Direct Numerical Simulation of Convective Turbulence

Transcription

1 Convective turbulence 1 Direct Numerical Simulation of Convective Turbulence Igor B. Palymskiy Modern Humanitarian University, Novosibirsk Branch, Russia palymsky@hnet.ru Turbulent convective flow of water in horizontal layer with free and rigid horizontal boundaries, arising at heating from below, is numerically simulated using the Boussinesq model without any semiempirical relationships (DNS) in 2-D case. It is shown that numerical results (free) have good agreement with experimental data and results of numerical simulations with rigid boundary conditions at enough high supercriticality and that such simulation reflects the transition to hard turbulence. We study also the role of boundary conditions (free-rigid). Key words: Numerical simulation, DNS, convective turbulence.

2 2 Title of the journal. Volume X no X/ Introduction Many researchers studied thermal Rayleigh-Benard convection using numerical simulation. As rule, they used spectral methods with periodic boundary conditions. In numerical simulations were revealed secondary stationary, periodic, quasiperiodic and stochastic regimes (Palymskiy 2003a). Some authors have performed numerical simulations for high supercriticality with free (Sirovich et al. 1989a,b; Cortese et al. 1993; Malevsky 1996) and rigid (Kerr 1996; Werne et al. 1990; Grotzbach 1982; Woerner 1997) boundary conditions on the horizontal planes. The results of correctly performed numerical simulations with rigid boundary conditions, as rule, have good agreement with experimental data. On the other hand, it seems natural that results of simulations with free and rigid boundary conditions must draw together at enough high supercriticality. The question of togetherness of solutions with free and rigid boundary conditions is practical, as using of free boundary conditions very simplifies the DNS of turbulent convection, simple and efficient numerical algorithms are created using the formulas of linear stability theory (Palymskiy 2000). It is said in work (Cortese 1993) that difference between solutions with free and rigid boundary conditions is in thin layers near bottom and upper boundaries, since rigid boundaries impose zero values of vertical vorticity on them. Moreover, using of free boundary conditions bring to decreasing of effective Prandtl number because of disregard of viscous boundary layer on the horizontal planes. It seems that the available data of numerical simulations with free boundary conditions shows a bad agreement with experimental data and data of numerical simulations with rigid boundary conditions. For instance, in table 1 we compare the calculating Nusselt numbers, here and below r = Ra/Ra cr is supercriticality, Ra and Ra cr are Rayleigh number and critical value of Rayleigh number, respectively. Kerr 1996, 3-D, rigid, air Werne et al. 1990, 2-D, rigid, water Sirovich et al. 1989a,b, 3-D, free, air Cortese et al. 1993, 3-D, free, air Malevsky 1996, 3-D, free, air Threlfall 1975, exp. in gaseous He Present, 2-D, free, water ± Table 1. Comparing of Nusselt Number at r = 9800 It is seen from table 1 that results of numerical simulations with free boundary conditions on the horizontal planes (Sirovich et al. 1989a,b; Cortese et al. 1993; Malevsky 1996) have a bad agreement, but the result of present work is close to experimental data and data of numerical simulations with rigid boundary conditions.

3 Convective turbulence 3 The same situation at r 1000 may be seen in work (Moore et al. 1973, 2-D, free, water), (Werne et al. 1990, 2-D, rigid, water) and (Chu et al. 1973, experiment, water). In our opinion, the possible reasons of solution deviations with free boundary conditions are insufficient space resolution and decreasing of effective Prandtl number at using of free boundary conditions in these simulations. It is shown by recently investigations (Palymskiy 2003b) that results of 2-D simulations with free boundary conditions on the horizontal planes are close to both the results of numerical simulations with rigid boundary conditions on the horizontal planes and experimental data at enough high values of supercriticality. The important questions are role of boundary conditions (free-rigid) at DNS of turbulent convection and comparison with experimental data. It is shown below that 2-D simulations with free boundary conditions on the horizontal planes reflect also transition from soft toward hard turbulence observed by Ghicago research group (Wu et al. 1992, Castaing et al. 1989). The aim of this work is the proof of the togetherness of solutions of convection problems with free and rigid boundary conditions and of experimental data at enough big value of supercriticality, the investigation of boundary conditions role (free-rigid) and study of transition from soft toward hard turbulence. 2. Problem Formulation Turbulent convectional flow of water in a horizontal layer heating from below numerically is simulated. The fluid is viscous and incompressible. The flow is timedependent and two-dimensional. Boundaries of a layer are isothermal and free (stress-free) or rigid (no-slip for velocity). The model Boussinesq is used without semiempirical relationships. The dimensionless system of equations in terms of deviations from an equilibrium solution, representation of problem solution in the form of sum of eigenfunctions of linear stability theory, the boundary conditions, the special numerical method, testing and the results of linear and non linear analysis (on model non linear system) for free boundary conditions are described in works (Palymskiy 2000, 2004). In our simulations with free boundary conditions we used up to 257*63 harmonics at supercriticality up to r = For simulations with rigid boundary conditions we used the spectral representation in x-direction and finite differences in y-direction with uniform mesh. We used up to 65 harmonics in x-direction and 65 points in y-direction. The simulations were performed for supercriticality up to 2000 for rigid boundary conditions. We simulated the convection flows for the Prandtl number Pr is equal to 2 or 10, for all simulations the dimensionless periodicity interval is equal to 2π, the dimensionless distance between the planes is equal to 1.

4 4 Title of the journal. Volume X no X/ DNS of Turbulent Convection and Role of Boundary Conditions At figs. 1-5 below y denotes transverse coordinate. Fig. 1 represents the average temperature profile. At fig. 1 sign denotes experimental results (Deardorff et al. 1967, r = 5900, air), dash line experimental results (Chu et al. 1973, r = 5500, water), solid line - results of present work with free boundary conditions (r = 6000, Pr = 10). Fig.2 represents the rms of vertical velocity pulsations. Here sign denotes the experimental results (Deardorff et al. 1967, r = 5900, air), sign box experimental result (Fitzjarrald 1976, r = 5900, air), solid line - results of present work (free) (r = 5500, Pr = 10). Figs. 1 and 2 demonstrate that results of numerical 2-D simulation with free boundary conditions on the horizontal planes are consistent with experimental data in water and air for supercriticality r Fig. 2 also shows, that such numerical simulation represents the profile of vertical velocity fluctuations and maximal value of one very well (for r 6000). Figure 1. Average temperature profile Figure 2. Rms of vertical velocity pulsations Fig.3 represents the profiles of rms temperature pulsations at r = 950, here solid line is simulation with rigid and points are simulation result with free boundary conditions, dashed line is theoretical law (Balachandar et al. 1991). We can see a good agreement. Fig.4 represents the rms of temperature pulsations at moderate supercriticality r = Here sign denotes the experimental results (Deardorff et al. 1967, r = 1470, air), sign experimental result (Fitzjarrald 1976, r = 1400, air), sign - experimental result of Somerscales, 1965 (r = 1170, data is from work (Deardorff et al. 1967)), sign experimental result (Wu et al. 1992, r = 1250, gaseous He), solid line - results of present work (r = 1250, Pr = 10, free). Fig. 5 represents the rms of temperature pulsations at high supercriticality r = Here sign denotes the experimental results (Deardorff et al. 1967, r = 5900, air), sign box experimental result (Fitzjarrald 1976, r = 5900, air), sign circle experimental result (Wu et al. 1992, r = 5900, gaseous He), solid line - result of present work (free) with [129*31] of space harmonics and dash line - result of present work (free) with [257*65] of space harmonics (r = 6000, Pr = 10). Figs.4 and 5 show a good agreement with

5 Convective turbulence 5 experimental data (Deardorff et al. 1967), some waviness of our numerical results is coupled possibly with Gibbs effect. Figure 3. Temperature pulsations at r = 950 Figure 4. Temperature pulsations at r = 1250 Using the calculated stream function field we calculated the rms value of wavelength. Fig. 6 represents rms of wavelength d versus supercriticality r up to r = 400. Here solid line is numerical result of present work (Pr = 2, free), dash line experimental results of the several authors assembled in work (Fitzjarrald 1976, air), dadot line - experimental results of the several authors assembled in work (Daly 1974, air), sign denotes experimental results (Farhadien et al. 1974, water) and sign denote the result of numerical simulation (Grotzbach 1982). Fig. 6 shows that our numerical simulation represents wavelength rightly. The deviations (for r 150) (up to 30 % at r 400) possibly are coupled with restriction of 2-D simulation. Figure 5. Temperature pulsations at r = 6000 Figure 6. Wavelength versus r

6 6 Title of the journal. Volume X no X/2002 The rms of temperature pulsations in the center (at y = 0.5) is plotted in fig. 7 versus supercriticality r up to Here is numerical result of present work (Pr = 10, free), solid line 3-D numerical simulation (Kerr 1996, air), dash line experimental result (Fitzjarrald 1976, air) and dadot line experimental result (Wu et al. 1992, gaseous He), sign x denotes experimental result (Thomas 1957, only single point, air), sign circle - experimental results (Deardorff et al. 1967, air), sign numerical result (Grotzbach 1982), sign black box present numerical result (Pr = 10, rigid) and sign - numerical result (Sirovich et al. 1989b) with free boundary conditions. Fig. 7 shows that data of our 2-D numerical simulation with free boundary conditions is consistent with experimental data and results of numerical simulations with rigid boundary conditions, a big scatter of numerical data up to r 1000 mirrors the existent scatter in experimental data (see for example (Deardorff et al. 1967). Possible, some accuracy loss of present numerical result for high supercriticality (r ) is coupled with restriction of 2-D simulation. Fig.8 represents the profiles of rms horizontal velocity pulsations at r = 950, here solid line is simulation with rigid and points are simulation result with free boundary conditions. The simulation results have reasonably good agreement, the differences at y = 0 and y = 1 are because of difference in boundary conditions. Such situation was observed up to r = Figure 7. Temperature pulsations versus r Figure 8. Pulsations of horizontal velocity Fig.9 represents the Nusselt numbers, here solid lines experimental result (Chu et al. 1973) (water), experimental result of Schmidt R.J. and Saunders D.A., 1938 (water, result is from work (Rossby 1969)), sign - 3-D numerical simulation (Kerr 1996, rigid, air), sign x 2-D numerical result (Werne et al. 1990, rigid, water), black point - present result with free boundary conditions, sign present results with rigid boundary conditions.

7 Convective turbulence 7 Fig. 10 represents Nusselt number versus supercriticality at 1000 r Here sign denotes the result of present work (Pr = 10, free), solid line - experimental result (Chu et al. 1973, water), sign - 3-D numerical simulation (Kerr 1996, air), dash line - the result of the experimental work (Fitzjarrald 1976, air), sign x - 2-D numerical simulation (Werne et al. 1990, water), sign - results of 3-D numerical simulations (Sirovich et al. 1989a,b; Cortese et al. 1993; Malevsky 1996) with free boundary conditions on the horizontal planes. Figure 9. Comparing of Nusselt numbe Figure 10. Nusselt number up to r = In table 2 we compare the calculating values of Nusselt number of present work (Pr = 10), results of 2-D simulation (Werne et al. 1990, spectral method, Pr = 7, up to 129*129 harmonics), results of 3-D simulation (Kerr 1996, spectral method, Pr = 0.71, up to 288*288*96 harmonics) and experimental data (Fitzjarrald 1976, air) and (Chu et al. 1973, water) for values of supercriticality 750 r Data of works (Kerr 1996, Fitzjarrald 1976, Chu et al. 1973) were calculated on the approximation curves of these works. The results have reasonably good agreement. r Present, 2-D, free Werne et al. 1990, 2-D, rigid Kerr 1996, 3-D, rigid Fitzjarrald 1976, exp. in air Chu et al. 1973, exp. in water Table 2. Comparing of Nusselt Number

8 8 Title of the journal. Volume X no X/2002 A least squares fit to Nusselt number versus r from our simulations (free) is Nu = r 0.302, this formula is right for r 700. This law practically coincides with experimental law from (Rossby 1969, Nu = r 0.3 ) and close to experimental law (Fitzjarrald 1976, Nu = r 0.3 ). Figs.1 10 show a good agreement of simulation results with free boundary conditions, experimental data and simulation results with rigid boundary conditions at r > 700. But, the profiles of rms vertical velocity pulsations have a big difference for simulations with free and rigid boundary conditions, it is approximately in Pr 1/3 times. Fig.11 represents the values of rms vertical velocity pulsations in centre between the planes, here solid lines experimental result of (Fitzjarrald 1976, air), result of 3-D numerical simulations (Kerr 1996, rigid, air), dash line - experimental result of (Garon et al. 1973, water, divided by (Pr/0.71) 1/3 = 1.978), sign - present numerical simulation (free), sign - present numerical simulation (rigid, divided by (Pr/0.71) 1/3 = 2.415) and sign experimental result of (Deardorff et al. 1967, air). Fig.11 shows that rms of vertical velocity pulsations depends from kind of boundary conditions (free-rigid). The results of present work (rigid) and experimental results in water (Garon et al. 1973) must be divided by (Pr/0.71) 1/3 for comparing with results in air and gaseous He. Figure 11. Vertical velocity pulsations We have compared the results of 2-D simulations with free and rigid boundary conditions of turbulent Rayleigh-Benard convection at Prandtl number is equal to 10. It is revealed that the profiles of mean temperature, rms of temperature and horizontal velocity pulsations are close. The Nusselt numbers are close too. But rms of vertical velocity pulsations strongly depends from kind of boundary conditions (free-rigid).

9 Convective turbulence 9 4. Soft and Hard Turbulence The works of Chicago research group (Wu et al. 1992, Castaing 1989) show that at supercriticality r = Ra/Ra cr is equal to approximately 7000 in experiments on turbulent Rayleigh-Benard convection the transition toward new mode of turbulent convection take place. The modes of turbulence were named as soft turbulence (at r < 7000) and hard turbulence (at r > 7000). After transition to hard turbulence, the form of probability density for temperature pulsations in the centre of cell changes, the gaussian distribution replaces by exponential. In this section all simulations were performed with free boundary conditions. Fig.12 represents the probability density function for temperature pulsations in the centre of cell at r = 6000 (points), we may see that this distribution is gaussian. Here c = is the value of rms temperature pulsations in the middle between the plates. Fig.13 represents also the probability density function for temperature pulsations in the centre of cell (points), but at r = 10000, we may see that this distribution is exponential, here c is equal to The result of 3-D numerical simulation in air at r = 9800 (dash line), experimental result of Chicago group (solid line 2) and theoretical result of Yakhot (solid line 1) are shown also on fig. 13, these curves are constructed from estimates (Sirovich et al. 1989b). We may see the differences only in neighbourhood of mean value of temperature pulsations. Figure 12. Probability density at r = 6000 Figure 13. Probability density at r = If the distributions the same shown in figs.12 and 13 is fitted by exponential distribution: α p ( T ') = exp( α T ' / c), 2 c

10 10 Title of the journal. Volume X no X/2002 then we obtain result shown on fig. 14, when we show α versus supercriticality r, here points present result of 2-D simulation with free boundary conditions, sign - numerical result (Sirovich et al. 1989b), sign numerical result (Werne et al. 1990) and dash line experimental result (Wu et al. 1992). Experimental result (Wu et al. 1992) was received for high values of supercriticality r Universality of probability density function (independence of α from the supercriticality) was denoted also in experimental work (Castaing 1989). The result of present calculations also shows approximately universality at r and is close to experimental value. Fig.15 represents the log-log dependence of temperature pulsations in centre between the planes versus supercriticality at 4000 r In this distribution the big kink at r = 7000 is neatly seen. Figure 14. Alpha versus supercriticality Figure 15. Temperature pulsations versus r Fig.16 represents the values of vertical velocity pulsations in centre between the planes versus supercriticality at 4000 r (points). This distribution also kink has at r = Here sign numerical result (Kerr 1996), sign numerical result (Sirovich et al. 1989b) and sign experimental result in air (Deardorff et al. 1967). Fig.17 represents the Nusselt number versus supercriticality at 4000 r Here sign present result, sign numerical result (Kerr 1996) (Nu ~ r ), sign - numerical result (Werne et al. 1990) (Nu ~ r ), solid lines are: 1 experimental result in air of W. Mull and H. Reiher, 1930 (Nu ~ r 1/3, result is in interpretation of works (Rossby 1969) and (Malkus 1954)), 2 experimental result in acetone (Malkus 1954) (Nu ~ r ), 3 experimental asymptotic derived in pressurizes gases (Fleischer 2002) (Nu ~ r ), dash line experimental result in gaseous He (Wu et al. 1992) (Nu ~ r ).

11 Convective turbulence 11 Figure 16. Vertical velocity pulsations Figure 17. Nusselt number versus supercriticality The distribution of Nusselt number versus supercriticality also shows the kink in area of transition point r = Fig. 17 clear shows that in experimental and numerical investigations the various Nu-Ra power laws are observed. The derived numerical asymptotic Nu~r 1/3 confirms by some experimental data, but some experimental data shows a power law close to Nu~r 2/7 and other (see (Fleischer 2002) for more detail consideration). 5. Conclusion In conclusion, we emphasize that results of our 2-D simulations with free boundary conditions on the horizontal planes are closely allied both with results simulations with rigid boundary conditions on the horizontal planes and experimental data at enough high values of supercriticality. In particular, the values of Nusselt number at r > 700 describe by formula: Nu = 1.223r 0.302, it is close to experimental data and data of numerical simulations with rigid boundary conditions. The profiles of mean temperature, rms of temperature and vertical velocity pulsations are typical for turbulent convection. For r 6000 these profiles and experimental data are close. Some loss of simulation accuracy at high supercriticality possibly is coupled with restriction of 2-D simulation. The scatter of numerical data (up to r 1000) mirror the existent scatter in experimental data. We have compared the results of 2-D simulations with free and rigid boundary conditions of turbulent Rayleigh-Benard convection at Prandtl number is equal to 10. It is revealed that the profiles of mean temperature, rms of temperature and horizontal velocity pulsations are close at r > 700. The Nusselt numbers are close

12 12 Title of the journal. Volume X no X/2002 too. But a big difference observes in rms of vertical velocity pulsations, here we observed dependence from kind of boundary conditions (free-rigid). We emphasize also, that our 2-D numerical simulation with free boundary conditions mirrors the replace of turbulence modes. After transition to hard turbulence, the form of probability density for temperature pulsations in the centre of cell changes, the gaussian distribution replaces by exponential. The distributions of Nusselt number, temperature and vertical velocity pulsations versus supercriticality show the kink in area of transition point. The place of transition point (r 7000) coincides with experimental value. 6. Bibliography Balachandar S., Sirovich L., Probability Distributions in Turbulent Convection, Phys. Fluids, 1991, vol. 3A no. 5, p Castaing B., Gunaratne G., Heslot F., Kadanoff L., Libchaber A., Thomae S., Wu X.-Z., Zaleski S., Zanetti G., Scaling of Hard Thermal Turbulence in Rayleigh-Benard Convection, J. Fluid Mech., vol. 204 no. 1, 1989, p Chu T.Y., Goldstein R.J., Turbulent Convection in a Horizontal Layer of Water, J. Fluid Mech., vol. 60 no. 1, 1973, p Cortese T., Balachandar S., Vortical Nature of Thermal Plumes in Turbulent Convection, Phys. Fluids, vol. 5A no. 12, 1993, p Daly B.J., Turbulence Transitions in Convective Flow. J. Fluid Mech., vol. 64, 1974, p Deardorff J.W., Willis G.E., Investigation of Turbulent Thermal Convection Between Horizontal Plates, J. Fluid Mech., vol. 28, 1967, p Farhadien R., Tankin R. S., Interferometric Study of Two-Dimensional Benard Convection, J. Fluid Mech., vol. 66 no. 4, 1974, p Fitzjarrald D.E., An Experimental Study of Turbulent Convection in Air, J. Fluid Mech., vol. 73, 1976, p Fleischer A.S., Goldstein R.S., High-Rayleigh-Number Convection of Pressurized Gases in a Horizontal Enclosure, J. Fluid Mechanics, vol. 469, 2002, p Garon A.M., Goldstein R.J., Velocity and Heat Transfer Measurements in Thermal Convection, Phys. Fluids, vol. 16 no. 11, 1973, p Grotzbach G., Direct Numerical Simulation of Laminar and Turbulent Benard Convection, J. Fluid Mech., vol. 119, 1982, p Kerr R.M., Rayleigh Number Scaling in Numerical Convection, J. Fluid Mech., vol. 310, 1996, p Malevsky A.V., Spline-Characteristic Method for Simulation of Convective Turbulence, J. Comput. Phys., vol. 123, 1996, p

13 Convective turbulence 13 Malkus W.V.R., Discrete Transitions in Turbulent Convection, Proc. Roy. Soc., vol. 225 A, 1954, p Moore D.R., Weiss N.O., Two-Dimensional Rayleigh-Benard Convection, J. Fluid Mech., vol. 58, 1973, p Palymskiy I.B., Metod Chislennogo Modelirovaniya Konvektivnykh Techeniy, Vychislitel nye texnologii, vol. 5 no. 6, 2000, p.53-61; Palymskiy I. B., Determinism and Chaos in the Rayleigh-Benard Convection, Proceeding of the Second International Conference on Applied Mechanics and Materials ICAMM 2003a, Durban, South Africa, January 2003, p ; Palymskiy I.B., Direct Numerical Simulation of Turbulent Rayleigh-Benard Convection, Proceedings of the International Conference: Advanced Problems in Thermal Convection, November 2003b, Perm, Russia, p ; full text of report on Palymskiy I.B., Linejnyj I Nelinejnyj Analiz Chislennogo Metoda Rasscheta Konvektivnykh Techenij, Sibirskij Zhurnal Vychislitel noj Matematiki, vol. 7 no.2, 2004, p ; Rossby H. T., A Study of Benard Convection With and Without Rotation, J. Fluid Mech., vol. 36, 1969, p Sirovich L., Balachandar S., Maxey M.R., Simulations of Turbulent Thermal Convection, Phys. Fluids, vol. 1A no. 12, 1989a, p Sirovich L., Balachandar S., Maxey M.R., Numerical Simulation of High Rayleigh Number Convection, J. Scientific Computing, vol. 4 no.2, 1989b, p Thomas D. B., Townsend A. A., Turbulent Convection over a Heated Horizontal Surface, J. Fluid Mech., vol. 2, 1957, p Threlfall D. C., Free Convection in Low-Temperature Gaseous Helium, J. Fluid Mech., vol. 67 no. 1, 1975, p Veronis G., Large-Amplitude Benard Convection, J. Fluid Mech., vol. 26 no. 1, 1966, p Werne J., DeLuca E.E., Rosner R., Numerical Simulation of Soft and Hard Turbulence: Preliminary Results for Two-Dimensional Convection, Phys. Rev. Lett., vol. 64 no. 20, 1990, p Woerner M., simulation/e_index.html, Wu X.-Z., Libchaber A., Scaling Relations in Thermal Turbulence: The Aspect-Ratio Dependence, Physical Review, vol. 45A no. 2, 1992, p

14 14 Title of the journal. Volume X no X/2002 Igor Palymskiy is Professor of Modern University for Humanities, Novosibirsk Branch, Mathematics Department. His main scientific interests are Direct Numerical Simulation of Turbulent Flows and Flows with Hydrodynamical Instabilities.