4 Mathematical Modeling of Wave Motions of Fluids
45
distinction of results may be explained by using different physical models. In SMIF
approach, the diffusion of stratification component (perturbation of salinity) is taken
into account. In CABARET approach, the diffusion of stratification component—
density is absent.
4.5 Conclusions
Two models of spot burst problem were considered: in the case of homogeneous
density spot located in sharply stratified medium and in the case of salinity homogeneous spot in seawater stratified by salinity. The comparison of given results was
carried out with fixed Froude numbers and Reynolds numbers. The case of homogeneous spot located inside viscous density-stratified medium was solved using
CABARET method. The case with salinity diffusion inside salinity stratified medium
was solved using SMIF method. Simulated domain in SMIF method is solved with
plane X = 0 defined as symmetry boundary condition. Compared quantities (height
and width of the spot time dependent) were dimensions of spot in horizontal and
vertical direction time depending.
As we see from graph dependents of vertical spot dimension, top and bottom
dimensions differ dynamically due to appearance of buoyancy force. Horizontal
dimension of spot fairly correlates with theoretical estimation of spot dimension.
From this, we can conclude that both methods are in sufficient agreement with other
authors’ works that deal with this task.
Acknowledgements This work was carried out in frame of the State Task of ICAD RAS.
References
1. Belotserkovskii, O., Gushchin, V., Shchennikov, V.: Use of the splitting method to solve problems of the dynamics of a viscous incompressible fluid. USSR Comput. Math. Math. Phys.
15(1), 190–200 (1975)
2. Gushchin, V.: Family of quasi-monotonic finite-difference schemes of the second-order of
approximation. Math. Models Comput. Simul. 8, 487–496 (2016)
3. Goloviznin, V., Samarskii, A.: Some characteristics of finite difference scheme “CABARET”.
Math. Mod. 10(1), 101–116 (1998) (in Russian)
4. Goloviznin, V., Samarskii, A.: Finite difference approximation of convective transport equation
with space splitting time derivative. Math. Mod. 10(1), 86–100 (1998) (in Russian)
5. Goloviznin, V., Karabasov, S., Kondakov, V.: Generalization of the CABARET scheme to
two-dimensional orthogonal computational grids. Math. Models Comput. Simul. 6(1), 56–79
(2014)
6. Glotov, V., Goloviznin, V.: CABARET scheme in velocity–pressure formulation for twodimensional incompressible fluids. Comput. Math. Math. Phys. 53(6), 721–735 (2013)
7. Gushchin, V., Matyushin, P.: Method SMIF for incompressible fluid flows modeling. In: Dimov,
I., Faragó, I., Vulkov, L. (eds.) Numerical Analysis and Its Applications. NAA 2012. LNCS,
vol. 8236, pp. 311–318. Springer, Berlin, Heidelberg (2013)
45
distinction of results may be explained by using different physical models. In SMIF
approach, the diffusion of stratification component (perturbation of salinity) is taken
into account. In CABARET approach, the diffusion of stratification component—
density is absent.
4.5 Conclusions
Two models of spot burst problem were considered: in the case of homogeneous
density spot located in sharply stratified medium and in the case of salinity homogeneous spot in seawater stratified by salinity. The comparison of given results was
carried out with fixed Froude numbers and Reynolds numbers. The case of homogeneous spot located inside viscous density-stratified medium was solved using
CABARET method. The case with salinity diffusion inside salinity stratified medium
was solved using SMIF method. Simulated domain in SMIF method is solved with
plane X = 0 defined as symmetry boundary condition. Compared quantities (height
and width of the spot time dependent) were dimensions of spot in horizontal and
vertical direction time depending.
As we see from graph dependents of vertical spot dimension, top and bottom
dimensions differ dynamically due to appearance of buoyancy force. Horizontal
dimension of spot fairly correlates with theoretical estimation of spot dimension.
From this, we can conclude that both methods are in sufficient agreement with other
authors’ works that deal with this task.
Acknowledgements This work was carried out in frame of the State Task of ICAD RAS.
References
1. Belotserkovskii, O., Gushchin, V., Shchennikov, V.: Use of the splitting method to solve problems of the dynamics of a viscous incompressible fluid. USSR Comput. Math. Math. Phys.
15(1), 190–200 (1975)
2. Gushchin, V.: Family of quasi-monotonic finite-difference schemes of the second-order of
approximation. Math. Models Comput. Simul. 8, 487–496 (2016)
3. Goloviznin, V., Samarskii, A.: Some characteristics of finite difference scheme “CABARET”.
Math. Mod. 10(1), 101–116 (1998) (in Russian)
4. Goloviznin, V., Samarskii, A.: Finite difference approximation of convective transport equation
with space splitting time derivative. Math. Mod. 10(1), 86–100 (1998) (in Russian)
5. Goloviznin, V., Karabasov, S., Kondakov, V.: Generalization of the CABARET scheme to
two-dimensional orthogonal computational grids. Math. Models Comput. Simul. 6(1), 56–79
(2014)
6. Glotov, V., Goloviznin, V.: CABARET scheme in velocity–pressure formulation for twodimensional incompressible fluids. Comput. Math. Math. Phys. 53(6), 721–735 (2013)
7. Gushchin, V., Matyushin, P.: Method SMIF for incompressible fluid flows modeling. In: Dimov,
I., Faragó, I., Vulkov, L. (eds.) Numerical Analysis and Its Applications. NAA 2012. LNCS,
vol. 8236, pp. 311–318. Springer, Berlin, Heidelberg (2013)