BLASIUS EQUATION Solution with Finite Difference Method

In summary, the Blasius equation is a nonlinear differential equation that describes the boundary layer near a flat plate in a fluid flow and is important in science for modeling various fluid flows. The Finite Difference Method is a numerical method used to solve the Blasius equation by approximating the solution and solving for unknown values at grid points. This method has the advantage of handling complex geometries and providing a quick and accurate solution, but it may be computationally expensive for large systems and introduce numerical errors. Other alternative methods for solving the Blasius equation include the Shooting Method, Finite Element Method, and Boundary Element Method, each with their own advantages and limitations. The most suitable method depends on the specific problem being solved.
  • #1
antiochos
3
0
1) Using a similarity variable, the boundary layer equations for a two-dimensional, incompressible flow over a flat plate can be written below:

2f'''+ff''=0


The boundary conditions are:

a) f ' (0) = 0, no slip at the wall
b) f(0)=0, solid wall
c) f ' (n) goes 1 as n goes infinity boundary layer solution merges into the inviscid solution.

I) using finite difference method, obtain a numerical solution of this equation. Plot f ' and f as a function n.
II) The shear stress on the wall requires f " (0) to be determined. From the numerical solution compute f " (0).
 
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  • #2
thank you
 
  • #3


I would first like to commend the use of the Blasius equation and the finite difference method to solve the boundary layer equations for a two-dimensional, incompressible flow over a flat plate. This is a highly relevant and important problem in fluid mechanics, and the use of numerical methods allows for a more accurate and efficient solution.

To obtain a numerical solution using the finite difference method, we first need to discretize the domain into a grid of points. Then, we can use the difference equations to approximate the derivatives in the Blasius equation. This will result in a system of algebraic equations, which can be solved using matrix methods.

Once the numerical solution is obtained, we can plot f ' and f as a function of n to visualize the behavior of the solution. This will provide valuable insights into the boundary layer thickness and the behavior of the flow near the wall.

Furthermore, the shear stress on the wall can be calculated by determining the second derivative of the solution at the wall, i.e. f " (0). This will give us a quantitative measure of the frictional forces acting on the wall, which is crucial in many practical applications.

Overall, the use of the finite difference method to solve the Blasius equation is a powerful and effective approach, and the results obtained can provide valuable information for understanding and predicting the behavior of boundary layer flows.
 

1. What is the Blasius equation and why is it important in science?

The Blasius equation is a nonlinear differential equation that describes the boundary layer near a flat plate in a fluid flow. It is important in science because it is used to model and understand the behavior of a wide range of fluid flows, including aerodynamics, hydrodynamics, and heat transfer.

2. How is the Blasius equation solved using the Finite Difference Method (FDM)?

The Finite Difference Method is a numerical method that approximates the solution to a differential equation by dividing the domain into a grid and solving for the unknown values at each grid point. To solve the Blasius equation using FDM, the equation is discretized, and the finite difference approximations of the derivatives are substituted in. The resulting system of linear equations is then solved using a matrix solver.

3. What are the advantages of using FDM to solve the Blasius equation?

One of the main advantages of using FDM is that it can handle complex geometries and boundary conditions, making it a versatile method for solving a wide range of differential equations. FDM is also relatively easy to implement and can provide a quick and accurate solution.

4. Are there any limitations to using FDM for solving the Blasius equation?

One limitation of FDM is that it can be computationally expensive for large systems of equations, as the grid size needs to be increased for accurate solutions. Additionally, FDM can introduce numerical errors, especially near sharp discontinuities or singularities in the solution.

5. Are there any alternative methods for solving the Blasius equation?

Yes, there are other numerical methods for solving the Blasius equation, such as the Shooting Method, the Finite Element Method, and the Boundary Element Method. Each method has its own advantages and limitations, and the choice of method depends on the specific problem at hand.

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