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TWiki> CfdTm Web>InternalSeminars>InternalSeminar002 (2011-03-02, StefanoRolfo)

TWiki> CfdTm Web>InternalSeminars>InternalSeminar002 (2011-03-02, StefanoRolfo)

Internal Seminar Series 2011, 2011-03-02

C18, 15:30

School of MACE, The University of Manchester

C18, 15:30

School of MACE, The University of Manchester

ruggero.poletto@postgrad.manchester.ac.uk

**Files:** Abstract: Presentation:

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 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.
### Numerical methods

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

### 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 (, inlet data
from \citet{moser_direct_1999}) it showed a shorter influence to
the calculated (see Fig. 1).

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.

(1) |

where is the number of eddies introduced into the SEM domain, is the location of the centre of the eddy, is a suitable shape function, are random numbers and are coefficients. Although this formulation does allow the desired Reynolds stress field to be prescribed (via the coefficients), the velocity field will not, in general, also satisfy continuity.

The above formulation, applied to the vorticity and solved (), leads, for a divergence free velocity field, to the following result:

(2) |

where the vector is the Biot-Savart kernel, defined as with a suitable shape function. 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 is taken as having components , where the are random numbers. If the coefficients are taken as , where 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 , the resulting velocity field can also be shown to satisfy the divergence-free condition.

In the present work the shape function has been taken as

(3) |

where and are prescribed eddy velocity and lengthscales respectively.

Figure 1: performed in Channel Flows, , 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**

I | Attachment | Action | Size | Date | Who | Comment |
---|---|---|---|---|---|---|

RPoletto_presentation.pdf | manage | 479.7 K | 2011-03-02 - 12:17 | StefanoRolfo | ||

Ruggero_Poletto.pdf | manage | 328.5 K | 2011-02-10 - 08:07 | StefanoRolfo | Abstract RPoletto | |

png | cf_comparison.png | manage | 8.8 K | 2011-02-10 - 11:35 | StefanoRolfo | Figures Ruggero 04 |

png | press_rms375.6.png | manage | 10.7 K | 2011-02-10 - 11:34 | StefanoRolfo | Figures Ruggero 03 |

gif | pressure_fluctuation_video.gif | manage | 880.9 K | 2011-03-02 - 12:18 | StefanoRolfo | |

png | pressure_fluctuations_along_channel.png | manage | 6.9 K | 2011-02-10 - 11:36 | StefanoRolfo | Figures Ruggero 05 |

png | pressure_iteration_comparison_rescaledDFSEM.png | manage | 7.2 K | 2011-02-10 - 11:34 | StefanoRolfo | Figures Ruggero 02 |

png | pressureiteration.png | manage | 7.3 K | 2011-02-10 - 11:36 | StefanoRolfo | Figures Ruggero 06 |

png | 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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