Spectral Element Direct Numerical Simulation of Sodium Flow over a Backward Facing Step

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1 Spectral Element Direct Numerical Simulation of Sodium Flow over a Backward Facing Step ABSTRACT Jure Oder, Jernej Urankar, Iztok Tiselj Jožef Stefan Institute Jamova cesta 39 SI-1000, Ljubljana, Slovenia jure.oder@ijs.si, iztok.tiselj@ijs.si In this paper we present the preliminary results of direct numerical simulations of a turbulent flow of a liquid metal past the backward-facing step (BFS) with finite dimensions. The BFS geometry can be visualised as a channel where one of the walls has a shape of a step. The flow is flowing from the narrower part to the wider part. The simulations are performed in three dimensions. For the inflow boundary condition over the BFS, a fully developed turbulent velocity field is used. To obtain this fully developed turbulent inflow, a recycling boundary condition is used with which a plane of velocity values from the middle of domain is copied and used to set the inflow boundary condition. Simulations are performed with the NEK5000 code. The most notable feature of this code is the use of spectral elements to solve for velocity, temperature and any other passive scalar. It is an open source code developed by the Argonne National Laboratory. Spectral element method is a hybrid method between finite element method and a collocation spectral method. The method divides the computational domain into finite elements, within which a spectral method is used to solve for variables. This method allows for the use of spectral method in irregularly shaped geometries and to perform direct numerical simulations in such geometries. We performed some preliminary analysis of the calculated friction Reynolds number and the velocity profile before the step. Some analysis of the reattachment of flow was also performed. The main purpose of this work is to test the numerical set-up to later perform calculations with temperature field as a passive scalar. Dimensional walls with internal heating will be added to simulate the heat production in the walls. This work is part of our contribution to the SESAME project of Horizon2020 research programme. 1 INTRODUCTION In the last decades Direct Numerical Simulation (DNS) became an important research tool of the turbulent heat transfer [1]. Results of the DNS studies attained a similar role as experimental results and are typically used for verification and validation of the CFD codes and turbulent models in these codes. Particular attention was paid to the DNS of the fully 511.1

2 511.2 developed turbulent channel flow, as it reveals the basic mechanisms of the convective heat transfer between the fluid and the solid wall. In the recent decade, this topic is becoming more important also in the field of nuclear engineering, where it is used to study thermal fatigue related problems [2], [3]. The first DNSs of heat transfer in the channel flow were made by Kim and Moin [4] in 1989 and Kasagi et al. [5] in They were followed by a significant number of simulations performed at various Reynolds and Prandtl numbers [6]. Various papers of simulating the BFS geometry can be found in the literature. The more interesting for our case are the simulations performed with the spectral element code performed by Patera in 1984 [7]. The simulations performed therein were two dimensional and in a laminar flow regime. More recently similar simulations were performed in 2009 by Barri et al. [8]. Biswas et al. [9] performed simulations of a three dimensional BFS geometry, but at lower Reynolds numbers and for various expansion ratios. This paper presents the middle stage in the development of simulations of turbulent heat transfer in the BFS geometry. The simulations with turbulent channel flow with heat transfer were performed and presented by Oder et al. [10]. The goal of that paper was to test the code and to learn how to use it to obtain a fully developed turbulent flow in an infinite channel. In this regard, the BFS geometry in this paper differs significantly from the previous case. Only the velocity part was simulated in this paper and the periodic boundary condition cannot be used in this geometry. To develop the turbulent inflow, we used the so called recycling boundary condition. The values of velocity from previous time-step in the middle of domain were saved and then used as the boundary condition for the inflow. Because only one domain was used for the whole BFS geometry, the forcing, which was used in the previous paper, cannot be used in a similar way. Instead, we prescribed the volumetric inflow rate through the narrow part of the BFS geometry. Simulations presented in this work were performed with the numerical code NEK5000 [11], which is based on spectral element method. The spectral element code was already validated with results from our previous simulations [10][12] obtained with the spectral method code. Implementation of the spectral element code is part of our efforts within the Horizon2020 project SESAME, where we intend to use spectral elements to analyse the conjugate heat transfer in the backward facing step geometry, where spectral methods limited to very simple geometries cannot be applied. 2 MATHEMATICAL MODEL A sketch of the BFS geometry is presented in Figure 1. The geometry is chosen so as to conform to the geometry of the BFS experiment that is scheduled to be performed within the SESAME project. The experimental setup is being developed by the Karlsruhe Institute of Technology (KIT). The expansion ratio of the BFS geometry is fixed and set to 2. This means that the height of the step is equal to the height of the channel before the step. The outflow plane has a shape of a square. Computational domain of the test section is shown in Figure 1. Dimensionless lengths are obtained with normalisation with the half-width of the channel in front of the step. The plane from which the velocities are copied to develop the fully turbulent inflow is also marked in Figure 1. The velocities in the previous time-step are copied (and distributed between parallel processes), scaled to enforce the constant volumetric flow and then set as a

3 511.3 boundary condition at the inflow. The outflow is set to have a pressure equal to zero, but we also check the velocity so as to not allow for any inflow at that boundary. The other boundary conditions are set to no-slip walls. Copy velocity field Origin Outflow 2 z y x 6 6 Flow direction Figure 1: A sketch of the BFS geometry with coordinate system and dimensionless lengths. The inflow boundary condition is recycled from the plane that is marked The dimensionless governing equations of the fluid, normalized with channel half width h and friction velocity u τ are similar to the equations found in the paper of Kim et al. [13]: u + = 0, (1) u + t = (u+ u + ) + 1 Re τ 2 u + p, (2) where u + = (u +, v +, w + ) is the dimensionless velocity vector, p the dimensionless pressure and Re τ is the friction Reynolds number. No gravity is present in the equation. At this moment, the gravity is not planned to be implemented and simulated in the BFS geometry. 2.1 SPECTRAL ELEMENTS To simulate the BFS geometry, the open source implementation of spectral element method NEK5000 is used. This code is developed by the Argonne National Laboratory. The spectral element method is a hybrid method between finite element method and a spectral method. Similar to the finite element method, the domain is divided into finite elements, but on the other hand, similar to the spectral method, the solution within each element is obtained with a spectral method. The method was introduced by Patera in the year 1984 [7]. The spectral element method combines the benefits of both parent methods. The spectral method contributes the quick convergence rate and because of the finite elements, irregularly shaped geometries can be simulated. NEK5000 uses the collocation spectral method to obtain the solution within the element. These specially chosen points in which the solution is calculated within the element, are in the case of NEK5000 the Gauss-Lobatto-Legendre (GLL) points. These points are the N + 1 roots of the Legendre polynomial of degree N. The points are not uniformly distributed, but are rather clustered near the border of the element. Similar to finite element method, the spectral element method solved differential equations by constructing the stiffness and mass matrix.

4 511.4 The BFS domain in our case was divided first into regions and then into around 100,000 elements. The spectral method used 7 collocation points within each element and in each direction and the whole domain used around collocation points. Figure 2 is showing the distribution of spectral elements in the z = 0 plane. The size of the spectral elements after the step was varied in such a way, that the lower part of the domain had bigger elements in the y direction. The details of the sizes and number of elements in each region of BFS are given in Table 1. The region marked as Box 1 covers all elements from inlet till the length 6. Region Box 2 entails all elements from length 6 till the step. In region Box 3 are all the elements from the step onwards that are in the upper part of the domain and region Box 4 all the elements below region Box 3. Figure 2: The distribution and sizes of elements in the plane where z = 0. The elements after the step in the lower part of domain are larger in y direction. Table 1: Details of the distribution of the elements and the size of regions. Region Dimensions No. of elements Box Box Box Box

5 RESULTS The initial conditions for the simulation of the turbulent flow were set so that the average dimensionless velocity in the channel was around 20 with sinusoidal perturbations. As mentioned above, the velocity at the inflow was copied from other part of domain and scaled with the average velocity in Box 1 region and then multiplied by This velocity is roughly the velocity obtained from previous simulations of channel flow. The initial perturbations evolved into a stationary turbulent flow in whole domain in about 10 5 time steps. The average velocity component in each direction was calculated in each box (see Figure 2) to determine if steady state was reached. After the initial time, the simulation was running for another time steps before averaging was started. The averaging was performed during the time steps or roughly, from simulation time 29 to 49. A slice of average velocity component in x direction through the middle of the domain is shown in Figure 3. The simulation was running in parallel on 256 processors for about a week. Figure 3: The slice at z = 2 of averaged velocity in x direction. Our attention is first focused on the channel before the step. This is necessary to establish the simulated Re τ value. Figure 4 shows the average x velocity component and RMS of velocity components. These variables are calculated by averaging over time and over x spatial coordinate from inlet up to length 6 of the channel, with z fixed to 2 (the middle of the channel). Figure 4(a) shows the average component in the direction of the flow. It can be seen that the distribution of velocity is not completely symmetric yet, thus our averaging interval must be extended. The dashed line shows the average velocity component mirrored across the middle of the channel. From the average velocity Re τ can be calculated. To obtain a value for it, a parabola (which was required to be zero at the edges of the channel) was fitted to the first and last few values of averaged velocity component in Figure 4(a). On the lower wall (left side in Figure 4(a)) of the channel (Box 1) Re τ was equal to 219. On the upper plate Re τ was equal to 218. This is a slight asymmetry which can also be observed in Figure 4(a). In Figure 4(b), (c) and (d) the RMS calculated in similar way as the average velocity is shown. Again, the dashed line shows the values mirrored across the middle of the channel. Here also a slight asymmetry can be observed due to the short averaging time.

6 511.6 (a) (b) (c) Figure 4: Figures show the average velocity component in the x direction (a), and three components of RMS values. The dashed line shows the values mirrored through the middle of the channel. There are interesting structures in the flow behind the step. There are two vortices below the channel. This can be observed in Figure 5 which is showing the streamlines of the averaged flow. The streamlines are of course three-dimensional, but the figure shows the projection onto a plane. Two vortices can be observed. Right after the step the vortex rotating clockwise can be observed. In the lower edge of the step a smaller, weaker vortex rotating in the counter-clockwise direction can be seen. Figure 6 shows time averaged profile in the middle of the channel behind the step. We can observe that the maximum velocity at the step is not exactly in the middle of the channel. This might also be the reason, why profiles in Figure 4 are not symmetric. (d) Figure 5: Streamlines of the flow behind the step in the BFS geometry. The figure shows the two-dimensional projection of three-dimensional streamlines. The color marks the velocity of the flow.

7 511.7 Figure 6: x velocity component profile in the middle of the channel behind the step (plane z = 0). In the large vortex in the lower part of the domain the component of velocity in the x direction is directed opposite to the main flow. After long enough distance behind the BFS the flow must again attain the same general direction as before the step. Between these two regions there lies a point where the flow reattaches to the lower wall behind the step. Figure 7 shows the time averaged velocity component in the x direction at the lower wall behind the step (the first computational point above the lowest wall). From it we can conclude that the reattachment happens at about 15 units behind the step. Figure 7: The velocity component in the x direction in the plane y = (first point above y = 2 which is the lowest point in the domain). The figure shows, where the velocity in the general flow direction changes the sign.

8 CONCLUSIONS We presented some preliminary results of DNS of flow in BFS geometry. The fully turbulent flow was developed using the recycling boundary condition technique in which data from the middle of the domain is taken, copied, scaled and imposed as a boundary condition at the inflow. The analysis shows that the friction Reynolds number Re τ calculated in the simulation is higher than aimed for. From the presented results we are making a conclusion, that the buffer domain Box 2 might not be large enough for this case. Our spatial resolution might not be entirely sufficient for accurate DNS since our local friction Reynolds numbers are higher than the friction Reynolds number of the channel flow, where this resolution was verified. Grid sensitivity analysis remains one of our future tasks. In the future we are planning to add the solid walls and the temperature equations. The solid wall, which will be located after the step at the lowest wall, will have a fixed thickness and will have a distributed constant volumetric heat source. ACKNOWLEDGMENTS This project has received funding from the Euratom research and training programme under grant agreement No and from the Ministry of Higher Education, Science and Technology, Republic of Slovenia, Research programme P REFERENCES [1] Moin, P., Mahesh, K., Direct Numerical Simulation: A Tool in Turbulence Research. Annual Review of Fluid Mechanics, Vol. 30, pp [2] Brillant, G., Husson, S., Bataille, F., Subgrid-scale diffusivity: wall behaviour and dynamic methods, J. Appl. Mech., Trans ASME, 73(3): , [3] Aulery, F., Toutant, A., Brillant, G., Monod, R., Bataille, F., Numerical simulations of sodium mixing in a T-junction, Appl. Therm. Eng., 37:38-43, [4] Kim, J., Moin, P., Transport of Passive Scalars in a Turbulent Channel Flow. Turbulent Shear Flows VI, Springer-Verlag, Berlin, pp. 85. [5] Kasagi, N., Tomita, Y., Kuroda, A., Direct Numerical Simulation of Passive Scalar Field in a Turbulent Channel Flow. Jornal of Heat Transfer -Transactions of ASME, Vol. 114, [6] Kawamura, H., Ohsaka, K, Abe, H., Yamamoto, K., DNS of Turbulent Heat Transfer in Channel Flow with low to medium-high Prandtl number fluid. International Journal of Heat and Fluid Flow, Vol. 19, [7] Patera, A.T A spectral element method for fluid dynamics: laminar flow in a channel expansion. Journal of computational Physics 54, 468. [8] Barri, M.; El Khoury, G. K.; Andersson, H. I. & Pettersen, B. DNS of backward-facing step flow with fully turbulent inflow. Int. J. Numer. Meth. Fluids, 2009, 64,

9 511.9 [9] Biswas, G.; Breuer, M.; Durst, F. Backward-Facing Step Flows for Various Expansion Ratios at Low and Moderate Reynolds Numbers. Journal of Fluids Engineering, 2004, 126, [10] Oder, J.; Urankar, J.; Iztok T. Jenčič, I. (Ed.) Spectral Element Direct Numerical Simulation of Heat Transfer in Turbulent Channel Sodium Flow 24th International Conference Nuclear Energy for New Europe - NENE 2015, Portorož, Slovenia, September 14-17, 2015 [11] Fischer, P.F., Lottes, J.W., Kerkemeier, S.G., Nek5000 web page. (accessed 27. August 2015) [12] Tiselj, I., Bergant, R., Mavko, B., Bajsič, I., Hetsroni, G., DNS of Turbulent Heat Transfer in Channel Flow With Heat Conduction in the Solid Wall. Journal of Heat Transfer - ASME, 123, [13] Kim, J.; Moin, P. & Moser, R. Turbulence statistics in fully developed channel flow at low Reynolds number. Journal of Fluid Mechanics 1987, 177,