Difference between revisions of "Helix and spiral"
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Corresponding Wikipedia article: [[wikipedia:Helicoidal flow|Helicoidal flow]] | Corresponding Wikipedia article: [[wikipedia:Helicoidal flow|Helicoidal flow]] | ||
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| − | A simple condition of a steady-state vortex follows from the | + | A simple condition of a steady-state vortex follows from the equation ([[Vortex#Eq-7|"Vortex", 7]]): |
| − | :<math>\mathbf{\overrightarrow{v}}\times curl\mathbf{\overrightarrow{v}}=0\label{ABC_1}</math> | + | :<math>\mathbf{\overrightarrow{v}}\times curl\,\mathbf{\overrightarrow{v}}=0\label{ABC_1}</math> |
| − | :<math>curl\mathbf{\overrightarrow{v}}=k\mathbf{\overrightarrow{v}}\label{ABC_2}</math> | + | :<math>curl\,\mathbf{\overrightarrow{v}}=k\mathbf{\overrightarrow{v}}\label{ABC_2}</math> |
The flow with such condition looks like a helix (spiral) called the [[wikipedia:Eugenio Beltrami|Beltrami]] or ABC (Arnold-Beltrami-Childress) flow. The helical flow, regardless of the listed authors, was better described in details by Gromeka<ref name="gromeka_fluid"></ref>. | The flow with such condition looks like a helix (spiral) called the [[wikipedia:Eugenio Beltrami|Beltrami]] or ABC (Arnold-Beltrami-Childress) flow. The helical flow, regardless of the listed authors, was better described in details by Gromeka<ref name="gromeka_fluid"></ref>. | ||
| − | The obvious properties of a helical flow: | + | The obvious properties of a helical flow are: |
* The [[wikipedia:Vorticity#Vortex_lines_and_vortex_tubes|vortex lines]] are matching the [[wikipedia:Streamlines, streaklines, and pathlines|streamlines]] (vectors are [[wikipedia:Collinearity|collinear]]). | * The [[wikipedia:Vorticity#Vortex_lines_and_vortex_tubes|vortex lines]] are matching the [[wikipedia:Streamlines, streaklines, and pathlines|streamlines]] (vectors are [[wikipedia:Collinearity|collinear]]). | ||
| − | * The constant in | + | * The constant in [[Continuum#Bernoulli's law|Bernoulli's law]] is the same over entire continuum, that is, the same Bernoulli's equation is valid over any line. This follows from the Gromeka-Lamb equation ([[Vortex#Eq-3|"Vortex", 3]]). |
| − | For an uniform helical flow (<math>k=const</math>), the condition \ref{ABC_2} subjected to the curl operator and the known [[Physical field#Vector analysis formulas|formula of vector analysis]] | + | For an uniform helical flow (<math>k=const</math>), the condition \ref{ABC_2} subjected to the curl operator and the known [[Physical field#Vector analysis formulas|formula of vector analysis]] becomes: |
| − | :<math>curl\;curl\mathbf{\overrightarrow{v}}=k\;curl\mathbf{\overrightarrow{v}}\label{Helmholtz_wave_2}</math> | + | :<math>curl\;curl\,\mathbf{\overrightarrow{v}}=k\;curl\,\mathbf{\overrightarrow{v}}\label{Helmholtz_wave_2}</math> |
| − | :<math>grad\;div\mathbf{\overrightarrow{v}}-\Delta\mathbf{\overrightarrow{v}}=k^2\mathbf{\overrightarrow{v}}\label{Helmholtz_wave_1}</math> | + | :<math>grad\;div\,\mathbf{\overrightarrow{v}}-\Delta\mathbf{\overrightarrow{v}}=k^2\mathbf{\overrightarrow{v}}\label{Helmholtz_wave_1}</math> |
| − | Hence, for a steady-state vortex (<math>div\mathbf{\overrightarrow{v}}=0</math>) of the incompressible or barotropic flow | + | Hence, the [[wikipedia:Helmholtz equation|Helmholtz wave equation]] is valid for a steady-state vortex (<math>div\,\mathbf{\overrightarrow{v}}=0</math>) of the incompressible or barotropic flow: |
:<math>\boxed{\Delta\mathbf{\overrightarrow{v}}+k^2\mathbf{\overrightarrow{v}}=0}\label{Helmholtz_wave}</math> | :<math>\boxed{\Delta\mathbf{\overrightarrow{v}}+k^2\mathbf{\overrightarrow{v}}=0}\label{Helmholtz_wave}</math> | ||
| − | This [[wikipedia:Elliptic partial differential equation|elliptic equation]] has many solutions, which describe something like a [[Waves|wave]] with | + | This [[wikipedia:Elliptic partial differential equation|elliptic equation]] has many solutions, which describe something like a [[Waves|wave]] with [[wikipedia:Wavenumber|wavenumber]] <math>\approx k</math>, which propagates with arbitrary velocity including zero. Such wave is called the [[wikipedia:Soliton|soliton]]. |
[[Image:Bessel_1.png|frame|[[wikipedia:Bessel function|Bessel function]] of the 1st kind]] | [[Image:Bessel_1.png|frame|[[wikipedia:Bessel function|Bessel function]] of the 1st kind]] | ||
For a helical flow along one axis, Gromeka had found a simple solution in the [[wikipedia:Cylindrical coordinate system|cylindrical coordinates]] <math>(r,\theta,z) </math><ref name="gromeka_fluid"></ref>: | For a helical flow along one axis, Gromeka had found a simple solution in the [[wikipedia:Cylindrical coordinate system|cylindrical coordinates]] <math>(r,\theta,z) </math><ref name="gromeka_fluid"></ref>: | ||
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<math>J_0, J_1</math> is the [[wikipedia:Bessel function|Bessel functions]] of 1st kind. | <math>J_0, J_1</math> is the [[wikipedia:Bessel function|Bessel functions]] of 1st kind. | ||
| − | The given example shows that the flow velocity not only decreases with distance from the vortex axis, but it can change own direction. | + | The given example shows that the flow velocity not only decreases with distance from the vortex axis, but it can also change own direction. |
The [[wikipedia:Vortex tube|Ranque-Hilsch tube]] shows the effect of the direction change. The fast cold air is directed to one side from the vortex center, and the slow hot air to the opposite direction from the vortex periphery. | The [[wikipedia:Vortex tube|Ranque-Hilsch tube]] shows the effect of the direction change. The fast cold air is directed to one side from the vortex center, and the slow hot air to the opposite direction from the vortex periphery. | ||
Latest revision as of 10:27, 29 August 2016
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Corresponding Wikipedia article: Helicoidal flow
A simple condition of a steady-state vortex follows from the equation ("Vortex", 7): \[\mathbf{\overrightarrow{v}}\times curl\,\mathbf{\overrightarrow{v}}=0\tag{1}\] \[curl\,\mathbf{\overrightarrow{v}}=k\mathbf{\overrightarrow{v}}\tag{2}\] The flow with such condition looks like a helix (spiral) called the Beltrami or ABC (Arnold-Beltrami-Childress) flow. The helical flow, regardless of the listed authors, was better described in details by Gromeka[1].
The obvious properties of a helical flow are:
- The vortex lines are matching the streamlines (vectors are collinear).
- The constant in Bernoulli's law is the same over entire continuum, that is, the same Bernoulli's equation is valid over any line. This follows from the Gromeka-Lamb equation ("Vortex", 3).
For an uniform helical flow (\(k=const\)), the condition (2) subjected to the curl operator and the known formula of vector analysis becomes: \[curl\;curl\,\mathbf{\overrightarrow{v}}=k\;curl\,\mathbf{\overrightarrow{v}}\tag{3}\] \[grad\;div\,\mathbf{\overrightarrow{v}}-\Delta\mathbf{\overrightarrow{v}}=k^2\mathbf{\overrightarrow{v}}\tag{4}\] Hence, the Helmholtz wave equation is valid for a steady-state vortex (\(div\,\mathbf{\overrightarrow{v}}=0\)) of the incompressible or barotropic flow: \[\boxed{\Delta\mathbf{\overrightarrow{v}}+k^2\mathbf{\overrightarrow{v}}=0}\tag{5}\] This elliptic equation has many solutions, which describe something like a wave with wavenumber \(\approx k\), which propagates with arbitrary velocity including zero. Such wave is called the soliton.
For a helical flow along one axis, Gromeka had found a simple solution in the cylindrical coordinates \((r,\theta,z) \)[1]:
\[v_r=0\]
\[v_\theta=CJ_1(kr)\]
\[v_z=CJ_0(kr)\]
where \(C\) is a constant, which depends on the fluid discharge;
\(J_0, J_1\) is the Bessel functions of 1st kind.
The given example shows that the flow velocity not only decreases with distance from the vortex axis, but it can also change own direction.
The Ranque-Hilsch tube shows the effect of the direction change. The fast cold air is directed to one side from the vortex center, and the slow hot air to the opposite direction from the vortex periphery.
The optimal twisting angle can minimize the flow velocity at the pipe walls, reducing the friction loss. Here are the examples:
- Flow twisting in the aorta reduces the heart workload.
- Helicoid, a helical profile pipe with the minimum flow resistance, which repeats the river bed shape, as the inventor V. Schauberger explains.
See also
References
- ↑ 1.0 1.1 Громека И.С., Некоторые случаи движения несжимаемой жидкости (докторская диссертация), Отд. изд. Казань, 1881, 107 стр.; Громека И.С., Собрание трудов, Москва: АН СССР, 1952, с.76-148.
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