vorticity dynamics

1. Vorticity Dynamics vortex

1 Vorticity Dynamics vortex

Vorticity_Dynamics.html

Vortex: a mass of air, water, etc. that spins around very fast and pulls things into its centre (Oxford Dictionary)
Vortex: ¹

vortex line: a line whose tangent is everywhere parallel to the local vorticity vector
vortex filament: a vortex tube whose cross-section is of infinitesimal dimensions (extremely small).

vortex can be quantified by:

circulation
Q-criterion
λ-2

1.1 Vorticity - vector, ω

Vorticity: twice the angular velocity, $ \nabla \times v $
Angular velocity: ω $ \omega = 0.5 \nabla \times v $

physical meaning:

vorticity meansures the solid body like rotation of a material point, p', that neighbors the primary material point, p (panton)

Features:

∇ ⋅ ω = 0 : vorticity is divergence free
∇ × ∇ φ =0
the existence of vorticity indicates that viscous effects are important

why does fluid particles rotate? it due to unbalanced shear stress

Roughly speaking, Vorticity dynamics offers a method to separate a flow into viscous and inviscid effects.

1.2 Q-criterion– Hunt, Wray & Moin 1988

velocity gradient, ∇ v ∇ v = S + Ω where, S is rate-of-strain tensor, and Ω is vorticity tensor

1.3 Vortex shedding from an airfoil in low Re flow

S. Yarusevych, 2009, J. fluid Mech. On vortex shedding from an airfoil in low-Reynolds-number flows

1.4 Vortex vs Eddy

vortices don't necessarily include turbulence
eddy is used to describe turbulence

https://www.researchgate.net/post/What_is_the_difference_between_a_vortex_and_an_eddy2

a good start for learning the subtleties of vortex flow, and the relation to coherent structures (CS).

Jeong and Hussain "On the identification of a vortex", J Fluid Mechanics, 285, p69-94 (1995)

1.5 2D vortex

V_θ = Γ /(2 π r) V_r = 0 V_z = 0

> Anderson 5.1

1.6 3D vortex filaments

vortex-filament.pdf

1.7 vortex flow : fourth elementary flow

all the streamlines are concentric circles about a given point

the velocity along any given circular streamline be constant
velocity vary from one streamline to another inversely with distance from the common center.

1.7.1 Feature/Property

physically possible incompressible at every point

\[ \nabla \dot \mathbf{v} = 0 \]

irrotational at every point except the origin

\[ \nabla \times \mathbf{v} = 0 \]

tangental velocity

\[ v_{\theta} = \frac{\Gamma}{2\pi r} \]

1.8 Irrotational(Free) vortex

potential (free) vortex flow: a flow with circular paths around a central point

such that the velocity distribution still satisfies the irrotational condition : curl(v)=0

$ \nabla \times \mathbf{v}=0 $ (i.e. the fluid particles don't rotate about their own centers but simply move on circular path.

no radial velocity

in a cylindrical coordinate ( r, θ, z) For 2D potential vortex, u_z =0, u_r =0 \[ u_{\theta} = \frac{1}{r} \frac{\partial \phi}{\partial \theta} =\frac{\partial \Psi}{\partial r} \]

$C:\Users\exw692\Dropbox\Emacs\figures\tangential_v_potential_vortex.png$

Figure 1: tangential velocity of potential vortex vs r

The origin (center) of the potential vortex is considered as a singularity point in the flow since the velocity goes to infinity at this point

If the contour encircles the potential vortex origin, the circulation will be non-zero.
If the contour does not encircle any singularities, however, the circulation will be zero.

1.9 Kelvin's Circulation Theorem

> 5.2, Kundu

1.10 Helmhotz's Vortex Theorems

1.11 Vorticity Equation in a Nonrotating Frame

1.12 Vortex Sheet

1.13 References

3.14 anderson
2.7 Pope
G.Haller,2005,J.Fluid Mech. , an objective definition of a vortex
Kundu, Pijush K., and Ira M. Cohen. Fluid Mechanics. 6th ed. Academic Press, 2015. ISBN: 9780124059351.
Chapter 5: Vorticity Dynamics 5.1: Introduction
5.3: Helmhotz's Vortex Theorems

5.4: Vorticity Equation in a Nonrotating Frame

5.8: Vortex Sheet

1.13.1 text books for vortex theory

Chapter 13, incompressible flow, Panton 3.14, 5.1 Fundamentals of Aerodynamics, John Anderson Milne-Thomsen, 1952

1.14 footnotes

Footnotes:

G.Haller,2005,J.Fluid Mech. , an objective definition of a vortex

Author: kaiming

Created: 2019-07-05 Fri 20:57

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