Vortex Flows Flashcards
What are vortices and what are two rotation modes for the fluid particles?
A vortex is a coherent rotational motion of fluid particles around a common axis.
Either the FP themselves rotate or do not rotate
What is Helmholtz Decomposition?
Decomposition of Vector Field into Rotational and Divergence Free Components
What is the formula for Vorticity?
/omega = /nabla x u
What are Some Vortex Models?
Solid Body Vortex: 2D velocity field / Fixed Relative Location of FP / Rotating About Common Axis
Potential Vortex: 2D Velocity Field / No Fixed Relative Location / Vorticity only on axis
Rankine Vortex: Solid Body Core + Potential Vortex Far Field / Inviscid Model with Kink
What is Circulation?
It is an alternate intensity measure to vorticity because of infinite vorticity on axis for Potential Vortices
Line integral of the velocity field of a fluid around a closed loop or contour
What are Three Additional quantities that help understand a vortex?
Vortex Lines, Vortex Surfaces, Vortex Tubes
VL: are Integral curves of the vorticity field
VS: is a collection of vo
What is a Biot-Savart Law?
It reconstructs a velocity field given the vortex lines for simply conned patch of vorticity
What is the mirror principle?
Creating a mirrored vortex across a wall to use the Biot Sarvat Law
What is a vortex sheet?
A generic model to investigate areas of large velocity gradients
What is conservation of circulation?
Temporal derivative of vorticity under certain idealizations, remains constant.
What is Helmholtz theorem 1?
Irrotational FM remain Irrotational under certain circumstances
What is Helmholtz theorem 2?
FP that at any time belong to a vortex line remain on that vortex line at all times
What is Helmholtz theorem 3?
For any two closed lines that enclose a vortex tube, circulation is constant
What is the formula for the induced velocity of a straight infinite vortex line?
circulation / (2 pi r)
What is the formula for circulation?
\int u ds
what is the relation between velocity and radius in a potential vortex?
u = 1/r * k
What are the differences between vorticity and velocity?
Vorticity:
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- Always divergence free
- Integral Curves are vortex lines
- Vortex tube has limiting surface generated by vortex lines
- Has Circulation
- Is Galilean Invariant
Velocity:
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- Divergence free only for incompressible flow
- Integral curves are streamlines
- Streamtubes has liming surface generated by streamlines
- Volume Flux
- Not Galilean invariant
How is the Biot Servant Law Obtained?
From Green Function of Poisson Equation
What wall boundary condition do Navier Stokes Equation demand for viscous and inviscid flow?
No slip condition for viscous flow, free slip condition for inviscid flow
What is the Kelvin Vortex Theorem?
In an inviscid and barotropic fluid flow where the acting volume force has a potential the circulation of a (closed) material curve remains constant in time
What are the shared conditions for Helmholtz and Kelvin theorem?
- Inviscid flow
- Volume force has a potential
- Barotropic fluid
What is the formula for the velocity field of a solid body vortex?
u = omega r
What is the formula for the velocity field of a potential vortex?
u = (circulation / 2 pi) * 1/r
How does the vorticity of solid-body, potential and Rankine vortex vary?
Solid body: constant
Potential: Infinity at 0, zero elsewhere
Rankine: Finite value at core, 0 elsewhere
