# Results for case Flow through a staggered tube bundle (FEDORA)

Code: Code_Saturne

Version: 2.0-beta2

Authors: D. Sénéchal

## Method and Numerical Options

All calculations have been run with a variable time-step, but constant in space.

## Models

Turbulence : %$k-\varepsilon$%, %$k-\varepsilon$% (LP), %$k-\omega$%, %$R_{ij}$% - %$\varepsilon$% (LRR), %$R_{ij}$% - %$\varepsilon$% (SSG)

## Mesh

The mesh used was created with the SIMAIL software. It is formed with hexahedra and no structured. This one allows a good discretization of the tubes surface. The figure 1 gives a good view of the mesh.

Figure 1: Mesh used for the different computations.

## Description of the results files

### No standard post-processing:

• For all the computation cases, time averages are activated to compute the velocity average as the kinetic energy of the unresolved structures. this energy is added to the "unresolved" energy contained in the model.

At first, we can compare the head loss computed during the crossing bundle with the one theoretical given by [1]. We recall below the expression for the tubes bundle staggered. The head loss is expressed as following in the case where %$\tfrac{S_1}{d} \ge 1$% and %$7 \le \tfrac{S_1 -d}{S_2 -d} \le 5.2$% (see the Figure 2 for the definitions of %$S_1$% and %$S_2$%):

%BEGINLATEX{label="pressure_delta}% $\Delta P \simeq \displaystyle \frac{\rho w^2_{\mbox{\small\sc moy}}}{2} \displaystyle \left[ 0.44\left( \frac{S_1 - d}{S_2 -d} +1\right)^2 Re^{-0.27}_{\mbox{\small\sc moy}}(n +1) \right]$ %ENDLATEX%

Figure 2: Definition of the different quantities.

The %$Re^{-0.27}_{\mbox{\small\sc moy}}$% is defined as %$\displaystyle Re^{-0.27}_{\mbox{\small\sc moy}}=\tfrac{w_{\mbox{\small\sc moy}}d}{\nu}$% where %$w_{\mbox{\small\sc moy}} = w_0\tfrac{S_1}{S_1-d}$% with %$w_0$% the bundle upstream velocity.
The results obtained with the different computations are given below inside the following table 1. The head loss is estimated on all the domain (between 0 and 180 mm) and the central part (between 45 and 135 mm). The results are globally the same order of magnitude and underestimate the head loss. The head loss between 45 and 135 mm is not quite good predicted, but this result is compared with the Idel'Cick 's formula, which assumes that the velocity strictly longitudinal at the entrance of the bundle. It is not exactly the case between 0 and 180 mm and not at all between 45 and 135 mm. It should be noted that the method used to compute the head loss differs from the 1.3 version (the mean pressure is here averaged along a line, assuming a constant mesh surface along this line which does not seem to be the case for the 45-135mm sections), this explain the differences.
Model version 0 - 180 mm 45 - 135 mm Error [0-180] Error [45 - 135]
Idel'Cik   3910 2445 0 %$\%$% 0%$\%$%
%$k-\varepsilon$% [v.1.3.1] 2905 1487 -26 %$\%$% -39%$\%$%
%$k-\varepsilon~\mbox{\small \sc pl}$% [v.1.3.1] 2923 1837 -25 %$\%$% -25%$\%$%
%$R_{ij}$% - %$\varepsilon~\mbox{\small \sc lrr}$% [v.1.3.1] 3432 1893 -12%$\%$% -22%$\%$%
%$R_{ij}$% - %$\varepsilon~\mbox{\small \sc ssg}$% [v.1.3.1] 3613 1817 -8%$\%$% -26%$\%$%
%$k-\omega$% [v.1.3.1] 2981 1664 -24 %$\%$% -32%$\%$%
%$k-\varepsilon$% [v.2.0-beta2] 2869 721 26.6 %$\%$% 70.5 %$\%$%
%$k-\varepsilon~\mbox{\small \sc pl}$% [v.2.0-beta2] 2770 789 -29.1 %$\%$% -67.7 %$\%$%
%$R_{ij}$% - %$\varepsilon~\mbox{\small \sc lrr}$% [v.2.0-beta2] 3370 976 -13.8 %$\%$% -60 %$\%$%
%$R_{ij}$% - %$\varepsilon~\mbox{\small \sc ssg}$% [v.2.0-beta2] 3611 1070 -7.6 %$\%$% -56.2 %$\%$%
%$k-\omega$% [v.2.0-beta2] 2779 760 -28.9 %$\%$% -68.9 %$\%$%
Table 1: head losses %$(Pa)$% obtained for different numerical computations.

### Computation/measure comparisons:

The Figures 3 to 8 show the mean velocities (U, V) and the mean turbulent kinetic energy (k) profiles for different planes x=constant which are compared to the experimental measurements. The plane locations are marked on the Figure 1.
The Figures 9 to 11 show the mean velocity fields (in streamwise and spanwise directions) and the turbulent kinetic energy for the different models tested.

Figure 3: Mean streamwise velocity profiles in the spanwise direction for different abscissas and three different turbulent models with 2-transport equations
(%$iturb20$%: k-%$\varepsilon$% standard model, %$iturb21$%: k-%$\varepsilon-(\mbox{\sc lp})$% model with linear production and %$iturb60$%: k-%$\omega$% model).

Figure 4: Mean streamwise velocity profiles in the spanwise direction for different abscissas and two second order
turbulent models (%$iturb30$%: %$R_{ij} - \varepsilon$% standard model, %$iturb31$%: %$R_{ij} - \varepsilon-(\mbox{\sc ssg})$% model).

For the mean streamwise velocity profiles (cf. Figures 1 and 2), we observed that the results are all globally right and close to the previous computational results obtained with the 1.3.1 version. The k - %$\varepsilon$% seems to be very slightly better that the other models with two equations. But probably for bad reasons since this case combines both break points and narrowing/widening zones where the k - %$\varepsilon$% is known as being notoriously unsuited (as we can see with the turbulent kinetic energy figures).

Figure 5: Mean spanwise velocity profiles in the spanwise direction for different abscissas and three different turbulent models with 2-transport equations
(%$iturb20$%: k-%$\varepsilon$% standard model, %$iturb21$%: k-%$\varepsilon-(\mbox{\sc lp})$% model with linear production and %$iturb60$%: k-%$\omega$% model).

Figure 6: Mean streamwise velocity profiles in the spanwise direction for different abscissas and two second order
turbulent models (%$iturb30$%: %$R_{ij} - \varepsilon$% standard model, %$iturb31$%: %$R_{ij} - \varepsilon-(\mbox{\sc ssg})$% model).

The standard %$R_{ij}$% - %$\varepsilon$%- (%$\mbox{\small \sc lrr}$%) and (%$\mbox{\small \sc ssg}$%) lead to similar results. And as regards the spanwise velocity, the results tend to also be similar, the profiles being in a better agreement with the experiment for the k - %$\varepsilon$% - (%$\mbox{\small \sc lp}$%) and the %$R_{ij}$% - %$\varepsilon$%-(%$\mbox{\small \sc ssg}$%).

Figure 7: Mean streamwise velocity profiles in the spanwise direction for different abscissas and three different turbulent models with 2-transport equations
(%$iturb20$%: k-%$\varepsilon$% standard model, %$iturb21$%: k-%$\varepsilon-(\mbox{\sc lp})$% model with linear production and %$iturb60$%: k-%$\omega$% model).

Figure 8: Mean streamwise velocity profiles in the spanwise direction for different abscissas and two second order
turbulent models (%$iturb30$%: %$R_{ij} - \varepsilon$% standard model, %$iturb31$%: %$R_{ij} - \varepsilon-(\mbox{\sc ssg})$% model).

Differences are observed with the mean turbulent kinetic energy curves. As usual the k -%$\varepsilon$% gives results which substantially overestimate the turbulence. For the other turbulent models of which the result is unsteady, we recall that we represent the sum of the average k%$_{\mbox{\small \sc moy}}$% of both the modelled turbulent kinetic energy and the energy of the resolved structures. This last contribution is in general largely predominant, as we can see on Figure 7, where k%$_{\mbox{\small \sc moy}}$% is also presented for the %$R_{ij}$% - %$\varepsilon$% model. It should also be noted that the results between version 1.3 and version 2.0 are, on some graphs, slightly different,

## Boundary conditions

• The computations with a second order model, which will give a unsteady behaviour, have been realized with periodicity conditions on the lateral faces.
• The computations with a first order model have been run with symmetry conditions.

## Reference Publications

### 3D visualization mean velocities and turbulent kinetic energy fields:

Figure 9: Mean streamwise and spanwise velocity fields and mean turbulent kinetic energy field for the k-%$\varepsilon$% (top) and k-%$\varepsilon-(\mbox{\sc lp})$% (bottom) models.

Figure 10: Mean streamwise and spanwise velocity fields and mean turbulent kinetic energy field for the %$R_{ij}-\varepsilon$% (top) and %$R_{ij}-\varepsilon-(\mbox{\sc ssg})$% (bottom) models.

Figure 11: Mean streamwise and spanwise velocity fields and mean turbulent kinetic energy field for the k-%$\omega$% model.

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cgns fedora.cgns manage 1596.0 K 2010-12-06 - 11:38 JuanUribe
png mesh.png manage 68.9 K 2010-11-03 - 07:17 DorotheeSenechal mesh - 2D plane
png pdc.png manage 6.3 K 2010-11-03 - 16:17 DorotheeSenechal head losses - definition
png results_K20_21_60_finish.png manage 17.4 K 2010-11-03 - 10:27 DorotheeSenechal Mean turbulent kinetic energy profiles
png results_K30_31_finish.png manage 15.5 K 2010-11-03 - 10:28 DorotheeSenechal Mean turbulent kinetic energy profiles
png results_U20_21_60_finish.png manage 14.7 K 2010-11-03 - 10:23 DorotheeSenechal Mean streamwise velocity profiles
png results_U30_31_finish.png manage 14.6 K 2010-11-03 - 08:26 DorotheeSenechal Mean streamwise velocity profiles
png results_V20_21_60_finish.png manage 19.4 K 2010-11-03 - 10:21 DorotheeSenechal Mean spanwise velocity profiles
png results_V30_31_finish.png manage 14.5 K 2010-11-03 - 10:22 DorotheeSenechal Mean spanwise velocity profiles
jpg visu3D_UVKmoy_iturb20.jpg manage 315.8 K 2010-11-03 - 12:39 DorotheeSenechal mean velocity and turbulent kinetic energy fields
jpg visu3D_UVKmoy_iturb21.jpg manage 286.7 K 2010-11-03 - 12:54 DorotheeSenechal mean velocity and turbulent kinetic energy fields
jpg visu3D_UVKmoy_iturb30.jpg manage 261.1 K 2010-11-03 - 12:42 DorotheeSenechal mean velocity and turbulent kinetic energy fields
jpg visu3D_UVKmoy_iturb31.jpg manage 263.2 K 2010-11-03 - 12:51 DorotheeSenechal mean velocity and turbulent kinetic energy fields
jpg visu3D_UVKmoy_iturb60.jpg manage 312.1 K 2010-11-03 - 12:55 DorotheeSenechal mean velocity and turbulent kinetic energy fields
Topic revision: r79 - 2018-11-28 - 22:01:27 - ConstantinosKatsamis
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19 Oct 2019

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