Integral Method For Non-Zero Pressure Gradient Flows
A wide variety of "integral methods" in this category have been discussed by Rosenhead . The Thwaites method is found to be a very elegant method, which is an extension of the method due to Holstein and Bohlen . We shall discuss the Holstein-Bohlen method in this section.
This is an approximate method for solving boundary layer equations for two-dimensional generalized flow. The integrated Eq. (29.14) for laminar flow with pressure gradient can be written as
or
(30.11)
The velocity profile at the boundary layer is considered to be a fourth-order polynomial in terms of the dimensionless distance , and is expressed as
The boundary conditions are
A dimensionless quantity, known as shape factor is introduced as
(30.12)
The following relations are obtained
Now, the velocity profile can be expressed as
(30.13)
where
The shear stress is given by
(30.14)
We use the following dimensionless parameters,
(30.15)
(30.16)
(30.17)
The integrated momentum Eq. (30.10) reduces to
(30.18)
The parameter L is related to the skin friction
The parameter K is linked to the pressure gradient.
If we take K as the independent variable . L and H can be shown to be the functions of K since
(30.19)
(30.20)
(30.21)
Therefore,
The right-hand side of Eq. (30.18) is thus a function of K alone. Walz pointed out that this function can be approximated with a good degree of accuracy by a linear function of K so that
[Walz's approximation]
Equation (30.18) can now be written as
Solution of this differential equation for the dependent variable
subject to the boundary condition U = 0 when x = 0 , gives
With a = 0.47 and b = 6. the approximation is particularly close between the stagnation point and the point of maximum velocity.
Finally the value of the dependent variable is
(30.22)
By taking the limit of Eq. (30.22), according to L'Hopital's rule, it can be shown that
This corresponds to K = 0.0783.
Note that is not equal to zero at the stagnation point. If is determined from Eq. (30.22), K(x) can be obtained from Eq. (30.16).
Table 30.1 gives the necessary parameters for obtaining results, such as velocity profile and shear stress The approximate method can be applied successfully to a wide range of problems.
Table 30.1 Auxiliary functions after Holstein and Bohlen
K
12
0.0948
2.250
0.356
10
0.0919
2.260
0.351
8
0.0831
2.289
0.340
7.6
0.0807
2.297
0.337
7.2
0.0781
2.305
0.333
7.0
0.0767
2.309
0.331
6.6
0.0737
2.318
0.328
6.2
0.0706
2.328
0.324
5.0
0.0599
2.361
0.310
3.0
0.0385
2.427
0.283
1.0
0.0135
2.508
0.252
0
0
2.554
0.235
-1
-0.0140
2.604
0.217
-3
-0.0429
2.716
0.179
-5
-0.0720
2.847
0.140
-7
-0.0999
2.999
0.100
-9
-0.1254
3.176
0.059
-11
-0.1474
3.383
0.019
-12
-0.1567
3.500
0
0
0
0
0
0.2
0.00664
0.006641
0.006641
0.4
0.02656
0.13277
0.13277
0.8
0.10611
0.26471
0.26471
1.2
0.23795
0.39378
0.39378
1.6
0.42032
0.51676
0.51676
2.0
0.65003
0.62977
0.62977
2.4
0.92230
0.72899
0.72899
2.8
1.23099
0.81152
0.81152
3.2
1.56911
0.87609
0.87609
3.6
1.92954
0.92333
0.92333
4.0
2.30576
0.95552
0.95552
4.4
2.69238
0.97587
0.97587
4.8
3.08534
0.98779
0.98779
5.0
3.28329
0.99155
0.99155
8.8
7.07923
1.00000
1.00000
As mentioned earlier, K and are related to the pressure gradient and the shape factor.
Introduction of K and in the integral analysis enables extension of Karman-Pohlhausen method for solving flows over curved geometry. However, the analysis is not valid for the geometries, where and
Point of Seperation
For point of seperation,
or,
or,
End of Lecture
30!
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, and is expressed as 
is given by 
subject to the boundary condition U = 0 when x = 0 , gives 
is not equal to zero at the stagnation point. If 
is determined from Eq. (30.22), K(x) can be obtained from Eq. (30.16).
The approximate method can be applied successfully to a wide range of problems. 
are related to the pressure gradient and the shape factor.
and 
