ADCP processing and the homogeneous flow assumption

This problem is designed to illustrate some of the difficulties that can arise if fundamental assumptions of instrument operation are violated.

The transformation equations developed in class (and shown in the handout) to convert horizontal flow to radial velocities (as measured by an ADCP) are given as:

Here, r₁ and r₂ are the radial velocities along each beam (at a given depth), Q is the angle of the beam relative to the vertical axis, and u and w are the horizontal and vertical velocities of the flow, respectively. The subscripts 1 and 2 refer to the velocities seen in each of the two beams. (In the questions below let u₁ = 50 cm/s.)

a. For a given u₁, w₁ = w₂ = 0.1·u₁ and Q = 20, plot r₁ and r₂ versus u₁ – u₂ letting u₂ range between +u₁ to – u₁.

b. For a given u₁, w₁ = w₂ = 0.1·u₁ and Q, plot u versus u₁ – u₂ letting u₂ range between +u₁ to – u₁. Plot w versus u₁ – u₂ on a separate graph.

import numpy as np

import matplotlib.pyplot as plt

from matplotlib.font_manager import FontProperties

from pylab import *

#Definition of variable

#r1, r2, are the radial velocities along each beam(at a given depth)

#THETA, is the angle of the beam relative to the vertical axis

#u and v are the horizontal and vertical velocities of the flow

Data=[]

THETA=20*np.pi/180

u1=50

w1=w2=0.1*u1

#creating the loop

for u2 in range(u1,-u1,-2):

r1=(-u1*np.sin(THETA))+(w1*np.cos(THETA))

r2=(u2*np.sin(THETA))+(w2*np.cos(THETA))

U=(0.5*(u1+u2))+((0.5*(w1-w2))/np.tan(THETA))

W=(0.5*(w1+w2))+((0.5*(u1-u2))*np.tan(THETA))

ur=u1-u2

mat=[r1,r2,u1,u2,ur,U,W]

Data.append(mat)

Datos=np.array(Data)

r1=Datos[0:,0]

r2=Datos[0:,1]

ur=Datos[0:,4]

U=Datos[0:,5]

W=Datos[0:,6]

subplot(221)

plot(ur, r1, 'b-', ur, r2, 'r-')

ylabel('r1 and r2')

xlabel('u1-u2')

suptitle('ADCP processing', fontsize=16)

legend( ('r1', 'r2'), loc='upper right')

subplot(223)

plot(ur,U,'r-')

xlabel('u1-u2')

ylabel('U')

subplot(224)

plot(ur,W,'r-')

xlabel('u1-u2')

ylabel('W')

show()

a. For a given u₁, w₁ = w₂ = 0.1·u₁,and u₂ = u₁/2, plot u versus Q, letting Q range from 0 to pi/2. Plot w versus Q on a separate graph.

import numpy as np

import matplotlib

import matplotlib.pyplot as plt

from pylab import *

#Definition of variable

Data=[]

u1=50

w1=w2=0.1*u1

u2=u1/2

#create the loop

for T in range(0,60,2):

r1=(-u1*np.sin(T*np.pi/180))+(w1*np.cos(T*np.pi/180))

r2=(u2*np.sin(T*np.pi/180))+(w2*np.cos(T*np.pi/180))

U=(0.5*(u1+u2))+((0.5*(w1-w2))/np.tan(T*np.pi/180))

W=(0.5*(w1+w2))+((0.5*(u1-u2))*np.tan(T*np.pi/180))

ur=u1-u2

theta=T*np.pi/180

mat=[r1,r2,u1,u2,ur,U,W,T,theta]

Data.append(mat)

Datos=np.array(Data)

r1=Datos[0:,0]

r2=Datos[0:,1]

ur=Datos[0:,4]

U=Datos[0:,5]

W=Datos[0:,6]

T=Datos[0:,7]

theta=Datos[0:,8]

plt.plot(T,W,'r')

subplot(121)

plot(T,U, 'b-')

ylabel('U')

xlabel('Theta')

suptitle('ADCP processing', fontsize=16)

subplot(122)

plot(T,W,'r-')

xlabel('Theta')

ylabel('W')

plt.show()