PROBLEM 4.E – Flat plate boundary layer with uniform suction – integral method
The boundary layer on a flat plate with uniform suction (ue(x) = U = constant; vw = constant <0)
displays a non–similar development. (see the figure below and section 4–5.2 in White).
Near the leading edge the boundary layer behaves like the Blasius solution, while it develops
towards the uniform suction solution for large distance downstream. Properties of these two
solutions according to the exact theory are:
1) Blasius solution: 2
2 0.66411w
x U U x
2) Suction profile: 1u( ) = e U
where: wyv
Here we apply an integral analysis to compute the (approximate) development of the boundary
layer, in particular of the momentum thickness. Under the considered conditions the integral
momentum equation is given by:
2
2
w w w v vU d + S + dx U where: =
wS U and with: ( 0) 0x
The shear–parameter S is only a function of the shape of the velocity profile and not of its thickness.
a. The effect of the (constant) parameters U, vw and ν can be conveniently incorporated in a
coordinate scaling:
2
wv= x Uand * wv= (remember that 0wv !)
– Verify that with these definitions the integral momentum equation can be written as:
2* 2 * d= Sd
with: *( 0) 0
– Determine from the exact solutions for the Blasius and suction boundary layer (given
above) what the solution for should be for small and large values of , respectively, and
check if these two (asymptotic) results are in agreement with the character of the differential
equation given here.
(1) VISCOUS FLOWS – AE 4120 5
b. The solution of the momentum integral equation requires the value of S, which is not a
constant for the entire boundary layer, because the shape of the velocity profile develops
with. We may attempt to simplify the problem by using a constant value of S.
– Determine from the exact solution data given above the values of S for the Blasius solution
and for the suction profile. Integrate the momentum integral equation (numerically) over the
domain0 5 , for each of these values of S.
– Plot the results graphically as 2* . vs and compare it to the accurate results of a finite–
difference solution of the problem, given in the table below. Also indicate in the graph the
asymptotes of the exact solution as determined in part (a). Comment on the accuracy of the
followed approach and the impact of the value of S?
c. As the shape variation has an impact on the computation of the boundary layer, it seems a
logical step to try to improve the method by introducing a shape parameter λ, similar as in
the method of Thwaites, such that:( ) S . Thwaites’method cannot be applied directly,
however, as it does not take the effect of suction (both on the shape and in the momentum
equation) into account.
– The original definition of the shape factor involves the curvature of the velocity profile
near the wall:
2 2
2e w
u
u y
Evaluate the (differential) x–momentum equation at the wall to show how λ for the general
case is related to the pressure gradient and the wall suction velocity. Verify that it reduces to
the familiar expression of Thwaites when vw = 0. Derive that for the present problem (where
ue(x) = U= constant) it can be directly related to S and *. Determine the values of λ for
the Blasius and the suction profiles.
– Assume that there is approximately a linear relation between S and λ: S A B, and
determine the constants such that the Blasius and suction–profile values are reproduced.
Show that for the current problem the result can be written as: *( ) S S .
– Repeat the numerical integration, now with this modified (adaptive) relation for S.
Compare the outcome with the previous results and comment if it matches your
expectations.
Table: Numerical results for the flat plate boundary layer with uniform suction.
ξ *S
0
0.1
0.2
0.3
0.4
0.5
0.7
1.0
1.4
2.0
3.0
4.0
5.0
0
0.1747
0.2293
0.2657
0.2930
0.3147
0.3475
0.3816
0.4113
0.4395
0.4652
0.4787
0.4864
0.2205
0.3008
0.3290
0.3487
0.3640
0.3766
0.3961
0.4172
0.4364
0.4555
0.4736
0.4835
0.4893
(1) VISCOUS FLOWS – AE 4120 6
PROBLEM 5.A – Heat transfer from an isothermal flat plate
Consider the development of the thermal boundary layer due to the heat transfer from a heated wall
(compressibility and viscous dissipation are neglected). For a flat–plate flow (i.e. constant velocity U in
the external flow) over an isothermal surface (i.e. constant wall temperature Tw) a self–similar thermal
boundary layer results, where the shape of the temperature profile still depends on the value of the
Prandtl number Pr.
The integral temperature equation, which reads as follows for the problem under consideration: 0
( ) Pr
w
p w
qd T u T T dydx c y
can be used for an approximate estimation of the thermal boundary layer development, when assumed
shapes of the velocity and temperature profiles are prescribed. Let now both profiles be approximated
by a linear relation, i.e.:
(0 )u y yU
and 1 ( )u yU 1 (0 )T
w T
T T y yT T
and 0 ( )T
w
T T yT T
For the case that δT δ it has been shown during the lectures that: 1/ 3 2 / 3/ ( ) 1Pr Pr/ Pr
w p wT
w
q c T Ts U
where s is the Reynolds–analogy factor. The first expression directly reveals that this situation
corresponds to the case that Pr 1.
a. Derive the corresponding expressions for ζ and s for the case that δT δ. Show that this
corresponds to Pr 1.
Hints: split the integral in two parts for 0 y δ and for δ y δT. Eliminate δT by introducing
ζ (= constant!); δ follows from the integral momentum equation.
b. Investigate the behaviour of both expressions for the limit when Pr goes to zero.
N.B.: by “behaviour” it is meant not just the limiting values, but in particular the way in which
ζ and s are related to Pr.
c. Give the results of the complete approximation, i.e. for Pr 1 and Pr 1 together, in the form
of a graph where 1/ζ and 1/s have been plotted against Pr, on the interval 0 Pr 5. Show that
for the limit of Pr going to 1 the results of both approximations match smoothly (continuous in
both value and first derivative).
N.B.: for this it is sufficient to prove that for ζ and dζ/dPr both limits (i.e. for Pr coming from
either side of 1) are equal.
(1) VISCOUS FLOWS – AE 4120 7
d. For Pr << 1 the exact solution for the temperature profile can be obtained analytically. As in
this case δ << δT, it can be assumed in approximation that the development of the thermal
boundary layer takes place completely in the inviscid outer flow. This allows the temperature
equation to be simplified to: 2
2Pr
T T U x y
– Derive, by means of a similarity transformation, that the above equation can be written as
follows: ” 2 ‘ 0
with Θ the transformed temperature profile and η the transformed y–coordinate. Give the
expression for η and determine the boundary conditions which θ has to satisfy.
N.B.: this η–scaling is not the usual Blasius scaling, which applies to the velocity profile!
– Show that the solution of this equation can be written in terms of the Gaussian error–function
(see remark below), and derive from this the heat transfer at the wall, expressed by the local
Nusselt number: Nu ( )
wx
w
q x
k T T
as function of Rex and Pr.
– Calculate from this the expression for the Reynolds analogy factor s, if given that the wall
shear stress according to the Blasius solution is: 0.332w
UU x
Compare this result to the approximation obtained in part (b).
Remark: the error function is: 2
0
2( )
x terf x e dt
, with ( ) 1erf .
(1) VISCOUS FLOWS – AE 4120 8
PROBLEM 6.B – Instability and transition estimates of laminar boundary layers
A rough estimate of the location of the point of instability (critical point) or transition can be obtained
from semi–empirical correlations, such as e.g.:
– for the point of instability: ,Re exp(26.3 8 )crit H (after Wieghardt)
– for the point of transition: 0.4, , Re 2.9(Re )trans x trans(after Michel)
a. Apply these correlations to determine the point of instability and the point of transition for the
following self–similar boundary layer flows.
Give in both cases the values for Rex as well as Reθ.
–i– flat plate flow: 1/ 22.591 0.664 RexH x
–ii– stagnation point flow: 1/ 22.216 0.292 RexH x
b. Based on these estimates, what can you conclude about the effect of the pressure gradient on
the stability of a laminar boundary layer?
c. The velocity profile of the asymptotic suction boundary layer is given by the exponential
function: 1 with: w
e
yvu eu
where wv is the normal velocity at the wall (for suction wv is negative).
Determine with the given correlation for the instability point, how large the suction velocity wv
must be chosen, in order to keep the boundary layer on the margin of stability.
(1) VISCOUS FLOWS – AE 4120 9
PROBLEM 7.C – The Clauser plot
The so–called Clauser–plot technique is used to determine the wall shear stress in a turbulent boundary
layer from the mean velocity profile, in conditions where an accurate, direct determination of the
velocity gradient at the wall is not possible (which is the common situation for turbulent boundary
layers).
The table given below represents the measured velocity profile, such as determined e.g. by means of a
traverse with a pitot tube or a hot–wire probe. As a function of the distance y from the wall (measured
in mm), the velocity is given in non–dimensional form as u/Ue, where u is the local (mean) velocity in
the boundary layer, and Ue the velocity in the external flow.
a. Plot the velocity profile on a semi–logarithmic scale, with u/Ue versus Re /y e yU .
b. Determine the value of the skin friction coefficient Cf by means of a curve fit of the law of the
wall to (the lower part of) the velocity profile.
c. Determine also the strength of the ‘wake–component’ of the velocity profile. For a measured
profile, this is defined as the maximum difference between u(y) and the law of the wall
expression.
Further data: = 15.0*10–6 m2/s
Ue = 9.804 m/s
y(mm) U/Ue
1 0.25 0.219
2 0.50 0.351
3 1.00 0.479
4 1.50 0.535
5 2.00 0.565
6 2.50 0.589
7 3.00 0.607
8 3.50 0.621
9 4.00 0.633
10 4.50 0.644
11 5.00 0.654
12 6.00 0.672
13 7.00 0.687
14 8.00 0.700
15 9.00 0.712
16 10.00 0.723
17 12.00 0.744
18 14.00 0.762
19 16.00 0.779
20 18.00 0.796
y(mm) U/Ue
21 20.00 0.813
22 22.00 0.830
23 24.00 0.847
24 26.00 0.864
25 28.00 0.881
26 30.00 0.899
27 32.00 0.915
28 34.00 0.931
29 36.00 0.946
30 38.00 0.960
31 40.00 0.973
32 42.00 0.983
33 44.00 0.992
34 46.00 0.998
35 48.00 1.000
36 50.00 1.000
37 52.00 1.000
(1) VISCOUS FLOWS – AE 4120 10
PROBLEM 8.A – Turbulence scaling in the wall region + damping functions
The wall region of a (2D, incompressible) turbulent boundary layer displays a universal structure,
which is known as the law of the wall. According to this concept, the properties of the flow can be
expressed uniquely in terms of the so–called wall scaling (also referred to as the use of ‘wall units’).
In addition, it is commonly assumed that across the wall region the total shear stress is approximately
constant: constantvisc turb w
One element of this law of the wall is that the velocity profile of the mean flow satisfies an expression
of the following type:
( ) u f y; where: *; with: * /* w
u yvu y vv
a. Derive from this wall scaling, by means of an elementary dimensional analysis, the scaling of
the following (kinematic) properties of the turbulence:
1. the (specific) turbulent shear stress: / ‘ ‘turb u v
2. the eddy viscosity: t
3. the mixing–length: mixl
In the scaling scale only the kinematic properties that are relevant for the wall region (*v and
) are used; in other words: determine for each of the above variables how they are to be non–
dimensionalised, such that: dimensionless property = function of y+.
b. Show for each of these properties how the corresponding “function of y+” is related to the
gradient of the velocity profile of the mean flow, that is given by: ‘( ) df duf y dy dy
c. Consider the behaviour of the expressions found in (b), for the limit y+>>1, which constitutes
the overlap region, in which the logarithmic velocity profile holds.
Use the results to show that in this overlap region the original (i.e. non–scaled) kinematic
turbulence properties mentioned in (a), are independent of viscosity.
(1) VISCOUS FLOWS – AE 4120 11
The effect of the viscosity on the turbulence in the wall region is often described by using so–called
‘damping functions’, which are defined as:
‘effective value’ = damping function * ‘fully turbulent value’
where the ‘fully turbulent value’ corresponds to the behaviour of a variable in the overlap region, where
the effect of the viscosity can be neglected.
In correspondence to what has been derived in (b), the damping functions for the eddy–viscosity and
for the mixing–length can be derived from the shape of the velocity profile.
d. An accurate description of the velocity profile in the entire wall layer, including the viscous
region, is given by Spalding’s implicit expression, which reads: 2 3 ( ) ( )1 2 6
B u u u y u e e u
Show that this expression satisfies the two familiar limits of the law of the wall:
y+<<1: u y (viscous sublayer)
y+>>1: 1 ln( )u y B
(overlap region)
e. – Derive the expressions for both damping functions (i.e., for the eddy viscosity and for the
mixing length), as they follow from Spalding’s law of the wall.
– Investigate the behaviour of both damping functions for small values of y+ (i.e. determine the
first term of the series expansion for small y+).
– Plot both damping functions for the interval 0 y+ 100. In the calculations use for the
constants in the law of the wall the standard values κ = 0.41 and B = 5.0.
Hint: maintain in the derivations u+ as the independent variable, and make use of the relation
that: 1/ /du dy dy du
