Understanding the 8: 1 cavity problem via scalable stability analysis algorithms

Salinger, Andrew G.; Lehoucq, Richard B.; Pawlowski, Roger P.; Shadid, John N.

doi:10.1016/b978-008043944-0/50949-5

Cited by 3 publications

(5 citation statements)

References 18 publications

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“…The stream-lines pattern obtained by solving the problem with the compressible solver is shown in Figure 7(b). The results match very well those reported by Salinger et al in [22], namely the stream-line pattern corresponding to the flow after the first Hopf bifurcation. However, in [22], the critical Rayleigh number was found to be Ra cr D 3.61 E05, which is larger than the ones reported by Xin and Le Quere [21] and Christon [18], that found Ra cr D 3.1 E05.…”

Section: Natural Convection In a Square Cavity At Low Prandtl Numbersupporting

confidence: 90%

“…The results match very well those reported by Salinger et al in [22], namely the stream-line pattern corresponding to the flow after the first Hopf bifurcation. However, in [22], the critical Rayleigh number was found to be Ra cr D 3.61 E05, which is larger than the ones reported by Xin and Le Quere [21] and Christon [18], that found Ra cr D 3.1 E05. The present results show that Ra cr D 3.4 E05 correspond to the over-critical regime.…”

Section: Natural Convection In a Square Cavity At Low Prandtl Numbercontrasting

confidence: 64%

“…The stabilization, having the energy equation uncoupled, can be carried out separately for the energy and the momentum-continuity equation, using classical stabilization techniques, such as Galerkin/least-squares, various variational multiscale methods Algebraic Subgrid Scale (ASGS), Orthogonal Subscale (OSS) [13,14] or finite calculus (FIC) [15]. 6 P. RYZHAKOV, R. ROSSI AND E. OÑATE Semi-explicit approach for the momentum-continuity system The time integration strategy has a major impact upon the computational efficiency of the method based upon Equation (22). A crucial choice consists in whether to derive an explicit or an implicit time integration scheme.…”

Section: Discrete Modelmentioning

confidence: 99%

“…In establishing this benchmark, Christon et al proposes Ra = 3.4 E 05 as it constitutes a slightly over‐critical Rayleigh number. All the investigations carried out announce critical Rayleigh number to be above 3.0 E 05. To ensure that in the under‐critical regime the flow converges to the steady behavior, we performed a series of tests with Ra = 2.8 E 05.…”

Section: Examplesmentioning

confidence: 99%

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An algorithm for the simulation of thermally coupled low speed flow problems

Ryzhakov

Rossi

Oñate

2011

Numerical Methods in Fluids

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SUMMARY In this paper, we propose a computational algorithm for the solution of thermally coupled flows in subsonic regime. The formulation is based upon the compressible Navier–Stokes equations, written in nonconservation form. An efficient modular implementation is obtained by solving the energy equation separately and then using the computed temperature as a known value in the momentum‐continuity system. If an explicit single‐step time integration scheme for the energy equation is used, the decoupling results to be natural. Integration of the momentum‐continuity system is carried out using a semi‐explicit method, combining Runge–Kutta and Backward Euler schemes for the momentum and continuity equations, respectively. Implicit treatment of pressure leads to favorable time step estimates even in the low Mach number (Ma ≪ 1) regimes. The numerical dissipation introduced by the Backward Euler scheme ensures absence of the spurious high frequencies in the numerical solution. The key point of the method is the assumption of linear variation of the temperature within a time step. Combined with a fractional splitting of the momentum‐continuity system, it allows to solve the continuity only once per time step. Omitting the necessity of solving for the pressure at every intermediate step of the Runge–Kutta scheme minimizes the computational cost associated to the implicit step and leads to an efficiency close to that of a purely explicit scheme. The method is tested using two benchmark examples.Copyright © 2011 John Wiley & Sons, Ltd.

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Section: Natural Convection In a Square Cavity At Low Prandtl Numbersupporting

confidence: 90%

Section: Natural Convection In a Square Cavity At Low Prandtl Numbercontrasting

confidence: 64%

Section: Discrete Modelmentioning

confidence: 99%

Section: Examplesmentioning

confidence: 99%

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An algorithm for the simulation of thermally coupled low speed flow problems

Ryzhakov

Rossi

Oñate

2011

Numerical Methods in Fluids

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“…This problem has been studied extensively and, in fact, was the topic of a series of special sessions at the First MIT Conference on Computational Fluid and Solid Mechanics (Christon et al, 2001). Stability analyses indicate that the problem will reach a quasi-steady state, that is, the solution will oscillate about a mean (Salinger et al, 2001). In the current results, a quasi-steady state is achieved after approximately 350 s when running the simulation from a quiescent state.…”

Section: Benchmark Problemmentioning

confidence: 63%

A vorticity method for the solution of natural convection flows in enclosures

Ingber

2003

International Journal of Numerical Methods for Heat &Amp; Fluid Flow

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Vorticity formulations for the incompressible Navier-Stokes equations have certain advantages over primitive-variable formulations including the fact that the number of equations to be solved is reduced through the elimination of the pressure variable, identical satisfaction of the incompressibility constraint and the continuity equation, and an implicitly higher-order approximation of the velocity components. For the most part, vorticity methods have been used to solve exterior isothermal problems. In this research, a vorticity formulation is used to study the natural convection flows in differentially-heated enclosures. The numerical algorithm is divided into three steps: two kinematic steps and one kinetic step. The kinematics are governed by the generalized Helmholtz decomposition (GHD) which is solved using a boundary element method (BEM) whereas the kinetics are governed by the vorticity equation which is solved using a finite element method (FEM). In the first kinematic step, vortex sheet strengths are determined from a novel Galerkin implementation of the GHD. These vortex sheet strengths are used to determine Neumann boundary conditions for the vorticity equation. (The thermal boundary conditions are already known.) In the second kinematic step, the interior velocity field is determined using the regular (non-Galerkin) form of the GHD. This step, in a sense, linearizes the convective acceleration terms in both the vorticity and energy equations. In the third kinetic step, the coupled vorticity and energy equations are solved using a Galerkin FEM to determine the updated values of the vorticity and thermal fields. Two benchmark problems are considered to show the robustness and versatility of this formulation including natural convection in an 8 £ 1 differentially-heated enclosure at a near critical Rayleigh number.

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