Dynamical and chaotic behaviors of natural convection flow in semi-annular cylindrical domains using energy-conserving low-order spectral models

Abstract

This paper presents a comprehensive theoretical study on the dynamical behavior of natural convection flow in the confined region between horizontal half-cylinders. For this purpose, a low-order spectral model with three modes will produce for the fluid flow system using the Galerkin technique. It proved that the generated model is physically meaningful, as it conserves energy in the dissipationless limit and has bounded solutions in the phase space. Analytical procedures indicate that the system has three stationary points and the onset of instability in the flow is when the Rayleigh number reaches a critical value. With an appropriate Lyapunov function is proved that for the Rayleigh numbers below the critical value, the flow is globally stable. As the Rayleigh number gets higher and reaches a fixed value, a Hopf bifurcation occurs, and chaotic motion appears in the system. The critical and Hopf Rayleigh numbers relation are derived parametrically based on dynamical system theories. Also, numerical simulations will carry on the presented low-order model. Different dynamical behaviors of this flow and its transition from regular to chaotic motion are explained, with phase portraits and velocity-temperature diagrams obtained by numerical solutions. This parametric study can pave the way for future researchers to determine at around values of critical parameters should an experiment or direct numerical simulation be performed to have more accurate data without resorting to tests at all operating conditions.

Introduction

The study of buoyancy-induced flows has long been of interest and continues to provide unresolved challenges. These types of flow exhibit complex behaviors due to confining walls, gravity, and coupled involvement of two vector fields – velocity and temperature [1]. To determine the fluid flow behavior and configure its different regimes, an accurate approach is Direct Numerical Simulations (DNS). Unfortunately, in the foreseeable future, this method is not a realistic possibility in most cases of study due to its cost and other limitations [2]; therefore, other modeling strategies should be taken in most practical studies. For studies around stability and transitional regimes, the strategy of isolation by spectral bands has received a lot of attention in recent years with the increasing interest in spectral models of fluid flow [3, 4]. In the research topics around buoyancy-induced flow transitions, this strategy is deeply entwined with the history of chaos theory [5]. Edward N. Lorenz is the official discoverer of this theory [6]. Section 1.1 begins with an explanation of the Lorenz system and reviews the research that has been done so far in this regard. At the end of Section 1.1, the research gap that exists in the literature and the novelty of the present study are clearly addressed. The objectives and the problem statement of this study are stated in Section 1.2.

In 1963, meteorologist E. N. Lorenz [6] suggested a three-parameter family of 3D Ordinary Differential Equations (ODE), which is now called the Lorenz system. He showed that a chaotic attractor exists in this low-dimensional system and this discovery stimulated the rapid development of the mathematical theory of chaos, which is still attracting attention from researchers in various fields [7]. The Lorenz equations are a model of a 2D dissipative Rayleigh-Bénard convection, which is a type of natural convection occurring in a plane horizontal layer of fluid heated from below [6, 8]. Lorenz derived his simplified mathematical model by considering three low-order modes in the double Fourier expansions of the stream function and the temperature variations of convection [6]. In this study, it was shown that at normalized Rayleigh numbers r greater than a critical value, the numerical results become chaotic and sensitive to initial conditions [6, 9]. Studies around the above 3D Lorenz system have shown that the transition to chaos and formation of strange attractors occurs via chaotic transients with both homoclinic and heteroclinic bifurcations, as well as homoclinic orbits, being present [10], [11], [12]. Various kinds of studies were performed to analyze different aspects of the original Lorenz system or investigate the possibility of extending the model to other relevant problems. Some of these studies are reviewed in the following. Ge and Li [13] studied the time-reversed Lorenz system (which is called the “historical system” in their original article). Various kinds of phenomena were investigated in such a system by phase portraits and bifurcation diagrams. Khan and Khan [14] studied the numerical solution of the Lorenz system by using a novel time-discretization approach, which is based on the Galerkin-Petrov time-discretization formulation. Their results showed that the presented method has a great potential for simulating chaotic dynamical systems and can predict different features of these systems. Numerous studies were done to find out whether the Lorenz system with a larger number of Fourier modes would have the same dynamical behavior as the 3D model. One such study is the work of Roy and Musielak [11], in which they have used the method of vertical mode truncation to select higher-order Fourier modes. In this method, the horizontal modes are kept fixed and only the vertical modes change. They have claimed that the vertical modes would play a dominant role in this flow because the motions of flow in thermal convection are primarily in the vertical direction. Their results showed that the lowest-order physically meaningful generalized Lorenz model obtained by the method of vertical mode truncation is a 9D system and its route to chaos is via chaotic transients, which is the same as the 3D Lorenz system. They have also concluded that a larger number of Fourier modes results in higher values of r required for the system to enter the fully developed chaotic regime. Shen [9, 15] analyzed the fifth- and sixth-order Lorenz systems and looked for critical r values beyond which chaos would ensue. It was concluded that both systems possess similar critical r values, which are greater than the critical r value of the original Lorenz model. However, critical r values also depend on other parameters of the system (such as the Prandtl number and aspect ratio) as well; therefore, it is better to study the behavior of generalized Lorenz models at pair of parameters, instead of r alone. Moon et al. [16] considered the original Lorenz model and five high-order Lorenz systems, ranging from order 3 to 11. They have plotted periodicity diagrams for all the six systems, based on different pairs of Prandtl and normalized Rayleigh number values. Results showed that the behavioral patterns of high-order systems in these diagrams differ from the original Lorenz system and one another. Even the behavioral changes are not monotonic to the order of the system, which means increasing the complexity of equations does not necessarily lead to an increase in the system's complexity. Another topic that has recently been explored on Lorenz models is the effect of boundary condition. Kanchana et al. [17] studied the dynamical behavior of the Rayleigh-Bénard convective system in the case of 16 possible boundary condition combinations by using a lower-order Fourier-Galerkin expansion. They achieved the Rayleigh numbers at which the transition from regular convective motion to chaotic, mildly chaotic, and periodic motions take place for all the 16 boundary conditions. Siddheshwar et al. [18] made a local nonlinear stability analysis for the flow of Rayleigh-Bénard convection with rigid isothermal boundaries. The possibility of different motions in their Lorenz-like system is shown by the plots of the maximum Lyapunov exponent and the bifurcation diagram. The effect of external rotation on thermal convection is another topic that has attracted significant theoretical interest. Stein [19] studied a four-mode model of convection in a rotating fluid layer with an extension of the Lorenz model. Actually, in this study, an extra mode is included in the Lorenz model for the regeneration of vorticity by rotation. The results showed that the first oscillatory solutions appear in a Hopf bifurcation from either the conduction solution or the stationary convective solution. After it has given way to chaos, further oscillatory solutions are generated at saddle-node bifurcations or period-doubling bifurcations. Recently, Gupta et al. [20] conducted a similar study for thermal convection in a rotating fluid layer subjected to gravity and heated from below. They obtained a low-dimensional, Lorenz-type model for the problem using Galerkin truncated approximation. Their general conclusion is that the transition from steady convection to chaos depends on the level of a parameter, called the Taylor number. Zhang [21] considered a Lorenz-like system, describing the rotating fluid convection. In this study, some properties of the global attractors of the three-mode model are presented based on the stability theory of dynamical systems. Porous media convection is another issue which has received a lot of attention among the equivalent problem for pure fluids. Theoretical studies [22], [23], [24], [25] demonstrate that the transition from steady to chaotic convection in saturated porous media subject to local thermal equilibrium yields a set of equations that is equivalent to the Lorenz system. The governing equations in this problem consist of the continuity, the extended Darcy, and the energy equations subject to the assumption of local thermal equilibrium and the Boussinesq approximation. Vadasz [22] has reviewed and compared the analytical, computational, and numerical solutions of the resulting set of equations. There are also several computational results on the effect of rotation in porous media. Vadasz and Olek [26] solved the nonlinear problem of centrifugally driven free convection in a rotating porous layer by a truncated Galerkin approximation. Their results demonstrate different transitions, e.g. from steady convection to a non-periodic regime via a Hopf bifurcation, and a further transition from chaos to periodic convection at significantly higher values of the centrifugal Rayleigh number. So far, articles were reviewed in which researchers have considered a geometry similar to Lorenz geometry, and by considering other side effects on this system (such as changing the boundary condition, adding rotation, etc.) examined the change in the dynamical behavior of the system as compared to the original Lorenz model. It should be mentioned that Lorenz modeled natural convection flow that is confined to a box whose boundaries are stress-free [3, 6]. One of the interesting topics that can be considered is changing the geometry to curved boundaries such as cylindrical region. Natural convection flow originating in the domain between two horizontal cylinders with differential heating is a fundamental case, which has received a lot of attention in recent years. Such flow is more complicated than the classical Rayleigh-Bénard convection since, due to the curvature of the cylindrical differentially heated surfaces, its phenomenology encompasses the features of both the Rayleigh-Bénard and the vertical enclosure systems [5]. Most of the works around investigating dynamical behavior of such systems and clarifying their flow regimes are done by numerical simulations. The main results of these studies are reviewed in Refs. [5, 27]. A very few experimental studies were also done to investigate the transition to chaos in such flows (see, e.g. [28, 29]). Although numerical simulations and experimental works in this regard are valuable, they are available only for a limited flow condition and it is not possible to determine the flow transition characteristics in all physical situations (such as different sizes of geometry, different fluids, etc.) using these two methods. This is because of the time-consuming nature of the numerical simulations (especially DNS studies to study transition and chaos processes in fluid systems) and the high cost of advanced facilities for capturing transition and chaos in experimental works. Therefore, there is a need to perform a theoretical parametric study to discuss the stability and dynamical properties of natural convection flow in systems with curved boundaries and find out the critical values of parameters that cause changes in their dynamical behavior. Such a study could pave the way for future researchers to determine at which values of critical parameters, the DNS or experimental tests should be performed to more accurately determine the system properties. In the present study, a Lorenz-type approach is followed to investigate the dynamical behaviors of such flows in a system with curved boundaries. The system considered has retained the basic tenets of the original Lorenz model except for the geometry which is a semi-annular cylindrical region. Consequently, a 2D approach is followed similar to what Lorenz did on a 2D Rayleigh-Bénard convection system and the main differences are elucidated vis-à-vis the same problem with flat boundaries.

As the literature review indicates, a comprehensive parametric study is missing to investigate the dynamical properties and transition process of natural convection flow of systems with curved boundaries of different sizes. On the other hand, given the classical problem of natural convection between two isothermal horizontal boundaries (the Rayleigh-Bénard convection) and the valuable studies of Lorenz which were later referenced by other researchers, it would be very interesting to extend the classical Rayleigh-Bénard convection to curved boundaries which is the main objective of the present study.

To begin with, consider a natural convective motion of fluid confined in a domain between horizontal concentric half-cylinders, as shown in Fig. 1. The radius of inner and outer cylinders are considered as $r_i^{*}$ and $r_o^{*}$, respectively. It is assumed that the inner and outer cylinders are maintained at constant temperatures $T_i^{*}$ and $T_o^{*}$ ($<T_i^{*}$), respectively and the other two boundaries are nonconducting. The coordinate system is also shown in Fig. 1, where r* is taken along the radial direction, and θ is measured counterclockwise from the right horizontal plane through the center of half-cylinders. The study is restricted to two-dimensions with $r_i^{*}\le r^{*}\le r_o^{*}$ and 0 ≤ θ ≤ π.

In summary, the objectives are as follows:

• Obtaining the low-order spectral model of the mentioned natural convective flow for various geometrical aspect ratios $r_o^{*}/r_i^{*}$.

• Analyzing the system of equations to investigate their dynamical properties and the possibility of transition to chaos.

The layout of the rest of the paper is as follows. In Section 2, the mathematical formulation of the problem and the steps to produce the low-order spectral model are introduced. In Section 3, the properties of the generated model are given. In Section 4, the dynamical and chaotic behaviors of the system are discussed by using analytical procedures and numerical simulations. Finally, the conclusion is presented in Section 5.