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fluid-dynamics
GitHub用于求解不可压缩和可压缩流的Navier-Stokes方程，涵盖 lid-driven cavity、通道流及涡动力学。支持拖曳/升力计算、湍流统计及流场可视化，采用谱法和有限差分法。
触发场景
2D lid-driven cavity simulation
computing drag and lift
turbulence statistics analysis
vortex dynamics simulation
安装
npx skills add synthetic-sciences/openscience --skill fluid-dynamics -g -y
SKILL.md
Frontmatter
{
"name": "fluid-dynamics",
"tags": [
"CFD",
"Navier-Stokes",
"Fluid Dynamics",
"Turbulence",
"Incompressible Flow"
],
"author": "Synthetic Sciences",
"license": "MIT",
"version": "1.0.0",
"category": "physics",
"description": "Computational fluid dynamics — Navier-Stokes solvers, lid-driven cavity, channel flow, vortex methods, turbulence statistics, drag\/lift computation. Spectral and finite-difference methods for incompressible and compressible flows.",
"dependencies": [
"scipy>=1.11.0",
"numpy>=1.24.0",
"matplotlib>=3.7.0"
]
}
Fluid Dynamics (CFD)
Overview
Solve the Navier-Stokes equations for incompressible and compressible flows. Covers lid-driven cavity, channel flow, flow past obstacles, and turbulence analysis. For pseudospectral methods on periodic domains, also consider the fluidsim skill.
When to Use
- Incompressible flow simulations (lid-driven cavity, channel flow, jets)
- Computing drag and lift on bodies
- Turbulence statistics (energy spectrum, Reynolds stresses)
- Vortex dynamics and wake analysis
- Flow visualization (streamlines, vorticity fields)
Do NOT Use When
- Periodic-domain turbulence at high resolution (use
fluidsim— optimized pseudospectral) - Compressible flow with shocks (need specialized Riemann solvers)
- Complex 3D geometries with unstructured meshes (use FEniCS or OpenFOAM)
Core Workflows
1. 2D Lid-Driven Cavity (Incompressible, Vorticity-Streamfunction)
import numpy as np
import matplotlib.pyplot as plt
def lid_driven_cavity(N=64, Re=100, dt=0.001, n_steps=50000):
"""
2D lid-driven cavity using vorticity-streamfunction formulation.
∂ω/∂t + u·∇ω = (1/Re)∇²ω
∇²ψ = -ω
u = ∂ψ/∂y, v = -∂ψ/∂x
"""
dx = 1.0 / (N - 1)
x = np.linspace(0, 1, N)
y = np.linspace(0, 1, N)
omega = np.zeros((N, N)) # vorticity
psi = np.zeros((N, N)) # streamfunction
# CFL check
U_lid = 1.0
CFL = U_lid * dt / dx
diff_num = dt / (Re * dx**2)
print(f"Grid: {N}x{N}, Re={Re}, dt={dt}")
print(f"CFL = {CFL:.4f}, Diffusion number = {diff_num:.4f}")
if CFL > 0.5 or diff_num > 0.25:
print("WARNING: stability may be marginal")
for step in range(n_steps):
# Solve Poisson: ∇²ψ = -ω (Jacobi iteration)
for _ in range(50):
psi[1:-1, 1:-1] = 0.25 * (
psi[2:, 1:-1] + psi[:-2, 1:-1] +
psi[1:-1, 2:] + psi[1:-1, :-2] +
dx**2 * omega[1:-1, 1:-1]
)
# BCs: ψ = 0 on all walls
psi[0, :] = 0; psi[-1, :] = 0
psi[:, 0] = 0; psi[:, -1] = 0
# Compute velocity from streamfunction
u = np.zeros((N, N))
v = np.zeros((N, N))
u[1:-1, 1:-1] = (psi[1:-1, 2:] - psi[1:-1, :-2]) / (2*dx)
v[1:-1, 1:-1] = -(psi[2:, 1:-1] - psi[:-2, 1:-1]) / (2*dx)
# Lid BC: u = 1 at y = 1 (top wall)
u[-1, :] = U_lid
# Vorticity on boundaries (Thom's formula)
omega[0, 1:-1] = -2*psi[1, 1:-1] / dx**2 # bottom
omega[-1, 1:-1] = -2*psi[-2, 1:-1] / dx**2 - 2*U_lid/dx # top (lid)
omega[1:-1, 0] = -2*psi[1:-1, 1] / dx**2 # left
omega[1:-1, -1] = -2*psi[1:-1, -2] / dx**2 # right
# Advection-diffusion of vorticity (FTCS)
domega_dx = (omega[2:, 1:-1] - omega[:-2, 1:-1]) / (2*dx)
domega_dy = (omega[1:-1, 2:] - omega[1:-1, :-2]) / (2*dx)
laplacian = (omega[2:, 1:-1] + omega[:-2, 1:-1] +
omega[1:-1, 2:] + omega[1:-1, :-2] -
4*omega[1:-1, 1:-1]) / dx**2
omega[1:-1, 1:-1] += dt * (
-u[1:-1, 1:-1] * domega_dx
-v[1:-1, 1:-1] * domega_dy
+ laplacian / Re
)
if (step+1) % 10000 == 0:
max_div = np.max(np.abs(
(u[1:-1, 2:] - u[1:-1, :-2])/(2*dx) +
(v[2:, 1:-1] - v[:-2, 1:-1])/(2*dx)
))
print(f"Step {step+1}: max|ω|={np.max(np.abs(omega)):.2f}, max|div|={max_div:.2e}")
return x, y, u, v, omega, psi
x, y, u, v, omega, psi = lid_driven_cavity(N=64, Re=100, n_steps=30000)
X, Y = np.meshgrid(x, y)
fig, axes = plt.subplots(1, 3, figsize=(18, 5))
# Streamfunction
ax = axes[0]
cs = ax.contour(X, Y, psi.T, levels=30, cmap='RdBu_r')
ax.set_title(r'Streamfunction $\psi$')
ax.set_xlabel('x'); ax.set_ylabel('y')
ax.set_aspect('equal')
plt.colorbar(cs, ax=ax)
# Vorticity
ax = axes[1]
cf = ax.contourf(X, Y, omega.T, levels=30, cmap='RdBu_r')
ax.set_title(r'Vorticity $\omega$')
ax.set_xlabel('x'); ax.set_ylabel('y')
ax.set_aspect('equal')
plt.colorbar(cf, ax=ax)
# Velocity vectors
ax = axes[2]
skip = 4
ax.quiver(X[::skip, ::skip], Y[::skip, ::skip],
u.T[::skip, ::skip], v.T[::skip, ::skip], scale=20)
ax.set_title('Velocity field')
ax.set_xlabel('x'); ax.set_ylabel('y')
ax.set_aspect('equal')
plt.suptitle(f'Lid-Driven Cavity (Re={100})', fontsize=14)
plt.tight_layout()
plt.savefig('cavity_flow.png', dpi=150, bbox_inches='tight')
2. Centerline Velocity Profiles (Ghia Benchmark)
# Compare with Ghia et al. (1982) benchmark data for Re=100
# Vertical centerline: u(x=0.5, y)
j_center = len(x) // 2
u_centerline = u[j_center, :]
fig, ax = plt.subplots(figsize=(6, 8))
ax.plot(u_centerline, y, 'b-', linewidth=2, label='Computed')
ax.set_xlabel('u')
ax.set_ylabel('y')
ax.set_title(f'Vertical centerline velocity (Re={100})')
ax.legend()
ax.grid(True, alpha=0.3)
plt.savefig('centerline.png', dpi=150, bbox_inches='tight')
3. Flow Past a Cylinder (Immersed Boundary)
def cylinder_flow_simple(Nx=200, Ny=100, Re=100, n_steps=10000):
"""
Simplified 2D flow past a cylinder using penalty method.
For production use, consider lattice Boltzmann or FEniCS.
"""
# This is a simplified placeholder — real cylinder flows need
# proper immersed boundary or body-fitted coordinates
dx = 1.0 / Ny
dt = 0.1 * dx / 1.0 # conservative CFL
# Initialize velocity field with uniform inflow
u = np.ones((Nx, Ny))
v = np.zeros((Nx, Ny))
# Cylinder mask
cx, cy = Nx//4, Ny//2
R = Ny // 10
Y_grid, X_grid = np.meshgrid(np.arange(Ny), np.arange(Nx))
mask = ((X_grid - cx)**2 + (Y_grid - cy)**2) < R**2
for step in range(n_steps):
# Force velocity to zero inside cylinder
u[mask] = 0
v[mask] = 0
# ... (full NS solver needed for proper simulation)
return u, v, mask
Flow Regime Reference
| Re | Flow Type | Characteristics |
|---|---|---|
| Re < 1 | Stokes (creeping) | Reversible, no inertia |
| 1 < Re < 40 | Steady laminar | Attached flow, twin vortices |
| 40 < Re < 200 | Periodic (Von Karman) | Vortex shedding |
| 200 < Re < 10⁵ | Turbulent wake | Broad spectrum |
| Re > 10⁵ | Fully turbulent | Boundary layer transition |
Validation Checklist
- Verify divergence-free condition: ∇·u = 0
- Check CFL condition: |u|dt/dx < 1
- Compare centerline profiles with Ghia et al. (1982) for cavity
- Verify mass conservation: ∫u·n dA = 0
- Check symmetry (if problem is symmetric)
Troubleshooting
| Symptom | Fix |
|---|---|
| Solution blows up | CFL too large, reduce dt |
| Checkerboard pattern | Pressure-velocity decoupling — use staggered grid |
| Poisson solver slow | Use SOR (ω ≈ 1.5-1.9) or FFT-based solver |
| Wrong drag coefficient | Insufficient resolution near body surface |
版本历史
- e9844a4 当前 2026-07-11 17:32
依赖关系
-
required
scipy>=1.11.0 -
required
numpy>=1.24.0 -
required
matplotlib>=3.7.0
