Vortex

In fluid dynamics, a vortex (plural vortices/vortexes) is a region in a fluid in which the flow revolves around an axis line, which may be straight or curved. Vortices form in stirred fluids, and may be observed in smoke rings, whirlpools in the wake of a boat, and the winds surrounding a tropical cyclone, tornado or dust devil. In fluid dynamics, a vortex (plural vortices/vortexes) is a region in a fluid in which the flow revolves around an axis line, which may be straight or curved. Vortices form in stirred fluids, and may be observed in smoke rings, whirlpools in the wake of a boat, and the winds surrounding a tropical cyclone, tornado or dust devil. Vortices are a major component of turbulent flow. The distribution of velocity, vorticity (the curl of the flow velocity), as well as the concept of circulation are used to characterize vortices. In most vortices, the fluid flow velocity is greatest next to its axis and decreases in inverse proportion to the distance from the axis. In the absence of external forces, viscous friction within the fluid tends to organize the flow into a collection of irrotational vortices, possibly superimposed to larger-scale flows, including larger-scale vortices. Once formed, vortices can move, stretch, twist, and interact in complex ways. A moving vortex carries with it some angular and linear momentum, energy, and mass. A key concept in the dynamics of vortices is the vorticity, a vector that describes the local rotary motion at a point in the fluid, as would be perceived by an observer that moves along with it. Conceptually, the vorticity could be observed by placing a tiny rough ball at the point in question, free to move with the fluid, and observing how it rotates about its center. The direction of the vorticity vector is defined to be the direction of the axis of rotation of this imaginary ball (according to the right-hand rule) while its length is twice the ball's angular velocity. Mathematically, the vorticity is defined as the curl (or rotational) of the velocity field of the fluid, usually denoted by ω → {displaystyle {vec {omega }}} and expressed by the vector analysis formula ∇ × u → {displaystyle abla imes {vec {mathit {u}}}} , where ∇ {displaystyle abla } is the nabla operator and u → {displaystyle {vec {mathit {u}}}} is the local flow velocity. The local rotation measured by the vorticity ω → {displaystyle {vec {omega }}} must not be confused with the angular velocity vector of that portion of the fluid with respect to the external environment or to any fixed axis. In a vortex, in particular, ω → {displaystyle {vec {omega }}} may be opposite to the mean angular velocity vector of the fluid relative to the vortex's axis. In theory, the speed u of the particles (and, therefore, the vorticity) in a vortex may vary with the distance r from the axis in many ways. There are two important special cases, however: In the absence of external forces, a vortex usually evolves fairly quickly toward the irrotational flow pattern, where the flow velocity u is inversely proportional to the distance r. Irrotational vortices are also called free vortices. For an irrotational vortex, the circulation is zero along any closed contour that does not enclose the vortex axis; and has a fixed value, Γ, for any contour that does enclose the axis once. The tangential component of the particle velocity is then u θ = Γ 2 π r {displaystyle u_{ heta }={ frac {Gamma }{2pi r}}} . The angular momentum per unit mass relative to the vortex axis is therefore constant, r u θ = Γ 2 π {displaystyle ru_{ heta }={ frac {Gamma }{2pi }}} . However, the ideal irrotational vortex flow is not physically realizable, since it would imply that the particle speed (and hence the force needed to keep particles in their circular paths) would grow without bound as one approaches the vortex axis. Indeed, in real vortices there is always a core region surrounding the axis where the particle velocity stops increasing and then decreases to zero as r goes to zero. Within that region, the flow is no longer irrotational: the vorticity ω → {displaystyle {vec {omega }}} becomes non-zero, with direction roughly parallel to the vortex axis. The Rankine vortex is a model that assumes a rigid-body rotational flow where r is less than a fixed distance r0, and irrotational flow outside that core regions. The Lamb–Oseen vortex model is an exact solution of the Navier–Stokes equations governing fluid flows and assumes cylindrical symmetry, for which