Three dimensional simulation of radiation induced convection
Introduction
Convection due to the selective absorption of radiation has been shown experimentally by Krishnamurti [1] for a viscous fluid. The convection mechanism is essentially a penetrative one (which occurs when buoyancy driven motion penetrates into stably stratified layers) effectively modeled via an internal heat source. Due to the wide range of industrial and geophysical applications, extensive literature has been recently produced on penetrative convection, see e.g. Carr [2], Carr and de Putter [3], Carr and Straughan [4], Hill [5], Aziz and Aziz [6], Cloete and Smit [7], Sharma [8], Hasan et al. [9], Singh et al. [10], Mahajan and Monika Arora [11], Pal et al. [12] and Srivastava et al. [13].
The experiment performed by Krishnamurti [1] involves the use of a layer of water containing thymol blue. Thymol blue is predominantly orange substance which becomes blue (the conjugated form) in a high pH. Introducing a positive electrode along the bottom layer of the system produces hydroxyl ions, which precipitates a high pH. Effectively there is diffusion, due to the pH differential, of the blue coloring into the orange from the bottom to top layer. The key part of the experiment is the introduction of a sodium lamp which emits orange radiation that is only significantly absorbed by the blue form of the fluid. This has the effect of causing the less dense blue fluid to rise, creating a convective motion.
Krishnamurti [1] developed a model to describe this experiment, where the internal heat source was assumed to be linearly proportional to the concentration field of the conjugated form of the thymol blue. Straughan [14] further developed the model by studying its linear instability and nonlinear stability. Straughan [14] also employed realistic boundary conditions, appropriate to fixed surfaces, which more accurately reflect the physical experiments. The results of [14] lend much credence to use of the model introduced in [1] for radiation induced convection. Wicks and Hill [15] studied the linear and nonlinear stability analyses of double-diffusive convection in a fluid layer, with a concentration-based internal heat source present. Chang [16] considers this system in a fluid overlying a porous medium. Olali [17] presented a linear instability analysis for the inception of double-diffusive convection with a concentration based internal heat source. The system encompasses a layer of fluid which lies above a porous layer saturated with the same fluid. Hill [5], [18], [19], [20], [21], [22] explores this system through the use of porous materials.
In this paper we investigate the onset of convection in Krishnamurti’s model [1] to isolate regions of subcritical instability (where the linear instability and nonlinear stability thresholds do not agree), and then explore these regions for the full three dimensional system (using a velocity–vorticity formulation in order to employ second order finite difference schemes). We use both implicit and explicit schemes to enforce the free divergence equation. The size of the physical space explored is evaluated according to the normal modes representation, where periodic boundary conditions for velocity, temperature, and concentration in the dimensions are employed. Standard indicial notation is employed throughout, where .
Section snippets
Governing equations
Let us consider a fluid layer bounded by two horizontal parallel planes. Let and be a Cartesian frame of reference with unit vectors and , respectively.
The momentum and conservation of mass equations for a linear viscous fluid [1], [14] arewhere is the Laplacian, and p, the velocity and pressure, is acceleration due to gravity, and the kinematic viscosity. Denoting T to be the temperature, we take the density
Linear and nonlinear energy stability theories
Linear instability results for stationary convection are obtained via the application of standard procedures to the linearized version of Eqs. (2.7), (2.8), (2.9), (2.10). Adopting normal mode representations, Straughan [14] found the critical Rayleigh number of linear theory by determining the lowest eigenvalue of the systemon . Here , , , , is a
Velocity–vorticity formulation
It is well known that there is a major difference between two and three dimensions for vorticity-based numerical methods. Most apparent of all is the fact that both vorticity and stream function become vector (instead of scalar) fields in 3D. At the same time, the stream function changes to a vector potential. Along with this is the necessity to enforce divergence free conditions for vorticity and vector potential. This turns out to be a major problem in designing efficient numerical methods in
Numerical schemes for 3D problems
The first step in the numerical computational is to give an initial values for the vorticity vectors . Next, the Poisson equations (4.5) are discretized in space using an implicit scheme as followswhere are the second-order central difference operators, which are define as
Numerical results and conclusions
In this section, , is the critical Rayleigh number for linear instability and is the global nonlinear stability threshold. The corresponding critical wavenumbers of the linear instability and global nonlinear stability will be denoted by and . In Table 1, we present the results of numerical results of linear instability and nonlinear stability analyses. The dimensions of the box which were calculated according to the critical wavenumber are showed in Table 1. In this table Lx and
Acknowledgments
This work was supported by a scholarship from the Iraqi ministry of higher education and scientific research. The author acknowledges the comments and suggestions of Prof. B. Straughan and Dr. Antony A. Hill, which led to improvements in the manuscript. Also, the author would like to thank anonymous referees for their comments that have led to improvements in the manuscript.
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