Stefan G. Llewellyn Smith

We investigate the competition between horizontal convection (HC) and Rayleigh–Bénard convection (RBC) in a fluid layer subject to a uniform destabilising buoyancy flux at the bottom and a horizontally varying buoyancy distribution at the top. The RBC forcing imposes negative horizontal mean vertical buoyancy gradients at the top and bottom of the fluid layer. But if the HC forcing is sufficiently strong then the volume-averaged vertical buoyancy gradient, left angle bracket b Subscript z Baseline right angle bracket <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content"> <mml:mo stretchy="false" fence="false">⟨</mml:mo> <mml:msub> <mml:mi>b</mml:mi> <mml:mi>z</mml:mi> </mml:msub> <mml:mo stretchy="false" fence="false">⟩</mml:mo> </mml:math> $\langle b_z \rangle$ , is positive i.e. opposite in sign to destabilising RBC buoyancy gradients at the boundaries. If left angle bracket b Subscript z Baseline right angle bracket greater than 0 <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content"> <mml:mo stretchy="false" fence="false">⟨</mml:mo> <mml:msub> <mml:mi>b</mml:mi> <mml:mi>z</mml:mi> </mml:msub> <mml:mo stretchy="false" fence="false">⟩</mml:mo> <mml:mo>&gt;</mml:mo> <mml:mn>0</mml:mn> </mml:math> $\langle b_z \rangle \gt 0$ we say that the layer has been ‘restratified’. Using scaling analysis based on power integrals together with two-dimensional direct numerical simulations at Rayleigh numbers up to 10 Superscript 10 <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content"> <mml:msup> <mml:mn>10</mml:mn> <mml:mrow> <mml:mn>10</mml:mn> </mml:mrow> </mml:msup> </mml:math> $10^{10}$ , we identify two cases: a neutral stratification state, in which HC first offsets RBC so that left angle bracket b Subscript z Baseline right angle bracket equals 0 <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content"> <mml:mo stretchy="false" fence="false">⟨</mml:mo> <mml:msub> <mml:mi>b</mml:mi> <mml:mi>z</mml:mi> </mml:msub> <mml:mo stretchy="false" fence="false">⟩</mml:mo> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:math> $\langle b_z \rangle = 0$ , and a strong stratification regime, in which HC dominates and left angle bracket b Subscript z Baseline right angle bracket <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content"> <mml:mo stretchy="false" fence="false">⟨</mml:mo> <mml:msub> <mml:mi>b</mml:mi> <mml:mi>z</mml:mi> </mml:msub> <mml:mo stretchy="false" fence="false">⟩</mml:mo> </mml:math> $\langle b_z \rangle$ is opposite in sign, and greater in magnitude, than the prescribed destabilising vertical buoyancy gradient at the layer boundaries. For the range of parameters explored in this study, we derive scaling laws for the onset of these regimes in terms of the horizontal and vertical flux Rayleigh numbers, upper R a Subscript upper H <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content"> <mml:mrow> <mml:mi mathvariant="italic" class="MJX-tex-mathit">R</mml:mi> <mml:msub> <mml:mi mathvariant="italic" class="MJX-tex-mathit">a</mml:mi> <mml:mi mathvariant="italic" class="MJX-tex-mathit">H</mml:mi> </mml:msub> </mml:mrow> </mml:math> $\mathit{Ra_H}$ and upper R a Subscript upper V <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core

We obtain the viscous and diffusive fundamental solution for monochromatic internal waves in a uniformly stratified medium and for anisotropic Brinkman flow. These solutions take the form of single integrals with logarithmic singularities, and can be computed numerically in an efficient manner for possible use in boundary integral methods. Far-field asymptotic results are obtained, giving solutions valid far from and inside a ``beam'' corresponding to the internal wave angle in the internal wave case, consistent with Thomas &amp; Stevenson (1972). For Prandtl numbers $\text{Pr} \gtrsim O(1)$, the wave field is given by a superposition of wave- and Stokeslet-like terms. Unlike previous studies, a uniform asymptotic expansion of the wave-field for $\text{Pr} \gtrsim O(1)$ can be computed rigorously. Density diffusion attenuates the wave amplitude as to $(1+\text{Pr}^{-1})^{-2/3}$ and broadens the beam width according to $(1+\text{Pr}^{-1})^{1/3}$. Evanescent waves in a stratified medium and anisotropic Brinkman flows have similar behaviour. Anisotropic Brinkman flow is purely real, dominated by a single circulation cell. As anisotropy increases, the flow becomes increasingly confined to the direction with least resistance. The stratified evanescent wave field has near-vertical cells in its real part, and a dominant single circulation cell in its imaginary part.

We obtain the viscous and diffusive fundamental solution for monochromatic internal waves in a uniformly stratified medium and for anisotropic Brinkman flow. These solutions take the form of single integrals with logarithmic singularities, and can be computed numerically in an efficient manner for possible use in boundary integral methods. Far-field asymptotic results are obtained, giving solutions valid far from and inside a ``beam'' corresponding to the internal wave angle in the internal wave case, consistent with Thomas & Stevenson (1972). For Prandtl numbers $\text{Pr} \gtrsim O(1)$, the wave field is given by a superposition of wave- and Stokeslet-like terms. Unlike previous studies, a uniform asymptotic expansion of the wave-field for $\text{Pr} \gtrsim O(1)$ can be computed rigorously. Density diffusion attenuates the wave amplitude as to $(1+\text{Pr}^{-1})^{-2/3}$ and broadens the beam width according to $(1+\text{Pr}^{-1})^{1/3}$. Evanescent waves in a stratified medium and anisotropic Brinkman flows have similar behaviour. Anisotropic Brinkman flow is purely real, dominated by a single circulation cell. As anisotropy increases, the flow becomes increasingly confined to the direction with least resistance. The stratified evanescent wave field has near-vertical cells in its real part, and a dominant single circulation cell in its imaginary part.

Equations of motion for a body moving through an ideal fluid when the flow is irrotational and incompressible are obtained taking account of embedded dipoles on the boundary and the Kutta – Chaplygin condition. We develop the embedded dipole model from the complex potential of a dipole on the boundary of a body, oriented so as to preserve no-penetration through the body, using a conformal mapping approach. The resulting hydrodynamic force and moment on the body depend on the dipoles’ strength and position along the body. Using the flat plate as a model geometry, we examine the evolution of the resulting system under the conditions of fixed and time-varying circulation with and without embedded dipoles. We assume two embedded dipoles symmetrically positioned about the center point of the plate, finding that the presence of the dipoles reduces the fluctuations of the angle of attack of the plate. We explore conserved quantities for the system and perform a linear stability analysis, which leads to a constraint on the dipole strength for stability of a plate moving at zero angle of attack with either circulation equal to zero or the Kutta – Chaplygin condition applied.

We investigate the competition between horizontal convection (HC) and Rayleigh-Bénard convection (RBC) in a fluid layer subject to a uniform destabilizing buoyancy flux at the bottom and a horizontally varying buoyancy distribution at the top. The RBC forcing imposes negative horizontal mean vertical buoyancy gradients at the top and bottom of the fluid layer. But if the HC forcing is sufficiently strong then the volume averaged vertical buoyancy gradient, $\langle b_z \rangle$, is positive i.e.~opposite in sign to destabilizing RBC buoyancy gradients at the boundaries. If $\langle b_z \rangle&gt;0$ we say that the layer has been ''restratified''. Using scaling analysis based on power integrals together with two-dimensional direct numerical simulations at Rayleigh numbers up to $10^{10}$, we identify two cases: a neutral stratification state, in which HC first offsets RBC so that $\langle b_z \rangle = 0$, and a strong stratification regime, in which HC dominates and $\langle b_z \rangle$ is opposite in sign, and greater in magnitude, than the prescribed destabilizing vertical buoyancy gradient at the layer boundaries. For the range of parameters explored in this study, we derive scaling laws for the onset of these regimes in terms of the horizontal and vertical flux Rayleigh numbers, $\RaH$ and $\RaV$, finding $\RaHN \sim \RaV^{4/5}$ for the neutral state and $\RaHstrg \sim \RaV$ for the onset of strong stratification. The results highlight the controlling role of the top boundary layer in setting the mean stratification and clarify the conditions under which HC suppresses RBC. These findings are relevant to geophysical environments such as subglacial lakes, and the oceans of Snowball Earth and icy moons, where bottom heating and horizontal buoyancy variations jointly shape ocean stratification.

We investigate the competition between horizontal convection (HC) and Rayleigh-Bénard convection (RBC) in a fluid layer subject to a uniform destabilizing buoyancy flux at the bottom and a horizontally varying buoyancy distribution at the top. The RBC forcing imposes negative horizontal mean vertical buoyancy gradients at the top and bottom of the fluid layer. But if the HC forcing is sufficiently strong then the volume averaged vertical buoyancy gradient, $\langle b_z \rangle$, is positive i.e.~opposite in sign to destabilizing RBC buoyancy gradients at the boundaries. If $\langle b_z \rangle>0$ we say that the layer has been ''restratified''. Using scaling analysis based on power integrals together with two-dimensional direct numerical simulations at Rayleigh numbers up to $10^{10}$, we identify two cases: a neutral stratification state, in which HC first offsets RBC so that $\langle b_z \rangle = 0$, and a strong stratification regime, in which HC dominates and $\langle b_z \rangle$ is opposite in sign, and greater in magnitude, than the prescribed destabilizing vertical buoyancy gradient at the layer boundaries. For the range of parameters explored in this study, we derive scaling laws for the onset of these regimes in terms of the horizontal and vertical flux Rayleigh numbers, $\RaH$ and $\RaV$, finding $\RaHN \sim \RaV^{4/5}$ for the neutral state and $\RaHstrg \sim \RaV$ for the onset of strong stratification. The results highlight the controlling role of the top boundary layer in setting the mean stratification and clarify the conditions under which HC suppresses RBC. These findings are relevant to geophysical environments such as subglacial lakes, and the oceans of Snowball Earth and icy moons, where bottom heating and horizontal buoyancy variations jointly shape ocean stratification.

Quasi-geostrophic flow is an asymptotic theory for flows in rotating systems that are in geostrophic balance to leading order. It is characterized by the conservation of (quasi-geostrophic) potential vorticity and weak vertical flows. Surface quasigeostrophy (SQG) is the special case when the flow is driven by temperature anomalies at a horizontal boundary. The next-order correction to QG, QG+, takes into account ageostrophic effects. We investigate point vortx dynamics in SQG+, building on the work of Weiss. The conservation laws for SQG point vortices that parallel the 2D Euler case no longer exist when ageostrophic effects are included. The trajectories of point vortices are obtained explicitly for the general two-vortex case in SQG and SQG+. For the three-vortex case, exact solutions are found for rigidly rotating and stationary equilibria consisting of regular polygons and collinear configurations. As in the 2D case, only certain collinear vortex configurations are rigid equilibria. Trajectories of passive tracers advected by point vortex systems are studied numerically, in particular their vertical excursions, which are non-zero because of ageostrophic effects. Surface trajectories can manifest local divergence even though the underlying fluid equations are incompressible.

We investigate the convective stability of a thin, infinite fluid layer with a rectangular cross-section, subject to imposed heat fluxes at the top and bottom and fixed temperature along the vertical sides. The instability threshold depends on the Prandtl number as well as the normalized flux difference ( $f$ ) and decreases with the aspect ratio ( $\epsilon$ ), following a $\epsilon f^{-1}$ power law. Using a three-dimensional (3-D) initial value and two-dimensional eigenvalue calculations, we identify a dominant 3-D mode characterized by two transverse standing waves attached to the domain edges. We characterize the dominant mode’s frequency and transverse wavenumber as functions of the Rayleigh number and aspect ratio. An analytical asymptotic solution for the base state in the bulk is obtained, valid over most of the domain and increasingly accurate for lower aspect ratios. A local stability analysis, based on the analytical base state, reveals oscillatory transverse instabilities consistent with the global instability characteristics. The source term for this most unstable mode appears to be interactions between vertical shear and horizontal temperature gradients.

The 2022 GFD Program theme was Data-Driven GFD with Professors Peter Schmid of King Abdullah University of Science and Technology (KAUST) and Laure Zanna of New York University serving as principal lecturers. Together they introduced the masked audience in the re-opened cottage and on the better-ventilated porch to a fascinating mixture of data-driven methods and their potential application to fluid mechanics in general and GFD in particular. The first ten chapters of this volume document these lectures, each prepared by teams of the summer’s GFD fellows.

Quasi-geostrophic flow is an asymptotic theory for flows in rotating systems that are in geostrophic balance to leading order. It is characterized by the conservation of (quasi-geostrophic) potential vorticity and weak vertical flows. Surface quasigeostrophy (SQG) is the special case when the flow is driven by temperature anomalies at a horizontal boundary. The next-order correction to QG, QG+, takes into account ageostrophic effects. We investigate point vortx dynamics in SQG+, building on the work of Weiss. The conservation laws for SQG point vortices that parallel the 2D Euler case no longer exist when ageostrophic effects are included. The trajectories of point vortices are obtained explicitly for the general two-vortex case in SQG and SQG+. For the three-vortex case, exact solutions are found for rigidly rotating and stationary equilibria consisting of regular polygons and collinear configurations. As in the 2D case, only certain collinear vortex configurations are rigid equilibria. Trajectories of passive tracers advected by point vortex systems are studied numerically, in particular their vertical excursions, which are non-zero because of ageostrophic effects. Surface trajectories can manifest local divergence even though the underlying fluid equations are incompressible.

A new transform-based approach is presented that can be used to solve mixed boundary value problems for Laplace’s equation in non-convex and other planar domains, specifically the so-called Lipschitz domains. This work complements Crowdy (2015, CMFT, 15, 655–687), where new transform-based techniques were developed for boundary value problems for Laplace’s equation in circular domains. The key ingredient of the present method is the exploitation of the properties of the Szegö kernel and its connection with the Cauchy kernel to obtain transform pairs for analytic functions in such domains. Several examples are solved in detail and are numerically implemented to illustrate the application of the new transform pairs.

Abstract Bottom Un-heading Devices (BUD’s) are often installed on coke drums to facilitate remote opening and closing of the bottom of the drum pair during petroleum coke production. The BUD removes the need for personnel to manually open the drum, which is a high-risk activity and has led to many process safety incidents across the refining industry, and in some cases may offer shortened time to close the drum, resulting in increased production time. These valves are bolted to the bottom head flange and typically only removed during scheduled maintenance intervals, often exceeding five years. During service, the bolted joint must maintain integrity while being subjected to repeated thermal transients during the drum cycle. Correct design and assembly of these critical joints is essential to avoid safety risks and expensive downtime resulting from operational leaks. This paper will show how the use of load indicating studs can achieve reliable assembly, optimize scheduled re-tightening activities and provide effective ongoing joint management. Three case studies will be presented detailing how to correctly assemble and manage a typical BUD joint throughout its life cycle involving the use of load indicating studs. In each case, the joints have maintained integrity during service and allowed the plant engineers to gather reliable data to ensure effective future joint management. Load indicating studs alone cannot guarantee leak free operation and should be used in conjunction with a properly designed joint. BUD anatomy will be explored with insights into the fundamentals of reliable joint from flange geometry, gasket behavior and a review of common assembly techniques.

Abstract The free fall of an ellipse in an infinite linearly-stratified fluid is investigated using a linear two-dimensional, Boussinesq, diffusionless, inviscid model. The oscillations of the ellipse decay because of radiation damping, but unlike the case of a circular cylinder, the ellipse can also rotate and move horizontally. The resulting equations are solved analytically for some simple cases for which there is little or no rotation. Motions with rotation are studied numerically using a spectral method to solve for the wave field in the fluid.

A new transform-based approach is presented that can be used to solve mixed boundary value problems for Laplace's equation in non-convex and other planar domains, specifically the so-called Lipschitz domains. This work complements Crowdy (2015, CMFT, 15, 655--687), where new transform-based techniques were developed for boundary value problems for Laplace's equation in circular domains. The key ingredient of the present method is the exploitation of the properties of the Szegő kernel and its connection with the Cauchy kernel to obtain transform pairs for analytic functions in such domains. Several examples are solved in detail and are numerically implemented to illustrate the application of the new transform pairs.

GRAMS (Gamma-Ray and AntiMatter Survey) is a next-generation balloon/satellite experiment utilizing a LArTPC (Liquid Argon Time Projection Chamber), to simultaneously target astrophysical observations of cosmic MeV gamma-rays and conduct an indirect dark matter search using antimatter. While LArTPCs are widely used in particle physics experiments, they have never been operated at balloon altitudes. An engineering balloon flight with a small-scale LArTPC (eGRAMS) was conducted on July 27th, 2023, to establish a system for safely operating a LArTPC at balloon altitudes and to obtain cosmic-ray data from the LArTPC. The flight was launched from the Japan Aerospace Exploration Agency's (JAXA) Taiki Aerospace Research Field in Hokkaido, Japan. The total flight duration was 3 hours and 12 minutes, including a level flight of 44 minutes at a maximum altitude of 28.9 km. The flight system was landed on the sea and successfully recovered. The LArTPC was successfully operated throughout the flight, and about 0.5 million events of the cosmic-ray data including muons, protons, and Compton scattering gamma-ray candidates, were collected. This pioneering flight demonstrates the feasibility of operating a LArTPC in high-altitude environments, paving the way for future GRAMS missions and advancing our capabilities in MeV gamma-ray astronomy and dark matter research.

Abstract A new technique is presented that can be used to solve mixed boundary value problems for Laplace’s equation and the complex Helmholtz equation in bounded convex planar domains. This work is an extension of Crowdy (2015, CMFT, 15, 655–687) where new transform-based techniques were developed for boundary value problems for Laplace’s equation in circular domains. The key ingredient of the method is the analysis of the so-called global relation, which provides a coupling of integral transforms of the given boundary data and of the unknown boundary values. Three problems which involve mixed boundary conditions are solved in detail, as well as numerically implemented, to illustrate how to apply the new approach.

A theoretical analysis is presented of peristaltic pumping down a narrow conduit with permeable walls, motivated by the flushing action of lugworms and other marine organisms in sandy burrows. Flow in the conduit is dealt with using lubrication theory; the leakage into the surrounding medium is taken into account by exploiting slender-body theory to solve the associated Darcy problem. By adopting a model for the local force balance on the pumping surface, we bridge between the limits in which the pump operates with either fixed load or displacement. In the latter limit we characterize peristaltic waves with either fixed form or ones that partially collapse the conduit. We construct pump characteristics (the relation between the mean flux and net pressure drop) when the burrow wall is impermeable and pressures are fixed at each end, and compare the results with existing laboratory experiments performed on lugworms. We then consider how the peristaltic dynamics is changed when the wall is made permeable. Last, we consider pumping along an impermeable burrow into a leaky head shaft. The results reveal that the permeability of the conduit wall or end can greatly impact the direction and strength of the recirculating flow.

A new transform pair which can be used to solve mixed boundary value problems for Laplace's equation and the complex Helmholtz equation in bounded convex planar domains is presented. This work is an extension of Crowdy (2015, CMFT, 15, 655--687) where new transform techniques were developed for boundary value problems for Laplace's equation in circular domains. The key ingredient of the method is the analysis of the so called global relation which provides a coupling of integral transforms of the given boundary data and of the unknown boundary values. Three problems which involve mixed boundary conditions are solved in detail, as well as numerically implemented, to illustrate how to apply the new approach.

Linear and nonlinear stability analyses are performed to determine critical Rayleigh numbers ( ${Ra}_{cr}$ ) for a Rayleigh–Bénard convection configuration with an imposed bottom boundary heat flux that varies harmonically in time with zero mean. The ${Ra}_{cr}$ value depends on the non-dimensional frequency $\omega$ of the boundary heat-flux modulation. Floquet theory is used to find ${Ra}_{cr}$ for linear stability, and the energy method is used to find ${Ra}_{cr}$ for two different types of nonlinear stability: strong and asymptotic. The most unstable linear mode alternates between synchronous and subharmonic frequencies at low $\omega$ , with only the latter at large $\omega$ . For a given frequency, the linear stability ${Ra}_{cr}$ is generally higher than the nonlinear stability ${Ra}_{cr}$ , as expected. For large $\omega$ , ${Ra}_{cr} \omega ^{-2}$ approaches an $O(10)$ constant for linear stability but zero for nonlinear stability. Hence the domain for subcritical instability becomes increasingly large with increasing $\omega$ . The same conclusion is reached for decreasing Prandtl number. Changing temperature and/or velocity boundary conditions at the modulated or non-modulated plate leads to the same conclusions. These stability results are confirmed by selected direct numerical simulations of the initial value problem.

We report a numerical study of horizontal convection (HC) at Prandtl number $Pr = 1$, with both no-slip and free-slip boundary conditions. We obtain 2D and 3D solutions and determine the relation between the Rayleigh number $Ra$ and the Nusselt number $Nu$. In 2D we vary $Ra$ between $0$ and $10^{14}$. In the range $10^6 \le Ra \le 10^{10}$ the $Nu$-$Ra$ relation is $Nu \sim Ra^{1/5}$. With $Ra$ greater than about $10^{11}$ we find a 2D regime with $Nu \sim Ra^{1/4}$ over three decades, up to the highest 2D $Ra$. In 3D, with maximum $Ra = 10^{11.5}$, we find only $Nu \sim Ra^{1/5}$. These results apply to both free slip and no slip boundary conditions. The $Nu \sim Ra^{1/4}$ regime has a double boundary layer (BL): there is a thin BL with thickness $\sim Ra^{-1/4}$ nested inside a thicker BL with thickness $\sim Ra^{-1/5}$. The $Ra^{-1/4}$ BL thickness, which determines $Nu$, coincides with the Kolmogorov and Batchelor scales of HC. Numerical and theoretical results indicate that 3D HC is qualitatively and quantitatively similar to 2D HC. At the same $Ra$, the 3D $Nu$ exceeds the 2D $Nu$ by less than $20$%, i.e., there is very little 3D enhancement of heat transport. Boundary conditions are more important than dimensionality: the 2D free-slip solutions have larger $Nu$ than 3D no-slip solutions. Using the mechanical energy power integral of HC we show that the mean square vorticity of 3D HC is nearly equal to that of 2D HC at the same $Ra$. Thus vorticity amplification by strain-mediated vortex stretching does not operate in 3D HC.