Finite element analysis of flow in a gas‐filled rotating annulus

Berger, Marsha

doi:10.1002/fld.1650070303

Cited by 6 publications

(3 citation statements)

References 10 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This equation models the flow in a gas centrifuge, a physical motivation can be found in [35]. First finite element discretizations of this equation are given in [6,17]. In [14], a tensor product finite element with B-spline basis functions is used.…”

Section: Remark 22mentioning

confidence: 99%

A Priori Error Estimates for Finite Element Methods forH^(2,1)-Elliptic Equations

Apel

Flaig

Nicaise

2013

Numerical Functional Analysis and Optimization

View full text Add to dashboard Cite

The convergence of finite element methods for linear elliptic boundary value problems of second and forth order is well understood. In this article, we introduce finite element approximations of some linear semi-elliptic boundary value problem of mixed order on a two-dimensional rectangular domain Q. The equation is of second order in one direction and forth order in the other and appears in the optimal control of parabolic partial differential equations if one eliminates the control and the state (or the adjoint state) in the first order optimality conditions. We establish a regularity result and estimate for the finite element error of conforming approximations of this equation. The finite elements in use have a tensor product structure, in one dimension we use linear, quadratic or cubic Lagrange elements in the other dimension cubic Hermite elements. For these elements, we prove the error bound O(h 2 + k ) in the energy norm and O((h 2 + k )(h 2 + )) in the L 2 (Q )-norm.Keywords Anisotropic interpolation error estimate; Finite element method; Parabolic optimal control problem; Semi-elliptic boundary value problem; a priori error estimate.Mathematics Subject Classification Primary 65N30; Secondary 65N15, 35H99, 49K20, 49M29.

show abstract

Section: Remark 22mentioning

confidence: 99%

A Priori Error Estimates for Finite Element Methods forH^(2,1)-Elliptic Equations

Apel

Flaig

Nicaise

2013

Numerical Functional Analysis and Optimization

View full text Add to dashboard Cite

show abstract

“…The usual one-dimensional, axially symmetric solution was obtained as a limiting case. Analysis of compressible, finite length, circular Couette flow was also reported in Berger (1987). The thermal and hydrodynamical analysis of a counter-current gas flow in a rotating cylinder was carried out by Andrade & Bastos (1998).…”

Section: Introductionmentioning

confidence: 99%

“…Axially linear inner and outer temperature distributions and velocity slip were investigated by Berger (1987). An existence condition for a class of non-trivial, one-dimensional solutions was given.…”

Section: Introductionmentioning

confidence: 99%

The generalized Onsager model for the secondary flow in a high-speed rotating cylinder

Pradhan

Kumaran

2011

J. Fluid Mech.

View full text Add to dashboard Cite

The generalizations of the Onsager model for the radial boundary layer and the Carrier-Maslen model for the end-cap axial boundary layer in a high-speed rotating cylinder are formulated for studying the secondary gas flow due to wall heating and due to insertion of mass, momentum and energy into the cylinder. The generalizations have wider applicability than the original Onsager and Carrier-Maslen models, because they are not restricted to the limit A 1, though they are restricted to the limit Re 1 and a high-aspect-ratio cylinder whose length/diameter ratio is large. Here, the stratification parameter A = mΩ 2 R 2 /(2k B T). This parameter A is the ratio of the peripheral speed, ΩR, to the most probable molecular speed, √ 2k B T/m, the Reynolds number Re = ρ w ΩR 2 /µ, where m is the molecular mass, Ω and R are the rotational speed and radius of the cylinder, k B is the Boltzmann constant, T is the gas temperature, ρ w is the gas density at wall, and µ is the gas viscosity. In the case of wall forcing, analytical solutions are obtained for the sixth-order generalized Onsager equations for the master potential, and for the fourthorder generalized Carrier-Maslen equation for the velocity potential. For the case of mass/momentum/energy insertion into the flow, the separation-of-variables procedure is used, and the appropriate homogeneous boundary conditions are specified so that the linear operators in the axial and radial directions are self-adjoint. The discrete eigenvalues and eigenfunctions of the linear operators (sixth-order and second-order in the radial and axial directions for the Onsager equation, and fourth-order and second-order in the axial and radial directions for the Carrier-Maslen equation) are determined. These solutions are compared with direct simulation Monte Carlo (DSMC) simulations. The comparison reveals that the boundary conditions in the simulations and analysis have to be matched with care. The commonly used 'diffuse reflection' boundary conditions at solid walls in DSMC simulations result in a non-zero slip velocity as well as a 'temperature slip' (gas temperature at the wall is different from wall temperature). These have to be incorporated in the analysis in order to make quantitative predictions. In the case of mass/momentum/energy sources within the flow, it is necessary to ensure that the homogeneous boundary conditions are accurately satisfied in the simulations. When these precautions are taken, there is excellent agreement between analysis and simulations, to within 10 %, even when the stratification parameter is as low as 0.707, the Reynolds number is as low as 100 and the aspect ratio (length/diameter) of the cylinder is as low as 2, and the secondary flow velocity is as high as 0.2 times the maximum base flow velocity. The predictions of the generalized models are also significantly better than those of the original Onsager † Email address for correspondence: kumaran@chemeng.iisc.ernet.in 110 S. Pradhan and V. Kumaran and Carrier-Maslen models, which are restricted to thin bou...

show abstract

A Guide to Literature Related to the Taylor-Couette Problem

Tagg

1992

NATO ASI Series

View full text Add to dashboard Cite

Finite element analysis of flow in a gas‐filled rotating annulus

Cited by 6 publications

References 10 publications

A Priori Error Estimates for Finite Element Methods forH^(2,1)-Elliptic Equations

A Priori Error Estimates for Finite Element Methods forH^(2,1)-Elliptic Equations

The generalized Onsager model for the secondary flow in a high-speed rotating cylinder

A Guide to Literature Related to the Taylor-Couette Problem

Contact Info

Product

Resources

About

Finite element analysis of flow in a gas‐filled rotating annulus

Cited by 6 publications

References 10 publications

A Priori Error Estimates for Finite Element Methods forH(2,1)-Elliptic Equations

A Priori Error Estimates for Finite Element Methods forH(2,1)-Elliptic Equations

The generalized Onsager model for the secondary flow in a high-speed rotating cylinder

A Guide to Literature Related to the Taylor-Couette Problem

Contact Info

Product

Resources

About

A Priori Error Estimates for Finite Element Methods forH^(2,1)-Elliptic Equations

A Priori Error Estimates for Finite Element Methods forH^(2,1)-Elliptic Equations