Physics of Buoyant Flows

The following sections are included:

• Dedication
• Reviews
• Foreword
• Preface
• Acknowledgments
• Notation
• Contents

Hydrodynamic flows exhibit turbulent behaviour which is not well understood till date. When a fluid is in a gravitational field, the flow becomes even more complex due to the gravity, confining walls, and two vector fields–velocity and temperature. Despite such complexities, researchers have been able to construct models and theories for buoyancy-driven flows that will be discussed in this book…

Buoyancy is induced by density variations that may occur due to gravity, as in planetary and stellar atmosphere; or due to temperature variation, as in thermal convection; or due to concentration gradient, as in oceans. In this chapter, we introduce the framework and equations of buoyancy-driven flows. In later chapters we will describe complex behaviour exhibited by such flows.

Fourier space description of a flow helps us quantify structures and their dynamics at different scales. In this chapter, we will discuss how to decompose the velocity, temperature, and density fields of buoyancy-driven flows in Fourier space…

In the last chapter we introduced the equations for the buoyancy-driven flows in Fourier space. We also wrote the equations for the modal energy and entropy. In this chapter, we will revisit the energy and entropy equations, and derive formulas for the energy and entropy transfers among the Fourier modes. Using these formulas we will derive diagnostic tools like the energy flux, shell-to-shell energy transfers, etc., which are very useful for understanding buoyancy-driven flows…

Stably stratified flows are linearly stable hence they support waves under linear perturbation. In this chapter we derive wave solution inside a stably stratified flow. These waves are called internal gravity waves.

When a fluid confined between two plates is subjected to an adverse temperature gradient, as shown in Fig. 2.3(c), the flow exhibits thermal instability beyond a critical temperature gradient. In this chapter we will derive instability criteria and associated flow properties. We will study thermal instability in the framework of RBC. In addition, we will also describe Rayleigh-Taylor instability…

In Chapter 6, we performed linear instability analysis of RBC and showed that for Rayleigh number beyond Ra_c, the flow becomes unstable leading to primary convective rolls. The amplitude of these rolls grow in time. However, when the amplitude of the roll becomes significant, nonlinearity starts to dominate the flow and it generates higher Fourier modes. These new modes do many things—saturate the growth of the primary mode, create stationary and time-dependent patterns, and exhibit chaos and turbulence. We will cover these topics in the present and subsequent chapters…

In Chapter 6 we described the thermal instability, and in Chapter 7 we showed how nonlinearity saturates the growth of the primary convective roll. In this chapter we will advance these discussions by showing how secondary modes are generated in RBC, and how patterns and chaos are created by these secondary modes. A further increase of Rayleigh number leads to a generation of turbulence that will be discussed in Part B of the book…

In RBC, a fluid is confined between two thermal plates, around which viscous and thermal boundary layers are formed. The region between the top and bottom boundary layers contains the bulk flow. In this chapter we will describe the properties of the boundary layers and temperature profile of RBC. We will also state several exact relations of RBC.

In this book we focus on the present understanding of buoyancy-driven turbulence. The present models of buoyancy-driven turbulence borrow concepts from hydrodynamic turbulence, for example the energy flux. Buoyancy-driven turbulence involve the density or temperature, hence, it has similarities with passive scalar turbulence. Due to these reasons, in this chapter we review some of the important ideas of hydrodynamic and passive scalar turbulence.

In this chapter we will quantify the large-scale quantities of RBC, namely the Nusselt and Reynolds numbers, and the viscous and thermal dissipation rates. We also describe the relative strengths of various terms of RBC equations, for example, the pressure gradient and buoyancy. In addition, we will review the phenomenologies of Kraichnan (1962), Shraiman and Siggia (1990), Grossmann and Lohse (2000), Grossmann and Lohse (2001), Doering et al. (2006), Pandey and Verma (2016), and others…

As discussed in Chapter 10, the kinetic energy (KE) spectrum of three-dimensional (3D) hydrodynamic turbulence, when forced at large scales, is

The equations of RBC and stably stratified flows are identical, as described in Chapter 2. The only difference between the two systems is the sign of density stratification $\overline{\rho }\left(z\right)$. In stably stratified turbulence (SST), $d\overline{\rho }/dz<0$, but in turbulent thermal convection, $d\overline{\rho }/dz>0$. Since the set of equations for the two systems are identical, one may expect Bolgiano-Obukhov or BO scaling to be applicable to both SST and RBC. Procaccia and Zeitak (1989), L’vov and Falkovich (1992), and Rubinstein (1994) employed field-theoretic techniques to thermal convection and concluded that BO scaling should hold for turbulent thermal convection…

The energetics arguments described in the last two chapters are quite general. For ows with stable stratification, in the inertial range, Π_u(k) decreases with k, i.e., Π_u(k) ∼ k^−ξ with ξ > 0. Hence, we expect the kinetic energy spectrum as

Buoyancy-driven flows are anisotropic due to the gravitational force. In Rayleigh- Bénard convection, buoyancy drives the hot plumes preferentially along the direction of gravity, thus u_‖ > u_⊥, where u_‖, u_⊥ are the velocity components parallel and perpendicular to the buoyancy direction. In stably stratified flows, u_‖ < u_⊥ due to the transfer of the kinetic energy to the potential energy. In real space, anisotropy can be quantified using directional measures of flow structures (such as flow alignment along the gravity direction) or by angular dependence of quantities like kinetic energy (u²)…

Direct numerical simulation of a turbulent flow is very expensive computationally due to the large degrees of freedom (e.g., number of u(k) modes) present in the system. Therefore, scientists often employ shell models that have far fewer variables. In this chapter we present shell models for buoyancy-driven flows. These shell models help us compute the spectra and fluxes of the velocity and temperature fields. The results of the shell model are consistent with the simulation results described in the earlier chapters. In this chapter we also compare various shell models proposed for buoyancy-driven flows…

In Chapters 6, 7, and 8 we discussed various structures and patterns in RBC near the onset of thermal instability. These patterns arose due to thermal instability, and nonlinear interactions among the large-scale Fourier modes. In the turbulent regime, one may expect these patterns to be washed out. But this is not the case; turbulent RBC too exhibits coherent structures. In this chapter we will discuss the nature of the structures in turbulent RBC, as well as a well-known phenomena called flow reversal…

Before we embark on our discussion on rotating RBC and rotating stratified flows, it is prudent to understand the basic properties of rotating flows. Here we will focus on the linear regime, and introduce Taylor-Proudman theorem and inertial waves. More complex topics such as rotating turbulence are beyond the scope of this book. Reader is referred to textbooks, e.g., Chandrasekhar (2013), Greenspan (1968) and Davidson (2013) for a detailed discussion…

In this chapter we discuss the effects of rotation on buoyancy-driven flows. First we will describe rotating stably stratified flow, and then rotating convection.

The magnetic field and rotation are present in many buoyant flows, e.g., in Earth’s outer core and in Sun’s convection zone. We discuss this topic in the next chapter after setting up the equations for magnetofluids and rotating magnetofluids, which are the subjects of the present chapter…

In this chapter we describe the effects of magnetic field on Stably stratified flows and on RBC. We limit ourselves to the linear limit.

In this chapter, we will perform instability analysis of magnetofluid which is under simultaneous influence of rotation and buoyancy.

In nature, we often encounter advection of two scalar fields. For example, in ocean, the two scalar fields are the temperature and salt concentration. In this chapter, we will briey discuss linear instability of such systems.

In this book we covered various aspects of buoyancy-driven flows—linear, weakly nonlinear, and turbulent. Some of these topics are well understood, while others are either unsolved or partially understood. Here we present a summary of the present status of the field…

The following sections are included:

• Appendix A: Thermal Parameters of Common Fluids
• Appendix B: Phase and Group Velocities of a Wave Packet
• Appendix C: Proper Orthogonal Decomposition of RBC
  • Proper Orthogonal Decomposition (POD)
  • POD vs. Fourier modes
• Bibliography
• Index