Main CfdTm Forum Saturne Aster ATAAC TWikiEditRaw editAttachPrint version

TWiki> Main Web>KarimOsman>ConvectiveHeatTransferOnADisruptedMesh (2011-08-17, KarimOsman)

TWiki> Main Web>KarimOsman>ConvectiveHeatTransferOnADisruptedMesh (2011-08-17, KarimOsman)

- : Tangential velocity.
- : Turbulent kinetic energy per unit of mass.
- : Dissipation of turbulent kinetic energy.
- : Temperature.
- : Shear stress at the wall.
- : Molecular kinematic viscosity.
- : Turbulent kinematic viscosity.
- : Wall temperature.
- : Heat flux at the wall.
- : Density.
- : Specific heat.
- : Molecular thermal conductivity.
- : Molecular thermal diffusivity.
- : Turbulent thermal diffusivity.
- : Distance to the wall.
- : Constant in the turbulent viscosity model.
- : Constant of the log law.
- Nu : Nusselt number.
- Pr : Prandtl number.
- : Friction coefficient.
- : Turbulent Prandtl number.
- : Production of turbulent kinetic energy.
- : Height of the channel.

In industrial studies, the simulation of turbulent flow with convective heat transfer is often performed using high Reynolds turbulence model. This induce using the so called “wall functions” in order to avoid the resolution of the near wall region. One of the requirement of the wall function approach is that the centre of the first cell to the wall is located in the log layer (namely 30 < y+ <300, where y+ is the distance of this cell centre normalised by the friction velocity and the dynamic molecular viscosity). When the geometry and/or the flow pattern are complex, it can be difficult to design a mesh that matches this criterion everywhere at the boundary. In order to investigate the consequences of not respecting the previous condition we decide to simulate an academic test case using a mesh which is disrupted in several locations. We choose the simulation of convective heat transfer in a turbulent channel because it is the most classical academic test case relevant to nuclear engineering that has been studied experimentally and numerically. We start by explaining the underlying assumption of the wall function approach then we present the test case and finally the results obtained and the conclusion of this study.

It has been now more that forty years since Bradshaw introduced the first wall function technique. During this time, several wall functions have been proposed, a good review is given by Craft and al. (2002). We will present the wall function formulation used in *Code_Saturne* which is much closer to the conventional one used nowadays.

The mean wall shear stress is imposed with the formula

If the wall temperature is fixed we impose a heat flux with the equation

is a law of the wall for temperature given by Arpaci and Larsen.

;

For the near wall treatment of the turbulent kinetic energy the production term is modify.

The boundary condition for the turbulent kinetic energy is a Dirichlet condition

For epsilon, the value at the face is extrapolated using a second order approximation

And the derivative at point M of dissipation is evaluated using the expression

The wall function approach strongly remain on the idea of scaling, we note u* the friction velocity, and T* the temperature scale.

All the variable are non-dimensional using u*, T*, and. , ,, , ,

The law of the wall is used to relate in a simple way the velocity at the first cell and the friction at the wall.

It can be derived using Prandtl equation and the mixing length hypothesis.

This relation is valid in the viscous sub layer and in the log region it is not valid in the buffer layer and in the wake region.

The Reichardt’s profile gives a better behaviour in the buffer layer.

It has been observed that it is physically more realistic to replace the velocity scale in the definition of y+ by a velocity scale coming from the turbulence. However this is valid only in the log layer.

Then using the law of the wall one can write the friction at the wall as

A similar procedure is used to link the near wall cell temperature and the heat flux coming from the wall. The profile of Arpaci and Larsen can be derived in a similar way and also contain a Prandtl number correction. Other profiles exist such as Kader’s law.

,

In order to obtain boundary conditions for turbulent quantities we assume that the turbulence is in an equilibrium state which means that the production of turbulent kinetic energy is equal to the dissipation of kinetic energy. We also assume that the viscous shear stress at the wall is equal to the turbulent shear stress at the center of the cell. Again theses assumptions are only valid in the log layer.

We simulate an incompressible turbulent flow between two parallel and infinite planes with a distance of H between them. The Reynolds number based on the friction velocity and the half width channel is Re* = 1020 and the Prandtl number is Pr = 0.71. **Table 2Descitption of the case**

The channel is heated with a uniform average heat flux qw on both sides, this introduce a constant linear increase of the average temperature. Indeed the energy balance of a piece of fluid between x and x+dx is

So we have With and

This test case has been studied numerically by Kawamura and al.

- The first disturbance is a curvature of the mesh longitudinal lines that modify the y+ distribution at the wall from 32 to 1.
- The second one is a non conformal refinement occurring with 3 stages modifying the y+ distribution from 32 to 4.
- The last one is a transition to triangular mesh but with a constant y+ distribution equal to 32.

After each disturbance we have put a regular channel to make the flow redevelop on the correct solution, this regular channel is preceded by a geometrical contraction which help the flow to come back to the previous solution by generating turbulence. The mesh contain in average 32 cells in the normal to the wall direction and 2400 cells in the stream wise direction. Sensitivity tests have been carried on different mesh to ensure that the numerical error is small enough to allow relevant comparison. One has to bear in mind that this kind of test have to be performed using a mesh size at the wall which is bigger than 60 in wall units in order to distinguish from the errors coming from the model.

**Table 3Descritption of the mesh**

At the inlet, we use the law of the wall to impose the velocity profile and the profile of Arpaci and Larsen to impose the temperature profile. For the outlet, the “free outlet” condition of *Code_Saturne* is used. At the wall we impose a friction deduced from the wall function formulation for the velocity component and a constant heat flux for temperature. This is done numerically by ensuring the following discretized conditions.

The main outputs of a turbulent flow simulation are usually friction and heat transfer coefficient which are defined as follow.

with

These coefficients are computed locally using the previous formula. On main interest is the computation of the wall temperature which has to be calculated using the thermal equivalent of the law of the wall (instead of using the face temperature from the code). For this purpose we use again the law of Arpaci and Larsen.

We also include comparison of the variable profile at the location of the disturbances.

**Table 4Y+ Distribution**

**Table** **5 Cf coefficient along the** **channel**

**Table 6Nusselt coefficient along the channel**

**Table 7Variable pr****ofiles fo****r standard**** k epsilon with wall function**

**Table 8 Variable profiles for standard k epsilon with scalable wall function**

**Table 9 Variable profiles for k omega SST**

We provide a new verification test case that can be used to test the mesh sensitivity of turbulence model in the near wall cell. As we have seen it is able to show clearly the difference of between turbulence models on an academic test case like the channel flow. The standard wall functions are proven to be extremely dependant of the position of the near wall cell centre as well as the k omega model. The scalable well function show better result and we recommend their use on every mesh were the y+ distribution is not well controlled. We are currently carrying on additional test about wall function and developing new idea that can help to reduce their mesh dependency. We are also testing the opposite procedure, studying the effect of a local coarsing of the near wall mesh when using a low Reynolds model.

- H. Abe, H. Kawamura, Y. Matsuo, Surface heat-flux fluctuations in a turbulent channel flow up to Re-tau=1020 with Pr=0.025 and 0.71
*, International Journal of Heat and Fluid Flow*, 25(3), 404-419 (2004). - Launder and Sharma, 1974. B.E. Launder and B.I. Sharma , Application of the energy-dissipation model of turbulence to the calculation of flow near a spinning disc.
*Lett. Heat Mass Transfer***1**(1974), pp. 131–138. - Menter, F.R. Zonal two-equation k-w turbulence model for aerodynamic flows. AIAA Paper 1993-2906, 1993.
- Grotjans H, Menter FR. Wall functions for industrial applications. In Proceedings of Computational Fluid Dynamics’98, ECCOMAS, 1(2), Papailiou KD (ed.). Wiley: Chichester, U.K., 1998; 1112–1117.
- Craft, T.J., Gerasimov, A.V., Iacovides, H., Launder, B.E.: Progress in the generalisation of wall-function treatments. Int. J. Heat Fluid Flow
**23**, 148–160 (2002)

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

png | image001.png | manage | 0.3 K | 2011-03-28 - 13:20 | KarimOsman | |

zip | mesh_complete.cgns.zip | manage | 3172.0 K | 2011-04-04 - 14:28 | KarimOsman | |

zip | mesh_complete.ensight.zip | manage | 2870.0 K | 2011-04-04 - 14:28 | KarimOsman | |

zip | mesh_complete.med.zip | manage | 5129.7 K | 2011-04-04 - 14:28 | KarimOsman | |

zip | mesh_part.zip | manage | 5666.0 K | 2011-04-04 - 14:28 | KarimOsman | |

zip | sources_mesh.zip | manage | 2220.0 K | 2011-04-04 - 14:29 | KarimOsman |

Topic revision: r5 - 2011-08-17 - 17:51:13 - KarimOsman

Code_Saturne

KNOO

FLOMANIA

Photo Wall

Upcoming Events

Add Event

Copyright © by the contributing authors. Unless noted otherwise, all material on this web site is the property of the contributing authors.