INVESTIGATION OF THE BOUNDARY LAYERS IN TURBULENT RAYLEIGH-BÉNARD CONVECTION

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1 Seoul, Korea, June 29 INVSTIGATION OF TH BOUNDARY LAYRS IN TURBULNT RAYLIGH-BÉNARD CONVCTION Matthias Kaczorowski Institute of Aerodynamics and Flow Technology German Aerospace Center (DLR) Bunsenstr., D3773 Göttingen, Germany Anna bert Dept. of Mechanical ngineering Technical University of Ilmenau.O. 565, D98684 Ilmenau, Germany Claus Wagner Institute of Aerodynamics and Flow Technology German Aerospace Center (DLR) Bunsenstr., D-3773 Göttingen, Germany André Thess Dept. of Mechanical ngineering Technical University of Ilmenau.O. 565, D98684 Ilmenau, Germany ABSTRACT In the present study the temperature proles of turbulent Rayleigh-Bénard convection obtained through DNS and microthermistor measurements in a rectangular container are compared. It is found that both DNS and experimental data are in good agreement when comparing temperature proles at topologically similar ow positions. It is observed that the data sets obtained at Ra 8 are in signicantly better agreement than at Ra 7. Finally, our results are compared to the analytical t function proposed by Shishkina & Thess (29) and nd that this function gives a very good representation of the DNS data when the temperature proles through the centre of the roll structures are evaluated. INTRODUCTION The accurate prediction of the heat transfer between a turbulently moving uid and a solid wall is despite its importance to engineering applications still a numerically challenging task. Reynolds Averaged Navier-Stokes (RANS) simulations which are typically performed to solve engineering problems suer from the inherent problem that they have to model all turbulent quantities, whereas Direct Numerical Simulations (DNS) are not normally applicable to complex geometries and highly turbulent ows. In order to be able to improve the modelling of the turbulent quantities it is essential to gain a deeper understanding of the underlying phenomena, where the turbulent boundary layers and their interaction with the bulk ow play a crucial role. xperimental investigations on the other hand have to cope with with problems of accessibility and (simultaneous) data acquisition at dierent locations. In fundamentel research the geometrically simple Rayleigh-Bénard experiment is often chosen to investigate the turbulent heat exchange between a thermally driven uid and a hot bottom and a cold top wall, respectively. The characteristic parameter of this natural convection phenomenon is the Rayleigh number Ra = ˆαĝĤ3 ˆT /(ˆνˆκ), which is a nondimensional measure of the buoyancy and diusive forces acting on the uid. Here ˆα, ˆν and ˆκ denote the thermal expansion coecient, kinematic viscosity and thermal diusivity, respectively. ˆT is the vertical temperature gradient between the two bounding surfaces, Ĥ the height of the uid layer and ĝ the gravitational acceleration; ˆ indicates dimensional quantities. Hölling & Herwig (26) theoretically derived a scaling for the temperature prole in turbulent Rayleigh-Bénard convection (RBC) with innite aspect ratio Γ = Ŵ /Ĥ, and hence no mean ow, in the limit of Ra. They found a linear scaling of the temperature near the wall (the conductive sublayer) and a logarithmic one well outside the mean boundary layer thickness δ θ. The shape of the boundary layer at various aspect ratios and Rayleigh numbers was investigated experimentally by du uits et al. (27) along the axis of a cylindrical cell showing that the boundary layers' shape hardly changes with the cell's aspect ratio. Maystrenko et al. (27) studied the structure of the thermal boundary layers of turbulent RBC in a long rectangular cell. They found signicant changes of the structure of the thermal boundary layer in the longitudinal direction which they concluded to be a result of the mean wind. bert et al. (28) observed a buer layer with a power-law behaviour in their Rayleigh-Bénard cell of nite aspect ratio. This led to a separation of the temperature prole into three regions: linear for x + 3.2, power law for.2 x+ 3 3 and logarithmic for x + 3 8, where x+ 3 is the wall distance normalised as follows from (4). In the following we compare the structure of the thermal boundary layer at dierent locations of a rectangular cell obtained through DNS from the onset of turbulence at Ra = until a fully developed turbulent thermal eld is achieved at Ra = The DNS data is compared to experimental results by Maystrenko et al. (27) and bert et al. (28), as well as to an approximation function proposed by Shishkina & Thess (29), who theoretically analysed the mean temperature proles of RBC and compared their results to the temporally and spatially averaged proles obtained through DNS of RBC in a cylindrical container (Γ = ) lled with water. NUMRICAL ROCDUR In the present numerical simulations the nondimensionalisation of the governing equations is carried out using x i = ˆx i /Ŵ, u i = û i /(ˆαĝ ˆT Ĥ)/2, θ = ( ˆT ˆT )/ ˆT, p = ˆp/(ˆρˆαĝĤ ˆT ) and t = ˆt(ˆαĝ ˆT Ĥ)/2 /Ŵ. Together with the Boussinesq approximation one obtains the 77

2 Seoul, Korea, June 29 Table : Simulation parameters of the DNS of turbulent RBC in the rectangular enclosure. N, N 2 and N 3 denote the number of grid points in the span wise, longitudinal and vertical direction, t avg. the time over which the statistics are averaged and Nu top,bottom the Nusselt number averaged over t avg. as well as the top and bottom plates. Ra N N 2 N 3 t avg. Nu top,bottom ± ± ± ±.35 incompressible Navier-Stokes equations given by () - (3) u i t + u u i j + p = x j x i u i x i = () θ t + u θ i = x i r 2 u i (Γ 3 Ra) /2 x 2 +θδ 3i (2) j 2 θ (Γ 3 Rar) /2 x 2, (3) i where u i are the velocity components in i =, 2, 3 direction, p and θ are the pressure and temperature, respectively. The simulations are conducted in a rectangular container with aspect ratios Γ 3 = Ŵ /Ĥ = and Γ 23 = ˆL/Ĥ = 5. The lateral walls are adiabatic and top and bottom walls isothermal with non-dimensional temperatures θ bottom = +.5 and θ top =.5, respectively. No-slip and impermeability conditions are applied to all walls and the grid spacing is gradually rened towards the rigid walls, so that the computational domain is discretised using non-equidistant meshes in all three coordinate directions. The volume balance procedure by Schumann et al. (979) is used for the integration over the uid cells and the solution is evolved in time by means of the explicit uler-leapfrog scheme. Spatial derivatives and cell face velocities are approximated by piece-wise integrated fourth order accurate polynomials, where the velocity components are stored on staggered grids as described in detail by Shishkina & Wagner (27). The velocity pressure coupling is carried out through the projection method by Chorin (968) which requires the solution of a oisson equation. Here, a direct solver is employed using a cyclic reduction algorithm. The system of equations is decoupled using the separation of variables method described in Kaczorowski et al. (28). RSULTS Global ow structure In gure isosurfaces of the temperature eld illustrate the turbulent RBC in a long rectangular enclosure with solid walls in all coordinate directions for Rayleigh numbers between Ra = and Ra = The illustration highlights how the large scale plumes forming the large scale circulation disintegrate into smaller plumes that are clustered in regions of rising or falling (not shown) uid. The two highest Rayleigh numbers of the DNS are close to two cases investigated experimentally by Maystrenko et al. (27) and bert et al. (28). It is observed that the ow elds obtained through the DNS reect four counter rotating convection rolls, whereas two rolls were observed (d) Figure : Isosurfaces of temperature for the hot uid (θ.) and Rayleigh numbers Ra = 3.5 5, 3.5 6, and (d) in the experiment by Maystrenko et al. (27). A four-rollstructure was also found for Ra 8 by means of LS in the same geometry conducted by Sergent & Le Quéré (28), whereas they found a two-roll-structure for Ra, and for Ra 8 when the Rayleigh number was approached from a higher Ra with two rolls. The four-roll-structure of the DNS is also reected by the variation of the thermal boundary layer thickness in longitudinal (x 2 -) direction. This is illustrated in gure 2 in terms of the local Nusselt number Nu loc = /(2δ θ ), where δ θ denotes the boundary layer thickness obtained through the maximum rms-value criterion (Belmonte et al., 994). Because of the global ow structure the temporally averaged temperature proles in the centre plane (x =.5) depend on their longitudinal (x 2 ) position, which is illustrated for Ra = in gure 3. This implies that a mean wind is formed in the rectangular convection cell, 78

3 Seoul, Korea, June 29 2 Nu loc x θ = Nu loc 2 x Nu loc 2 x Figure 2: Local Nusselt number Nu loc = /(2δ θ ) in the centre plane (x =.5) for Ra = and Ra = in the hot ( ) and cold ( ) boundary layer. The solid line indicates the global mean Nusselt number averaged over both horizontal surfaces. so that in regions with predominantly rising or falling uid the centre of the cell has temperatures θ c > or θ c <, respectively. This behaviour is also reected by the experimentally measured temperature proles near the top and bottom plates by Maystrenko et al. (27), where the mean temperature does not always match the arithmetic mean temperature of heating and cooling plates. Maystrenko et al. (27) as well consider this to be an eect of the largescale ow structures. At Ra = the numerically obtained temperature proles show a signicantly reduced dependency on the x 2 -position, which reects a more homogeneous mixing for this case. Analysis of the mean temperature proles In order to compare the DNS data with the experimental results both are normalised as x + 3 = 2Nu loc (x 3 x 3,wall ) (4) and = 2 θ, (5) where the horizontal hot or cold walls with a temperature of = are located at x + 3 = and x+ 3 = represents the edge of the thermal boundary layer at that location. The arithmetic mean temperature of the uid is then =. The temperature proles measured by bert et al. (28) in their periferical position (x 2.2) for Ra = 6. 7 are plotted in gure 4 together with the DNS data for Ra = In gure 5 experimental data for Ra = x 3 Figure 3: Temporally averaged temperature proles in the centre plane (x =.5) at various longitudinal positions for Ra = is compared to DNS data at Ra = It is observed that especially for the lower Ra comparison there are clear discrepancies between the DNS and the experimental data that cannot be related to the small dierence in Rayleigh number. specially the cold wall temperature proles dier signicantly. It is obvious that the experimentally measured temperature proles are signicantly more symmetrical than those obtained through the DNS which is believed to be a result of the dierent global ow structres. Better agreement for the higher Ra case is probably due to the fact that the large-scale structures vanish with increasing Ra. In order to investigate the theoretically predicted and experimentally measured temperature proles and to analyse whether the prole follows a power-law (6) or an exponential scaling (7), the diagnostic functions = (ln θ+ ) (ln x + 3 ) (6) and = (ln x + 3 ) (7) can be used. It has to be pointed out that the temperature proles at the higher Ra are in signicantly better agreement, especially in the near-wall region, where the DNS predict a lower gradient than the experimentally measured proles suggest. It follows from gure 4 that the powerlaw diagnostic function falls below the DNS data for x + 3. the higher Ra case data plotted in gure 5, however, is found to be in very good agreement. The exponential diagnostic functions plotted in gure 4 and 5 on the other hand reveal signicant dierences for x + 3, where the DNS data show a slower decrease of for x + 3 >. In gures 6 and 7 experimental and numerical proles are not compared at the same longitudinal position but for positions through the centre of a roll structure. for this case experimental and numerical data are found to be in excellent agreement as far as the power-law diagnostic function is concerned. A non-negligible oset is found for the exponential diagnostic function. However, the qualitative behaviour is found to be identical. Self-similar temperature proles In gures 4 to 7 we also compare our results to the ap- 79

4 Seoul, Korea, June 29 x x x x x x Figure 4: Analysis of the temporally averaged thermal boundary layers (BL) at the position x 2.2 and x =.5: temperature prole, power-law and exponential Ra = is compared to experimental data (cold BL:, hot BL: ) for Ra = 6 7 and the analytical function exp( x + 3.5(x+ 3 )2 ). Figure 5: Analysis of the temporally averaged thermal boundary layers (BL) at the position x 2.2 and x =.5: temperature prole, power-law and exponential Ra = is compared to experimental data (cold BL:, hot BL: ) for Ra = 6 8 and the analytical function exp( x + 3.5(x+ 3 )2 ). 8

5 Seoul, Korea, June 29 x x x x x x Figure 6: Analysis of the temporally averaged thermal boundary layers (BL) at the position x 2 2. and x =.5: temperature prole, power-law and exponential Ra = is compared to experimental data (cold BL:, hot BL: ) for Ra = 6 7 and the analytical function exp( x + 3.5(x+ 3 )2 ). Figure 7: Analysis of the temporally averaged thermal boundary layers (BL) at the position x 2 2. and x =.5: temperature prole, power-law and exponential Ra = is compared to experimental data (cold BL:, hot BL: ) for Ra = 6 8 and the analytical function exp( x + 3.5(x+ 3 )2 ). 8

6 Seoul, Korea, June 29 Furthermore, we showed that the temperature proles of the DNS obtained in a closed rectangular cell can (within reasonable accuracy) seem to collapse onto the approximation function proposed by Shishkina and Thess (29), irrespective of the longitudinal position if the local Nusselt number (thermal boundary layer thickness) is used for the normalisation of the proles. However, dierences between the temperature proles and the approximation function are observed when the diagnostic functions are evaluated. We therefore conclude that the approximation (8) gives an excellent approximation of temperature proles if the mean ow is parallel the the heated surfaces. x Figure 8: Normalised temperature proles of the DNS at Ra = plotted in inner coordinates for 2 longitudinal positions in the centre plane x =.5 (cold BL:, hot BL: ). proximation function exp( x + 3.5(x+ 3 )2 ) (8) for the temperature proles in RBC proposed by Shishkina & Thess (29). It can be observed from gures 4 and 5 that the function (8) gives a good approximation of the DNS data, even though the cold boundary layer prole in gure 4 does not follow this approximation. However, gure 8 illustrates that the function (8) gives a reasonable approximation for most temperature proles when they are normalised with Nu loc as suggested by bert et al. (28). The comparison of the diagnostic functions with the results obtained from (8) reveals that the approximation function approaches a value of zero much to rapidly in this case, since the DNS data derease almost linearly for x + 2 >. On the other hand the temperature proles through the centre of the roll structures show an excellent agreement with the function (8) for both diagnostic functions. We therefore conclude that despite the fact that the plots of seem to match for most of the temperature proles, the diagnostic functions reveal signicant dierences that reect a dependency of the prole on the longitudinal position. Therefore the approximation function (8) applies only to temperature proles with horizontal mean ow. CONCLUSIONS A comparison of Rayleigh-Bénard convection in a long rectangular enclosure obtained through DNS and detailed experimental measurements of the thermal boundary layers has been conducted. It was shown that the topological structure of the ow eld diers when comparing our experimental and numerical data. However, a very good agreement was observed for the temperature proles through the centre of the roll structures, which is reected by a good agreement of the diagnostic functions (6) and (7). However, the exponential diagnostic function reveals an oset between DNS and experimental data. It is also observed that both data sets are in better agreement at the higest Rayleigh numbers compared. RFRNCS Belmonte, A., Tilgner, A. and Libchaber, A., 994, Temperature and velocity boundary layers in turbulent convection, hys. Rev., 5(), pp Chorin, A.J., 968, Numerical solution of the Navier- Stokes equations, Math. Comp., Vol. 22, pp Du uits, R., Resagk, C., Tilgner, A., Busse, F.H. and Thess, A., 27, Structure of thermal boundary layers in turbulent Rayleigh-Bénard convection, J. Fluid Mech., Vol. 572, pp bert, A., Resagk, C. and Thess, A., 28, xperimental study of the temperature distribution and local heat ux for turbulent Rayleigh-Bénard convection of air in a long rectangular enclosure, Int. J. Heat and Mass Tranf., Vol. 5, pp Hölling, M. and Herwig, H., 26, Asymptitic analysis of heat transfer in turbulent Rayleigh-Bénard convection, Int. J. Heat and Mass Transf., Vol. 49, pp Kaczorowski, M., Shishkin, A., Shishkina, O. and Wagner, C., 28, Developement of a Numerical rocedure for direct simulations of turbulent convection in a closed rectangular cell, New Results in Numerical and xperimental Fluid Mechanics VI, Vol. 96, pp Kaczorowski, M. and Wagner, C., 29, Analysis of the thermal plumes in turbulent Rayleigh-Bénard convection based on well-resolved numerical simulations, J. Fluid Mech., vol. 68, pp Maystrenko, A., Resagk, C., and Thess, A., 27, Structure of the thermal boundary layer for turbulent Rayleigh- Bénard convection of air in a long rectangular enclosure, hys. Rev., Vol. 75(6), Schumann, U. Grötzbach, G. and Kleiser, L., 979, Direct numerical simulations of turbulence, rediction methods for turbulent ows, VKI-lecture series, Brussels. Sergent, A. and Le Quéré,., 28, Large-scale patterns in a rectangular Raylegh-Bénard cell Direct and Large ddy Simulations 7, Trieste, Italy Shishkina, O. and Wagner, C., 27, Boundary and interior layers in turbulent thermal convection in cylindrical containers, Int. J. Sci. Comp. Math., Vol. (2/3/4), pp Shishkina, O. and Thess, A., 29, Mean temperature proles in turbulent Rayleigh-Bénard convection of water, J. Fluid Mech., submitted. 82