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Blasius Flow Over A Flat Plate

  • The classical problem considered by H. Blasius was
    1. Two-dimensional, steady, incompressible flow over a flat plate at zero angle of incidence with respect to the uniform stream of velocity U.
    2. The fluid extends to infinity in all directions from the plate. 

 

 

  • Blasius wanted to determine 
    (a) the velocity field solely within the boundary layer, 
    (b) the boundary layer thickness(δ), 
    (c) the shear stress distribution on the plate, and 
    (d) the drag force on the plate. 
  • The Prandtl boundary layer equations in the case under consideration are

Introduction to Laminar Boundary Layers - 2 | Fluid Mechanics for Mechanical Engineering

The boundary conditions are

 

Introduction to Laminar Boundary Layers - 2 | Fluid Mechanics for Mechanical Engineering  

  • Note that the substitution of the term  Introduction to Laminar Boundary Layers - 2 | Fluid Mechanics for Mechanical Engineering  in the original boundary layer momentum equation in terms of the free stream velocity produces Introduction to Laminar Boundary Layers - 2 | Fluid Mechanics for Mechanical Engineering which is equal to zero.
  • Hence the governing Eq. (28.15) does not contain any pressure-gradient term.
  • However, the characteristic parameters of this problem are U∞, x, v,y  that is, u = u U∞, x, v,y 
  • This relation has five variables U∞, x, v,y .
  • It involves two dimensions, length and time. 
  • Thus it can be reduced to a dimensionless relation in terms of (5-2) =3 quantities ( Buckingham Pi Theorem)
  • Thus a similarity variables can be used to find the solution 
  • Such flow fields are called self-similar flow field .

Law of Similarity for Boundary Layer Flows

  • It states that the component of velocity with two velocity profiles of u(x,y) at different locations differ only by scale factors in and .  
  • Therefore, the velocity profiles u(x,y) at all values of can be made congruent if they are plotted in coordinates which have been made dimensionless with reference to the scale factors.
  • The local free stream velocity U(x) at section is an obvious scale factor for u, because the dimensionless u(x) varies between zero and unity with at all sections. 
  • The scale factor for , denoted by g(x) , is proportional to the local boundary layer thickness so that itself varies between zero and unity. 
  • Velocity at two arbitrary locations, namely x1 and x2 should satisfy the equation 

Introduction to Laminar Boundary Layers - 2 | Fluid Mechanics for Mechanical Engineering                                  (28.17) 

  • Now, for Blasius flow, it is possible to identify g(x) with the boundary layers thickness δ we know

 

Introduction to Laminar Boundary Layers - 2 | Fluid Mechanics for Mechanical Engineering

 

Thus in terms of x we get       

Introduction to Laminar Boundary Layers - 2 | Fluid Mechanics for Mechanical Engineering

i.e

Introduction to Laminar Boundary Layers - 2 | Fluid Mechanics for Mechanical Engineering                                           ( 28.18)

 

where  Introduction to Laminar Boundary Layers - 2 | Fluid Mechanics for Mechanical Engineering

 

or more precisely,

 

  Introduction to Laminar Boundary Layers - 2 | Fluid Mechanics for Mechanical Engineering                                                                 (28.19)

 

Introduction to Laminar Boundary Layers - 2 | Fluid Mechanics for Mechanical Engineering

The stream function can now be obtained in terms of the velocity components as

 

Introduction to Laminar Boundary Layers - 2 | Fluid Mechanics for Mechanical Engineering

Introduction to Laminar Boundary Layers - 2 | Fluid Mechanics for Mechanical Engineering                                (28.20)

 

where D is a constant. Also  Introduction to Laminar Boundary Layers - 2 | Fluid Mechanics for Mechanical Engineering   and the constant of integration is zero if the stream function at the solid surface is set equal to zero.

Now, the velocity components and their derivatives are: 

 

Introduction to Laminar Boundary Layers - 2 | Fluid Mechanics for Mechanical Engineering                             (28.21a)


 

 

Introduction to Laminar Boundary Layers - 2 | Fluid Mechanics for Mechanical Engineering

or 

Introduction to Laminar Boundary Layers - 2 | Fluid Mechanics for Mechanical Engineering                                                       (28.21b)                

 

Introduction to Laminar Boundary Layers - 2 | Fluid Mechanics for Mechanical Engineering                     (28.21c)      

Introduction to Laminar Boundary Layers - 2 | Fluid Mechanics for Mechanical Engineering                                 (28.21d)      

Introduction to Laminar Boundary Layers - 2 | Fluid Mechanics for Mechanical Engineering                                              (28.21e)      

 

Substituting (28.2) into (28.15), we have

Introduction to Laminar Boundary Layers - 2 | Fluid Mechanics for Mechanical Engineering

or


Introduction to Laminar Boundary Layers - 2 | Fluid Mechanics for Mechanical Engineering                                                                         (28.22)

where 

Introduction to Laminar Boundary Layers - 2 | Fluid Mechanics for Mechanical Engineering

and 
Introduction to Laminar Boundary Layers - 2 | Fluid Mechanics for Mechanical Engineering
This is known as Blasius Equation .

 

Contd. from Previous Slide

The boundary conditions as in Eg. (28.16), in combination with Eg. (28.21a) and (28.21b) become

Introduction to Laminar Boundary Layers - 2 | Fluid Mechanics for Mechanical Engineering                                      (28.23)            

 

 

 

Equation (28.22) is a third order nonlinear differential equation .

  • Blasius obtained the solution of this equation in the form of series expansion through analytical techniques 
  • We shall not discuss this technique. However, we shall discuss a numerical technique to solve the aforesaid equation which can be understood rather easily. 
  • Note that the equation for f does not contain x .  
  • Boundary conditions at x = 0 and y = ∞ merge into the condition Introduction to Laminar Boundary Layers - 2 | Fluid Mechanics for Mechanical Engineering . This is the key feature of similarity solution. 
  • We can rewrite Eq. (28.22) as three first order differential equations in the following way

f' = G                            ( 28.24a)

G = H                           (28.24b)

Introduction to Laminar Boundary Layers - 2 | Fluid Mechanics for Mechanical Engineering                    (28.24c)

 

  • Let us next consider the boundary conditions. 

 

  1. The condition f (0)  = 0 remains valid. 
  2. The condition  f' (0) = 0  means that  G (0) = 0   . 
  3. The condition   f'' (∞)  = 1   gives us G (∞) = 1   . 

Note  that the equations for and have initial values. However, the value for H(0) is not known. Hence, we do not have a usual initial-value problem.

Shooting Technique 

We handle this problem as an initial-value problem by choosing values of H(0) and solving by numerical methods F (n) , G (n)  and H(n) . 

In general, the condition G (∞) = 1. will not be satisfied for the function G  arising from the numerical solution. 
We then choose other initial values of H so that eventually we find an H(0) which results in G (∞) = 1  . 
This method is called the shooting technique .

  • In Eq. (28.24), the primes refer to differentiation wrt. the similarity variable n. The integration steps following Runge-Kutta method are given below.

 

Introduction to Laminar Boundary Layers - 2 | Fluid Mechanics for Mechanical Engineering                                   (28.25a)

 

Introduction to Laminar Boundary Layers - 2 | Fluid Mechanics for Mechanical Engineering                                 (28.25b)

Introduction to Laminar Boundary Layers - 2 | Fluid Mechanics for Mechanical Engineering                        (28.25c)

 

  • One moves from nn +1 =nn . A fourth order accuracy is preserved if is constant along the integration path, that is, nn +1 - n =h for all values of . The values of k , l and are as follows. 
  • For generality let the system of governing equations be

Introduction to Laminar Boundary Layers - 2 | Fluid Mechanics for Mechanical Engineering

Introduction to Laminar Boundary Layers - 2 | Fluid Mechanics for Mechanical Engineering

 

In a similar way  k3 , l3, mand k4 , l4, mmare calculated following standard formulae for the Runge-Kutta integration. For example, k3 is given by  Introduction to Laminar Boundary Layers - 2 | Fluid Mechanics for Mechanical Engineering  The functions F1, F2and F3 are G, H , - f/H2 respectively. Then at a distance Δn from the wall, we have

 

Introduction to Laminar Boundary Layers - 2 | Fluid Mechanics for Mechanical Engineering                                                                (28.26d)

 

 

  • As it has been mentioned earlier F'(0) = H (0) = λ is unknown. It must be determined such that the condition  F' (0) = G (∞) = 1 is satisfied.

The condition at infinity is usually approximated at a finite value of n  (around n= 10 ). The process of obtainingIntroduction to Laminar Boundary Layers - 2 | Fluid Mechanics for Mechanical Engineering accurately involves iteration and may be calculated using the procedure described below.

  • For this purpose, consider Fig. 28.2(a) where the solutions of G versus n for two different values of H(0) are plotted. 
    The values of G(∞) are estimated from the G curves and are plotted in Fig. 28.2(b). 
  • The value of 
  • H(0) now can be calculated by finding the value Introduction to Laminar Boundary Layers - 2 | Fluid Mechanics for Mechanical Engineering at which the line 1-2 crosses the line G(∞) = 1 By using similar triangles, it can be said that  Introduction to Laminar Boundary Layers - 2 | Fluid Mechanics for Mechanical Engineering . By solving this, we get  Introduction to Laminar Boundary Layers - 2 | Fluid Mechanics for Mechanical Engineering  .
  • Next we repeat the same calculation as above by using Introduction to Laminar Boundary Layers - 2 | Fluid Mechanics for Mechanical Engineering and the better of the two initial values of H(0) . Thus we get another improved value Introduction to Laminar Boundary Layers - 2 | Fluid Mechanics for Mechanical Engineering . This process may continue, that is, we use Introduction to Laminar Boundary Layers - 2 | Fluid Mechanics for Mechanical Engineeringand Introduction to Laminar Boundary Layers - 2 | Fluid Mechanics for Mechanical Engineering as a pair of values to find more improved values for H(0) , and so forth. The better guess for  H (0) can also be obtained by using the Newton Raphson Method. It should be always kept in mind that for each value of H(0) , the curve G(n) versus n is to be examined to get the proper value of  G(∞) .
  • The functions  f(n), f' (n) = G and  f" (n) =H are plotted in Fig. 28.3.The velocity components, and inside the boundary layer can be computed from Eqs (28.21a) and (28.21b) respectively.
  • A sample computer program in FORTRAN follows in order to explain the solution procedure in greater detail. The program uses Runge Kutta integration together with the Newton Raphson method

Introduction to Laminar Boundary Layers - 2 | Fluid Mechanics for Mechanical Engineering

Introduction to Laminar Boundary Layers - 2 | Fluid Mechanics for Mechanical Engineering

Introduction to Laminar Boundary Layers - 2 | Fluid Mechanics for Mechanical Engineering

  • Measurements to test the accuracy of theoretical results were carried out by many scientists. In his experiments, J. Nikuradse, found excellent agreement with the theoretical results with respect to velocity distribution  (u/ U) within the boundary layer of a stream of air on a flat plate.
  • In the next slide we'll see some values of the velocity profile shape  f' (n) = (u/ U) = G and f" (n) = H in tabular format.

Values of the velocity profile shape  f' (n) = (u/ U) = G and f" (n) = H 

Introduction to Laminar Boundary Layers - 2 | Fluid Mechanics for Mechanical Engineering

Introduction to Laminar Boundary Layers - 2 | Fluid Mechanics for Mechanical Engineering 

The document Introduction to Laminar Boundary Layers - 2 | Fluid Mechanics for Mechanical Engineering is a part of the Mechanical Engineering Course Fluid Mechanics for Mechanical Engineering.
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FAQs on Introduction to Laminar Boundary Layers - 2 - Fluid Mechanics for Mechanical Engineering

1. What is a laminar boundary layer?
Ans. A laminar boundary layer refers to a thin layer of fluid that develops along a solid surface when the fluid flows over it. In this layer, the fluid particles move smoothly and parallel to the surface, with minimal mixing and turbulence.
2. How does a laminar boundary layer differ from a turbulent boundary layer?
Ans. Unlike a laminar boundary layer, a turbulent boundary layer is characterized by chaotic fluid motion with high levels of mixing and turbulence. In a turbulent boundary layer, fluid particles move in random directions and exhibit large velocity fluctuations.
3. What factors influence the development of a laminar boundary layer?
Ans. Several factors affect the development of a laminar boundary layer, including the viscosity of the fluid, the velocity of the flow, and the roughness of the surface. Higher viscosity and lower flow velocities tend to promote laminar flow, while rough surfaces can disrupt the smooth flow and lead to turbulent boundary layers.
4. What are the advantages of a laminar boundary layer in engineering applications?
Ans. A laminar boundary layer offers several advantages in engineering applications. It typically results in lower skin friction drag, which can reduce energy consumption in fluid flow systems. Additionally, laminar flow is more predictable and easier to model, making it advantageous for design and analysis purposes.
5. Can a laminar boundary layer transition to a turbulent boundary layer?
Ans. Yes, a laminar boundary layer can transition to a turbulent boundary layer under certain conditions. This transition is influenced by factors such as the Reynolds number, which represents the ratio of inertial forces to viscous forces in the fluid flow. As the Reynolds number increases, the laminar boundary layer becomes more susceptible to turbulence, and transition can occur.
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