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VORTEX DYNAMICS
1. Introduction
A vortex is commonly associated with the rotating motion of °uid around a common centerline. It is de¯ned by thevorticityin the °uid, which measures the rate of local °uid rotation. Typically, the °uid circulates around the vortex, the speed increases as the vortex is approached and the pressure decreases. Vor- tices arise in nature and technology in a large range of sizes as illustrated by the examples given in Table 1. The next section presents some of the mathematical background necessary to understand vortex formation and evolution. Section 3 describes sample °ows, including important instabilities and reconnection pro- cesses. Section 4 presents some of the numerical methods used to simulate these
°ows.
Vortex Diameter
super°uid vortices 10 ¡8 cm (= 1ºA) trailing vortex of Boeing 727 1{2 m dust devils 1{10 m tornadoes 10{500 m hurricanes 100{2000 km
Jupiter's Red Spot 25,000 km
spiral galaxies thousands of light years
TABLE 1: Sample vortices and typical sizes.
2. Background
LetDbe a region in 3D space containing a °uid, and letx=(x;y;z) T be a point inD. The °uid motion is described by its velocityu(x;t)=u(x;t)i+ v(x;t)j+w(x;t)k, and depends on the °uid density½(x;t), temperatureT(x;t), gravitational ¯eldgand other external forces possibly acting on it. The °uid vorticity is de¯ned by!!=r£u. The vorticity measures the local °uid rotation about an axis, as can be seen by expanding the velocity nearx=x 0 u(x)=u(x 0 )+D(x 0 )(x¡x 0 )+1 2!!(x 0 )£(x¡x 0 )+O(jx¡x 0 j 2 ) (1) where D(x 0 )=1
2(ru+ru
T );ru=" u x u y u z v x v y v z w x w y w z :(2)
The ¯rst termu(x
0 ) corresponds to translation: all °uid particles move with constant velocityu(x 0 ). The second termD(x 0 )(x¡x 0 ) corresponds to a strain ¯eld in the three directions of the eigenvectors of the symmetric matrix D. If the eigenvalue corresponding to a given eigenvector is positive, the °uid is stretched in that direction, if it is negative, the °uid is compressed. Note that in incompressible °owr¢u= 0, so the sum of the eigenvalues ofDequals zero. 1 (a)(b) Figure 1: Strain¯eld. (a) Two positive eigenvalues, sheet formation. (b) One positive eigenvalue, tube formation. Thus at least one eigenvalue is positive and one negative. If the third eigenvalue is positive, °uid particles move towards sheets (Fig. 1a). If the third eigenvalue is negative, °uid particles move towards tubes (Fig. 1b). The last term in Eq. (1), 1 2 !!(x 0 )£(x¡x 0 ), corresponds to a rotation: near a point with!!(x 0 )6=0, the °uid rotates with angular velocityj!!j=2 in a plane normal to the vorticity vector!!. Fluid for which!!=0is said to beirrotational. Avortex lineis an integral curve of the vorticity. For incompressible °ow, r¢!!=r¢(r£u) = 0 which implies that vortex lines cannot end in the interior of the °ow, but must either form a closed loop or start and end at a bounding surface. In 2D °ow,u=ui+vjand the vorticity is!!=!k, where!=v x ¡u y is thescalar vorticity. Thus in 2D, the vorticity points in thez-direction and the vortex lines are straight lines normal to thex-yplane. Avortex tubeis a bundle of vortex lines. Thestrengthof a vortex tube is de¯ned as thecirculationR C u¢dsabout a curveCenclosing the tube. By Stokes' Theorem, Z C u¢ds=ZZ A !!¢ndS ;(3) and thus the circulation can also be interpreted as the °ux of vorticity through a cross section of the tube. In inviscid incompressible °ow of constant density, Helmholtz' Theoremstates that the tube strength is independent of the curve C, and is therefore a well-de¯ned quantity, andKelvin's Theoremstates that a tube's strength remains constant in time. Avortex ¯lamentis an idealization in which a tube is represented by a single vortex line of nonzero strength. The evolution equation for the °uid vorticity, as derived from the Navier-
Stokes Equations, isd!!
dt=!!¢ru+º¢!!(4) where d=dt=@=@t+u¢ris the total time derivative. Equation (4) states that the vorticity is transported by the °uid velocity (¯rst term), stretched by the °uid velocity gradient (second term), and di®used by viscosityº(last term). These equations are usually nondimensionalized and written in terms of the Reynolds number, a dimensionless quantity inversely proportional to viscosity. 2 (a)(b) rdistance speed |u| Figure 2: Flow induced by a point vortex. (a) Streamlines. (b) Speedjujvs. distancer. To understand high Reynolds number °ow it is of interest to study the inviscid Euler Equations. The corresponding vorticity evolution equation in 2D isd!! dt=0;(5) which states that 2D vortex ¯laments in inviscid °ow move with the °uid veloc- ity. Furthermore, in incompressible °ow the °uid velocity is determined by the vorticity, up to an irrotational far-¯eld componentu 1 , through theBiot-Savart law, u(x)=¡1
4¼Z
(x¡x 0 )£!!(x 0 jx¡x 0 j 3 dx 0 +u 1 :(6)
In planar 2D °ow, Eq (6) reduces to
u(x)=K 2d 2d (x)=1
2¼¡yi+xjjxj
2 (7) and!(x) is the scalar vorticity. Equations (4,5) and (6,7) are the basis of the numerical methods discussed inx4. Avortexis typically de¯ned by a region in the °uid of concentrated vorticity. A simple model is apoint vortexin 2D °ow, which corresponds to a straight vortex ¯lament of unit circulation. The associated scalar vorticity is a delta function in the plane, and the induced velocity is obtained from the Biot-Savart law. For a point vortex at the origin this reduces to the radial velocity ¯eld u(x)=K 2d 2d (x). Corresponding particle trajectories are shown in Fig.
2(a). The particle speedjuj=1=rincreases unboundedly as the vortex center is
approached, and vanishes asr!1(Fig. 2b). In general, the far ¯eld velocity of a concentrated vortex behaves similarly to the one of a point vortex, with speeds decaying as 1=r. Near the vortex center, the velocity typically increases in magnitude and as a result, the °uid pressure decreases (Bernoulli's Theorem). A vortex of arbitrary shape can be approximated by a sum of point vortices (in
2D) or vortex ¯laments (in 3D), as is often done for simulation purposes.
Vorticity can be generated by a variety of mechanisms. For example, vortic- ity can be generated by density gradients, which in turn are induced by spatial 3 y wUu y d(a) (b) o Figure 3: Velocity and vorticity in boundary layer near a °at wall. temperature variations. This mechanism explains the formation of warm-air vortices when a layer of hot air is trapped underneath cooler air. Vorticity is also generated near solid walls in the form ofboundary layerscaused by viscos- ity. To illustrate, imagine horizontal °ow with speedU o moving past a solid wall at rest (Fig. 3a). Since in viscous °ow the °uid sticks to the wall (the no-slip boundary condition), the °uid velocity at the wall is zero. As a result, there is a thin layer near the wall in which the horizontal velocity varies greatly while the vertical velocity gradients are small, yielding large negative vorticity values !=v x ¡u y (Fig. 3b). Similarity solutions to the approximatingPrandtl bound- ary layer equationsshow that the boundary layer thicknessdgrows proportional top twheretmeasures the time from the beginning of the motion. Boundary layers can separate from the wall at corners or regions of high curvature andquotesdbs_dbs8.pdfusesText_14
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