22
P. Liu
gas transition from non-equilibrium state to equilibrium state and explained
the second law of thermodynamics from the statistical significance. The
Boltzmann equation is an equation describing the motion of rarefied gases.
In 1872, the British scientist Rankine proposed the famous Rankine vortex
model (as shown in Fig. 1.25) for the steady concentrated vortex and its
induced flow field. The model established a combination model of a free
vortex and forced vortex. The results show that the flow field in the vortex
core is a circular cylinder with equal vorticity, and the flow field outside the
vortex core is a free vortex-induced flow field. The obtained velocity and
pressure fields are as follows:
In the vortex core, the flow field with equal vorticity (because the deformation rate is zero, in fact, it is also inviscid and rotational motion), and its
circumferential velocity satisfies the rotation law of the rigid body around the
spin axis, namely
u θ =
Γ
2π R 2 r
where u θ is the velocity in the circumferential direction, R is the radius
of the vortex core, and Γ is the vortex intensity (velocity circulation). The
corresponding static pressure at arbitrary radius r is
p = p c +
1
2
ρu
2
θ
Fig. 1.25 Rankin eddy current model
P. Liu
gas transition from non-equilibrium state to equilibrium state and explained
the second law of thermodynamics from the statistical significance. The
Boltzmann equation is an equation describing the motion of rarefied gases.
In 1872, the British scientist Rankine proposed the famous Rankine vortex
model (as shown in Fig. 1.25) for the steady concentrated vortex and its
induced flow field. The model established a combination model of a free
vortex and forced vortex. The results show that the flow field in the vortex
core is a circular cylinder with equal vorticity, and the flow field outside the
vortex core is a free vortex-induced flow field. The obtained velocity and
pressure fields are as follows:
In the vortex core, the flow field with equal vorticity (because the deformation rate is zero, in fact, it is also inviscid and rotational motion), and its
circumferential velocity satisfies the rotation law of the rigid body around the
spin axis, namely
u θ =
Γ
2π R 2 r
where u θ is the velocity in the circumferential direction, R is the radius
of the vortex core, and Γ is the vortex intensity (velocity circulation). The
corresponding static pressure at arbitrary radius r is
p = p c +
1
2
ρu
2
θ
Fig. 1.25 Rankin eddy current model