Lock Exchange Problem

Lock Exchange Problem#

This example simulates a 2D lock exchange problem using mimetic methods. The lock exchange is a classic fluid dynamics problem where two fluids of different densities initially separated by a vertical barrier (lock) are allowed to flow into each other when the barrier is removed. This creates complex flow patterns including gravity currents and Kelvin-Helmholtz instabilities.

Governing Equations#

The simulation uses the Boussinesq approximation, which accounts for density variations only in the buoyancy term:

\[\frac{\partial \mathbf{u}}{\partial t} + (\mathbf{u} \cdot \nabla) \mathbf{u} = -\frac{1}{\rho_0}\nabla p + \nu \nabla^2 \mathbf{u} + \mathbf{g}\alpha(T-T_0)\]
\[\nabla \cdot \mathbf{u} = 0\]
\[\frac{\partial T}{\partial t} + \mathbf{u} \cdot \nabla T = \kappa \nabla^2 T\]

where:

  • \(\mathbf{u}\) is the velocity field

  • \(p\) is the pressure

  • \(T\) is the temperature

  • \(\rho_0\) is the reference density

  • \(\nu\) is the kinematic viscosity

  • \(\alpha\) is the thermal expansion coefficient

  • \(\mathbf{g}\) is the gravitational acceleration

  • \(T_0\) is the reference temperature

Domain and Initial Conditions#

The simulation is conducted on a rectangular domain \([0, 100] \times [0, 20]\) meters. Initially, fluids of different densities (or temperatures) are separated at the mid-point of the domain with a narrow transition region.

Numerical Method#

The equations are solved using a fractional step method (projection method):

  1. Predictor Step: Compute an intermediate velocity field \(\mathbf{u}^*\) without enforcing incompressibility

  2. Pressure Solution: Solve a Poisson equation for pressure to enforce incompressibility

  3. Corrector Step: Project the velocity field to be divergence-free

  4. Temperature Advection: Update temperature using upwind/downwind differencing

Spatial discretization uses mimetic operators:

  • Divergence operator (\(D\))

  • Gradient operator (\(G\))

  • Laplacian operator (\(L = DG\))

This example is implemented in:

Results#

The simulation captures the gravity currents that form as the denser fluid flows beneath the lighter fluid. Depending on the Reynolds number, Kelvin-Helmholtz billows may develop along the interface. Due to the relatively coarse grid, numerical diffusion limits the formation of sharp billows, but increasing the resolution would allow for more detailed structures to emerge.

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