Stream Function Vorticity Method

L26: 2D Convection-Diffusion using Stream Function-Vorticity

Lecture 26: Two-Dimensional Convection-Diffusion using Stream Function-Vorticity Method

Good morning everyone. Building on our earlier discussion, today we focus on solving unsteady 2D convection-diffusion problems using the stream function-vorticity formulation for incompressible flows.

1. Governing Equations (Primitive Variables)

For unsteady 2D incompressible flow, the equations are:

  • Continuity Equation:
    \( \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} = 0 \)
  • x-Momentum:
    \( \frac{\partial u}{\partial t} + u \frac{\partial u}{\partial x} + v \frac{\partial u}{\partial y} = -\frac{1}{\rho} \frac{\partial p}{\partial x} + \nu \left( \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} \right) \)
  • y-Momentum:
    \( \frac{\partial v}{\partial t} + u \frac{\partial v}{\partial x} + v \frac{\partial v}{\partial y} = -\frac{1}{\rho} \frac{\partial p}{\partial y} + \nu \left( \frac{\partial^2 v}{\partial x^2} + \frac{\partial^2 v}{\partial y^2} \right) \)

2. Stream Function–Vorticity Transformation

  • Vorticity (\( \zeta \)):
    \( \zeta = \frac{\partial v}{\partial x} - \frac{\partial u}{\partial y} \)
  • Stream Function (\( \psi \)):
    \( u = \frac{\partial \psi}{\partial y}, \quad v = -\frac{\partial \psi}{\partial x} \)
  • Vorticity Transport Equation:
    \( \frac{\partial \zeta}{\partial t} + u \frac{\partial \zeta}{\partial x} + v \frac{\partial \zeta}{\partial y} = \nu \left( \frac{\partial^2 \zeta}{\partial x^2} + \frac{\partial^2 \zeta}{\partial y^2} \right) \)
  • Poisson Equation:
    \( \nabla^2 \psi = -\zeta \)

3. Advantages

  • Eliminates pressure from momentum equations
  • Simplifies to a parabolic and an elliptic PDE
  • Reduces computational complexity for 2D flows

4. Numerical Solution Steps

  1. Initialize \( \zeta \) and \( \psi \) at \( t = 0 \)
  2. Solve vorticity transport equation to advance \( \zeta \) in time
  3. Solve Poisson equation for \( \psi \) using updated \( \zeta \)
  4. Compute velocities from \( \psi \): \( u = \frac{\partial \psi}{\partial y}, v = -\frac{\partial \psi}{\partial x} \)
  5. Update vorticity on boundaries using velocity or \( \psi \)
  6. Repeat until convergence criteria met

5. Discretisation Aspects

  • Time Discretisation:
    • Explicit (FTCS) – simple, but limited by stability
    • Implicit (Crank-Nicolson, Laasonen) – stable for larger time steps
  • Spatial Discretisation:
    • Diffusion terms – use central difference (CDS)
    • Convection terms – use:
      • CDS (2nd-order) – can cause oscillations
      • UDS (1st-order) – stable, but diffusive
      • Higher-order upwind and TVD schemes for improved results
  • Poisson Solver:
    • Jacobi, Gauss-Seidel, SOR, or Multigrid methods

6. Boundary Conditions

  • Stream Function (\( \psi \)): Dirichlet on walls (e.g., \( \psi = 0 \))
  • Vorticity (\( \zeta \)): Derived using Taylor expansions and wall conditions. Example at wall \( y = 0 \):
    \( \zeta = -\frac{\partial^2 \psi}{\partial y^2} \)
  • Inflow/outflow – specify gradients or values based on flow physics

Conclusion

The stream function-vorticity method effectively decouples the incompressible Navier-Stokes equations into manageable PDEs. This makes it especially suitable for 2D flow analysis using finite difference methods. The method avoids pressure computations and provides insights into numerical schemes like CDS, UDS, and their impact on stability and accuracy.