International Conference Nuclear Energy for New Europe 22 Kranjska Gora, Slovenia, September 9-2, 22 www.drustvo-js.si/gora22 INFLUENCE OF THE BOUNDARY CONDITIONS ON A TEMPERATURE FIELD IN THE TURBULENT FLOW NEAR THE HEATED WALL ABSTRACT Robert Bergant, Iztok Tiselj Jožef Stefan Institute Reactor Engineering Division Jamova 39, SI- Ljubljana, Slovenia robert.bergant@ijs.si, iztok.tiselj@ijs.si Direct Numerical Simulation (DNS of the fully developed velocity and temperature fields in the two-dimensional turbulent channel flow was performed for friction Reynolds number Re τ = 5 and Prandtl number Pr =.7. Two thermal boundary conditions (BCs, isothermal and isoflux, were carried out. The main difference between two ideal types of boundary conditions is in temperature fluctuations, which retain a nonzero value on the wall for isoflux BC, and zero for isothermal BC. Very interesting effect is seen in streamwise temperature auto-correlation functions. While the auto-correlation function for isothermal BC decreases close to zero in the observed computational domain, the decrease of the auto-correlation function for the isoflux BC is slower and remains well above zero. Therefore, another DNS at two times longer computational domain was performed, but results did not show any differences larger than the statistical uncertainty. INTRODUCTION Fully developed channel or flume flow is of great importance from the scientific and engineering point of view. Several experimental and numerical studies have been carried out to increase the understanding of the mechanics of the near-wall turbulent flows. Among them, the Direct Numerical Simulation (DNS became an important research tool of the turbulent heat transfer in the last 5 years. DNS means precise solving of Navier-Stoke s equations without any extra turbulent models. It is capable to describe all relevant scales in the velocity and thermal fields (Kolmogorov micro scales []. Some reviews of DNS results are presented by Kim&Moin [2] and Kasagi et al. [3] at moderate Reynolds number and Prandtl numbers of about one or less. Later Kawamura, et al. [4], Na and Hanratty [5] performed the DNS of the turbulent channel at Prandtl numbers up to ten. All DNSs mentioned above were performed at isothermal wall boundary condition for dimensionless temperature. Fig. shows fluid section bounded by bottom and top walls. Walls are heated by a constant heat source and cooled by a flow, which is assumed to be incompressible. In this paper two different thermal boundary conditions on the walls, isothermal and isoflux BCs, are considered. If a flow of air (low density, low heat capacity and low thermal conductivity is heated by a thick metal wall, the influence of the turbulence on the wall is practically negligible due to the high density, high heat capacity and high conductivity of the wall [6]. In such case the first ideal boundary condition (isothermal BC is approached. If water flow is 26.
26.2 heated by a tiny metal foil, the influence of the turbulence on the wall is strong because of the small wall thickness and relatively high density, high heat capacity and high thermal conductivity of water in comparison with density, thermal conductivity and heat capacity of the foil. The second boundary condition (isoflux BC is almost reached in such case. 2 EQUATIONS AND NUMERICAL PROCEDURE The flow in the channel is assumed to be fully developed. Both walls of the channel are heated by constant heat source, while the fluid flows between them, as shown in Fig.. L 3 - HEATED WALL 2 FLOW L2=2h=2 L Z HEATED WALL X Y, Figure. Computational domain. The dimensionless equations normalized with flume width h, friction velocity u τ kinematic viscosity ν, and friction temperature T τ = qw ( u τ ρ f c pf can be found in the papers of Kasagi [3] or Kawamura [4]: u v = v u v = u t v v 2 v ( u u p lx θ t = Re τ v 2 ux ( u θ θ Re τ Pr u B ( (2 (3 Reτ = uτ h / ν is the friction Reynolds number and Pr is Prandtl number. Terms l v x (unit vector in streamwise direction and u x u B appear in the equations (4 and (5 due to the numerical scheme that requires boundary conditions in streamwise and spanwise directions. u B is bulk mean velocity averaged over the time and entire volume. Therefore, the Reynolds number Re can be calculated as Re = ub Reτ. Dimensionless wall temperature difference is defined as θ ( x y, z, t ( x, y, z t Tw T,, =. (4 Tτ As can be seen from Eqs. (-3 temperature is assumed to be a passive scalar. This assumption introduces two approximations: neglected buoyancy, 2 neglected temperature dependence of the material properties - especially viscosity and heat conductivity. Results of Proceedings of the International Conference Nuclear Energy for New Europe, Kranjska Gora, Slovenia, Sept. 9-2, 22
26.3 the present study are thus very accurate only for the systems, where the temperature differences are not too large, while some caution is required for the systems, where the temperature differences are not negligible. The outer sides of the walls are adiabatic, which mean that heat released in the walls is entirely transferred to the fluid. As mentioned before, two ideal boundary conditions are considered. The isothermal boundary condition at the wall is θ =, (5 while the isoflux boundary condition is imposed with the mean dimensionless temperature at heated wall fixed to zero and the ( = ± = θ y (6 dθ dy ( y = ± =. (7 It is evident that velocity at the interface wall-fluid is zero. The equations are solved with pseudo-spectral scheme using Fourier series in x and z directions and Chebyshev polynomials in the wall-normal y direction. Numerical procedure and the code of Gavrilakis [7] modified by Lam and Banerjee [8], and Lam [9] is used to solve the continuity and momentum equations. Equations (-3 are periodic in streamwise (x and spanwise (z directions. All our DNS simulations were performed at Re = 458 (Re τ = 5 and Pr =.7 in two different size of computational domains, shorter and longer domains (see table. Fig. 6 shows that the length of the shorter domain might not be long enough. Therefore another DNS with two times longer domain but with the same grid spacing in all three directions was performed to check the differences. It should be stressed that resolution that was chosen, is capable to describe all velocity and thermal structures according to the Kolmogorov theory []. Distances in wall units instead of IS units, are useful because turbulent flows with different Reynolds numbers can be easily compared in the near wall region. In wall units the height of the channel is equal to the two times friction Reynolds number. In our case the height is 3 wall units, which is according to the Fig. equal to 2 units of the computational box (units that are used in the computer code. The averaging was performed over time interval t =36 respectively t =9 after the fully developed flow was achieved (6 respectively time steps. Table : Summary of DNS calculations. time simulation grid comp. size resolution averaging Pr=.7 Re τ =5 (x, z, y x y z step time shorter domain 28x97x64 5π x 2 x π 8.4.8-4.9 7.4.9 36 longer domain 256x97x64 π x 2 x π 8.4.8-4.9 7.4.9 9 Proceedings of the International Conference Nuclear Energy for New Europe, Kranjska Gora, Slovenia, Sept. 9-2, 22
26.4 RESULTS AND DISCUSSION The mean temperature profiles are given in Fig. 2. As can be seen, the type of thermal boundary condition does not affect the mean temperature profiles. Comparison of shorter and longer domain shows that two times longer length of the computational size does not give us different results. θ 8 6 4 2 8 6 4 short_q long_q short_t long_t 2, y Figure 2: Profiles of mean temperature (Q - isoflux BC, T - isothermal BC. The differences between isothermal and isoflux boundary conditions can be clearly seen in Fig. 3. The thermal boundary condition affects only the temperature fluctuations, so the velocity fluctuations are not shown in this paper. In the case of isoflux thermal boundary condition the θ RMS remains constant across the viscous sublayer. In both cases there are no differences due to the different computational domain. Differences, which are not larger than 2% can be treated as statistical uncertainties. 3 RMS_θ 2,5 2,5 short_q short_t long_q long_t,5, y Figure 3: Profiles of RMS temperature fluctuations. Fig. 4 shows the turbulent axial heat flux versus the dimensionless distance from the wall. The location of the peak turbulent axial heat flux values is approximately at 5 wall units. The minor differences between thermal BCs appear near the wall (see Tiselj et. al. [] for details. The location of a peak is at the same distance from wall, the only visible difference is the height of heat flux peak. The peak in the case of isoflux BC is about 5% Proceedings of the International Conference Nuclear Energy for New Europe, Kranjska Gora, Slovenia, Sept. 9-2, 22
26.5 higher than in the case of isothermal BC. If we compare heat flux for shorter and longer computational domain, one can see that there are no significant differences between the DNSs of different streamwise lengths. 7 6 5 θu 4 3 2 short_q short_t long_q long_t y 2 3 4 Figure 4: Profiles of the turbulent axial heat flux. The profiles of the normal heat fluxes are plotted in Fig. 5. The distinct feature is that the thermal wall boundary condition does not affect turbulent normal heat flux. The differences between different computational lengths are also negligible. However, the longer computational domain does not give any improvements in the meaning of heat fluxes.,8,7,6 θw,5,4,3,2 short_q short_t long_q long_t, y 2 3 4 Figure 5: Profiles of the turbulent wall normal heat flux. Fig. 6 shows two-point auto-correlation functions of temperature (for two boundary conditions and streamwise velocity in streamwise direction calculated at distance y =5. from the heated wall. Auto-correlation function in the numerical simulation is used to find out if the dimensions of the channel are suitable. Because of the periodical boundary conditions the fluid must be well mixed in the first half of the channel, which means that the situation in the middle of the channel has nothing to do with the situation at the entrance (or end of the channel. The points at the end of the channel are neighbors to the points at the entrance of the channel due to the periodic boundary conditions. This is the reason why the auto-correlation functions are calculated only in the first half of the channel. The influence of the periodic Proceedings of the International Conference Nuclear Energy for New Europe, Kranjska Gora, Slovenia, Sept. 9-2, 22
26.6 boundary conditions on the DNS are avoided, when the periodicity length ensures that autocorrelation functions of all fields fall to zero in the directions in which the periodic boundary conditions are imposed. An important difference is seen between two different boundary conditions (Fig. 6. The auto-correlation function for isothermal BC decreases close to zero in the observed computational domain, while the auto-correlation function for the isoflux BC decreases slower and remains well above zero. Therefore, the goal of the present study is to verify if the length of the computational domain for the isoflux BC should be longer to satisfy the generally accepted criteria for DNS about the sufficient length of the computational domain. For this reason auto-correlation function of isoflux BC at two times longer computational domain is added. In this concrete example (longrtt_q in Fig. 6 function decreases to zero and satisfies the sufficient length of the computational domain. The physical background of the differences between the auto-correlation functions (Fig. 6 at different thermal boundary conditions is not completely clear yet. We attempted to get a clearer physical picture with a new DNS study, which has been performed at Pr=.7 and with two times longer computational domain (2359 vs. 472 wall units but with the same grid resolution. Besides thermal BC comparison, also DNSs at "standard" and two times longer domain was performed (see Figs. -5. These results lead to next conclusions: The differences between the DNS in the "standard" length computational domain and the DNS in the extended computational domain did not show any differences larger than the statistical uncertainty for first-order statistics and also for the power spectra. 2 The periodicity length, which is long enough for the velocity field as an origin of the turbulence (Fig. 6, is long enough also for the passive scalar fields, despite the behavior of the streamwise two-point correlation at isoflux BC shown in Fig. 6. RR,2,8,6,4,2 shortruu shortrtt_q shortrtt_t longrtt_q -,2 5 5 2 25 x Figure 6: Two-point auto-correlation in streamwise direction at y =5.: temperature (isothermal, isoflux and streamwise velocity. Fig. 7 shows the spanwise two-point auto-correlations for temperatures at two different BCs and for streamwise velocity. The position of the minimum in the streamwise velocity correlation at approximately 5 wall units shows the presence of the low speed streaks. This minimum shows that certain flow structures - coherent structures exist close to the wall. These structures are the streamwise vortices that can be measured with the position of the first minimum. The first minimum shows the average distance between the high speed and low speed regions. Temperature minimums in fig. 7 show that they are slightly shifted to the right comparing to the velocity minimum. The temperature, which is a passive scalar do not have influence on the turbulence and the coherent structures close to the walls. Proceedings of the International Conference Nuclear Energy for New Europe, Kranjska Gora, Slovenia, Sept. 9-2, 22
26.7,2,8,6 Ruu Rtt_Q Rtt_T RR,4,2 -,2 -,4 5 5 2 25 z Figure 7: Two-point auto-correlation in spanwise direction at y =5.: temperature (isothermal, isoflux and streamwise velocity. 3 CONCLUSIONS A direct numerical simulation of the fully developed channel flow was performed at friction Reynolds number Re τ = 5, and Prandtl number Pr =.7 (air. The fluid temperature was a passive scalar and therefore did not have the influence on the turbulence. Two different thermal boundary conditions at the heated wall were carried out. The auto-correlation functions are used to verify if the dimensions of the computational domain are suitable. Because of the periodical boundary conditions the fluid must be well mixed in the first half of the channel, where the auto-correlation functions must fall to zero. The temperature streamwise auto-correlation function at isoflux BC remained well above zero, therefore another DNS at two times longer domain was performed. The results did not show any differences larger than statistical error. On the other side, spanwise auto-correlation functions are used also to show the presence of the coherent structures of the near wall flow. These structures are streamwise vortices with the characteristic diameter of approximately wall units. Thermal boundary conditions did not affect the mean temperature profiles. The main difference in the first order statistics was seen in temperature fluctuations near the wall. Temperature fluctuations retained a nonzero value on the wall for isoflux boundary condition, and zero for isothermal wall boundary condition. The minor differences were seen in the turbulent axial heat flux profiles, where the peak in the case of isoflux BC is about 5% higher than in the case of isothermal BC. NOMENCLATURE h k L,L 3 p Pr q w Re τ Rtt channel half height wave number streamwise and spanwise length of turbulent box pressure Prandtl number wall-to-fluid heat flux friction Reynolds number auto-correlation function for temperature Proceedings of the International Conference Nuclear Energy for New Europe, Kranjska Gora, Slovenia, Sept. 9-2, 22
26.8 Ruu auto-correlation function for streamwise velocity t time T τ = qw /( u τ ρ f c pf friction temperature u,v,w velocity components in x, y and z directions u τ = τ w / ρ friction velocity u B bulk mean velocity x streamwise distance y distance from the wall z spanwise distance Greek α = λ / ρc p thermal diffusivity ϑ = ( T w T / Tτ dimensionless temperature difference λ thermal conductivity ν kinematic viscosity ρ density r x unit vector in x direction (,, Subscripts and superscripts ( f fluid ( normalized by u τ, T τ, ν REFERENCES [] Tennekes, H., Lumley, J.L., 972, "A First Course in Turbulence", MIT Press, Cambridge, MA. [2] Kim, J., Moin, P., 989, "Transport of Passive Scalars in a Turbulent Channel Flow", Turbulent Shear Flows VI, Springer-Verlag, Berlin, pp. 85. [3] Kasagi, N., Tomita, Y., Kuroda, A., 992, "Direct Numerical Simulation of Passive Scalar Field in a Turbulent Channel Flow", Journal of Heat Transfer -Transactions of ASME, Vol. 4, 598-66. [4] Kawamura, H., Ohsaka, K, Abe, H., Yamamoto, K., 998, "DNS of Turbulent Heat Transfer in Channel Flow with low to medium-high Prandtl number fluid", International Journal of Heat and Fluid Flow, Vol. 9, 482-49. [5] Na, Y., Hanratty, T.J., 2, "Limiting Behavior of Turbulent Scalar Transport Close to a Wall", International Journal of Heat and Mass Transfer, Vol. 43, pp. 749-758. [6] Tiselj, I., Bergant, R., Mavko, B., Bajsič, I., Hetsroni, G., October 2, ýdns of turbulent heat transfer in Channel Flow with Heat Conduction in the Solid Wallý, Journal of Heat Transfer, Vol. 23, pp.: 849-857. [7] Gavrilakis S., Tsai, H.M., Voke, P.R., Leslie, D.C., 986, "Direct and Large Eddy Simulation of Turbulence", Notes on Numerical Fluid Mechanics Vol. 5, ed. U. Schumann, R. Friedrich, Vieweg, Braunschweig, D.B.R., pp. 5. [8] Lam, K.L., Banerjee, S., 988, "Investigation of Turbulent Flow Bounded by a Wall and a Free Surface", Fundamentals of Gas-Liquid Flows, Edited by Michaelides, Sharma, Vol. 72, 29-38, ASME, Washington DC. [9] Lam, K.L., 989, "Numerical Investigation of Turbulent Flow Bounded by a Wall and a Free-Slip Surface", Ph.D. Thesis, Univ. Calif. Santa Barbara. [] Tiselj, I., Pogrebnyak, E., C. Li, Mosyak, A., Hetsroni, G., 2, "Effect of wall boundary condition on scalar transfer in a fully developed turbulent flume", Physics of Fluids, 3 (4, pp.28-39. Proceedings of the International Conference Nuclear Energy for New Europe, Kranjska Gora, Slovenia, Sept. 9-2, 22