Download Problem I

Fluid Mechanics II
Laminar Boundary Layer
Prof. António Sarmento
Tel. 21 8417405
Email: [email protected]
Problem I
Consider a permanent flow over a horizontal flat plate with a leading edge at x=0, where the
undisturbed velocity U is parallel to the plate. It can be verified that due to the no slip
condition, the flow suffers a decrease of velocity close to the plate, originating a slow
divergence of the streamlines from the plate with the longitudinal direction x. The thickness of
the flow region in which the velocity is affected by the wall (called boundary layer – where
viscous effects occur) varies with the distance x to the leading edge of the plate and is called
the boundary layer thickness, usually represented by . Outside the boundar layer, the velocity
is equal to the undisturbed velocity U.
y
U
U
x
(x)
a) Show that the displacement * (the displacement thickness – and represented by d in
Fluid Flow (Sabersky)) with respect to the undisturbed conditions suffered by the
external streamlines (those further away from the wall than (x)) is given by

u

*
   1  dy
U
0
b) Show that the drag (friction force) acting on the plate between the leading edge and an
arbitrary distance x is given by
Fat  U 2 ,
where , the momentum thickness (represented by m in Fluid Flow), is given by

1
  2  U  u udy .
U 0
Based on your results, how do you physically interpret ?
Consider a laminar flow of air (with properties =1,2 kg/m3, =1,5  10-5 m2/s), where the
undisturbed velocity, parallel to the plate, is U=10 m/s. In these conditions, the boundary
layer thickness varies with the distance x to the leading edge of the plate according to
 5
x
U
.
Under these conditions the velocity profile can be approximated by
 y  y 2 
u  U 2    
     
if y
and
u=U
if y>U
c) For the given velocity profile, relate * with .
d) At what distance from the leading edge does the streamline that sits at 1mm from the
plate at the undisturbed region enter the boundary layer (i.e., it becomes closer to the
plate than (x))?
NOTES

 Recall that the mass flow is calculated using m    V .n ds and that the momentum

flow is given by M   V V .n ds
S
S

Recall that the mass flow equation applied to a control volume CV can be written as
dM CV
 m in  m out ,
dt
where M CV represents the mass within the CV at any time t.
 Recall that the momentum flow equation applied to a CV can be written as


dKCV
 m in  m out  FCV ,
dt


where K CV represents the momentum within the CV at any time t and FCV the force
exerted over the CV.
METODOLOGY

In order to solve question a) consider a streamline at a distance b0 from the plate in
an undisturbed section and at distance b>b0 at an arbitrary x section. Afterwards
calculate the momentum flow around the CV formed by that same streamline and the
plate, and by the transverse sections upstream the leading edge and x section. Make
sure you understand that the velocity profile is uniform for the section upstream the
leading edge and outside the boundary layer (this is, for b>y>) at the x section.


In order to solve question b) calculate the momentum flow balance on the CV
considered in the previous question. Try to introduce the definition of displacement
thickness in your solution for the friction force over the plate
In order to solve question d) use the answers from questions a) and c) and the
boundary layer momentum integral.
Fluid Flow 8.3
Consider a stationary and laminar flow of an incompressible fluid over a flat plate, parallel to
the undisturbed flow. The plate is porous and the fluid is aspirated through it at a constant
rate, such that the limit layer thickness does not vary in the flow direction, u x  0 .
a)
Using the limit layer equations, show that

 v y 
u y   u 0 1  exp  0  ,
  

where v0 is the aspiration velocity (constant along the x direction and negative
along the y direction).
b) Calculate the displacement thickness and the momentum flow.
(Answer: -/v0 and -2/v0 , note that since v0 is negative, the result turns out being
positive)
c) Calculate the plate drag coefficient (consider only one of the sides, and length L).
(Answer: -Uv0L)
NOTES
 The limit layer is the flow region where viscous effects are present, meaning
u y  0 . By definition, this region corresponds to the region where the fluid’s
velocity is under 99% of the undisturbed velocity (or outside velocity) U. It is also
the region 0<y<,  being the solution to y()=0,99U, named limit layer thickness.
Above, y represents the distance to the plate.
 The limit layer is known as slim when (x)<<x, where x is the tangent direction to
the plate. In these conditions the streamlines are almost parallel to the plate,
and p y  0 .
 The limit layer equations are as follow:
u v
o Continuity equation:

0
x y
u
u
1 dp
 2u
v
  u e  2
o Boundary layer equation: u
x
y
 dx
y
where pe is the pressure outsider the limit layer.


u
u


 By definition the displacement thickness is  d   1  dy   1  dy . The
U
U
0
0
displacement thickness can also be represented by *.



uu
uu


By definition the momentum thickness is  m   1   dy   1   dy . The
U U
U U
0
0
momentum thickness can also be written as .
METODOLOGY

In order to solve question a) simplify the continuity equation and come to the
conclusion that the fluid velocity v is constant and equal to v0 throughout the whole flow;
following that, simplify the momentum flow and integrate it, obtaining the velocity
profile. In order to integrate use the Separation of variables method, and note that
dx
1
 a  bx  b ln a  bx  .
Use the appropriate boundary conditions at y=0 and y=.
 In order to calculate the displacement thickness and momentum flow, one only needs
to be aware of the definitions and the velocity profile deduced.
 The drag is a result of the friction force on the plate. To obtain it is necessary to
calculate the shear stress on the wall and integrate it along the length of the plate.
Fluid Flow 8.6
Calculate the shear stress on the plate, the boundary layer thickness and drag over one of the
side of the plate, of length l, admitting the limit layer is laminar throughout the whole plate,
using the following approximations for the velocity profile: i) sinusoidal; ii) parabolic; iii)
linear. Compare your results using the Blasius equation. Using your results for a sinusoidal
velocity profile, calculate the numerical values for drag and limit layer thickness at the end of
the plate, for a plate with 0,3m in length, 0,3m in width, and immersed in a water flow
(T=20ºC, 1000 kg/m3, =1,1310-3 Pa.s) if U=7 m/s .
(Answer:
1
0,656
4,789 x
1
1,312
, 
, D  U 2 l
U 2
2
2
Re x
Re x
Re l
1
0,729
5,484 x
1
1,558
ii)  0  U 2
, 
, D  U 2 l
2
2
Re x
Re x
Re l
1
0,578
1
1,156
3,461x
iii)  0  U 2
, 
, D  U 2 l
2
2
Re x
Re l
Re x
i)  0 
D=0,066 N, =3,3 mm.)
NOTES
 As shown in Fluid Flow, admitting the velocity profile is given by u U  f  y  
one obtains the following results for the laminar limit layer:
2 a
2 2 a
0
2 x
D
, 
and C D 
cf 


1
1
a Re x
Re x
Re l
U 2
U 2 A
2
2
where
1
 d u U  
u  u  y


  
and a   1   d  
U U  
 d  y    y 0
0

The approximated velocity profile has to be in agreement with the following
u
 y 
 sin 
conditions: u(0)=0 and u()=U, from where
 for the sinusoidal case,
U
2
2
u
y  y
u
y
 for the linear case.
 2    for the parabolic case and
U
  
U 
 The values calculated from the Blasius equation are as follows:
0
D
1,328
0,6641
5x

, 
and C D 
cf 

1
1
Re l
Re x
Re x
U 2 A
U 2
2
2
METODOLOGY
 Start by calculating the values for  and a for each of the profiles.
 Use the general equations for cf,  and cD in order to calculate the expressions for
each parameter.
 Replace the values given in the deduced expressions in order to obtain the numerical
values asked for.
Problem II
Consider a flat thin plate (this is, of negligible thickness), very wide, and of length L=2 m,
aligned with an air flow (=1,2 kg/m3, =1,810-5 Pa.s) for which the undisturbed velocity is
U=2 m/s. Over the plate, the longitudinal pressure gradient is 0. Admit the transition from
laminar to turbulent flow happens for a Reynolds number Rex=106. Use the following
17
equation when the flow is turbulent, u U   y   .
a) Using the Blasius equation when appropriate, determine the limit layer thickness on
sections S1 and S2, each at distances x1=0,75 m and x2=1,5 m. Verify this is a slim limit
layer.
(Answer: 0,0119 m e 0,0168 m)
b) Calculate the mass flow and momentum flow across the limit layer in these two
sections. How do you stand by the difference in the values obtained?
(Answer: 0,0187 kg/s/m e 0,0264 kg/s/m; 0,02976 N/m e 0,04224 N/m)
c) Calculate the y coordinate at x1 of the streamline passing through x2=1,5 and y=.
What significance do you give to the vertical displacement for that streamline between
the two sections? (Answer: 0,0151 m)
d) Estimate the distributed load along the width between the sections S1 and S2.
(Answer: 0,000031 N/m)
e) Refer to the following statement as true or false, and justify: ‘For the given conditions,
if the plate is infinitely long, after the laminar-to-turbulent flow transition, the limit
layer would eventually separate itself.’
NOTES
 The exact solution for the continuity and momentum flow equations of a stationary
two dimension laminar limit layer of an incompressible fluid is given by the Blasius
solution (see notes for problem 8.6)
 The mass flow through a transvers section of the limit layer is given by U(-d).
Note that the Blasius solution, shows that d=0,334.
 The momentum flow through a transverse section of the limit layer is given by
U2(-d-m). Note that the Blasius solution shows that m=0,133 .
METODOLOGY
 Start by verifying that the flow is laminar in the two sections, by comparing the local
Reynolds number with the transition one.
 Use the Blasius solution results presented in the last note from problem 8.6 on
question a).
 Use the given equations to answer to question b).
 Note that the divergence of the outside streamlines is given by the displacement
thickness of the limit layer in the answer to question c).
 Use carefully the result for CD from the Blasius solution (check the last note from
problem 8.6) when calculating the answer for question d).