Integral Method For Non-Zero Pressure Gradient Flows

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,