4 Mathematical Modeling of Wave Motions of Fluids
37
• Intermediate stationary stage: The motive force is mainly counterbalanced by the
resistance of shape and the wave resistance due to the radiation of the internal
waves. The increase of the horizontal size of the spot proceeds almost as a linear
function of time, that is, the acceleration is negligibly small.
• Concluding viscous stage: The motive force is mainly counterbalanced by viscous
drag. The horizontal size of the spot changes only slightly.
Later as a result of diffusion, the spot is mixed with the surrounding fluid and
vanishes. Since that time, the simulation of such flows was undergone a lot of changes
[11]. New physical and mathematical models have been proposed [17, 18], and the
quality of methods designed for solving such problems has significantly improved
[2, 9]. Furthermore, the progress in computing has been amazing. In this chapter, we
want to adapt the mathematical model proposed in [17] and used for calculating the
flow around a sphere and circular cylinder [17, 18] to the problem of spot collapse,
which was earlier solved without taking into account the diffusion of the stratifying
component [11, 19].
Consider the flat nonstationary problem about the flow occurring when a homogeneous fluid region A surrounded by a stably and continuously density-stratified fluid
(for definiteness, the stratification is assumed to be linear) collapses in the vertical
direction (Fig. 4.1). The flow develops in the homogeneous gravity field with the
acceleration due to gravity g. The undisturbed linear density distribution [17]
ρ(x, y) = ρ 0 (1 −
y
+ s(x, y))
(4.1)
is characterized by the stratification scale =
1
ρ 0
∂ρ
∂ y
−1
, the buoyancy frequency
N =
√
g//, the buoyancy period T b = 2π/N , C = /R 0 is the ratio of scales, R 0
Fig. 4.1 Initial and
boundary conditions for spot
problem
37
• Intermediate stationary stage: The motive force is mainly counterbalanced by the
resistance of shape and the wave resistance due to the radiation of the internal
waves. The increase of the horizontal size of the spot proceeds almost as a linear
function of time, that is, the acceleration is negligibly small.
• Concluding viscous stage: The motive force is mainly counterbalanced by viscous
drag. The horizontal size of the spot changes only slightly.
Later as a result of diffusion, the spot is mixed with the surrounding fluid and
vanishes. Since that time, the simulation of such flows was undergone a lot of changes
[11]. New physical and mathematical models have been proposed [17, 18], and the
quality of methods designed for solving such problems has significantly improved
[2, 9]. Furthermore, the progress in computing has been amazing. In this chapter, we
want to adapt the mathematical model proposed in [17] and used for calculating the
flow around a sphere and circular cylinder [17, 18] to the problem of spot collapse,
which was earlier solved without taking into account the diffusion of the stratifying
component [11, 19].
Consider the flat nonstationary problem about the flow occurring when a homogeneous fluid region A surrounded by a stably and continuously density-stratified fluid
(for definiteness, the stratification is assumed to be linear) collapses in the vertical
direction (Fig. 4.1). The flow develops in the homogeneous gravity field with the
acceleration due to gravity g. The undisturbed linear density distribution [17]
ρ(x, y) = ρ 0 (1 −
y
+ s(x, y))
(4.1)
is characterized by the stratification scale =
1
ρ 0
∂ρ
∂ y
−1
, the buoyancy frequency
N =
√
g//, the buoyancy period T b = 2π/N , C = /R 0 is the ratio of scales, R 0
Fig. 4.1 Initial and
boundary conditions for spot
problem