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Concept
Vorticity equation
Summary
The vorticity equation of fluid dynamics describes the evolution of the vorticity ω of a particle of a fluid as it moves with its flow; that is, the local rotation of the fluid (in terms of vector calculus this is the curl of the flow velocity). The governing equation is:where D/Dt is the material derivative operator, u is the flow velocity, ρ is the local fluid density, p is the local pressure, τ is the viscous stress tensor and B represents the sum of the external body forces. The first source term on the right hand side represents vortex stretching.
The equation is valid in the absence of any concentrated torques and line forces for a compressible, Newtonian fluid. In the case of incompressible flow (i.e., low Mach number) and isotropic fluids, with conservative body forces, the equation simplifies to the vorticity transport equation:
where ν is the kinematic viscosity and is the Laplace operator. Under the further assumption of two-dimensional flow, the equation simplifies to:
The term Dω/Dt on the left-hand side is the material derivative of the vorticity vector ω. It describes the rate of change of vorticity of the moving fluid particle. This change can be attributed to unsteadiness in the flow (∂ω/∂t, the unsteady term) or due to the motion of the fluid particle as it moves from one point to another ((u ∙ ∇)ω, the convection term).
The term (ω ∙ ∇) u on the right-hand side describes the stretching or tilting of vorticity due to the flow velocity gradients. Note that (ω ∙ ∇) u is a vector quantity, as ω ∙ ∇ is a scalar differential operator, while ∇u is a nine-element tensor quantity.
The term ω(∇ ∙ u) describes stretching of vorticity due to flow compressibility. It follows from the Navier-Stokes equation for continuity, namely where v = 1/ρ is the specific volume of the fluid element. One can think of ∇ ∙ u as a measure of flow compressibility. Sometimes the negative sign is included in the term.
The term 1/ρ2∇ρ × ∇p is the baroclinic term. It accounts for the changes in the vorticity due to the intersection of density and pressure surfaces.
Official source
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