# COMPUTATIONAL METHOD;

COMPUTATIONAL METHOD;

PERFORMANCE CRITERIA:

TARGETED LEARNING OUTCOMES

Students are able to understand the complex numerical solution algorithms behind the commercial CFD code, while they are developing CFD skills by using commercially

available software for research, development and design tasks in industry. Students are able to communicate and interpret CFD efficiently/precisely to achieve

meaningful results in complex situations.

Important Information – Please Read Before Completing Your Work

All students should submit their work by the date specified using the procedures specified in the Student Handbook. An assessment that has been handed in after this

deadline will be marked initially as if it had been handed in on time, but the Board of Examiners will normally apply a lateness penalty.

Your attention is drawn to the Section on Academic Misconduct in the Student’s Handbook.

All work will be considered as individual unless collaboration is specifically requested, in which case this should be explicitly acknowledged by the student within

their submitted material.

Any queries that you may have on the requirements of this assessment should be e-mailed to jyao@lincoln.ac.uk. No queries will be answered after respective submission

dates.

You must ensure you retain a copy of your completed work prior to submission.

SCHOOL of ENGINEERING

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COURSEWORK BRIEF:

NUMERICAL CALCULATION IS EXPECTING TO BE PERFORMED USING FINITE CENTRAL DIFFERENCE AND FINITE VOLUME METHODS. THE COMPUTATIONAL SOLUTION IS GOING TO BE COMPARED WITH

ANALYTIC SOLUTION.

COURSEWORK INCLUDES 4 QUESTIONS.

THE FIRST AND SECOND QUESTIONS ARE INVOLVING IN NUMERICAL SCHEME – FORWARD AND BACKWARD DIFFERENCE FIRST AND SECOND ORDER APPROXIMATION. THE THIRD QUESTION INVOLVS

CENTRAL DIFFERENCE APPROXIMATION. FORTH QUESTION INVOLVS FINITE VOLUME METHOD.

THIS IS A INDVIDUAL COURSE. STUDENTS ARE EXPECTING TO COMPLETE THE ALL COURSEWORK INDEPENDENTLY AND ACCURATELY. FOR A LATER SUBMISSON, A PERCENTAGE OF MARK WILL BE

DEDUCTED. THIS IS FOLLOWED BY UNIVERISYT AND SCHOOL POLICY FOR A LATER SUBMISSION.

Coursework 2

1. U(x) = x2, find the first order backward difference approximations to Ux(4) using : a) h =0.01, (h) h = 0.4, (c) h = 0.8

[20 Marks]

2. U(x) = x2, find the first order forward difference approximations to Ux(2) using : a) h =0.1, (h) h = 0.6, (c) h = 0.36

[20 Marks]

3. U(x) = x2, using Central difference approximation and show that is (a) first order accurate to Ux(2), (b) second order accurate to UXX(2), while h = 0.04, h = 0.02

and h = 0.03

[30 Marks]

4. Now discuss a problem that includes sources other than those arising from boundary conditions. Following figure shows a large plate of thickness L = 3 cm with

constant thermal conductivity k = 1 W/mK and uniform heat generation q = 1000 kW/m3. The face A and B are at temperatures of 150 oC and 350 oC respectively. Assuming

that the dimensions in the y – and z – direction are so large that temperature gradients are significant in the x – directions only, calculate the steady state

temperature distribution. Compare the numerical result with the analytical solution. The governing equation is: 0=+??????qdxdTkdxd

Surface area of plate is A= 2 m2

Comparison with the analytical solution

The analytical solution to this problem can be obtained by integration twice with respect to x and by subsequent application of the boundary conditions. This gives:

SCHOOL of ENGINEERING

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()AABTxxLkqLTTT+??????-+-=2

Make a comparison of analytical solution with finite volume method and plot diagram in order to show the accuracy using two methods. Show percentage of error at each

point in x –direction.

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