A collocation method based on the influence matrix technique for Navier-Stokes problems in annular domains

Chaouche, A.; Randrianmampianina, A.; Bontoux, Patrick

doi:10.1016/0045-7825(90)90027-j

Cited by 8 publications

(6 citation statements)

References 7 publications

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“…The results are compared with available theoretical [32], experimental [33] and numerical [34] solutions. The latter have been obtained with a pseudospectral collocation-Chebyshev method, associated with the same time scheme used here, under vorticity-streamfunction formulation and using an influence matrix technique [35].…”

Section: Resultsmentioning

confidence: 99%

An improved projection scheme applied to pseudospectral methods for the incompressible Navier-Stokes equations

Hugues

Randriamampianina

1998

Int. J. Numer. Meth. Fluids

146

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An improved projection scheme is proposed and applied to pseudospectral collocation‐Chebyshev approximation for the incompressible Navier–Stokes equations. It consists of introducing a correct predictor for the pressure, one which is consistent with a divergence‐free velocity field at each time step. The main objective is to allow a time variation of the pressure gradient at boundaries. From different test problems, it is shown that this method, associated with a multistep second‐order time scheme, provides a time accuracy of the same order as the temporal scheme used for the pressure, and also improves the prediction of the velocity slip. Moreover, it does not exhibit any numerical boundary layer mentioned as a drawback of fractional steps algorithm, and does not require the use of staggered grids for the velocity and the pressure. Its effectiveness is validated by comparison with a previous time‐splitting algorithm proposed by Goda (K. Goda, J. Comput. Phys., 30, 76–95 (1979)) and implemented by Gresho (P. Gresho, Int. j. numer. methods fluids, 11, 587–620 (1990)) to finite element approximations. Steady and unsteady solutions for the regularized driven cavity and the rotating cavity submitted to throughflow are also used to assess the efficiency of this algorithm. © 1998 John Wiley & Sons, Ltd.

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Section: Resultsmentioning

confidence: 99%

An improved projection scheme applied to pseudospectral methods for the incompressible Navier-Stokes equations

Hugues

Randriamampianina

1998

Int. J. Numer. Meth. Fluids

146

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“…For the axisymmetric regimes (∂/∂φ = 0), a vorticity (ζ)-streamfunction (ψ) formulation with the azimuthal velocity (V φ ) was preferred to the use of primitive variables (Chaouche, Randriamampianina & Bontoux 1990).…”

Section: Mathematical Modelmentioning

confidence: 99%

Spatio-temporal behaviour in an enclosed corotating disk pair

2001

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We present a numerical investigation of the flow between corotating disks with a stationary outer casing – the enclosed corotating disk pair configuration. It is known that in such a geometry, axisymmetric and three-dimensional flow regimes develop depending on the value of the rotation rate. The three-dimensional flow is always unsteady flowing to its wavy structure in the radial-tangential plane. Axisymmetric regimes exhibit first a pitchfork bifurcation, characterized by a symmetry breaking with respect to the inter-disk midplane, before a Hopf bifurcation is established. The regime diagrams for these bifurcations are given in the (Re, G)-plane, where Re(= Ωb2/ν) is the rotational Reynolds number and G(= s/(b−a)) is the gap ratio. For values of G smaller than a critical limit Gc ∼ 0.26, there exists a range of rotation rates where the motion becomes time-dependent before bifurcating to a steady symmetry breaking regime. It is shown that for G [ges ] Gc the transition to unsteady three-dimensional flow occurs after the pitchfork bifurcation, and the flow structure is characterized by a shift-and-reflect symmetry. The transition to three-dimensional flow is consistent with experimental observations made by Abrahamson et al. (1989) where multiple solutions develop (known as the intransitivity phenomenon) with the presence of quasi-periodic behaviour resulting from successive vortex pairings. On the other hand, for smaller values of gap ratio, the three-dimensional flow shows a symmetry breaking. Finally, it is found that the variation of torque coefficient as a function of the rotation rate is the same for both the axisymmetric and three-dimensional solutions.

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“…(2.10) 2.1. Numerical method The system is solved numerically using a combination of a pseudo-spectral approach in space and a second-order finite difference scheme in time (Chaouche 1990;Chaouche, Randriamampianina & Bontoux 1990). Any flow variable f and its derivative (f stands for y , 5 and v) are developed in truncated series of the Chebyshev polynomials at the Gauss-Lobatto points,…”

Section: Formulation Of the Problemmentioning

confidence: 99%

Spatio—temporal behaviour in a rotating annulus with a source—sink flow

et al. 1996

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The axisymmetric flows arising in a rotating annulus with a superimposed forced flow are investigated with a pseudo-spectral numerical method. The flow enters the annulus at the inner radius with a radial velocity, then develops into a geostrophic flow azimuthally directed and flanked by two Ekman (nonlinear) boundary layers, and finally exits the outer radius, with a radially directed velocity. In this study the rotation rate of the cavity is fixed and very high. When the forced flow is weak, the flow is steady. On increasing the mass flow rate, the flow evolves to a chaotic temporal behaviour through several bifurcations, which perturbs the basic spatial configuration of the flow. The first bifurcation drives the steady state into an oscillatory regime, associated with a break of symmetry with respect to the midheight of the annulus. The entry flow travels radially through the cavity as in the steady flow, but it wavers and then is alternately sucked towards each Ekman layer. The frequency of this oscillation is close to the rotation rate frequency of the cavity, which is characteristic of inertial waves in rotating flows. A second transition to a quasi-periodic regime is characterized by the appearance of a second frequency. Further increases in the flow rate lead to a period-five state, via a locking of both frequencies, and then to a chaotic motion. This second frequency is of the order of the inverse of the Ekman spin-up characteristic time, suggesting that this instability is originated by the relaxation of the perturbations in the flow field. These perturbations of the unsteady flow field are corotating vortices along the rigid boundary walls. They are excited by the entry flow and their strength diminishes with increasing radius due to the low value of the Reynolds number. The parameters characterizing the unstable flows are also consistent with this explanation. The conclusion is that in this configuration, the origin of the described dynamical behaviour is not the instability of the Ekman boundary layers, as could be expected, but the instability of the entry flow. The reason is the importance of the nonlinear inertial terms in cavities with small radius of curvature.

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A collocation method based on the influence matrix technique for Navier-Stokes problems in annular domains

Cited by 8 publications

References 7 publications

An improved projection scheme applied to pseudospectral methods for the incompressible Navier-Stokes equations

An improved projection scheme applied to pseudospectral methods for the incompressible Navier-Stokes equations

Spatio-temporal behaviour in an enclosed corotating disk pair

Spatio—temporal behaviour in a rotating annulus with a source—sink flow

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