Karman Pohlhausen Approximate Method

# Karman Pohlhausen Approximate Method | Fluid Mechanics for Mechanical Engineering 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 direction momentum equation (boundary layer momentum equation) with respect to from the wall (at y = 0) to a distance δX which is assumed to be outside the boundary layer. Using this notation, we can rewrite the Karman momentum integral equation as

• 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   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

(30.2)

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

(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

• 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 Tw in Eq. (30.5) we get,

whereC1 is any arbitrary unknown constant.

• The condition at the leading edge
Finally we obtain,

(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 --

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 n = 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 is related to the skin friction
• If we take as the independent variable and can be shown to be the functions of since

(30.19)

(30.20)

(30.21)

Therefore

• 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

• 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 = 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

(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 τω The approximate method can be applied successfully to a wide range of problems.

• As mentioned earlier, K and λ are related to the pressure gradient and the shape factor.
• Introduction of K and λ 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 λ  < - 12 and  λ  > +12

Point of Seperation

For point of seperation, τω =  0

The document Karman Pohlhausen Approximate Method | Fluid Mechanics for Mechanical Engineering is a part of the Mechanical Engineering Course Fluid Mechanics for Mechanical Engineering.
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## FAQs on Karman Pohlhausen Approximate Method - Fluid Mechanics for Mechanical Engineering

 1. What is the Karman Pohlhausen approximate method in mechanical engineering?
The Karman Pohlhausen approximate method is a widely used analytical technique in mechanical engineering for solving boundary layer problems. It provides an approximate solution to the Navier-Stokes equations by making simplifying assumptions and using empirical correlations. The method is particularly useful for calculating the velocity and temperature profiles in laminar boundary layers.
 2. When is the Karman Pohlhausen approximate method used in mechanical engineering?
The Karman Pohlhausen approximate method is commonly used in mechanical engineering when analyzing flow over a flat plate or a similar surface. It is especially applicable to laminar boundary layers, where the fluid flow near the surface is smooth and well-behaved. The method is less accurate for turbulent boundary layers, where the fluid flow is more chaotic.
 3. How does the Karman Pohlhausen approximate method work?
The Karman Pohlhausen approximate method works by assuming that the velocity and temperature profiles in the boundary layer can be approximated using a simple polynomial function. This function is then substituted into the governing equations, resulting in a set of ordinary differential equations. These equations can be solved numerically to obtain the velocity and temperature profiles in the boundary layer.
 4. What are the limitations of the Karman Pohlhausen approximate method?
While the Karman Pohlhausen approximate method is a useful tool in mechanical engineering, it has some limitations. Firstly, it is only applicable to laminar boundary layers and may give inaccurate results for turbulent flows. Secondly, the method assumes that the fluid properties remain constant along the boundary layer, which is not always the case in practical situations. Finally, the method neglects the effects of pressure gradients and three-dimensional flow, limiting its applicability in certain scenarios.
 5. Are there any alternative methods to the Karman Pohlhausen approximate method?
Yes, there are alternative methods to the Karman Pohlhausen approximate method for solving boundary layer problems in mechanical engineering. Some commonly used techniques include the integral method, the similarity solution method, and computational fluid dynamics (CFD) simulations. These methods may provide more accurate results for turbulent flows or complex geometries, but they often require more computational resources and expertise to implement.

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