Vorticity equation
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The vorticity equation of
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
where ν is the kinematic viscosity and is the Laplace operator. Under the further assumption of two-dimensional flow, the equation simplifies to:
Physical interpretation
- 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 convectionterm).
- 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.
- The term ∇ × (∇ ∙ τ/ρ), accounts for the diffusion of vorticity due to the viscous effects.
- The term ∇ × B provides for changes due to external body forces. These are forces that are spread over a three-dimensional region of the fluid, such as electromagnetic forces. (As opposed to forces that act only over a surface (like drag on a wall) or a line (like surface tension around a meniscus).
Simplifications
- In case of conservative body forces, ∇ × B = 0.
- For a barotropic fluid, ∇ρ × ∇p = 0. This is also true for a constant density fluid (including incompressible fluid) where ∇ρ = 0. Note that this is not the same as an incompressible flow, for which the barotropic term cannot be neglected.
- For inviscidfluids, the viscosity tensor τ is zero.
Thus for an inviscid, barotropic fluid with conservative body forces, the vorticity equation simplifies to
Alternately, in case of incompressible, inviscid fluid with conservative body forces,
For a brief review of additional cases and simplifications, see also.[2] For the vorticity equation in turbulence theory, in context of the flows in oceans and atmosphere, refer to.[3]
Derivation
The vorticity equation can be derived from the Navier–Stokes equation for the conservation of angular momentum. In the absence of any concentrated torques and line forces, one obtains:
Now, vorticity is defined as the curl of the flow velocity vector; taking the curl of momentum equation yields the desired equation. The following identities are useful in derivation of the equation:
where is any scalar field.
Tensor notation
The vorticity equation can be expressed in
In specific sciences
Atmospheric sciences
In the
Here, η is the polar (z) component of the vorticity, ρ is the atmospheric
See also
References
- ISBN 978-0-486-43261-8.
- ^ Burr, K. P. "Marine Hydrodynamics, Lecture 9" (PDF). MIT Lectures.
- ^ Salmon, Richard L. "Lectures on Geophysical Fluid Dynamics, Chapter 4" (PDF). Oxford University Press; 1 edition (February 26, 1998).
Further reading
- Manna, Utpal; Sritharan, S. S. (2007). "Lyapunov Functionals and Local Dissipativity for the Vorticity Equation in Lp and Besov spaces". Differential and Integral Equations. 20 (5): 581–598. S2CID 50701138.
- Barbu, V.; Sritharan, S. S. (2000). "M-Accretive Quantization of the Vorticity Equation" (PDF). In Balakrishnan, A. V. (ed.). Semi-Groups of Operators: Theory and Applications. Boston: Birkhauser. pp. 296–303.
- Krigel, A. M. (1983). "Vortex evolution". Geophysical & Astrophysical Fluid Dynamics. 24 (3): 213–223. .
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