MAE561 Fall 2013 HW2
Prob 1(a) : The results are similar to those shown in the Tutorial and are not repeated here. As explained in
class, minor discrepancies between your plots and those in the Tutorial are acceptable.
Prob 1(b) : The results are actually
Spring 2015
AEE 471 / MAE 561 Computational Fluid Dynamics
Some conventions and definitions
y (usually cross-stream
or
wall-normal)
we usually work in Cartesian coordinates:
some variable names:
~ = ~ = (u, v, w) = ui
v u
velocity:
density:
pressure
Spring 2015
AEE471/MAE561 Computational Fluid Dynamics
Missing piece: Boundary Conditions (BC)
Geometry of the problem dictates the type of boundary
I) Solid surfaces
~
n
vn : surface normal velocity
vn v
t ~
t
vn = 0 : no flow though surface
vt
: surfa
AEE471/MAE561 Computational Fluid Dynamics
Spring 2015
Recap from last class:
@
(. . .) + spatial derivatives = 0
@t
approximate spatial derivatives
,
fi0
,
fi0
fi0
fi+1 fi
=
+ O(h)
h
=
=
fi
fi
h
fi+1
1
fi
2h
Forward difference: 1st order
+ O(h)
Backwar
Spring 2015
AEE471/MAE561 Computational Fluid Dynamics
4th-order PADE
0
fi 1
+
0
4fi
+
0
fi+1
3
= (fi+1
h
fi
1)
4
+ O(h )
only 2 slight problems:
- to get fi, we need fi-1 and fi+1
coupled system
implicit
- to get f0, we need f-1, and to get fN we need
Derivative Approximation by Finite Differences
David Eberly
Geometric Tools, LLC
http:/www.geometrictools.com/
c 1998-2015. All Rights Reserved.
Copyright
Created: May 30, 2001
Last Modified: April 25, 2015
Contents
1 Introduction
2
2 Derivatives of Univ
Chapter 9
Convection Equations
A physical system is usually described by more than one equation. Typical is the system of equations for an ideal gas or fluid. This requires equation for density , velocity u, and pressure p. In
one dimension these equation
EP711 Supplementary Material
Thursday, September 18, 2014
The Lax-Wendro Method And
Multi-Dimensional Problems
Jonathan B. Snively
!Embry-Riddle Aeronautical University
EP711 Supplementary Material
Contents
Thursday, September 18, 2014
Lax-Wendro, Revis
Crank Nicolson Solution to the Heat Equation
ME 448/548 Notes
Gerald Recktenwald
Portland State University
Department of Mechanical Engineering
gerry@pdx.edu
ME 448/548: Crank-Nicolson Solution to the Heat Equation
Overview
1. Use finite approximations to
Numerical Solutions to
Partial Differential Equations
Zhiping Li
LMAM and School of Mathematical Sciences
Peking University
More on Consistency, Stability and Convergence
Modified Equation Analysis
Modified Equation of a Difference Scheme
What is a Modifi
MAE561/471, Fall 2013 HW4 solutions (prepared by HPH)
Prob 1a Example of Matlab code
clear
C = 1; D = 0.5; dx = 0.02; dy = 0.02; dt = 0.01;
A1 = 1-(C*dt/dx)-(D*dt/dy); A2 = C*dt/dx; A3 = D*dt/dy;
NSTEP = 600; NOUT = 200;
x = [0:dx:5]; y = [0:dy:5];
N = le
HW1 All solutions prepared by HPH
Prob 1a code
(the codes for the other problems are not provided as they can be easily modified from this one)
clear
dx = 0.05; dt = 0.01; C = 1; A1 = 1-C*dt/dx; A2 = C*dt/dx;
x = [0:dx:20];
N = length(x);
for k = 2:N
if (
MAE561/471 HW3 solutions (prepared by HPH)
Prob 1a Matlab code for the first case (x = 0.01, t = 0.002)
clear;
dx = 0.01; dt = 0.002; A = dt/(2*dx);
x = [-1:dx:3]; N = length(x);
% - initial condition
for i = 1:N
if (x(i) <= 0)
u(i) = 2;
elseif (x(i) <= 1
MAE561/471, Fall 2013 HW5 solutions (prepared by HPH)
Prob 1a
(Contour interval is 0.03 for all plots. Dark blue contours are negative. The zero contour is suppressed.)
Prob 1b
Prob 2
(Contour interval is 0.03 for all plots. Dark blue contours are negativ
MAE561/471, Fall 2013, HW6 discussion
Case (a), (b), and (c1) : We have discussed in class about the dependence of the solution on the detailed
parameter setting in ANSYS-Fluent. We have also discussed the reason why the calculation by the
steady-state so