Periodic orbits in tall laterally heated rectangular cavities

Net, Marta; Umbría, Juan Sánchez

doi:10.1103/physreve.95.023102

Cited by 7 publications

(9 citation statements)

References 26 publications

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“…where C is independent of ε. Then, by expressing u h − u = e h +û h − u, θ h − θ = η h −θ h − θ and recalling bounds (11)(12)(13)(14)(15)(16)(17)(18) we arrive to the following result.…”

Section: Error Bounds For the Velocity And The Temperaturementioning

confidence: 99%

“…For example, in rectangular cavities of size l x × l y , θ b = −e 1 · x/l x for natural convection with constant temperatures on vertical walls and isolated horizontal walls (e.g., [14]), or θ b = −e d · x/l y for Rayleigh-Bénard convection with constant temperatures in horizontal walls and isolated lateral walls. With this value of θ notice that f u and c are given by…”

Section: Preliminaries and Notationmentioning

confidence: 99%

“…In this case, the analysis in the present section shows that one can expect O(h) and O(h 2 ) errors for elements of degree k ≥ 1 and k ≥ 2, respectively. Notice however that the above-mentioned nonlocal compatibility conditions are indeed satisfied if the initial velocity is on a periodic orbit or in an invariant torus, a very common situation in practical numerical studies of thermal convection problems modelled by (1), [14], [17].…”

Section: Error Bounds For the Velocity And The Temperaturementioning

confidence: 99%

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Grad-div stabilization for the time-dependent Boussinesq equations with inf-sup stable finite elements

Frutos

García-Archilla

Novo

2019

Applied Mathematics and Computation

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In this paper we consider inf-sup stable finite element discretizations of the evolutionary Boussinesq equations with a grad-div type stabilization. We prove error bounds for the method with constants independent on the Rayleigh numbers Keywords Boussinesq equations; inf-sup stable finite element methods; grad-div stabilization; High Rayleigh number flows * Instituto de Investigación en Matemáticas (IMUVA), Universidad de Valladolid, Spain. of heat. The functionp is given byp = p + ρ 0 gz /ρ, wherep is the pressure and −ρ 0 gz the hydrostatic pressure. For simplicity we only consider no slip boundary conditions for the velocityũ = 0, on ∂Ω, and for the temperature, both prescribed values and and no flux,whereñ denotes the exterior unit normal vector and ∂Ω = ∂Ω D ∪ ∂Ω N .In this paper we consider inf-sup stable mixed finite element approximations to the model (2) with grad-div stabilization. Our aim is to prove error bounds with constants independent on inverse powers of the viscosity ν and the thermal diffusivity κ > 0, or independent of Prandl and Rayleigh numbers (see (3) below). To this end we add to the Galerkin formulation a grad-div stabilization term. In [10] error bounds for the time-dependent Navier-Stokes equations with only grad-div stabilization and constants independent on inverse powers of the viscosity are obtained. The idea is to extend those bounds to the natural convection flow described by (2). The role of the grad-div stabilization term is explained in detail in the error analysis in the present paper (see Remark 1 below).There are several related works that should be mentioned. In [9] the authors prove error bounds for inf-sup stable mixed finite element approximations to the model (2) with both grad-div stabilization and local projection stabilization (LPS) of streamline-upwind (SUPG)-type for both the velocity and the temperature. The authors prove bounds for the velocity and the temperature with constants independent on inverse powers of ν and κ. The main drawback of [9] is that for proving the error bounds of the method the assumption ∇u h ∞ bounded (u h being the approximation to the velocity) is needed and no proof for this a priori bound is included. In [10] weaker assumptions (norms u h ∞ and ∇ · u h L 2d/(d−1) bounded) are needed to get the error bounds for the pressure once the bounds for the velocity are already obtained. However in [10] these a priori bounds are proved. Let us observe that bounds on the velocity are obtained in [10] whenever finite elements other than linears are used for the velocity in the case d = 3 so that the assumption ∇u h ∞ bounded for h → 0 appearing in [9] may not be valid for lower order finite elements. Error bounds for the pressure are missing in reference [9]. In this paper we get error bounds for the velocity, temperature and the pressure, without assuming a priori bounds for the velocity approximations and with only grad-div stabilization. The rate of convergence we prove is the same obtained in [9] for a scheme that apart from grad-div stabilizat...

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Section: Error Bounds For the Velocity And The Temperaturementioning

confidence: 99%

Section: Preliminaries and Notationmentioning

confidence: 99%

Section: Error Bounds For the Velocity And The Temperaturementioning

confidence: 99%

See 1 more Smart Citation

Grad-div stabilization for the time-dependent Boussinesq equations with inf-sup stable finite elements

Frutos

García-Archilla

Novo

2019

Applied Mathematics and Computation

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“…Then, the steady flows lose stability giving rise to multi-vortex oscillations. The study of the transition from the steady to oscillatory flows for Pr = 0.71 and 7 Γ 11 showed the coexistence of various branches of periodic orbits consisting of waves travelling along the boundary layer, and a stable core of fluid almost horizontally stratified [15,8,16,17], consequently the global circulation is maintained.…”

Section: Introductionmentioning

confidence: 99%

“…New computations by using continuation methods with Pr as continuation parameter, and stability analysis, are presented. As in [17], a rectangular domain of Γ = 8 is taken because it is tall enough as to allow to capture the dynamics of the convection in very tall systems. In the range 0.2 < Pr < 0.3 time periodic solutions, not described before in the literature, consisting in transients between a time global circulation filling the slot and the formation of isolated vortices have been found.…”

Section: Introductionmentioning

confidence: 99%

Prandtl number dependence of convective fluids in tall laterally heated slots

Umbría

Net

2018

Eur. Phys. J. Spec. Top.

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The influence of the Prandtl number on the stable stationary and periodic flows of the fluids contained in laterally heated slots under realistic conditions, namely non-slip boundaries, insulated top and bottom horizontal limits and perfectly conducting lateral sides, is analyzed by using continuation methods. The branches of solutions are computed by decreasing the Prandtl number, for four Rayleigh numbers. The dynamical behavior depends strongly on both parameters. For a Rayleigh number, Ra = 10 3 , the steady flow remains stable in the wide range of Prandtl numbers computed. At Ra = 10 4 and O(10 5) the first bifurcations are of Hopf type giving rise to a type of oscillations that affects the bulk of the fluid, alternating from a general circulation to multi-vortex solutions, or the boundary layer, respectively. However, it is found that, in any case, the location of the shear determines the type of the oscillations. Moreover, at Ra = 10 4 the critical multipliers at the secondary bifurcations on the main branch of POs are real, giving rise to different kinds of very bounded stable periodic states of different symmetries and periods. At Ra of order 10 5 the instability of the periodic orbits gives rise directly to quasi-periodic flows.

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Stationary Flows and Periodic Dynamics of Binary Mixtures in Tall Laterally Heated Slots

Umbría

Net

2018

Computational Methods in Applied Sciences

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Periodic orbits in tall laterally heated rectangular cavities

Cited by 7 publications

References 26 publications

Grad-div stabilization for the time-dependent Boussinesq equations with inf-sup stable finite elements

Grad-div stabilization for the time-dependent Boussinesq equations with inf-sup stable finite elements

Prandtl number dependence of convective fluids in tall laterally heated slots

Stationary Flows and Periodic Dynamics of Binary Mixtures in Tall Laterally Heated Slots

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