Nomenclature

  • image001.png : Tangential velocity.
  • image002.png : Turbulent kinetic energy per unit of mass.
  • image003.png : Dissipation of turbulent kinetic energy.
  • image004.png: Temperature.
  • image005.png: Shear stress at the wall.
  • image006.png: Molecular kinematic viscosity.
  • image007.png: Turbulent kinematic viscosity.
  • image008.png: Wall temperature.
  • image009.png : Heat flux at the wall.
  • image010.png : Density.
  • image011.png: Specific heat.
  • image012.png : Molecular thermal conductivity.
  • image013.png: Molecular thermal diffusivity.
  • image014.png : Turbulent thermal diffusivity.
  • image015.png: Distance to the wall.
  • image016.png: Constant in the turbulent viscosity model.
  • image017.png: Constant of the log law.
  • Nu : Nusselt number.
  • Pr : Prandtl number.
  • image018.png: Friction coefficient.
  • image019.png: Turbulent Prandtl number.
  • image020.png: Production of turbulent kinetic energy.
  • image021.png: Height of the channel.

Objectives of the study

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.

Wall function approach

Current formulation

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.

image022.jpg

The mean wall shear stress is imposed with the formula image023.png image024.png image025.png

If the wall temperature is fixed we impose a heat flux with the equation image027.png

image029.png is a law of the wall for temperature given by Arpaci and Larsen.

image028.png

image030.png;image031.png image032.pngimage033.png

For the near wall treatment of the turbulent kinetic energy the production term is modify. image034.png

The boundary condition for the turbulent kinetic energy is a Dirichlet condition image036.png

For epsilon, the value at the face is extrapolated using a second order approximation image035.png image037.png

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

image038.png

Underlying assumption

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. image039.png, image040.png,image041.png, image042.png, image043.png, image044.png

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.

image045.png

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.

image046.png

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

image047.png

image048.png

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.

image049.png image050.png image051.png, image052.png

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.

Cross plot comparison

Table 1Apriori calculation on wall function assumption
image053.jpg image054.jpg
image055.jpg image056.jpg
image057.jpg

Other methods

Scalable wall function

The scalable wall function has been introduced in Grotjans and Menter in 1998. This method allows using the wall functions on a mesh regardless of the size of the cells close to the boundary. The formulation simply said that we limit the value of y+ to 11, which mean that if the y+ value is below 11 we set it to 11 otherwise it is left like it is. The rest of the formulation of the standard wall function is unchanged. Obviously this provides a wrong behaviour inside the buffer layer and the viscous sub layer but it is coherent with the solution in the outer part and hence it reduce massively the mesh dependency of the formulation. This method can be seen as starting the calculation on the edge of the viscous sub layer everywhere on the boundary.

K Omega SST

This well known model has been introduced by Menter in 1993. This model is based on the k omega model of Wilcox (1988) but it switches to a k epsilon model in the free-stream in order to reduce the free-stream sensitivity of k omega model. Because it consists of a switch of high Reynolds model and a low Reynolds model, we can expect from this one a good behaviour of a mesh with huge variation of y+.

Simulation of turbulent convective heat transfer in a channel flow

Description of the flow physics

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 image058.jpg

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

image059.png

So we have image060.png With image061.png and image062.png

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

Simulation procedure

Description of the mesh

In order to quantify the effect of a local refinement we design a mesh for a channel flow simulation containing three perturbation of the regular mesh that can be experience when meshing complex domain.
  • 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 image063.jpg

Boundary conditions

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.

image064.png image065.png

Post-processing

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

image066.png

image067.png with image068.pngimage069.png

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.

image070.png

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

Result of the simulation

Comparison of the result

As it can be seen on the following result, perturbations in the mesh may induce large variation in the prediction of the friction and heat transfer coefficients. For standard wall function Nusselt number and friction coefficient is over predicted up to more than 100% depending on the perturbation. This is mainly due to the error on the evaluation of the turbulent quantities near the wall. Indeed one can observe that the velocity and temperature profiles are not so much disrupted but the turbulent kinetic energy and the dissipation are far from their expected values. This result in an over estimation of the turbulent diffusion (for velocity and temperature because of simple gradient hypothesis) and then to error in the prediction of both friction and heat flux at the wall. K omega model allow limiting this effect but still and error of 50% is observed this is mainly due to the fact that even if the y+ distribution is close to 1 in the disrupted region, the boundary layer doesn’t contain enough cell to fully resolve this region. The scalable wall functions show a good behaviour on this mesh, the perturbation makes the solution getting closer to the reference for every case. This is mainly due to the compatible treatment of both velocity and turbulent variable that allow achieving a better mesh independence. Namely, in the viscous and buffer layer the solution is wrong but it is coherent with the outer solution and the variables are coherent between themselves.

Cross plot figures

Table 4Y+ Distribution

image072.jpgimage073.jpg

image074.jpgimage075.jpg

Table 5 Cf coefficient along the channel

Table 6Nusselt coefficient along the channel

image083.jpgimage080.jpg

image081.jpg image082.jpg

Table 7Variable profiles for standard k epsilon with wall function

image084.jpgimage085.jpg

image086.jpgimage087.jpg

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

image088.jpgimage089.jpg

image090.jpgimage091.jpg

Table 9 Variable profiles for k omega SST

Conclusions of the study

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.

Bibliography

  • 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)

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