Editing Computational fluid dynamics (section)

====PDF methods====

[[Probability density function]] (PDF) methods for turbulence, first introduced by [[Thomas S. Lundgren|Lundgren]],<ref name="Lundgren_1969">{{cite journal|title=Model equation for nonhomogeneous turbulence|author=Lundgren, T.S.|journal=Physics of Fluids A|volume=12|issue=3|year=1969|pages=485–497|doi=10.1063/1.1692511|bibcode = 1969PhFl...12..485L }}</ref> are based on tracking the one-point PDF of the velocity, <math>f_{V}(\boldsymbol{v};\boldsymbol{x},t) d\boldsymbol{v}</math>, which gives the probability of the velocity at point <math>\boldsymbol{x}</math> being between <math>\boldsymbol{v}</math> and <math>\boldsymbol{v}+d\boldsymbol{v}</math>.  This approach is analogous to the [[kinetic theory of gases]], in which the macroscopic properties of a gas are described by a large number of particles.  PDF methods are unique in that they can be applied in the framework of a number of different turbulence models; the main differences occur in the form of the PDF transport equation.  For example, in the context of [[large eddy simulation]], the PDF becomes the filtered PDF.<ref name="Colucci_1998">{{cite journal|title=Filtered density function for large eddy simulation of turbulent reacting flows|author1=Colucci, P.J.|author2=Jaberi, F.A|author3=Givi, P.|author4=Pope, S.B.|journal=Physics of Fluids A|year=1998|volume=10|issue=2|pages=499–515|doi=10.1063/1.869537|bibcode = 1998PhFl...10..499C }}</ref>  PDF methods can also be used to describe chemical reactions,<ref name="Fox_2003">{{cite book|author=Fox, Rodney|title=Computational models for turbulent reacting flows|year=2003|publisher=Cambridge University Press|isbn=978-0-521-65049-6}}</ref><ref name="Pope_1985">{{cite journal|title=PDF methods for turbulent reactive flows|author=Pope, S.B.|journal=Progress in Energy and Combustion Science|year=1985|volume=11|pages=119–192|bibcode = 1985PrECS..11..119P|doi=10.1016/0360-1285(85)90002-4|issue=2 }}</ref> and are particularly useful for simulating chemically reacting flows because the chemical source term is closed and does not require a model.  The PDF is commonly tracked by using Lagrangian particle methods; when combined with large eddy simulation, this leads to a [[Langevin equation]] for subfilter particle evolution.