Vortex equation

Whirlwind equation (whirlwind evolution equation) is a partial differential equation that describes the evolution in space and time of the whirlwind of the flow velocity of a liquid or gas . A speed swirl ( vorticity ) is a speed rotor. ${\omega \equiv \operatorname {rot} ~v}\equiv \nabla \times v$ ${\ displaystyle {\ omega \ equiv \ operatorname {rot} ~ v} \ equiv \ nabla \ times v}$ ${\ displaystyle {\ omega \ equiv \ operatorname {rot} ~ v} \ equiv \ nabla \ times v}$ . The vortex equation is used in hydrodynamics , geophysical hydrodynamics , astrophysical hydrodynamics , in numerical weather prediction .

Content

Vortex equation of an ideal fluid

Liquid (or gas), in which the effects associated with internal friction ( viscosity ) and heat transfer are negligible, is called “ ideal ” . The dynamics of an ideal fluid obeys the Euler equation ^[1] (1755). If we write this equation in the absence of external forces in the form of Gromeki-Lamb

\ {\frac {\partial v}{\partial t}}-[v\times [\nabla \times v]]=-{\frac {\nabla p}{\rho }}-\nabla \left({\frac {v^{2}}{2}}\right),

{\ displaystyle \ {\ frac {\ partial v} {\ partial t}} - [v \ times [\ nabla \ times v]] = - {\ frac {\ nabla p} {\ rho}} - \ nabla \ left ({\ frac {v ^ {2}} {2}} \ right),}

(one)

Where $\ v$ ${\ displaystyle \ v}$ - velocity vector, $\ p$ ${\ displaystyle \ p}$ - pressure, $\ {\rho }$ ${\ displaystyle \ {\ rho}}$ - density, accept the condition of incompressibility $\ \rho =\operatorname {const} ,(\nabla \cdot v)=0$ ${\ displaystyle \ \ rho = \ operatorname {const}, (\ nabla \ cdot v) = 0}$ and apply the operation to both sides of this equation $\ \operatorname {rot}$ ${\ displaystyle \ \ operatorname {rot}}$ , taking into account the known properties of this operator, we obtain the vortex equation of an ideal incompressible fluid

{\begin{aligned}{\frac {\partial \omega }{\partial t}}+(v\cdot \nabla )\omega -(\omega \cdot \nabla )v=0\end{aligned}}.

{\ displaystyle {\ begin {aligned} {\ frac {\ partial \ omega} {\ partial t}} + (v \ cdot \ nabla) \ omega - (\ omega \ cdot \ nabla) v = 0 \ end {aligned }}.}

(2)

The integral form of this equation corresponds to the Helmholtz – Kelvin theorem on the conservation of circulation velocity in a barotropic fluid ^[2] ^[3] . Equation (2) is called the Helmholtz equation .

With irrotational fluid movement (also called “potential”) $\ \omega =0$ ${\ displaystyle \ \ omega = 0}$ . From equation (2) it follows that if at the initial moment of time the motion is irrotational, then it will remain so in the future.

Vortex equation of a viscous incompressible fluid

If in the equation (1) we also take into account the force of internal friction ( viscosity ), then instead of equation (2) we will have

${\begin{aligned}{\frac {\partial \omega }{\partial t}}+(v\cdot \nabla )\omega -(\omega \cdot \nabla )v=\nu \Delta \omega \end{aligned}},$ ${\ displaystyle {\ begin {aligned} {\ frac {\ partial \ omega} {\ partial t}} + (v \ cdot \ nabla) \ omega - (\ omega \ cdot \ nabla) v = \ nu \ Delta \ omega \ end {aligned}},}$ (3)

Where $\nu$ ${\ displaystyle \ nu}$ - kinematic viscosity ^[4] .

Vortex equation of baroclinic inviscid fluid

The condition of the absence of heat exchange (that is, adiabaticity ) of incompressible inviscid fluid flow is equivalent to the condition of constant entropy (that is, isentropicity ) ^[1] . If we abandon this restriction, then equation (2) will be replaced by a more general

${\begin{aligned}{\frac {\partial \omega }{\partial t}}+(v\cdot \nabla )\omega \ -(\omega \cdot \nabla )v+\omega (\nabla \cdot v)={\frac {1}{\rho ^{2}}}\left[\nabla \rho \times \nabla p\right]\end{aligned}},$ ${\ displaystyle {\ begin {aligned} {\ frac {\ partial \ omega} {\ partial t}} + (v \ cdot \ nabla) \ omega \ - (\ omega \ cdot \ nabla) v + \ omega (\ nabla \ cdot v) = {\ frac {1} {\ rho ^ {2}}} \ left [\ nabla \ rho \ times \ nabla p \ right] \ end {aligned}},}$ (four)

taking into account the effect of baroclinicity . The right side of this equation is zero if $\ \nabla \rho \sim \nabla p$ ${\ displaystyle \ \ nabla \ rho \ sim \ nabla p}$ , that is, if the isopycnic surface is parallel to the isobaric. Otherwise, the vector product of the density gradient and the pressure gradient is different from zero, which leads to a change in vorticity due to the influence of baroclinicity. The influence of baroclinicity on the evolution of the vortex was established by Wilhelm Bjerknes ^[5] ^[6] . This equation revealed the important role of baroclinic effects in the formation and development of eddies in the atmosphere and ocean.

Friedman equation

( Friedmann's equation also exists in cosmology. See Friedmann's equation ).

In the general case, the motion of a Newtonian fluid obeys the Navier-Stokes equations . Unlike the above form of the Euler equation for an incompressible fluid, it takes into account the effects of compressibility and internal friction. Applying a differential operator to the Navier-Stokes equation $\ \operatorname {rot}$ ${\ displaystyle \ \ operatorname {rot}}$ , we obtain the equation of A. A. Friedman ^[7] ^[8] .

$\ \operatorname {helm} \omega ={\frac {1}{\rho ^{2}}}\left[\nabla \rho \times \nabla p\right]+\left[\nabla \times \left({\frac {f}{\rho }}\right)\right],$ ${\ displaystyle \ \ operatorname {helm} \ omega = {\ frac {1} {\ rho ^ {2}}} \ left [\ nabla \ rho \ times \ nabla p \ right] + \ left [\ nabla \ times \ left ({\ frac {f} {\ rho}} \ right) \ right],}$ (five)

Where $\ \operatorname {helm} \omega \equiv {\frac {\partial \omega }{\partial t}}+(v\cdot \nabla )\omega -(\omega \cdot \nabla )v+\omega (\nabla \cdot v)$ ${\ displaystyle \ \ operatorname {helm} \ omega \ equiv {\ frac {\ partial \ omega} {\ partial t}} + (v \ cdot \ nabla) \ omega - (\ omega \ cdot \ nabla) v + \ omega (\ nabla \ cdot v)}$ - Helmholtzian differential operator, $\ f$ ${\ displaystyle \ f}$ - density of molecular viscosity.

The hydrodynamic meaning of a Helmholtzian is that equality $\ \operatorname {helm} \omega =0$ ${\ displaystyle \ \ operatorname {helm} \ omega = 0}$ means "frozen in" vector field $\ \omega$ ${\ displaystyle \ \ omega}$ into a moving fluid, understood in the sense that each vector line of this field (i.e., the line tangent to which at any point has the direction of the vector $\ \omega$ ${\ displaystyle \ \ omega}$ at this point) is preserved , that is, all the time consists of the same liquid particles, and the intensity of the vortex tubes (the walls of which consist of vortex lines), that is, flows $\ \int \limits _{\sigma }(\omega \cdot d\sigma )$ ${\ displaystyle \ \ int \ limits _ {\ sigma} (\ omega \ cdot d \ sigma)}$ of vector $\ \omega$ ${\ displaystyle \ \ omega}$ through any sections $\ \sigma$ ${\ displaystyle \ \ sigma}$ these tubes do not change with time ^[9] .

The effect of gravity does not change the form of equations (2) - (5) because this force is potential.

The Friedmann equation is the basic equation of geophysical hydrodynamics. It builds a theory of numerical weather prediction .

Turbulent Fluid Vortex Equation

The Friedman equation also applies to turbulent flows. But in this case, all the values included in it should be understood as averaged (in the sense of O. Reynolds ). However, it should be borne in mind that such a generalization is not sufficiently accurate here. The fact is that in deriving Eq. (5), the density vector of the turbulent momentum was not taken into account (because of relative smallness). $\ S={\overline {\rho ^{'}v^{'}}}$ ${\ displaystyle \ S = {\ overline {\ rho ^ {'} v ^ {'}}}}$ where the bar above is the averaging sign, the bar indicates deviations from the mean. This circumstance was manifested in the fact that the Friedman equation was incapable of explaining the phenomenon of the index cycle ( vascullation ), in which there is a reversible barotropic exchange of energy and angular momentum between ordered and turbulent motions.

Denote by $\ c=S/\rho$ ${\ displaystyle \ c = S / \ rho}$ - "vector of turbulent transfer velocity". Of course, $\ \left|c\right|\ll \left|v\right|$ ${\ displaystyle \ \ left | c \ right | \ ll \ left | v \ right |}$ Nevertheless, the neglect of turbulent transport in problems of geophysical and astrophysical hydrodynamics leads to a loss of effects that manifest themselves in slow, but developing processes. The evolution equation of a vortex, free from such a restriction, was proposed by A. M. Krigel^[10]^[11] :

$\ \operatorname {helm} \psi =\nabla \times \left[c\times \omega +c\left(\nabla \cdot c\right)+F\right]+{\frac {1}{\rho ^{2}}}\left[\nabla \rho \times \left(\nabla p-\rho \nabla c^{2}\right)\right],$ ${\ displaystyle \ \ operatorname {helm} \ psi = \ nabla \ times \ left [c \ times \ omega + c \ left (\ nabla \ cdot c \ right) + F \ right] + {\ frac {1} { \ rho ^ {2}}} \ left [\ nabla \ rho \ times \ left (\ nabla p- \ rho \ nabla c ^ {2} \ right) \ right],}$ (6)

Where $\ \psi \equiv \nabla \times \left(v+c\right)$ ${\ displaystyle \ \ psi \ equiv \ nabla \ times \ left (v + c \ right)}$ - “pseudovector of full speed vortex”, $\ \rho F$ ${\ displaystyle \ \ rho F}$ - the density of the total friction force (molecular and turbulent). If we omit the baroclinicity and viscosity effects in this equation, then the right-hand side remains, generally speaking, non-zero. In this case, as it is easy to show,theHelmholtz –Kelvinvelocity circulation conservationtheoremdoes not hold , despite the fact that the flow isbarotropic . This conclusion is a consequence of the non-potentiality of the“density of the turbulent Coriolis force” $\ 2\rho \left[c\times \omega \right]$ ${\ displaystyle \ 2 \ rho \ left [c \ times \ omega \ right]}$ . In equation (6), an additional mechanism appeared that influences the evolution of the vortex, opening the way to understanding the natureof the index cycle .

Literature

↑¹²Landau LD ,Lifshits E.M. Hydrodynamics (Theoretical Physics. T.VI). —M .: Nauka. — 1988. — 736 p. —ISBN 5-02-013850-9 .
↑Helmholtz H. Uber integrals der hydrodynamischen Gleichungen, welche den Wirbewegungen entsprechen // Crelle J. — 1858.—55 .
↑Thomson W. On vortex motion // Trans.Roy.Soc.Edinburgh. — 1869.—25 . —Pt.1. — pp.217—260.
↑J. Batchelor. Introduction to fluid dynamics.M.: Mir. —1973. — 760 p.
JBjerknes V. Geofysiske publikationer. — 1921. —2. —No 4. — 88p.
↑Bjerknes V. , Bjerknes J., Solberg H., Bergeron T. Physicalische hydrodynamik. — Berlin. — 1933.
↑Friedman A. A. The theory of motion of a compressible fluid and its application to the movement of the atmosphere // Geophysical Sbornik. — 1927. —5. — P.16—56 (Friedman A. A. Selected Works. M .: Nauka. — 1966. —С.178–226).
↑Friedman A. А.Experience of hydromechanics of compressible fluid .L. — M .: ONTI. — 1934. — 370 p.
↑Monin, AS, Theoretical Foundations of Geophysical Hydrodynamics. —L .: Hydrometeoizdat., 1988. — P.17.
А. Kriegel A. M. On nonconservation of the velocity circulation in a turbulent rotating fluid // Letters to the Journal of Technical Physics. — 1981.—7. — Issue 21. — С.1300—1303.
↑Krigel AM Vortex evolution // Geophys.Astrophys.Fluid Dynamics. — 1983. —24. — Pp.213—223.

[LL-1] ¹²Landau LD ,Lifshits E.M. Hydrodynamics (Theoretical Physics. T.VI). —M .: Nauka. — 1988. — 736 p. —ISBN 5-02-013850-9 .

[2] Helmholtz H. Uber integrals der hydrodynamischen Gleichungen, welche den Wirbewegungen entsprechen // Crelle J. — 1858.—55 .

[3] Thomson W. On vortex motion // Trans.Roy.Soc.Edinburgh. — 1869.—25 . —Pt.1. — pp.217—260.

[4] J. Batchelor. Introduction to fluid dynamics.M.: Mir. —1973. — 760 p.

[5] JBjerknes V. Geofysiske publikationer. — 1921. —2. —No 4. — 88p.

[6] Bjerknes V. , Bjerknes J., Solberg H., Bergeron T. Physicalische hydrodynamik. — Berlin. — 1933.

[7] Friedman A. A. The theory of motion of a compressible fluid and its application to the movement of the atmosphere // Geophysical Sbornik. — 1927. —5. — P.16—56 (Friedman A. A. Selected Works. M .: Nauka. — 1966. —С.178–226).

[8] Friedman A. А.Experience of hydromechanics of compressible fluid .L. — M .: ONTI. — 1934. — 370 p.

[9] Monin, AS, Theoretical Foundations of Geophysical Hydrodynamics. —L .: Hydrometeoizdat., 1988. — P.17.

[10] А. Kriegel A. M. On nonconservation of the velocity circulation in a turbulent rotating fluid // Letters to the Journal of Technical Physics. — 1981.—7. — Issue 21. — С.1300—1303.

[11] Krigel AM Vortex evolution // Geophys.Astrophys.Fluid Dynamics. — 1983. —24. — Pp.213—223.