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'''Vorticity confinement (VC)''', a physics-based [[computational fluid dynamics]] model analogous to [[shock capturing methods]], was invented by Dr. [[John Steinhoff]], professor at the University of Tennessee Space Institute, in the late 1980s
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<ref>{{Cite book
  | author=John Steinhoff
| chapter=Vorticity Confinement: A New Technique for Computing Vortex Dominated Flows
| title=Frontiers of Computational Fluid Dynamics
| year=1994
| publisher=John Wiley & Sons | ISBN=0-471-95334-2 
| url=http://www.alibris.com/search/books/isbn/9780471953340
}}</ref>
to solve [[vortex]] dominated flows.
It was first formulated to capture concentrated vortices shed from the wings, and later became popular in a wide range of research areas.<ref>{{Citation
| last=Grossman | first=Bernard | last2=Hu | first2=Guangchu
| title=The computation of massively separated flows using compressible vorticity confinement methods
| journal=Computers and fluids | volume=35
| issue =7 | year=2006 | pages=781&ndash;789
| url=http://cat.inist.fr/?aModele=afficheN&cpsidt=17795373
}}.</ref> During the 1990s and 2000s, it became widely used in the field of engineering.<ref>{{Citation
  | last=Wenren  | first=Y. | last2=Fan | first2=M.
| last3=Dietz | first3=W. | last4=Hu | first4=G.
| last5=Braun | first5=C. | last6=Steinhoff | first6=J.
| last7=Grossman | first7=B.
| title=Efficient Eulerian Computation of Realistic Rotorcraft Flows Using Vorticity Confinement
| journal=AIAA | year=2001-0996
| url=http://www.flowanalysis.com/techinfo/aiaa-2001-0996.pdf
}}.</ref><ref>{{Citation
| last=Murayama | first=M. | last2=Nakahashi | first2=K.
| last3=Obayashi | first3=S. | last4=Kato | first4=T.
| title=Numerical Simulation of Vortical Flows Using Vorticity Confinement Coupled with Unstructured Adaptive Grid Refinement
| journal=Computational Fluid Dynamics JOURNAL
| volume=10 | issue=1 | pages=28&ndash;36
| year=2001
| url=http://www.ad.mech.tohoku.ac.jp/murayama/Papers/AIAA2001-0606.pdf
}}.</ref>
 
==The method==
VC has a basic familiarity to [[soliton|solitary wave]] approach which is extensively used in many [[condensed matter physics]] applications.<ref>{{Citation
| last=Bishop | first=A.R. | last2=Krumhansl | first2=J.A.
  | last3=Trullinger | first3=S.E.
| title=Solitons in Condensed Matter: A Paradigm
| journal=Physica D: Nonlinear Phenomena
| volume=1 | issue =1 | year=1980 | pages=1&ndash;44
| url=http://adsabs.harvard.edu/abs/1980PhyD....1....1B
| bibcode=1980PhyD....1....1B | doi=10.1016/0167-2789(80)90003-2
}}.</ref> The effect of VC is to capture the small scale features over as few as 2 grid cells as they convect through the flow. The basic idea is similar to that of compression [[discontinuity (mathematics)|discontinuity]] in Eulerian [[shock capturing methods]]. The internal structure is maintained thin and so the details of the internal structure may not be important.
 
==Example==
<!-- Deleted image removed: [[File:rotor flow.jpeg|thumb|right|Simulation of vortex sheets shed from helicopter rotor blades. {{Deletable image-caption|date=May 2012}}]] -->Consider 2D [[Euler equations]], modified using the confinement term, F:
:<math>
\frac{\partial \bold u}{\partial t} + \bold u \cdot \nabla \bold u + \nabla\frac{P}{\rho}=F_D (\bold u) -F_C (\bold u)
</math>
 
The discretized Euler equations with the extra term can be solved on fairly coarse grids, with simple low order accurate numerical methods, but still yield concentrated vortices which convect without spreading. VC has different forms, one of which is VC1. It involves an added dissipation,<math> F_D </math>,to the [[partial differential equation]], which when balanced with inward convection, <math> F_C </math>, produce stable solutions. Another form is termed as VC2 in which [[dissipation]] is balanced with nonlinear anti-diffusion to produce stable [[soliton|solitary wave]]-like solutions.
:<math>
F_D </math>  : Dissipation
:<math>
F_C</math>  : Inward convection for VC1 and nonlinear anti-diffusion for VC2
The main difference between VC1 and VC2 is that in the latter the centroid of the [[vortex]] follows the local velocity [[moment (physics)|moment]] weighted by vorticity. This should provide greater accuracy than VC1 in cases where the convecting field is weak compared to the self-induced [[velocity]] of the vortex. One drawback is that VC2 is not as robust as VC1 because while VC1 involves convection like inward propagation of vorticity balanced by an outward second order diffusion, VC2 involves a second order inward propagation of [[vorticity]] balanced by 4th order outward [[dissipation]]. This approach has been further extended to solve [[wave equation]] and is called [[Wave confinement]] (WC).
 
==Immersed boundary==
 
<!-- Deleted image removed: [[File:with VC.jpeg|thumb|right|Simulation of Jet plume with VC. {{Deletable image-caption|date=May 2012}}]]  --><!-- Deleted image removed: [[File:without VC.jpeg|thumb|right|Simulation of Jet plume without VC. {{Deletable image-caption|date=May 2012}}]] -->To enforce no-slip boundary conditions on immersed surfaces, first, the surface is represented implicitly by a smooth “level set” function, “f”, defined at each grid point. This is the (signed) distance from each grid point to the nearest point on the surface of an object – positive outside, negative inside. Then, at each time step during the solution, velocities in the interior are set to zero. In a computation using VC, this results in a thin vortical region along the surface, which is smooth in the tangential direction, with no “staircase” effects.<ref>{{Citation
| last=Wenren | first=Y. | last2=Fan | first2=M. | last5=Steinhoff
| last3=Wang  | first3=J. | last4=Xiao | first4=M.
| title=Application of Vorticity Confinement to Prediction of the Flow over Complex Bodies
| journal=AIAA Journal
| volume=41 | issue=5 | year=2003 | pages=809&ndash;816
| url=http://adsabs.harvard.edu/abs/2003AIAAJ..41..809W
| bibcode=2003AIAAJ..41..809W |doi= 10.2514/2.2042
}}.</ref> The important point is that no special logic is required in the “cut” cells, unlike many conventional schemes: only the same VC equations are applied, as in the rest of the grid, but with a different form for F. Also, unlike many conventional immersed surface schemes, which are inviscid because of cell size constraints, there is effectively a no-slip boundary condition, which results in a boundary layer with well-defined total vorticity and which, because of VC, remains thin, even after separation. The method is especially effective for complex configurations with separation from sharp corners. Also, even with constant coefficients, it can approximately treat separation from smooth surfaces. General blunt bodies, which typically shed turbulent vorticity that induces a velocity around an upstream body. It is inconsistent to use body fitted grids as the vorticity convects through a non fitted grid.
 
==Applications==
VC is used in many applications including rotor wake computations, computation of wing tip vortices, drag computations for vehicles, flow around urban layouts, smoke/contaminant propagation and special effects. Also, it is used in wave computations for communication purposes.
 
==References==
{{reflist}}
 
{{Numerical PDE}}
 
[[Category:Numerical differential equations]]
[[Category:Computational fluid dynamics]]

Latest revision as of 23:30, 22 February 2014

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