Internal Seminar Series 2011, 2011-03-02
C18, 15:30
School of MACE, The University of Manchester



The new DF-SEM to generate inlet condition for embedded LES

Ruggero Poletto

  ruggero.poletto@postgrad.manchester.ac.uk  

Files: Abstract: pdf Presentation: ppt

Abstract



Jarrin et al. (2009) developed the Synthetic Eddy Method (SEM) as a quasi-particle based method to generate synthetic turbulence conditions for embedded LES. The method essentially involves the superposition of a (large) number of random eddies, which are convected through a domain of rectangular cross-section. The resultant, time-dependent, flow-field from a cross-section of this SEM domain is extracted and imposed as inlet conditions for the LES. Using this approach Jarrin et al. (2009) found that LES of a channel flow at $Re_{\tau}=395$ required a distance of around 10--12 channel half-widths to become fully-developed. Some further improvements were achieved by Pamies et al. (2009), by specifically tuning the shape functions associated with the eddy representations for a channel flow. Although they did report a decrease in the required development length, the form adopted would appear to be rather specific to the application.

One of the perceived weaknesses of the above SEM methods is that the imposed inlet flow-field does not satisfy the divergence-free condition. As a consequence of this the LES tends to introduce significant pressure fluctuations close to the inlet (in order to adapt the velocity field to something that does satisfy continuity), and this adds to the required development length. In the present work, we therefore explore a method of extending the SEM approach in order to produce a suitable inlet velocity field that does satisfy continuity.

Numerical methods

In the SEM developed by Jarrin et al. (2006) and Jarrin et al. (2009), the fluctuating velocity field is generated by taking

  \begin{equation*} \mathbf{u'(\mathbf{x})}=\frac{1}{\sqrt{N}}\sum_{k=1}^{N}a_{ij}\varepsilon_{j}^{k}f_{\sigma}(\mathbf{x}-\mathbf{x^{k}}) \end{equation*} (1)

where $N$ is the number of eddies introduced into the SEM domain, $\mathbf{x}_{k}$ is the location of the centre of the $k^{th}$ eddy, $f_{\sigma}(\mathbf{x})$ is a suitable shape function, $\varepsilon_{j}^{k}$ are random numbers and $a_{ij}$ are coefficients. Although this formulation does allow the desired Reynolds stress field to be prescribed (via the $a_{ij}$ coefficients), the velocity field will not, in general, also satisfy continuity.

The above formulation, applied to the vorticity and solved ($\nabla^{2}\mathbf{u}=\nabla\cdot\mathbf{u}-\nabla\times\boldsymbol{\omega}$), leads, for a divergence free velocity field, to the following result:

  \begin{equation*} \mathbf{u'}(\mathbf{x})=\sqrt{\frac{1}{N}}\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 the vector $\mathbf{K}_{\sigma}(\mathbf{y})$ is the Biot-Savart kernel, defined as $\mathbf{K}_{\sigma}(\mathbf{y})=\frac{q_{\sigma}(|\mathbf{y}|)}{|\mathbf{y}|^{3}}\mathbf{y}$ with $q_{\sigma}(|\mathbf{y}|)$ a suitable shape function. $R_{L}^{G}$ is a transformation matrix to rotate the desired Reynolds stress tensor to its principle axes (essentially, therefore, consisting of the three eigenvectors of the Reynolds stress tensor). Finally, the vector $(\boldsymbol{\alpha}^{k})^{L}$ is taken as having components $\alpha_{i}^{k}=\{C_{i}\varepsilon_{i}\}^{k}$, where the $\varepsilon_{i}$ are random numbers. If the coefficients $C_{i}$ are taken as $C_{i}=\sqrt{\lambda_{max}-2\lambda_{i}}$, where $\lambda$ are the eigenvalues of the Reynolds stress tensor, then it can be shown that the form of Eq. (2) does return the desired Reynolds stress statistics. As a result of the form of Eq. (2), with certain conditions placed on the form of the shape function $q_{\sigma}$, the resulting velocity field can also be shown to satisfy the divergence-free condition.

In the present work the shape function $q_{\sigma}$ has been taken as

  \begin{equation*} q_{\sigma}(y)=\sqrt{\frac{16V_{B}}{15\pi\sigma^{3}}}\left(\sin(\pi y)\right)^{2}y \end{equation*} (3)

where $V_{B}$ and $\sigma$ are prescribed eddy velocity and lengthscales respectively.

Results

The method here described improves the performances of many other generative methods. Compared to some of the most well-known and used methods in a Channel Flow simulations ($Re_{\tau}=395$, inlet data from \citet{moser_direct_1999}) it showed a shorter influence to the calculated $C_{f}$ (see Fig. 1).

Figure 1: $C_{f}$ performed in Channel Flows, $Re_{\tau}=395$, using different inlet methods



Figure 2: Pressure fluctuations

Regardless this positive result, a further problem rose: because of its stochastic nature, the method prescribes a constant average mass flow rate, but istanteneously it might vary whitin a range. This variation causes a gradient pressure fluctuations (Fig. 2). In fact, Navier-Stokes equations links the pressure gradient to the velocity time variations which is, in this case, imposed by the inlet method. A modification is then necessary: a mass flow rescalation is imposed at every time step in order to keep it constant. This modification slightly affects the divergence free condition of the method (the rescaling coefficient differs from 1 by less then 1% - Fig. 3) but highly decreases the pressure gradient fluctuations (Fig. 4).



Figure 3: Rescaling coefficient used in the modified version of the DF-SEM



Figure 4: Pressure fluctuations after mass flow rescaling


References



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pdfpdf RPoletto_presentation.pdf manage 479.7 K 2011-03-02 - 12:17 StefanoRolfo  
pdfpdf Ruggero_Poletto.pdf manage 328.5 K 2011-02-10 - 08:07 StefanoRolfo Abstract RPoletto
pngpng cf_comparison.png manage 8.8 K 2011-02-10 - 11:35 StefanoRolfo Figures Ruggero 04
pngpng press_rms375.6.png manage 10.7 K 2011-02-10 - 11:34 StefanoRolfo Figures Ruggero 03
gifgif pressure_fluctuation_video.gif manage 880.9 K 2011-03-02 - 12:18 StefanoRolfo  
pngpng pressure_fluctuations_along_channel.png manage 6.9 K 2011-02-10 - 11:36 StefanoRolfo Figures Ruggero 05
pngpng pressure_iteration_comparison_rescaledDFSEM.png manage 7.2 K 2011-02-10 - 11:34 StefanoRolfo Figures Ruggero 02
pngpng pressureiteration.png manage 7.3 K 2011-02-10 - 11:36 StefanoRolfo Figures Ruggero 06
pngpng rescalingcoeff.png manage 5.1 K 2011-02-10 - 11:33 StefanoRolfo Figures Ruggero 01
Topic revision: r7 - 2011-03-02 - 12:19:23 - StefanoRolfo
 

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