vorticity dynamics

vorticity dynamics

Table of Contents

• 1. Vorticity Dynamics   vortex
  • 1.1. Vorticity - vector, ω
  • 1.2. Q-criterion– Hunt, Wray & Moin 1988
  • 1.3. Vortex shedding from an airfoil in low Re flow
  • 1.4. Vortex vs Eddy
  • 1.5. 2D vortex
  • 1.6. 3D vortex filaments
  • 1.7. vortex flow : fourth elementary flow
    • 1.7.1. Feature/Property
  • 1.8. Irrotational(Free) vortex
  • 1.9. Kelvin's Circulation Theorem
  • 1.10. Helmhotz's Vortex Theorems
  • 1.11. Vorticity Equation in a Nonrotating Frame
  • 1.12. Vortex Sheet
  • 1.13. References
    • 1.13.1. text books for vortex theory
  • 1.14. footnotes

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 \[ u_{\theta} = \frac{1}{r} \frac{\partial \phi}{\partial \theta} =\frac{\partial \Psi}{\partial r} \]

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:

¹

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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