Lamb–Oseen vortex

Lamb–Oseen vortex. The Lamb–Oseen vortex is an exact, closed-form solution to the Navier–Stokes equations that describes the decay of a two-dimensional, axisymmetric, line vortex in a viscous fluid. It is a fundamental model in vortex dynamics and provides a complete temporal description, from an initial singular vorticity distribution to its eventual viscous dissipation. The solution is named for the physicists Horace Lamb and Carl Wilhelm Oseen, who made seminal contributions to its development and analysis.

Mathematical description

The model describes the azimuthal velocity field \( v_{\theta}(r, t) \) and the vorticity field \( \omega(r, t) \) as functions of radial distance \( r \) and time \( t \). The characteristic velocity profile is given by \( v_{\theta}(r, t) = \frac{\Gamma}{2 \pi r} \left( 1 - e^{-r^2/(4 \nu t)} \right) \), where \( \Gamma \) is the total circulation and \( \nu \) is the kinematic viscosity. The corresponding vorticity distribution is Gaussian: \( \omega(r, t) = \frac{\Gamma}{4 \pi \nu t} e^{-r^2/(4 \nu t)} \). This formulation satisfies the diffusion equation for vorticity, with the Reynolds number implicitly governing the decay process. The solution's self-similar nature is a key feature, with the vortex core radius growing as \( \delta \sim \sqrt{\nu t} \).

Physical interpretation

Physically, the solution represents the viscous spreading of a concentrated vortex filament, such as those shed from aircraft wingtips or observed in draining bathtub vortices. Initially, the vorticity is concentrated at a point, approximating an ideal potential vortex except at the origin. Viscosity then acts to diffuse vorticity outward, smoothing the velocity profile and removing the singular behavior at the core. The model accurately captures the transition from an inviscid-like flow at large distances to a rigid-body rotation near the center. This process is central to understanding vortex stretching in more complex three-dimensional flows and phenomena like wake turbulence.

Derivation from the Navier–Stokes equations

The derivation begins with the incompressible Navier–Stokes equations in cylindrical coordinates for an axisymmetric, swirl-only flow with no radial or axial velocity components. The azimuthal momentum equation reduces to a linear diffusion equation for the angular momentum, \( \frac{\partial}{\partial t}(r v_{\theta}) = \nu \frac{\partial}{\partial r} \left[ \frac{\partial}{\partial r}(r v_{\theta}) - 2 v_{\theta} \right] \). By introducing the circulation \( K = r v_{\theta} \), this equation simplifies further. The Gaussian vorticity solution is obtained using a similarity transform or by employing the Green's function for the two-dimensional diffusion equation, with the initial condition of a Dirac delta function in vorticity representing a point vortex.

Applications and examples

The Lamb–Oseen model is extensively used as a benchmark in computational fluid dynamics for validating numerical schemes for vortex-dominated flows. It provides the theoretical foundation for analyzing the decay of trailing vortices behind aircraft, a critical factor in determining safe wake turbulence separation standards set by organizations like the Federal Aviation Administration. The solution also models the behavior of isolated vortices in geophysical flows, such as oceanic mesoscale eddies or atmospheric tornado cores in simplified form. In industrial contexts, it informs the design of vortex generators and the analysis of mixing in stirred tank reactors.

Relationship to other vortex models

The Lamb–Oseen vortex generalizes the inviscid Rankine vortex, which has a rigidly rotating core and an irrotational outer flow, by including viscous diffusion. It is a special case of the Burgers vortex, which adds an axial strain field to achieve a steady state, and is distinct from the Taylor–Green vortex, an exact periodic solution used in turbulence research. Compared to the Kármán vortex street, which models an array of alternating vortices, the Lamb–Oseen solution describes an isolated structure. Its linear diffusive nature contrasts with nonlinear models like the Lundgren spiral vortex, which incorporates stretching effects.

Category:Fluid dynamics Category:Vortices Category:Equations of fluid dynamics