Calculations on a heated cylinder case

Calculations on a heated cylinder case J. C. Uribe and D. Laurence 1 Introduction In order to evaluate the wall functions in version 1.3 of Code Saturne, a heated cylinder case has been chosen. The case has been treated experimentally by Scholten and Murray [6] and numerically by Szczepanik et al. [7]. This is essentially an unsteady flow. In the case of the heated cylinder, plots of the Nusselt number show an impingement zone at the front of the cylinder where the heat transfer is high and again at the rear of the cylinder where the two vortices that are shed from the cylinder meet. The first part of the cylinder has a laminar region which extends up to around α = 8 9 (α = corresponds to the impingement point). The effects of increasing the Reynolds number are to bring forward the separation point and increase the heat transfer [4]. In this laminar region, the Nusselt number depends on the square root of the Reynolds number. Sanitjai and Goldstein [5] showed that the heat transfer is strongly dependent on the Prandtl number, by examining different gases and liquids with.7 < P r < 176. Another important parameter that influences the flow is the turbulence intensity at the inlet. The Reynolds number chosen for the present study is Re = 36 based on the diameter of the cylinder and bulk velocity at the inlet. The turbulence intensity is set.34% to match the experiments in Scholten and Murray [6]. The fluid is air at a temperature of 293K with P r =.7. 2 Computational Parameters Three meshes have been generated for this study. They all have the same domain, the axes of which coincide with the centre of the cylinder. The inlet plane is located at x = 8D where D is the cylinder diameter. The outlet plane is located at x = 2D. The top and bottom boundaries are set as inlet planes at y = ±8D. The first mesh (Mesh A) has 268 cells in total, with 164 cells around the cylinder. The first cell was found to have a minimum y + 1, and therefore another mesh was created to ensure a minimum value of y + 11.8. Mesh B has 22824 cells, with 192 cells around the cylinder periphery and with a minimum y + = 17.7 (see figure 1). The cells are clustered in the rear part of the cylinder as can be seen in figure 2. The cells are equally spaced in the direction normal to the cylinder surface, and extend to a radius of 5D and then are stretched towards the outer 1

boundaries. Mesh C has been created to obtain a low Reynolds model solution having 6 5 4 y + 3 2 1 1 Figure 1: Distribution of y + for Mesh B Figure 2: Close-up of Mesh B y + < 1 in the first cell along the cylinder wall. This has 2916 cells with 192 cells around the cylinder wall and clustered in the wall normal direction. All solutions using Code Saturne have been obtained using a centred scheme for velocity, turbulent variables and temperature. The standard k ε model is used for the high Reynolds meshes and the SST model is used for the low Reynolds mesh. The version of Code Saturne used was v1.3g. The effects of different numerical parameters such as convection scheme, inlet conditions and solution algorithm have been studied. Air was used in all calculations, with the following properties: ρ = 1.25 Kg/m 3, µ =.1511x1 4 Kg/ms, Cp = 15 J/KgK, P r =.7. A turbulent Prandltl number of.9 was used. There main calculations were performed on this case. The k ε model with standard wall function in both StarCd and Code Saturne, the modified turbulence wall function in Code Saturne (see section 3.2) and the SST model in both codes. Additionally some other studies were conducted in Code Saturne such as the effect of numerical parameters including inlet conditions, numerical schemes and the introduction of scalable wall functions and linear production modifications. 3 Boundary Conditions 3.1 Inlet The inlet velocity was calculated from the Reynolds number and the air viscosity yielding U ref =.45957 m/s. The freestream Temperature was set to T = 293.15 K. The inlet turbulence intensity (T u) was set to.34% according to the experiment therefore obtaining a value for the kinetic energy from k = 1.5T u 2 Uref 2. In order to obtain a complete set of inlet conditions, a value for ε needs to be prescribed. This can be done by prescribing a 2

length scale l and computing the dissipation as: ε = C3/4 µ k 3/2 l Is then up to the user to impose the turbulent length scale l. Another way is to impose a turbulence viscosity ratio (ν t /ν) and obtain the dissipation as: (1) ε = ν ν t C µ k 2 (2) The two methods present a problem since there is no reference to either the length scale or turbulence viscosity ratio from the experiment. The only length scale available is the cylinder diameter, D. It is known that in most wind tunnels the turbulent viscosity ration can vary from 5 to 5 depending on the level of turbulence. As it will be shown, the effect of the choice of value of dissipation greatly influences the solution. Whereas in StarCd equation (1) is used with the value of l prescribed by the user, Code Saturne computes the dissipation as: ε = 1 C3/4 µ k 3/2 (3) κl Equating equation (1) and (3) after inserting lengths l 1 and l 3 respectively leads to: l 1 = κl 3 /1. We see that in equation (3) the length scale corresponds to the edge of the Log layer when l 3 corresponds to the diameter of a pipe (the mixing-length increases as κy until about 1% of the diameter after which it stays constant). Similarly for a bluff body, the integral scale in the wake is about 5% of the body diameter. For the present case, setting l 1 = D (i.e. using equation (1)) yields a value of ε about twenty five times larger than setting l 3 = D (using equation (3)) which would then say that eddies are very small compared to the cylinder. There is not enough information on the experimental set up to determine which choice is the best, but by using equation (3) the turbulent viscosity ratio is ν t /ν = 2.84 which is a more realistic value for the low turbulence case treated here. The effect of the choice of dissipation value at the inlet can be seen in section 4.4 3.2 Wall 3.2.1 Standard Wall Function. At the wall three different approaches were tested. First the standard wall function, which is implemented differently in Code Saturne and in StarCD. In Code Saturne the value at the face has to be imposed, for the kinetic energy a value of k = u 2 k /C1/2 µ is prescribed, whereas for ε a constant flux condition is imposed so that the gradient calculated by the code at the first cell is equal to the theoretical value resulting in: u 3 k ε F = ε I + d (4) κ(d/2) 2 3

where F is the face centre, I is projection of the cell centre on the normal vector of the wall face and d is the distance between these two points. In StarCD the treatment of the wall boundary condition is different. According to the users manual the velocity is calculated from: u + = u u F (τ w /ρ) 1/2 (5) and the turbulence variables are treated as follows: k is solved at near-wall cells, after appropriate modifications. The latter consist of: setting the diffusive flux of k at the wall to zero (i.e. assuming k n = (6) evaluating the turbulence generation on the assumptions that the dominant strain rate is u/ y, evaluated from (5) and the dominant shear stress is the tangential component, which is uniform in the cell and equal to τ w (these assumptions are, of course, part of the underlying set on which the wall functions are based); evaluating the total turbulence dissipation by volume integration of the wall function distribution of ε, given by ε + = C 3/4 µ /κ By contrast, the transport equation for ε or ω is not employed at boundary cells; rather, the nodal values of these quantities are obtained directly from the algebraic wall-function relation given by equations ε + = εy k 3/2 (7) ω + = ωy u τ (8) This procedure whereby the value of epsilon is modified at the cell centre instead of applying a boundary condition at the cell face is in fact very common. 3.2.2 Modified Wall Function in Code Saturne. In order to compare the wall function implementation, an attempt was made to implement the boundary conditions for k and ε in a similar way as in StarCD. Although the manual does not describe the exact way the kinetic energy equation terms are represented at the first cell, a simple approach was taken here (see [2] for several possible ways to do this). The production term is calculated as: P k = τ w C1/4 µ k 1/2 κy (9) 4

and the boundary conditions are set as: k n = (1) ε = C3/4 µ k 3/2 κy note that the ε equation is still solved inside the wall boundary cell but the boundary conditions on ε is of Dirichlet type with a face value of ε = u 3 k /κy.the value of y has been chosen as the distance from the cell centre to the boundary face. This is arbitrary and could also be chosen as 1/2 or 1/4 of that value as in some cell vertex discretisation codes (N3S, ULYSSE). However it must be noted that a real Neumann condition on k as in (1) can now be applied instead of k = u 2 k /C1/2 µ which is what is currently coded. Finally the scalable wall function approach was also tested. 4 Results 4.1 Velocity Field The pressure coefficient around the cylinder can be seen in figure 3 and the friction coefficient in figure 4. From the figures it can be seen how the standard wall function approach (11) 2 1 Cp Saturne Cp StarCD Cp SST Saturne Cp Saturne Modified WF C p -1-2 -3 5 1 15 Figure 3: Pressure Coefficient Figure 4: Friction coefficient produces late separation compared to the low Reynolds solution (SST). By imposing a modified boundary condition (see section 3.2, equations (9), (1) and (11)), the separation point is still the same as with the standard wall function, but the pressure coefficient is closer to the SST solution. It is important to note that due to the limitations on the StarCD user subroutines, the friction coefficient was calculated by approximating τ w ν(u I /d) and although when using wall functions is it not a reliable approximation, it gives an idea of where the separation point lays. The averaged streamlines are shown in figures 5, 6, 7 and 8. It can be seen how the vortices predicted by the SST model on the fine mesh are larger since the separation point 5

is near = 9. It is also interesting to note that although the difference on the separation point between Code Saturne and StarCD is not very large (about 1 ) the recirculation zone is different not only on length (Code Saturne: 1.1D, StarCD:.7D) but also in the angle of the separated stream lines. The effect of applying a modified boundary condition is not very noticeable in terms of recirculation length which is only.1d larger than with the standard boundary condition in Code Saturne. In contrast to the velocity field, by Figure 5: k ε, standard wall functions, Figure 6: Code Saturne. StarCD. k ε, standard wall functions, Figure 7: SST fine mesh, Code Saturne. Figure 8: k ε, Modified WF, Code Saturne. changing the turbulent boundary condition, the behaviour for k and ε at the first cell around the cylinder is similar to the StarCD solution, as it can be seen from figures 9 and 1. The k profile around the cylinder is higher and the ε lower (when compared to StarCD) which in turn produces a large value for the turbulent viscosity. This difference could be due to the different implementation of the ε equation as explain in sections 3.2. It is also important to note than in this study, only the wall boundary condition for the turbulent variables has been modified, no attempt has been made to try to change the velocity wall function. 6

.2.3.15 k StarCD k Saturne k Saturne Modified WF.25 ε StarCD ε Saturne ε Saturne Modified WF.2.1.15.1.5.5 5 1 15 5 1 15 Figure 9: k at the first cell around the cylinder. 4.2 Temperature Field Figure 1: ε at the first cell around the cylinder In figure 11 the Nusselt number around the cylinder is shown. Both codes produce very similar SST-low-Re-mesh results, close to the experimental value except after separation where the plateau is over predicted. The use of wall functions yields different predictions. StarCD predicts a maximum which is not located at the impingement points at the front or the back but close to the 9 angle. This is a feature difficult to explain from the physical point of view. On the other hand, Code Saturne predicts correct levels at the impingement but the minimum (separation point) is wrongly estimated. The use of the modified wall function reproduces the shape of predicted by StarCD but at higher level. This could be because of the implementation of the turbulent wall functions. In StarCD the ε equation is not solved at the first cell, whereas in Code Saturne the face value is imposed as u 3 k /κy. Another explanation could be the formulation of the production and dissipation terms in the k equation which is not know in StarCD. It is clear that the treatment of the turbulent variables at the first cell is very important for this case. 4.3 Effects of the convection scheme To test the sensitivity of the solution to the convetion scheme, another calculation was computed with an upwind scheme for the turbulent variables. Even though the velocities had a centered scheme, the resulting Nusselt number is greatly affected. This is not the case for the SST calculation on the fine mesh where the use of different schemes for the turbulent variables practically does not affect the results. 4.4 Effect of the inlet length scale choice This particular case is sensitive to the length scale used to compute the dissipation at the inlet as it can be seen from figure 13. Here the results of using l 1 = D, that is equation (1) or l 3 = D in (3) can be seen. By using equation (1) the value of ε at the inlet is smaller 7

4 3 Exp k-ε Saturne k-ε Star CD SST Saturne SST StarCD k-ε Saturned Modified WF Nu 2 1 5 1 15 Figure 11: Nusselt number distribution 3 25 Exp k-ε Saturne k-ε Saturne upwind k-ε StarCD k-ε StarCD upwind 2 Nu 15 1 5 5 1 15 2 Figure 12: Effect of the convection scheme on Nusselt number distribution. 8

therefore is producing a higher viscosity ratio. At the impingement point, the use of a higher length scale (equation (1)) produces a value for the Nusselt number 25% higher for StarCd and 62% for Code Saturne. This illustrates the high sensitivity of a key parameter to a basically unknown inlet condition, something users should be made aware of in e.g. tutorials. 3 25 Exp k-ε Saturne k-ε Star CD StarCD hi length scale Saturne hi length scale 2 Nu 15 1 5 5 1 15 2 Figure 13: Effect of the inlet length scale on Nusselt number distribution. Using equation (3): StarCD, Code Saturne. Using equation (1): Star CD Code Saturne 4.5 Other computations Another two possibilities available in Code Saturne have been studied, namely the use of the scalable wall function in the fine mesh with the k ε and the linear production model on the coarse mesh. The scalable wall function was design to allow the high Reynolds models to use a fine with y + 11 The linear production model [3] reduces the production of k at stagnation points by making it proportional to the strain instead of the quadratic proportionality of the traditional k ε models. The effect of these two approaches can be seen in figure 14. The corresponding streamlines can be seen in figures 15 and 16. The scalable wall function produces a much larger over prediction of the Nusselt number, similar to what is expected from a low Reynolds k ε model without corrections. The recirculation length is smaller than the one produced by the standard wall function. The linear production model reduces the Nusselt number at the impingement point as it would be expected, therefore laminarising the flow around the front of the cylinder. This laminar region is present until the separation point which is earlier than the standard wall function approach. 9

8 6 Exp k-ε Saturne standard k-ε Saturne Scalable WF k-ε Saturne Linear production Nu 4 2 5 1 15 2 Figure 14: Nusselt number with Scalable wall function and linear production. Figure 15: k ε, scalable wall functions. Figure 16: k ε, linear production. 1

4.6 Higher Reynolds number The same case was treated at a higher Reynolds number (Re = 89) to compare with the experiments of Sanitjai and Goldstein [5]. The same mesh was used and with the nondimensional distance varying as 48 < y + < 18. In the experiment the measure turbulence intensity was T u =.3% which is comparable wit the previous case at lower Reynolds number (T u =.34%). Equation (3) was used to compute the dissipation at the inlet with a length scale of 1D. The resulting Nusselt number predictions can be seen in figure 17. When using the presently modified or more universal boundary conditions for k and ε as explained in section 3.2 both Code Saturne and StarCD have similar patterns as in the previous case but this time there is a large overprediction of the Nusselt number. The original Code Saturne boundary condition now also gives a severe overestimation of the heat transfer, in particular at the stagnation point. This also shows that the occasional agreement at this stagnation point in previous graphs was coincidental and due to a fortunate chose of value of epsilon at the inlet. Here again the low Reynolds number model captures the impingement and separation quite well but overpredicts the heat transfer at the rear of the cylinder. 6 5 Exp k-ε StarCD SST Saturne k-ε Saturne k-ε Saturne modified WF Nu 4 3 2 1 1 Figure 17: Nusselt number with Scalable wall function and linear production. 5 Conclusions The use of the wall function in this case yields unsatisfactory results. Mesh requirements have been followed to ensure that the value of y + of the first cell is inside the log law region, and the circumferential resolution has been demonstrated to be sufficient. By comparing the results with StarCD, the question of the behaviour of the wall function cannot be answered, since both codes yield different results. By resesting the bound- 11

ary conditions on k and ε in a similar way to what is the commonly agreed norm for wall functions, better agreement between the codes can be achieved. The results with the scalable wall function differ from the standard wall function, implying that it might be a low Reynolds number effect, therefore needing more points after the large first cell. This case has been proven to be very sensitive to different parameters. The choice of lengthscale at the inlet boundary is crucial for the prediction of flow and thermal fields but since there is no information about the experimental set up it remains an open question. The original boundary conditions for Code Saturne seem to be more sensitive to the inlet values or the Re number than show StarCD and Code Saturne with modified boundary conditions. The numerical parameters such as the convection scheme and resolution method also play an important role and need to be carefully taken into account. Although the results with standard wall function in Code Saturne differ from the StarCD, it can be concluded that the main difference arise from the implementation of the k and ε wall functions. Unfortunately this is not a case that can yield to a decision of which way of implementing the these boundary conditions is better since both results are not acceptable and only a low Reynolds model seems to be able to capture the main important features such as separation point and heat transfer distribution. Another inconvenience of this case is the lack of data, such as reliable inlet conditions and data about the flow field such as friction coefficient and nature of the flow in the wake of the cylinder. [1] presents an interesting set of heat transfer measurements for a small tube bundle in cross-flow (4 rows of tubes) which might be interesting to consider. This could also be accompanied by a refined LES with heat transfer to provide combined velocity, turbulence and temperature data, thus resolving a number of unknowns. References [1] E. Buyuk. Heat transfer and flow structures around circular cylinders in cross flow. Journal of Engineering and environmental science, 23:299 315, 1999. [2] S. Gant. Development and Application of a New Wall Function for Complex Turbulent Flows. PhD thesis, UMIST, 22. [3] V. Guimet and D. Laurence. A linearised turbulent production in the k ε model for engineering applications. In Engineering Turbulence and Measurements 5, page 11, 22. [4] H. Nakamura and T. Igarashi. Unsteady heat transfer from a circular cylinder for Reynolds numbers from 3 to 15. International Journal of Heat and Fluid Flow, 25:741 748, 24. [5] S. Sanitjai and R. J. Goldstein. Forced convection heat transfer from a circular cylinder in crossflow to air and liquids. International Journal of Heat and Mass Transfer, 47: 4795 485, 24. 12

[6] J. W. Scholten and D. B. Murray. Unsteady heat transfer and velocity of a cylinder in cross flow I. low freestream turbulence. International Journal of Heat and Mass Transfer, 41(1):1139 1148, 1998. [7] K. Szczepanik, A. Ooi, L. Aye, and G. Rosengarten. A numerical study of heat transfer from a cylinder in cross flow. In 15 th Australasian Fluids Mechanics Conference. The university of Sydney, 24. 13