Numerical study of an unsteady confined thermal plume under the influence of gas radiation
Introduction
A thermal plume is a buoyancy-induced flow arising from a local heat source. In non-confined spaces, similarity solutions have been used in early theoretical studies to describe such natural convective flows [[1], [2], [3]]. At the same time, many experiments have been carried out and confirmed the laminar theory of plume [[1], [2], [3]]. Gebhart et al. [4] pointed out that free plumes are much less stable compared to flows adjacent to surface which can damp disturbances. The experiment of Forstrom and Sparrow [5], performed on a buoyant plume above a heated horizontal wire, showed that the laminar plume exhibits a slow, regular swaying motion in the plane perpendicular to the heater. Later on, Pera and Gebhart [6] investigated numerically the stability of a laminar plume above a linear heat source and confirmed these stability predictions. The water experiments of Eichhorn and Vedhanayagam [7], as well as the spindle oil experiments of Urakawa et al. [8] showed that the buoyant plume not only sways in the plane perpendicular to the heater but also meanders along the heater direction. In addition, Urakawa et al. found that the meandering wave shape is stable when the heater length is an integral multiple of half the wave length. This type of open flow is less investigated numerically, due to the complexity to impose proper boundary conditions at the bounds of the computational domain, specifically in the case of unsteady flows. Some efforts have been dedicated to reduce the computational cost of the simulations by imposing analytical solutions at the outer boundary [9] or by proposing new outer boundary conditions for a limited computational domain [10,11].
Plumes developing in confined geometries do not have this difficulty for numerical modelling. Moreover, confined flows are relevant for many engineering applications, such as building thermal design, heat storage in boilers or electronic cooling. The buoyant flow triggered by the presence of a plume inside the cavity fills the enclosure. Consequently, it is sensitive to both the boundary conditions at walls and the resulting thermal stratification of the far-field fluid, which can generate strong intermittency [12,13]. Because of the confinement, it is quite difficult to find a general analytical approach. Therefore, numerical simulations or experiments are more commonly used to study such flows. Desrayaud and Lauriat [14] investigated numerically plumes in air above a linear heat source in a 2D enclosure with various aspect ratios of the cavity and altitudes of the source. They identified the sequence of bifurcations leading to the weakly turbulent motion. Moreover, in the case of a square vessel and a heat source located near the bottom wall, they highlighted a transition of the 2D plume from steady state to mono periodic motion through a Hopf bifurcation. They determined the critical Rayleigh number to be close to by linear extrapolation of the amplitudes of velocity and heat flux fluctuations. They showed that a swaying motion of the plume establishes above the heat source, first with a periodic regime associated to high fundamental frequency, followed by a two-frequency locked regime and finally leading to the appearance of the chaotic state. In the same configuration, Bastiaans et al. [12] obtained a more accurate value for the first critical Rayleigh number () of the 2D plume with a spectral element method. They also carried out both direct numerical simulations (DNS) and large eddy simulations (LES) for a 3D turbulent plume.
Following these studies, Fiscaletti et al. [15] investigated the transition to unsteadiness of a plume produced by a horizontal cylindrical heat source within a water-filled tank by means of experiment and numerical simulations. They described the change of the flow structure throughout the first bifurcation and they illustrated the swaying motion by 2D visualization. Later, Hernández [13] studied numerically the steady and periodic regimes of thermal plumes in a slender air cavity with a linear source located on the floor of the cavity. They observed that the 3D plume oscillates with the same spatial phase in all the transverse planes along the linear source direction, with a slight modulation of the vertical expansion of the plume. They suggested that this oscillation in phase of the plume plane can be broken for higher Rayleigh numbers or higher depth aspect ratios of the cavity.
However, the above-mentioned works only consider the development of thermal plumes in cavities filled with a transparent gas. If the working fluid is a radiative participating medium, the radiation-convection coupling modifies the distribution of the thermal transfer modes and consequently the flow field. To the authors knowledge, there is no systematic study of the influence of gas radiation on plumes development. This is not the case for other natural convection configurations, such as the differentially heated cavity in which the influence of participating radiative media has been extensively studied.
The pioneering work of Lauriat [16] and the further study of Draoui et al. [17] using the gray gas assumption proved that gas radiation strongly modifies the temperature field and alters the fluid motion. Later on, Lari et al. [18] studied variations in temperature and velocity distributions by increasing the optical thickness. Recently, many works [[19], [20], [21], [22]] considered real gas radiative models for humid air in a differentially heated cavity. In these studies, the Radiative Transfer Equation (RTE) is solved by using real gas spectra. These studies showed that gas radiation delays the transition to turbulence by decreasing the temperature gradients, thickens the boundary layers and reduces the central thermal stratification within the cavity [20].
In the present work, we aim to simulate a thermal plume above a linear heat source in a 3D cavity filled with a gas. The objective is to highlight the influence of gas radiation on the development of the plume and on the far-field flow, by considering gradually the gas as a transparent medium, a gray gas with increasing optical thickness and finally a real gas modelling humid air. For this purpose, we first describe the spatio-temporal dynamics of the flow for a transparent medium considering three particular Rayleigh numbers corresponding to different flow regimes (, and ). The paper is organized as follows: section 2 introduces the physical problem and the model equations. In section 3, we present the numerical methods and discuss the code validation in a pure 2D convection situation by comparison against literature data. 3D results are finally presented in section 4, first focusing on pure convection in section 4.1, then considering gas radiation in section 4.2 at a particular Rayleigh number. The effects of gas radiation on flow regime, heat transfer, temperature and velocity fields are analyzed for various radiative gas models. We conclude with a summary in section 5.
Section snippets
Problem description
The considered geometry is presented in Fig. 1. It corresponds to an air-filled cubic cavity of 1-m side . A plume is produced by a linear heat source located along the line , as indicated by red line in Fig. 1. The heat source is assumed to be intangible and produces a power by unit length. A characteristic temperature difference can be derived from the heat source as , being the thermal conductivity of the fluid. The top and the bottom
Numerical methods
The governing equations are solved with the computational software SUNFLUIDH developed at LIMSI laboratory [26], coupled with a module of radiative heat transfer calculation previously developed during a joint research project [27,28]. The set of equations (1)–(3) is solved using a finite volume approach. The spatial discretization is performed on a staggered grid with second-order central differences. The time marching uses a semi-implicit scheme combining a second-order backward Euler scheme
Results
We consider now a 3D plume in the cubic cavity described in section 2.1. First, we describe the spatio-temporal structure of a pure convective plume without any radiation coupling (section 4.1), second we consider the radiation effects in section 4.2. Based on the validation results above, we consider a 1293 grid resolution as a reasonable grid size in terms of accuracy versus computational effort.
Conclusions
Thermal plumes generated by a linear heat source in a confined cavity are numerically investigated. In order to validate the numerical code, 2D simulations were first performed, and the critical Rayleigh number for transition from steady to periodic flow is estimated at which is in good agreement with literature [12]. The study is then extended to 3D configuration. For the pure convective situation (neither gas nor wall radiation), results are provided for different regimes. For
Declaration of competing interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgements
This work is financed by the China Scholarship Council (CSC) and the Civil Aviation University of China (CAUC). It was granted access to the HPC resources of IDRIS under allocation 2a0326 made by GENCI. The authors thank Dr. Y. Fraigneau for his help in using the code SUNFLUIDH, developed at LIMSI, and Dr. L. Cadet for the radiative module, co-developed in the LaSIE, LIMSI and Institut PPRIME laboratories.
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