def test_solver():
"""
In this test we compare the velocities from the simulation for the last
time step with the terminal velocity.
"""
# This is incredibly tedious : (every constant....)
v0 = -10 # start velocity (already falling)
T = 20 # set a relatively large simulation
dt = 0.01
C_D = 0.45 # Sphere
rho = 1.2 # density of air
rho_b = 6.8 # density of body -> for V = 4.716m^3; m = 6.8*4.716 = 32.07 kg
A = 3.40 # Area of the sphere pi*r^2
V = 4.716 # Volume of the sphere (4/3)*pi*r^3
d = 2.08 # Diameter of the sphere 2*r
mu = 1.8 * 10**-5 # dynamic viscosity
g = 9.81 # Gravitational constant
a_s = 3 * pi * rho * d * mu / (
rho_b * V) # Constant to be used for the Stokes model
a_q = 0.5 * C_D * rho * A / (
rho_b * V) # Constant to be used for the quadratic model
b = g * (-1 + rho / rho_b
) # A common constant for both Stokes and quad. model
rdm = rho * d / mu # A constant related to the Reynolds number
t, v = vm.solver(T, dt, a_s, a_q, b, v0, rdm)
v_terminal = -sqrt(fabs(b) / a_q)
if v[-1] > 0: # can be risky to base the terminal velocity on the simulation
v_terminal = -1 * v_terminal
nt.assert_almost_equal(v[-1], v_terminal, delta=1E-7)
def main():
v0 = 0 # start velocity
T = 23 # here this time is chosen for falling 5000 m when only gravity is applied
dt = 0.01
C_D = 1.2 # Coeffi
rho = 0.79 # density of air (assumed constant)
rho_b = 1003.0 # kgm^3;
m = 80 # Mass of the jumper and parachute in kg
R = 0.5
A = pi * R**2 # Area of the sphere pi*r^2
V = m / rho_b # Volume of the falling object
d = 2.08 # Diameter of the sphere 2*r
mu = 1.8 * 10**-5 # dynamic viscosity
g = 9.81 # Gravitational constant
a_s = 3 * pi * rho * d * mu / (
rho_b * V) # Constant to be used for the Stokes model
a_q = 0.5 * C_D * rho * A / (
rho_b * V) # Constant to be used for the quadratic model
b = g * (-1 + rho / rho_b
) # A common constant for both Stokes and quad. model
rdm = rho * d / mu # A constant related to the Reynolds number
t, v = vm.solver(T, dt, a_s, a_q, b, v0, rdm)
plot(t, v, xlabel='t', ylabel='v', title='Parachute Jumper')
raw_input('Press any key.')
def test_solver(): """ In this test we compare the velocities from the simulation for the last time step with the terminal velocity. """ # This is incredibly tedious : (every constant....) v0 = -10 # start velocity (already falling) T = 20 # set a relatively large simulation dt = 0.01 C_D = 0.45 # Sphere rho = 1.2 # density of air rho_b = 6.8 # density of body -> for V = 4.716m^3; m = 6.8*4.716 = 32.07 kg A = 3.40 # Area of the sphere pi*r^2 V = 4.716 # Volume of the sphere (4/3)*pi*r^3 d = 2.08 # Diameter of the sphere 2*r mu = 1.8*10**-5 # dynamic viscosity g = 9.81 # Gravitational constant a_s = 3*pi*rho*d*mu/(rho_b*V) # Constant to be used for the Stokes model a_q = 0.5*C_D*rho*A/(rho_b*V) # Constant to be used for the quadratic model b = g*(-1 + rho/rho_b) # A common constant for both Stokes and quad. model rdm = rho*d/mu # A constant related to the Reynolds number t,v = vm.solver(T,dt,a_s,a_q,b,v0,rdm) v_terminal = -sqrt(fabs(b)/a_q) if v[-1]>0: # can be risky to base the terminal velocity on the simulation v_terminal = -1*v_terminal nt.assert_almost_equal(v[-1],v_terminal,delta=1E-7)
def main():
v0 = 0 # start velocity
T = 23 # here this time is chosen for falling 5000 m when only gravity is applied
dt = 0.01
C_D = 1.2 # Coeffi
rho = 0.79 # density of air (assumed constant)
rho_b = 1003.0 # kgm^3;
m = 80 # Mass of the jumper and parachute in kg
R = 0.5
A = pi*R**2 # Area of the sphere pi*r^2
V = m/rho_b # Volume of the falling object
d = 2.08 # Diameter of the sphere 2*r
mu = 1.8*10**-5 # dynamic viscosity
g = 9.81 # Gravitational constant
a_s = 3*pi*rho*d*mu/(rho_b*V) # Constant to be used for the Stokes model
a_q = 0.5*C_D*rho*A/(rho_b*V) # Constant to be used for the quadratic model
b = g*(-1 + rho/rho_b) # A common constant for both Stokes and quad. model
rdm = rho*d/mu # A constant related to the Reynolds number
t,v = vm.solver(T,dt,a_s,a_q,b,v0,rdm)
plot(t,v,xlabel = 't',ylabel = 'v',title = 'Parachute Jumper')
raw_input('Press any key.')
def calculate_forces(v0, mu, density_m, CD, diameter_b, \
area_b, volume_b, density_b, \
dt, T):
"""
Calculate the gravity force, buoyancy force and
the drag force of a falling body in a viscous
fluid.
"""
# Gravitational const. m/s^2
g = 9.81
# Proportionality constant for
# Reynolds number
Re_const = diameter_b*density_m/mu
a_s = 3*math.pi*diameter_b*mu/(density_b*volume_b)
a_q = 0.5*CD*density_m*area_b/(density_b*volume_b)
b = g*(density_m/density_b - 1.0)
# Numerical solution gives velocity as
# a function of time.
v, t = vm.solver(v0, a_s, a_q, b, Re_const, T, dt)
# Initialize vectors
Fg = zeros(len(v))
Fb = zeros(len(v))
Fd = zeros(len(v))
# Loop over time steps
for n in range(0, len(v)):
# Evaluate Reynolds number
Re = Re_const*v[n]
# Gravity force
Fg[n] = -density_b*volume_b*g
# Bouyancy force
Fb[n] = density_m*g*volume_b
# Drag force
if abs(Re) < 1:
# If Re < 1, use Stokes' drag force
Fd[n] = -3.0*math.pi*diameter_b*mu*v[n]
else:
# If Re >= 1, use the quadratic
# drag force
Fd[n] = -0.5*CD*density_m*area_b*abs(v[n])*v[n]
return Fg, Fb, Fd, t
def calculate_forces(v0, mu, density_m, CD, diameter_b, \
area_b, volume_b, density_b, \
dt, T):
"""
Calculate the gravity force, buoyancy force and
the drag force of a falling body in a viscous
fluid.
"""
# Gravitational const. m/s^2
g = 9.81
# Proportionality constant for
# Reynolds number
Re_const = diameter_b * density_m / mu
a_s = 3 * math.pi * diameter_b * mu / (density_b * volume_b)
a_q = 0.5 * CD * density_m * area_b / (density_b * volume_b)
b = g * (density_m / density_b - 1.0)
# Numerical solution gives velocity as
# a function of time.
v, t = vm.solver(v0, a_s, a_q, b, Re_const, T, dt)
# Initialize vectors
Fg = zeros(len(v))
Fb = zeros(len(v))
Fd = zeros(len(v))
# Loop over time steps
for n in range(0, len(v)):
# Evaluate Reynolds number
Re = Re_const * v[n]
# Gravity force
Fg[n] = -density_b * volume_b * g
# Bouyancy force
Fb[n] = density_m * g * volume_b
# Drag force
if abs(Re) < 1:
# If Re < 1, use Stokes' drag force
Fd[n] = -3.0 * math.pi * diameter_b * mu * v[n]
else:
# If Re >= 1, use the quadratic
# drag force
Fd[n] = -0.5 * CD * density_m * area_b * abs(v[n]) * v[n]
return Fg, Fb, Fd, t
def test_terminal_velocity_q():
#
# In the quadratic drag regime the final
# velociy should be v = -sqrt(abs(a_q)/b).
#
a_s = 1E-6
a_q = 0.0044
b = -9.80
Re_const = 21944.0
T = 100.0
dt = 0.1
v0 = 0.0
v, t = vm.solver(v0, a_s, a_q, b, Re_const, T, dt)
vmax = -np.sqrt(abs(b)/a_q)
print 'vmax = ', vmax
diff = abs(vmax - v[len(v)-1])
nt.assert_almost_equal(diff, 0, delta=1E-12)
def test_terminal_velocity_q():
#
# In the quadratic drag regime the final
# velociy should be v = -sqrt(abs(a_q)/b).
#
a_s = 1E-6
a_q = 0.0044
b = -9.80
Re_const = 21944.0
T = 100.0
dt = 0.1
v0 = 0.0
v, t = vm.solver(v0, a_s, a_q, b, Re_const, T, dt)
vmax = -np.sqrt(abs(b) / a_q)
print 'vmax = ', vmax
diff = abs(vmax - v[len(v) - 1])
nt.assert_almost_equal(diff, 0, delta=1E-12)
def test_heavy_slow():
#
# Assumption: density of body is very large,
# density of medium is very small.
# This should give linear velocity
# according to v(t) = v0 - g*t.
#
g = 9.81
v0 = 0.0
# Solve numerically
v, t = vm.solver(v0, a_s=0.0, a_q=0.0, b=-g, \
Re_const=0.0, T=2.0, dt=0.001)
v_a = np.zeros(len(t))
# The analytic velocity
for n in range(0, len(t)):
v_a[n] = v0 - g*t[n]
diff = np.abs(v-v_a).max()
nt.assert_almost_equal(diff, 0, delta=1E-12)
def test_heavy_slow():
#
# Assumption: density of body is very large,
# density of medium is very small.
# This should give linear velocity
# according to v(t) = v0 - g*t.
#
g = 9.81
v0 = 0.0
# Solve numerically
v, t = vm.solver(v0, a_s=0.0, a_q=0.0, b=-g, \
Re_const=0.0, T=2.0, dt=0.001)
v_a = np.zeros(len(t))
# The analytic velocity
for n in range(0, len(t)):
v_a[n] = v0 - g * t[n]
diff = np.abs(v - v_a).max()
nt.assert_almost_equal(diff, 0, delta=1E-12)
def plotter(v0,T,dt,C_D,rho,rho_b,A,V,d,mu):
g = 9.81 # Gravitational constant
a_s = 3*pi*rho*d*mu/(rho_b*V) # Constant to be used for the Stokes model
a_q = 0.5*C_D*rho*A/(rho_b*V) # Constant to be used for the quadratic model
b = g*(-1 + rho/rho_b) # A common constant for both Stokes and quad. model
rdm = rho*d/mu # A constant related to the Reynolds number
t,v = vm.solver(T,dt,a_s,a_q,b,v0,rdm)
F_b = rho*g*V*ones(len(v))
F_g = -g*rho_b*V*ones(len(v))
rdm = rho*d/float(mu)
Re = rdm*fabs(v0)
F_d_s = -a_s*v
F_d_q = -a_q*v*fabs(v)
# The following code attempts to create a force vector based on the appropriate
# Reynolds value :
F_d = zeros(len(v))
R_e = rdm*fabs(v0)
for n in range(len(v)) :
if Re < 1 :
F_d[n] = F_d_s[n]
else :
F_d[n] = F_d_q[n]
# Update Re :
R_e = rdm*fabs(v[n])
plot(t,F_b,
t,F_g,
xlabel='t',ylabel='F',
legend=('Buouncy Force','Gravity Force'),
title='Forces acting on a sphere')
hold('on')
plot(t,F_d,legend='Stokes and Quad for spesific Re')
raw_input('Press any key.')
def test_solver(): """ In this test we neglect air resistance and buoyancy force. To accomplish this we can effectively set the density of air to be zero. As a result we can compare the result from the solver with the known physical formula v = v0 - g*t. """ # This is annoying v0 = 0 # starts at rest rho = 0 # Neglect the buoyancy force as well as air resistance T = 8 # Choose a sufficiently large simulation dt = 0.1 # Apparently for smaller time steps the error increases slightly g = -9.81 # Constant of gravity a_s = 0 # Constant to be used for the Stokes model a_q = 0 # Constant to be used for the quadratic model b = g # Common constant for both Stokes and quadratic rdm = 0 # Constant used to assess the Reynolds number t,v = vm.solver(T,dt,a_s,a_q,b,v0,rdm) v_est = v0 + g*T # v = v0 + g*t nt.assert_almost_equal(v[-1],v_est,delta=1E-15)
def test_solver():
"""
In this test we neglect air resistance and buoyancy force.
To accomplish this we can effectively set the density of air
to be zero. As a result we can compare the result from the solver
with the known physical formula v = v0 - g*t.
"""
# This is annoying
v0 = 0 # starts at rest
rho = 0 # Neglect the buoyancy force as well as air resistance
T = 8 # Choose a sufficiently large simulation
dt = 0.1 # Apparently for smaller time steps the error increases slightly
g = -9.81 # Constant of gravity
a_s = 0 # Constant to be used for the Stokes model
a_q = 0 # Constant to be used for the quadratic model
b = g # Common constant for both Stokes and quadratic
rdm = 0 # Constant used to assess the Reynolds number
t, v = vm.solver(T, dt, a_s, a_q, b, v0, rdm)
v_est = v0 + g * T # v = v0 + g*t
nt.assert_almost_equal(v[-1], v_est, delta=1E-15)
