Flow past a lifting cylinder

Superposing a uniform stream of speed \(U_\infty\) with a doublet of strength \(\kappa = 2\pi U_\infty R^2\) places a circular streamline of radius \(R\) exactly at the origin – a solid cylinder, in potential-flow terms. Adding a point vortex of circulation \(\Gamma\) at the same location keeps that circle intact (a vortex centered on the circle induces purely tangential velocity there) while breaking its front-back symmetry, generating lift with no viscosity anywhere in the calculation. This example builds that flow with flow_past_cylinder(), plots its surface pressure coefficient

\[C_p = 1 - \frac{u^2+v^2}{U_\infty^2},\]

and checks the resulting lift against the Kutta-Joukowski theorem,

\[L' = -\rho\,U_\infty\,\Gamma,\]

(\(\Gamma>0\) counterclockwise; the clockwise \(\Gamma<0\) used here speeds up the flow over the top and lifts upward, like an airfoil) (kutta_joukowski_lift()), by directly integrating that same pressure around the surface.

import matplotlib.pyplot as plt
import numpy as np

from physicskit.fluids.systems.potential_flow import flow_past_cylinder, kutta_joukowski_lift
from physicskit.fluids.visualizers.flow_fields import plot_streamlines
from physicskit.fluids.visualizers.potential_flow import plot_pressure_coefficient

Build the flow

Circulation is chosen large enough to make the effect obvious; too much circulation would pull the (physically required) stagnation points off the cylinder surface entirely.

U_inf, R, rho = 1.0, 1.0, 1.0
Gamma = -3.0  # clockwise: faster flow over the top, upward lift
flow = flow_past_cylinder(U_inf=U_inf, radius=R, circulation=Gamma)

x = np.linspace(-3, 3, 300)
X, Y = np.meshgrid(x, x, indexing="ij")
u, v = flow.velocity(X, Y)
inside = X**2 + Y**2 < R**2
u[inside] = np.nan
v[inside] = np.nan

fig, ax = plot_streamlines(X, Y, u, v)
theta_circle = np.linspace(0, 2 * np.pi, 200)
ax.plot(R * np.cos(theta_circle), R * np.sin(theta_circle), color="black", lw=1.5)
ax.set_title(f"Cylinder with circulation $\\Gamma$ = {Gamma}")
fig.tight_layout()

Surface pressure coefficient

Circulation shifts the two stagnation points (where \(C_p = 1\)) away from the front/back symmetry point they’d sit at with no circulation, and makes the low-pressure suction peak on top stronger than the one on the bottom – exactly the pressure asymmetry that produces lift.

theta = np.linspace(0, 2 * np.pi, 200)
x_surface, y_surface = R * np.cos(theta), R * np.sin(theta)
Cp = flow.pressure_coefficient(x_surface, y_surface)

fig, ax = plot_pressure_coefficient(theta, Cp)
fig.tight_layout()

Cross-checking the Kutta-Joukowski theorem

Integrating the surface pressure directly should reproduce kutta_joukowski_lift()’s closed-form answer, \(L' = -\rho U_\infty \Gamma\).

theta_fine = np.linspace(0, 2 * np.pi, 4000, endpoint=False)
x_fine, y_fine = R * np.cos(theta_fine), R * np.sin(theta_fine)
Cp_fine = flow.pressure_coefficient(x_fine, y_fine)
p_fine = Cp_fine * (0.5 * rho * U_inf**2)
dtheta = theta_fine[1] - theta_fine[0]
lift_numeric = -np.sum(p_fine * np.sin(theta_fine) * R) * dtheta  # y-component of -oint p n dA
lift_theory = kutta_joukowski_lift(rho=rho, U_inf=U_inf, circulation=Gamma)

print(f"lift from pressure integral: {lift_numeric:.4f}")
print(f"lift from Kutta-Joukowski theorem (-rho * U_inf * Gamma): {lift_theory:.4f}")

plt.show()

lift from pressure integral: 3.0000
lift from Kutta-Joukowski theorem (-rho * U_inf * Gamma): 3.0000