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Direct Numerical Simulation and Rans Modeling of Turbulent Natural Convection for Low Prandtl Number Fluids

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Direct Numerical Simulation and Rans Modeling of Turbulent Natural Convection for Low Prandtl Number Fluids March 19, 2004 10:27 Proceedings of ASME/JSME Symposium 2004 5th International Bi-Annual ASME/JSME Symposium on computational technology for fluid / thermal / chemical / stressed systems with industrial applications San Diego / La Jolla, California, USA, July 25-29, 2004 DIRECT NUMERICAL SIMULATION AND RANS MODELING OF TURBULENT NATURAL CONVECTION FOR LOW PRANDTL NUMBER FLUIDS I. Otic´ G. Grotzbac¨ h Institute for Nuclear and Energy Technologies Institute for Nuclear and Energy Technologies Forschungszentrum Karlsruhe Forschungszentrum Karlsruhe 76021 Karlsruhe, Germany 76021 Karlsruhe, Germany Email: [email protected] Email: [email protected] ABSTRACT Prt Turbulent Prandtl number. gβ∆TD3 Results of direct numerical simulation (DNS) of turbulent Ra Rayleigh number, Ra ¡ νκ . ¢ Rayleigh-Benar´ d convection for a Prandtl number Pr ¡ 0 025 p Pressure. 5 and a Rayleigh number Ra ¡ 10 are used to evaluate the turbu- t Time. lent heat flux and the temperature variance. The DNS evaluated T Temperature. turbulent heat flux is compared with the DNS based results of a ∆T Temperature difference. standard gradient diffusion turbulent heat flux model and with ui Velocity in i- direction. ¤ 1 ¦ 2 the DNS based results of a standard algebraic turbulent heat flux u0 Velocity scale, gβ∆TD ¥ . model. The influence of the turbulence time scales on the predic- ¤ § ¥ Xi Coordinates in horizontal i ¡ 1 2 tions by the standard algebraic heat flux model at these Rayleigh- ¤ ¥ and vertical i ¡ 3 directions. and Prandtl numbers is investigated. A four equation algebraic Greek turbulent heat flux model based on the transport equations for β Thermal expansion coefficient. the turbulent kinetic energy k, for the dissipation of the turbu- ∆x Laplace operator. ε θ2 lent kinetic energy , for the temperature variance , and for δi j Kronecker delta. the temperature variance dissipation rate εθ is proposed. This ε Dissipation rate of the turbulent kinetic energy. model should be applicable to a wide range of low Prandtl num- εθ Dissipation rate of the temperature variance. ber flows. θ2 Temperature variance. κ Thermal diffusivity. ν Kinematic viscosity. NOMENCLATURE ρ Density. D Channel height. τ Time scale. f Normalization factor. N Superscripts g Gravitational acceleration. ¤ ¥ Time averaged quantity k Turbulent kinetic energy. ν Pr Molecular Prandtl number, Pr ¡ κ . £ Address all correspondence to this author. 1 Copyright ¨ c 2004 by ASME 1 INTRODUCTION 2 DNS and statistical analysis of Rayleigh-Benar´ d convection A simple physical model for the investigation of heat trans- Turbulent heat transfer in several technical processes, like fer by natural convection is the Rayleigh-Benard´ convection. It in steel casting, semiconductor production, and cooling of new is given by an infinite fluid layer which is confined by two rigid liquid metal cooled nuclear reactors is of basic interest. Com- horizontal isothermal walls. The lower one is heated and the putational fluid dynamics (CFD) is widely applied to get more upper one is cooled. The physical problem is characterized by gβ∆TD3 insight, to design, and to improve such processes. Common tur- two dimensionless numbers: The Rayleigh number Ra ¡ νκ , ν β bulent heat flux models are using the simple concept of the tur- and the Prandtl number Pr ¡ , where g is the gravity; is the ¤ κ bulent Prandtl number Prt ¥ based on the Reynolds analogy to thermal expansion coefficient; ∆T is the wall temperature differ- model the turbulent heat transfer. This widely used k ε Prt ence; D is the distance between the two horizontal walls; ν is model may result in very unsatisfactory predictions, in partic- the kinematic viscosity; and κ is the thermal diffusivity. An ad- Ra ular for low Prandtl number flows (see e.g. [1], [2], [3]). To ¡ ditional dimensionless number is the Grashof number Gr Pr . obtain more realistic predictions for the turbulent heat transfer, Direct numerical simulation is a method in which the three di- especially for flows in which buoyancy is included, models are mensional conservation equations for mass, momentum and en- needed, which are based on the transport equations for the turbu- ergy are solved numerically such that all relevant physical pro- lent heat fluxes and for the temperature variance [3]. cesses are resolved by the grid and by the computational domain. The development of improved statistical heat flux models This means that the mesh size is fine enough to resolve the small- for engineering applications, in particular for development and est scales of turbulence and to resolve the viscous and thermal analysis of the lead-bismuth cooled Accelerator-Driven-System boundary layers in the near wall region. In particular the period- (ADS) reactor concept, requires data from measurements of the icity lengths which define the size of the computational domain cross-correlations between velocity and temperature fluctuations. must be large enough to resolve the largest scales of turbulence. ¤ ¡ ¥ Experimental studies in liquid lead-bismuth Pr 0 ¢ 025 are Meeting both requirements determines the huge computational an ongoing project in the Karlsruhe lead laboratory (KALLA) effort for such DNS, especially for convection in liquid metals. which will provide a data base for a model development of tur- The simulations of Rayleigh-Benard´ convection are per- bulent convection in PbBi, which is required for the design of the formed with the TURBIT code (Grotzbach¨ [5], Worner¨ [6]). It spallation target, see Knebel et al. [4]. However, due to the spe- is a finite volume code which allows for direct numerical sim- cific properties of PbBi, corrosion for example, and certain de- ulations of turbulent heat and mass transfer in simple channel ficiencies in the PbBi technology, the experimental thermal and geometries. The governing equations are solved in dimension- hydraulic investigations at the present stage are rather limited. less form where the following normalization is used: channel ¢ 2 £ Therefore, numerical investigations by means of direct numeri- ¡ β∆ ρ height D, velocity u0 g TD, time D u0, pressure u0, and cal simulation (DNS) are necessary in order to get more insight difference between the temperatures of the two walls ∆T. The into the physical mechanisms of turbulent convection in PbBi, to boundary conditions are periodic in both horizontal directions, gain some of the required turbulence data, and to complement whereas at the lower and upper wall the no slip condition and the experimental data basis. constant wall temperatures are specified. In this paper, the turbulent heat flux and the temperature In the following analysis we will use data from a new DNS ¢ ¡ § variance for the natural convection are analyzed using the re- for Pr ¡ 0 025 and Ra 100 000. Simulations for this flow at 5 ¢ ¡ sults of the new DNS for Pr ¡ 0 025 (lead-bismuth) at Ra 10 . sufficiently large turbulence levels became only recently feasi- The DNS evaluated turbulent heat flux is compared with the ble because this flow requires the resolution of very small ve- DNS based results of a standard gradient diffusion turbulent heat locity scales with the need for recording long-wave structures flux model and with the DNS based results of a standard alge- for the slow changes in the convective temperature field. The braic turbulent heat flux model. The influence of the turbulence new simulation is started from an earlier simulation by Bunk time scales on the predictions by the standard algebraic heat flux and Worner¨ [7]. The simulation is performed on a mesh with model at these Rayleigh- and Prandtl numbers is investigated. 400x400x75 cells within the horizontally periodic domain of size An improved statistical turbulent heat flux model is developed. 8x8x1on a VPP5000 computer. It covers about 24 000 statisti- This new modeling approach is based on the transport equations cally relevant time-steps for analyzing the results. In the follow- for the turbulent kinetic energy k, for the dissipation of the tur- ing we assign a time averaged quantity with X. Numerically X is bulent kinetic energy ε, for the temperature variance θ2, for the determined by averaging the data over both homogeneous hori- temperature variance dissipation rate εθ, and on the new model- zontal directions and over time. ing formulation for the turbulent diffusion in θ2 developed in [2] Fig. 1 shows the vertical profiles of the temperature root- which explicitly considers molecular fluid properties. mean-square values and of the turbulent heat flux. The results 2 Copyright ¨ c 2004 by ASME where δi j is the Kronecker delta. 0,25 The simplest form of two equation eddy diffusivity model is 0,20 2 Cµ k ∂T § uiθ ¡ (2) 0,15 Prt ε ∂xi 0,10 ¢ ¡ ¢ here the standard coefficients Cµ ¡ 0 09 and Prt 0 9 are used. Unlike in isothermal flows, the application of the model (2) to 0,05 buoyancy driven flows is insufficient, since it fails to account for the intensive turbulent heat flux in conditions where the mean 0,00 temperature may be uniform, or its gradient may have the same 0,0 0,2 0,4 0,6 0,8 1,0 x direction as the heat flux vector, as pointed out by Hanjalic´ [9]. 3 In the following we assign the model (2) as simple gradient dif- fusion model (SGD). ¢ θ2 The buoyancy term in the heat flux equations includes the Figure 1. DNS EVALUATED VERTICAL PROFILES OF - - AND θ θ2 ¡ ¢ ¡ § ui - ¡ -, at Pr 0 025, Ra 100 000. temperature variance . In analogy to the turbulent kinetic en- ergy k the temperature variance θ2 is a measure for the temper- ature fluctuations. Gibson and Launder [10] formulated an alge- braic heat flux model, which includes buoyancy effects to over- come a deficiency of the gradient models for buoyant flows. This indicate thick thermal boundary layers and the dominance of the model assumes a constant turbulence time scale ratio to predict 2 § ¡ ¢ large structures at Ra ¡ 100 000, Pr 0 025.
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