Simulation of the Turbulent Rayleigh-benard Problem Using a Spectral/finite Difference Technique

The three-dlmensional, incompressible Navler-Stokes and energy equations with the Bousslnesq assumption have been directly simulated at a Rayleigh number of 3.8 x 105 and a Prandtl number of 0.76. In the vertical direction, wall boundaries were used and in the horizontal, periodic boundary conditions were applied. A spectral/finite difference numerical method was used to simulate the flow. At these conditions the flow is turbulent, and a sufficiently fine mesh was used to capture all relevant flow scales. The results of the simulation are compared to experimental data to justify the conclusion that the small scale motion was adequately resolved. Research was supported by the National Aeronautics and Space Administration under NASA Contract Nos. NASI-17070 and NASI-18107 while the second author was in residence at ICASE, NASA Langley Research Center, Hampton, VA 23665-5225. I° INTRODUCTION Direct simulation of turbulent fluid flows is now possible with the large vector computers that have become available [1,2]. Prediction of low-order flow statistics is definitely within current capabilities, and some results have already been published which show predictions of small scale turbulent features which are consistent with experimental observations [3-5]. The current study was undertaken to explore the quality of information that can be extracted from a direct flow simulation (DFS) of turbulence on a sufficiently fine mesh. The turbulent Rayleigh-Benard problem (natural convection)was chosen for study since it is a simple turbulent flow for which a good body of experimental measurements exists. Moreover, some DFS and large-eddy simulations (LES) of this problem have been published albeit on coarser meshes. While experimental data do exist, measurements of velocity, where no mean flow exists, are difficult. Hence, there is much to be learned about turbulent natural convection from an accurate simulation. The two requirements for conducting such a study are a hlgh-speed computer and an efficient, accurate flow simulation code. The CYBER-205 computer with a 16 mega-word memory provides sufficient computation power. This current code has been extensively tested, and various versions of it have been used to study transition in channel flow [6]. The version used in this study includes the addition of the energy equation and a modified vertical momentum equation that includes buoyancy consistent with the Boussinesq assumption. A simulation of a turbulent flow was then conducted, and these data as well as a discussion of the code will be presented in this paper. Though the overall goal of this work is an in-depth examination of the quality and type of information that can be extracted from such a simulation, the purpose of this paper is to document the basic simulation. The simulation results will be compared with experimental mean measurements as well as previous DFS results. The increased resolution of this work over previous DFS resulted in an improvement in the prediction of the Nusselt number; it was sufficiently close to experimental results to suggest that in addition to a good prediction of the large-scale flow, the small-scale features are accurately represented. Grotzbach discusses this connection extensively [7,8]. Comparisons with experimental data which are more dependent on the small scale components of the flow will also be presented to justify further this conclusion. II. RAYLEIGH-BENARD PROBLEM The Rayleigh-Benard problem is a simple geometry, laboratory-type problem used to study natural convection (Figure I). Chandrasekhar [9] and Busse [I0] have described the basic problem and discuss both the stability analysis and some experimental results. Krishnamurti [II,12] has summarized much experimental data and developed a map showing the qualitative flow at different values of Ra and Pr, the principal independent problem parameters (defined below). For Pr = 0.76 (air) and Ra = 3.8 × 105 the motion is turbulent, although it should possibly be qualified as low Reynolds number turbulence. Several experimental studies [13-18] and numerical simulations [7,19,20] (both LES and DFS) have been completed in the qualitatively similar Pr-Ra region. The flow at these values of Pr and Ra consists of a core flow (a horizontal layer in the middle 80% of the fluid layer) and a boundary region near each plate. The turbulence is statistically homogeneous in the horizontal directions for both layers. In the core the vertical variation of most statistical quantities is small. In the boundary layer there is a transition from molecular dominated physical processes near the wall to the fully turbulent core flow. This flow is described by the incompressible Navier-Stokes equations modified to include the effect of temperature-induced density variations on the buoyancy force (Boussinesq assumption) plus the temperature equation. These equations, when non-dimensionalized by _, h, and AT, are aui a(uiu j) aP a2ui --+ = ---+ Pr----_+ Pr Ra T_i3 , (la) at ax. axi ax. 3 3 aT a(Tuj) a2T --+ 2 + u3 ' (Ib) at ax. ax. 3 J and au. .__i= O. (lc) ax. 3 The temperature and pressure in Equations (la,b,c) are the difference between the actual temperature, Ta, and pressure, Pa' and the values due to the static temperature gradient only. These are defined as follows: T (x,t)= T x3 + T(x,t) a _ O = T + T (x,t) o r --