Spectral analysis of turbulent structure properties in thermal convection based on direct numerical simulation

Abstract

The differentially heated walls of the vertical channel pose as sources of aiding and opposing buoyant forces altering the amplification and suppression of turbulent motions. In comparison with isothermal channel flow as published by Kim, Moin & Moser in 1987 [2], this offers the possibility for analysing amplification and surpression of turbulent fluctuations close to the heated and cooled walls, respectively. In 1997, Kasagi & Nishimura (KN) [3] published a variety of statistical results they obtained with DNS of vertical channel flow including stress balances and budgets of the turbulent kinetic energy. They found that the isothermal heating and cooling lead to an asymmetric flow structure. Towards the heated channel wall the buoyancy aids and augments the mean flow rate while it opposes it near the cooled wall. The locally higher Reynolds number in the aiding flow should lead to increased turbulence intensities. Instead, the turbulence is suppressed there and augmented on the opposing flow side. They also presented coarsely resolved depictions of streaky structures in the streamwise velocity component near the walls which they compared to the ones observed in channel flows altered by uniform suction and blowing on the walls as well as channel flows influenced by magnetohydrodynamic forces. Direct numerical simulations of asymmetrically heated turbulent vertical channel flow at Reτ≈300 and Pr=0.71 are performed for different Grashof numbers (Gr) from 0 up to 1.6•10^6 by solving the Navier-Stokes equations together with the Boussinesq approximation for the bouyancy as well as the thermal energy equation for the temperature field with a finite volume method. The latter is based on a fourth order accurate interpolation scheme in space, which has been adapted to cartesian grids from the cylindrical version used in Feldmann & Wagner. A second order accurate Euler-Leapfrog scheme is applied in time. At the STAB workshop we will present a spectral analysis of the turbulent velocity fluctuations in the vertical channel setup and compare it to the plane isothermal channel. All simulations are conducted for the same Reb=4328 and are rescaled to their respective Reτ in order to compare the resulting statistical profiles in local viscous coordinates. Figure 1 shows a depiction of the streamwise velocity fluctuations near the heated and cooled channel walls at a distance of z+=15. The fluctuations organize in the well-known streaky structures and appear more elongated and smooth near the heated wall while they are shorter and more disrupted near the cooled wall. This behaviour is reflected in the turbulent kinetic energy (TKE) profile, which shows lower values of TKE in the aiding flow. The results from an earlier analysis of the full budget equations of the Reynolds stresses revealed, that the buoyancy term has no significant direct influence on the turbulent stresses. Thus, in this presentation, we will focus on the spectral analysis of the Reynolds stresses and will show, which structure sizes are mainly influenced by a change in Gr and thus buoyant forces. Figure 2 shows a comparison of the power density spectrum of the streamwise Reynolds stress at a wall distance z+=15 from the heated and cooled wall compared for the different Grashof numbers. There, it is clearly visible, that all wavenumbers and such structures of all sizes are influenced by the buoyant forces. In the presentation, we will present a detailed analysis of which structure sizes are energetically dominant and how their spectral statistics can help build an understanding of the decay in Reynolds stresses for increasing Gr.