Numerical study of swirl instabilities in Boussinesq Navier-Stokes models with geophysical applications

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(1)UNIVERSIDAD DE CASTILLA-LA MANCHA Departamento de Matemáticas. Numerical study of swirl instabilities in Boussinesq Navier-Stokes models with geophysical applications. A thesis presented in partial fulfilment of the requirements for the degree of Doctor of Philosophy at the University of Castilla-La Mancha. by. Damián Castaño Torrijos Supervised by Dr. Henar Herrero Sanz (UCLM) Dr. Marı́a Cruz Navarro Lérida (UCLM). Ciudad Real, 2016.

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(3) Esta tesis ha sido posible gracias a la ayuda y dedicación de Marı́a Cruz Navarro y Henar Herrero. Siempre agradeceré vuestros consejos y vuestra amistad. Gracias a Fran por los ratos de ocio compartidos estos años. Y gracias a mis padres por haberme dado la motivación y el apoyo necesarios para poder alcanzar esta meta..

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(5) A mis padres.

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(7) Contents Introduction. 1. 1 Cylindrical annulus setup. 9. 1.1. 1.2. 1.3. The model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 9. 1.1.1. Physical setup . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 9. 1.1.2. General equations of the problem . . . . . . . . . . . . . . . . .. 10. 1.1.3. Change of variables . . . . . . . . . . . . . . . . . . . . . . . . .. 14. 1.1.4. Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . .. 15. Axisymmetric stationary approach . . . . . . . . . . . . . . . . . . . .. 17. 1.2.1. Basic state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 17. 1.2.2. Axisymmetry assumption . . . . . . . . . . . . . . . . . . . . .. 18. 1.2.3. Linear stability analysis . . . . . . . . . . . . . . . . . . . . . .. 18. 1.2.4. Numerical implementation . . . . . . . . . . . . . . . . . . . . .. 20. Numerical results for the axisymmetric stationary approach . . . . . . .. 21. 1.3.1. Set of parameters . . . . . . . . . . . . . . . . . . . . . . . . . .. 21. 1.3.2. Non-rotating basic states when ā is varied . . . . . . . . . . . .. 23. 1.3.3. Stability of the non-rotating basic states . . . . . . . . . . . . .. 23. 1.3.4. Vortices when ā is varied . . . . . . . . . . . . . . . . . . . . . .. 24. 1.3.5. Vorticity and angular momentum . . . . . . . . . . . . . . . . .. 29. 1.3.6. Stability of the vortices . . . . . . . . . . . . . . . . . . . . . . .. 30. 1.3.7. Trajectories inside the vortex when ā is varied . . . . . . . . . .. 31. 1.3.8. Contraction and stabilization of the RMAV . . . . . . . . . . .. 33. 1.3.9. Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 34. i.

(8) ii. CONTENTS 1.4. 1.5. 3D temporal approach . . . . . . . . . . . . . . . . . . . . . . . . . . .. 35. 1.4.1. First assumptions . . . . . . . . . . . . . . . . . . . . . . . . . .. 35. 1.4.2. Temporal discretization and projection scheme . . . . . . . . . .. 36. 1.4.3. Spatial discretization . . . . . . . . . . . . . . . . . . . . . . . .. 39. 1.4.4. Final equations and boundary conditions . . . . . . . . . . . . .. 39. 1.4.5. Validation of the code . . . . . . . . . . . . . . . . . . . . . . .. 43. Numerical results for the 3D temporal approach . . . . . . . . . . . . .. 44. 1.5.1. Thermoconvective bifurcation diagram . . . . . . . . . . . . . .. 44. 1.5.2. Secondary whirl . . . . . . . . . . . . . . . . . . . . . . . . . . .. 45. 1.5.3. Route to chaos . . . . . . . . . . . . . . . . . . . . . . . . . . .. 50. 1.5.4. Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 67. 2 Cylinder setup. 69. 2.1. Axisymmetric stationary approach . . . . . . . . . . . . . . . . . . . .. 70. 2.2. Numerical results for the axisymmetric stationary approach . . . . . . .. 71. 2.2.1. Formation of the vortex and stability analysis . . . . . . . . . .. 71. 2.2.2. Formation of the eye . . . . . . . . . . . . . . . . . . . . . . . .. 76. 2.2.3. Tilting of the axis/eye . . . . . . . . . . . . . . . . . . . . . . .. 80. 2.2.4. Trajectories of particles . . . . . . . . . . . . . . . . . . . . . . .. 81. 2.2.5. Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 82. 3D temporal approach . . . . . . . . . . . . . . . . . . . . . . . . . . .. 83. 2.3.1. Computational requirements at r = 0 . . . . . . . . . . . . . . .. 83. 2.3.2. Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . .. 89. Rotating cylinder setup . . . . . . . . . . . . . . . . . . . . . . . . . . .. 91. 2.4.1. Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 92. 2.4.2. Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . .. 93. Numerical results for the rotating cylinder setup . . . . . . . . . . . . .. 94. 2.5.1. E > 0.056 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 95. 2.5.2. 0.027 ≤ E ≤ 0.056 . . . . . . . . . . . . . . . . . . . . . . . . .. 97. 2.5.3. 0.007 ≤ E < 0.027 . . . . . . . . . . . . . . . . . . . . . . . . . 111. 2.3. 2.4. 2.5.

(9) CONTENTS. iii. 2.5.4. Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129. 3 Conclusions and future work. 131. A Chebyshev polynomials. 137.

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(11) Introduction Vertical vortices and rotating flows have been of great interest for centuries. They are present at all scales in nature. There are different atmospheric phenomena with the common factor of vortex dynamics, from small events such as dust devils and tornadoes, to large events as hurricanes and cyclones [2, 24, 106, 111]. In all these atmospheric phenomena there exists a rich and complex spectrum of fluid dynamical processes involved in their formation and evolution. Thermal convection is one of these relevant processes [23, 100]. In thermal convection, the fluid motion is caused by buoyancy forces that results from the density variations due to thermal gradients in the fluid. The simplest case of thermal convection is the well-known Rayleigh Bénard convection where cells or rolls appears in a horizontal fluid layer heated from below [6]. Instability occurs at the minimum temperature gradient at which a balance can be steadily maintained between the kinetic energy dissipated by viscosity and the internal energy released by the buoyancy flow [15]. Dust devils are widely recognized as examples of convective phenomena. They are columnar, ground-based whirlwinds, made by dust, common in dry regions [7, 32, 40, 112] (Figure 1). The conditions under which a dust devil appears are a surface interface region particle loaded and a strong horizontal thermal gradients on areas with anomalously high soil temperature [46, 110, 111]. They are formed when surface isolation leads to a superadiabatic lapse rate, causing an unstable stratified atmosphere and strong convection [2, 29, 111]. The dust devil structure is formed by an inflow towards the center, a near vertical column of rotating dust and the top part where the rotation decays [111, 112]. It is known that there are several morphologies for a dust 1.

(12) 2. INTRODUCTION. devil [14, 49, 112, 116]. The first one is a narrow tightly defined column with a single cell where the vertical velocity is high in the center and the fluid rotates around this axis [14, 116]. The second one is a V-shaped less defined column formed by two cells with a flow with central downdrafts [14, 112]. These downdrafts form the characteristic eye in dust devils. It is also known from observations in dust devils that when a defined columnar core is present, it often tilts towards the direction of motion by about 10o [68, 69, 112]. Dust devils frequently contain subvortices, small “parasites” or secondary circulations embedded in the primary whirl (dust devil) that normally form near the center of the dust devil and follow essentially concentric circular paths about the dust devil center [2, 72, 108, 112, 122]. The diameter of a dust devil can varies from less than 1 m to generally less than 100 m [116] and their height ranges from a few meters to a few kilometers [68]. The horizontal velocity in dust devils varies from 10 to 25 m/s [2, 111], and the vertical velocities tend to be of the same order of magnitude as the horizontal swirling velocities, varying from 2 to 15 m/s [108, 112]. Dust devils are characterized by a warmer area close to the dust devil core that coincides with a drop of pressure . The temperature difference with the ambient varies between 1o C and 10o C [73, 119] and the difference in pressure between 0.3 and 7 mb [101, 112, 119]. Most of the authors emphasized the importance to dust devil formation of intense surface heating, which leads to high surface air temperatures and superadiabatic lapse rate, but a crucial question remaining is how they acquire rotation, how and when the eye is created and why dust devils tilt towards the direction of motion [68, 69, 112]. There are several theories for the source of vorticity in dust devils. Some works explain it as resulting from interactions with terrain features or wind fluctuations [14, 110], but many dust devils occur in flat and calm zones. Planetary vorticity is another explanation, but dust devils can rotate in both senses [14, 111]. Convective cell circulations are the accepted theory. Several authors [14, 51, 109] have proposed or provided evidence that larger-scale convective circulations, that are not initially rotational, can provide vertical and/or tiltable horizontal vorticity. Other atmospheric vortices with similar characteristics to dust devils are tornadoes, cyclones and hurricanes. As in the case of dust devils, planetary rotation effects should.

(13) INTRODUCTION. 3. Fig. 1: An intense dust devil on a beach just southwest of Boulder City, Nevada, USA. (June 26, 2010). This vortex held together for approximately 15 minutes. Courtesy of Timothy I. Michaels.. not be relevant a priori for tornadoes [17, 106]. Hurricanes are more complex atmospheric vortices involving many fluid-dynamical processes, including rotation, stratified flow dynamics, boundary layers, convection and air-sea interaction [21, 22]. Most of the laboratory experiments in the field of vertical vortex dynamics are those in so-called vortex chambers [16, 17, 26, 121]. The first vortex chamber was the Ward-type chamber, which consists of an open cylinder with an imposing rotating screen and an upward-directed fan producing a volume flow rate through an orifice which generates the vortex. The presence of open boundaries at which the flow is not known before the experiment presents a special challenge. For this reason, Fiedler [26] suggested experiments in a closed cylindrical domain with ambient rotation. Different mathematical models for atmospheric phenomena can be found in literature, depending mainly on the atmospheric scale and the degree of turbulence. For atmospheric microscale (up to 2 Km) [95], the incompressible Navier-Stokes model under the Boussinesq approximation, and turbulence models can be used. Incompressible Boussinesq Navier-Stokes is valid for a wide range of atmospheric conditions for which.

(14) 4. INTRODUCTION. the Reynolds numbers correspond to non turbulent regimes [1, 34, 54, 87]. This approximation has been used in Refs. [58, 113] with applications to dust-devil dynamics in a non-rotating frame but considering rotation on the lower portion of the sidewall, and in Refs. [26, 27, 28, 88, 91, 92, 103, 104, 105] in a rotating frame of reference. These numerical experiments produce a range of vortex types as a function of the swirl ratio, essentially the ratio between the rotation strength and the updraft strength, analogous to that produced in laboratory experiments [121]. Turbulence models, mainly large eddy simulations (LES), have been considered in Refs. [59, 60] in a rotation frame of reference, and in Refs. [50, 51] in a non-rotating frame of reference, where the turbulent convective pattern appears to be relevant in the formation of vertical vortices. The bifurcation and instability analysis has been introduced in the research field of vertical vortices in Refs. [66, 82, 83]. Finite differences are mainly used to solve these partial differential equations problems [26, 27, 28, 51, 88, 91, 92, 103, 104, 105, 113], althought finite elements, finite volumes, and spectral methods are used in atmospheric models [13, 115, 124]. Spectral codes are widely used to solve incompressible Boussinesq Navier-Stokes equations under the Rayleigh-Bénard instability perspective [4, 8, 36, 66, 78, 118]. A lot of research is being done currently to improve the efficiency of the numerical codes. These methods have advanced while computer facilities, from stationary two dimensional codes to three dimensional time dependent codes, reduced models, domain decomposition or continuation techniques [19, 20, 39, 71, 85, 117]. In this thesis the mathematical modeling is performed on Newtonian fluids obeying the incompressible Navier Stokes equations under the Boussinesq approximation coupled with the heat equation. We solve numerically the problem considering a nonhomogeneous heating from below without and with ambient rotation, for a better understanding of the relevance that thermal gradients have in the generation and evolution of vertical vortices and its connection to atmospheric vortices. For a fully 3D temporal numerical simulation, a second-order time-splitting method described in Ref. [71] has been implemented. This time stepping method was proposed by Hugues and Randriamampianina [45]. A pseudo-spectral method is used for the spatial discretiza-.

(15) INTRODUCTION. 5. tion, with a Fourier expansion in the azimuthal coordinate φ and Chebyshev collocation in r and z. In the first part of the thesis we consider a cylindrical annulus configuration under the presence of horizontal temperature gradients in a non-rotating frame of reference. There are many works on modeling thermal convection when a horizontal temperature gradient is stablished at the bottom [12, 36, 43, 63, 64, 66, 78, 79, 81] but not on thermally driven vortices, and most of them impose artificially a rotating fluid [77, 94, 113]. In Ref [82] authors show numerically that an axisymmetric vortex can appear spontaneously after a thermal bifurcation of a stationary axisymmetric non rotating convective flow, when two temperature gradients (vertical and horizontal) come into play. We analyse here the influence of the size of the radius of the inner cylinder on the structure and intensity of the vortex developed. We also show that a thermoconvective instability is responsible for the formation of secondary whirls embedded in the primary axisymmetric vortex and study the influence of the inner radius and the profile of the temperature at the bottom on these parasitic whirls. Moreover, we show that a contraction is observed in the radius of maximum azimuthal velocity (RMAV) when the vortex is intensified by thermal mechanism. These results agree with observations of the RMW in hurricanes [53]. In a second part we extend the study to a cylinder setup. We prove that under particular thermal conditions, vertical and horizontal temperature gradients and geometrical conditions, a vortex can be generated by a convective instability. Different morphologies for the axisymmetric vortex can be developed depending on the heat profile at the bottom and the thermal gradients, morphologies similar to those observed in real dust devils without and with a central eye [49, 112]. We prove that thermoconvection is the responsible for the tilting of the vertical vortex, results that connect to observations in dust devils that often tilt towards the direction of motion by about 10o [35, 72, 123]. We complete the study by considering an ambient rotation in the system and analizing the effect that rotation, combined with the thermal gradients, have in the vortices developed. For moderate rotation rates we show the transition from vortices without a.

(16) 6. INTRODUCTION. central eye to single eyed-vortices, a change in the vortex type observed in laboratory tornadoes [77, 103, 121]. For larger rotation rates we report the formation of double vortices, result that also connects with laboratory tornadoes, where multiple vortices are found, and with real observations as the interesting formation of a double-eyed vortex at Venus’s south pole [97]. Another key point of the thesis is the study of the transition to chaos from the axisymmetric vertical vortices. In closed systems like Rayleigh-Benard convection, transition to chaos or turbulence is a subject of great interest from both theoretical and experimental points of view. As the Rayleigh number is increased a sequence of instabilities takes place, leading the flow from the laminar state to chaos. Different routes to chaos are well known. The flow may be chaotic after three incommensurate bifurcations (Ruelle-Takens-Newhouse scenario) [86, 107], after an infinite sequence of period-doubling bifurcations (Feigenbaum scenario) [25], or after an intermittency regime (Manneville and Pomeau scenario) [65]. The transition scenario depends on the geometry, initial conditions, and other specific characteristics of the flow. There are many studies on the transition to chaos in Rayleigh-Benard convection for small rectangular cavities, both experimentally [33] and numerically [11, 74, 75, 76]. In Ref. [33] authors identified experimentally different routes to turbulence in Rayleigh-Benard convection: quasiperiodic flows with two or three frequencies, as well as periodic-doubling bifurcations preceding chaos. In Ref. [120] authors observed quasiperiodic flows with four or five independent frequencies. In horizontal cylindrical annuli, very few studies of the transition to chaos of buoyancyinduced flows have been carried out. A complete transition path was experimentally performed in Ref. [55], identifying a Pomeau-Manneville route to chaos. A very thorough experimental analysis of the transition to chaos with a route characterized by a sequence of Hopf transitions, similar to the Ruelle-Takens scenario, is found in Ref. [56]. In this thesis we study the evolution from the vertical axisymmetric vortex to a chaotic flow as the Rayleigh number is increased, showing that a scenario similar to the Ruelle-Takens-Newhouse scenario is verified. The appearance, evolution, and disappearance of periodic and quasiperiodic dynamics of fluid flows is reported. In.

(17) INTRODUCTION. 7. the transition to chaos for the cylindrical annulus setup we find the appearance of subvortices embedded in the primary axisymmetric vortex, flows where the subvortical structure strengthens and weakens, that almost disappears before reforming again, leading to a more disorganized flow to a final chaotic regime. Results are remarkable as they connect to observations describing formation, weakening, and virtual disappearance before revival of subvortices in some atmospheric swirls such as dust devils [2, 35, 72, 108, 111, 122]. For the cylinder setup, changes in the size and intensity of the tilted vortex developed are reported as well as the transition from the double vortex structure to a single-eyed vortex, depending on the rotation rate considered. The thesis is organized as follows: chapter 1 is enterely dedicated to the cylindrical annulus setup. Chapter 2 includes the results on the cylinder setup without and with ambient rotation, and in chapter 3 conclusions and future work are presented..

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(19) Chapter 1 Cylindrical annulus setup In a first approach, we consider a cylindrical annulus setup as the domain. This approach is interesting as a starting point because it is easier to deal with compared to the cylinder setup as it avoids the difficulty of treating the origin and because the inner cylinder can artificially acts as an eyewall.. 1.1. The model. We describe the physical setup used, and the equations and boundary conditions considered.. 1.1.1. Physical setup. The physical setup considered consists of a horizontal fluid layer in a container bounded by two concentric cylinders of radii a and a+l (r coordinate) and depth d (z coordinate). The domain is called D, and the analytical expression of it is: D = {(x, y, z) ∈ R3 : a <. p x2 + y 2 < a + l, 0 < z < d}.. A key point in the model is the inclusion of horizontal temperature gradients, shown to be relevant for the generation of vertical atmospheric vortices [23, 106, 112]. Therefore, the fluid is heated from below with an imposed Gaussian temperature profile,. 9.

(20) 10. CHAPTER 1. CYLINDRICAL ANNULUS SETUP. g. a. l. T0 ∆ Tv. d ez eφ. er Tmax ∆ Th β. β T. min. Fig. 1.1: Physical setup.. an easy way to introduce the non-homogeneity, which takes the value Tmax at r = a and the value Tmin at the outer part (r = a + l). The upper surface is open to the atmosphere where the temperature is kept fixed at T0 . At the bottom plate, we have the following temperature distribution: . T (r) = Ae where A =. ∆Th. 1 e β2. −1. , = Tmin −. ∆Th 1. (r−a)2 1 − 2 β l β. 2. + ,. and β is a parameter that measures the sharpness. e β 2 −1. of the Gaussian profile. We define the horizontal temperature gradient as ∆Th = Tmax − Tmin and the vertical temperature gradient as ∆Tv = Tmax − T0 . Figure 1.1 provides a scheme of the physical setup considered.. 1.1.2. General equations of the problem. The equations governing the problem are basically three: the equation of motion (Navier-Stokes), the continuity equation and the heat equation. These are the equations which model the hydrodynamical flow of a viscous fluid with varying density and.

(21) 1.1. THE MODEL. 11. temperature. The above equations relate the fluid velocity u = (u1 , u2 , u3 ) with the temperature T , the pressure p, and the density ρ. All of these fields are a function of the position x = (x1 , x2 , x3 ) = (x, y, z) and of the time t. We can find a detailed explanation of these equations in [3] and [15]. According to the notation used in [3], and adopting the usual summation convention for the Cartesian tensors, we get the basic hydrodynamic equations: a) The equation of motion (Navier-Stokes) 2 ρ∂t ui + ρuj ∂xj ui = ρFi −∂ xi p + ∂xj µ ∂xj ui + ∂xi uj − µ∂xk uk , 3. 1 ≤ i ≤ 3, (1.1). where F = (F1 , F2 , F3 ) is the external force acting on the fluid per unit volume and µ is the coefficient of viscosity which is considered as a constant. This equation expresses the conservation of momentum. b) The equation of continuity ∂t ρ + ∇ · (ρu) = 0,. (1.2). which expresses the conservation of mass. c) The equation of heat ρ∂t (cV T ) + ρu · ∇ (cV T ) = ∂xj κ̄∂xj T − p∂xj uj + Φ,. (1.3). where κ̄ is the coefficient of heat conduction, cV is the specific heat at constant volume, Φ indicates the rate at which energy is dissipated, irreversibly, by viscosity in each element of volume of the fluid and the nonlinear term ρu · ∇ (cV T ) represents the energy transport in the fluid. The Oberbeck-Boussinesq approximation Equations (1.1)-(1.3) describe the behavior of a fluid in many different situations. However, they are complicated to solve. Oberbeck in 1879 [93] and Boussinesq in 1903 [9], shown that if the temperature variations are small, the variations of the thermodynamic properties, such as viscosity, thermal diffusivity, density and specific heat of a fluid are also small and the fluid is approximately incompressible..

(22) 12. CHAPTER 1. CYLINDRICAL ANNULUS SETUP In fluid dynamics, the Oberbeck-Boussinesq approximation is used in the field of. buoyancy-driven flow. It ignores density differences except where they appear in the buoyancy term in the vertical momentum equation, as the acceleration resulting in the term of the external force ρFi can be larger than, for example, the acceleration due to the inertial term uj ∂xj ui in the equation of motion. The approximation consists of a linear dependence of the density with the temperature in the external force term, ρ = ρ0 [1 − α (T − T0 )] ,. (1.4). where ρ0 is the mean density at temperature T0 and α is the coefficient of volume expansion, and a constant density equal to the mean density at temperature T0 in the rest of the terms in the equations. In the atmosphere, the basic state density varies across the lowest kilometer by only about 10%, and the fluctuating component of density deviates from the basic state by only a few percentage points. These circumstances might suggest that boundary layer dynamics could be modeled by setting density constant and using the theory of the homogeneous incompressible fluids. However, density fluctuations cannot be totally neglected because they are essential for representing the buoyancy force, therefore the Oberbeck-Boussinesq approximation can be considered [41]. This approximation is valid for a wide variety of atmospheric conditions in non turbulent regimes [34, 54]. The first consequence from the Oberbeck-Boussinesq approximation is that the equation of continuity (1.2) becomes the incompressibility condition ∇ · u = 0.. (1.5). Using the equation (1.5), the equation of motion (1.1) transforms into ρ0 ∂t ui + ρ0 uj ∂xj ui = ρFi −∂ xi p + µ∂xj ∂xj ui ,. 1 ≤ i ≤ 3.. If only gravity is considered as an external force, i.e., F = −ge3 , where g is the gravity constant and e3 is the unit vector in the vertical direction, we get ∂t ui + uj ∂xj ui = −. ρ 1 µ ge3i − ∂xi p + ∂xj ∂xj ui , ρ0 ρ0 ρ0.

(23) 1.1. THE MODEL. 13. which is expressed in vectorial form as follows ∂t u + (u · ∇)u = −. ρ 1 ge3 − ∇p + ν∆u, ρ0 ρ0. (1.6). where ν = µ/ρ0 is the kinematic viscosity. By considering cv and κ̄ as constants, and defining κ =. κ̄ ρ 0 cv. as the thermal diffu-. sivity, equation (1.3) reduces to ∂t T + u · ∇T = κ∆T.. (1.7). The term Φ has been ignored as it is negligible compared to the other terms in the equation [34]. Dimensionless equations In most works, it is usual to solve the equations in dimensionless form. We can dimensionless equations (1.5)-(1.7) by the change of variables x0 =. x , d. t0 =. κ t, d2. u0 =. d u, κ. Θ0 =. T − T0 , ∆Tv. p0 =. d2 p. ρ0 κν. (1.8). This change transforms the original domain D into (primes are omitted) D∗ = {(x, y, z) ∈ R3 : ā < with ā =. a d. p x2 + y 2 < ā + Γ, 0 < z < 1},. and Γ = dl , which is called aspect ratio.. The state equations in dimensionless form are: ∇ · u = 0,. (1.9). ∂t u + (u · ∇) u = Pr (−∇p + ∆u + RΘe3 ) , ∂t Θ + u · ∇Θ = ∆Θ,. (1.10) (1.11). where two new dimensionless numbers appear R=. αd3 g∆Tv , κν. Pr =. ν . κ. The first number, R, is the Rayleigh number [99], which represents the buoyant effect, and P r is the Prandtl number, the ratio of kinematic viscosity to thermal diffusivity. For small values of P r thermal diffusivity dominates, whereas for large values of P r momentum diffusivity determines more dominantly the behavior of the fluid..

(24) 14. CHAPTER 1. CYLINDRICAL ANNULUS SETUP Theoretical studies for the existence and characterization of the solutions in func-. tional spaces can be found in Refs. [61, 62, 98].. 1.1.3. Change of variables. Transformation required by the shape of the domain Due to the shape of the domain, a change of coordinates from cartesian to cylindrical is required. Considering    x = r cos φ,   1 x2 = r sin φ,     x = z, 3.    e = cos φe1 + sin φe2 ,   r eφ = − sin φe1 + cos φe2 ,     e =e , z. 3. where {e1 , e2 , e3 } refers to the canonical basis of R3 and {er , eφ , ez } is the reference basis in cylindrical coordinates (see Figure 1.1), equations (1.9)-(1.11) are transformed into 1 1 ∂r (rur ) + ∂φ uφ + ∂z uz = 0, r r u2φ 1 −1 Pr (∂t ur + ur ∂r ur + uφ ∂φ ur + uz ∂z ur − ) = r r ur 2 = −∂r p + ∆c ur − 2 − 2 ∂φ uφ , r r 1 ur uφ −1 Pr (∂t uφ + ur ∂r uφ + uφ ∂φ uφ + uz ∂z uφ + )= r r 1 2 uφ = − ∂φ p + ∆c uφ + 2 ∂φ ur − 2 , r r r 1 −1 Pr (∂t uz + ur ∂r uz + uφ ∂φ uz + uz ∂z uz ) = −∂z p + ∆c uz + RΘ, r 1 ∂t Θ + ur ∂r Θ + uφ ∂φ Θ + uz ∂z Θ = ∆c Θ, r where 1 1 2 2 ∆c = ∂r (r∂r ) + 2 ∂φφ + ∂zz , r r and u = (ur , uφ , uz ) is referred to the basis {er , eφ , ez }. The domain becomes now Ω̄ = (ā, ā + Γ) × (0, 2π) × (0, 1).. (1.12) (1.13). (1.14). (1.15) (1.16).

(25) 1.1. THE MODEL. 15. Transformation required by the numerical method The numerical method, which will be explained in the next section, requires a new change in the variables as the integration of the equations needs to be done in the domain Ω = (−1, 1) × (0, 2π) × (−1, 1). The change is the following r0 =. 2(r − ā) − 1, Γ. φ0 = φ,. z 0 = 2z − 1.. (1.17). Equations (1.12)-(1.16) transform into Gur + A∂r ur + G∂φ uφ + 2∂z uz = 0,. (1.18). Pr −1 (∂t ur + Aur ∂r ur + Guφ ∂φ ur + 2uz ∂z ur − Gu2φ ) =. (1.19). = −A∂r p + ∆H ur − G2 ur − 2G2 ∂φ uφ , Pr −1 (∂t uφ + Aur ∂r uφ + Guφ ∂φ uφ + 2uz ∂z uφ + Gur uφ ) =. (1.20). = −G∂φ p + ∆H uφ + 2G2 ∂φ ur − G2 uφ , Pr −1 (∂t uz + Aur ∂r uz + Guφ ∂φ uz + 2uz ∂z uz ) = −2∂z p + ∆H uz + RΘ, (1.21) ∂t Θ + ur A∂r Θ + Guφ ∂φ Θ + 2uz ∂z Θ = ∆H Θ,. (1.22). where we define a new technical function G and the constant A by G (r) =. 2 , 2ā + Γ(1 + r). A=. 2 , Γ. (1.23). and the operator 2 2 2 ∆H = A2 ∂rr + GA∂r + G2 ∂φφ + 4∂zz .. 1.1.4. Boundary conditions. To complete the problem statement, conditions satisfied by the fluid on the top, the bottom, the outer lateral wall and the inner lateral wall of the domain have to be fixed. For convenience, we write the boundaries Γ = ∂Ω = Γ1 ∪ Γ2 ∪ Γ3 ∪ Γ4 in dimenssionless.

(26) 16. CHAPTER 1. CYLINDRICAL ANNULUS SETUP. form and in cylindrical coordinates, where Γ1 = {(r, φ, z) ∈ {−1} × [0, 2π] × [−1, 1]}, Γ2 = {(r, φ, z) ∈ {1} × [0, 2π] × [−1, 1]}, Γ3 = {(r, φ, z) ∈ [−1, 1] × [0, 2π] × {−1}}, Γ4 = {(r, φ, z) ∈ [−1, 1] × [0, 2π] × {1}}. In the lateral wall of the inner cylinder, Γ1 , the velocity is considered to be zero. In dust devils with a proper eye formed [49], radial and azimuthal velocities are very small inside the core and it is observed updraft (positive vertical velocity) in the region close (but out) of the inner core and downdraft inside the core (negative vertical velocity). An insulating wall is also considered, ur = uφ = uz = ∂r Θ = 0, on Γ1 .. (1.24). The outer cylinder is open. We stablish that the fields do not change radially at Γ2 , and we consider also an insulated boundary ∂r ur = ∂r uφ = ∂r uz = ∂r Θ = 0, on Γ2 .. (1.25). The fact that whirlwinds move as coherent structures suggests the consideration of a solid boundary in the case of a moving contact line, that corresponds to slip boundary conditions [5]. Therefore, at the bottom we consider ∂z ur = ∂z uφ = uz = 0, on Γ3 .. (1.26). A key point in the study is the presence of horizontal temperature gradients. A Gaussian profile is imposed at the bottom boundary 1 2 1 2 r+1 1 2 1 Θ(r) = 1 − δ e( β ) − e( β −( 2 ) β ) / e( β ) − 1 , on Γ3 ,. (1.27). where δ = ∆Th /∆Tv . If δ > 0 the center of the cell at the bottom is hotter than the outer part. Small values of β correspond to sharp profiles while large β correspond to widespread inhomogeneities along the bottom boundary..

(27) 1.2. AXISYMMETRIC STATIONARY APPROACH. 17. On the top surface, Γ4 , free slip boundary conditions are considered, and the temperature is fixed to T = T0 , that after rescaling becomes ∂z ur = ∂z uφ = uz = Θ = 0, on Γ4 .. (1.28). Regarding the boundary conditions for the pressure, and following Ref. [78], we impose the continuity equation at the bottom of the domain Γ3 , ∂r p = 0 is imposed in the wall of the inner cylinder Γ1 , and the normal component of momentum equation is imposed at the top of the domain Γ4 an in the wall of the outer cylinder Γ2 . The pressure always appears under a gradient, therefore, the matrix associated with the linear algebraic system is singular. In order to avoid this problem, the node (1,z4 ) is replaced by a Dirichlet condition for the pressure (i.e., p = 0). In this way, the pressure with the imposed conditions will be defined up to a constant.. 1.2. Axisymmetric stationary approach. In this section we look for axisymmetric stationary solutions of equations (1.18)-(1.22) and boundary conditions (1.24)-(1.28), and we study their linear stability analysis. The method used is described in detail in Ref. [43]. Here we show the most relevant aspects.. 1.2.1. Basic state. When a horizontal temperature gradient is produced at the bottom (δ 6= 0), generates a stationary convective motion, which can be named basic state. It is a time-independent solution to the stationary problem obtained from equations (1.9)-(1.11), i.e., a solution of ∇ · u = 0, (u · ∇) u = Pr (−∇p + ∆u + RΘe3 ) , u · ∇Θ = ∆Θ, with the boundary conditions (1.24)-(1.28).. (1.29) (1.30) (1.31).

(28) 18. CHAPTER 1. CYLINDRICAL ANNULUS SETUP. 1.2.2. Axisymmetry assumption. The basic state is considered to be axisymmetric and therefore depends only on r − z coordinates, i.e., all φ derivatives are zero. This is not a strong restriction in our interest on developing vertical vortices, as real atmospheric vortices (dust devils, tornadoes, hurricanes) present in many occasions an axisymmetric structure [23, 106, 112]. With this assumption, equations (1.18)-(1.22) change to Gur + A∂r ur + 2∂z uz = 0,. (1.32). Pr −1 (Aur ∂r ur + 2uz ∂z ur − Gu2φ ) = −A∂r p + ∆B ur − G2 ur , Pr −1 (Aur ∂r uφ + 2uz ∂z uφ + Gur uφ ) = ∆B uφ − G2 uφ , Pr −1 (Aur ∂r uz + 2uz ∂z uz ) = −2∂z p + ∆B uz + RΘ, Aur ∂r Θ + 2uz ∂z Θ = ∆B Θ,. (1.33) (1.34) (1.35) (1.36). where 2 2 ∆B = A2 ∂rr + GA∂r + 4∂zz ,. and the following boundary conditions z = −1. z=1. r = −1. r=1. ∂z ur = 0. ∂z ur = 0. ur = 0. ∂r ur = 0. ∂z uφ = 0. ∂z uφ = 0. uφ = 0. ∂r uφ = 0. uz = 0. uz = 0. uz = 0. ∂r uz = 0. Θ = (1.27). Θ=0. ∂r Θ = 0. ∂r Θ = 0. (1.37). The equations (1.32)-(1.36) and the boundary conditions (1.37) contain six external parameters (R, Γ, δ, β, ā, P r).. 1.2.3. Linear stability analysis. Once a basic state is known, it is interesting to perform its linear stability analysis. In considering the stability of the system, we essentially seek to determine the reaction of the system to small perturbances [15]. Specifically, if the system is perturbed, the system is stable with respect to the particular disturbance if the disturbance gradually.

(29) 1.2. AXISYMMETRIC STATIONARY APPROACH. 19. dies down. The system is unstable if the disturbance grows in amplitude in such a way that the system progressively departs from the initial state and never reverts it. In the space of the parameters it can be got a region where the system is stable and other where it is not. The locus which separates the two classes of states defines the states of marginal stability of the system and it is defined by an equation of the form λ(R, Γ, δ, β, ā, P r) = 0. States of marginal stability can be one of two kinds depending on the way in which the amplitude of a small disturbance can grow: if the amplitude of the perturbation grows in a constant way, the transition from stability to instability takes place via a state exhibiting a stationary pattern of motion (it is said in that case that a stationary bifurcation takes place). However, if the amplitude of the perturbation is periodic, the transition takes place via a state exhibiting oscillatory motion with a certain definite characteristic frequency (oscillatory bifurcation or Hopf bifurcation). The stability of this basic state is studied by perturbating it with a vector field depending on the r, φ and z coordinates, in a fully 3D analysis. Fixed (Γ, δ, β, ā, P r), the solution U (r, φ, z, t) = (u, Θ, p)(r, φ, z, t) of the problem at given R is expressed as U (r, φ, z, t) = U b (r, z) + Ũ (r, z)eikφ+λt ,. (1.38). where U b (r, z) is the base flow for the given (R, Γ, δ, β, ā, P r), and Ũ (r, z) = (ũ, Θ̃, p̃)(r, z) refers to the perturbation. We have considered Fourier mode expansions in the angular direction, because along it boundary conditions are periodic. Introducing (1.38) into equations (1.9)-(1.11), with the corresponding boundary conditions, and linearizing the resulting system, an eigenvalue problem in λ is obtained ∆k ũr − A∂r p̃ − G2 ũr − 2G2 ikũφ − P r−1 (Aubr ∂r ũr + Aũr ∂r ubr + Gikubφ ũr +. (1.39). 2ubz ∂z ũr + 2ũz ∂z ubr − 2Gubφ ũφ ) = P r−1 λũr , ∆k ũφ − Gik p̃ + 2G2 ikũr − G2 ũφ − P r−1 (Aubr ∂r ũφ + Aũr ∂r ubφ + Gikubφ ũφ + 2ubz ∂z ũφ. +. 2ũz ∂z ubφ. +. Gubr ũφ. +. Gubφ ũr ). = P r λũφ ,. ∆k ũz − 2∂z p̃ + RΘ̃ − P r−1 (Aubr ∂r ũz + Aũr ∂r ubz + Gikubφ ũz + 2ubz ∂z ũz + 2ũz ∂z ubz ). −1. = P r λũz ,. (1.40). −1. (1.41).

(30) 20. CHAPTER 1. CYLINDRICAL ANNULUS SETUP ∆k Θ̃ − Aũr ∂r Θb − Aubr ∂r Θ̃ − Gikubφ Θ̃ − 2ubz ∂z Θ̃ − 2ũz ∂z Θb = λΘ̃, Gũr + A∂r ũr + Gikũφ + 2∂z ũz = 0,. (1.42) (1.43). where 2 2 ∆k = A2 ∂rr + GA∂r − k 2 G2 + 4∂zz ,. and the following boundary conditions for the perturbations: ũr = ũφ = ũz = ∂r Θ̃ = 0, on r = −1,. (1.44). ∂r ũr = ∂r ũφ = ∂r ũz = ∂r Θ̃ = 0, on r = 1,. (1.45). ∂z ũr = ∂z ũφ = ũz = Θ̃ = 0, on z = 1,. (1.46). ∂z ũr = ∂z ũφ = ũz = Θ̃ = 0, on z = −1.. (1.47). The instability is achieved when the real part of the eigenvalue with maximum real part, λmax (R), changes from a negative value to a positive one as R increases, for a specific wavenumber k. The critical value of R for which λmax (R, k) = 0 is denoted by Rc and the critical wavenumber, minimum k for which the bifurcation occurs, by kc .. 1.2.4. Numerical implementation. The Chebyshev collocation method has been demonstrated to be efficient and useful for solving numerically thermoconvective problems [18, 42, 70]. This numerical method applied to the incompresible Navier-Stokes equations and the heat equation in a cylindrical configuration is described in detail and tested in Refs. [42, 78]. The nonlinearities appearing in equations (1.32)-(1.36) are treated with a Newton-like iterative method. Starting from an initial approximation, we solve at each iteration the problem coming from the linearization around the solution at the previous step. The convergence criterion taken to stop the iterative method is that the l2 norm of U = U i+1 − U i should. be less than a tolerance tol, which in our case is tol = 10−9 . This iterative method is. described in detail in Ref. [78]. For the basic state, at each step of Newton’s method, the fields are expanded in Chebyshev polynomials. x=. nX r −1 n z −1 X l=0 n=0. xln Tl (r)Tn (z),. (1.48).

(31) 1.3. RESULTS FOR THE AXISYMMETRIC STATIONARY APPROACH. 21. where Tn is the nth Chebyshev polynomial. The evaluations are done at the GaussLobatto collocation points [13]. A more detailed description of the Chebyshev polynomials and Gauss-Lobatto collocation points is included in Appendix A. The eigenvalue problem derived from the linear stability analysis is also discretized by expanding perturbations Ū (r, z) in a truncated series of orthonormal Chebyshev polynomials as done for the basic state. Equations for the perturbation (1.39)-(1.47) lead to a discrete eigenvalue problem AW = λBW solved (after a Cayley transformation [80]) with eigs, the MATLAB implementation of ARPACK. λ is the set of eigenvalues and the W are coefficients in the Chebyshev basis of the corresponding eigenfunctions. We consider nr = 33 and nz = 21 in our computations.. 1.3. Numerical results for the axisymmetric stationary approach. Previous work developed by the group reveal the important role of thermoconvection in the generation of vertical axisymmetric vortices and the influence of the thermal parameters on the structure and the intensity of the vortex developed [81, 82, 83, 84]. Under certains thermal conditions (on vertical and horizontal temperature gradients) and conditions on the annular geometry, a stable vortical structure can be generated from a convective instability. A non-rotating axisymmetric state destabilizes into a vortex after a stationary bifurcation of wavenumber k = 0. In this section we extend the results reported in Refs. [81, 82] and analyze the influence of the size of the radius of the inner cylinder on the structure and intensity of the vortex developed.. 1.3.1. Set of parameters. In our numerical experiments, we will consider that a + l = d, i.e., we fix ā + Γ to 1. Therefore, the domain is Ω̄ = [ā, 1] × [0, 2π] × [0, 1]. The Prandtl number P r is set to 0.7 (as in the air), β is set to 0.5, and ā can vary from 0 to 1. We consider 0.02 ≤ ā ≤ 0.75 for numerical restrictions. Γ changes its value with ā. Regarding δ, we.

(32) 22. CHAPTER 1. CYLINDRICAL ANNULUS SETUP. are interested in δ > 0, i.e., the bottom part of the inner cylinder is hotter than that for the outer cylinder. The range 0 < δ ≤ 30 is studied, and the parameter R will be the control parameter to detect bifurcations.. a. f. 1. z. z. 0.5. 0 0.15. b. 0.6 r. g z. c. 0.6 r. z. h z. d. 0.6 r. z. i. 0 0.15. e. 0.6 r. j. 1. z. 0.8 r. 1. 0.8 r. 1. 1. z 0.5. 0.5. 0 0.15. 1. 0.5. 0 0.6. 1. 0.8 r. 1. z. 0.5. 1. 0.5. 0 0.6. 1. 1. 0.8 r. 1. 0.5. 0 0.15. 1. 0.5. 0 0.6. 1. 1. 0.8 r. 1. 0.5. 0 0.15. 0.5. 0 0.6. 1. 1. z. 1. 0.6 r. 1. 0 0.6. Fig. 1.2: Basic states at δ = 4 for ā = 0.15 and ā = 0.6. The values of R are Rc = 463.2 and Rc = 2815.2, respectively. (a)-(f) Isotherms of Θ; (b)-(g) contour plot of the pressure p; (c)-(h) contour plot of the radial velocity component ur ; (d)-(i) contour plot of the vertical velocity component uz ; (e)-(j) velocity field. The contours correspond to equally spaced values within their ranges, for ā = 0.15 and ā = 0.6, respectively, of [-3:1] and [-3:1] for Θ, [-5.5:0.55] × 102 and [-35:4.04] × 102 for p, [-4:2.6] and [-10.4:4.16] for ur and [0:4.3] and [0:8.9] for uz . The pressure p is determined up to a constant..

(33) 1.3. RESULTS FOR THE AXISYMMETRIC STATIONARY APPROACH. 1.3.2. 23. Non-rotating basic states when ā is varied. Figure 1.2 shows two non-rotating basic states (uφ = 0) for δ = 4 and different values of the inner radius, ā = 0.15 (small inner cylinder) and ā = 0.6 (large inner cylinder), at the values of the Rayleigh number Rc = 463.2 and Rc = 2815.2, respectively. The non-rotating basic states are very similar for both, small and large values of the inner radius, as it can be observed in Figure 1.2. A layer of cold air is formed between two warmer layers, as shown in Figures 1.2 (a) and 1.2 (f). The pressure decreases with height and a slight pressure drop is found in the lower part of the cell close to the inner cylinder as appreciated in Figures 1.2 (b) and 1.2 (g). The radial component of the velocity, ur , is characterized by positive values in the upper part of the cell (outward flow in this region) and negative values in the lower part of the domain (flow towards the inner cylinder) as shown in Figures 1.2 (c) and 1.2 (h). Figures 1.2 (d) and 1.2 (i) show the profile of the vertical component uz . It is positive everywhere and it presents its highest values in the central part of the cylindrical annulus, close to the inner cylinder, which implies upward motion in this zone. Figures 1.2 (e) and 1.2 (j) show the combined effect of ur and uz .. 1.3.3. Stability of the non-rotating basic states. The linear stability of the non-rotating basic states has been studied numerically depending on parameters δ and ā, following the explanations in Section 1.2.3. This analysis reveals that for any ā ∈ [0.02, 0.75] and δ ≥ 0.4 the non-rotating basic state destabilizes through a stationary bifurcation with critical wavenumber kc = 0 and ũφ 6= 0 when R goes beyond a certain critical threshold. The new steady flow emerging from this convective instability is computed numerically with an analogous procedure to that explained before for the non-rotating basic state in Section 1.2.3. As it was shown in [82], where the problem is studied for a fixed ā = 0.06, this new basic state presents a non zero azimutal velocity component uφ 6= 0, that does not change its sign, as the main feature. The fluid inside the annulus moves in the azimuthal direction, rotating around the inner cylinder. A vortex is formed. Circulation could take either.

(34) 24. CHAPTER 1. CYLINDRICAL ANNULUS SETUP. sign (via a pitchfork bifurcation) and both uφ and −uφ are possible. Counter-clockwise vortices have been considered in our numerical experiments (uφ > 0).. 1.3.4. Vortices when ā is varied. In this section, we describe the structure of the vortex and its properties when the radius of the inner cylinder ā is varied. Figure 1.3 shows the profiles of temperature, pressure and velocity components corresponding to the vortex for δ = 4 and R = 10000 for ā = 0.15 (Figs. 1.3 (a) to 1.3 (f)) and ā = 0.6 (Figs. 1.3 (g) to 1.3 (l)). The temperature fields (Figs. 1.3 (a) and 1.3 (g)) are similar to that observed for the basic state shown in Figure 1.2: two warm zones, one localized at the bottom of the annulus close to the inner cylinder and another one extended on the top of the domain, with a cold zone between them. The pressure decreases with height and a pressure drop is found in the lower part of the cell, close to the inner cylinder, as shown in Figures 1.3 (b) and 1.3 (h). Figures 1.3 (c) and 1.3 (e) and 1.3 (i) and 1.3 (k) show the contours of the velocity components ur and uz for ā = 0.15 and ā = 0.6, respectively. The behavior observed is similar to that found for the basic state: inward flow near the bottom plate, upward motion closer to the inner cylinder and outward flow in a layer near the top of the structure formed. No significant differences are observed for the different values (small or large) of the radius ā of the inner cylinder. The meridional velocity (ur ,uz ) is depicted in Figures 1.3 (f) and 1.3 (l). The azimuthal component of the velocity uφ , does not change its sign and the fluid spins around the inner cylinder. It reaches its highest values in the lower part of the cell, close to the inner cylinder for small values of ā (small inner cylinder) as shown in Figure 1.3 (d) and extended along the cell for large values of ā (large inner cylinder) as shown in Figure 1.3 (j). No relation exists between the intensity of the vortex and the radius of the inner cylinder as it can be deduced from data in caption of Figure 1.3, where the magnitudes of every field are included. In Figure 1.4, it is shown the maximum absolute values of the three components of the velocity field for the vortex at δ = 4 and R = 10000 when ā is varied. As observed, no significant variations on the intensity of the vortex are found when the.

(35) 1.3. RESULTS FOR THE AXISYMMETRIC STATIONARY APPROACH. a. g. 1. z 0.5. 0 0.15. 1. z 0.5. 0.6. 0 0.6. 1. r. b. h. 1. c. 0.6 r. i. 0.6 r. j. 0.6 r. k. 0.6 r. 1. 0.8 r. 1. 0.8 r. 1. 1. 0 0.6. 1. l. 1. 1. z 0.5. z 0.5. 0 0.15. 0.8 r. z 0.5. z 0.5. f. 1. 1. 0 0.6. 1. 1. 0 0.15. 0.8 r. z 0.5. z 0.5. e. 1. 1. 0 0.6. 1. 1. 0 0.15. 0.8 r. z 0.5. z 0.5. d. 1. 1. 0 0.6. 1. 1. 0 0.15. 0.8 r. z 0.5. z 0.5. 0 0.15. 25. 0.6 r. 1. 0 0.6. Fig. 1.3: Vortices at δ = 4 and R = 10000 for ā = 0.15 and ā = 0.6. (a)-(g) Isotherms of Θ; (b)-(h) contour plot of the pressure p; (c)-(i) contour plot of the radial velocity component ur ; (d)-(j) contour plot of the azimuthal velocity component uz ; (e)-(k) contour plot of the vertical velocity component uz ; (f)-(l) velocity field. The contours correspond to equally spaced values within their ranges, for ā = 0.15 and ā = 0.6, respectively, of [-3:1] and [-3:1] for Θ, [-12.9:1.43] × 103 and [-11.9:1.42] × 103 for p, [-17.5:4.3] and [-17.2:5.3] for ur , [0:68.8] and [0:61.2] for uφ , and [-1.5:12.6] and [-0.93:14.4] for uz . The pressure p is determined up to a constant..

(36) 26. CHAPTER 1. CYLINDRICAL ANNULUS SETUP. 80 max |ur| max |u | Φ max |uz|. 70 60 50 40 30 20 10 0 0.11. 0.2. 0.3. 0.4. 0.5. 0.6. 0.7. ā. Fig. 1.4: Maximum absolute values of three components of the velocity field for the vortex at δ = 4 and R = 10000 as a function of ā.. radius of the inner cylinder ā is varied in [0.11, 0.6]. For ā > 0.6, the vortex has not gained enough intensity yet and the reason for this is that the values of Rc for the first bifurcation, after which the spin motion appears, increases substantially for ā > 0.6 (see Figure 1.8) so it is needed an increase of R to intensify the strength of the vortical structure. Every vortex developed, independently of ā, behaves in a similar way: once it is developed it can be intensified just by thermal mechanism. Figure 1.5 displays the radial profile of the velocity components at z-level z5 = 0.09 for uφ and uz and z5 = 0.09 and z10 = 0.39 for ur , for four different values of ā: 0.15, 0.3, 0.45 and 0.6. Figure 1.5 (a) shows radial inflow at lower levels (ur < 0) and radial outflow in the upper levels (ur > 0). No significant differences are found on these profiles when ā is varied. The profile of the vertical velocity component uz , displayed in Figure 1.5 (b), shows that the maximum of uz is reached near the inner cylinder and as r increases, uz rapidly falls. This behavior is found for every value of ā noting that as larger ā is (larger inner cylinder), closer to the inner cylinder are the highest values of uz . Regarding the azimuthal velocity component depicted in Figure 1.5 (c), it is observed that for small values of ā (small inner cylinder, e.g., ā = 0.15).

(37) 1.3. RESULTS FOR THE AXISYMMETRIC STATIONARY APPROACH. 27. a 5 ur (r,z10) 0 −5. ur (r,z5). −10 0. b. ā = ā = ā = ā =. 0.1. 0.15 0.30 0.45 0.6. 0.2. 0.3. 0.4. 0.5 r. 0.6. 0.7. 0.8. 1. 15 ā = ā = ā = ā =. 10 uz (r,z5). 0.9. 0.15 0.3 0.45 0.6. 5 0 0. 0.1. 0.2. 0.3. 0.4. 0.5 r. 0.6. 0.7. 0.8. 0.9. 1. c 60 40 uΦ (r,z5). ā = ā = ā = ā =. 20 0 0. 0.1. 0.2. 0.3. 0.4. 0.5 r. 0.6. 0.7. 0.8. 0.15 0.30 0.45 0.6. 0.9. 1. Fig. 1.5: (a) Radial profile of the velocity component ur at z-levels z5 = 0.09 and z10 = 0.39 for the vortex at different values of ā; (b) radial profile of the velocity component uz at z5 for the vortex at different values of ā; (c) radial profile of the velocity component uφ at z5 for the vortex at different values of ā. The parameters are δ = 4 and R = 10000. (r, z) in the domain [ā,1]×[0,1].. the profile appears similar to that which is a distinguishing feature of the tangential velocity in dust devils, the so-called Rankine combined vortex velocity distribution. It is also needed relative large values of δ to obtain such a profile. A true Rankine vortex has uniform angular velocity (uφ /r = constant) in the central core (region between the center and the point where uφ is maximum). Outside of this region, the uφ field is essentially inversely proportional to the radius, i.e., uφ r = constant..

(38) 28. CHAPTER 1. CYLINDRICAL ANNULUS SETUP. a uΦ(r,z1). 200. v(r)=c1/(r−0.15)+c2 v(r)=c(r−0.15). uΦ. 100. 0 0.2. 0.3. 0.4. 0.5. 0.6. 0.7. 0.8. 0.9. 1. 0.6. 0.7. 0.8. 0.9. 1. 0.6. 0.7. 0.8. 0.9. 1. r. b 1000 u (r,z ) / r Φ. 1. 500 0 0.2. c. 0.3. 0.4. 0.5 r. 2000. Axial Vorticity 1000. 0 0.2. 0.3. 0.4. 0.5 r. Fig. 1.6: (a) Radial profile of the azimuthal velocity component at z1 = 0.01; (b) radial profile of the angular velocity uφ /r at z1 = 0.01 ; (c) radial profile of the axial vorticity ξ at z1 = 0.01. The parameters are δ = 20, ā = 0.15 and R = 14000.. In Figure 1.6 (a), we display the radial uφ profile for R = 14000, ā = 0.15, and δ = 20 together with a true Rankine vortex. As appreciated, uφ = 0 at the inner cylinder wall, and the maximum values of uφ is reached not far from that wall. In this region, the dependence of uφ on r is almost linear. Outside of this zone, uφ is essentially inversely proportional to the radius. Plots of the angular velocity uφ /r (Fig. 1.6 (b)) and axial vorticity ξ = 1r ∂r (ruφ ) (Fig. 1.6 (c)) justify the analogy. In Figure 1.6 (b), it is observed an almost uniform behavior of the angular velocity in the core and in Figure 1.6 (c) the uniform behavior of the axial vorticity in the outer region..

(39) 1.3. RESULTS FOR THE AXISYMMETRIC STATIONARY APPROACH. 1.3.5. 29. Vorticity and angular momentum. In this section, we analyze the vorticity and the angular momentum of the vortices developed and the influence of the thermal parameters on their distribution. Figure 1.7 displays the axial vorticity ξ = 1r ∂r (ruφ ), angular momentum m = ruφ diffusive flux of m due to viscosity F = −νr2 ∇(m/r2 ) and the local source/sink of angular momentum due to viscosity −∇ · F in dimensionless form, for the vortex developed at R = 10000 and ā = 0.15 for δ = 0.5 (Fig. 1.7 (a) and (d)) and δ = 10 (Fig. 1.7 (e) and (h)), respectively. Figure 1.7 (a) shows the contour of the axial vorticity ξ for δ = 0.5. As it can be seen, it is generated at the bottom close to the inner cylinder, where radial warm inflow gets into the rising plume. The contour of the angular momentum m is displayed in Figure 1.7 (b). It decreases from the lower right corner towards the interior of the flow. a. c 1. b 1. 1. d1 0. 0 0. z 0.5. 0 0.15. e. z 0.5. 0.6 r. 0 0.15. 1. f. 1. z 0.5. z 0.5. 0.6 r. 0 0.15. 1. g. 1. z 0.5. z 0.5. 0.6 r. h. 1. 0. z 0.5. 0.6 r. 1. 0 0.15. 0.6 r. 1. 0 0.15. 0.6 r. 1. 0.6 r. 1. 1. z 0.5. 0 0. 0. 0 0.15. 0 0.15. 1. 0.6 r. 1. 0 0.15. Fig. 1.7: (a) Contour plot of ξ (normalized by the maximum value, ξmax = 188.31) for δ = 0.5; (b) contour plot of m (normalized by the maximum value, mmax = 24.67) for δ = 0.5; (c) contour of the local torque −∇ · F for δ = 0.5 (equally spaced values in [-301:28]). The contour line where ∇ · F = 0 is included; (d) diffusive flux F of m for δ = 0.5; (e) contour plot of ξ (normalized by the maximum value, ξmax = 1260) for δ = 10; (f) contour plot of m (normalized by the maximum value, mmax = 64.77) for δ = 10; (g) contour of the local torque −∇ · F for δ = 10 (equally spaced values in [-3985:185]). The contour line where ∇ · F = 0 is included; (h) diffusive flux F of m for δ = 10. The fields are represented in dimensionless form. The rest of parameters are ā = 0.15 and R = 10000..

(40) 30. CHAPTER 1. CYLINDRICAL ANNULUS SETUP The distribution of the local torque on the fluid (−∇ · F), shown in Figure 1.7. (c) for δ = 0.5, exhibits a pronounced sink of m at the bottom boundary closer to the inner cylinder. The viscous torque integrates different from zero with respect to volume, due to the open boundaries. |F | is seen to be largest at both sides of this sink as can be appreciated from Figure 1.7 (d). The effect of increasing the thermal parameter δ at fixed R = 10000 (increase the horizontal temperature gradient at the bottom boundary) has been studied. Figures 1.7 (e) and (h) show the case δ = 10. The region of largest vorticity is contracted towards the lower left corner (region at the bottom close to the inner cylinder) and its magnitude grows (Fig. 1.7 (e)). The angular momentum also grows and clearly diffuses to the left in the lower levels (Fig. 1.7 (f)). As it can be appreciated from Figure 1.7 (g) a slight source of m appears at both sides of the sink of m and the sink is displaced towards the inner cylinder (see Figure 1.7 (h)). There is no influence of the inner radius on the distribution of vorticity and angular momentum, finding the same behavior for different values of the inner radius as described above.. 1.3.6. Stability of the vortices. In this section, we analyze the influence of the radius of the inner cylinder ā on the stability of the vortex developed. Figure 1.8 shows the stability region of the vortex for δ = 4, when ā varies. The dashed curve represents the first bifurcation (the critical threshold Rc at which the non-rotating basic state becomes unstable with kc = 0 and ũφ 6= 0) when ā is varied in [0.02, 0.75]. The critical Rayleigh number Rc decreases for 0.02 ≤ ā ≤ 0.16 and it increases with ā after that. The solid curve indicates the secondary bifurcation (the critical threshold Rc at which the vortex becomes unstable) which is oscillatory with kc = 1 for any value of ā. As can be appreciated in Figure 1.8, the critical Rayleigh number Rc increases for 0.02 ≤ ā < 0.29. At ā = 0.29, the R threshold tends to grow so much that we have not found it, what suggests that there is an asymptote. Then Rc decreases for 0.29 < ā ≤ 0.34, increasing afterwards in the range 0.34 < ā ≤ 0.38,.

(41) 1.3. RESULTS FOR THE AXISYMMETRIC STATIONARY APPROACH. 31. 4. 8. x 10. 7. 1st bifurcation (kc = 0) 2nd bifurcation (k = 1) c. 6 5 R. c. 4 3 6000 4000 2000 0 0. 2 1 0 0. 0.2. 0.4. 0.05. 0.1. 0.6. 0.75. ā. Fig. 1.8: (a) Critical Rayleigh number Rc and critical wavenumber kc as a function of ā for δ = 4. Empty circles correspond to real eigenvalues while filled ones stand for complex eigenvalues.. and after that another asymptote is found. Rc decreases again for 0.38 < ā ≤ 0.44 to finally increases till ā = 0.62, where another asymptote is localized. Beyond this value of ā, the vortex is stable for any value of R. Note that for very small values of ā the axisymmetric vortex is stable for a very small region of values of R.. 1.3.7. Trajectories inside the vortex when ā is varied. The track of a particle in the vortex can be obtained by integrating the evolution of the element of fluid which follows the velocity field dr = ur (r, z), dt. dφ = uφ (r, z), dt. dz = uz (r, z). dt. (1.49). In Figure 1.9, it is compared the trajectory of a particle in the fluid for δ = 4 and R = 10000 at ā = 0.15 (small inner cylinder) and ā = 0.6 (large inner cylinder). The starting point is localized at the bottom of the cylindrical annulus (r = 0.95, z = 0.05, φ = 0) in Ω̄ = [ā, 1] × [0, 2π] × [0, 1]. We observe, for both cases, a spiral up motion of the particle goes up, moves towards the inner cylinder and rotates around it. The combination of these movements gives the spiral trajectorie shown in Figure.

(42) 32. CHAPTER 1. CYLINDRICAL ANNULUS SETUP. a. b t = 0.8. t = 0.8 t=0. t=0. Fig. 1.9: Track of a particle in the fluid for δ = 4 and R = 10000 at different ā. The starting point at (r = 0.95, z = 0.05, φ = 0) in the domain Ω̄ = [ā, 1] × [0, 2π] × [0, 1]; (a) ā = 0.15; (b) ā = 0.6.. 1.9 (a) at ā = 0.15 and in Figure 1.9 (b) at ā = 0.6. This behavior is observed for any value of the radius of the inner cylinder ā. As appreciated in the trajectory plots, the intensity of the spin motion is similar in both cases, small and large inner cylinder, as already pointed out in Section 1.3.4, which implies that intense vortices can be found independently of the size of the inner cylinder. 180 δ = 20. 160 140 δ = 10. 120 uΦ (r,z1). 100 80. δ=4 δ=3 δ=2 δ=1 δ = 0.5. 60 40 20 0 0. 0.1. 0.2. 0.3. 0.4. 0.5. 0.6. 0.7. 0.8. 0.9. 1. r. Fig. 1.10: Radial profile of the azimuthal velocity uφ at z1 = 0.01 for the vortex with ā = 0.15 and R = 10000 for different values of δ. Black dots indicate the location of the maximum value of uφ ..

(43) 1.3. RESULTS FOR THE AXISYMMETRIC STATIONARY APPROACH a. b. δ = 0.5 3000. δ=1 4000. Pressure term − Inertial term. 33. Pressure term − Inertial term. 3000. 2000. 2000 1000 1000 0. 0 −1000. −1000 0.2. 0.4. 0.6 r. 0.8. 1. δ=4. c. 0.4. 0.6 r. 0.8. 1. δ = 10. 4. d. x 10 5. Pressure term − Inertial term. 15000. 0.2. Pressure term − Inertial term. 4. 10000. 3. 5000. 2 1. 0 0.2. 0.4. 0.6 r. 0.8. 1. 0 0.2. 0.4. 0.6. 0.8. 1. r. Fig. 1.11: Radial profile at z1 = 0.01 of the radial pressure gradient and radial inertial force for ā = 0.15 and R = 10000, for different values of δ. (a) δ = 0.5; (b) δ = 1; (c) δ = 4; (d) δ = 10.. 1.3.8. Contraction and stabilization of the RMAV. In Figure 1.10, it is shown the radial profile of the velocity component uφ at z-level z1 = 0.01 (where the maximum value of uφ is reached) for the vortex at ā = 0.15 and R = 10000 for different values of δ. As observed from Figure 1.10, uφ grows (the spin motion is intensified) as δ is increased (the horizontal temperature gradient is increased for a fixed R). The radius of maximum azimuthal velocity (RMAV) contracts when δ varies from δ = 0.5 to approximately δ = 4 and it stabilizes afterwards even though δ keeps increasing, i.e., the vortex keeps intensifying. Looking at the equation for the radial component of the velocity field ur , three terms appear: inertial forces, presure gradient forces and viscous forces. Viscous forces compensates inertial and pressure gradient forces. There are two main processes that govern the contraction of the radius of maximum azimuthal velocity RMAV: the inward advection by the radial inflow (generated by the pressure drop on the center of the cell.

(44) 34. CHAPTER 1. CYLINDRICAL ANNULUS SETUP. due to the horizontal temperature gradient) that tends to shift the RMAV inward, and the inertial force. The first appears to dominate for smaller values of δ (see Figures 1.11 (a) and (c)) while the second becomes significant near the inner core region after the vortex intensity becomes sufficiently strong, when δ is sufficiently large (see Figure 1.11 (d)). The inertial force is the main factor responsible for preventing the collapse of the RMAV to the center. As soon as the inertial force can balance the inward advection by the radial inflow (by balancing the radial pressure gradient), the contraction stops and the RMAV maintains a stationary value after that although the vortex keeps intensifying. In Figure 1.11, the profiles of the radial pressure gradient and the inertial force for different values of δ are shown. From δ = 4, the inertial force balances the radial pressure gradient and the contraction of the RMAV is broken. In Figure 1.10, it is appreciated how for δ > 4 the RMAV is stabilized.. 1.3.9. Discussion. The vortical structure we found numerically presents qualitative similarities with dust devils [112]: a low-pressure region in the center which coincides with a warm core, a radial inflow at lower levels and radial outflow in the upper levels, a vertical velocity which reaches the highest values on either sides of the center and then falls of rather rapidly as the radius is increased and a tangential velocity that presents a distinguishing feature of a Rankine vortex. Other more complex meteorological phenomena such as tropical cyclones or hurricanes also presented these structural characteristics. It is known that the center (eye) of a tropical cyclone is the area of lowest atmospheric pressure in the region, which corresponds to a warm core [24]. Regarding motion in cyclones, it is observed inward flow next to the surface, strong upward motion in the eyewall and outflow in a layer near the top of the storm [12]. A counter-clockwise motion (clockwise in the southern hemisphere) is observed around the center of the storm, stronger just above the surface in a ring around the center and weaker as we go up from the surface [24]. The radius of the maximum tangential wind (RMW) associated with the hurricane primary circulation undergoes continuous contraction during the hurricane development. This contraction.

(45) 1.4. 3D TEMPORAL APPROACH. 35. appears to slow down abruptly at the middle of the hurricane intensification, and the RMW becomes nearly stationary subsequently, despite the rapidly strengthening rotational flows [53]. We have also observed a contraction in the radius of maximum azimuthal velocity when the vortex intensifies (which agrees with the diffusion of angular momentum), with a later stop contraction even though the azimuthal velocity keeps increasing as the vortex continues intensifying. The balance of the inward advection by the inertial forces is the mechanism behind the break of the contraction. In Ref. [53] the role of frictional dissipation in preventing the collapse of the hurricane radius of maximum wind RMW is demonstrated.. 1.4. 3D temporal approach. Interesting results have been obtained with the axisymmetric stationary approach and the linear stability analysis, but a key point of interest is in the 3D temporal behavior of the stationary axisymmetric vortex when the Rayleigh number keeps increasing. For a fully 3D temporal numerical simulation, the governing equations (1.18)-(1.22) have been solved using the second-order time-splitting method described and tested in the cylindrical configuration by Mercader et al. [71]. This time stepping method was proposed by Hugues and Randriamampianina [45]. The fractional steps consist of a predictor for the pressure, directly derived from the Navier-Stokes equations with the Neumman boundary conditions [52]; a predictor for an intermediate velocity field from the momentum equation, which takes into account the predicted pressure obtained from the previous time level, and finally a projection step with an explicit evaluation of the final divergence-free velocity field.. 1.4.1. First assumptions. The introduction of new variables combining ur and uφ , the radial and azimuthal components of the velocity field, as we will detail in the next section, requires that they satisfy boundary conditions of the same type, i.e.,.

(46) 36. CHAPTER 1. CYLINDRICAL ANNULUS SETUP. aur + b∂n ur = hur , auφ + b∂n uφ = huφ ,. (1.50). where a, b, hur and huφ are constants. The boundary conditions considered in our problem, shown in (1.37), are compatible with restriction (1.50).. 1.4.2. Temporal discretization and projection scheme. For the time discretization a second order stiffly-stable scheme is used [52, 71]: ∇ · un+1 = 0,. (1.51). 3un+1 − 4un + un−1 = − 2N L(un ) + N L(un−1 ) 2∆t + P r(−∇p. n+1. 2. n+1. +∇ u. (1.52) n+1. + RaΘ. ez ),. 3Θn+1 − 4Θn + Θn−1 = − 2N L(un , Θn ) + N L(un−1 , Θn−1 ) + ∇2 Θn+1 , 2∆t. (1.53). where 1 N L(u, Θ) = ur ∂r Θ + uφ ∂φ Θ + uz ∂z Θ, r. . 1 u ∂ u r φ φ r. 1 2 u r φ. ur ∂r ur + + uz ∂z ur −   N L(u) =  ur ∂r uφ + 1r uφ ∂φ uφ + uz ∂z uφ + 1r ur uφ  ur ∂r uz + 1r uφ ∂φ uz + uz ∂z uz.     . and 1 2 1 2 + ∂zz . ∇2 = ∂r (r∂r ) + 2 ∂φφ r r The fractional steps are the following: 1. Θn+1 is obtained from the Helmholtz-type problem 3 4Θn − Θn−1 2 ∇ − Θn+1 = 2N L(un , Θn ) − N L(un−1 , Θn−1 ) − . (1.54) 2∆t 2∆t.

(47) 1.4. 3D TEMPORAL APPROACH. 37. 2. Applying ∇ to equation (1.52), and using (1.51), a preliminary pressure field p̄ is obtained from the Navier-Stokes equations ∇2 p̄n+1 =. 1 4un − un−1 ∇(−2N L(un ) + N L(un−1 ) + P rRaΘn+1 ez + ), (1.55) Pr 2∆t. with a boundary condition for the pressure obtained from equation (1.52) and the corresponding boundary conditions for the velocity field. 3. A predictor velocity field u∗ = (u∗r , u∗φ , u∗z ) is calculated from the Navier-Stokes equation by including the predictor pressure p̄ with the actual boundary conditions ∇2 −. 3 2P r∆t. . 1 u = Pr ∗. . n. n−1. 2N L(u ) − N L(u. + ∇p̄. n+1. n+1. − RaΘ. 4un + un−1 )+ 2∆t. (1.56). ez .. As the radial and azimuthal velocity components (ur , uφ ) are coupled in the linear viscous term, 2 (∇ u )r = − 2 ∂φ u∗φ − r 2 (∇2 u∗ )φ = ∇2 u∗φ + 2 ∂φ u∗r − r 2 ∗. ∇2 u∗r. u∗r , r2 u∗φ . r2. (1.57) (1.58). We introduce new complex functions. Following Ref. [96], we define u∗+ = u∗r + iu∗φ ,. u∗− = u∗r − iu∗φ ,. (1.59). u∗+ − u∗− . 2i. (1.60). and therefore u∗r =. u∗+ + u∗− , 2. u∗φ =. Substituting (1.60) in (1.57)-(1.58), we obtain. (∇2 u∗ )r =. ∇2 u∗+ ∇2 u∗− u∗ u∗ 1 1 + − 2 ∂φ u∗+ + 2 ∂φ u∗− − +2 − −2 , 2 2 ir ir 2r 2r. (∇2 u∗ )φ =. ∇2 u∗+ ∇2 u∗− u∗ u∗ 1 1 − + 2 ∂φ u∗+ + 2 ∂φ u∗− − +2 + −2 . 2i 2i r r 2ir 2ir.

(48) 38. CHAPTER 1. CYLINDRICAL ANNULUS SETUP It is straightforward that 2i ∂φ u∗+ − r2 2i (∇2 u∗ )r − i(∇2 u∗ )φ = ∇2 u∗− − 2 ∂φ u∗− − r (∇2 u∗ )r + i(∇2 u∗ )φ = ∇2 u∗+ +. 1 ∗ u , r2 + 1 ∗ u . r2 −. Under the new unknowns, the system (1.56) gives rise to decoupled equations for (u∗+ , u∗− , u∗z ). ∇2 u∗+ +. 1 3 2i u∗ = (RHS)r + i(RHS)φ , ∂φ u∗+ − 2 u∗+ − 2 r r 2P r∆t +. ∇2 u∗− −. 2i 1 ∗ 3 ∗ u∗− = (RHS)r − i(RHS)φ , ∂ u − u − φ − − 2 2 r r 2P r∆t 3 2 ∇ − u∗z = (RHS)z , 2P r∆t. (1.61) (1.62) (1.63). where RHS is the right hand side of equation (1.56). Due to the fact that equations (1.61) and (1.62) are conjugated, and so are u∗+ and u∗− , only system (1.61)-(1.63) or (1.62)-(1.63) needs to be solved to obtain the predictor for the velocity fields (u∗r , u∗φ , u∗z ). 4. Finally, in the correction step, the system 3 un+1 − u∗ = −∇ pn+1 − p̄n+1 , 2P r∆t. (1.64). ∇ · un+1 = 0,. (1.65). is solved with (1.65) satisfied in the interior as well as in the boundary, with the correct boundary condition satisfying the restriction (1.50). Defining Φ = 2P r∆ n+1 (p 3. − p̄n+1 ), a Poisson equation of variable Φ is found, with Neumann. boundary condition. ∂Φ ∂n. = 0. Finally, the pressure field pn+1 , and the velocity. fields un+1 , are calculated from the definition of Φ and equation (1.65): pn+1 = p̄n+1 +. 3Φ , 2∆t. un+1 = u∗ − ∇Φ.. (1.66) (1.67).

(49) 1.4. 3D TEMPORAL APPROACH. 1.4.3. 39. Spatial discretization. A pseudo-spectral method is used for the spatial discretization, with a Fourier expansion in the azimuthal coordinate φ and Chebyshev collocation in r and z. Each field is then expanded as follows nφ /2−1. x(r, φ, z) =. X. Fk (r, z)eikφ ,. (1.68). k=−nφ /2. where F0 (r, z) and F−nφ /2 (r, z) are real and F−k (r, z) = F̄k (r, z) for k = 1 : nφ /2 − 1, and where F̄k denotes the complex conjugare of Fk . The coefficients Fk are expanded in Chebyshev polynomials. Fk (r, z) =. nz nr X X. fln Tl (r)Tn (z),. (1.69). l=0 n=0. where Tn is the nth Chebyshev polynomial. The evaluations are done at the GaussLobatto collocation points (see Appendix A). In our computations we use nr = 33, nz = 21 and nφ = 64, solving the problem in 47872 nodes. A 4GHz Intel Core i7 processor, with a RAM of 32 Gb 1867MHz has been used. Each iteration has a computational cost of 1.518 s for nφ = 64 and 0.8247 s for nφ = 32. Computations with nφ = 32 give reliable results but nφ = 64 has been used for a better graphical resolution.. 1.4.4. Final equations and boundary conditions. After the change of variables (1.17), equations become: Equation for the temperature Θ 3 4Θn − Θn−1 2 n+1 n n n n n n ∇ − Θ =2Aur ∂r Θ + 2Guφ ∂φ Θ + 4uz ∂z Θ − 2∆t 2∆t n−1 ∂r Θn−1 + Gun−1 + 2uzn−1 ∂z Θn−1 . − Aun−1 r φ ∂φ Θ Note that. (1.70).

(50) 40. CHAPTER 1. CYLINDRICAL ANNULUS SETUP. nφ /2−1. Θ(r, φ, z) =. X. Fk (r, z)eikφ. k=−nφ /2. implies ∂ 2Θ (r, φ, z) = ∂φ2. nφ /2−1. X k=−nφ /2. −k 2 Fk (r, z)eikφ .. Therefore, 3 2 2 2 ∇rz − G k − Fk (r, z) = (RHST )k (r, z), 2∆t. k = 0, 1, ...,. nφ −1, −nφ /2, (1.71) 2. 2 2 and (RHST )k is the corresponding k-Fourier coef+ GA∂r + 4∂zz where ∇2rz = A2 ∂rr. ficient of the right hand side of equation (1.70), which is known as it involves fields at previous time steps. Boundary conditions for temperature in terms of the Fourier coefficients are summarized in Figure 1.12. Fk = 0. Θ=0. ∂r Θ = 0. ∂r Θ = 0. Fourier coefficients. Θ = Gaussian profile. ∂r Fk = 0. ∂ r Fk = 0. {. F0 = Gaussian profile Fk = 0, if k = 0. Fig. 1.12: Boundary conditions for the temperature Θ.. Equation for the preliminary pressure p̄ Equation (1.55) gives ∇2rz − G2 k 2 Fk (r, z) = (RHSP )k (r, z). k = 0, 1, ...,. nφ − 1, −nφ /2, 2. (1.72). where (RHSP )k is the corresponding k-Fourier coefficient of the right hand side of the equation (1.55)..