# Synthetic Eddy Method

## Motivations

This research was motivated by the growing interest of the engineering community in LES and hybrid RANS-LES methods, and the lack of a cost effective, robust and accurate method of generation of inflow data for LES.

The Synthetic Eddy Method (SEM) is a stochastic algorithm that generates instantaneous velocity fluctuations with prescribed mean velocity, Reynolds stresses, length and time scales distributions. The method is based on the classical view of turbulence as a superposition of eddies. The velocity signal is thus expressed as a sum of synthetic eddies with random position and intensity. The characteristics of the synthetic eddies are calculated from input statistical quantities typically available from a RANS solution, and determine the characteristics of the synthetized signal.

A brief description of the method as it is implement in its latest version is provided below, followed by an summary of the main results obtained in the validation process of the method. Compact implementations (in F77) of the SEM for initialization of flow field or generation of inlet conditions for LES can be downloaded at the bottom of this page.

Fig. 1: Zonal RANS-LES coupling of the flow over an airfoil trailing edge using the SEM at the RANS to LES interface

## Description of the method

We begin by taking a finite set %$S \subset \mathbb{R}^3$% of points %$S=\left\lbrace \mathbf{x}_1, \mathbf{x}_2,\cdots, \mathbf{x}_{s} \right\rbrace$% on which we want to generate synthetic velocity fluctuations with the SEM. We assume for now that the mean velocity %$\mathbf{U}$%, the Reynolds stresses %$R_{ij}$% and a characteristic length scale of the flow %$\sigma$% are available for the set of points considered.

###### Definition of the box of eddies

The first step is to create a box of eddies %$B$% which is going to contain the synthetic eddies. It is defined by

%BEGINLATEX{label="eq:boxsem1"}% \begin{equation*} B=\{ (x_1,x_2,x_3)\in \mathbb{R}^3:\ \ \ x_{i,\min} < x_i < x_{i,\max}, i=\{1,2,3\} \} \end{equation*} %ENDLATEX%

where

%BEGINLATEX{label="eq:boxsem2"}% \begin{equation*} x_{i,\min} = \min_{\mathbf{x}\in S} (x_i-\sigma(\mathbf{x})) \quad \text{ and } \quad x_{i,\max} = \max_{\mathbf{x}\in S} (x_i+\sigma(\mathbf{x})) \end{equation*} %ENDLATEX%

The volume of the box of eddies is noted %$V_B$%.

###### Computation of the velocity signal

In the synthetic eddy method, the velocity fluctuations generated by %$N$% eddies have the representation

%BEGINLATEX{label="eq:basesem1"}% \begin{equation*} u_i = U_i + \frac{1}{\sqrt{N}} \sum_{k=1}^{N} a_{ij} \varepsilon^k_j f_{\sigma(\mathbf{x})}(\mathbf{x}-\mathbf{x}^k). \end{equation*} %ENDLATEX%

where the %$\mathbf{x}^k=(x^k,y^k,z^k)$% are the locations of the %$N$% eddies, the %$\varepsilon^k_j$% are their respective intensities and %$a_{ij}$% is the Cholesky decomposition of the Reynolds stress tensor

%BEGINLATEX{label="eq:cholsem"}% \begin{eqnarray*} \left( \begin{array}{ccc} \sqrt{R_{11}} & 0 & 0 \\ R_{21} / a_{11} & \sqrt{R_{22}-a_{21}^2} & 0 \\ R_{31} / a_{11} & (R_{32}-a_{21}a_{31}) / a_{22} & \sqrt{R_{33}-a_{31}^2-a_{32}^2} \\ \end{array} \right) , \end{eqnarray*} %ENDLATEX%

%$f_{\sigma(\mathbf{x})}(\mathbf{x}-\mathbf{x}^k)$% is the velocity distribution of the eddy located at %$\mathbf{x}^k$%. We assume that the differences in the distributions between the eddies depend only on the length scale %$\sigma$% and define %$f_\sigma$% by

%BEGINLATEX{label="eq:defdistrosem"}% \begin{equation*} f_\sigma (\mathbf{x}-\mathbf{x}_k) = \sqrt{V_B}\ \sigma^{-3}\ f(\frac{x-x^k}{\sigma}) \ f(\frac{y-y^k}{\sigma}) \ f(\frac{z-z^k}{\sigma}) \end{equation*} %ENDLATEX%

where the shape function %$f$% is common to all eddies. %$f$% has compact support %$[-\sigma,\sigma]$% and has the normalization %$\| f \|_2 = 1$%. In the subroutine available at the bottom of the page, the shape function is taken as a tent function, %$f(x) = \sqrt{\frac{3}{2}} \ \left( 1-|x|\right)$% , if %$\quad x<1$% and %$0$% otherwise.

The position of the eddies %$\mathbf{x}^k$% before the first time step are independent from each other and taken from a uniform distribution %$U(B)$% over the box of eddies %$B$% and %$\varepsilon^k_j$% are independent random variables taken from any distribution with zero mean and unite variance. In all simulations carried out in this thesis we choose %$\varepsilon^k_j \in \left\lbrace -1, 1 \right\rbrace$% with equal probability to take one value or the other. We choose this distribution because it has a lower flatness than any other distribution. The advantages of this argument will become clearer later when some exact results concerning the intermittency of the signal are established.

###### Convection of the population of eddies

The eddies are convected through the box of eddies %$B$% with a constant velocity %$\mathbf{U}_c$% characteristic of the flow. In our case it is straight forward to compute %$\mathbf{U}_c$% as the averaged mean velocity over the set of points %$S$%. At each iteration, the new position of eddy %$k$% is given by

%BEGINLATEX{label="convsem"}% \begin{equation*} \mathbf{x}^k(t+dt) = \mathbf{x}^k(t) + \mathbf{U}_c dt. \end{equation*} %ENDLATEX%

where %$dt$% is the time step of the simulation. If an eddy %$k$% is convected out of the box through face %$F$% of %$B$%, then it is immediately regenerated randomly on the inlet face of %$B$% facing %$F$% with a new independent random intensity vector %$\mathbf{\varepsilon}^k_j$% still taken from the same distribution.

## References

A Synthetic-Eddy Method for Generating Inflow Conditions for LES, Jarrin, N., Benhamadouche, S., Laurence, D. and Prosser, R., International Journal of Heat and Fluid Flow, Vol. 27, pp. 585-593, (2006) Download

Reconstruction of Turbulent Fluctuations for Hybrid RANS/LES Simulations Using a Synthetic-Eddy Method, Jarrin, N., Revell, A., Prosser, R. and Laurence, D., to appear in the Proceedings of the 7th International ERCOFTAC Symposium on Engineering Turbulence Modelling and Measurements (ETMM7), Limassol, Cyprus, 4-6 June, (2008) Download

Synthetic Inflow Boundary Conditions for the Numerical Simulation of Turbulence, Jarrin, N., PhD Thesis (2008) Download

Here are some compact implementations of the SEM (in F77) to be downloaded. More precisely, "semoutsidecs.tar" is a directory with fortrans of the SEM to generate inlet conditions for LES on any kind of inlet mesh. For any enquiry about the SEM or the fortrans provided and its usage, please contact Nicolas Jarrin or Ruggero Poletto at the following addresses:

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tar semoutsidecs.tar manage 1670.0 K 2008-02-08 - 10:19 NicolasJarrin SEM for LES inflow data in F77 (standalone fortran)
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Topic revision: r20 - 2012-10-17 - 17:51:46 - DominiqueLaurence
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