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TWiki> CfdTm Web>InternalSeminars>InternalSeminar004 (2011-03-16, StefanoRolfo)

Internal Seminar Series 2011, 2011-03-16

C18, 15:30

School of MACE, The University of Manchester

C18, 15:30

School of MACE, The University of Manchester

dalila.ammour@postgrad.manchester.ac.uk

**Files:** Abstract: Presentation:

The reliable computation of buoyant flows is important in a number
of engineering sectors, including the nuclear one. Here, turbulent
natural convection of air in both two dimensional vertical and inclined
rectangular cavities are investigated numerically by means of an unstructured
finite volume code (*Code_Saturne*) for an aspect ratio of %$H/L=28.6$%, Rayleigh
number of %$0.86\cdot10^{6}$%. The two opposing long walls are maintained
at uniform and different temperatures. The flow within the cavity
is computed, first, using several Unsteady-Reynolds-Averaged-Navier-Stokes
(URANS) models, high-Reynolds-number models, such as the Standard
%$k-\varepsilon$% and Reynolds Stress model %$R_{ij}-\varepsilon\, SSG$%
and low-Reynolds-number models, like the Shear Stress Transport %$k-\omega SST$%
and the %$\upsilon^{2}-f$% model. The results obtained are compared
to two recent experimental data \citep{Betts2000,Craft2009}. The
cases of vertical and inclined cavity at %$60^{o}$%, heated from the
upper side, give similar results of temperature and velocity fields
and most of the models are in good agreement with the experiments.
On the other hand, once the 2-D cavity is inclined at %$15^{o}$% to
the horizontal, heated from the lower side, under unstable stratification,
the majority of RANS models over-predict velocity and temperature
because of the three-dimensional nature of the flow with the presence
of four longitudinal vortices. The 3D RANS simulation is then adopted
for accurate prediction of flow details and wall heat transfer. Now,
High-Reynolds-number models seem to capture the flow pattern, however,
low-Reynolds-number models disagree with measurements and they tend
to capture only one recirculation cell. Finally, LES prediction
is also presented in order to provide a validation data-set to the
previous URANS computation.
### Numerical methods

The present computational study is carried out using a finite volume
*Code_Saturne*. In natural convection, the flow fluid arises naturally from
the effect of density differences, resulting from variation in temperature
or concentration in a force field such as gravity. It means that the
incompressible Navier-Stokes equations in natural convection cases
can be written as:
### Results

In Fig. %REFLATEX{fig:fig1}%, time-averaged
temperature normalised by the temperature difference %$\triangle T$%,
obtained from the LES computational study, is compared to experimental
data of \citet{Craft2009}. Time-averaged temperature plotted in the
central plane of the cavity at different heights, is fairly well predicted
by LES. When the mixing of the thermal field is strong the convective
effects become higher in the flow which make the temperature in the
core of the cavity isothermal and invariant across the height of the
cavity. The cavity herein is inclined at %$15^{0}$% to the horizontal
and heated from the lower side, according to the qualitative results
shown in Fig. %REFLATEX{fig:fig1}%, the temperature
distribution correspond with those of Benard convection, which indicates
a two boundary layer flows along the hot and cold walls.

%BEGINFIGURE{label="fig:fig1" caption="Time-averaged temperature profiles at two different heights inside the cavity (inclination of the cavity %$15^{0}$%)" span="twocolumn"}% %ENDFIGURE%

%BEGINLATEX{label="eq1"}% \begin{equation*} \rho\frac{\partial U_{i}}{\partial t}+U_{j}\rho\frac{\partial U_{i}}{\partial x_{j}}=-\frac{\partial P}{\partial x_{i}}-\frac{\partial}{\partial x_{j}}\left(\mu\frac{\partial U_{i}}{\partial x_{j}}-\rho\overline{u_{i}u_{j}}\right)+\rho g_{i} \end{equation*} %ENDLATEX%

The density changes only in the form of a gravity force $\rho g_{i}$. The changes in density can be related to changes in temperature via the thermal expansion coefficient $\beta$ defined as:

%BEGINLATEX{label="eq2"}% \begin{equation*} \beta=-\frac{1}{\rho}\frac{\partial\rho}{\partial T} \end{equation*} %ENDLATEX%

In order to use the eddy viscosity models in buoyant cases, the production of kinetic energy must be altered by adding a gravity term. That is:

%BEGINLATEX{label="eq3"}% \begin{equation*} P=\mu_{t}\frac{\partial U_{i}}{\partial x_{j}}\left(\frac{\partial U_{i}}{\partial x_{j}}+\frac{\partial U_{j}}{\partial x_{i}}\right)+G \end{equation*} %ENDLATEX%

where

%BEGINLATEX{label="eq4"}% \begin{equation*} G=\rho\beta g_{i}(T-T_{o}) \end{equation*} %ENDLATEX%

Inclination of the cavity means that the gravity vector is divided into two components along the two directions %$x$% and %$y$%. The components of gravity are:

%BEGINLATEX{label="eq5"}% \begin{equation*} g_{x}=g\cos\left(\alpha\right) \end{equation*} %ENDLATEX%

%BEGINLATEX{label="eq6"}% \begin{equation*} g_{y}=-g\sin\left(\alpha\right) \end{equation*} %ENDLATEX%

%$\alpha$% is the angle of inclination of the cavity.

%BEGINFIGURE{label="fig:fig1" caption="Time-averaged temperature profiles at two different heights inside the cavity (inclination of the cavity %$15^{0}$%)" span="twocolumn"}% %ENDFIGURE%

%BEGINFIGURE{label="fig:fig2" caption="Iso-Q contours coloured by the
temperature (inclination of the cavity %$15^{0}$%)" span="twocolumn"}%
%ENDFIGURE%

In order to identify coherent vortices and distinguish them from the mean shear, the structure parameter %$Q$% is calculated and plotted in Fig. %REFLATEX{fig:fig2}%. Using this parameter it is possible to observe the turbulent structures in the core of the heated cavity.

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

DAmmour_Abstract.pdf | manage | 440.7 K | 2011-03-02 - 01:07 | StefanoRolfo | ||

jpg | Q-LES.jpg | manage | 168.3 K | 2011-03-02 - 01:04 | StefanoRolfo | |

ppt | pres_internal-seminars1.ppt | manage | 5740.5 K | 2011-03-16 - 14:13 | StefanoRolfo | |

jpg | t-les-poster.jpg | manage | 22.8 K | 2011-03-02 - 01:04 | StefanoRolfo |

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