42
V. A. Gushchin et al.
Finally, the scheme ends by calculating the flow variables on the new time layer
in the form of Eq. 4.17, where S is the index of the face of the flux variable, C is the
index of the adjacent cell, from where the flow goes in the direction of the face of S,
and S op is the index of the face of the opposite face of S and belonging to cell C.
ψ
n+1
S
= 2ψ
n+1/2
C
− ψ
n
Sop
ψ =
⎛
⎝
ρ
u
v
⎞
⎠
(4.17)
Now let us consider the application of these two approaches for investigation of
spot collapse in stratified fluid.
4.4 Test Problems
Hereinafter, the spot problem by SMIF method and CABARET method is presented
in Sects. 4.4.1 and 4.4.2, respectively. Section 4.4.3 provides a comparison of results
using these both methods.
4.4.1 Spot Problem by SMIF Method
Using SMIF method, the calculations were carried out in the field with dimensions
X = 10, Y = 5, R 0 = 1 with the following coefficients and parameters: μ/ρ 0 =
0.01 cm
2 /s
−1 , k s = 1.41 × 10
–5 cm
2 /s
–1 , N = 1 s
–1 , T b = 2π s, = 10 cm, C = 10,
Re = 100, Fr = 0.1, Sc = 709.2 that is close to the laboratory experimental conditions.
As boundary conditions on the top, bottom, and right borders of the computational
domain chosen resting state, i.e., u = v = s = 0. The computational domain was
covered with a uniform grid with steps in both directions δx = δy = 0.1. With a view
to verifying the correctness of program, the calculations in the absence of stain and
on different grids were performed.
The time dependences of horizontal and vertical sizes of a spot are shown in
Fig. 4.2.
4.4.2 Spot Problem by CABARET Method
In [12], authors used CABARET method for studying the problem of spot dynamics
in a fluid that is stably stratified by density. In contrast SMIF method, a new difference
scheme CABARET is solved by direct calculation elliptic equations for pressure by
fast direct method [22].
V. A. Gushchin et al.
Finally, the scheme ends by calculating the flow variables on the new time layer
in the form of Eq. 4.17, where S is the index of the face of the flux variable, C is the
index of the adjacent cell, from where the flow goes in the direction of the face of S,
and S op is the index of the face of the opposite face of S and belonging to cell C.
ψ
n+1
S
= 2ψ
n+1/2
C
− ψ
n
Sop
ψ =
⎛
⎝
ρ
u
v
⎞
⎠
(4.17)
Now let us consider the application of these two approaches for investigation of
spot collapse in stratified fluid.
4.4 Test Problems
Hereinafter, the spot problem by SMIF method and CABARET method is presented
in Sects. 4.4.1 and 4.4.2, respectively. Section 4.4.3 provides a comparison of results
using these both methods.
4.4.1 Spot Problem by SMIF Method
Using SMIF method, the calculations were carried out in the field with dimensions
X = 10, Y = 5, R 0 = 1 with the following coefficients and parameters: μ/ρ 0 =
0.01 cm
2 /s
−1 , k s = 1.41 × 10
–5 cm
2 /s
–1 , N = 1 s
–1 , T b = 2π s, = 10 cm, C = 10,
Re = 100, Fr = 0.1, Sc = 709.2 that is close to the laboratory experimental conditions.
As boundary conditions on the top, bottom, and right borders of the computational
domain chosen resting state, i.e., u = v = s = 0. The computational domain was
covered with a uniform grid with steps in both directions δx = δy = 0.1. With a view
to verifying the correctness of program, the calculations in the absence of stain and
on different grids were performed.
The time dependences of horizontal and vertical sizes of a spot are shown in
Fig. 4.2.
4.4.2 Spot Problem by CABARET Method
In [12], authors used CABARET method for studying the problem of spot dynamics
in a fluid that is stably stratified by density. In contrast SMIF method, a new difference
scheme CABARET is solved by direct calculation elliptic equations for pressure by
fast direct method [22].