Kelvin's vortex law

The Kelvin's vortex law , Thomson's vortex law or Kelvin's circulation law by William Thomson, 1st Baron Kelvin is a statement of fluid mechanics about the speed in a barotropic , frictionless fluid under the influence of a conservative gravity field. If it is possible to mark all fluid elements in a flow on a closed curve under the conditions mentioned and to determine the circulation of the speed along this curve, which is swimming with the flow, then the circulation will always be the same. This circulation is equal to the intensity of the vortex tube with the cross-sectional area enclosed by the curve. Accordingly, the intensity of a vortex tube is constant for all times.

The assumption of freedom from friction, apart from hydrodynamic boundary layers, fits well with flows of fluids with low viscosity . For real gases at low pressures and high temperatures, freedom from friction is a proven assumption. The Earth's gravitational field is an example of a conservative gravitational field. Nevertheless, the prerequisites in real fluids are only approximately given, so that the circulation actually decreases over time due to the dissipation not taken into account in the sentence .

The Kelvin's theorem of vertebrae was historically formulated according to the Helmholtz 's theorem of vertebrae , but today it serves to prove the latter. Even if the prerequisites of the theorem are only approximately given under real conditions, some remarkable properties of flows can be explained with the vortex theorems.

Preliminary remarks

A material curve of fluid elements is defined and the circulation of velocity along this curve is followed over time. It turns out that the time derivative of the circulation in a barotropic fluid depends on two non-rotating fields. Because the circulation along the curve can also be calculated from the rotation of the fields according to Stokes' theorem , the time derivative of the circulation vanishes and it is therefore constant over time.

The time derivative of the integral of a field quantity along a moving path required for the set and the required properties of barotropic fluids are provided below.

Time derivative of a path integral along a moving path

The circulation is the curve integral of the speed along a time-dependent path. In order to be able to calculate the time derivative of the circulation, the analogue of Reynolds' transport theorem is required for curve integrals. The following applies:

${\displaystyle {\frac {\mathrm {D} }{\mathrm {D} t}}\oint _{b}{\vec {f}}\cdot \mathrm {d} {\vec {b}}=\oint _{b}({\dot {\vec {f}}}+{\vec {f}}\cdot \operatorname {grad} {\vec {v}})\cdot \mathrm {d} {\vec {b}}=\oint _{b}{\dot {\vec {f}}}\cdot \mathrm {d} {\vec {b}}+\oint _{b}{\vec {f}}\cdot (\mathrm {d} {\vec {b}}\cdot \nabla ){\vec {v}}}$

In it is

• the differential operator D / Dt and the added point the substantial time derivative ,
• ${\displaystyle b\colon [0,1)\mapsto v}$the curve along which the spatial, vectorial line element is integrated and which runs in the volume v occupied by the fluid at time t, ${\displaystyle \mathrm {d} {\vec {b}}}$
• ${\displaystyle {\vec {f}}}$ a field size transported by the fluid,
• ${\displaystyle {\vec {v}}}$ the flow rate of the fluid and
• degree the gradient .

The vector gradient is a different way of writing the product ${\displaystyle (\mathrm {d} {\vec {b}}\cdot \nabla ){\vec {v}}}$ ${\displaystyle \operatorname {grad} ({\vec {v}})\cdot \mathrm {d} {\vec {b}}\,.}$

proof

Given is a curve with a material, vectorial line element in the volume V occupied by the fluid at time t 0. Time t 0 is the beginning of the observation and is fixed in time, see figure. ${\displaystyle B\colon [0,1)\mapsto V}$ ${\displaystyle \mathrm {d} {\vec {B}}}$ The fluid elements that are on curve B are (mentally) labeled with the material coordinates so that they can be clearly identified in the fluid. The motion function gives the spatial position at time t of a fluid element . As is common in continuum mechanics , uppercase letters indicate material variables and lowercase letters indicate spatial variables. At a later time t, the fluid elements lie on the curve with a spatial, vectorial line element in the volume v occupied by the fluid at time t. The field to be integrated is present in spatial (left) and material representation (right). We are looking for the time derivative of the curve integral of the field size along the path b, which can be expressed as the path integral along the unchangeable path B: This is the deformation gradient that converts the line elements into one another and with which the spatial velocity gradient is formed. The substantial time derivative can be calculated from the above equation , because curve B was defined in a time-invariant manner: If the variables are again expressed spatially, the desired transport sentence is created: ${\displaystyle {\vec {X}}}$ ${\displaystyle {\vec {\chi }}({\vec {X}},t)={\vec {x}}}$ ${\displaystyle {\vec {x}}}$ ${\displaystyle {\vec {X}}}$ ${\displaystyle b\colon [0,1)\mapsto v}$ ${\displaystyle \mathrm {d} {\vec {b}}}$ ${\displaystyle {\vec {f}}({\vec {x}},t)={\vec {f}}({\vec {\chi }}({\vec {X}},t),t)=:{\vec {F}}({\vec {X}},t)\,,}$ ${\displaystyle {\vec {f}}}$ ${\displaystyle \oint _{b}{\vec {f}}\cdot \mathrm {d} {\vec {b}}=\oint _{B}{\vec {F}}\cdot \mathbf {F} \cdot \mathrm {d} {\vec {B}}\,.}$ ${\displaystyle \mathbf {F} }$ ${\displaystyle \mathbf {F} \cdot \mathrm {d} {\vec {X}}=\mathrm {d} {\vec {x}}}$ ${\displaystyle {\dot {\mathbf {F} }}\cdot \mathbf {F} ^{-1}=\operatorname {grad} {\vec {v}}}$ ${\displaystyle {\frac {\mathrm {D} }{\mathrm {D} t}}\oint _{B}{\vec {F}}\cdot \mathbf {F} \cdot \mathrm {d} {\vec {B}}=\oint _{B}({\dot {\vec {F}}}\cdot \mathbf {F} +{\vec {F}}\cdot {\dot {\mathbf {F} }})\cdot \mathrm {d} {\vec {B}}=\oint _{B}({\dot {\vec {F}}}+{\vec {F}}\cdot \underbrace {{\dot {\mathbf {F} }}\cdot \mathbf {F} ^{-1}} _{=\operatorname {grad} {\vec {v}}})\cdot \underbrace {\mathbf {F} \cdot \mathrm {d} {\vec {B}}} _{=\mathrm {d} {\vec {b}}}\,.}$ ${\displaystyle {\frac {\mathrm {D} }{\mathrm {D} t}}\oint _{b}{\vec {f}}\cdot \mathrm {d} {\vec {b}}=\oint _{b}({\dot {\vec {f}}}+{\vec {f}}\cdot \operatorname {grad} {\vec {v}})\cdot \mathrm {d} {\vec {b}}\,.}$

Flow (dark blue) with a material curve (blue) that borders a surface (yellow)

Barotropic fluids

In a barotropic fluid, the density is a function of the pressure alone. Then there is a function P with the property

${\displaystyle P=\int {\frac {\mathrm {d} p}{\rho (p)}}\quad \Leftrightarrow \quad \mathrm {d} P={\frac {\mathrm {d} p}{\rho (p)}}\quad \Leftrightarrow \quad \operatorname {grad} P={\frac {1}{\rho (p)}}\operatorname {grad} p\,.}$

Hence the Euler equations in a conservative gravitational field for a barotropic fluid read : ${\displaystyle {\vec {k}}=-\operatorname {grad} V}$

${\displaystyle {\frac {\partial {\vec {v}}}{\partial t}}+({\vec {v}}\cdot \nabla ){\vec {v}}={\frac {\mathrm {D} {\vec {v}}}{\mathrm {D} t}}={\dot {\vec {v}}}=-{\frac {1}{\rho }}\operatorname {grad} p+{\vec {k}}=-\operatorname {grad} (P+V)\,.}$

The gradient is the substantial acceleration in a barotropic, frictionless fluid moving in a conservative acceleration field, i.e. rotation-free. ${\displaystyle {\dot {\vec {v}}}}$

Proof of Kelvin's vortex law

Given is a closed curve with a vectorial line element in the volume v occupied by the fluid at time t. Then the circulation Γ of the velocity along the spatial curve b is the curve integral ${\displaystyle b\colon [0,1)\mapsto v}$ ${\displaystyle \mathrm {d} {\vec {b}}}$ ${\displaystyle {\vec {v}}}$

${\displaystyle \Gamma =\oint _{b}{\vec {v}}\cdot \mathrm {d} {\vec {b}}\,.}$

The substantial time derivative can be calculated from this with the transport rate for line integrals given above :

${\displaystyle {\begin{aligned}{\dot {\Gamma }}={\frac {\mathrm {D} }{\mathrm {D} t}}\oint _{b}{\vec {v}}\cdot \mathrm {d} {\vec {b}}=\oint _{b}({\dot {\vec {v}}}+{\vec {v}}\cdot \operatorname {grad} {\vec {v}})\cdot \mathrm {d} {\vec {b}}=\oint _{b}\left({\dot {\vec {v}}}+{\frac {1}{2}}\operatorname {grad} ({\vec {v}}\cdot {\vec {v}})\right)\cdot \mathrm {d} {\vec {b}}\,,\end{aligned}}}$

because according to the product rule

${\displaystyle \sum _{i=1}^{3}{\frac {\mathrm {d} (v_{i}v_{i})}{\mathrm {d} x_{j}}}=2\sum _{i=1}^{3}v_{i}{\frac {\mathrm {d} v_{i}}{\mathrm {d} x_{j}}}\quad {\text{für}}\,j=1,2,3\quad \Leftrightarrow \quad \operatorname {grad} ({\vec {v}}\cdot {\vec {v}})=2{\vec {v}}\cdot \operatorname {grad} ({\vec {v}})\,.}$

According to Stokes' theorem , the time derivative of the circulation can also be expressed as an area integral of the rotation of the integrand over a surface a bordered by curve b, but otherwise arbitrary , and according to its vectorial surface element ${\displaystyle \mathrm {d} {\vec {a}}}$

${\displaystyle {\dot {\Gamma }}=\oint _{b}\left({\dot {\vec {v}}}+{\frac {1}{2}}\operatorname {grad} ({\vec {v}}\cdot {\vec {v}})\right)\cdot \mathrm {d} {\vec {b}}=\int _{a}\operatorname {rot} \left({\dot {\vec {v}}}+{\frac {1}{2}}\operatorname {grad} ({\vec {v}}\cdot {\vec {v}})\right)\cdot \mathrm {d} {\vec {a}}=\int _{a}\operatorname {rot} ({\dot {\vec {v}}})\cdot \mathrm {d} {\vec {a}}}$

be calculated. The circulation is consequently constant precisely when the substantial acceleration is rotation-free. This statement is also called the general Thomson vortex law . Because the substantial acceleration in a frictionless, barotropic fluid, which moves under the influence of a conservative gravity field, as a gradient field, as shown above, is actually rotation-free, the special statement formulated at the beginning follows . ${\displaystyle {\dot {\vec {v}}}}$

Individual evidence

1. ^ H. Altenbach: Continuum Mechanics . Springer, 2012, ISBN 978-3-642-24118-5 , pp. 179 .
2. ↑ Kameier 2013, p. 268f.

See also

• Vortex (fluid dynamics)
• Euler equations (fluid mechanics)
• Tensoranalysis formula collection

literature

• Ludwig Prandtl: Guide through fluid mechanics . Ed .: Herbert Oertel. Vieweg, 2008, ISBN 978-3-8348-0430-3 .
• F. Kameier, CO Paschereit: Fluid Mechanics . Walter de Gruyter, 2013, ISBN 978-3-11-029221-3 .

Web links

• NA Adams: Fluid Mechanics 2. (PDF; 2.0 MB) Introduction to the dynamics of fluids. 2015, accessed August 29, 2015 .