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2014-02-19T06:00:09Z

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2014-01-23T17:50:52Z

59.92.62.71: /* Bilinear blending */

The '''viscous vortex domains (VVD) method''' is a [[Meshfree methods|mesh-free]] method of [[computational fluid dynamics]] for directly numerically solving 2D [[Navier-Stokes equations]] in [[Lagrangian and Eulerian specification of the flow field|Lagrange coordinates]]
<ref name="vvd_dan">{{cite journal|last=Dynnikova|first=G. Ya.|title=The Lagrangian approach to solving the time-dependent Navier-Stokes. equations|journal=[[Doklady Physics]]|date=1 November 2004|volume=49|issue=11|pages=648–652|doi=10.1134/1.1831530|bibcode = 2004DokPh..49..648D }}</ref><ref name="vvd_eccm">{{cite journal|last=Dynnikova|first=G.Ya.|title=The Viscous Vortex Domains (VVD) method for non-stationary viscous incompressible flow simulation|journal=Proceedings of IV European Conference on Computational Mechanics, Paris, France|date=16–21 May 2010|url=http://www.eccm2010.org/complet/fullpaper_193.pdf}}</ref>
It doesn't implement any [[turbulence model]] and free of arbitrary parameters.
The main idea of this method is to present [[vorticity]] field with discrete regions (domains), which travel with diffusive velocity relatively to fluid and conserve their [[Circulation (fluid dynamics)|circulation]]. The same approach was used in Diffusion Velocity method of Ogami and Akamatsu
,<ref name="dvm">{{cite journal|last=Ogami|first=Yoshifumi|coauthors=Akamatsu, Teruaki|title=Viscous flow simulation using the discrete vortex model—the diffusion velocity method|journal=Computers & Fluids|date=31 December 1990|volume=19|issue=3-4|pages=433–441|doi=10.1016/0045-7930(91)90068-S}}</ref> but VVD uses other discrete formulas

==Features==

The VVD method deals with [[viscous]] [[Incompressible flow|incompressible]] fluid. The viscosity and density of fluid is considered to be constant. Method can be extended for simulation of heat conductive fluid flows ([[viscous vortex-heat domains method]])

The main features are:
* Direct solving Navier-Stokes equations ([[Direct numerical simulation|DNS]])
* Calculation of the friction force at the body surfaces
* Proper description of the [[boundary layers]] (even turbulent)
* Infinite computation region
* Convenient simulation of deforming boundaries<ref name=airfoil_fd>{{cite journal|last=Guvernyuk|first=S. V.|coauthors=Dynnikova, G. Ya.|title=Modeling the flow past an oscillating airfoil by the method of viscous vortex domains|journal=Fluid Dynamics|date=31 January 2007|volume=42|issue=1|pages=1–11|doi=10.1134/S0015462807010012}}</ref>
* Investigation of the flow-structure interaction,<ref name=plate_fd>{{cite journal|last=Andronov|first=P. R.|coauthors=Grigorenko, D. A., Guvernyuk, S. V., Dynnikova, G. Ya.|title=Numerical simulation of plate autorotation in a viscous fluid flow|journal=Fluid Dynamics|date=1 October 2007|volume=42|issue=5|pages=719–731|doi=10.1134/S0015462807050055|bibcode = 2007FlDy...42..719A }}</ref> even in case of zero mass
* Estimated numerical diffusion and stability criteria <ref name=vvd_eccm_stability>{{cite journal|last=Dynnikov|first=Ya. A.|coauthors=Dynnikova, G. Ya.|title=Numerical stability and numerical viscosity in certain meshless vortex methods as applied to the Navier-Stokes and heat equations|journal=Computational Mathematics and Mathematical Physics|date=12 October 2011|volume=51|issue=10|pages=1792–1804|doi=10.1134/S096554251110006X|bibcode = 2011CMMPh..51.1792D }}</ref>

==Governing equations==
[[File:Notations in method of viscous vortex domains (VVD).png|thumb|Scheme of VVD method]]

The VVD method is based on a theorem,<ref name=vvd_dan /> that circulation in viscous fluid is conserved on contours travelling with speed
:<math>\mathbf{u} = \mathbf{V} + \mathbf{V}_d; ~~~\mathbf{V}_d = -\nu \dfrac{\nabla\mathbf{\Omega}}{|\mathbf{\Omega}|}; ~~~\mathbf{\Omega} = [\nabla \times \mathbf{V}]</math>,
where '''V''' is fluid velocity, '''V'''d — diffusion velocity, ν — [[Kinematic_viscosity#Kinematic_viscosity|kinematic viscosity]].
This theorem shows resemblance with [[Kelvin's circulation theorem]], but it works for viscid flows.

Basing on this theorem, flow region with non-zero circulation is presented with number of domains (small regions with finite volumes), which move with velocity '''u''' and thus their circulation <math>\gamma</math> remains constant. The actual boundaries of every domain are not tracked, but coordinates of the only tracking point in every domain is saved. Array of domains' coordinates and circulations is known either from [[boundary conditions]] or from [[initial conditions]]. Such a motion results in vorticity evolution and satisfies Navier-Stokes equations.

==Discrete formulas==
[[File:Calculation of diffusion speed in VVD method.png|thumb|Diffusive vortex-vortex interaction]]
[[File:Calculation of body diffusion speed in VVD method.png|thumb|Diffusive body-vortex interaction]]

Fluid velocity '''V''' in point '''r''' can be calculated with help of [[Biot-savart law]]
:<math>\mathbf{V}(\mathbf{r}) = \dfrac{1}{2\pi} \sum_i \gamma_i \cdot \left[\mathbf{e}_z \times \dfrac{\mathbf{r}-\mathbf{r}_i}{(\mathbf{r}-\mathbf{r}_i)^2 + \delta^2}\right]</math>
where ''i'' indexes domains in flow, '''r'''i — tracking point of domain and γi — his circulation.
δ is a so-called "radius of discreteness" — small value that smooths the vortex and helps to get rid of singularity in the domain tracking point.<ref name=vvd_eccm_stability/> It equals to mean distance between domains.

Calculation of diffusion velocity is more difficult<ref name=vvd_dan/><ref name=airfoil_fd/>
:<math>\mathbf{V}_d(\mathbf{r}) = \nu\left( \dfrac{\mathbf{I}_2(\mathbf r)}{I_1(\mathbf r)} + \dfrac{\mathbf{I}_3(\mathbf r)}{2\pi\varepsilon^2 - I_0(\mathbf r)} \right)</math>

First fraction produces vortex-vortex interaction (''i'' — vortex index).
:<math>\mathbf{I}_2(\mathbf r) = \sum\limits_i
\dfrac{\mathbf{r}-\mathbf{r}_i}{\varepsilon \left|\mathbf{r}-\mathbf{r}_i\right|}
\cdot \gamma_i \cdot \exp(-\left|\mathbf{r}-\mathbf{r}_i\right|/\varepsilon)</math>
:<math>I_1(\mathbf r) = {\sum\limits_i \gamma_i \cdot \exp(-\left|\mathbf{r}-\mathbf{r}_i\right|/\varepsilon)}</math>

And second fraction represents vortex-boundary repulsion. It helps to calculate ∇Ω near body surface and properly describe boundary layer.
:<math>\mathbf{I}_3(\mathbf r) = {\sum\limits_k d\mathbf S_k\cdot \exp(-\left|\mathbf{r}-\mathbf{r}_k\right|/\varepsilon)}</math>
:<math>I_0(\mathbf r) = {\varepsilon^2\sum\limits_k \dfrac{\left|\mathbf{r}-\mathbf{r}_k\right| /\varepsilon +1}{(\mathbf{r}-\mathbf{r}_k)^2}
\cdot((\mathbf{r}-\mathbf{r}_k) \cdot d\mathbf S_k)\cdot \exp(-\left|\mathbf{r}-\mathbf{r}_k\right|/\varepsilon)}</math>
Here ''k'' indexes boundary segments, '''r'''k — its center, d'''S'''k — its [[surface normal|normal]].

==References==
{{Reflist}}

==External links==
* [http://www.youtube.com/user/rosikru YouTube channel with some VVD computations]

[[Category:Computational fluid dynamics]]

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2013-06-04T10:14:18Z

59.92.62.247: /* Others */

In [[game theory]], '''differential games''' are a group of problems related to the modeling and analysis of conflict in the context of a [[dynamical system]]. The problem usually consists of two actors, a pursuer and an evader, with conflicting goals. The dynamics of the pursuer and the evader are modeled by systems of differential equations.

Differential games are related closely with [[optimal control]] problems. In an optimal control problem there is single control <math>u(t)</math> and a single criterion to be optimized; differential game theory generalizes this to two controls <math>u(t),v(t)</math> and two criteria, one for each player. Each player attempts to control the state of the system so as to achieve his goal; the system responds to the inputs of both players.

The first to study differential games was [[Rufus Isaacs (game theorist)|Rufus Isaacs]] (1951, published 1965)<ref>Rufus Isaacs, ''Differential Games'', Dover, 1999. ISBN 0-486-40682-2 [http://books.google.com/books?id=XIxmMyIQgm0C Google Books]</ref> and one of the first games analyzed was the [[Homicidal chauffeur problem|'homicidal chauffeur game']].

Differential games have been applied to economics. Recent developments include adding [[stochasticity]] to differential games and the derivation of the [[stochastic feedback Nash equilibrium]] (SFNE). A recent example is the stochastic differential game of capitalism by Leong and Huang (2010).<ref>{{cite doi|10.1016/j.jmateco.2010.03.007}}</ref>

==Applications==
For a survey of [[pursuit-evasion]] differential games see Pachter.<ref>[http://med.ee.nd.edu/MED10/pdf/477.pdf Meir Pachter: Simple-motion pursuit-evasion differential games, 2002]</ref>

==Notes==
{{reflist}}

===Textbooks and general references===

* {{Citation |last4=Sorger | first4=Gerhard |last3=Long | first3=Ngo Van |last2=Jorgensen | first2=Steffen | last1=Dockner | first1=Engelbert | title=Differential Games in Economics and Management Science | publisher=Cambridge University Press | isbn=978-0-521-63732-9 | year=2001}}
* {{Citation | last1=Petrosyan | first1=Leon | title=Differential Games of Pursuit (Series on Optimization, Vol 2) | publisher=World Scientific Publishers | isbn=978-981-02-0979-7 | year=1993}}

==External links==
* [http://faculty.gvsu.edu/aboufade/web/szurley.htm An overview of differential games]

{{DEFAULTSORT:Differential Game}}
[[Category:Control theory]]
[[Category:Game theory]]

{{gametheory-stub}}

Subderivative

2012-08-17T19:22:42Z

59.92.179.71: /* Definition */

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