scholarly journals A variational formulation of vertical slice models

2013 ◽  
Vol 469 (2155) ◽  
pp. 20120678 ◽  
Author(s):  
C. J. Cotter ◽  
D. D. Holm
Keyword(s):  
Three Dimensional ◽  
Boussinesq Equations ◽  
Subsequent Evolution ◽  
Boussinesq Model ◽  
Low Mach Number ◽  
Low Mach Number Limit ◽  
Variational Framework ◽  
Vertical Slice ◽  
Error Testing ◽  
Circulation Theorem

A variational framework is defined for vertical slice models with three-dimensional velocity depending only on x and z . The models that result from this framework are Hamiltonian, and have a Kelvin–Noether circulation theorem that results in a conserved potential vorticity in the slice geometry. These results are demonstrated for the incompressible Euler–Boussinesq equations with a constant temperature gradient in the y -direction (the Eady–Boussinesq model), which is an idealized problem used to study the formation and subsequent evolution of weather fronts. We then introduce a new compressible extension of this model. Unlike the incompressible model, the compressible model does not produce solutions that are also solutions of the three-dimensional equations, but it does reduce to the Eady–Boussinesq model in the low Mach number limit. Hence, the new model could be used in asymptotic limit error testing for compressible weather models running in a vertical slice configuration.

scholarly journals Stochastic Wave–Current Interaction in Thermal Shallow Water Dynamics

2021 ◽  
Vol 31 (2) ◽  
Author(s):  
Darryl D. Holm ◽  
Erwin Luesink
Keyword(s):  
Fluid Dynamics ◽  
Shallow Water ◽  
Asymptotic Expansions ◽  
Three Dimensional ◽  
Water Dynamics ◽  
Nonlinear Wave Propagation ◽  
Entire Family ◽  
Variational Framework ◽  
Current Interaction ◽  
Circulation Theorem

AbstractHolm (Proc R Soc A Math Phys Eng Sci 471(2176):20140963, 2015) introduced a variational framework for stochastically parametrising unresolved scales of hydrodynamic motion. This variational framework preserves fundamental features of fluid dynamics, such as Kelvin’s circulation theorem, while also allowing for dispersive nonlinear wave propagation, both within a stratified fluid and at its free surface. The present paper combines asymptotic expansions and vertical averaging with the stochastic variational framework to formulate a new approach for developing stochastic parametrisation schemes for nonlinear waves in fluid dynamics. The approach is applied to two sequences of shallow water models which descend from Euler’s three-dimensional fluid equations with rotation and stratification under approximation by asymptotic expansions and vertical averaging. In the entire family of nonlinear stochastic wave–current interaction equations derived here using this approach, Kelvin’s circulation theorem reveals a barotropic mechanism for wave generation of horizontal circulation or convection (cyclogenesis) which is activated whenever the gradients of wave elevation and/or topography are not aligned with the gradient of the vertically averaged buoyancy.

Laminar natural convection about an isothermally heated sphere at small Grashof number

1968 ◽  
Vol 34 (1) ◽  
pp. 163-176 ◽  
Author(s):  
Francis E. Fendell
Keyword(s):  
Natural Convection ◽  
Three Dimensional ◽  
Perturbation Expansion ◽  
Outer Sphere ◽  
Convective Transport ◽  
Convection Flow ◽  
Constant Density ◽  
Boussinesq Model ◽  
Grashof Number ◽  
Recirculating Flow

The flow induced by gravity about a very small heated isothermal sphere introduced into a fluid in hydrostatic equilibrium is studied. The natural-convection flow is taken to be steady and laminar. The conditions under which the Boussinesq model is a good approximation to the full conservation laws are described. For a concentric finite cold outer sphere with radius, in ratio to the heated sphere radius, roughly less than the Grashof number to the minus one-half power, a recirculating flow occurs; fluid rises near the inner sphere and falls near the outer sphere. For a small heated sphere in an unbounded medium an ordinary perturbation expansion essentially in the Grashof number leads to unbounded velocities far from the sphere; this singularity is the natural-convection analogue of the Whitehead paradox arising in three-dimensional low-Reynolds-number forced-convection flows. Inner-and-outer matched asymptotic expansions reveal the importance of convective transport away from the sphere, although diffusive transport is dominant near the sphere. Approximate solution is given to the nonlinear outer equations, first by seeking a similarity solution (in paraboloidal co-ordinates) for a point heat source valid far from the point source, and then by linearization in the manner of Oseen. The Oseen solution is matched to the inner diffusive solution. Both outer solutions describe a paraboloidal wake above the sphere within which the enthalpy decays slowly relative to the rapid decay outside the wake. The updraft above the sphere is reduced from unbounded growth with distance from the sphere to constant magnitude by restoration of the convective accelerations. Finally, the role of vertical stratification of the ambient density in eventually stagnating updrafts predicted on the basis of a constant-density atmosphere is discussed.

The eruptive regime of mass-transfer-driven Rayleigh–Marangoni convection

2016 ◽  
Vol 791 ◽  
Author(s):  
Thomas Köllner ◽  
Karin Schwarzenberger ◽  
Kerstin Eckert ◽  
Thomas Boeck
Keyword(s):  
Concentration Distribution ◽  
Marangoni Convection ◽  
Experimental Studies ◽  
Three Dimensional ◽  
Boussinesq Equations ◽  
Navier Stokes ◽  
Flow Structures ◽  
Rayleigh Convection ◽  
Marangoni Effects ◽  
Validation Experiments

The transfer of an alcohol, 2-propanol, from an aqueous to an organic phase causes convection due to density differences (Rayleigh convection) and interfacial tension gradients (Marangoni convection). The coupling of the two types of convection leads to short-lived flow structures called eruptions, which were reported in several previous experimental studies. To unravel the mechanism underlying these patterns, three-dimensional direct numerical simulations and corresponding validation experiments were carried out and compared with each other. In the simulations, the Navier–Stokes–Boussinesq equations were solved with a plane interface that couples the two layers including solutal Marangoni effects. Our simulations show excellent agreement with the experimentally observed patterns. On this basis, the origin of the eruptions is explained by a two-step process in which Rayleigh convection continuously produces a concentration distribution that triggers an opposing Marangoni flow.

scholarly journals Low Mach number limit for a model of accretion disk

2018 ◽  
Vol 38 (7) ◽  
pp. 3239-3268 ◽  
Author(s):  
Donatella Donatelli ◽  
◽  
Bernard Ducomet ◽  
Šárka Nečasová ◽  
◽  
...  
Keyword(s):  
Mach Number ◽  
Accretion Disk ◽  
Low Mach Number ◽  
Low Mach Number Limit

scholarly journals Exact solution and the multidimensional Godunov scheme for the acoustic equations

2022 ◽  
Author(s):  
Wasilij Barsukow ◽  
Christian Klingenberg
Keyword(s):  
Exact Solution ◽  
Mach Number ◽  
Euler Equations ◽  
Two Dimensions ◽  
Finite Volume Scheme ◽  
Godunov Scheme ◽  
Low Mach Number ◽  
Low Mach Number Limit ◽  
Acoustic Equations ◽  
Spatial Dimensions

The acoustic equations derived as a linearization of the Euler equations are a valuable system for studies of multi-dimensional solutions. Additionally they possess a low Mach number limit analogous to that of the Euler equations. Aiming at understanding the behaviour of the multi-dimensional Godunov scheme in this limit, first the exact solution of the corresponding Cauchy problem in three spatial dimensions is derived. The appearance of logarithmic singularities in the exact solution of the 4-quadrant Riemann Problem in two dimensions is discussed. The solution formulae are then used to obtain the multidimensional Godunov finite volume scheme in two dimensions. It is shown to be superior to the dimensionally split upwind/Roe scheme concerning its domain of stability and ability to resolve multi-dimensional Riemann problems. It is shown experimentally and theoretically that despite taking into account multi-dimensional information it is, however, not able to resolve the low Mach number limit.

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