Uniform flow past a doublet…

Flow over a cylinder

A doublet alone does not give so much information about how it can be used to represent a practical flow pattern in aerodynamics. But let’s use our superposition powers: our doublet in a uniform flow turns out to be a very interesting flow pattern. Let’s first define a uniform horizontal flow.

u_inf = 1.0        # freestream speed

Now, we can calculate velocities and stream function values for all points in our grid. And as we now know, we can calculate them all together with one line of code per array.

u_freestream = u_inf * numpy.ones((N, N), dtype=float)
v_freestream = numpy.zeros((N, N), dtype=float)

psi_freestream = u_inf * Y

Below, the stream function of the flow created by superposition of a doublet in a free stream is obtained by simple addition.

The plot shows that this pattern can represent the flow around a cylinder with center at the location of the doublet. All the streamlines remaining outside the cylinder originated from the uniform flow. All the streamlines inside the cylinder can be ignored and this area assumed to be a solid object. This will turn out to be more useful than you may think.

# superposition of the doublet on the freestream flow
u = u_freestream + u_doublet
v = v_freestream + v_doublet
psi = psi_freestream + psi_doublet

# plot the streamlines
width = 10
height = (y_end - y_start) / (x_end - x_start) * width
pyplot.figure(figsize=(width, height))
pyplot.xlabel('x', fontsize=16)
pyplot.ylabel('y', fontsize=16)
pyplot.xlim(x_start, x_end)
pyplot.ylim(y_start, y_end)
pyplot.streamplot(X, Y, u, v,
                  density=2, linewidth=1, arrowsize=1, arrowstyle='->')
pyplot.contour(X, Y, psi,
               levels=[0.], colors='#CD2305', linewidths=2, linestyles='solid')
pyplot.scatter(x_doublet, y_doublet, color='#CD2305', s=80, marker='o')

# calculate the stagnation points
x_stagn1, y_stagn1 = +math.sqrt(kappa / (2 * math.pi * u_inf)), 0.0
x_stagn2, y_stagn2 = -math.sqrt(kappa / (2 * math.pi * u_inf)), 0.0

# display the stagnation points
pyplot.scatter([x_stagn1, x_stagn2], [y_stagn1, y_stagn2],
               color='g', s=80, marker='o');

Bernoulli’s equation and the pressure coefficient

A very useful measurement of a flow around a body is the coefficient of pressureCp. To evaluate the pressure coefficient, we apply Bernoulli’s equation for ideal flow and with simple mathematical steps:

\(C_{p}=1-\left(\frac{U}{U_{\infty}}\right)^{2}\)

In an incompressible flow, Cp=1 at a stagnation point. Let’s plot the pressure coefficient in the whole domain.

# compute the pressure coefficient field
cp = 1.0 - (u**2 + v**2) / u_inf**2

# plot the pressure coefficient field
width = 10
height = (y_end - y_start) / (x_end - x_start) * width
pyplot.figure(figsize=(1.1 * width, height))
pyplot.xlabel('x', fontsize=16)
pyplot.ylabel('y', fontsize=16)
pyplot.xlim(x_start, x_end)
pyplot.ylim(y_start, y_end)
contf = pyplot.contourf(X, Y, cp,
                        levels=numpy.linspace(-2.0, 1.0, 100), extend='both')
cbar = pyplot.colorbar(contf)
cbar.set_label('$C_p$', fontsize=16)
cbar.set_ticks([-2.0, -1.0, 0.0, 1.0])
pyplot.scatter(x_doublet, y_doublet,
               color='#CD2305', s=80, marker='o')
pyplot.contour(X,Y,psi,
               levels=[0.], colors='#CD2305', linewidths=2, linestyles='solid')
pyplot.scatter([x_stagn1, x_stagn2], [y_stagn1, y_stagn2],
               color='g', s=80, marker='o');
pyplot.streamplot(X, Y, u, v,
                  density=2, linewidth=1, arrowsize=1, arrowstyle='->')

Below, we report the complete Python code:

import math as mt
import numpy as np
from matplotlib import pyplot as pyp

N = 50                                # Number of points in each direction
x_start, x_end = -2.0, 2.0            # x-direction boundaries
y_start, y_end = -1.0, 1.0            # y-direction boundaries
x = np.linspace(x_start, x_end, N)    # creates a 1D-array for x
y = np.linspace(y_start, y_end, N)    # creates a 1D-array for y
X, Y = np.meshgrid(x, y)              # generates a mesh grid

kappa = 1.0                        # strength of the doublet
x_doublet, y_doublet = 0.0, 0.0    # location of the doublet

def get_velocity_doublet(strength, xd, yd, X, Y):
    
    u = (- strength / (2 * mt.pi) *
         ((X - xd)**2 - (Y - yd)**2) /
         ((X - xd)**2 + (Y - yd)**2)**2)
    v = (- strength / (2 * mt.pi) *
         2 * (X - xd) * (Y - yd) /
         ((X - xd)**2 + (Y - yd)**2)**2)
    
    return u, v

def get_stream_function_doublet(strength, xd, yd, X, Y):
    
    psi = - strength / (2 * mt.pi) * (Y - yd) / ((X - xd)**2 + (Y - yd)**2)
    
    return psi

# compute the velocity field on the mesh grid
u_doublet, v_doublet = get_velocity_doublet(kappa, x_doublet, y_doublet, X, Y)

# compute the stream-function on the mesh grid
psi_doublet = get_stream_function_doublet(kappa, x_doublet, y_doublet, X, Y)

# plot the streamlines
width = 10
height = (y_end - y_start) / (x_end - x_start) * width
pyp.figure(figsize=(width, height))
pyp.xlabel('x', fontsize=16)
pyp.ylabel('y', fontsize=16)
pyp.xlim(x_start, x_end)
pyp.ylim(y_start, y_end)
pyp.streamplot(X, Y, u_doublet, v_doublet,
                  density=2, linewidth=1, arrowsize=1, arrowstyle='->')
pyp.scatter(x_doublet, y_doublet, color='#CD2305', s=80, marker='o');



u_inf = 1.0        # freestream speed

u_freestream = u_inf * np.ones((N, N), dtype=float)
v_freestream = np.zeros((N, N), dtype=float)

psi_freestream = u_inf * Y

# superposition of the doublet on the freestream flow
u = u_freestream + u_doublet
v = v_freestream + v_doublet
psi = psi_freestream + psi_doublet

# plot the streamlines
width = 10
height = (y_end - y_start) / (x_end - x_start) * width
pyp.figure(figsize=(width, height))
pyp.xlabel('x', fontsize=16)
pyp.ylabel('y', fontsize=16)
pyp.xlim(x_start, x_end)
pyp.ylim(y_start, y_end)
pyp.streamplot(X, Y, u, v,
                  density=2, linewidth=1, arrowsize=1, arrowstyle='->')
pyp.contour(X, Y, psi,
               levels=[0.], colors='#CD2305', linewidths=2, linestyles='solid')
pyp.scatter(x_doublet, y_doublet, color='#CD2305', s=80, marker='o')

# calculate the stagnation points
x_stagn1, y_stagn1 = +mt.sqrt(kappa / (2 * mt.pi * u_inf)), 0.0
x_stagn2, y_stagn2 = -mt.sqrt(kappa / (2 * mt.pi * u_inf)), 0.0

# display the stagnation points
pyp.scatter([x_stagn1, x_stagn2], [y_stagn1, y_stagn2],
               color='g', s=80, marker='o');

# compute the pressure coefficient field
cp = 1.0 - (u**2 + v**2) / u_inf**2

# plot the pressure coefficient field
width = 10
height = (y_end - y_start) / (x_end - x_start) * width
pyp.figure(figsize=(1.1 * width, height))
pyp.xlabel('x', fontsize=16)
pyp.ylabel('y', fontsize=16)
pyp.xlim(x_start, x_end)
pyp.ylim(y_start, y_end)
contf = pyp.contourf(X, Y, cp,
                        levels=np.linspace(-2.0, 1.0, 100), extend='both')
cbar = pyp.colorbar(contf)
cbar.set_label('$C_p$', fontsize=16)
cbar.set_ticks([-2.0, -1.0, 0.0, 1.0])
pyp.scatter(x_doublet, y_doublet,
               color='#CD2305', s=80, marker='o')
pyp.contour(X,Y,psi,
               levels=[0.], colors='#CD2305', linewidths=2, linestyles='solid')
pyp.scatter([x_stagn1, x_stagn2], [y_stagn1, y_stagn2],
               color='g', s=80, marker='o');
pyp.streamplot(X, Y, u, v,
                  density=2, linewidth=1, arrowsize=1, arrowstyle='->')


pyp.show()

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