Computational Fluid Dynamics

Running HEAD: COMPUTATIONAL FLUID DYNAMICS 1
Computational fluid dynamics.
Name
Institute of affiliation
Date

COMPUTATIONAL FLUID DYNAMICS 2
KC = T Um
D
U= Water’s velocity in x direction
ρ: Water’s density
Cd: Drag force coefficient
Cm: Mass coefficient
A: = Area
Fx ( t )=D ( t )
¿ Uρ ∀Cm+ 0.5Cd AU |U |..............................................................Equation 1
velocity is defined as follows
u=U0 +sin ( 2 πt
T )Um ..................................................................... ...Equation 2
U0= Steady flow velocity.
Um= flow oscillatory velocity
T= time period
Given that

COMPUTATIONAL FLUID DYNAMICS 3
P= 1
T ∫
0
T
c V 2 tⅆ ...........................................................................Equation 3
where P= power stored, and V is the Vibration of the system.
And
KC =U m T
D .......................................................................................Equation 4
It then follows that the power can be derived as follows
P=N−1
∑
n=1
N
C ( V n )
2......................................................................Equation 5
Power from the water is as follows
Fw=md CA
d V r
dt +0.5 ρ A projected CD V r∨V r∨¿.....................................................Equation
6
Total force Ftacting is sum of the force acting on the spring Fs, damper Fd and the water FW
Ft=FS + Fd + Fw ..............................................................................Equation 7
Ft=−kx−c dx
dt +0.5 ρ Cd A projected|V r|V r +C A md
d2 x
d t2 ............................Equation 8
Since awater=¿ d2 x
d t2 equation above becomes
Ft=−kx−c dx
dt +C A md awater + 1
2 ρ Cd A p |V w−V a|(V ¿¿ w−V a )¿.........................Equation 9
Differentiating equation 2 with respect t gives
aw=ω U m cos (ωt )................................................................................Equation 10
awater with be added into the f t equation¿ substitute other terms
Developing the 4th Runge kurta term
dx
dt =v..........................................................................................Equation 11
dv
dt =f ( x , v )....................................................................................Equation 12

COMPUTATIONAL FLUID DYNAMICS 4
Where,
f ( x , v , t )=
kx + (c + 1
2 ρ Cd Ap |v|)v
C A md + m
......................................................Equation 13
x(n+ 1)∆ t =xn ∆ t + ∆ t
6 ( kx 1 +2 kx 2 +2 k x3 +k x 4 ) .............................Equation 14
v(n+1) ∆ t= ( kv 1 +2 kv 2 +2 kv 3 +k v 4 ) ∆t
6 + vn ∆ t ..............................Equation 15
Where,
k xa
=vn ∆ t.......................................................................................Equation 16
k xb
=vn ∆ t + ∆ t
2 k v1...........................................................................Equation 17
k xc
=vn ∆ t + ∆ t
2 k v2..........................................................................Equation 18
k xd
=vn ∆ t + ∆ t k v3...........................................................................Equation 19
k va
=f (xn ∆ t , vn ∆ t ) ..........................................................................Equation 20
k vb
=f ( xn ∆ t + ∆ t
2 k x 1 , vn ∆ t + ∆ t
2 kv 1 ) .................................................Equation 21
k vc
=f (xn ∆ t + ∆ t
2 k x2 , vn ∆ t + ∆ t
2 kv 2) ...............................................Equation 22
k vd
=f (xn ∆ t + ∆ t k x 3 , vn ∆ t + ∆t kv 3 ) ......................................................Equation 23
In order to calculate the power from the equations above, the code below
was developed.
clear all
% Runge-Kutta method
j=7; % last digit of ID
U_m = 1+ (j/10); % Flow velocity Amplitude
U_0= 0.1 * U_m; %Velocity of steady flow
T = .2; % oscillatory flow period
D = (10+j)/100; % Diameter of cylinder
% Cylinder length
Length_cyl = 1;
mass_c = 50; % cylinder mass
rho_water = 1000; % water density