Turbulence modeling

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

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

The DNS is limited due to their extreme computational effort for the foreseeable future to low (and thus for practical applications often less interesting ) Reynolds numbers. For this reason, different strategies to reduce the amount of computation can be used. The more assumptions are made here, the lower the computing time usually and the greater the uncertainty about the outcome.

Statistical Modeling

The currently most widely used modeling strategy is the statistical modeling (also: Reynolds Averaged Navier Stokes ( RANS ) ). The turbulent flow model in the simplest case as a time mean value and variance of the velocity and pressure. Repeating this model assumption in the Navier -Stokes equations and averages this time, the Reynolds -averaged Navier -Stokes equations arise. These include the Reynolds stress tensor

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 equation system is no longer closed. The closure is achieved by additional assumptions for the components of the Reynolds stress tensor in the form of equations. These additional equations are called turbulence model. Since turbulence is essentially not understood, these are usually based on heuristics. For the validation data are from experiments. A distinction is made between zero -, one-and two-equation models, as well as closure approaches second order.

Eddy viscosity models

The eddy viscosity models of the Reynolds stress tensor is approximated by the hypothesis of Boussinesq. Here, the Reynolds stresses in analogy to be treated to the induced molecular viscosity voltages:

The quantity is called turbulent eddy viscosity and describes the increase in viscosity due to turbulent fluctuation movements. Usually the molecular weight viscosity exceeds significantly. The root of the turbulent kinetic energy is a typical velocity measure the turbulent fluctuating motions Represents the symbol denotes the Kronecker delta. The term is a " turbulent pressure term ", which is necessary to apply the equation for normal voltages.

For dimension reasons, the turbulent eddy viscosity with a turbulent length dimension and a turbulent velocity measure can be expressed according to. Through the Boussinesq approach the closing of the Reynolds -averaged Navier -Stokes equations by determining the eddy viscosity and the associated therewith turbulent length and velocity measurements succeed.

The eddy viscosity models according to the number of independent variables turbulence, which are used to calculate and distinguished.

Zero equation models

Algebraic or zero equation models use the closure only algebraic relations. These include the Baldwin - Lomax model and the turbulence model according Cebeci and Smith.

Eingleichungsmodelle

Eingleichungsmodelle use an additional transport equation for the determination of. The most common dates of Spalart and Allmaras Eingleichungsmodell, which introduces an additional transport equation for the turbulent viscosity ajar to the auxiliary variable. Except near the wall is consistent with the turbulent viscosity match:

The two terms behind the square brackets describe the turbulence and destruction of the turbulence production. The disadvantage of this turbulence model, the inability rapid changes in the turbulent length dimension, as they occur during the transition of a boundary layer in a free shear layer to predict correctly

Two-equation models

Two-equation turbulence models are a closed approach, which consists of the solution of two coupled transport equations. One distinguishes the models based on the turbulence parameters used. Two large groups are, for example, the turbulence models and turbulence models.

Standard k- ε turbulence model

The turbulence model is a widely used two-equation model. It describes two partial differential equations, the development of the turbulent kinetic energy dissipation rate and the isotropic. The equations are:

And

In the equations above are some, incorporated in part considerably simplifying model assumptions. This limits the scope and thus the scope of a much. In the equations still appear unknown coefficients. These are calculated by considering a simple flow fields. The parameter is calibrated by a homogeneous shear in the equilibrium state. The size follows from the decay of homogeneous grid turbulence. The turbulent Prandtl number is derived from an analysis of the logarithmic region of a plane turbulent wall boundary layer. The anisotropy arises from a dimensional analysis of eddy viscosity. It immediately follows. The observation of a turbulent wall boundary layer then provides a value for.

For the standard model it is often found in the literature:

The method of determining the constants referred to the flow field, in which the model should provide good agreement with measurements.

Non-linear k- ε turbulence models

The Standard Model has some major drawbacks. The normal stresses are the same size calculated by the Boussinesq approximation of the Reynolds stress tensor in all spatial directions → isotropic turbulence. However, this means that flow fields, where the velocity vector is greatly affected by the normal stresses can be mapped inaccurately. This is the case in transfer areas Rezirkulationsbereichen and secondary flows. A way out is an extension of the Boussinesq approximation. This results in additional non- linear terms in the model equations are non-linear in the gradient of the mean velocity. These nonlinear terms allow a more accurate calculation of normal stresses.

V2F turbulence model

The turbulence in the vicinity of the walls is characterized by inhomogeneity and anisotropy. The two-equation models, such as and, close to the wall using the assumption of homogeneous, isotropic turbulence. Damping functions are inserted into these models in order to correct these false assumptions. Damping functions are designed such that a particular solution can be represented 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 dissipation and an equation for the speed measurement to be normal to the wall, and the production rate achieved by normalized. The equations for and are identical to those of the standard model.

And

For the wall- normal velocity measure the additional equation is

Formulated. The term represents the source and can be interpreted as re-distribution of the turbulence intensity of the flow of the parallel component. The non-local effects are mathematically represented by an elliptical Relaxationsgleichung for:

Appearing within the model length and time dimensions are:

With

And

Is the coefficient in the expression was determined by means of direct numerical simulation. The eddy viscosity is given by:

.

The model should be constant depending on the distance to the wall according to the literature between, far away from the wall, and, in an attached boundary layer lie. is the equation

Interpolating between these two values. The other model constants are given by:

K- ω turbulence model

Another widely used two-equation turbulence model is that of Wilcox specified model. There are solved here for a transport equation and a transport equation for the characteristic frequency, which energiedissipierenden vertebrae. After Wilcox is the transport equation for:

Corresponds to the models, the constants for the closure of the system were determined in a manner analogous to the model and are given by Wilcox with:

The model reduces the turbulent length dimension automatically near the wall. Another advantage is the robust formulation of the viscous sublayer. A disadvantage is the dependence of the calculated boundary layer edge of the Freiströmbedingung for which is specified by the user. This behavior is known in the literature as a " free stream" sensitivity.

K- ω -SST turbulence model

The model offers advantages in near-wall regions of the flow field, whereas the model in areas remote from the wall gives good results. The union of the advantages of these two models provides developed by Menter SST turbulence model.

Step into the flow additional phenomena (combustion, particles, drops, supersonic, etc. ), so must also the related variables ( eg, density, temperature, mass fractions, etc.) are averaged. In the associated transport equations these encounter analogous closure problems.

Large Eddy Simulation

Instead of the time-averaging their temporal and spatial low-pass filtering at the Large Eddy Simulation. As a result, the large-scale phenomena to be simulated transient, while the contribution of the small-scale phenomena must continue to be modeled.

Although related modeling problems, the LES promises at a higher computational cost, a better description of the turbulence as the statistical methods, because at least a part of the turbulent fluctuations is reproduced.

Detached Eddy Simulation

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

The DES replaced the wall distance, which occurs as a variable in the Spalart - Allmaras model, in areas far away from the wall by the greatest width of a grid cell. By this formulation, a LES -like behavior of the invoice can be achieved in the wall remote areas. In effect, one obtains as a RANS formulation in the boundary layer and LES formulation in the free stream, so that in each area the most appropriate method ( in terms of accuracy and computational effort ).

Since RANS and LES place different demands on the grid, creating a suitable, divided into zones corresponding lattice has a large impact on the success of the bill. The same applies to the use of numerical methods. However, these are usually forced in the entire computational domain the same, which partly leads to compromises in accuracy.