Optimisation of Hermitian methods for Navier-Stokes equations in the vorticity and stream-function formulation

Roux, B.; Bontoux, Patrick; Loc, Ta Phuoc; Daube, Olivier

doi:10.1007/bfb0086923

Cited by 11 publications

(8 citation statements)

References 20 publications

(13 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The block-tridiagonal matrix inversion algorithm (Thomas algorithm), resulting from the use of high order Hermitian finite-difference relationships, was employed for (2). The vorticity at the boundary was calculated with the third-order relationship (known in the literature as Hirsh's relationship) and already used for natural convection problems by Roux et al (1979). We included a compatibility condition for variables on boundaries a t the intermediate time level (Fairweather & Mitchell 1967); and an iterative process at each time step.…”

Section: Numerical Finite-difference Methodsmentioning

confidence: 99%

Thermocapillary convection in long horizontal layers of low-Prandtl-number melts subject to a horizontal temperature gradient

Hadid

Roux

1990

J. Fluid Mech.

View full text Add to dashboard Cite

Thermocapillary convection arising in small-depth layers (long horizontal cavities) subject to a horizontal temperature gradient is studied numerically. A broad range of values of the Reynolds-Marangoni number, Re, is considered for three values of the aspect ratio (A = length/height). For the largest aspect ratio considered, A = 25, the fully developed Poiseuille-Couette solution is reached, but only for moderate Re. The limiting Re value for the observability of such a fully developed solution is derived as a function of A(Re [les ] 20A). For Re [les ] 20A, the flow exhibits three distinct regimes, in the upwind, central and downwind regions, respectively. The Poiseuille-Couette solution (when it exists) fills the central region, and the flow is accelerated, in the upwind region, to reach this Poiseuille-Couette solution at a distance that is proportional to Re. In the downwind region, where the flow is deflected by the endwall, a multi-roll structure is exhibited for Re [ges ] 1330. The number of rolls increases with Re. When Re > 20A, the upwind and downwind regions coalesce and some of the downwind rolls can be suppressed. Most of the computations concern interfacial conditions (with fixed temperature distribution) for which the dynamical solution is decoupled from the thermal one. A few thermal solutions are given herein, for Pr = 0.015 only.

show abstract

Section: Numerical Finite-difference Methodsmentioning

confidence: 99%

Thermocapillary convection in long horizontal layers of low-Prandtl-number melts subject to a horizontal temperature gradient

Hadid

Roux

1990

J. Fluid Mech.

View full text Add to dashboard Cite

show abstract

“…The governing equations (1)-(3) are solved using an ADL (alternating direction implicit) technique with a finite-difference method involving forward differences for time derivatives and Hermitian relationships for spatial derivatives with a truncation error of O(At2, Ax4, Ay4) (see Hirsh 1975; Roux et al 1979). Boundary vorticity was updated with an equation exhibiting fourth-order accuracy.…”

Section: Numerical Proceduresmentioning

confidence: 99%

“…Boundary vorticity was updated with an equation exhibiting fourth-order accuracy. The derivation of the discretized equations for the two-dimensional problem is not given here but it follows the approach described in Roux et al (1979) and Ben Hadid (1989). A block tridiagonal matrix inversion algorithm (Thomas algorithm) is employed for the finite-difference form of (1)-(3).…”

Section: Numerical Proceduresmentioning

confidence: 99%

Buoyancy- and thermocapillary-driven flows in differentially heated cavities for low-Prandtl-number fluids

Hadid

Roux²

1992

J. Fluid Mech.

View full text Add to dashboard Cite

The influence of thermocapillary forces on buoyancy-driven convection is numerically simulated for shallow open cavities with differentially heated endwalls and filled with low-Prandtl-number fluid. Calculations are carried out by solving two-dimensional Navier-Stokes equations coupled to the energy equation, for three aspects ratios A = (length/height) = 4, 12.5 and 25, and several values of the Grashof number (up to 6 × 104) and Reynolds number (|Re| ≤ 1.67 × 104). Thermocapillarity can have a quite significant effect on the stability of a primarily buoyancy-driven flow. The result of the combination of the two basic mechanisms (thermo-capillarity) and buoyancy) depends on whether their effects are additive (positive Re) or opposing (negative Re); counter-acting mechanisms yield more complex flow patterns. The critical Grashof number Grc for the onset of the unsteady regime is found to decrease substantially within a small range of negative Re, and to increase for positive Re (and also for large negative Re). For Gr = 4 × 104, A = 4 and small negative Reynolds numbers, −2.4 × 103 ≤ Re < 0, mono-periodic and bi- or quasi-periodic regimes are shown to exist successively, followed by a reverse transition. The development of the instabilities from an initial steady-state regime has been investigated by varying Re for Gr = 1.5 × 104 (below Grc at Re = 0); the onset of buoyant instabilities is enhanced in a narrow range of Re only (-1200 < Re < -200). It is also noteworthy that for small enough Grashof numbers (e.g. Gr = 3 × 103), a steady-state solution prevails over the whole range of Reynolds numbers investigated. This means that a critical Grashof number exists below which the effect of the thermocapillary forces is no longer destabilizing.

show abstract

“…The governing equations of incompressible flows are the Navier-Stokes equations. In two-dimensional problems the vorticity and stream-function formulation has the advantage that it not only eliminates the pressure variable entirely, but also ensures a divergence-free velocity field (mass conservation, i.e., ∇ • u = 0), if the Poisson equation (2) properly satisfied, see [6] and [7]. One encounters two scalar valued quantities, i.e., the vorticity ω and the stream-function ψ, instead of the velocity vector and the pressure field, thus it makes the computations more efficient.…”

Section: Governing Equations Of Incompressible Flowmentioning

confidence: 99%

“…Among finite difference methods high order compact methods [5,10] are more advantageous in terms of accuracy and reasonable cost. We refer to [19] and [35] for high-order compact discretizations of the incompressible Navier-Stokes equations in primitive variables and to [6]- [7] and [12] for the vorticity and stream-function formulation. Solving the incompressible Navier-Stokes equations typically implies an elliptic Poisson equation which is the most time consuming part of the algorithm.…”

Section: Introductionmentioning

confidence: 99%

Simulation of forced deformable bodies interacting with two-dimensional incompressible flows: Application to fish-like swimming

Ghaffari

Viazzo

Schneider

et al. 2015

International Journal of Heat and Fluid Flow

View full text Add to dashboard Cite

International audienceWe present an efficient algorithm for simulation of deformable bodies interacting with two-dimensional incompressible flows. The temporal and spatial discretizations of the Navier-Stokes equations in vorticity stream-function formulation are based on classical fourth-order Runge-Kutta and compact finite differences, respectively. Using a uniform Cartesian grid we benefit from the advantage of a new fourth-order direct solver for the Poisson equation to ensure the incompressibility constraint down to machine zero. For introducing a deformable body in fluid flow, the volume penalization method is used. A Lagrangian structured grid with prescribed motion covers the deformable body interacting with the surrounding fluid due to the hydrodynamic forces and moment calculated on the Eulerian reference grid. An efficient law for curvature control of an anguilliform fish, swimming to a prescribed goal, is proposed. Validation of the developed method shows the efficiency and expected accuracy of the algorithm for fish-like swimming and also for a variety of fluid/solid interaction problems

show abstract

Optimisation of Hermitian methods for Navier-Stokes equations in the vorticity and stream-function formulation

Cited by 11 publications

References 20 publications

Thermocapillary convection in long horizontal layers of low-Prandtl-number melts subject to a horizontal temperature gradient

Thermocapillary convection in long horizontal layers of low-Prandtl-number melts subject to a horizontal temperature gradient

Buoyancy- and thermocapillary-driven flows in differentially heated cavities for low-Prandtl-number fluids

Simulation of forced deformable bodies interacting with two-dimensional incompressible flows: Application to fish-like swimming

Contact Info

Product

Resources

About