Advanced Thermal and Fluid Engineering

Running head: ADVANCED THERMAL AND FLUID ENGINEERING
ADVANCED THERMAL AND FLUID ENGINEERING
Name of the Student
Name of the University
Author Note

ADVANCED THERMAL AND FLUID ENGINEERING1
Part one: Simple initial value problem
j = 9
Given,
Um= 1 + 9/10 = 1.9 m/s
U0 = 0.1*1.9 = 0.19 m/s
D = 10 + 9 = 19 cm = 19*10^(-2) m
Mass of cylinder m = 50 kgs
Water density ρ = 1024 kg/m^3
Stiffness of spring K = 200 N/m
Damping coefficient c = 10 N*m/sec
md = 1 and CA = CD = 1.
The KC number is given by,
KC = Um*T/D
Hence, for KC = 2 => T = 2*D/Um = 0.2 sec.
Time period is divided in small time intervals.
Flow velocity V r =u−V
Now, initially, V r (0) = V(0) = U0 and V r (end) = U0
Now, the Morrison equation is given by,
Fwater=C A∗md∗( d V r
dt )+ ( ½ )∗ρ∗CD∗A P∗|V r |∗V r

ADVANCED THERMAL AND FLUID ENGINEERING2
Now, Fwater = m*g = 1024*(2*pi*D/2)*L*9.8 = 5990 N.
Now, this force will balance the hydrodynamic force and hence will cancel out
Hence, in finite difference method the Morrison equation will be given by,
CA∗md∗Vr ( t +1 ) −Vr ( t )
dt + ( ½ )∗ρ∗CD∗AP∗|V r ( t )|∗V r ( t ) =0
 Vr ( t +1 )= (− ( ½ )∗ρ∗C D∗AP∗|V r ( t )|∗V r ( t ) ) dt /(C A∗md )+ Vr ( t )
Now, after obtaining V r (t), V(t) can be found for entire time period of T.
Then the power P can be calculated by numerical integration with the formula
P = 1
N ∑
n=1
N
c∗(V ¿¿ n)2 ¿
Where, N = the number of points T is divided.
MATLAB code for Vibration speed:
function P = question1(KC)
j = 9;
Um = 1 + (j/10); % amplitude of oscillatory flow in m/s
U0 = 0.1*Um; % steady state flow in m/s
D = (10 + j)*1e-2; % diameter in m
L = 1;
m = 50;
rho = 1024;
K = 200;

ADVANCED THERMAL AND FLUID ENGINEERING3
c = 10;
Ap = 2*pi*(D/2)*L; % projected area is the surface area of the cylinder
md = 1; % added mass coefficient
CA = 1.8; CD = 1.8; T = KC*D/Um;
Fwater = 1024*(2*pi*D/2)*L*9.8;
t = linspace(0,T,150);
dt = t(2)- t(1);
Vr = U0 + Um.*sin(2*pi*t/T);
Vr(1) = U0; Vr(end) = U0;
%%% solving Vr(t) by FDM
for idt = 2:length(t)-2
Vr(idt+1) = (- (1/2)*rho*CD*Ap*abs(Vr(idt))*Vr(idt))*dt/(CA*md) + Vr(idt);
end
V = Vr;
plot(t,V,'b-')
xlabel('Time t in secs')
ylabel('Vibration speed in m/sec')
title('Vibration speed with respect to time')