Karman-Pohlhausen Approximate Method For Solution Of Momentum Integral Equation Over A Flat Plate Mechanical Engineering Notes | EduRev

Fluid Mechanics

Civil Engineering (CE) : Karman-Pohlhausen Approximate Method For Solution Of Momentum Integral Equation Over A Flat Plate Mechanical Engineering Notes | EduRev

The document Karman-Pohlhausen Approximate Method For Solution Of Momentum Integral Equation Over A Flat Plate Mechanical Engineering Notes | EduRev is a part of the Civil Engineering (CE) Course Fluid Mechanics.
All you need of Civil Engineering (CE) at this link: Civil Engineering (CE)

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 direction momentum equation (boundary layer momentum equation) with respect to 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

Karman-Pohlhausen Approximate Method For Solution Of Momentum Integral Equation Over A Flat Plate Mechanical Engineering Notes | EduRev                          (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 varies from the leading edge, the velocity profile u / U remains geometrically similar. 
  • We choose a velocity profile in the form

Karman-Pohlhausen Approximate Method For Solution Of Momentum Integral Equation Over A Flat Plate Mechanical Engineering Notes | EduRev                         (30.2)

In order to determine the constants a0,a1, a2, and  a3 we shall prescribe the following boundary conditions 

Karman-Pohlhausen Approximate Method For Solution Of Momentum Integral Equation Over A Flat Plate Mechanical Engineering Notes | EduRev                         (30.3a)

 

Karman-Pohlhausen Approximate Method For Solution Of Momentum Integral Equation Over A Flat Plate Mechanical Engineering Notes | EduRev                         (30.3b)

Karman-Pohlhausen Approximate Method For Solution Of Momentum Integral Equation Over A Flat Plate Mechanical Engineering Notes | EduRev                         (30.3c)

Karman-Pohlhausen Approximate Method For Solution Of Momentum Integral Equation Over A Flat Plate Mechanical Engineering Notes | EduRev                         (30.3d)

These requirements will yield   Karman-Pohlhausen Approximate Method For Solution Of Momentum Integral Equation Over A Flat Plate Mechanical Engineering Notes | EduRev    respectively
Finally, we obtain the following values for the coefficients in Eq. (30.2), 

Karman-Pohlhausen Approximate Method For Solution Of Momentum Integral Equation Over A Flat Plate Mechanical Engineering Notes | EduRev and the velocity profile becomes

Karman-Pohlhausen Approximate Method For Solution Of Momentum Integral Equation Over A Flat Plate Mechanical Engineering Notes | EduRev                         (30.4)

For flow over a flat plate,  Karman-Pohlhausen Approximate Method For Solution Of Momentum Integral Equation Over A Flat Plate Mechanical Engineering Notes | EduRevand the governing Eq. (30.1) reduces to

Karman-Pohlhausen Approximate Method For Solution Of Momentum Integral Equation Over A Flat Plate Mechanical Engineering Notes | EduRev                        (30.5)

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

Karman-Pohlhausen Approximate Method For Solution Of Momentum Integral Equation Over A Flat Plate Mechanical Engineering Notes | EduRev

The wall shear stress is given by

Karman-Pohlhausen Approximate Method For Solution Of Momentum Integral Equation Over A Flat Plate Mechanical Engineering Notes | EduRev

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

Karman-Pohlhausen Approximate Method For Solution Of Momentum Integral Equation Over A Flat Plate Mechanical Engineering Notes | EduRev

Karman-Pohlhausen Approximate Method For Solution Of Momentum Integral Equation Over A Flat Plate Mechanical Engineering Notes | EduRev                      (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,

Karman-Pohlhausen Approximate Method For Solution Of Momentum Integral Equation Over A Flat Plate Mechanical Engineering Notes | EduRev                      (30.7)

Karman-Pohlhausen Approximate Method For Solution Of Momentum Integral Equation Over A Flat Plate Mechanical Engineering Notes | EduRev                      (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 --

Karman-Pohlhausen Approximate Method For Solution Of Momentum Integral Equation Over A Flat Plate Mechanical Engineering Notes | EduRev                      (30.9)

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

Karman-Pohlhausen Approximate Method For Solution Of Momentum Integral Equation Over A Flat Plate Mechanical Engineering Notes | EduRev

  • This yields the constants as Karman-Pohlhausen Approximate Method For Solution Of Momentum Integral Equation Over A Flat Plate Mechanical Engineering Notes | EduRev . Finally the velocity profile will be   

Karman-Pohlhausen Approximate Method For Solution Of Momentum Integral Equation Over A Flat Plate Mechanical Engineering Notes | EduRev

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

Karman-Pohlhausen Approximate Method For Solution Of Momentum Integral Equation Over A Flat Plate Mechanical Engineering Notes | EduRev                      (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

 

Karman-Pohlhausen Approximate Method For Solution Of Momentum Integral Equation Over A Flat Plate Mechanical Engineering Notes | EduRev
or

Karman-Pohlhausen Approximate Method For Solution Of Momentum Integral Equation Over A Flat Plate Mechanical Engineering Notes | EduRev                   (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

Karman-Pohlhausen Approximate Method For Solution Of Momentum Integral Equation Over A Flat Plate Mechanical Engineering Notes | EduRev

The boundary conditions are

Karman-Pohlhausen Approximate Method For Solution Of Momentum Integral Equation Over A Flat Plate Mechanical Engineering Notes | EduRev

  • A dimensionless quantity, known as shape factor is introduced as

Karman-Pohlhausen Approximate Method For Solution Of Momentum Integral Equation Over A Flat Plate Mechanical Engineering Notes | EduRev                   (30.12)

  • The following relations are obtained

Karman-Pohlhausen Approximate Method For Solution Of Momentum Integral Equation Over A Flat Plate Mechanical Engineering Notes | EduRev

  • Now, the velocity profile can be expressed as

Karman-Pohlhausen Approximate Method For Solution Of Momentum Integral Equation Over A Flat Plate Mechanical Engineering Notes | EduRev                   (30.13)

where

Karman-Pohlhausen Approximate Method For Solution Of Momentum Integral Equation Over A Flat Plate Mechanical Engineering Notes | EduRev

  • The shear stress Karman-Pohlhausen Approximate Method For Solution Of Momentum Integral Equation Over A Flat Plate Mechanical Engineering Notes | EduRev is given by

Karman-Pohlhausen Approximate Method For Solution Of Momentum Integral Equation Over A Flat Plate Mechanical Engineering Notes | EduRev                   (30.14)

  • We use the following dimensionless parameters,

Karman-Pohlhausen Approximate Method For Solution Of Momentum Integral Equation Over A Flat Plate Mechanical Engineering Notes | EduRev                   (30.15)

Karman-Pohlhausen Approximate Method For Solution Of Momentum Integral Equation Over A Flat Plate Mechanical Engineering Notes | EduRev                   (30.16)

Karman-Pohlhausen Approximate Method For Solution Of Momentum Integral Equation Over A Flat Plate Mechanical Engineering Notes | EduRev                   (30.17)

  • The integrated momentum Eq. (30.10) reduces to

Karman-Pohlhausen Approximate Method For Solution Of Momentum Integral Equation Over A Flat Plate Mechanical Engineering Notes | EduRev                   (30.18)

  • The parameter is related to the skin friction 
  • The parameter is linked to the pressure gradient.
  • If we take as the independent variable and can be shown to be the functions of since

Karman-Pohlhausen Approximate Method For Solution Of Momentum Integral Equation Over A Flat Plate Mechanical Engineering Notes | EduRev                   (30.19)

Karman-Pohlhausen Approximate Method For Solution Of Momentum Integral Equation Over A Flat Plate Mechanical Engineering Notes | EduRev                   (30.20)

Karman-Pohlhausen Approximate Method For Solution Of Momentum Integral Equation Over A Flat Plate Mechanical Engineering Notes | EduRev                   (30.21)

Therefore,

Karman-Pohlhausen Approximate Method For Solution Of Momentum Integral Equation Over A Flat Plate Mechanical Engineering Notes | EduRev

  • The right-hand side of Eq. (30.18) is thus a function of alone. Walz  pointed out that this function can be approximated with a good degree of accuracy by a linear function of so that Karman-Pohlhausen Approximate Method For Solution Of Momentum Integral Equation Over A Flat Plate Mechanical Engineering Notes | EduRev

 

  • Equation (30.18) can now be written as

Karman-Pohlhausen Approximate Method For Solution Of Momentum Integral Equation Over A Flat Plate Mechanical Engineering Notes | EduRev

Solution of this differential equation for the dependent variable Karman-Pohlhausen Approximate Method For Solution Of Momentum Integral Equation Over A Flat Plate Mechanical Engineering Notes | EduRev subject to the boundary condition  U = 0 when x = 0 , gives

Karman-Pohlhausen Approximate Method For Solution Of Momentum Integral Equation Over A Flat Plate Mechanical Engineering Notes | EduRev

  • With = 0.47 and = 6. the approximation is particularly close between the stagnation point and the point of maximum velocity.
  • Finally the value of the dependent variable is

Karman-Pohlhausen Approximate Method For Solution Of Momentum Integral Equation Over A Flat Plate Mechanical Engineering Notes | EduRev                   (30.22)

  • By taking the limit of Eq. (30.22), according to L'Hopital's rule, it can be shown that

Karman-Pohlhausen Approximate Method For Solution Of Momentum Integral Equation Over A Flat Plate Mechanical Engineering Notes | EduRev

This corresponds to K = 0.0783.

  • Note that Karman-Pohlhausen Approximate Method For Solution Of Momentum Integral Equation Over A Flat Plate Mechanical Engineering Notes | EduRev is not equal to zero at the stagnation point. If Karman-Pohlhausen Approximate Method For Solution Of Momentum Integral Equation Over A Flat Plate Mechanical Engineering Notes | EduRev 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 τw The approximate method can be applied successfully to a wide range of problems. 

Table 30.1    Auxiliary functions after Holstein and Bohlen

Karman-Pohlhausen Approximate Method For Solution Of Momentum Integral Equation Over A Flat Plate Mechanical Engineering Notes | EduRev

Karman-Pohlhausen Approximate Method For Solution Of Momentum Integral Equation Over A Flat Plate Mechanical Engineering Notes | EduRev

Karman-Pohlhausen Approximate Method For Solution Of Momentum Integral Equation Over A Flat Plate Mechanical Engineering Notes | EduRev

Karman-Pohlhausen Approximate Method For Solution Of Momentum Integral Equation Over A Flat Plate Mechanical Engineering Notes | EduRev

Karman-Pohlhausen Approximate Method For Solution Of Momentum Integral Equation Over A Flat Plate Mechanical Engineering Notes | EduRev

  • As mentioned earlier, and Karman-Pohlhausen Approximate Method For Solution Of Momentum Integral Equation Over A Flat Plate Mechanical Engineering Notes | EduRev are related to the pressure gradient and the shape factor. 
  • Introduction of and Karman-Pohlhausen Approximate Method For Solution Of Momentum Integral Equation Over A Flat Plate Mechanical Engineering Notes | EduRev 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, whereKarman-Pohlhausen Approximate Method For Solution Of Momentum Integral Equation Over A Flat Plate Mechanical Engineering Notes | EduRev

Point of Seperation

For point of seperation

Karman-Pohlhausen Approximate Method For Solution Of Momentum Integral Equation Over A Flat Plate Mechanical Engineering Notes | EduRev

Karman-Pohlhausen Approximate Method For Solution Of Momentum Integral Equation Over A Flat Plate Mechanical Engineering Notes | EduRev

Offer running on EduRev: Apply code STAYHOME200 to get INR 200 off on our premium plan EduRev Infinity!

Related Searches

ppt

,

past year papers

,

Karman-Pohlhausen Approximate Method For Solution Of Momentum Integral Equation Over A Flat Plate Mechanical Engineering Notes | EduRev

,

Semester Notes

,

MCQs

,

Exam

,

mock tests for examination

,

study material

,

Karman-Pohlhausen Approximate Method For Solution Of Momentum Integral Equation Over A Flat Plate Mechanical Engineering Notes | EduRev

,

shortcuts and tricks

,

practice quizzes

,

Extra Questions

,

Viva Questions

,

Previous Year Questions with Solutions

,

Free

,

Karman-Pohlhausen Approximate Method For Solution Of Momentum Integral Equation Over A Flat Plate Mechanical Engineering Notes | EduRev

,

Objective type Questions

,

pdf

,

video lectures

,

Important questions

,

Summary

,

Sample Paper

;