DNS of Buoyancy Driven Flow Inside a Horizontal Coaxial Cylinder

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1 DNS of Buoyancy Driven Flow Inside a Horizontal Coaxial Cylinder Imama Zaidi 1, Yacine Addad 2, Dominique Laurence 1 1 The University of Manchester, School of Mechanical, Aerospace and Civil Eng., M60 1QD, U.K. (Imama.Zaidi@postgrad.manchester.ac.uk, Dominique.Laurence@manchester.ac.uk ) 2 The University of Khalifa of Science, Technology and Research, P.O. Box , Abu Dhabi, U.A.E. (Yacine.addad@kustar.ac.ae)

2 Abstract A direct numerical simulation of the buoyancy driven turbulent flow inside a horizontal annular cavity at higher Rayleigh number, Ra = 1.18x10 9, and the cylinders ratio of 4.87 has been carried out using the commercial code STAR-CCM+. Kinetic energy budgets have been calculated to verify the accuracy of the unstructured finite volume code on polyhedral cells in Direct Numerical Simulation (DNS) mode. Comparison of DNS results with wall resolved Unsteady RANS (URANS) models shows that the later models are able to capture the general flow features but fail to predict the large unsteadiness and high turbulence levels in the plume. However local heat transfer rates along the inner and outer cylinder walls are on average of acceptable accuracy for engineering purposes. 1. Introduction Natural convection in annular cavities bounded by co-axial, horizontal cylinders has many practical applications ranging from nuclear reactors, dry casks for nuclear waste storage, underground cables, to thermal storage systems. The complex flow patterns inside the annuli and most challenging features for simulation of this flow are due to the coexistence of turbulent and stagnation regions along with the large recirculation and a wide range of timescales, from boundary layer heat transfer to global cavity thermal equilibrium. Majority of the earlier studies were dedicated to the lower Rayleigh numbers (Ra), onset of the laminar and transitional flows. Earlier experimental and numerical work carried out by Keun et al. 1 and Padilla et al. 2 has shown that at Ra=10 4 a thermal plume appears above the inner cylinder, which tends to become thinner as the Ra number is increased. To examine the turbulent flow regime, Bishop 3 performed an experimental study for Ra ranging from 8x10 6 to 2x10 9 for a ratio of the outer to inner cylinder diameters of It has been claimed that this range of Ra was well above the transitional flow range. Later, McLeod et al. 4 extended the previous study with the annuli geometry of diameter ratio These two studies provided a baseline of experimental data for the heat transfer within the annular cavity which 1

3 was later on used for comparing computational results. Miki et al. 5 performed an LES of this flow for Ra ranging from 2.51x10 6 to 1.18x10 9 with different values of Smagorinsky model constant (Cs), but surprisingly with Cs=0, he was unable to obtain fully developed flow in the cavity at Ra=10 9. Additionally, the results showed a dependence on the value of the sub-grid model constant. They concluded that a lower value of Cs (compared to the common value Cs=0.1) had to be used along with a fine grid to take into account the viscous effects introduced by the boundary layer at the inner and outer cylinder surfaces. In the THMT conference series, Addad et al. 6 successfully performed an LES with the code STAR-CD for this coaxial cavity at Ra = and using Cs = A comparison with RANS models showed rather large differences for temperature distribution, and it has been concluded that additional detailed data was needed to draw definite conclusions. To the Author s knowledge, no DNS has been performed for Ra 10 9 since the common structured polar meshes would be refined on the inner cylinder instead of the outer one where fully turbulent flow occurs. Furthermore, the absence of a detailed reference data and recent demonstrations of unstructured finite-volume codes can be used for direct numerical simulations (Fig. 1 shows STAR-CCM+ and polyhedral cells results for channel flow DNS) has motivated the present numerical study using the STAR-CCM+ code (CD-Adapco) to provide detailed and additional reference data for the Verification and Validation (V&V) of the existing RANS turbulence models. 2. Case Description: The cavity bounded by inner and outer cylinder is filled with incompressible Newtonian fluid. Different isothermal boundary conditions are set on the inner (hot) and the outer (cold) cylinders. A buoyancy plume flow forms on top of the hotter inner cylinder, then impinges and later cools down in a wall-jet boundary layer flow along the outer cold cylinder. Due to the small temperature difference the Boussinesq assumption is applied, i.e. thermal density 2

4 variation is only accounted for in the buoyancy force while fluid properties are set constant in all other terms. The diameters of inner and outer cylinder are D i and D o. The dimensionless 9 parameters considered in this case are Ra , D / D 4.87 and Pr Numerical Procedure: In the present study, the unstructured finite volume code (STAR-CCM+, v.5.04 & 7.04) is used for the DNS. The fundamental equations governing the flow of incompressible fluid are solved. As highlighted above, the polyhedral cells feature in this case is ideal, since the active flow and fine-scale turbulence is limited to the narrow plume and cooling along the outer wall boundary layers in the top half of the cavity. Hence a Cartesian polar mesh (as used for classic DNS codes) would be very wasteful. A very gradually refined adaptive polyhedral o i cells mesh has been used (Fig. 2 and 3). The Kolmogorov length scale, 3 1/4 ( / ) distribution was estimated from a precursor RANS calculation, then local mesh refinement for the DNS was chosen using STAR-CCM+ macros such that: x 2.5, y 2.5, z 5, using here local Cartesian mesh notations for a better understanding. The mesh is polyhedral in the 2D plane and extruded in homogeneous direction, resulting in a total of 6 million cells. Bounded central differencing scheme (BCD) is employed for the spatial discretization to enhance stability, whilst for time discretization, the implicit three level second order scheme has been used. This simulation was left to run for 120 seconds (487 L ref /V ref ), starting statistical sampling from 70 sec when probes seemed to exhibit a self-similar turbulent signal (no mean drift). Prior to this cavity simulation, a DNS mode validation test case for the code and schemes has been performed by simulating the well-known turbulent channel flow at Re This case also served for testing and validating the user-coded Java scripts which are used to compute the turbulent energy budgets, and the numerical schemes of the STAR-CCM+ 3

5 selected for the DNS simulations; see Fig. 1 showing the mean flow profiles and budget of turbulent kinetic energy (TKE). It has to be noted that the DNS was run by selecting LES mode in STAR-CCM+ and setting Cs=0 (LES sub-grid scale model set to zero). Note that dissipation is recomputed here from its definition rather than as a balance of all other terms, hence matching the reference DNS dissipation profile will demonstrate that even Kolmogorov scales are fairly well resolved. Turbulent transport is only in qualitative agreement, possibly due to insufficient time averaging as it is related to larger scales. Nevertheless, sum of terms in the budget is everywhere nearly zero. Lref 0.5( Do D i ) Table 1: Dimensionless parameters definitions V g TL ref ref 4 Ra Pr g TL v T ref Nu R ln( R / R )( dt / dn ) R* ( R R ) / ( R R ) T* ( T T ) / ( T T ) o 4. Cavity Flow Results: i i o i i o i The Iso-Q contours, coloured by temperature, in Fig. 5 show highly turbulent activity in the plume and outer cylinder boundary layer whereas the inner cylinder boundary layer remains laminar. Compared to Addad et al. s earlier LES, even smaller structures are captured on this finer mesh. In good agreement with previous experiments and numerical simulations, the lower region of the cavity is stagnant below the inner cylinder. The lower value of Q highlights the large structures which are observed to penetrate further down along the outer cylinder wall, while in the upper part the turbulent activity extends far outside the plume and boundary layers. Fig. 4 represents an instantaneous snap-shot of temperature distribution. Animations corresponding to fig. 4 shown that the plume at 0 angle undergoes a drastic change in both; the laminar plume length and angle of its inclination with the vertical line. The latter oscillatory motion closely matches the flow structure postulated by McLeod et al. 8 and is canceled out in the long time-averaged distribution (Fig. 6). Nevertheless, this low frequency motion motivated the use of (URANS) approach in the present study in contrast to

6 the steady calculations reported in previous numerical work. Fig. 7 and 8 illustrate the qualitative comparison made of the predicted flow patterns from URANS and DNS. Fig 9, streamlines of velocity coloured by temperature show a pair of elliptical recirculation bubbles in the upper region, entrained by only the top half of the plume, as it becomes fully turbulent, whereas there is a weak horizontal entrainment of the flow in the plume from stratified lower temperature region, connecting to the lower point of the inner cylinder. These features consistent with the velocity magnitude map, illustrated in fig. 8. Fig 10 shows terms of the TKE budget. Turbulence is mainly produced on either sides of the plume. Buoyancy production is positive at the centre then negative on the sides of the plume as turbulence works against gravity to entrain upwards cooler fluid coming from the sides, as seen on streamlines in fig. 9. Total dissipation is nicely symmetrical as this is made up of small-scale eddies which come in very large numbers even in short statistical samples. More dissipation occurring along the outer cylinder boundary layer is present. In order to confirm the validity of the DNS performed, fig. 11 gives the ratio of the Kolmogrov length scale to the current grid spacing, on the plume axis, i.e. at 0 0. This figure shows that the maximum ratio of grid spacing to the Kolmogrov scale does not exceed 5 for the whole domain and for natural convection problems this ratio has been used safely 9. Note that near the walls there is a prism layer hence cells are very fine in the boundary layer. Therefore, this might give some confidence that the grid resolution used herein is sufficient to resolve all length scales. In addition, the precursor RANS simulation, even if qualitative, has been found very helpful to guide the mesh refinement, in opposition to classic DNS benchmarks for which the flow length scales are known in advance. Fig. 12 & 13 illustrate instantaneous velocity history and the TKE spectral density curve at mid height of the plume axis at 0 0. First peak in the curve at 0.35 frequency corresponds to the plume large scale instability with 5

7 a period of 3 (s) also very visible in fig. 4 which is showing a quasi-sinusoidal signal superimposed with smaller amplitude turbulence. The Nusselt number (Nu) distribution is the principle objective of many engineering CFD studies, but is not available from the experiments, hence the main motivation for the present DNS. Fig. 14a provides Nu on the inner cylinder, a clear laminar boundary-layer growth is observed along the inner cylinder wall followed by a sharp decrease around the 10 angle due to the presence of a small separation bubble (the flow direction is from 180 at the bottom to 0 at the top where the plume is generated). Both low-re RANS models are expected perform well in this laminar flow region (if this was an isolated cylinder). On the outer cold cylinder, the plume behaves like an impingement jet developing into a wall jet, although here the DNS Nu profile is flatter, i.e. with a lower and wider peak. This is due to the plume s axial offset instability. The inertia of the wall jet allows it to overshoot the stratification level down to 120 angle, as seen previously on fig. 6. Both 2 f and k sst model fail to predict the heat transfer in the plume impingement region i.e. they produce a high and narrow peak around to 0-30 more similar to a steady impinging jet on a flat plate for which they are known to produce accurate results. On average and through integration and error compensation, the predictions of the global heat transfer prediction from inner to outer cylinder should be quite satisfactory for engineering purposes. Figure 15 now shows comparisons with the available experimental temperature profiles. The DNS is in fairly good agreement at all angles, particularly along the plume axis and as concerns the bulk temperature in the cavity. It does differ however significantly in the outer cylinder wall jet region where the DNS shows a much thinner thermal boundary layer. Fig. 17 represents the turbulent kinetic energies in the annular cavity. Rms values are shown in fig. 18. The origin of the plume is nearly a one dimensional turbulence with high values of v rms and rms, i.e. a series of puffs before evolving to more isotropic turbulence. 6

8 with: The balance of kinetic energy equation for buoyancy driven flow is: tk Ui ik P G VD (1) P= u u U G g v = ( pu ) / i k k i i i = u u u / 2 VD = k = u u k i i k k i k i (2) where P is shear production, G is buoyancy production, turbulent transport, pressure diffusion, VD viscous diffusion and is dissipation. Fig 19 shows the turbulence kinetic energy budgets at 0 0 and At the top of the inner cylinder, i.e. at 0 0, due to recirculation; production term shows a double peak, then buoyancy dominates, followed by convection. Near the inner wall pressure diffusion dominates, balancing viscous dissipation and convection. On the outer cylinder, the more classical wall jet budget is observed, with production occurring mainly in the large free-shear layer side of the jet while it drops in the very narrow near wall shear layer (see fig 16). Buoyancy contribution is also significant. The balance is nearly zero which shows that the numerical schemes implemented in STAR-CCM+ are conserved. 5. Conclusions: Buoyancy driven turbulent flow inside a horizontal annular cavity at high Rayleigh number, free from inlet condition uncertainties, is a very apt but challenging test case for CFD. Thanks to a precursor RANS simulation providing Kolmogorov scales, an optimal polyhedral cells mesh for DNS using the commercial STAR-CCM+ code was generated, and checked a posterior using the turbulent kinetic energy budget imbalance. The boundary layer flow on the inner hot cylinder remains laminar, but the plume above it starts off with an unconventional quasi 1D turbulence or series of hot puffs motion, later developing into a more classical buoyant jet type flow which impinges on the cold outer cylinder summit, followed by a pair of curved hot wall jets overshooting the stratified lower half of the cavity. 7

9 The k sst,and even more so the 2 f model, overestimate the maximum heat flux where the hot plume impinges on the cold container cylinder summit, but overall, global, and in most region local, heat transfer rates along the inner and outer cylinder walls are of acceptable accuracy for engineering purposes. 6. References: 1. Kuehn, T. & Goldstien, R. An experimental study of natural convection heat transfer in concentric and eccentric horizontal cylindrical annuli" J. Heat trans. 100, (1978). 2. Padilla, E. & Silveira-Neto, A. Large eddy simulation of transition to turbulence in natural convection in horizontal annular cavity. IJHMT (2008). 3. Bishop, H. Heat transfer by natural convection of Helium between horizontal isothermal concentric cylinders at cryogenic temperatures. IJHMT (1988). 4. McLeod, A. E. & Bishop, E. H. Turbulent natural convection of gases in horizontal cylindrical annuli at cryogenic temperatures. IJHMT 32, (1989). 5. Miki, Y., Fukuda, K. & Taniguchi, N. Large eddy simulation of turbulent natural convection in concentric horizontal annuli. IJHFF 14, (1993). 6. Addad, Y., Laurence, D. & Rabbitt, M. Turbulent Natural Convection in Horizontal Coaxial Cylindrical Enclosures: LES and RANS Models. ICHMT 10, (2006). 7. Kawamura, H., Abe, H. & Matsuo, Y. DNS of turbulent heat transfer in channel flow with respect to Reynolds and Prandtl number effects. IJHFF 20, (1999). 8. Mcleod, A. & Bishop, H. Turbulent natural convection of gases in horizontal cylindrical annuli at cryogenic temperature. IJHMT (1989). 9. Kunnen, R. P. J., Geurts, B. J. & Clercx, H. J. H. Turbulence statistics and energy budget in rotating Rayleigh Bénard convection. Eur. J. Mech. - B/Fluids 28, (2009). 8

Fig. 1: Channel flow DNS at Re 395 ; mean

10 Fig. 1: Channel flow DNS at Re 395 ; mean velocity and temperature (left); TKE budgets (right), black symbols=ref. DNS, open symbols = polyhedral mesh STAR-CCM+ DNS. Fig. 2: Coaxial cylinder grid. Fig. 3: Near-wall refinement. Fig. 4: Instantaneous iso-temperatures illustrating transient behavior of the plume (left T= 84.1s, center = 85.5s and right = 86.05s). Fig. 5: Iso values of Q =1 coloured by mean T*. Fig. 6: DNS mean temp. 9

11 Fig. 7: Iso values of mean TKE* (left =DNS, center = k sst model, right = 2 f model). Fig. 8: Iso values of mean velocity (left =DNS, center = k sst model, right = 2 f model). Fig. 9: Streamlines of mean velocity vectors coloured by temperature. (a) (b) (c) (d) (e) (f) Lref Fig. 10: Budgets of TKE normalized by 3 V turbulence transport, (d) viscous diffusion, (e) convection and (f) dissipation. 10 ref (a) shear production, (b) buoyancy production, (c)

Fig. 11: Ratio of / x. Fig. 12: Instantaneous velocity at center of plume. Fig. 13: Spectral density curve at the center of plume. Fig. 14: Nu number; left: inner hot cylinder, right: outer cold cylinder.

12 Fig. 11: Ratio of / x. Fig. 12: Instantaneous velocity at center of plume. Fig. 13: Spectral density curve at the center of plume. Fig. 14: Nu number; left: inner hot cylinder, right: outer cold cylinder. (a) (b) (c) Fig. 15: Mean temperature at different angles (a) 0, (b) 90, (c) 30, and (d) 60 ). (d) 11

13 (a) (b) (c) (d) Fig. 16: Mean vertical component of velocity ((a) 0, (b) 90, (c) 30 and (d) 60 ). (a) (b) (c) (d) Fig. 17: Turbulent kinetic energy ((a) 0, (b) 90, (c) 30 and (d) 60 ). 12

14 (a) (b) (c) Fig. 18: Rms values of u, v w and t ((a) 0, (b) 90, (c) 30 and (d) 60 ) (d) Lref Fig. 19: Budgets of kinetic energy (left 0 and right 30) normalized by 3 V ref 13

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