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Computational Fluid Dynamics Worked Examples

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One - Dimensional Convection - Conduction - Upwind Differencing

Example 15.1 - One - Dimensional Convection - Conduction - Upwind Differencing

A scalar property φ is transported by an incompressible flow, as shown in Figure(14.2)

Find the distribution of φ as a function of x using upwind method for the convective terms in the following cases:

u = 0.2m/s

u = 2.5m/s

compare the results with the analytical solution

The Grid

We use a five equally spaced grid as shown in Figure 14.3.1

Discretization

1. Inner Nodes: Defined by

   2 ≥ i ≥ I_max − 2

   Using the upwind method explained in Chapter 15, we have

   

2. West: defined by

   i=1

   The conservation law for this control volume would be

   

   Using the Dirichlet boundary condition on the west face, we get

   

   Then

   

3. East: defined by

   i=I_max

   The balance equation is

   

   Then

   

Results

• Case 1: u= 0.1 m/s

  The obtained results vs the exact values can be plotted as shown in Figure 15.1

  
• Case 2: u= 2.5 m/s

  The obtained results vs the exact values can be plotted as shown in Figure 15.2

  

It is obvious that with the increase in the velocity the result does not show any oscillations. However, here we see that the sharp gradient of the exact solution has been smeared out by the upwind method. The amount of smearing depends on many factors, including the grid size. If we reduce Δx, the smearing will decrease as shown in Figure 15.3 and will be discussed in the next chapter.