Instability of a boundary layer flow on a vertical wall in a stably stratified fluid

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Internal gravity wave

The internal gravity waves are transverse waves as the movement of the fluid parcels is perpendicular to the direction of wave propagation. For a freely-propagating internal gravity wave, there is a special characteristic : the group velocity is perpendicular to the phase velocity. As shown in figure 1.7(a), internal gravity waves can be generated experimentally in a uniformly stratified fluid (constant N), and the energy is propagating away along surfaces of constant phase (Mowbray & Rarity 1967). More features and behaviours of internal gravity waves can be described with the help of linear theories, as reviewed by Fritts & Alexander (2003). The influences and effects of these waves are evaluated in these theories.
In nature, internal gravity waves also become visible in the satellite era, for example, figure 1.6 and 1.8. Internal waves exist both in the atmosphere and in the ocean, where the density stratification is stable and continuous. They are significant in vertical transfer and mixing.
In the circulation system of the ocean, the internal gravity waves transport micro-organisms, plankton, and nutrients. However, the physics of mixing in ocean interior is not well understood, and the internal mixing problem should be considered (Ivey et al. 2008). Although existing in the interior, internal gravity waves can alter the sea surface currents and make a difference. Where these currents converge, the surface is more turbulent and brighter. Where they diverge, the surface is smoother and darker, and thus, creating a zone called “slicks”. The slicks in the Sulu sea appear as dark bands in the centre of figure 1.6. Faint but visible ripples of internal gravity waves appear in figure 1.8 with the help of sunlight. In the ocean, the corresponding waves are tens to hundreds of meters beneath the sea surface and propagate slowly there. The comparisons of the scales in the mixing processes are shown in figure 1.7(b). Phenomena that mix the ocean are hard to be resolved in general ocean circulation models because the scales of relevant turbulent eddies are typically smaller than the grid spacing. In both ocean and atmosphere, this kind of small eddies should be examined so that their effects can be parametrized (Garrett 2003).
In the atmosphere, internal waves mainly appear in the troposphere where the den-sity is relatively high. Due to a decrease in density, the amplitude of these waves has a quasi-exponential growth when propagating in the vertical direction. Hence, at a certain altitude, the energy and momentum of these waves are transported and spread as turbu-lence mixing. Although the gravity waves themselves are not visible in the atmosphere, the marks of their propagation are left behind and represented by the cloud. The special structures appear on satellite photos, for example, figure 1.8, where the uncommon clouds are marked as atmospheric waves.
The gravity waves in the atmosphere are usually caused by the mountain waves (Lee waves), which have been studied for a long time through multiple observations and analysis Fritts & Alexander (2003). We often see the waves patterns of the clouds when the dry air moves towards the much moister air. The dry air pushes the moist air higher in the atmosphere, causing water vapour to amass into clouds. When the moister air rises, the gravity pulls it down since it is heavier in density. This procedure repeats and forms the wave patterns in the clouds. However, the source of the atmospheric waves shown in figure 1.8 is not clear. Because the wave patterns are far from the inland, and Western Australia is relatively flat, it is unlikely that these patterns are formed by mountain waves. The studies of internal gravity waves require further attentions because of their importance in global energy circulation and energy budget.
Numerous studies have also examined the effects of stable stratification in the context of atmospheric flows (Mahrt 2014). The internal gravity waves are fully three-dimensional in space, and they are relatively small compared to the normal scales in climate models. With the influence of some instabilities, the exchange of energy and momentum by internal gravity wave can lead to turbulence, mixture and dissipation (Staquet & Sommeria 2002).
To have a better understanding of mechanisms and mixing effects, we need to take the localized dissipation of energy and small-scale processes into account. The onset of turbulent mixing is related to the instability of fluid. To uncover the basic physical mecha-nisms of instabilities, we need to simplify the realistic complex context while keeping the essential characteristics of the flow. Researches on the instability start with homogeneous fluids, both experimentally and theoretically.

Modal instability in stratified shear flow

The instability of shear flows has been a classic subject in fluid dynamics since the theoretical studies of Rayleigh (1880) and the experiments by Reynolds (1883) at the end of 19th century, and it is covered by several textbooks (e.g. Drazin & Reid 1981; Schmid & Henningson 2001). For the instability problem, small amplitude perturbations and linearised Navier-Stokes equations are useful approximations and methods in the research field. With the exactly parallel base flow profiles, the linearized instability can be determined by the eigenvalue problem of Orr-Sommerfeld equation and the critical Reynolds number can be defined by the solutions. Some of the well-known instabilities are reviewed in this section.

Kelvin-Helmholtz instability

The instability associated with the inflexion point of the velocity profile (Kelvin-Helmholtz instability) is the one of the first instabilities discovered in fluid mechanics. The Kelvin-Helmholtz instability may appear in the atmosphere, as outlined by the clouds in figure 1.9. The question of in what circumstances, this instability occurs in shear flows has occupied many generations of scientists. First works have concerned homogeneous fluids. Rayleigh (1880) demonstrated that a necessary condition for a parallel inviscid shear flow to become unstable is the existence of inflexion point in the velocity profile. Wavelike disturbances appear in this kind of unstable flow and grow exponentially with time. Fjørtoft (1950) later complemented that the inflexion point has to correspond to a maximum (rather than a minimum) of the velocity shear. Another general stability condition is called Howard’s semicircle theorem (Howard 1961). This condition bounds the complex phase velocity of the disturbance in a semicircle, whose diameter is the difference between the largest and smallest velocity in the parallel shear flow.
The stable stratification is intuitively believed to have a stabilising effect. In a stably stratified fluid, Miles (1961) and Howard (1961) showed that the criterion of Kelvin-Helmholtz instability for two-dimensional inviscid parallel shear flow can be determined by the local Richardson number (Ri), which is the square of the ratio between buoyancy frequency and the vertical velocity shear rate. The Miles-Howard theorem states that the flow is stable when the local Richardson number is everywhere greater than or equal to 0.25. The stable stratification is thus believed to have a stabilisation effects on Kelvin-Helmholtz instability and can even eliminate the turbulence when the stratification is strong enough. However, the numerical simulations (e.g. Riley & De Bruyn Kops 2003; Lindborg 2006), laboratory experiments (e.g. Spedding et al. 1996; Augier et al. 2014) and ocean observations (e.g. Mack & Schoeberlein 2004; Polzin & Ferrari 2004) showed unexpected turbulence in strongly stratified fluid (Ri > 0.25). These facts suggest that other mechanisms might be substantial in stratified fluids, and further research is required in the regions beyond the applicability of Miles-Howard theorem.
Little work has been done when shear and stratification are not aligned in the same direction, but such kind of cases occur in nature (e.g. Figure 1.10). The Kelvin-Helmholtz instability is modified in such cases as studied by Deloncle et al. (2007), Candelier et al. (2011) and Arratia (2011).

Viscous instability (Tollmien-Schlichting wave)

In the subject of instability in fluids, the viscosity is well-known to have dual roles. Most studies about viscous instability depend on the exact parallel shear flows. The plane Poiseuille flow is one of the simplest shear flows, but its instability and transition to turbulence have proved to be subtle. As a result, this flow is widely used to illustrate the fundamentals of the stability theory (Drazin & Reid 1981).
In the context of an unstratified fluid, Heisenberg (1924) first demonstrated the vis-cous instability mechanism for plane Poiseuille flow. As the velocity profile of base flow has no inflexion point, the instability must be related to viscosity. The Reynolds number of the base flow can estimate the effects of viscosity, and instability appears when the Rey-nolds number exceeds a specific critical value (the critical Reynolds number). The viscous instability is commonly referred to as Tollmien-Schlichting (TS) wave (Tollmien 1935; Schlichting 1933) in honour of the researchers who first predicted that Orr-Sommerfeld equation has unstable modes for flows without inflexion points. The Squire’s theorem (Squire 1933) gives the assertion that it is sufficient to consider two-dimensional modes to determine the critical Reynolds number. In other words, the most unstable mode is always two-dimensional. Experimental observations of TS wave were obtained by Klebanoff et al. (1962).
For unstratified plane Poiseuille flow, Orszag (1971) obtained the critical Reynolds number Rec = 5772.22 from the accurate solution of the Orr-Sommerfeld stability equa-tion. The definition of the Reynolds number is based on the half width of the channel, the centre-plane velocity, and the kinematic viscosity.
Experimental results of Nishioka et al. (1975) show good agreement that the small perturbations behave as predicted by the linear stability theory, in a condition of a very low background turbulence level. However, the transition in plane Poiseuille flow is sensitive to ambient disturbances. Carlson et al. (1982) observed that both natural and artificially triggered transition can occur for Reynolds numbers slightly greater than 1000, and the most amplified wave is three dimensional. More experimental results (e.g. Nishioka & Asai 1985; Alavyoon et al. 1986) and the considerable variations of different critical Reynold numbers indicate that the three-dimensional effects are important in the transition of plane Poiseuille flow, which is opposed to the Squire’s theorem. This reduction of critical Reynolds number is related to the so-called subcritical instability. The subcritical insta-bility exists in a fluid with a Reynolds number below the critical one. Experiments in boundary layer flows (e.g. Klebanof 1971; Kendall 1985; Matsubara & Alfredsson 2001) also demonstrate that transition is usually preceded by streamwise motion in the form of streaks rather than the TS waves predicted by modal stability analysis. This phenomenon is especially distinct in the presence of natural background disturbance.
The limitation of Squire’s theorem was further revealed by Wu & Luo (2006), who examined the influence of small imperfections on the stability of plane Poiseuille flow. The classical Squire’s transformation was also extended by Jerome & Chomaz (2014).
In a stratified fluid, the Squire’s theorem is not applicable, but Ohya & Uchida (2003) experimentally proved the existence of TS waves in stratified boundary layers. However, the effect of stratification on TS waves is less understood, and it could be destabilising as indicated by Wu & Zhang (2008a). Although the Squire’s theorem applies strictly to linear normal-mode analysis for a homogeneous viscous fluid, both the 3D and 2D TS waves should be examined in a stably stratified fluid.

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Radiative instability

Although it may be true to a degree that stratification can stabilise the fluid (Miles 1961; Howard 1961), instability still exist even without inflexion points in the velocity profiles of the flow. For instance, Churilov (2005, 2008) analysed the stability of the flows with inflection-free velocity profiles in a stratified fluid. Stratified fluids can support inter-nal gravity waves. Together with shear effects, small scale unstable waves are generated and thus can promote turbulent mixing in stratified fluids.
When shear and stratification are not aligned, an inflection-free boundary layer can become unstable. The coupling between shear effects and internal gravity waves becomes the strongest when the inclination angle between shear and stratification is π/2, as de-monstrated by Candelier et al. (2012). The responsible instability for this phenomenon is called radiative instability. The word ‘radiative’ describing this instability is borrowed from the quantum mechanics because the demonstrated phenomenon is analogous to the radioactive decay of nuclei (Le Dizès & Billant 2009).
The radiative instability is a hydrodynamic instability that is much less known than the shear instability. However, it has been a subject of many works for decades, and it has also shown its significant influences with some specific conditions. Broadbent & Moore (1979) first found the radiative instability in the context of vortices in a compressible fluid. This instability appears in various contexts, in the shallow waters associated with surface gravity waves, in compressible fluids with acoustic waves and in linearly stratified flows with internal gravity waves. The genesis of radiative instability is reviewed in Riedinger (2009).
With the help of a WKBJ analysis for large axial wave number, Le Dizès & Billant (2009) and Billant & Le Dizès (2009) showed that the radiative instability could be vie-wed as a result of an over-reflection process at a critical level. The relationship between instability and over-reflection mechanism has been studied for a long time (e.g. Lindzen & Tung 1978; Lindzen & Barker 1985). Further details about this mechanism will be explained later in section 1.3.1.
In recent researches, Riedinger et al. (2010a) obtained radiative instability for the Lamb–Oseen vortex and analysed the Froude and Reynolds number effects. Experimental results were first provided in Riedinger et al. (2010b). In figure 1.11, Riedinger et al. (2011) also demonstrated, both experimentally and numerically, the occurrence of the radiative instability generated by a rotating cylinder in the stratified flow. In Taylor–Couette and Keplerian flows, Le Dizès & Riedinger (2010) analysed the transition of the instability for a finite-gap, which is associated with a mechanism of resonance, into the radiative instability for an infinite gap. Park & Billant (2013b) studied the stability of stratified Taylor-Couette flow and its relationship to radiative instability. Park & Billant (2012, 2013a) also investigated the radiative instability of a columnar Rankine vortex in a stra-tified rotating fluid.
In addition to the stratified rotating flows, the radiative instability can also be found in supersonic flows (Mack 1990; Parras & Le Dizès 2010), atmospheric jets, boundary layers (Candelier 2010; Candelier et al. 2012) and stratified shallow waters (Satomura 1981; Riedinger & Gilbert 2014).

Mechanisms of modal instability in stratified fluids

Because of the abundant literature about the radiative instability and its universal characters, the mechanism was rediscovered several times in history by different communi-ties, and was explained in terms of over-reflection phenomenon (Grimshaw 1979; Lindzen
& Barker 1985; Takehiro & Hayashi 1992), negative energy waves (Kópev & Leontev 1983; Schecter & Montgomery 2004) or spontaneous wave emission (Plougonven & Zeitlin 2002; Le Dizès & Billant 2009).

Over-reflection phenomenon

Over-reflection is a phenomenon that the incident waves in a fluid can be reflected with a consequent increase in amplitude. In other words, the reflection coefficient in some circumstances can be greater than 1. This mechanism is found in numerous situations and physical models. The over-reflection phenomenon was first encountered in the field of sound waves by Miles (1957) and Ribner (1957). Acheson (1976) later summarised the conditions for the occurrences of over-reflection. He also attempted to clarify the way in which the excess reflected waves extracted energy from the mean flow motion, and the sense that the transmitted wave can be viewed as a carrier of ‘negative energy’. The over-reflection process was thus explained through the conservation of wave action (Acheson 1976).
The summary by Acheson (1976), however, is not sufficient. The connections between over-reflection and unstable modes are discovered in different context (Lindzen & Tung 1978; Lindzen & Rambaldi 1986). Lindzen (1988) summarised and gave the statement that the mode which satisfies ‘quantization’ conditions can become unstable through the over-reflection process. Then the unstable mode determines the wave geometry.
A quantization zone is an oscillating area confined either by two turning levels or by one turning level together with a perfectly reflecting wall. Perturbations in this area will propagate, be reflected by over-reflection and thus increase. The reflection is always accompanied by a transmitted wave in the external transmission region. This reflection satisfies the conservation law of momentum for three waves : incident, transmitted and reflected waves. However, the conservation law of energy is different in this case, as part of the energy is drawn from the mean flow motion.
A perturbation in the quantization zone can be modelled as a wave propagating in this domain to illustrate the mechanism. A simple example of a wave packet is shown in figure 1.12. The phase velocity and group velocity may be different and can move towards separate directions.
The comparison between a conventional reflection case and an over-reflection case is illustrated in figure 1.13. In the conventional reflection case (figure 1.13 (a)), for example, the radioactive emission through a potential barrier, there is no critical level inside the evanescent region between two turning levels. The phase velocity and group velocity of the incident wave in the conventional case are in the same direction. The intermediate evanes-cent region partially reflects the incident wave with an amplitude Ai. The amplitudes of the reflected and transmitted waves are Ar and At, respectively. The classical relationship (Ai = Ar + At) for the amplitudes of these three waves applies to this conventional case.

Table of contents :

1 Introduction 
1.1 Basic concepts about stratified fluid
1.1.1 Buoyancy frequency
1.1.2 Internal gravity wave
1.2 Modal instability in stratified shear flow
1.2.1 Kelvin-Helmholtz instability
1.2.2 Viscous instability (Tollmien-Schlichting wave)
1.2.3 Radiative instability
1.3 Mechanisms of modal instability in stratified fluids
1.3.1 Over-reflection phenomenon
1.3.2 Resonance phenomenon
1.4 Transient growth and optimal perturbation
1.4.1 Non-normality and finite time intervals
1.4.2 Optimal perturbation and adjoint equations
1.5 Mechanism of transient growth
1.5.1 Orr mechanism
1.5.2 Lift-up mechanism
1.5.3 Combination of Orr and lift-up mechanisms
1.6 Motivation and purpose
1.7 Summary of this thesis
2 Method 
2.1 Base flow and perturbation equations
2.2 Optimal perturbations
2.3 Numerical method for eigenvalues
2.3.1 Eigenvalues for plane Poiseuille flow
2.3.2 Eigenvalues for boundary layer flow
3 Instability of a boundary layer flow on a vertical wall in a stably stratified fluid 
3.1 Introduction
3.2 Temporal stability results
3.2.1 Boundary layer instability (Tollmien-Schlichting waves)
3.2.2 Radiative instability
3.2.3 Competition between radiative instability and viscous instability
3.3 Discussion
4 Instability of plane Poiseuille flow in a stably stratified fluid 
4.1 Introduction
4.1.1 Modal stability of stratified Poiseuille flow
4.1.2 Non-modal stability of stratified Poiseuille flow
4.1.3 Effect of horizontal shear and vertical stratification
4.2 Modal stability analysis
4.2.1 Tollmien-Schlichting waves
4.2.2 The gravity mode in the presence of stratification
4.2.3 Resonance mechanism
4.2.4 Instability contours in wavenumber plane
4.3 Non-modal stability analysis
4.3.1 Verification for unstratified fluid
4.3.2 Eigenfunctions and optimal perturbations
4.3.3 Velocity field of transient growth
4.3.4 Stratification effects on transient growth
4.4 Discussion
5 Conclusion and perspective

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