The splitting method for investigating flows of a stratified liquid with a free surface

Belot︠s︡erkovskiĭ, O. M.; Gushchin, V. A.; Konshin, V. N.

doi:10.1016/0041-5553(87)90175-3

Cited by 53 publications

(20 citation statements)

References 8 publications

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“…A detailed description of this method and various examples of its use for two-and three-dimensional nonstationary stratified fluid flows, including free-surface flows, are given in [8,9]. Results of calculations and experimental studies of attached internal waves past a cylinder are in good agreement [10].…”

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confidence: 83%

Numerical and experimental study of the fine structure of a stratified fluid flow over a circular cylinder

Gushchin

Mitkin

Rozhdestvenskaya

et al. 2007

J Appl Mech Tech Phys

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The fine structure of the flow field of a continuously stratified fluid around a circular cylinder for small values of the Froude number was investigated in laboratory and numerical experiments. The parameters of the leading perturbation, the internal-wave field, and the cylinder wake were calculated using a two-dimensional model. The existence of the previously experimentally observed high-gradient density layers in the wake that are parallel to the flow axis was for the first time confirmed by numerical calculations. Results of the numerical and experimental studies are in good agreement with each other and with analytical models for small values of the Froude number.Introduction. The space-time characteristics of continuously stratified fluid flows over two-dimensional obstacles, their stability, and critical restructuring conditions are of scientific and practical interest because in nature and in technological devices, fluids are stratified due to nonuniform distributions of temperature and concentration of dissolved or suspended materials.In an analysis of stratified fluid flows taking into account stratification, the equations of hydrodynamics are supplemented by a buoyancy force and the diffusion equation for the stratifying component. As a result, along with a boundary layer, a trailing wake, bottom vortices, and a vortex trail, the flow structure contains internal waves and a leading perturbation. It should be noted that a stratified fluid is a nonequilibrium medium, in which even in the absence of external destabilizing factors, boundary flows with individual velocity and density scales are formed on impermeable surfaces because of the interruption of the flow of the stratifying component [1]. In this case, the solution of the problem is considerably complicated by the occurrence of new small and large parameters due to the weakness of the stratification and the smallness of the terms due to diffusion.The most widely used stratification models are the two-layer fluid model, for which calculations are performed by analogy with surface waves, and the exponentially stratified model. Because of the weak stratification in the formulation of the problem, the Navier-Stokes equation are written in the Boussinesq approximation, in which the density changes are considered negligibly small in all terms of the equation, except in the term containing gravity that takes into account buoyancy.There have been extensive experimental studies of linearly stratified fluid flows around a horizontal cylinder (exponential and linear density distributions are indistinguishable within the experimental error in the case of weak stratification). In most of these studies, the flow structure was visualized by optical methods and geometrical characteristics of waves, vortices, and wakes were determined. The phenomenological classification of flow regimes [2] was extended and supplemented in [3] taking into account high-gradient layers in wakes. New types of instability of laminar flow past a cylinder at low Froude number were found...

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mentioning

confidence: 83%

Numerical and experimental study of the fine structure of a stratified fluid flow over a circular cylinder

Gushchin

Mitkin

Rozhdestvenskaya

et al. 2007

J Appl Mech Tech Phys

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“…= 0. The requirement that the scheme viscosity should be a minimum, as can readily be seen from equation (11), impose on α the following condition: = -γ (the modified upwind difference scheme (MUDS)).…”

Section: Numerical Methods Smifmentioning

confidence: 99%

“…Let us investigate the class of the difference scheme which can be written in the form of the two-parameter family which depends on the parameters and in the following manner: (10) In this case the first differential approximation for equation (9) has the form , xx x t f u h uf f (11) where h u C C and 1 5 . 0…”

Section: Numerical Methods Smifmentioning

confidence: 99%

“…For solving of the Navier-Stokes equations (1) -(3) the Splitting on physical factors Method for Incompressible Fluid flows (SMIF) has been used [11][12].…”

Section: Numerical Methods Smifmentioning

confidence: 99%

“…The classification of the regimes of the 2D stratified viscous fluid flows around a square cylinder obtained by the numerical method SMIF[11][12].…”

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confidence: 99%

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Mathematical modeling of the incompressible fluid flows

Gushchin¹,

Matyushin²

2014

AIP Conference Proceedings

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The aim of the present paper is the demonstration of the opportunities of the mathematical modeling of the separated flows of the river and the sea water around blunt bodies on the basis of the Navier-Stokes equations (NSE) in the Boussinesq approximation. In the river water only the wakes of the obstacles are observed. In the sea water along with wakes the internal waves (IWs) are generated. The 3D homogeneous (river) and density stratified (sea) incompressible viscous fluid flows around a sphere and a square cylinder have been investigated by means of the direct numerical simulation on supercomputers and the visualization of the 3D vortex structures in the wake. For solving of NSE the Splitting on physical factors Method for Incompressible Fluid flows (SMIF) with hybrid explicit finite difference scheme (second-order accuracy in space, minimum scheme viscosity and dispersion, capable for work in wide range of the Reynolds (Re) and the internal Froude (Fr) numbers and monotonous) has been developed and successfully applied. The different transitions in sphere wakes (2D-3D, laminar-turbulent and others) with increasing of Re (1 < Re < 500000) and decreasing of Fr (0.005 < Fr < 100) have been investigated in details. Thus the classifications of the viscous fluid flow regimes around a sphere have been refined. The original classification of the 2D stratified viscous fluid flow regimes around a square cylinder has been obtained at Re < 200, 0.1 < Fr < 100. At Fr =0.1, Re = 50 the formation process of the two symmetric wavy hanging (soaring) sheets of density (starting from two hanging vortices) is demonstrated. It was found also that the values of the maximum phase difference along the circular cylinder axis are approximately equal to 0.1-0.2 Tf (for mode A, 191 < Re ≤ 300) and 0.015-0.030 Tf (for mode B, 300 ≤ Re ≤ 400), where the time Tf is the period of the flow.

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Numerical simulation of the separated fluid flows at large Reynolds numbers

Gushchin

Konshin

1992

Numerical Meth Engineering

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The variety of flow regimes (steady separated, periodically separated‐‘Karman vortex street’, unsteady turbulent) and their characteristic peculiarities (separation and reattachment points, secondary separation, boundary layer, instability of the shear mixing layer, etc.) require the construction of effective numerical methods, which will be able to simulate adequately the considered flows. MERANGE  SMIF–a splitting method for physical factors of incompressible fluids1‐is used for calculations of the steady and unsteady fluid flows past a circular cylinder in a wide range of Reynolds numbers (10° < Re < lo6). The finite‐difference scheme for this method is of second order accuracy in the space variables, has minimal numerical viscosity and is also monotonic. Use of the Navier‐Stokes equations with the corresponding transformation of Cartesian co‐ordinates allows the calculations to be made by one algorithm both in a boundary layer and out of it. The method allows calculations at Re = ∞ cc and simulation of d‘Alembert’s paradox. Some results on the classical problem of the flow around a circular cylinder for a wide range of Reynolds numbers are discussed. The crisis of the total drag coefficient and the sharp rise of the Strouhal number are simulated numerically (without any turbulence models) for the critical Reynolds numbers (Re ≈ 4 × 105), and are in a good agreement with experimental data.

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The splitting method for investigating flows of a stratified liquid with a free surface

Cited by 53 publications

References 8 publications

Numerical and experimental study of the fine structure of a stratified fluid flow over a circular cylinder

Numerical and experimental study of the fine structure of a stratified fluid flow over a circular cylinder

Mathematical modeling of the incompressible fluid flows

Numerical simulation of the separated fluid flows at large Reynolds numbers

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