52
F. A. Maksimov
between the cylinders. If the calculation for a certain L value is completed, then this
solution can be used as initial data for some new value of L* that is sufficiently close
to the initial one. In this case, the values of the gas-dynamic functions in the nodes
of the computational grid are assumed to be equal at the corresponding nodes for the
previously calculated variant. This is the second calculation variant.
If the Reynolds number is rather small, then the Couette flow is the solution
to the problem: This is a plane-parallel flow with a known velocity profile in the
circumferential direction [12]. If to consider the conditions, in which the Couette
flow is unstable [1], then the flat solution is to be destroyed due to instability in
the first variant of the calculation, and then a stable regular solution with the Taylor
vortices is to be arranged. In the second variant of the calculation, the flow structure
with the Taylor vortices is actually set in the initial field, the solution is established
in connection with the change in the periodicity size, and the stability of the flow
structure to the change in parameter L is checked.
The inner cylinder rotates with the angular velocity ω, and the Reynolds number
is Re =
ω·r ·(R−r )
ν
= 200, where ν is the kinematic viscosity coefficient. The linear
velocity of the surface of the inner cylinder corresponded to the velocity with the
Mach number ≈0.1, which does not lead to a significant change in the density in the
flow and to the appearance of the compressibility effects. With the given Re number,
the Taylor vortices are formed in the experiments [13]. The calculations for various
preset values of the periodicity length L were made. As well as in the experiments,
the Taylor vortices are formed in the numerical simulation with an adequate choice
of parameter L. Figure 5.2 demonstrates an example of the solution with the Taylor
vortices, where L = 0.835.
The flow is represented by the distribution of velocity along the axis of the cylinders in the plane passing through the axis of the cylinders. This velocity component is zero (w = 0), and the flow is plane in the Couette flow. When the Taylor
vortices form, a “chess” structure with a periodic change of a sign along the axis of
the cylinders and along the radius appears in the velocity distribution w. The flow
is axisymmetric. Figure 5.2a demonstrates the distribution of the velocity w in the
cross-section. It can be seen from Fig. 5.2b that the additionally superimposed isosurface of a constant value of this velocity is axisymmetric. Figure 5.2c demonstrates
the spatial streamlines visualizing two Taylor vortices in a pair.
Figure 5.3 shows the number of pairs N of the Taylor vortices depending on the
assigned periodicity size L. The set of the results obtained when using a flat field
with the velocity discontinuity as the initial data corresponds to data 1 marked by
large markers in the form of a circle. At L < 0.35, the Taylor vortices do not form,
and the flow remains plane. At 0.4 < L < 1.2, one pair of the vortices is formed; at L
> 1.2, two pairs of the vortices are formed in the considered range up to L = 2.
The set of the results 2 corresponds to the case of motion with an increase of the
size L, when using a solution with the Taylor vortices obtained at lower values of
L as initial data. In this case, a solution with one pair of the Taylor vortices can be
obtained for L < 1.5. This leads to significantly increase the range L, at which one
pair of the Taylor vortices is formed.
F. A. Maksimov
between the cylinders. If the calculation for a certain L value is completed, then this
solution can be used as initial data for some new value of L* that is sufficiently close
to the initial one. In this case, the values of the gas-dynamic functions in the nodes
of the computational grid are assumed to be equal at the corresponding nodes for the
previously calculated variant. This is the second calculation variant.
If the Reynolds number is rather small, then the Couette flow is the solution
to the problem: This is a plane-parallel flow with a known velocity profile in the
circumferential direction [12]. If to consider the conditions, in which the Couette
flow is unstable [1], then the flat solution is to be destroyed due to instability in
the first variant of the calculation, and then a stable regular solution with the Taylor
vortices is to be arranged. In the second variant of the calculation, the flow structure
with the Taylor vortices is actually set in the initial field, the solution is established
in connection with the change in the periodicity size, and the stability of the flow
structure to the change in parameter L is checked.
The inner cylinder rotates with the angular velocity ω, and the Reynolds number
is Re =
ω·r ·(R−r )
ν
= 200, where ν is the kinematic viscosity coefficient. The linear
velocity of the surface of the inner cylinder corresponded to the velocity with the
Mach number ≈0.1, which does not lead to a significant change in the density in the
flow and to the appearance of the compressibility effects. With the given Re number,
the Taylor vortices are formed in the experiments [13]. The calculations for various
preset values of the periodicity length L were made. As well as in the experiments,
the Taylor vortices are formed in the numerical simulation with an adequate choice
of parameter L. Figure 5.2 demonstrates an example of the solution with the Taylor
vortices, where L = 0.835.
The flow is represented by the distribution of velocity along the axis of the cylinders in the plane passing through the axis of the cylinders. This velocity component is zero (w = 0), and the flow is plane in the Couette flow. When the Taylor
vortices form, a “chess” structure with a periodic change of a sign along the axis of
the cylinders and along the radius appears in the velocity distribution w. The flow
is axisymmetric. Figure 5.2a demonstrates the distribution of the velocity w in the
cross-section. It can be seen from Fig. 5.2b that the additionally superimposed isosurface of a constant value of this velocity is axisymmetric. Figure 5.2c demonstrates
the spatial streamlines visualizing two Taylor vortices in a pair.
Figure 5.3 shows the number of pairs N of the Taylor vortices depending on the
assigned periodicity size L. The set of the results obtained when using a flat field
with the velocity discontinuity as the initial data corresponds to data 1 marked by
large markers in the form of a circle. At L < 0.35, the Taylor vortices do not form,
and the flow remains plane. At 0.4 < L < 1.2, one pair of the vortices is formed; at L
> 1.2, two pairs of the vortices are formed in the considered range up to L = 2.
The set of the results 2 corresponds to the case of motion with an increase of the
size L, when using a solution with the Taylor vortices obtained at lower values of
L as initial data. In this case, a solution with one pair of the Taylor vortices can be
obtained for L < 1.5. This leads to significantly increase the range L, at which one
pair of the Taylor vortices is formed.