{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

# HW4 - I STANBUL TECHN ICAL UN IVERSITY FACULTY OF MECHAN...

This preview shows pages 1–4. Sign up to view the full content.

ISTANBUL TECHNICAL UNIVERSITY FACULTY OF MECHANICAL ENGINEERING MAK 475E Numerical Fluid Dynamics Homework #4 Can Tümer 030050282 Doç. Dr. Hasan Güne ş 08.01.2008

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
1. A wave is propagating in a closed-end tube. Compute the wave propagation up to t =0.15 sec by solving the first-order wave equation; a > 0. where a, the speed of sound, is selected to be 200 m/s. Assume that at time t = 0, a disturbance of a triangular shape has been generated as shown below. Solve the problem using Lax-Wendroff method. Three sets of step sizes are specified as follows: I) dx = 1.0, dt = 0.005 II) dx = 1.0, dt = 0.0025 III) dx = 1.0, dt = 0.00125 Print and plot the solution at intervals of 0.025 sec up to t = 0.15. Solution: The Theory; From the Taylor Expansion, we can write; ( , + )= , +∂ ∂ * +∂ * !+ u x t ∆t ux t u t ∆t 2u t2 ∆t22 O∆t3 At this point, all we have to do is the determination of the velocity derivatives by; ∂ ∂ =- *∂ ∂ u t a u x =- *∂ ∂ ∂ ∂ =- *∂ ∂ ∂ ∂ = (∂ ) 2u t2 a u t u x a u x u t a2 u2 x2
With the adaptation of these derivatives to our expansion equation; ( , + )≅ , - *∂ ∂ * + (∂ )* u x t ∆t ux t a u x ∆t a2 u2 x2 ∆t22 After obtaining this, we implement the central difference formulations; + = - * + - - * + * * * + - * + - uin 1 uin a ui 1n ui 1n2 ∆t 12 a2 ∆t2 ui 1n 2 uin ui 1n∆x2 Now, with all the nth order terms known, this FDE represents itself as an implicit solution to a time propagating problem. However we need to check if it is stable in order to commence the codification of the algorithm. = , ; = * = * , = For ∆t 0 005s C a ∆t∆x 200 0 0051 1 so the algorithm is stable since C 1 for = , . . ∆t 0 005 s and all the other step sizes are smaller The MATLAB Code; clear all clc %The velocity of the sound wave at these conditions.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern