies in x- direction along the surface.

There is no pressure variation in the y- direction within the boundarylayer. Hence pressure in boundary layer is equal to that just outsideit.

As this pressure just outside of a boundary layer varies along x axisthat inside the boundary layer also varies along x axis


7/12

Pressure Gradients

0dx

dp 0"dx

dp

Favorable pressuregradient

Adverse pressuregradient


8/12

In a situation where pressure increases down stream the fluidparticles can move up against it by virtue of its kinetic energy.

Inside the boundary layer the velocity in a layer could reduce somuch that the kinetic energy of the fluid particles is no longeradequate to move the particles against the pressure gradient.

This leads to flow reversal.

Since the fluid layer higher up still have energy to mover forward arolling of fluid streams occurs, which is called separation

Flow Separation


9/12

Influence of a

strongpressuregradient on aturbulent flow:(a) a strongnegative

pressuregradient mayre-laminarizea flow;(b) a strongpositive

pressuregradientcauses astrongboundary layertop thicken.

(Photographby R.E. Falco)

(a)

(b)

Pressure Gradient in Turbulent B.L.


10/12

Separation starts with zero velocity gradient at the wall

Flow reversal takes place beyond separation pointdP/dx>0

Adverse pressure gradient is necessary for separation

There is no pressure change after separation So,pressure in the separated region is constant.

Fluid in turbulent boundary layer has appreciably moremomentum than the flow of a laminar B.L. Thus a turbulent B.Lcan penetrate further into an adverse pressure gradientwithout separation

Separation


11/12

Smooth ball Rough ball


12/12

Flows With Pressure Gradient: Falkner-Skan Equation

2

2

y

u

dx

du

uy

u

vx

u

ue

ex

x

!x

x

x

x

QVVV

s

s

yue

RL

2!Transformations: 'f

f

u

u

e

!x

x!

L

Resulting ODE: f f + f + [1-(f)2]F = 0

whereds

du

u

s e

e

2!F represents the pressure gradient term

For F negative, dp/dx>0 (decelerating flow)For F positive, dp/dx

l8 ch4 boundary layers

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