Hamel-Oseenscher vortex

The Hamel-Oseen or Lamb-Oseen vortex (by Carl Wilhelm Oseen , Georg Hamel , Horace Lamb , hereinafter simply Oseen vortex ) is a mathematical model of a vortex flow of a linearly viscous , incompressible fluid . The velocity field of flows of such fluids is described in fluid mechanics with the Navier-Stokes equations , which are exactly fulfilled by the Oseen vortex. The fluid flows in a purely circular but time-dependent, unsteady manner around the vortex center. The viscosity consumes the kinetic energy of the eddy over time, especially in the expanding central region of the eddy, and the flow velocity decreases monotonically over time.

At the beginning of the movement or in the limit of vanishing viscosity, the vortex is a potential vortex . Otherwise, the speed profile of the Oseen vortex is limited and corresponds to a Rankine vortex in the center of the vortex and in the outer area .

Peripheral speed

Circumferential velocities in the Oseen vortex at different times

Circumferential speed of the Oseen vortex in comparison with the rigid rotation and the potential vortex

In the Oseen's vortex, the fluid elements in the vortex plane move in a circle around the vortex center. The two figures on the right give an impression of the speed distribution as a function of the distance from the center. The picture above shows the speed distribution at different times as a function of the radius ( see below). The black dotted curve ("vmax") connects the points with the maximum circumferential speed that mark the core radius . The peripheral speed decreases over time, especially within twice the core radius. Kinetic energy is mainly dissipated within this core region, which expands over time . Outside the core radius, the Oseen vortex changes into the stationary potential vortex of frictionless fluids (black curve in the picture), where no dissipation takes place. With twice the core radius, the speed deviation from the potential vortex has already shrunk to 2%. ${\displaystyle \Gamma _{0}=2\pi \,,\;\nu =1/4\,,}$

A cylindrical coordinate system is used for the mathematical description of the Oseen vortex . The flow then only depends on the radial coordinate r and the time t and has the circumferential speed:

${\displaystyle v_{\varphi }:={\frac {\Gamma _{0}}{2\pi r}}\left(1-e^{-{\frac {r^{2}}{4\nu t}}}\right)={\frac {\Gamma _{0}}{4\pi {\sqrt {\nu t}}}}{\frac {1-e^{-q^{2}}}{q}}\quad {\text{mit}}\quad q:={\frac {r}{r_{0}}}\,,\;r_{0}:=2{\sqrt {\nu t}}\,.}$

The material parameter ν is the kinematic viscosity ( dimension L ² T ⁻¹ , air 14 mm² / s, water 1 mm² / s), a parameter with the same dimension that controls the flow velocity, the denominator is the core radius of the Rankine vortex , which clings to the Oseen's vortex at a given time t, and e ^x denotes the e-function . The velocity distributions of the rigid rotation, the potential vortex - which together result in the Rankine vortex - and the Oseen's vortex are shown in the figure below on the right for the case . ${\displaystyle \Gamma _{0}}$ ${\displaystyle r_{0}}$ ${\displaystyle r_{0}={\frac {\Gamma _{0}}{2\pi }}=1}$

The core radius of a vortex is the radius at which the maximum velocity occurs. In the velocity maximum, the derivative must be at a certain time t

${\displaystyle {\frac {\partial }{\partial q}}{\frac {1-e^{-q^{2}}}{q}}={\frac {2q^{2}e^{-q^{2}}+e^{-q^{2}}-1}{q^{2}}}}$

disappear, which is approximately the case with. The maximum speed ${\displaystyle q=1{,}121}$

${\displaystyle v_{\varphi \max }=0{,}051{\frac {\Gamma _{0}}{\sqrt {\nu t}}}=0{,}120{\frac {\Gamma _{0}}{r_{0}}}}$

occurs in the radius . This is the core radius of the Oseen's vortex. The limit values ${\displaystyle r_{k}=1{,}121r_{0}=2{,}242{\sqrt {\nu t}}}$

${\displaystyle \lim _{q\to 0}{\frac {1-e^{-q^{2}}}{q}}=0\quad {\text{und}}\quad \lim _{q\to 0}{\frac {2q^{2}e^{-q^{2}}+e^{-q^{2}}-1}{q^{2}}}=1}$

exist and therefore increases linearly with the radius at a certain time in the center of the vortex : ${\displaystyle v_{\varphi }}$

${\displaystyle \left.{\frac {\partial v_{\varphi }}{\partial r}}\right|_{r=0}=\left.{\frac {\partial v_{\varphi }}{\partial q}}\right|_{q=0}{\frac {\mathrm {d} q}{\mathrm {d} r}}={\frac {\Gamma _{0}}{4\pi {\sqrt {\nu t}}}}{\frac {1}{r_{0}}}={\frac {\Gamma _{0}}{8\pi \nu t}}\quad \Rightarrow \quad v_{\varphi s}:={\frac {\Gamma _{0}}{8\pi \nu t}}r={\frac {\Gamma _{0}}{2\pi r_{0}^{2}}}r}$

The speed distribution corresponds to a rigid rotation. At a greater distance from the center ( ), the peripheral speed is roughly that of the potential vortex: ${\displaystyle v_{\varphi s}}$ ${\displaystyle r\gg r_{0}}$

${\displaystyle v_{\varphi p}:={\frac {\Gamma _{0}}{2\pi r}}\,.}$

At this point the circumferential speeds of the rigid rotation and the potential vortex are the same and this point is - as mentioned above - the core radius of the Rankine vortex. Taking the units into account, a core radius and a circulation result in a maximum circumferential speed , so that the fluid elements circle the center once per second. At a distance of 50 centimeters, the circumferential speed would have already decreased to, so that fluid elements at this distance only circle the center once every seven seconds. ${\displaystyle r=r_{0}}$ ${\displaystyle r_{k}=0{,}16\mathrm {m} }$ ${\displaystyle \Gamma _{0}=1{,}4{\frac {\mathrm {m} ^{2}}{\mathrm {s} }}}$ ${\displaystyle v_{\varphi \max }=1{\frac {\mathrm {m} }{\mathrm {s} }}}$ ${\displaystyle v_{\varphi }\approx v_{\varphi p}(0{,}5\mathrm {m} )=0{,}46{\frac {\mathrm {m} }{\mathrm {s} }}}$

Vortex strength

Vortex strength across the radius at different times; the vortex strength of the Rankine vortex belonging to time t = 1 is shown in black.

The vortex strength in a planar flow is twice the angular velocity of the fluid elements around themselves. In a planar flow, the vortex strength has only one component perpendicular to the plane and thus it can be treated as a scalar field. In the Oseen vortex, the vortex strength is:

${\displaystyle \omega ={\frac {\Gamma _{0}}{4\pi \nu t}}e^{-{\frac {r^{2}}{4\nu t}}}={\frac {\Gamma _{0}}{\pi r_{0}^{2}}}e^{-{\frac {r^{2}}{r_{0}^{2}}}}\,.}$

This results from the speed and its rotation in cylinder coordinates: ${\displaystyle {\vec {v}}=v_{\varphi }{\hat {e}}_{\varphi }}$

${\displaystyle {\begin{aligned}\operatorname {rot} (v_{\varphi }(r,t){\hat {e}}_{\varphi })=&{\frac {1}{r}}{\frac {\partial }{\partial r}}\left(r\,v_{\varphi }\right){\hat {e}}_{z}=\left({\frac {v_{\varphi }}{r}}+{\frac {\partial v_{\varphi }}{\partial r}}\right){\hat {e}}_{z}=:\omega {\hat {e}}_{z}\\\rightarrow \omega =&{\frac {\Gamma _{0}}{2\pi r^{2}}}\left(1-e^{-{\frac {r^{2}}{4\nu t}}}\right)-{\frac {\Gamma _{0}}{2\pi r^{2}}}\left(1-e^{-{\frac {r^{2}}{4\nu t}}}\right)+{\frac {\Gamma _{0}}{2\pi r}}{\frac {2r}{4\nu t}}e^{-{\frac {r^{2}}{4\nu t}}}={\frac {\Gamma _{0}}{4\pi \nu t}}e^{-{\frac {r^{2}}{4\nu t}}}\,.\end{aligned}}}$

For the vortex strength changes into the Dirac delta , which fits into a potential vortex. The derivative of the vortex strength according to the radius is calculated as follows: ${\displaystyle \nu \,,\;t\rightarrow 0}$

${\displaystyle {\frac {\partial \omega }{\partial r}}=-{\frac {\Gamma _{0}r}{8\pi \nu ^{2}t^{2}}}e^{-{\frac {r^{2}}{4\nu t}}}\,.}$

At this derivation disappears and the vortex strength is equal to twice the rotational speed in the center. In the center there is a rigid rotation. For the vortex strength approaches zero, which is why the potential and Rankine vortices cling to the Oseen vortex here as well. ${\displaystyle r=0}$ ${\displaystyle \partial v_{\varphi }/\partial r|_{r=0}}$ ${\displaystyle r\to \infty }$

pressure

Pressure distribution in the Oseen's vortex at ${\displaystyle p_{\infty }=0}$

The pressure gradient in a circularly flowing vortex just balances out the centrifugal force so that the fluid elements flow in a circle, which results in cylinder coordinates from the Navier-Stokes equations (see below) and in the Oseen vortex on the condition

${\displaystyle {\frac {\partial p}{\partial r}}=\rho {\frac {v_{\varphi }^{2}}{r}}={\frac {\rho \Gamma _{0}^{2}}{4\pi ^{2}}}{\frac {1}{r^{3}}}\left(1-e^{-{\frac {r^{2}}{4\nu t}}}\right)^{2}}$

runs out. Using the integral exponential function Ei with the properties

${\displaystyle \mathrm {Ei} (x):=\int _{-\infty }^{x}{\frac {e^{s}}{s}}\,\mathrm {d} s\quad \Rightarrow \quad \mathrm {d} \mathrm {Ei} (x)={\frac {e^{x}}{x}}\,\mathrm {d} x\quad \Leftrightarrow \quad {\frac {\mathrm {d} \mathrm {Ei} (x)}{\mathrm {d} x}}={\frac {e^{x}}{x}}}$

the above derivation can be integrated closed with the result:

${\displaystyle p=p_{z}\left[-{\frac {4\nu t}{r^{2}}}\left(1-e^{-{\frac {r^{2}}{4\nu t}}}\right)^{2}+2\mathrm {Ei} \left(-{\frac {r^{2}}{4\nu t}}\right)-2\mathrm {Ei} \left(-2{\frac {r^{2}}{4\nu t}}\right)\right]+p_{\infty }\quad {\text{mit}}\quad p_{z}:={\frac {\rho \Gamma _{0}^{2}}{32\pi ^{2}\nu t}}}$

The constant of integration is the pressure in the (infinitely) distant outer area. The pressure is in the center ${\displaystyle p_{\infty }}$

${\displaystyle p(r=0)=p_{\infty }-2\ln(2)p_{z}\,.}$

Proof:

With the abbreviations and ${\displaystyle r_{0}^{2}:=4\nu t}$ ${\displaystyle q:={\frac {r^{2}}{4\nu t}}={\frac {r^{2}}{r_{0}^{2}}}\rightarrow q'={\frac {2r}{r_{0}^{2}}}\rightarrow {\frac {q'}{2q^{2}}}={\frac {r_{0}^{2}}{r^{3}}}}$ the pressure is written as It is calculated For small arguments , the value of the integral exponential function with its series expansion results in where the Landau symbol is for values ​​that do not grow significantly faster than x and that compared to a constant, e.g. B. the Euler-Mascheroni constant can be neglected. This is how it is calculated: ${\displaystyle {\frac {p}{2p_{z}}}=-{\frac {1}{2q}}\left(1-e^{-q}\right)^{2}+\mathrm {Ei} \left(-q\right)-\mathrm {Ei} \left(-2q\right)+{\frac {p_{\infty }}{2p_{z}}}\,.}$ ${\displaystyle {\begin{aligned}\\{\frac {1}{2p_{z}}}{\frac {\partial p}{\partial r}}=&{\frac {q'}{2q^{2}}}\left(1-e^{-q}\right)^{2}\underbrace {-{\frac {1}{q}}\left(1-e^{-q}\right)q'e^{-q}+{\frac {e^{-q}}{-q}}\cdot (-q')-{\frac {e^{-2q}}{-2q}}\cdot (-2q')} _{=0}={\frac {r_{0}^{2}}{r^{3}}}\left(1-e^{-{\frac {r^{2}}{r_{0}^{2}}}}\right)^{2}\\\rightarrow {\frac {\partial p}{\partial r}}=&2p_{z}{\frac {r_{0}^{2}}{r^{3}}}\left(1-e^{-{\frac {r^{2}}{r_{0}^{2}}}}\right)^{2}={\frac {\rho \Gamma _{0}^{2}}{4\pi ^{2}r^{3}}}\left(1-e^{-{\frac {r^{2}}{4\nu t}}}\right)^{2}\,.\end{aligned}}}$ ${\displaystyle x\ll 1}$ ${\displaystyle \mathrm {Ei} (x)=\gamma +\ln |x|+\sum _{k=1}^{\infty }{\frac {x^{k}}{k!\cdot k}}=\gamma +\ln |x|+{\mathcal {O}}(x)\,,}$ ${\displaystyle {\mathcal {O}}(x)}$ ${\displaystyle x\ll 1}$ ${\displaystyle \gamma \approx 0{,}5772}$ ${\displaystyle {\begin{aligned}\lim _{q\to 0}[\mathrm {Ei} (-q)-\mathrm {Ei} (-2q)]=&\lim _{q\to 0}[\gamma +\ln |q|-\gamma -\ln |2q|+{\mathcal {O}}(q)]\\=&\lim _{q\to 0}[\ln |q|-\ln 2-\ln |q|]=-\ln 2\,.\end{aligned}}}$ The pressure in the center follows with the limit value . ${\displaystyle \lim _{q\to 0}(1-e^{-q})^{2}/q=0}$ ${\displaystyle p(r=0)=p_{\infty }-2\ln(2)p_{z}\,.}$

The picture shows the pressure distribution with a vanishing external pressure. The factor is the pressure in the potential vortex that clings to the Oseen's vortex (blue curve): ${\displaystyle -{\frac {4\nu t}{r^{2}}}p_{z}}$

${\displaystyle v_{\varphi p}={\frac {\Gamma }{2\pi r}}\quad \rightarrow \quad {\frac {\partial p_{p}}{\partial r}}=\rho {\frac {v_{\varphi p}^{2}}{r}}={\frac {\rho \Gamma ^{2}}{4\pi ^{2}r^{3}}}\quad \rightarrow \quad p_{p}-p_{\infty }=-{\frac {\rho \Gamma ^{2}}{8\pi ^{2}r^{2}}}=-{\frac {4\nu t}{r^{2}}}p_{z}=-p_{z}{\frac {r_{0}^{2}}{r^{2}}}\,.}$

Again, the pressure in the center indicates a rigid rotation, because with this the peripheral speed is proportional to the radius ${\displaystyle v_{\varphi s}}$

${\displaystyle v_{\varphi s}={\frac {\Gamma _{0}}{2\pi r_{0}^{2}}}r\quad \rightarrow \quad {\frac {\partial p_{s}}{\partial r}}=\rho {\frac {v_{\varphi s}^{2}}{r}}={\frac {\rho \Gamma _{0}^{2}}{4\pi ^{2}r_{0}^{4}}}r\quad \rightarrow \quad p_{s}-p_{0}={\frac {\rho \Gamma _{0}^{2}}{8\pi ^{2}r_{0}^{2}}}{\frac {r^{2}}{r_{0}^{2}}}=p_{z}{\frac {r^{2}}{r_{0}^{2}}}\,,}$

which is why the pressure curve is then parabolic over the radius (red curve in the picture).

Kinetic energy

Kinetic energy of the fluid as a function of the multiplicity of the core radius

The kinetic energy of the fluid within a multiple of the core radius of the Rankine vortex is dependent neither on the core radius nor on the time as long as the multiplicity is maintained:

${\displaystyle E_{\text{k}}(nr_{0}):=\int _{0}^{nr_{0}}{\frac {\rho }{2}}v_{\varphi }^{2}\,2\pi r\mathrm {d} r={\frac {\rho \Gamma _{0}^{2}}{8\pi }}\left[\ln \left({\frac {n^{2}}{2}}\right)-2\mathrm {Ei} (-n^{2})+\mathrm {Ei} (-2n^{2})+\gamma \right]\,.}$

The value in the square brackets is only a function of the factor n with the Euler-Mascheroni constant , see the figure on the right. The kinetic energy of the fluid within the expanding radius is therefore constant over time if the ratio n is fixed. Conversely, this means: The kinetic energy of the fluid elements newly occupied in a time interval by a circle with n-times the core radius is dissipated within the circle in this time interval. ${\displaystyle \gamma \approx 0{,}5772}$ ${\displaystyle nr_{0}=2n{\sqrt {\nu t}}}$

Proof:

With the peripheral speed ${\displaystyle v_{\varphi }={\frac {\Gamma _{0}}{2\pi r}}\left(1-e^{-{\frac {r^{2}}{4\nu t}}}\right)}$ the kinetic energy of the fluid results within a multiple of the Rankine core radius because the function is actually the antiderivative you are looking for : At the lower limit, the value of the integral exponential function with its series expansion results for the required, small, quadratic arguments where the Landau symbol is for values ​​that grow at not significantly faster than x and that are compared to a constant, e.g. B. the Euler-Mascheroni constant can be neglected. This results in: The value of the antiderivative at is due to ${\displaystyle E_{\text{k}}(nr_{0})=\int _{0}^{nr_{0}}{\frac {\rho }{2}}v_{\varphi }^{2}\,2\pi r\mathrm {d} r={\frac {\rho \Gamma _{0}^{2}}{4\pi }}\int _{0}^{nr_{0}}{\frac {1}{r}}\left(1-e^{-{\frac {r^{2}}{r_{0}^{2}}}}\right)^{2}\,\mathrm {d} r={\frac {\rho \Gamma _{0}^{2}}{4\pi }}\left[\ln \left({\frac {r}{r_{0}}}\right)-\mathrm {Ei} \left(-{\frac {r^{2}}{r_{0}^{2}}}\right)+{\frac {1}{2}}\mathrm {Ei} \left(-2{\frac {r^{2}}{r_{0}^{2}}}\right)\right]_{0}^{nr_{0}}}$ ${\displaystyle f(r):=\ln \left({\frac {r}{r_{0}}}\right)-\mathrm {Ei} \left(-{\frac {r^{2}}{r_{0}^{2}}}\right)+{\frac {1}{2}}\mathrm {Ei} \left(-2{\frac {r^{2}}{r_{0}^{2}}}\right)}$ ${\displaystyle {\frac {\mathrm {d} }{\mathrm {d} r}}f(r)={\frac {r_{0}}{r}}{\frac {1}{r_{0}}}-{\frac {e^{-{\frac {r^{2}}{r_{0}^{2}}}}}{-{\frac {r^{2}}{r_{0}^{2}}}}}\cdot \left(-2{\frac {r}{r_{0}^{2}}}\right)+{\frac {1}{2}}{\frac {e^{-2{\frac {r^{2}}{r_{0}^{2}}}}}{-2{\frac {r^{2}}{r_{0}^{2}}}}}\cdot \left(-4{\frac {r}{r_{0}^{2}}}\right)={\frac {1}{r}}\left(1-e^{-{\frac {r^{2}}{r_{0}^{2}}}}\right)^{2}\,.}$ ${\displaystyle \mathrm {Ei} (x)=\gamma +\ln |x|+\sum _{k=1}^{\infty }{\frac {x^{k}}{k!\cdot k}}=\gamma +\ln |x|+{\mathcal {O}}(x)\,,}$ ${\displaystyle {\mathcal {O}}(x)}$ ${\displaystyle x\ll 1}$ ${\displaystyle \gamma \approx 0{,}5772}$ ${\displaystyle {\begin{aligned}\lim _{r\to 0}f(r)=&\lim _{r\to 0}\left[\ln \left({\frac {r}{r_{0}}}\right)-\mathrm {Ei} \left(-{\frac {r^{2}}{r_{0}^{2}}}\right)+{\frac {1}{2}}\mathrm {Ei} \left(-2{\frac {r^{2}}{r_{0}^{2}}}\right)\right]\\=&\lim _{r\to 0}\left[\ln \left({\frac {r}{r_{0}}}\right)-\gamma -\ln \left|{\frac {r^{2}}{r_{0}^{2}}}\right|+{\frac {1}{2}}\gamma +{\frac {1}{2}}\ln \left|2{\frac {r^{2}}{r_{0}^{2}}}\right|+{\mathcal {O}}(r^{2})\right]\\=&{\frac {1}{2}}(\ln 2-\gamma )+\lim _{r\to 0}{\bigg [}\underbrace {\ln \left({\frac {r}{r_{0}}}\right)-2\ln \left|{\frac {r}{r_{0}}}\right|+\ln \left|{\frac {r}{r_{0}}}\right|} _{=0}{\bigg ]}\\\Rightarrow f(0)=&{\frac {1}{2}}(\ln 2-\gamma )\,.\end{aligned}}}$ ${\displaystyle r=nr_{0}}$ ${\displaystyle f(nr_{0})=\ln(n)-\mathrm {Ei} (-n^{2})+{\frac {1}{2}}\mathrm {Ei} (-2n^{2})}$ only a function of the factor n. With these results the kinetic energy is calculated - as announced - to ${\displaystyle E_{\text{k}}={\frac {\rho \Gamma _{0}^{2}}{4\pi }}[f(r)]_{0}^{nr_{0}}={\frac {\rho \Gamma _{0}^{2}}{4\pi }}[f(nr_{0})-f(0)]={\frac {\rho \Gamma _{0}^{2}}{8\pi }}\left[\ln \left({\frac {n^{2}}{2}}\right)-2\mathrm {Ei} (-n^{2})+\mathrm {Ei} (-2n^{2})+\gamma \right]\,.}$

circulation

Circulation in the Oseen's vortex

A measure of the rotational speed in a fluid is the circulation , which is the curve integral of the speed along a closed path. Along a circle K with radius r the following is calculated:

${\displaystyle \Gamma :=\oint _{K}{\vec {v}}\cdot \mathrm {d} {\vec {r}}=\int _{0}^{2\pi }v_{\varphi }\,r\mathrm {d} \varphi =2\pi rv_{\varphi }=\Gamma _{0}\left(1-e^{-{\frac {r^{2}}{4\nu t}}}\right)\,.}$

The course of the function is shown in the picture on the right ( ) Far from the vortex center ( ), the circulation approaches the parameter Γ ₀ , which is the constant circulation of the potential vortex over the radius (blue line), which is located on the outside of the Oseen vortex hugs. At a distance of twice the core radius, the circulation only deviates from the parameter by 2% . The time dependency of the circulation contradicts Kelvin's vortex law for frictionless fluids and this contradiction also resolves . ${\displaystyle r_{0}=2{\sqrt {\nu t}}\,.}$ ${\displaystyle r/r_{0}\to \infty }$ ${\displaystyle \Gamma _{0}}$ ${\displaystyle \nu \to 0}$

In the center the speed is proportional to the radius and then the circulation is:

${\displaystyle v_{\varphi s}={\frac {\Gamma _{0}}{8\pi \nu t}}r\quad \rightarrow \quad \Gamma _{s}=\int _{0}^{2\pi }v_{\varphi s}\,r\mathrm {d} \varphi =2\pi {\frac {\Gamma _{0}}{8\pi \nu t}}r^{2}=\Gamma _{0}{\frac {r^{2}}{r_{0}^{2}}}\,.}$

It is drawn in red in the picture. At time t = 0, the vortex starts with the circulation Γ ₀ , which tends to zero at a given interval as time progresses because the viscosity - especially in the vortex core - consumes the kinetic energy and the core radius r ₀ expands over time. If the ratio is fixed , the circulation is constant over time, or - in other words - if the circulation is fixed, the circles expand over time like the core radius. ${\displaystyle n:=r/r_{0}}$

Shear distortion speed

Circumferential and distortion speed in the Oseen's vortex

The shear strain velocity in the fluid results from the strain velocity tensor d , which is the symmetrical component of the velocity gradient . In the cylindrical coordinates used here, the gradient is calculated as: ${\displaystyle {\dot {\gamma }}_{r\varphi }=2{\hat {e}}_{r}\cdot \mathbf {d} \cdot {\hat {e}}_{\varphi }}$ ${\displaystyle \operatorname {grad} {\vec {v}}}$

${\displaystyle {\begin{aligned}\operatorname {grad} {\vec {v}}=&{\hat {e}}_{\varphi }\otimes {\frac {\partial v_{\varphi }}{\partial r}}{\hat {e}}_{r}-{\frac {v_{\varphi }}{r}}{\hat {e}}_{r}\otimes {\hat {e}}_{\varphi }\\=&{\hat {e}}_{\varphi }\otimes \left[-{\frac {\Gamma _{0}}{2\pi r^{2}}}\left(1-e^{-{\frac {r^{2}}{4\nu t}}}\right)-{\frac {\Gamma _{0}}{2\pi r}}e^{-{\frac {r^{2}}{4\nu t}}}\cdot \left(-{\frac {2r}{4\nu t}}\right)\right]{\hat {e}}_{r}-{\frac {\Gamma _{0}}{2\pi r^{2}}}\left(1-e^{-{\frac {r^{2}}{4\nu t}}}\right){\hat {e}}_{r}\otimes {\hat {e}}_{\varphi }\\\rightarrow \mathbf {d} :=&{\frac {1}{2}}[\operatorname {grad} ({\vec {v}})+\operatorname {grad} ({\vec {v}})^{\top }]=\left[{\frac {\Gamma _{0}}{8\pi \nu t}}\left(1+{\frac {4\nu t}{r^{2}}}\right)e^{-{\frac {r^{2}}{4\nu t}}}-{\frac {\Gamma _{0}}{2\pi r^{2}}}\right]({\hat {e}}_{r}\otimes {\hat {e}}_{\varphi }+{\hat {e}}_{\varphi }\otimes {\hat {e}}_{r})\\\rightarrow {\dot {\gamma }}_{r\varphi }=&2{\hat {e}}_{r}\cdot \mathbf {d} \cdot {\hat {e}}_{\varphi }={\frac {\Gamma _{0}}{4\pi \nu t}}\left(1+{\frac {4\nu t}{r^{2}}}\right)e^{-{\frac {r^{2}}{4\nu t}}}{\underline {-{\frac {\Gamma _{0}}{\pi r^{2}}}}}={\frac {\Gamma _{0}}{4\pi \nu t}}\left[\left(1+{\frac {r_{0}^{2}}{r^{2}}}\right)e^{-{\frac {r^{2}}{r_{0}^{2}}}}-{\frac {r_{0}^{2}}{r^{2}}}\right]\,.\end{aligned}}}$

The superscript marks the transposition and the arithmetic symbol " " forms the dyadic product . In the case of rigid rotation, there is no shear distortion and the term underlined in the above formula is the shear distortion speed in the potential vortex, see picture. ${\displaystyle \top }$ ${\displaystyle \otimes }$ ${\displaystyle ({\dot {\gamma }}_{r\varphi s}=0)}$ ${\displaystyle {\dot {\gamma }}_{r\varphi p}}$

The maximum shear strain rate occurs where its slope is zero:

${\displaystyle {\frac {\mathrm {d} }{\mathrm {d} r}}{\dot {\gamma }}_{r\varphi }{\stackrel {\displaystyle !}{=}}0\quad \rightarrow \quad q(q+1)e^{-q}+e^{-q}-1=0\quad {\text{mit}}\quad q:={\frac {r^{2}}{r_{0}^{2}}}\,.}$

This is approximately the case with. The maximum shear distortion speed is thus shown at about 1.2 times the core radius ${\displaystyle r_{m}:=1{,}339\,r_{0}=2{,}678{\sqrt {\nu t}}}$ ${\displaystyle r_{k}=1{,}121\,r_{0}\,.}$

comment

The skew symmetrical part of the velocity gradient is the vortex tensor.

${\displaystyle \mathbf {w} :={\frac {1}{2}}[\operatorname {grad} ({\vec {v}})-\operatorname {grad} ({\vec {v}})^{\top }]=-{\frac {\Gamma _{0}}{8\pi \nu t}}e^{-{\frac {r^{2}}{4\nu t}}}({\hat {e}}_{r}\otimes {\hat {e}}_{\varphi }-{\hat {e}}_{\varphi }\otimes {\hat {e}}_{r})\,,}$

whose dual vector - defined by - is the angular velocity or half the vortex strength: ${\displaystyle {\vec {\Omega }}}$ ${\displaystyle {\vec {\Omega }}\times {\vec {u}}=\mathbf {w} \cdot {\vec {u}}\quad \forall {\vec {u}}}$

${\displaystyle {\vec {\Omega }}=-{\frac {1}{2}}\left[-{\frac {\Gamma _{0}}{8\pi \nu t}}e^{-{\frac {r^{2}}{4\nu t}}}({\hat {e}}_{r}\times {\hat {e}}_{\varphi }-{\hat {e}}_{\varphi }\times {\hat {e}}_{r})\right]={\frac {\Gamma _{0}}{8\pi \nu t}}e^{-{\frac {r^{2}}{4\nu t}}}{\hat {e}}_{z}={\frac {1}{2}}{\vec {\omega }}\,.}$

Time courses

Velocity, vortex strength, pressure and shear distortion velocity of a fluid element over time

In the previous sections, the course of the variables at a certain time was examined as a function of the radius. In this section, the course of time at a certain radius is to be examined.

The core radius expands over time. Be

${\displaystyle t_{r}:={\frac {r^{2}}{4\nu }}}$

the core time that elapses until the Rankine core radius has increased to a predetermined size r. The core time increases with the square of the radius.

The velocity of a fluid element at a certain distance r from the center is

${\displaystyle v_{\varphi }={\frac {\Gamma _{0}}{2\pi r}}\left(1-e^{-{\frac {r^{2}}{4\nu t}}}\right)=v_{\varphi p}\left(1-e^{-{\frac {t_{r}}{t}}}\right)\quad \rightarrow \quad {\frac {v_{\varphi }}{v_{\varphi p}}}=1-e^{-{\frac {t_{r}}{t}}}\,.}$

Until the circumferential speed deviates by a maximum of 2% from that of the potential vortex. Then the speed decreases rapidly, see the red curve in the picture. The ratio of the circulation to has the same time course for a given radius. ${\displaystyle t=t_{r}/4}$ ${\displaystyle \Gamma _{0}}$

With a fixed radius r, the vortex strength first increases and then decreases again and passes through a maximum in between. Initially the rotation is less because the fluid element moves like in a rotation-free potential vortex, then it increases due to friction effects and later, when the fluid element is within the core radius, the vortex strength decreases again due to the consumption of kinetic energy. At the maximum the time derivative of the vortex strength vanishes : ${\displaystyle {\dot {\omega }}}$

${\displaystyle {\dot {\omega }}=-{\frac {\Gamma _{0}}{4\pi \nu t^{2}}}e^{-{\frac {r^{2}}{4\nu t}}}+{\frac {r^{2}}{4\nu t^{2}}}{\frac {\Gamma _{0}}{4\pi \nu t}}e^{-{\frac {r^{2}}{4\nu t}}}={\frac {\Gamma _{0}}{4\pi \nu t^{2}}}e^{-{\frac {r^{2}}{4\nu t}}}\left({\frac {r^{2}}{4\nu t}}-1\right)=0\quad \rightarrow \quad {\frac {r^{2}}{4\nu t}}={\frac {t_{r}}{t}}=1\,.}$

The vortex strength can be expressed in terms of the core time and its maximum can be represented in this way:

${\displaystyle \omega ={\frac {\Gamma _{0}}{4\pi \nu t}}e^{-{\frac {r^{2}}{4\nu t}}}={\frac {\Gamma _{0}}{4\pi \nu t_{r}}}{\frac {t_{r}}{t}}e^{-{\frac {t_{r}}{t}}}={\frac {\Gamma _{0}}{\pi r^{2}}}{\frac {t_{r}}{t}}e^{-{\frac {t_{r}}{t}}}\quad \rightarrow \quad \omega _{\max }={\frac {\Gamma _{0}}{\pi er^{2}}}}$

The ratio of the vortex strength to its maximum at a given radius r (green curve in the picture) is therefore:

${\displaystyle {\frac {\omega }{\omega _{\max }}}={\frac {\Gamma _{0}}{\pi r^{2}}}{\frac {\pi er^{2}}{\Gamma _{0}}}{\frac {t_{r}}{t}}e^{-{\frac {t_{r}}{t}}}={\frac {t_{r}}{t}}e^{1-{\frac {t_{r}}{t}}}\,.}$

When the fluid element lies on the Rankine core radius, it rotates most rapidly around itself. ${\displaystyle t=t_{r}\,,}$

The pressure-time curve (blue curve in the picture) results from

${\displaystyle {\frac {p_{\infty }-p}{p_{r}}}=\left(1-e^{-{\frac {t_{r}}{t}}}\right)^{2}-2{\frac {t_{r}}{t}}\mathrm {Ei} \left(-{\frac {t_{r}}{t}}\right)+2{\frac {t_{r}}{t}}\mathrm {Ei} \left(-2{\frac {t_{r}}{t}}\right)\quad {\text{mit}}\quad p_{r}={\frac {\rho \Gamma _{0}^{2}}{8\pi ^{2}r^{2}}}={\frac {4\nu t}{r^{2}}}p_{z}={\frac {t}{t_{r}}}p_{z}}$

For follows from the series expansion of the integral exponential function ${\displaystyle t\to \infty }$

${\displaystyle {\begin{aligned}\mathrm {Ei} (x)=&\gamma +\ln |x|+\sum _{k=1}^{\infty }{\frac {x^{k}}{k!\cdot k}}\\\Rightarrow \lim _{x\to 0}[\mathrm {Ei} (2x)-\mathrm {Ei} (x)]=&\lim _{x\to 0}\left[\gamma +\ln |2x|+\sum _{k=1}^{\infty }{\frac {(2x)^{k}}{k!\cdot k}}-\gamma -\ln |x|-\sum _{k=1}^{\infty }{\frac {x^{k}}{k!\cdot k}}\right]\\=&\lim _{x\to 0}[\ln 2+\ln |x|-\ln |x|]=\ln 2\,.\end{aligned}}}$

Therefore, the pressure difference tends to zero over time. ${\displaystyle p_{\infty }-p}$

The speed of shear distortion over time (orange curve in the picture) results in:

${\displaystyle {\dot {\gamma }}_{r\varphi }={\frac {\Gamma _{0}}{\pi r^{2}}}\left[{\frac {t_{r}}{t}}\left(1+{\frac {t}{t_{r}}}\right)e^{-{\frac {t_{r}}{t}}}-1\right]\quad \rightarrow \quad -{\frac {{\dot {\gamma }}_{r\varphi }}{{\dot {\gamma }}_{r\varphi p}}}=1-{\frac {t_{r}}{t}}\left(1+{\frac {t}{t_{r}}}\right)e^{-{\frac {t_{r}}{t}}}\,.}$

Navier-Stokes equations

The fact that the model equations of the Oseen vortex satisfy the Navier-Stokes equations can be demonstrated in cylinder coordinates using the equations for a density-stable fluid without a gravitational field. Under these circumstances the Navier-Stokes equations are, if all variables depend only on radius or time and the movement is purely circular ( ): ${\displaystyle {\vec {v}}=v_{\varphi }(r,t){\hat {e}}_{\varphi }}$

${\displaystyle {\begin{aligned}-{\frac {v_{\varphi }^{2}}{r}}=&-{\frac {1}{\rho }}{\frac {\partial p}{\partial r}}\\{\frac {\partial v_{\varphi }}{\partial t}}=&\nu \left(\Delta v_{\varphi }-{\frac {v_{\varphi }}{r^{2}}}\right)={\frac {\nu }{r}}\,{\frac {\partial v_{\varphi }}{\partial r}}+\nu \,{\frac {\partial ^{2}v_{\varphi }}{\partial r^{2}}}-\nu {\frac {v_{\varphi }}{r^{2}}}\end{aligned}}}$

The pressure was calculated from the first equation above. The second equation is with the specified velocity field

${\displaystyle v_{\varphi }={\frac {\Gamma _{0}}{2\pi r}}\left(1-e^{-{\frac {r^{2}}{4\nu t}}}\right)}$

met what with

${\displaystyle {\begin{aligned}{\frac {\partial v_{\varphi }}{\partial t}}=&-{\frac {\Gamma _{0}r}{8\pi \nu t^{2}}}e^{-{\frac {r^{2}}{4\nu t}}}\\{\frac {\partial v_{\varphi }}{\partial r}}=&-{\frac {\Gamma _{0}}{2\pi r^{2}}}\left(1-e^{-{\frac {r^{2}}{4\nu t}}}\right)+{\frac {\Gamma _{0}}{4\pi \nu t}}e^{-{\frac {r^{2}}{4\nu t}}}\\{\frac {\partial ^{2}v_{\varphi }}{\partial r^{2}}}=&{\frac {\Gamma _{0}}{\pi r^{3}}}\left(1-e^{-{\frac {r^{2}}{4\nu t}}}\right)-{\frac {\Gamma _{0}}{4\pi \nu rt}}e^{-{\frac {r^{2}}{4\nu t}}}-{\frac {\Gamma _{0}r}{8\pi \nu ^{2}t^{2}}}e^{-{\frac {r^{2}}{4\nu t}}}\end{aligned}}}$

can be proven.

In cylindrical coordinates results from

${\displaystyle \operatorname {div} ({\vec {v}})=\operatorname {div} (v_{\varphi }{\hat {e}}_{\varphi })={\frac {1}{r}}{\frac {\partial }{\partial r}}(rv_{r})+{\frac {1}{r}}{\frac {\partial v_{\varphi }}{\partial \varphi }}+{\frac {\partial v_{z}}{\partial z}}=0}$

the freedom from divergence of the eddy flow, which, via the mass balance, results in a density that is constant over time, which in turn is consistent with the incompressibility of the fluid. ${\displaystyle {\dot {\rho }}+\rho \operatorname {div} {\vec {v}}={\dot {\rho }}=0}$

See also

• Strain tensor

Footnotes

1. ↑ In cylindrical coordinates the gradient of a vector field is calculated according to and the gradient of a scalar field is calculated with it
   ${\displaystyle \operatorname {grad} ({\vec {f}})={\hat {e}}_{r}\otimes \operatorname {grad} (f_{r})+{\frac {f_{r}}{r}}{\hat {e}}_{\varphi }\otimes {\hat {e}}_{\varphi }+{\hat {e}}_{\varphi }\otimes \operatorname {grad} (f_{\varphi })-{\frac {f_{\varphi }}{r}}{\hat {e}}_{r}\otimes {\hat {e}}_{\varphi }+{\hat {e}}_{z}\otimes \operatorname {grad} (f_{z})}$
   
   ${\displaystyle \operatorname {grad} (f)={\frac {\partial f}{\partial r}}{\hat {e}}_{r}+{\frac {1}{r}}{\frac {\partial f}{\partial \varphi }}{\hat {e}}_{\varphi }+{\frac {\partial f}{\partial z}}{\hat {e}}_{z}}$

Individual evidence

1. ↑ Bestehorn, 2006, p. 87
2. ^ M. Bestehorn: hydrodynamics and structure formation . Springer, 2006, ISBN 978-3-540-33796-6 , pp. 380 .

literature

• F. Kameier, CO Paschereit: Fluid Mechanics . Walter de Gruyter, 2013, ISBN 978-3-11-018972-8 , p. 274 ff .

Web links

• Thomas Fischer: Oseenscher vortex. University of Stuttgart, accessed on September 17, 2015 .