7ENT1007 CFD Techniques Assignment Problem

Reference no: EM132398274

7ENT1007 CFD Techniques - University of Hertfordshire

Learning Outcomes:

- Examine the numerical methods underpinning CFD;
- Assess different meshing procedures and discretisation methods.

Assignment Brief:

This assignment requires you to answer the questions given in this briefing. Note that some of the questions are numerical and you will have your own data which is specified in the tables included.

You will also need to make use of the three Excel spreadsheets uploaded with this assignment.

CFD Techniques - Coursework 1

Q1. Consider the partial differential equation given by:

A ∂²φ/∂x² + B ∂²φ/∂x∂y + C ∂²φ/∂y² + D∂φ/∂x + E∂φ/∂y + F = 0

where D = 3.0, E = 2.0 and F = 3.5.

Classify this partial differential equation (hyperbolic, parabolic or elliptic) according to the values of A, B, C for your case as shown in Table 1. Show your calculation to justify your classification.

Q2. A variable u is a function of x. Show that a third order accurate finite difference approximation for ∂u/∂x at x = x_i is:

(∂u/∂x)_i = (2u_i+1+ 3u_i - 6u_i-1 + u_i-2) /6Δx + O(Δx³)

where Δx denotes the increment between points x_i-1, x_i, x_i+1 etc.

Q3. Let u be given by:

u = ae^bx + csin(dx)

where a, b, c, d, x should be chosen according to your case in Table 2.

(i) Determine ∂u/∂x for your values of a, b, c, d, x by differentiating u.

(ii) Estimate ∂u/∂x using the formula in Q2 for ?x = 0.01 and ?x = 0.02.
(iii) Estimate ∂u/∂x for ?x = 0.01 and ?x = 0.02 using the finite difference formula:

(∂u/∂x)i = (u_i+1 - u_i)/?x

(iv) Comment on the accuracy of your results.

Q4. Use von Neumann stability analysis to investigate the numerical stability of the Euler- Explicit scheme:

u_iⁿ⁺¹ = u_iⁿ - 1/2 v(uⁿ_i+1 - uⁿ_i-1)

Q5. Practical Study of Finite Difference Schemes

In this study, we consider solutions to the hyperbolic problem:

∂u/∂t + c∂u/∂x = 0

subject to the initial conditions:

u(x, 0) = 0 for x< 0

u(x, 0) = 1 for x ≥ 0

This question will involve the use of the three Excel spreadsheets uploaded with this assignment. They all take the same form. Sheet 1 gives the Finite Difference solution, while sheet 2 gives the exact solution. The comparison between the two is plotted in Sheet 1. On sheet 1, the initial condition is indicated in GREEN and the solution after the defined number of steps is highlighted in BLUE.

The only thing you will need to change is the Courant Number, which is highlighted in YELLOW on sheets 1 and 2.

Investigate the numerical stability of the following schemes:

(i) Euler Explicit Method (which is considered in Question 4)
(ii) The Lax-Wendroff Scheme
(iii) The Beam Warming Scheme

In each case, vary the Courant number as necessary and use the resulting plots to draw conclusions on the stability of the schemes.

It is essential to include some plots in your report as evidence for your conclusions.

Reference no: EM132398274

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