AdvektionDiffusionsgleichungen

Advektion-Diffusion, usually referred to as the advection-diffusion equation, is a partial differential equation used to describe the transport of a scalar field, such as a concentration or temperature, within a moving fluid. The equation captures two main processes: advection, the transport by the bulk velocity field u, and diffusion, the spreading driven by concentration gradients. A common form for incompressible flow with constant diffusivity D is ∂C/∂t + u · ∇C = D ∇^2 C + S, where C is the scalar quantity and S represents sources or sinks. More general versions allow variable diffusivity and tensorial diffusion.

The advection-diffusion model is widely used across science and engineering to predict how substances move in fluids. A key dimensionless parameter is the Peclet number, Pe = UL/D, where U and L are characteristic velocity and length scales. Pe quantifies the relative importance of advection to diffusion: large Pe indicates advection-dominated transport, while small Pe indicates diffusion-dominated transport. Solutions depend on boundary and initial conditions and can be analyzed in steady or transient regimes. In one-dimensional, steady form with constant U and D, the equation reduces to U dC/dx = D d^2C/dx^2 + S.

Numerical methods are typically employed for complex geometries or time-varying flows, including finite difference, finite element, and finite volume approaches. Special schemes, such as upwind discretization, may be used to handle advection-dominated cases, often with implicit time-stepping for stability. The advection-diffusion framework underpins models of pollutant transport in rivers and groundwater, atmospheric and oceanic dispersion, heat transfer in fluids, and various chemical engineering processes, making it a foundational tool in environmental science and engineering.