36
V. A. Gushchin et al.
and in the cell centers, the integral variables of density, pressure, salinity, temperature,
etc. are defined. Second method is based on CABARET [3, 4] scheme or balancecharacteristic approach. In contrast to SMIF, CABARET scheme uses a double set of
variables both in faces and in the centers of cells. This allows CABARET scheme to
use positive properties of characteristic approach such as solution of shock waves and
rarefaction waves in the computational domain without introducing any restrictors
and monotonizers, and the correct accounting of the flow at the boundary allows
the schemes to remain conservative throughout the entire calculation time. These
two methods are widely used in different domains of mathematical simulations:
modeling of transport equations [4], equations of compressible gas [5], equations
of incompressible flows in closed regions [6, 7], the Navier–Stokes equations of
multicomponent gas dynamics [8], etc.
For the problems with a free surface, approximations of the Navier–Stokes equations in arbitrary Lagrange-Euler variables were obtained earlier for both SMIF
scheme [9] and CABARET scheme [10]. Satisfactory results were also obtained
in solving problems with stable stratification using the example of the problem of
collapse of a spot [11–15] of a homogeneous fluid in the thick of a stratified medium.
In this chapter, we partially present the results of comparison of the application of
two different numerical approaches for solving the problem of spot collapse: SMIF
method and CABARET method in Sects. 4.2 and 4.3, respectively. Test problems
are discussed in Sect. 4.4. Section 4.5 concludes the chapter.
4.2 SMIF Method
Numerical method for solving the problem of the dynamics of a spot (collapse)
in a stably density-stratified fluid was discussed in [11]. This method can be used
to investigate the flows of an inhomogeneous incompressible viscous fluid. The
possibility of specifying the stratification of density and viscosity either by analytic
formulas or by tables obtained by processing experimental data is foreseen that
considerably widens the range of laminar flows considered. According to the model
proposed in [16], the origin and development of turbulence in a stably densitystratified fluid are inseparable from internal waves and proceed as follows. Under
the action of external forces, the internal waves of large size arise in the stratified
fluid. As a result of their nonlinear interaction and subsequent breaking up or loss of
stability, the regions of mixed fluid (spots) arise. These spots of mixed-up turbulent
fluid evolve, gradually flattening (the collapse of turbulent spots), which in turn leads
to the formation of new spots, and so on.
In the evolution of a spot, it is natural to consider three basic stages [16]:
• Initial stage: The motive force acting on the fluid particles situated inside a spot
considerably exceeds the resistive forces. Intense internal waves are produced by
the spot.
V. A. Gushchin et al.
and in the cell centers, the integral variables of density, pressure, salinity, temperature,
etc. are defined. Second method is based on CABARET [3, 4] scheme or balancecharacteristic approach. In contrast to SMIF, CABARET scheme uses a double set of
variables both in faces and in the centers of cells. This allows CABARET scheme to
use positive properties of characteristic approach such as solution of shock waves and
rarefaction waves in the computational domain without introducing any restrictors
and monotonizers, and the correct accounting of the flow at the boundary allows
the schemes to remain conservative throughout the entire calculation time. These
two methods are widely used in different domains of mathematical simulations:
modeling of transport equations [4], equations of compressible gas [5], equations
of incompressible flows in closed regions [6, 7], the Navier–Stokes equations of
multicomponent gas dynamics [8], etc.
For the problems with a free surface, approximations of the Navier–Stokes equations in arbitrary Lagrange-Euler variables were obtained earlier for both SMIF
scheme [9] and CABARET scheme [10]. Satisfactory results were also obtained
in solving problems with stable stratification using the example of the problem of
collapse of a spot [11–15] of a homogeneous fluid in the thick of a stratified medium.
In this chapter, we partially present the results of comparison of the application of
two different numerical approaches for solving the problem of spot collapse: SMIF
method and CABARET method in Sects. 4.2 and 4.3, respectively. Test problems
are discussed in Sect. 4.4. Section 4.5 concludes the chapter.
4.2 SMIF Method
Numerical method for solving the problem of the dynamics of a spot (collapse)
in a stably density-stratified fluid was discussed in [11]. This method can be used
to investigate the flows of an inhomogeneous incompressible viscous fluid. The
possibility of specifying the stratification of density and viscosity either by analytic
formulas or by tables obtained by processing experimental data is foreseen that
considerably widens the range of laminar flows considered. According to the model
proposed in [16], the origin and development of turbulence in a stably densitystratified fluid are inseparable from internal waves and proceed as follows. Under
the action of external forces, the internal waves of large size arise in the stratified
fluid. As a result of their nonlinear interaction and subsequent breaking up or loss of
stability, the regions of mixed fluid (spots) arise. These spots of mixed-up turbulent
fluid evolve, gradually flattening (the collapse of turbulent spots), which in turn leads
to the formation of new spots, and so on.
In the evolution of a spot, it is natural to consider three basic stages [16]:
• Initial stage: The motive force acting on the fluid particles situated inside a spot
considerably exceeds the resistive forces. Intense internal waves are produced by
the spot.