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Rayleigh-B´enard-Poiseuille/Couette flows
Motivation
It is clear that modal stability analysis offers good predictions for the onset of instability in Rayleigh-B´enard convection problem. It fails, however, in the case of many simple yet common wall-bounded shear flows, for example, plane Couette flow and plane Poiseuille flow. In the study of turbulent shear flows, plane Poiseuille flow and plane Couette flow are prototypes in which perturbations exhibit large transient growth in disturbance kinetic energy O(Re 2) via the mechanisms discussed earlier. It is reasonable to assume that the presence of a cross-stream temperature gradient in a parallel shear flow would influence transient growth. If so, what are the dominant physical mechanisms of tran-sient growth in such flows? Is lift-up dominant at all Rayleigh and Prandtl numbers? It is the aim of the present dissertation to examine thoroughly the influence of buoy-ancy induced by an adverse cross-stream temperature gradient on the transient growth phenomenon in plane Couette flow and plane Poiseuille flow.
Indeed, there are simple stationary solutions to the Navier-Stokes equations for fluid motion between two infinitely long, rigid walls (moving/fixed) maintained at different temperatures (cold upper wall or vice versa) with no-slip boundary condition. If the temperature difference between the walls is small enough then conduction would be the only means of heat transfer and one expects a linear temperature variation between both walls to prevail. The base flow, under the assumption that the buoyancy force is the only temperature effect in the momentum equation, could be plane Poiseuille or plane Couette flow depending on whether the walls are stationary or moving relative to each other. Hereafter, the former is referred to as Rayleigh-B´enard-Poiseuille flow (RBP ) and the latter is referred to as Rayleigh-B´enard-Couette flow (RBC). From an experimental as well as a theoretical point of view, the Rayleigh-B´enard convection problem is the simplest and most easily accessible case, in which the onset of instabilities can be readily studied. Plane Couette and plane Poiseuille flow represent prototype shear flows in which the onset of instabilities depends strongly on the initial conditions and hence, on the background disturbance field. Thus, the linear stability analysis of Rayleigh-B´enard-Poiseuille and Rayleigh-B´enard-Couette flows is expected to be of fundamental interest in hydrodynamic stability.
Moreover, this type of fluid motion is commonly encountered in various forms in geo-physical flows, heat exchangers, electroplating, chemical vapor deposition, etc. Thermal convection in the presence of the atmospheric boundary layer that leads to the alignment of clouds in the lower atmosphere is a well-known example of thermal instability in the presence of a shear flow [52]. The study of such flows is useful in fields often remote from fluid dynamics such as plankton research. For example, the motion and spatial distribution of phytoplankton and algal suspensions are affected by convection in the presence of shear flows in ice-covered lakes [46, 14]. It is not surprising that shear flows involving thermal convection are fundamental to flow situations arising in astrophysics, meteorology, and many engineering applications.
A brief history of RBP and RBC flows: Modal stability
The thermal instability leading to secondary motion in the form of Rayleigh-B´enard con-vection rolls has been well-known for more than a century. If the domain is large enough compared to the depth of the fluid layer, the convection rolls do not have a preferential orientation. In the presence of a shear flow, however, modal stability theory predicts that streamwise-uniform convection rolls would occur. In fact, the preference for streamwise-uniform convection rolls (called longitudinal rolls in the literature) in unstably stratified (adverse temperature gradient) shear flows was first discovered by Idrac [41]. It is in-teresting to note that Idrac was interested mainly in possible atmospheric applications, including migration patterns of birds from Europe to Africa. The phenomenon, however, has long since intrigued a number of researchers. In the case of a static fluid heated from below, convection develops in the form of locally two-dimensional rolls. If allowed to develop in a sufficiently large aspect ratio apparatus and from background disturbances instead of controlled initial disturbances, rolls can have a random orientation. Thus, in Idrac’s experiments, the emergence of a well-ordered motion from a seemingly random state is fascinating to say the least. Terada [42] made quantitative observations of these rolls in shear flows. For example, he noted that, for small values of the gap height h, the wavelength of the rolls was about 2h. Later, the linear stability analysis of RBP and RBC flows was motivated by the observation of cloud streets [2, 3, 52] and sand dunes in deserts [33]. The convection rolls in the lower atmosphere tend to align in the direction of the atmospheric boundary layer. The moisture in the up-flowing warm air of these rolls condenses to form clouds that are aligned in the streamwise direction, thereby leading to the formation of cloud streets [52] (see figure 1.6). It was not until Chandra in 1938 [9] that experimentalists claimed that shear does not affect the critical Rayleigh number for plane Couette flow. However, definitive evidence was not available until Ingersoll [43] showed via heat flux measurements that this is true. Ingersoll [43] used two horizontal concentric discs with fluid contained between them. Such a set-up allows for reduced end effects. The upper disc is rotated at a constant angular velocity while the lower disc is kept at rest. The gap between the discs is maintained small. The results from Ingersoll’s experiments are displayed in figure 1.7. It shows Nusselt number N u which is the ratio of heat transports with and without convection as a function of Rayleigh number for various Reynolds numbers. The Reynolds numbers (evaluated at the outer radius) are denoted by various symbols. Since the symbols collapse on a single curve, it is concluded that the critical Rayleigh number of thermal convection in Couette flow is independent of Reynolds number. For the sake of completeness, it is here mentioned that Akiyama, Hwang and Cheng [1] determined experimentally that the longitudinal rolls form near the Rayleigh number Ra = 1708 (based on the channel width) for the case of fully developed, plane Poiseuille flow. This experimental result is close to the theoretical value RaRBc = 1707.78. In fact Akiyama et. al. [1] observed weak convection in the form of rolls for Ra < 1708 and non-zero Reynolds numbers Re > 0. It was, however, attributed to the subcritical instability arising from non-Boussinesq effects. Thus, the experimental results predominantly indicate that longitudinal rolls oc-cur at a fixed Rayleigh number, independent of Reynolds number, and that the onset of secondary motion occurs via streamwise convective motion. However, in order to demonstrate that spanwise-uniform rolls are more stable than streamwise-uniform rolls, the linearized equations should be solved subject to appropriate boundary conditions. The earliest known stability analysis of plane Poiseuille flow with unstable thermal strat-ification in a Boussinesq fluid is due to Gage and Reid [26]. If ReTc S (≈ 5772.2) is the Reynolds number (based on the channel half-width) at which Tollmien-Schlichting waves (T S) become unstable in plane Poiseuille flow without temperature effects, Gage and Reid showed that for all Reynolds numbers less than a critical value, approximately equal to ReTc S , the dominant eigenmode of RBP is in the form of streamwise-uniform convection rolls due to the Rayleigh-B´enard instability (RBI) above a critical Rayleigh number RaRBc = 1707.78 (based on the channel width). This value is independent of both Reynolds number and Prandtl number. It was concluded that the effect of a shear flow on the linear stability of a fluid subjected to unstable cross-stream temperature gra-dient is only to align the rolls along the streamwise direction. Furthermore, the effect of the cross-stream temperature gradient on the Tollmien-Schlichting instability (T SI) is negligible for all Ra < RaRBc : the critical Reynolds number for the onset of T S waves in RBP remains very close to ReTc S ≈ 5772.2.
A brief history of RBP and RBC flows: Non-modal stability
For the last two decades, non-modal stability analysis has been an active area of research in hydrodynamic stability and it has been recognized as fundamental to the understand-ing of shear flow instabilities. Many new concepts and techniques have been developed and successfully employed in various flows [76]. Recently, there had been an increased interest in the Rayleigh-B´enard-Poiseuille/Couette system as a prototype problem for transient growth mechanisms related to buoyancy forces [5, 74, 55], absolute and con-vective instabilities involving two propagation directions [7, 66, 29], three-dimensional global modes [8, 57, 56, 58], etc. The system has so far motivated many researchers in flow instabilities because of its simplicity and fascinating properties. Biau and Bottaro [5] investigated the effect of stable thermal stratification, solely induced by buoyancy, on the spatial transient growth of energy in RBP flow. The analysis showed that the presence of stable stratification reduces the optimal transient growth of perturbations. Perhaps the most akin to this thesis is the article by Sameen and Govindarajan [74] who studied the effect of heat addition on the transient growth and secondary instability in plane channel flow. According to their study, the effect of heating may be split into three contributions: the first one is due to the generation of buoyancy forces as in classical Rayleigh-B´enard convection, the second one is asso-ciated with the temperature-dependent base flow viscosity, and the third one results from viscosity variations induced by temperature perturbations. The computations re-vealed that heat addition gives rise to very large optimal growth. For various control parameter settings, it was demonstrated that viscosity stratification had a very small effect on transient growth. At moderately large Reynolds number (= 1000), the opti-mal disturbances could be either streamwise-uniform vortices (as in pure shear flows) or spanwise-uniform vortices, largely depending on Prandtl number and Grashof number. However, the transient growth mechanisms related to such optimal initial disturbances, and their corresponding response were not examined. Finally, cross-stream viscosity stratification was determined to have a destabilizing influence on the secondary insta-bility of T S waves.
Table of contents :
I Transient Growth in Rayleigh-B´enard-Poiseuille/Couette flows
1 Introduction
1.1 Basic concepts
1.1.1 Method of normal modes
1.1.2 Transient growth analysis
1.2 Rayleigh-B´enard-Poiseuille/Couette flows
1.2.1 Motivation
1.2.2 A brief history of RBP and RBC flows: Modal stability
1.2.3 A brief history of RBP and RBC flows: Non-modal stability
1.3 Objective
2 Linear stability analysis of RBP and RBC flows
2.1 Base flow
2.2 Governing Equations
2.3 Squire’s transformation
2.4 Adjoint equations and the norm
3 Modal Stability Analysis
3.1 Formulation
3.2 Squire’s Theorem in RBP and RBC flows
3.3 Numerical Technique
3.3.1 Chebyshev discretization of the governing equations
3.3.2 Validation
3.4 Modal stability characteristics
3.4.1 Road to the stability diagram
3.4.2 Dominant modal instability
4 Non-modal Stability Analysis
4.1 Road to Santiago
4.1.1 Choice of an appropriate norm
4.1.2 Computational method
4.1.3 Validation
4.2 Results: Non-modal stability analysis
4.2.1 Effect of varying Rayleigh number at constant Reynolds number .
4.2.2 Effect of varying Reynolds number at constant Rayleigh number
4.2.3 Domain of Transient Growth
4.3 Transient growth of streamwise-uniform disturbances in RBP and RBC flows
4.3.1 Lift-up Mechanism in the presence of temperature perturbations .
4.3.2 Short-time dynamics
4.3.3 Reynolds number scaling for Gmax(α, β;Re,Ra, Pr)
4.3.4 Long-time Optimal Response
4.3.5 Transient Growth at arbitrary time
4.3.6 Effect of Prandtl number
4.3.7 Effect of the norm kqk
4.4 Conclusion
II Consequences of the Squire transformation on 3D Optimal Perturbations
5 Squire’s transformation and 3D disturbances
5.1 Introduction
5.2 Governing equations
5.3 Consequences on the eigenfunctions
5.4 Consequences on long-time optimal gains
5.4.1 Case (1): Streamwise-uniform disturbances (α = 0)
5.4.2 Case (2): 3D long-time optimal perturbations
5.5 Discussion
5.6 Conclusion
6 Remarks and Perspectives
Bibliography
