Karman-Pohlhausen Approximate Method For Solution Of Momentum Integral Equation Over A Flat Plate | Fluid Mechanics for Civil Engineering - Civil Engineering (CE) PDF Download

Karman-Pohlhausen Approximate Method For Solution Of Momentum Integral Equation Over A Flat Plate
 

• The basic equation for this method is obtained by integrating the x direction momentum equation (boundary layer momentum equation) with respect to y from the wall (at y = 0) to a distance δ(x) which is assumed to be outside the boundary layer. Using this notation, we can rewrite the Karman momentum integral equation as

                          (30.1)

• The effect of pressure gradient is described by the second term on the left hand side. For pressure gradient surfaces in external flow or for the developing sections in internal flow, this term contributes to the pressure gradient. 
• We assume a velocity profile which is a polynomial of . η = y / δ, η being a form of similarity variable , implies that with the growth of boundary layer as distance x varies from the leading edge, the velocity profile u / U_∞ remains geometrically similar. 
• We choose a velocity profile in the form

                        (30.2)

In order to determine the constants a_0,a₁, a₂, and  a₃ we shall prescribe the following boundary conditions 

                        (30.3a)

                        (30.3b)

                        (30.3c)

                        (30.3d)

These requirements will yield       respectively
Finally, we obtain the following values for the coefficients in Eq. (30.2), 

 and the velocity profile becomes

                         (30.4)

For flow over a flat plate,  and the governing Eq. (30.1) reduces to

                        (30.5)

Again from Eq. (29.8), the momentum thickness is

The wall shear stress is given by

• Substituting the values of δ^** and τ_w in Eq. (30.5) we get,   

                      (30.6)

where C₁ is any arbitrary unknown constant.

• The condition at the leading edge (at x = 0, δ = 0) yields C₁ = 0
  Finally we obtain,

                      (30.7)

                      (30.8)

• This is the value of boundary layer thickness on a flat plate. Although, the method is an approximate one, the result is found to be reasonably accurate. The value is slightly lower than the exact solution of laminar flow over a flat plate . As such, the accuracy depends on the order of the velocity profile. We could have have used a fourth order polynomial instead --

                      (30.9)

• In addition to the boundary conditions in Eq. (30.3), we shall require another boundary condition at

• This yields the constants as  . Finally the velocity profile will be   

• Subsequently, for a fourth order profile the growth of boundary layer is given by

                      (30.10)

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 η = y / δ, 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 

• 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 τ_w The approximate method can be applied successfully to a wide range of problems. 

Table 30.1    Auxiliary functions after Holstein and Bohlen

• 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

Point of Seperation

For point of seperation