RANS/LES coupling with the SEM method

Researcher: Ruggero Poletto

Supervisor(s): Dr. A. Revell, Dr. T. Craft
Sponsor: EDF
Start Date: September 2009 End Date: September 2012
Keywords: SEM, RANS, LES, synthetic turbulence, embedded LES

Overall Research Aim

This research aims to improve the SEM (synthetic eddy method) suggested by N. Jarrin (2009) in order to decrease the pressure fluctuations and the developing length noticed.

Research Progress

STEP 1 - Introduction

SEM method, created and implemented by N. Jarrin (2009), is one way used to create inlet conditions for a LES directly from a RANS solution. This method is based on some eddies scattered in a box. The main equation of this method is (1), which defines the velocity as a sum of an averaged one, obtained from the RANS solution, and a fluctuating one, defined by the SEM method:


Embedded LES simulation

  \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*}(1)

The inlet is always followed by a transition zone, where the LES results are not so accurate and, in order to reduce it, a divergence free development of the SEM is here proposed. This new method must be:

  1. Computationally efficient.
  2. Easy to be implemented.
  3. Good approximation of a turbulent inlet velocity field.

STEP 2 - DFSEM: a divergence free method

The application of (1) to the vorticity field leads to a fluctuating vorticity field. This generates a divergence free velocity field, after solving the following equation: $ \nabla \times \boldsymbol{\omega} = \nabla (\nabla \cdot \mathbf{u}) - \nabla^2 \mathbf{u} $. The final solution, obtained using Green Function in order to solve the Poisson equation, is:

 \begin{equation*} \mathbf{u'}(\mathbf{x}) = \sqrt{\frac{V_b k}{N \sigma^3}} \sum_{k=1}^{N} \mathbf{K}_{\sigma}(\frac{\mathbf{x}-\mathbf{x}^k}{\sigma}) \times \{ R_L^G ( \boldsymbol{\alpha}^k)^L \} \end{equation*}(2)

where:

  • $\mathbf{K}_{\sigma}(\frac{\mathbf{x}-\mathbf{x}^k}{\sigma}) = \frac{q_\sigma(|\mathbf{y}|)}{|\mathbf{y}|^3}\mathbf{y} $ is the Biot-Savart kernel with $q_\sigma(|\mathbf{y}|)$ new shape function of the DFSEM;
  • $ \alpha_i^k = \{ C_i \varepsilon_i \}^k$ is the intensity vector of the k-th eddy, where $ C_i = \sqrt{\lambda_k - 2 \lambda_i} $ and $ \varepsilon_i $ is a random number with $<\varepsilon_i>=0$ and $ <\varepsilon_i^2>=1 $. Furthermore $ \lambda_i $ is the eigenvalue of the Reynolds stress tensor at $ \mathbf{x}^k $ location;
  • $ R_L^G $ is the rotational matrix from the main reference system of the local Reynolds stress tensor ( L ) to the global reference system ( G ). It consists of three eigenvectors of the local Reynolds stress tensor.


Lumley triangle
Red line - Turbulence reproducibility limit of DF-SEM.
- DNS channel flow solution.

The main limit of the method comes from the condition over $ C_i $, where the square root argument must be positive, which is a mathematical restriction of the method.

STEP 3 - Some results

cf_comparison.pngcf_comparison.png
The $ C_f $ obtained with the DF-SEM reaches a fully developed value in a shorter length compared to many other generative method. Furthermore, although the computational time of the method does not affect appreciably the whole simulation costs, the pressure equation is solved more quickly than the previous SEM, because of the decreased pressure fluctuations imposed by the inlet itself.

STEP 4 - Implementation under Code_Saturne

NEW subroutines

memsyn.F -> memory management
ussyint.F -> Set-up subroutine where must be defined:
syntur.F -> subroutine which implements the synthetic turbulences
sytusu.F -> repository of some functions and subroutines used by syntur
varsem.h -> all the variables used in every subroutines

MODIFIED subroutines

caltri.F -> memory management call
tridim.F -> main method subroutines call
usini1.F -> method activation


All these subroutines are available in the DF-SEM subroutines page.



Last Modification: r18 - 2011-02-03 - 14:20:56 - RuggeroPoletto


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Topic revision: r18 - 2011-02-03 - 14:20:56 - RuggeroPoletto
 

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