# Turbulence model

A turbulence model is used in numerical flow simulation to close the underlying system of equations.

Since turbulence occurs both spatially and temporally on very different and, above all, very small scales, extremely fine grids and time steps are required for the correct resolution of all phenomena, as is done in direct numerical simulation (DNS).

Due to its extreme computational effort, DNA is limited to low Reynolds numbers (and therefore often of little interest for practical applications) for the foreseeable future . For this reason, different strategies are used to reduce the computational effort. The more assumptions are made, the shorter the computing time and the greater the uncertainties regarding the result.

## Statistical modeling

The currently most widespread modeling strategy is statistical modeling (also: Reynolds Averaged Navier Stokes (RANS) ). In the simplest case, the turbulent flow is modeled as a temporal mean value and variance of the speed and pressure . ${\ displaystyle {\ overline {\ bullet}}}$ ${\ displaystyle {\ bullet '}}$ ${\ displaystyle u_ {i}}$ ${\ displaystyle p}$ If this model assumption is introduced into the Navier-Stokes equations and averaged over time, the Reynolds- averaged Navier-Stokes equations arise . These contain with the Reynolds stress tensor

${\ displaystyle {- \ tau _ {ij}} = \ rho \, {\ overline {u_ {i} 'u_ {j}'}} = \ rho {\ begin {pmatrix} {\ overline {u_ {1} 'u_ {1}'}} & {\ overline {u_ {1} 'u_ {2}'}} & {\ overline {u_ {1} 'u_ {3}'}} \\ {\ overline {u_ { 2} 'u_ {1}'}} & {\ overline {u_ {2} 'u_ {2}'}} & {\ overline {u_ {2} 'u_ {3}'}} \\ {\ overline { u_ {3} 'u_ {1}'}} & {\ overline {u_ {3} 'u_ {2}'}} & {\ overline {u_ {3} 'u_ {3}'}} \ end {pmatrix }}}$ an additional term with additional variables , the Reynolds stresses. The diagonal elements of the tensor represent normal stresses, while the remaining elements are shear stresses. The system of equations is thus no longer closed. The closure is achieved through additional assumptions for the components of the Reynolds' stress tensor in the form of equations. These additional equations are called the turbulence model. Since turbulence is still largely not understood, it is mostly based on heuristics . Data from experiments are used for validation. A distinction is made between zero, one and two equation models, as well as 2nd order closure approaches. ${\ displaystyle \ rho \, {\ overline {u_ {i} 'u_ {j}'}}}$ ### Vortex viscosity models

In the eddy viscosity models, the Reynolds stress tensor is approximated using the Boussinesq approximation. The Reynolds stresses are treated in analogy to the stresses caused by molecular viscosity:

${\ displaystyle \ rho {\ overline {u_ {i} 'u_ {j}'}} = - \ mu _ {t} \ left ({\ frac {\ partial {\ bar {u}} _ {i}} {\ partial x_ {j}}} + {\ frac {\ partial {\ bar {u}} _ {j}} {\ partial x_ {i}}} - {\ frac {2} {3}} {\ frac {\ partial {\ bar {u}} _ {k}} {\ partial x_ {k}}} \ delta _ {ij} \ right) + {\ frac {2} {3}} \ rho k \ delta _ {ij}}$ .

The variable is called turbulent eddy viscosity and describes the increase in viscosity due to turbulent fluctuations. As a rule, the molecular viscosity exceeds significantly. The root of the turbulent kinetic energy represents a typical measure of the speed of the turbulent fluctuations. The symbol denotes the Kronecker delta . The term is a "turbulent pressure term" that is necessary to be able to apply the equation to normal stresses. ${\ displaystyle \ mu _ {t} = \ rho \ cdot \ nu _ {t}}$ ${\ displaystyle \ mu _ {t}}$ ${\ displaystyle k = {\ overline {u_ {i} 'u_ {i}'}} / 2}$ ${\ displaystyle \ delta _ {ij}}$ ${\ displaystyle {\ frac {2} {3}} \ rho k \ delta _ {ij}}$ For dimensional reasons, the turbulent eddy viscosity can be expressed with a turbulent length measure and a turbulent speed measure according to. The Boussinesq approach enables the Reynolds-averaged Navier-Stokes equations to be closed by determining the eddy viscosity or the associated turbulent length and velocity measures. ${\ displaystyle \ nu _ {t}}$ ${\ displaystyle L_ {t}}$ ${\ displaystyle U_ {t}}$ ${\ displaystyle \ nu _ {t} \ sim L_ {t} \ cdot U_ {t}}$ ${\ displaystyle \ nu _ {t}}$ The vortex viscosity models are differentiated according to the number of independent turbulence variables that are used to calculate and . ${\ displaystyle L_ {t}}$ ${\ displaystyle U_ {t}}$ #### Zero equation models

Algebraic or null equation models only use algebraic relationships for closure. These include the Baldwin-Lomax model and the turbulence model according to Cebeci and Smith.

#### Equation models

One-equation models use an additional transport equation to determine . The most common equation model comes from Spalart and Allmaras, which introduces an additional transport equation for the auxiliary variable based on the turbulent viscosity . Except in the vicinity of the wall, this corresponds to the turbulent viscosity : ${\ displaystyle \ nu _ {t}}$ ${\ displaystyle {\ tilde {\ nu}}}$ ${\ displaystyle {\ tilde {\ nu}}}$ ${\ displaystyle \ nu _ {t}}$ ${\ displaystyle \ rho {\ frac {\ partial {\ tilde {\ nu}}} {\ partial t}} + \ rho {\ frac {\ partial \ left ({\ tilde {\ nu}} u_ {i} \ right)} {\ partial x_ {i}}} = {\ frac {1} {\ sigma _ {\ tilde {\ nu}}}} \ left [{\ frac {\ partial} {\ partial x_ {j }}} \ left \ {\ left (\ mu + \ rho {\ tilde {\ nu}} \ right) {\ frac {\ partial {\ tilde {\ nu}}} {\ partial x_ {j}}} \ right \} + C_ {b2} \ rho \ left ({\ frac {\ partial {\ tilde {\ nu}}} {\ partial x_ {j}}} \ right) ^ {2} \ right] -C_ {\ omega 1} \ rho f _ {\ omega} \ left ({\ frac {\ tilde {\ nu}} {d}} \ right) ^ {2} + C_ {b1} \ rho {\ tilde {S} } {\ tilde {\ nu}}}$ The two terms behind the square brackets describe the turbulence destruction and the turbulence production. A disadvantage of this turbulence model is the inability to correctly predict rapid changes in the turbulent length dimension, such as those that occur when a boundary layer changes into a free shear layer.

#### Two equation models

Two-equation turbulence models are a closure approach that consists of the solution of two coupled transport equations. The models are differentiated based on the turbulence sizes used. Two large groups are e.g. B. the turbulence models and the turbulence models. ${\ displaystyle k {\ text {-}} \ varepsilon}$ ${\ displaystyle k {\ text {-}} \ omega}$ ##### Standard k-ε turbulence model

The turbulence model is a widely used two-equation model. It describes the development of the turbulent kinetic energy and the isotropic dissipation rate with two partial differential equations . The equations are: ${\ displaystyle k {\ text {-}} \ varepsilon}$ ${\ displaystyle k}$ ${\ displaystyle \ varepsilon = \ nu {\ overline {(\ partial u_ {i} '/ \ partial x_ {k}) (\ partial u_ {i}' / \ partial x_ {k})}}}$ ${\ displaystyle \ rho {\ frac {\ partial k} {\ partial t}} + \ rho {\ bar {u}} _ {j} {\ frac {\ partial k} {\ partial x_ {j}}} = C _ {\ mu} \ rho \ mu _ {t} \ left ({\ frac {\ partial {\ bar {u}} _ {i}} {\ partial x_ {j}}} + {\ frac {\ partial {\ bar {u}} _ {j}} {\ partial x_ {i}}} \ right) {\ frac {\ partial {\ bar {u}} _ {i}} {\ partial x_ {j} }} - \ rho \ varepsilon + {\ frac {\ partial} {\ partial x_ {j}}} \ left [\ left (\ mu + {\ frac {\ mu _ {t}} {\ sigma _ {k }}} \ right) {\ frac {\ partial k} {\ partial x_ {j}}} \ right]}$ and

${\ displaystyle \ rho {\ frac {\ partial \ varepsilon} {\ partial t}} + \ rho {\ bar {u}} _ {j} {\ frac {\ partial \ varepsilon} {\ partial x_ {j} }} = C _ {\ varepsilon 1} {\ frac {\ varepsilon} {k}} \ tau _ {ij} {\ frac {\ partial {\ bar {u}} _ {i}} {\ partial x_ {j }}} - C _ {\ varepsilon 2} {\ frac {\ varepsilon ^ {2}} {k}} C _ {\ mu} \ rho \ mu _ {t} \ left ({\ frac {\ partial {\ bar {u}} _ {i}} {\ partial x_ {j}}} + {\ frac {\ partial {\ bar {u}} _ {j}} {\ partial x_ {i}}} \ right) { \ frac {\ partial {\ bar {u}} _ {i}} {\ partial x_ {j}}} - C _ {\ varepsilon 2} \ rho {\ frac {\ varepsilon ^ {2}} {k}} + {\ frac {\ partial} {\ partial x_ {j}}} \ left [\ left (\ mu + {\ frac {\ mu _ {t}} {\ sigma _ {\ varepsilon}}} \ right) {\ frac {\ partial \ varepsilon} {\ partial x_ {j}}} \ right]}$ Some model assumptions, some of which are considerably simplified, have been incorporated into the above equations. This significantly limits the area of ​​validity and thus the area of ​​application. Unknown coefficients appear in the equations. These are determined by considering simple flow fields. The parameter is calibrated by a homogeneous shear in the equilibrium state. The size follows from the decay behavior of homogeneous lattice turbulence. The turbulent Prandtl number results from an analysis of the logarithmic range of a flat, turbulent wall boundary layer. The anisotropy results from a dimensional analysis of eddy viscosity: . It follows immediately . The consideration of a turbulent wall boundary layer then provides a value for . ${\ displaystyle C _ {\ varepsilon 1}}$ ${\ displaystyle C _ {\ varepsilon 2}}$ ${\ displaystyle \ sigma _ {\ varepsilon}}$ ${\ displaystyle C _ {\ mu}}$ ${\ displaystyle \ nu _ {t} \ sim k \ cdot (k / \ varepsilon)}$ ${\ displaystyle \ nu _ {t} = C _ {\ mu} (k ^ {2} / \ varepsilon)}$ ${\ displaystyle C _ {\ mu}}$ For the standard model one often finds in the literature: ${\ displaystyle k {\ text {-}} \ varepsilon}$ ${\ displaystyle {\ begin {matrix} C _ {\ mu} & = & 0 {,} 09 \\ C _ {\ varepsilon 1} & = & 1 {,} 44 \\ C _ {\ varepsilon 2} & = & 1 {,} 92 \\\ sigma _ {\ varepsilon} & = & 1 {,} 3 \\\ sigma _ {k} & = & 1 \\\ end {matrix}}}$ The way in which the constants are determined describes the flow fields in which the model should provide good agreement with measurements.

##### Nonlinear k-ε turbulence models

The standard model has some serious disadvantages. The normal stresses are calculated using the Boussinesq approximation of Reynolds' stress tensor in all spatial directions with the same size. This must not be confused with the classical definition of the isotropy of a second order tensor (Reynolds stress tensor). Isotropy would make Reynolds' shear stresses disappear ( isotropic turbulence ), but this is not automatically mapped by the Boussinesq approximation , which is part of the k-ε turbulence model. This means, however, that flow fields in which the velocity vector is largely influenced by normal stresses can only be mapped imprecisely. This is the case in separation areas, recirculation areas and secondary flows . An extension of the Boussinesq approximation offers a way out. This introduces additional nonlinear terms into the model equations that are nonlinear in the mean velocity gradient. These non-linear terms allow a more precise calculation of the normal stresses. ${\ displaystyle k {\ text {-}} \ varepsilon}$ ##### V2F turbulence model

The turbulence near walls is characterized by inhomogeneity and anisotropy. The two- equation models , such as and , use the assumption of homogeneous, isotropic turbulence near the wall. Damping functions are inserted into these models to correct these false assumptions. Damping functions are designed in such a way that certain solutions can be reproduced by the model. In other cases, incorrect predictions are made. The V2F turbulence model is an extension of the turbulence model. In addition to the transport equations for the turbulent kinetic energy and the dissipation rate, an equation for the velocity measure normal to the wall and its production rate normalized with it are solved. The equations for and are identical to those of the Standard Model . ${\ displaystyle k {\ text {-}} \ varepsilon}$ ${\ displaystyle k {\ text {-}} \ omega}$ ${\ displaystyle k {\ text {-}} \ varepsilon}$ ${\ displaystyle {\ overline {v '^ {2}}}}$ ${\ displaystyle k}$ ${\ displaystyle f}$ ${\ displaystyle k}$ ${\ displaystyle \ varepsilon}$ ${\ displaystyle k {\ text {-}} \ varepsilon}$ ${\ displaystyle \ rho {\ frac {\ partial k} {\ partial t}} + \ rho {\ bar {u}} _ {j} {\ frac {\ partial k} {\ partial x_ {j}}} = C _ {\ mu} \ rho \ mu _ {t} \ left ({\ frac {\ partial {\ bar {u}} _ {i}} {\ partial x_ {j}}} + {\ frac {\ partial {\ bar {u}} _ {j}} {\ partial x_ {i}}} \ right) - \ rho \ varepsilon + {\ frac {\ partial} {\ partial x_ {j}}} \ left [ \ left (\ mu + {\ frac {\ mu _ {t}} {\ sigma _ {k}}} \ right) {\ frac {\ partial k} {\ partial x_ {j}}} \ right]}$ and

${\ displaystyle \ rho {\ frac {\ partial \ varepsilon} {\ partial t}} + \ rho {\ bar {u}} _ {j} {\ frac {\ partial \ varepsilon} {\ partial x_ {j} }} = C _ {\ varepsilon 1} {\ frac {\ varepsilon} {k}} \ tau _ {ij} {\ frac {\ partial {\ bar {u}} _ {i}} {\ partial x_ {j }}} - C _ {\ varepsilon 2} {\ frac {\ varepsilon ^ {2}} {k}} C _ {\ mu} \ rho \ mu _ {t} \ left ({\ frac {\ partial {\ bar {u}} _ {i}} {\ partial x_ {j}}} + {\ frac {\ partial {\ bar {u}} _ {j}} {\ partial x_ {i}}} \ right) { \ frac {\ partial {\ bar {u}} _ {i}} {\ partial x_ {j}}} - C _ {\ varepsilon 2} \ rho {\ frac {\ varepsilon ^ {2}} {k}} + {\ frac {\ partial} {\ partial x_ {j}}} \ left [\ left (\ mu + {\ frac {\ mu _ {t}} {\ sigma _ {\ varepsilon}}} \ right) {\ frac {\ partial \ varepsilon} {\ partial x_ {j}}} \ right]}$ For the normal wall velocity the additional equation

${\ displaystyle {\ frac {\ partial {\ overline {v '^ {2}}}} {\ partial t}} + u \ cdot \ nabla {\ overline {v' ^ {2}}} = kf_ {22 } - {\ overline {v '^ {2}}} {\ frac {\ varepsilon} {k}} + \ nabla \ left [\ left (\ nu + {\ frac {\ nu _ {t}} {\ sigma _ {k}}} \ right) \ nabla {\ overline {v '^ {2}}} \ right]}$ formulated. The term represents the source for and can be interpreted as a redistribution of turbulence intensity from the component parallel to the flow. The non-local effects are represented mathematically by an elliptic relaxation equation for : ${\ displaystyle kf_ {22}}$ ${\ displaystyle {\ overline {v '^ {2}}}}$ ${\ displaystyle f_ {22}}$ ${\ displaystyle L ^ {2} \ nabla ^ {2} f_ {22} -f_ {22} = \ left (1-C_ {1} \ right) {\ frac {\ left [{\ frac {2} { 3}} - {\ frac {\ overline {v '^ {2}}} {k}} \ right]} {T}} - {\ frac {C_ {2}} {k}} \ nu _ {t } \ left ({\ frac {\ partial u_ {j}} {\ partial x_ {i}}} + {\ frac {\ partial u_ {i}} {\ partial x_ {j}}} \ right) {\ frac {\ partial u_ {j}} {\ partial x_ {i}}}}$ The length and time dimensions occurring in the model are:

${\ displaystyle L = C_ {L} l}$ With

${\ displaystyle l ^ {2} = \ max \ left [{\ frac {k ^ {3}} {\ varepsilon ^ {2}}}, c _ {\ eta} ^ {2} \ left ({\ frac { \ nu ^ {3}} {\ varepsilon}} \ right) ^ {\ frac {1} {2}} \ right]}$ and

${\ displaystyle T = \ max \ left [{\ frac {k} {\ varepsilon}}, 6 \ left ({\ frac {\ nu} {\ varepsilon}} \ right) ^ {\ frac {1} {2 }} \ right]}$ The coefficient in the expression for was determined with the help of direct numerical simulation. The eddy viscosity is given by: ${\ displaystyle 6}$ ${\ displaystyle T}$ ${\ displaystyle \ nu _ {t} = C _ {\ mu} {\ overline {v '^ {2}}} T}$ .

According to the literature , the model constant should lie between , far away from the wall, and , in an adjacent boundary layer, depending on the distance to the wall . becomes with the equation ${\ displaystyle C _ {\ varepsilon 1}}$ ${\ displaystyle 1 {,} 3}$ ${\ displaystyle 1 {,} 55}$ ${\ displaystyle C _ {\ varepsilon 1}}$ ${\ displaystyle C _ {\ varepsilon 1} = 1 {,} 3+ \ left [{\ frac {0 {,} 25} {1+ \ left ({\ frac {d} {2l}} \ right) ^ { 8}}} \ right]}$ interpolated between these two values. The other model constants are given by:

${\ displaystyle {\ begin {matrix} C _ {\ mu} & = & 0 {,} 19 \\ C _ {\ varepsilon 2} & = & 1 {,} 9 \\ C_ {1} & = & 1 {,} 4 \ \ C_ {2} & = & 0 {,} 3 \\ C_ {L} & = & 0 {,} 3 \\ C _ {\ eta} & = & 70 {,} 0 \\\ sigma _ {\ varepsilon} & = & 1 {,} 3 \\\ sigma _ {k} & = & 1 {,} 0 \\\ end {matrix}}}$ ##### k-ω turbulence model

Another popular two-equation turbulence model is the Wilcox model. A transport equation for and a transport equation for the characteristic frequency ,, of the energy-dissipating vortices are solved here. According to Wilcox, the transport equation for : ${\ displaystyle k {\ text {-}} \ omega}$ ${\ displaystyle k}$ ${\ displaystyle \ omega = {\ frac {1} {C _ {\ mu}}} {\ frac {\ varepsilon} {k}}}$ ${\ displaystyle k}$ ${\ displaystyle \ rho {\ frac {\ partial k} {\ partial t}} + \ rho {\ bar {u}} _ {j} {\ frac {\ partial k} {\ partial x_ {j}}} = C _ {\ mu} \ rho \ mu _ {t} \ left ({\ frac {\ partial {\ bar {u}} _ {i}} {\ partial x_ {j}}} + {\ frac {\ partial {\ bar {u}} _ {j}} {\ partial x_ {i}}} \ right) {\ frac {\ partial {\ bar {u}} _ {i}} {\ partial x_ {j} }} - \ beta ^ {*} \ rho k \ omega + {\ frac {\ partial} {\ partial x_ {j}}} \ left [\ left (\ mu + \ sigma ^ {*} \ mu _ { t} \ right) {\ frac {\ partial k} {\ partial x_ {j}}} \ right]}$ ${\ displaystyle \ rho {\ frac {\ partial \ omega} {\ partial t}} + \ rho {\ bar {u}} _ {j} {\ frac {\ partial \ omega} {\ partial x_ {j} }} = \ alpha {\ frac {\ omega} {k}} C _ {\ mu} \ rho \ mu _ {t} \ left ({\ frac {\ partial {\ bar {u}} _ {i}} {\ partial x_ {j}}} + {\ frac {\ partial {\ bar {u}} _ {j}} {\ partial x_ {i}}} \ right) {\ frac {\ partial {\ bar { u}} _ {i}} {\ partial x_ {j}}} - \ beta \ rho \ omega ^ {2} + {\ frac {\ partial} {\ partial x_ {j}}} \ left [\ left (\ mu + \ sigma \ mu _ {t} \ right) {\ frac {\ partial \ omega} {\ partial x_ {j}}} \ right]}$ ${\ displaystyle \ beta ^ {*}}$ corresponds to that of the models. The constants for closing the system were determined in a manner analogous to the model and are given by Wilcox as: ${\ displaystyle C _ {\ mu}}$ ${\ displaystyle k {\ text {-}} \ varepsilon}$ ${\ displaystyle k {\ text {-}} \ varepsilon}$ ${\ displaystyle \ alpha = {\ frac {5} {9}}, \ beta = {\ frac {3} {40}}, \ beta ^ {*} = {\ frac {9} {100}}, \ sigma = {\ frac {1} {2}}, \ sigma ^ {*} = {\ frac {1} {2}}}$ The model automatically reduces the turbulent length near the wall. Another advantage is the robust formulation of the viscous underlayer. The disadvantage is the dependence of the calculated boundary layer edge on the free flow condition for which is specified by the user. This behavior is referred to in the literature as "free stream" sensitivity. ${\ displaystyle k {\ text {-}} \ omega}$ ${\ displaystyle L = {\ frac {k ^ {\ frac {1} {2}}} {\ omega}}}$ ${\ displaystyle \ omega}$ ##### k-ω-SST turbulence model

The model offers advantages in areas of the flow field close to the wall, whereas the model provides good results in areas remote from the wall. The SST turbulence model developed by Menter combines the advantages of these two models. ${\ displaystyle k {\ text {-}} \ omega}$ ${\ displaystyle k {\ text {-}} \ varepsilon}$ If additional phenomena ( combustion , particles , droplets , supersonic etc.) occur in the flow , the associated variables (e.g. density , temperature , mass fractions etc.) must also be averaged. Analog closure problems arise in the associated transport equations.

## Large eddy simulation

Instead of time averaging, the Large Eddy Simulation uses time and space low-pass filtering. This has the consequence that the large-scale phenomena are simulated transiently, while the contribution of the small-scale phenomena must still be modeled.

Although related modeling problems arise, the LES promises a better description of the turbulence than the statistical methods when the computational effort is greater, because at least some of the turbulent fluctuations are reproduced.

## Detached Eddy Simulation

The Detached Eddy Simulation (DES) was first published in 1997 by P. Spalart. In its original form, it is based on the Spalart-Allmaras turbulence model (a transport equation), but research is also being carried out into its application in conjunction with other models.

The DES replaces the wall distance, which occurs as a variable in the Spalart-Allmaras model, in areas far from the wall by the largest width of a grid cell. With this formulation, an LES-like behavior of the calculation can be achieved in the areas remote from the wall. In fact, you get a RANS formulation in the boundary layer and an LES formulation in the free flow, i.e. the most suitable method in the respective area (in terms of accuracy and computational effort).

Since RANS and LES have different requirements for the grid, the creation of a suitable grid divided into appropriate zones has a major influence on the success of the calculation. The same applies to the numerical methods used. However, these are mostly necessarily the same in the entire computation area, which sometimes leads to compromises in terms of accuracy.

## Individual evidence

1. For example in: JH Ferziger, M. Perić: Computational Methods for Fluid Dynamic , 3rd Edition, 2002
2. Florian Menter: Improved Two-Equation k-omega Turbulence Models for Aerodynamic Flows In: NASA Technical Memorandum 103975, 1992