6.2 Kelvin's Circulation Theorem and Helmholtz Vortex Theorems
5 min read•august 16, 2024
and Helmholtz Vortex Theorems are key concepts in fluid dynamics. They explain how behaves in ideal fluids, showing that stays constant as fluid moves and vortex tubes keep their strength.
These theorems help us understand vortex behavior in real-world situations. They're useful for studying things like hurricanes, aircraft wake turbulence, and even blood flow in the heart. Knowing these principles is crucial for grasping vorticity dynamics in incompressible flows.
Kelvin's Circulation Theorem
Theorem Statement and Proof
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Kelvin's circulation theorem states in a barotropic, inviscid fluid with conservative body forces, the circulation around any closed material contour remains constant as the contour moves with the fluid
Circulation Γ defined as the of velocity around a closed contour Γ=∮Cv⋅dl
Proof involves applying material derivative to circulation integral and using equations of motion for inviscid fluid
Key assumptions include absence of viscosity, barotropic conditions (density as function of pressure only), and conservative body forces
Theorem expressed mathematically as DtDΓ=0, where D/Dt represents material derivative
Closely related to vorticity conservation in inviscid flows
Implications for vortex behavior in ideal fluids include conservation of vortex strength and impossibility of spontaneous vortex generation
Example application demonstrates conservation of circulation for a vortex ring expanding in an inviscid fluid
Mathematical Formulation and Assumptions
Circulation defined mathematically as Γ=∮Cv⋅dl=∫Aω⋅dA (using Stokes' theorem)
Barotropic condition expressed as ρ=f(p), where ρ is density and p is pressure
Conservative body forces derived from a potential function F=−∇Φ
Euler equations for used in proof: ∂t∂v+(v⋅∇)v=−ρ1∇p+F
Material derivative of circulation expressed as DtDΓ=dtd∮Cv⋅dl=∮CDtDv⋅dl
Proof relies on cancellation of pressure gradient and body force terms due to conservative nature
Example demonstrates application of theorem to analyze circulation around a deforming material contour in a tornado-like vortex
Vorticity Dynamics in Inviscid Flows
Vorticity Concepts and Conservation
Vorticity ω defined as curl of velocity field ω=∇×v
Kelvin's theorem implies vortex lines move with fluid in inviscid flows, preserving strength and topology
Theorem used to analyze evolution of vortex rings, vortex filaments, and other vortical structures
In 2D inviscid flows, leads to conservation of vorticity for each fluid particle
Explains phenomena such as vortex stretching and tilting in 3D inviscid flows
Provides basis for understanding persistence of large-scale atmospheric and oceanic vortices (hurricanes, ocean eddies)
Used to derive other important results like Biot-Savart law for induced velocity fields
Example demonstrates vorticity conservation in the core of a growing tornado
Applications and Analysis Techniques
Vortex stretching in 3D flows described by DtDω=(ω⋅∇)v
Kelvin's theorem applied to analyze vortex ring dynamics, including ring expansion and translation
Conservation of vorticity used to study 2D vortex merger processes (tropical cyclone interactions)
Biot-Savart law derived from Kelvin's theorem: v(x)=4π1∫V∣x−x′∣3ω(x′)×(x−x′)dV′
Theorem applied to explain formation and persistence of von Kármán vortex streets behind obstacles
Analysis of vortex filament behavior using local induction approximation
Example calculation shows induced velocity field around a vortex ring using Biot-Savart law
Helmholtz Vortex Theorems
Fundamental Statements and Implications
Helmholtz vortex theorems consist of three fundamental statements about vortex behavior in inviscid, barotropic fluids
First theorem states vortex lines and tubes move with fluid, maintaining identity and strength over time
Second theorem asserts strength (circulation) of vortex tube remains constant along its length
Third theorem states vortex tube cannot end within fluid; must extend to boundaries or form closed loop
Imply vortex lines cannot be created or destroyed in inviscid flows, leading to vortex line conservation
Provide framework for understanding behavior of vortex rings, vortex filaments, and coherent vortical structures
Important applications in study of atmospheric and oceanic vortices, as well as analysis of turbulent flows
Example demonstrates application of theorems to predict behavior of a smoke ring in still air
Mathematical Formulation and Extensions
Vortex tube strength expressed mathematically as Γ=∫Aω⋅dA=constant
Conservation of vortex line topology formulated using frozen-in field theory
Kelvin-Helmholtz instability analyzed using vortex sheet model derived from Helmholtz theorems
Helmholtz decomposition of vector fields: v=∇φ+∇×A, separating irrotational and solenoidal components
Theorem extensions to compressible flows using Ertel's theorem for potential vorticity
Relation to Taylor-Proudman theorem in rotating fluids
Example calculation shows conservation of circulation for a vortex tube undergoing stretching
Conservation of Circulation and Vortex Tubes
Problem-Solving Techniques
Apply Kelvin's circulation theorem to calculate change in vortex strength due to stretching or compression of vortex tubes
Use concept of circulation conservation to analyze behavior of vortex rings in various flow configurations
Solve problems involving interaction of multiple vortices (motion of vortex pairs, merger of vortex rings)
Apply Helmholtz's theorems to predict evolution of vortex filaments in 3D flows
Analyze behavior of vortices near solid boundaries using method of image vortices
Calculate induced velocity field around vortex structure using Biot-Savart law
Solve problems involving conservation of helicity in inviscid flows, related to linking of vortex lines
Example problem demonstrates calculation of vortex ring velocity using circulation conservation