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 Why should vorticity be introduced into hydrodynamics? - Zhu Hui's answer - Know https://www.zhihu.com/question/31159018/answer/51047540
 Vorticity distribution like Karman vortex street produced by droplet hitting water surface http://blog.sciencenet.cn/blog-739225-889268.html
 Why is the curl of the linear velocity of a rigid body equal to twice the angular velocity https://baijiahao.baidu.com/s?id=1659243638317087666
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Vortex https://en.wikipedia.org/wiki/Vortex
 Chapter 7 of fluid mechanics of Dalian University of Technology https://www.icourse163.org/course/DUT-1002043010?tid=1450232493
 Karman vortex street experiment https://www.youtube.com/watch?v=Eh_vOcXazaU
 Kaman vortex street simulation https://www.youtube.com/watch?v=IDeGDFZSYo8
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Whats the differences between vortex and turbulence in fluid dynamics? https://www.quora.com/Whats-the-differences-between-vortex-and-turbulence-in-fluid-dynamics
free vortex https://www.ramauniversity.ac.in/online-study-material/fet/me/btech/iiisemester/fluidmechanics/lecture-41.pdf
 vortex https://baike.baidu.com/item/%E6%B6%A1%E6%97%8B/734600?fr=aladdin
 vortex https://zh.wikipedia.org/wiki/%E6%B8%A6%E6%97%8B
Group1 L3 Free and Forced Vortex https://www.youtube.com/watch?v=IVBOmnx1qk0

Vorticity

Vorticity vorticity
Vortex vortex (whirl pool vortex, with three points of water, refers to the vortex of water)
the vorticity is the curl of the flow velocity
w = w x i + w y j + w z k w = w_x i + w_y j + w_z k w=wx​i+wy​j+wz​k
r = x i + y j + z k r = x i + y j + z k r=xi+yj+zk
v = w × r v = w \times r v=w×r
Ω = ∇ × v = 2 w \Omega = \nabla \times v = 2w Ω=∇×v=2w
Vorticity is defined as the angular velocity whose curl result is 2 times, so vorticity can be used to reflect the angular velocity.

Vortices are divided into forced vortices and free vortices,
Forced vortex: stirring the fluid or rotating the container can make the fluid rotate, but there is no relative motion between the micro clusters. The fluid is rotated by consuming the energy provided by the outside world.
Free vortex: it rotates due to inertia and does not consume external energy. For example, opening the waterproof at the bottom of the pool will form a vortex.

Free vortex example:
1. Flow around a circular bend.
2. A whirlpool in a river.
3. Flow of liquid in a centrifugal pump casing after it has left the impeller.
4. Flow of water in a turbine casing before it enters the guide vanes.
5. Flow of liquid through a hole/outlet provided at the bottom of a shallow vessel (e.g., wash basin, bath tub, etc.)

Velocity circulation

Vorticity is equivalent to the circulation per unit area
The velocity circulation of the irrotational flow field is 0, and the velocity circulation of 0 is not necessarily irrotational (there may be two vortex tubes with the same strength but in the opposite direction)
Vortex strength is the area fraction of vorticity. The velocity circulation along any closed curve is equal to the vortex strength passing through any surface bounded by the curve.
The relationship between vortex strength and velocity circulation is established according to Stokes formula. (Stokes formula transforms solving surface integral into solving curve integral. Green's formula is a special case of Stokes formula, which was derived before Stokes formula.)
∮ v d l = ∬ Ω d A \oint v dl = \iint \Omega dA ∮vdl=∬ΩdA
∮ v x d x + v y d y + v z d z = ∬ Ω x d y d z + Ω y d x d z + Ω z d x d y \oint v_x dx + v_y dy + v_z dz = \iint \Omega_x dydz + \Omega_y dxdz + \Omega_z dxdy ∮vx​dx+vy​dy+vz​dz=∬Ωx​dydz+Ωy​dxdz+Ωz​dxdy

Kelvin's velocity circulation theorem

Barotropic fluid: a fluid in which the pressure at any point in the interior is only a function of density
Ideal fluid: inviscid fluid

Kelvin's velocity circulation theorem: also known as Thomson's theorem, the conservation theorem of velocity circulation along the fluid line
Lagrange's theorem: in an ideal barotropic fluid, and the mass force has potential. If there is no vortex before, there will be no vortex after; If there is a vortex, the vortex will not disappear by itself.

Helmholtz theorem

The first theorem: the vortex tube cannot start or end in the fluid. At the same moment, the Vortex Flux of each section of the vortex tube is the same. (the sea dragon absorbs water, starting at the sea and ending at infinity)
The second theorem - vortex tube conservation theorem: in an ideal barotropic fluid with potential mass force, the vortex tube always remains a vortex tube composed of the same fluid particles.
The third theorem: in an ideal barotropic fluid with potential mass force, the vortex tube strength does not change with time.

Induced velocity of vortex

The vortex field has an effect on the velocity field, and the velocity field has an effect on the vortex field
Biot Savart theorem: relationship between induced velocity and velocity loop

Binary vortex

Rankine Vortex: incompressible fluid, with cylindrical fluid rotating around the center in the middle, and the fluid outside the cylinder moves without rotation (the principle of cyclone separator) (Rankine is also famous in rotor dynamics)

irrotational flow without swirling flow: the angular velocity of micro cluster is 0, Ω = ∇ × v = 0 \Omega = \nabla \times v = 0 Ω=∇ × v=0, there is a mathematical definition: the curl of the gradient of the scalar function is 0, so the gradient of the scalar function can be used to define the velocity field without swirl, v = ∇ ϕ v = \nabla \phi v=∇ ϕ， Non swirling flow is also called potential flow. (the gradient of multivariate function is a vector function), the vortex without swirl is called irrotational vortex or free vortex or potential vortex.

What are the applications of Rankine vortex?

Powerful

Conservative force: F = F ( r ) F=F(r) F=F(r)， ∮ F d r = 0 \oint F dr = 0 ∮ Fdr=0, the cycle integral of the conservative force is 0.
Powerful: F = F ( r , r ˙ , t ) , F δ r = − δ U F=F(r,\dot{r},t), F \delta r = -\delta U F=F(r,r ˙, t),F δ r=− δ U. The virtual work done by a force is equal to the negative value of the generalized potential energy variation.

conservative force: gravity, buoyancy, elasticity, electrostatic force, universal gravitation, etc
Non conservative force: friction force, inelastic material stress, air resistance, water resistance, resistance, etc

kinetic energy,potential energy,work
The external work done on an elastic member in causing it to distort from its unstressed state is transformed into strain energy which is a form of potential energy.

other

Wake vortex wake vortex
trapped vortex
Starting vortex
Batchelor vortex
Burgers vortex
Horseshoe vortex
Kaufmann vortex
Lamb–Oseen vortex
Wingtip vortex
Vortex lift
vortex shedding frequency
Vortex induced vibration

In 1911, von Karman studied the stability of vortex patterns that form behind stationary bodies in flowing fluids ("K á rm á n vortex street")

Passive field
Active field
Irrotational field
Rotating field
Conformal transformation
Induction speed