Oseen looked for a solution for the Navier–Stokes equations in cylindrical coordinates with velocity components of the form
where is the circulation of the vortex core. Navier-Stokes equations lead to
which, subject to the conditions that it is regular at and becomes unity as , leads to[3]
where is the kinematic viscosity of the fluid. At , we have a potential vortex with concentrated vorticity at the axis; and this vorticity diffuses away as time passes.
The only non-zero vorticity component is in the direction, given by
The generalized Oseen vortex may be obtained by looking for solutions of the form
that leads to the equation
Self-similar solution exists for the coordinate , provided , where is a constant, in which case . The solution for may be written according to Rott (1958)[5] as
where is an arbitrary constant. For , the classical Lamb–Oseen vortex is recovered. The case corresponds to the axisymmetric stagnation point flow, where is a constant. When , , a Burgers vortex is a obtained. For arbitrary , the solution becomes , where is an arbitrary constant. As , Burgers vortex is recovered.
^Oseen, C. W. (1912). Uber die Wirbelbewegung in einer reibenden Flussigkeit. Ark. Mat. Astro. Fys., 7, 14–26.
^Saffman, P. G.; Ablowitz, Mark J.; J. Hinch, E.; Ockendon, J. R.; Olver, Peter J. (1992). Vortex dynamics. Cambridge: Cambridge University Press. ISBN0-521-47739-5. p. 253.
^Drazin, P. G., & Riley, N. (2006). The Navier–Stokes equations: a classification of flows and exact solutions (No. 334). Cambridge University Press.
^G.K. Batchelor (1967). An Introduction to Fluid Dynamics. Cambridge University Press.
^Rott, N. (1958). On the viscous core of a line vortex. Zeitschrift für angewandte Mathematik und Physik ZAMP, 9(5-6), 543–553.
April 10, 2024
lamb, oseen, vortex, fluid, dynamics, models, line, vortex, that, decays, viscosity, this, vortex, named, after, horace, lamb, carl, wilhelm, oseen, vector, plot, velocity, field, evolution, real, time, free, floating, test, particles, reveal, velocity, vortic. In fluid dynamics the Lamb Oseen vortex models a line vortex that decays due to viscosity This vortex is named after Horace Lamb and Carl Wilhelm Oseen 1 2 Vector plot of the Lamb Oseen vortex velocity field Evolution of a Lamb Oseen vortex in air in real time Free floating test particles reveal the velocity and vorticity pattern scale image is 20 cm wide Contents 1 Mathematical description 2 Generalized Oseen vortex 3 See also 4 ReferencesMathematical description editOseen looked for a solution for the Navier Stokes equations in cylindrical coordinates r 8 z displaystyle r theta z nbsp with velocity components vr v8 vz displaystyle v r v theta v z nbsp of the form vr 0 v8 G2prg r t vz 0 displaystyle v r 0 quad v theta frac Gamma 2 pi r g r t quad v z 0 nbsp where G displaystyle Gamma nbsp is the circulation of the vortex core Navier Stokes equations lead to g t n 2g r2 1r g r displaystyle frac partial g partial t nu left frac partial 2 g partial r 2 frac 1 r frac partial g partial r right nbsp which subject to the conditions that it is regular at r 0 displaystyle r 0 nbsp and becomes unity as r displaystyle r rightarrow infty nbsp leads to 3 g r t 1 e r2 4nt displaystyle g r t 1 mathrm e r 2 4 nu t nbsp where n displaystyle nu nbsp is the kinematic viscosity of the fluid At t 0 displaystyle t 0 nbsp we have a potential vortex with concentrated vorticity at the z displaystyle z nbsp axis and this vorticity diffuses away as time passes The only non zero vorticity component is in the z displaystyle z nbsp direction given by wz r t G4pnte r2 4nt displaystyle omega z r t frac Gamma 4 pi nu t mathrm e r 2 4 nu t nbsp The pressure field simply ensures the vortex rotates in the circumferential direction providing the centripetal force p r rv2r displaystyle partial p over partial r rho v 2 over r nbsp where r is the constant density 4 Generalized Oseen vortex editThe generalized Oseen vortex may be obtained by looking for solutions of the form vr g t r v8 G2prg r t vz 2g t z displaystyle v r gamma t r quad v theta frac Gamma 2 pi r g r t quad v z 2 gamma t z nbsp that leads to the equation g t gr g r n 2g r2 1r g r displaystyle frac partial g partial t gamma r frac partial g partial r nu left frac partial 2 g partial r 2 frac 1 r frac partial g partial r right nbsp Self similar solution exists for the coordinate h r f t displaystyle eta r varphi t nbsp provided ff gf2 a displaystyle varphi varphi gamma varphi 2 a nbsp where a displaystyle a nbsp is a constant in which case g 1 e ah2 2n displaystyle g 1 mathrm e a eta 2 2 nu nbsp The solution for f t displaystyle varphi t nbsp may be written according to Rott 1958 5 as f2 2aexp 2 0tg s ds ctexp 2 0ug s ds du displaystyle varphi 2 2a exp left 2 int 0 t gamma s mathrm d s right int c t exp left 2 int 0 u gamma s mathrm d s right mathrm d u nbsp where c displaystyle c nbsp is an arbitrary constant For g 0 displaystyle gamma 0 nbsp the classical Lamb Oseen vortex is recovered The case g k displaystyle gamma k nbsp corresponds to the axisymmetric stagnation point flow where k displaystyle k nbsp is a constant When c displaystyle c infty nbsp f2 a k displaystyle varphi 2 a k nbsp a Burgers vortex is a obtained For arbitrary c displaystyle c nbsp the solution becomes f2 a 1 be 2kt k displaystyle varphi 2 a 1 beta mathrm e 2kt k nbsp where b displaystyle beta nbsp is an arbitrary constant As t displaystyle t rightarrow infty nbsp Burgers vortex is recovered See also editThe Rankine vortex and Kaufmann Scully vortex are common simplified approximations for a viscous vortex References edit Oseen C W 1912 Uber die Wirbelbewegung in einer reibenden Flussigkeit Ark Mat Astro Fys 7 14 26 Saffman P G Ablowitz Mark J J Hinch E Ockendon J R Olver Peter J 1992 Vortex dynamics Cambridge Cambridge University Press ISBN 0 521 47739 5 p 253 Drazin P G amp Riley N 2006 The Navier Stokes equations a classification of flows and exact solutions No 334 Cambridge University Press G K Batchelor 1967 An Introduction to Fluid Dynamics Cambridge University Press Rott N 1958 On the viscous core of a line vortex Zeitschrift fur angewandte Mathematik und Physik ZAMP 9 5 6 543 553 Retrieved from https en wikipedia org w index php title Lamb Oseen vortex amp oldid 1187180101, wikipedia, wiki, book, books, library,