Vorticityyhtälö

The vorticity equation is a fundamental equation in fluid dynamics that describes the evolution of vorticity in a fluid flow. Vorticity itself is a vector quantity that represents the local rotation of a fluid parcel. The equation is derived from the Navier-Stokes equations and provides a powerful tool for analyzing and understanding various fluid phenomena, from atmospheric circulation to the formation of turbulent eddies.

The vorticity equation typically takes the form of a partial differential equation. It includes terms that describe how vorticity changes over time due to advection (the transport of vorticity by the fluid flow itself), stretching and tilting of vortex lines, diffusion (viscous dissipation of vorticity), and sources or sinks of vorticity, such as those arising from baroclinicity (density gradients) or external forces.

In inviscid flows, the vorticity equation simplifies significantly. A key result in this regime is the Kelvin's circulation theorem, which states that the circulation around any closed contour that moves with the fluid remains constant if the flow is inviscid and irrotational. For a rotational inviscid flow, the vorticity equation highlights that vorticity can be generated or destroyed only through baroclinic effects or external forces.

The vorticity equation is widely used in computational fluid dynamics (CFD) for simulating complex flows. By solving this equation alongside other governing equations like mass conservation and momentum conservation, researchers can predict the behavior of fluids under various conditions. Its applications span a broad range of fields, including meteorology, oceanography, aerospace engineering, and even biofluid mechanics.