Editing Computational fluid dynamics (section)

===Turbulence models===

In computational modeling of turbulent flows, one common objective is to obtain a model that can predict quantities of interest, such as fluid velocity, for use in engineering designs of the system being modeled.  For turbulent flows, the range of length scales and complexity of phenomena involved in turbulence make most modeling approaches prohibitively expensive; the resolution required to resolve all scales involved in turbulence is beyond what is computationally possible.  The primary approach in such cases is to create numerical models to approximate unresolved phenomena.  This section lists some commonly used computational models for turbulent flows.

Turbulence models can be classified based on computational expense, which corresponds to the range of scales that are modeled versus resolved (the more turbulent scales that are resolved, the finer the resolution of the simulation, and therefore the higher the computational cost). If a majority or all of the turbulent scales are not modeled, the computational cost is very low, but the tradeoff comes in the form of decreased accuracy.

In addition to the wide range of length and time scales and the associated computational cost, the governing equations of fluid dynamics contain a [[Nonlinear system|non-linear]] convection term and a non-linear and non-local pressure gradient term.  These nonlinear equations must be solved numerically with the appropriate boundary and initial conditions.

==== Reynolds-averaged Navier–Stokes ====
{{main|Reynolds-averaged Navier–Stokes equations}}
[[File:DrivAer SST-URANS-DDES Comparison.png|thumb|258x258px|External aerodynamics of the [https://www.mw.tum.de/en/aer/research-groups/automotive/drivaer/ DrivAer] model, computed using [[Reynolds-averaged Navier–Stokes equations|URANS]] (top) and [[Detached eddy simulation|DDES]] (bottom)]]

[[File:Verus Engineering Porsche 987.2 Ventus 2 Package.png|thumb|A simulation of aerodynamic package of a [[Porsche Cayman|Porsche Cayman (987.2)]]]]

[[Reynolds-averaged Navier–Stokes equations|Reynolds-averaged Navier–Stokes]] (RANS) equations are the oldest approach to turbulence modeling. An ensemble version of the governing equations is solved, which introduces new ''apparent stresses'' known as [[Reynolds stresses]]. This adds a second-order tensor of unknowns for which various models can provide different levels of closure. It is a common misconception that the RANS equations do not apply to flows with a time-varying mean flow because these equations are 'time-averaged'. In fact, statistically unsteady (or non-stationary) flows can equally be treated. This is sometimes referred to as URANS. There is nothing inherent in Reynolds averaging to preclude this, but the turbulence models used to close the equations are valid only as long as the time over which these changes in the mean occur is large compared to the time scales of the turbulent motion containing most of the energy.

RANS models can be divided into two broad approaches:

; [[Turbulence modeling|Boussinesq hypothesis]]: This method involves using an algebraic equation for the Reynolds stresses which include determining the turbulent viscosity, and depending on the level of sophistication of the model, solving transport equations for determining the turbulent kinetic energy and dissipation. Models include k-ε ([[Brian Launder|Launder]] and [[Brian Spalding|Spalding]]),<ref>{{cite journal|last=Launder|first=B.E.|author2=D.B. Spalding|year=1974|title= The Numerical Computation of Turbulent Flows|journal=Computer Methods in Applied Mechanics and Engineering|pages=269–289|doi = 10.1016/0045-7825(74)90029-2|bibcode = 1974CMAME...3..269L|volume=3|issue=2 }}</ref> Mixing Length Model ([[Ludwig Prandtl|Prandtl]]),<ref name=wilcox>{{cite book|last=Wilcox|first=David C.|title=Turbulence Modeling for CFD|year=2006|publisher=DCW Industries, Inc.|isbn=978-1-928729-08-2|edition=3}}</ref> and Zero Equation Model (Cebeci and [[Apollo M. O. Smith|Smith]]).<ref name=wilcox /> The models available in this approach are often referred to by the number of transport equations associated with the method. For example, the Mixing Length model is a "Zero Equation" model because no transport equations are solved; the <math>k-\epsilon</math> is a "Two Equation" model because two transport equations (one for <math>k</math> and one for <math>\epsilon</math>) are solved.
; [[Reynolds stress model]] (RSM): This approach attempts to actually solve transport equations for the Reynolds stresses. This means introduction of several transport equations for all the Reynolds stresses and hence this approach is much more costly in CPU effort.{{Citation needed|date=November 2010}}

====Large eddy simulation====
{{main|Large eddy simulation}}

[[Image:LESPremixedFlame.jpg|thumb|right|250px|Volume rendering of a non-premixed swirl flame as simulated by LES]]
[[Large eddy simulation]] (LES) is a technique in which the smallest scales of the flow are removed through a filtering operation, and their effect modeled using subgrid scale models.  This allows the largest and most important scales of the turbulence to be resolved, while greatly reducing the computational cost incurred by the smallest scales. This method requires greater computational resources than RANS methods, but is far cheaper than DNS.

====Detached eddy simulation====
{{main|Detached eddy simulation}}

[[Detached eddy simulation]]s (DES) is a modification of a RANS model in which the model switches to a subgrid scale formulation in regions fine enough for LES calculations. Regions near solid boundaries and where the turbulent length scale is less than the maximum grid dimension are assigned the RANS mode of solution. As the turbulent length scale exceeds the grid dimension, the regions are solved using the LES mode. Therefore, the grid resolution for DES is not as demanding as pure LES, thereby considerably cutting down the cost of the computation. Though DES was initially formulated for the Spalart-Allmaras model (Philippe R. Spalart et al., 1997), it can be implemented with other RANS models (Strelets, 2001), by appropriately modifying the length scale which is explicitly or implicitly involved in the RANS model. So while Spalart–Allmaras model based DES acts as LES with a wall model, DES based on other models (like two equation models) behave as a hybrid RANS-LES model. Grid generation is more complicated than for a simple RANS or LES case due to the RANS-LES switch. DES is a non-zonal approach and provides a single smooth velocity field across the RANS and the LES regions of the solutions.
[[File:Cp IDDES.gif|thumb|IDDES Simulation of the Karel Motorsports BMW. This is a type of DES simulation completed in OpenFOAM. The plot is coefficient of pressure.]]

====Direct numerical simulation====
{{main|Direct numerical simulation}}

[[Direct numerical simulation]] (DNS) resolves the entire range of turbulent length scales.  This marginalizes the effect of models, but is extremely expensive.  The computational cost is proportional to <math>Re^{3}</math>.<ref name="Pope_2000">{{cite book|title=Turbulent Flows|author=Pope, S.B.|publisher=Cambridge University Press|year=2000|isbn=978-0-521-59886-6}}</ref> DNS is intractable for flows with complex geometries or flow configurations.

====Coherent vortex simulation====

The coherent vortex simulation approach decomposes the turbulent flow field into a coherent part, consisting of organized vortical motion, and the incoherent part, which is the random background flow.<ref name="Farge_2001">{{cite journal|title=Coherent Vortex Simulation (CVS), A Semi-Deterministic Turbulence Model Using Wavelets|last1=Farge | first1= Marie | author1-link = Marie Farge|author2=Schneider, Kai|journal=Flow, Turbulence and Combustion|volume=66|issue=4|pages=393–426|doi=10.1023/A:1013512726409|year=2001|bibcode=2001FTC....66..393F |s2cid=53464243 }}</ref>  This decomposition is done using [[wavelet]] filtering.  The approach has much in common with LES, since it uses decomposition and resolves only the filtered portion, but different in that it does not use a linear, low-pass filter.  Instead, the filtering operation is based on wavelets, and the filter can be adapted as the flow field evolves. [[Marie Farge|Farge]] and Schneider tested the CVS method with two flow configurations and showed that the coherent portion of the flow exhibited the <math>-\frac{40}{39}</math> energy spectrum exhibited by the total flow, and corresponded to coherent structures ([[vortex stretching|vortex tubes]]), while the incoherent parts of the flow composed homogeneous background noise, which exhibited no organized structures.  Goldstein and Vasilyev<ref name="Goldstein_2004">{{cite journal|author1=Goldstein, Daniel|author2=Vasilyev, Oleg|title=Stochastic coherent adaptive large eddy simulation method|journal=Physics of Fluids A|year=1995|volume=24|page=2497|doi=10.1063/1.1736671|bibcode = 2004PhFl...16.2497G|issue=7 |citeseerx=10.1.1.415.6540}}</ref> applied the FDV model to large eddy simulation, but did not assume that the wavelet filter eliminated all coherent motions from the subfilter scales.  By employing both LES and CVS filtering, they showed that the SFS dissipation was dominated by the SFS flow field's coherent portion.

====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.

====Vorticity confinement method====

{{Main|Vorticity confinement}}

The [[vorticity confinement]] (VC) method is an Eulerian technique used in the simulation of turbulent wakes. It uses a solitary-wave like approach to produce a stable solution with no numerical spreading. VC can capture the small-scale features to within as few as 2 grid cells. Within these features, a nonlinear difference equation is solved as opposed to the [[finite difference equation]]. VC is similar to [[shock capturing methods]], where conservation laws are satisfied, so that the essential integral quantities are accurately computed.

====Linear eddy model====

The Linear eddy model is a technique used to simulate the convective mixing that takes place in turbulent flow.<ref>{{cite journal|last=Krueger|first=Steven K.|title=Linear Eddy Simulations Of Mixing In A Homogeneous Turbulent Flow|journal=Physics of Fluids|year=1993|volume=5|issue=4|pages=1023–1034|bibcode = 1993PhFlA...5.1023M |doi = 10.1063/1.858667 |url=https://zenodo.org/record/1232081}}</ref> Specifically, it provides a mathematical way to describe the interactions of a scalar variable within the vector flow field. It is primarily used in one-dimensional representations of turbulent flow, since it can be applied across a wide range of length scales and Reynolds numbers. This model is generally used as a building block for more complicated flow representations, as it provides high resolution predictions that hold across a large range of flow conditions.