58
F. A. Maksimov
Regardless of the geometry’s parameter, when setting a sufficiently small size
of the periodicity, the Couette flow is formed in the calculation. In this case, the
calculated value of the friction moment is consistent with the theoretical value in
accordance with the Couette solution (C m = 0.0533, C m = 0.036, and C m = 0.0278
for R = 2.0, R = 1.5, and R = 1.25, respectively).
If the size of a pair of the vortices with the distance between the cylinders is
correlated, then the size of a pair of the Taylor vortices can be from 1.0 to 2.2 of
the distance between the cylinders. The estimate of the minimum size of a pair of
the vortices decreases to 0.8 if to increase R. The estimate of the maximum size of
a pair of the vortices is the same for three calculated geometry variants. Artificial
pulling of the flow structure is possible. In this case, the size of a pair of the Taylor
vortices can be up to 2.8–3.2 of the distance between the cylinders. Artificial pulling
of the flow structure with the formation of the anomalously large Taylor vortices can
be established, for example, by decreasing the distance between the cylinders in the
already formed flow with the Taylor vortices.
The Taylor vortices, in fact, allow to actualize the maximum friction within the
regular laminar flow. In accordance with this fact, the choice of the solution with
the maximum friction, which is determined by the maximum of the friction, is the
most correct one for the real flow. In accordance with the calculation results, this
maximum exists, and in this case, the optimal size of a pair of the vortices is from 1.4
to 1.8 of the distance between the cylinders. As the distance between the cylinders
decreases, the optimal size of the vortex pair increases.
In accordance with the experimental studies [13], the value of the coefficient of
the friction moment at Re = 200 is C m ≈ 0.095 for R = 2, and the calculation results
satisfactorily conform to the experiment.
5.4 The Heat Exchange Between Rotating Cylinders
The flow of viscous gas between the rotating cylinders with different temperature is
considered. The outer cylinder is heated, and therefore, the gas near its surface has a
lower density. Under the influence of centrifugal force in the system of the rotating
cylinders, the Rayleigh–Taylor instability will develop.
The radius R of the outer cylinder is two times larger than the radius r of the inner
cylinder that rotates with the angular velocity ω. The Reynolds number is determined
by the expression Re = ωr · (R − r )/ν. The Re critical number, at which the Couette
flow reforms into the flow with the Taylor vortices, is approximately Re* ≈ 70 [13]
for a fixed and unheated external cylinder for the given geometry. For the formation
of the Rayleigh–Taylor instability, the temperature of the outer cylinder was set two
times higher than the temperature of the inner cylinder. The rotation of the outer
cylinder varied from the rest state (the outer cylinder does not rotate) to the rotation
with the same angular velocity as the inner cylinder, while the linear velocity of
the surface of the outer cylinder was two times higher than the linear velocity of
the inner cylinder. One of the intermediate rotation variants of the outer cylinder
F. A. Maksimov
Regardless of the geometry’s parameter, when setting a sufficiently small size
of the periodicity, the Couette flow is formed in the calculation. In this case, the
calculated value of the friction moment is consistent with the theoretical value in
accordance with the Couette solution (C m = 0.0533, C m = 0.036, and C m = 0.0278
for R = 2.0, R = 1.5, and R = 1.25, respectively).
If the size of a pair of the vortices with the distance between the cylinders is
correlated, then the size of a pair of the Taylor vortices can be from 1.0 to 2.2 of
the distance between the cylinders. The estimate of the minimum size of a pair of
the vortices decreases to 0.8 if to increase R. The estimate of the maximum size of
a pair of the vortices is the same for three calculated geometry variants. Artificial
pulling of the flow structure is possible. In this case, the size of a pair of the Taylor
vortices can be up to 2.8–3.2 of the distance between the cylinders. Artificial pulling
of the flow structure with the formation of the anomalously large Taylor vortices can
be established, for example, by decreasing the distance between the cylinders in the
already formed flow with the Taylor vortices.
The Taylor vortices, in fact, allow to actualize the maximum friction within the
regular laminar flow. In accordance with this fact, the choice of the solution with
the maximum friction, which is determined by the maximum of the friction, is the
most correct one for the real flow. In accordance with the calculation results, this
maximum exists, and in this case, the optimal size of a pair of the vortices is from 1.4
to 1.8 of the distance between the cylinders. As the distance between the cylinders
decreases, the optimal size of the vortex pair increases.
In accordance with the experimental studies [13], the value of the coefficient of
the friction moment at Re = 200 is C m ≈ 0.095 for R = 2, and the calculation results
satisfactorily conform to the experiment.
5.4 The Heat Exchange Between Rotating Cylinders
The flow of viscous gas between the rotating cylinders with different temperature is
considered. The outer cylinder is heated, and therefore, the gas near its surface has a
lower density. Under the influence of centrifugal force in the system of the rotating
cylinders, the Rayleigh–Taylor instability will develop.
The radius R of the outer cylinder is two times larger than the radius r of the inner
cylinder that rotates with the angular velocity ω. The Reynolds number is determined
by the expression Re = ωr · (R − r )/ν. The Re critical number, at which the Couette
flow reforms into the flow with the Taylor vortices, is approximately Re* ≈ 70 [13]
for a fixed and unheated external cylinder for the given geometry. For the formation
of the Rayleigh–Taylor instability, the temperature of the outer cylinder was set two
times higher than the temperature of the inner cylinder. The rotation of the outer
cylinder varied from the rest state (the outer cylinder does not rotate) to the rotation
with the same angular velocity as the inner cylinder, while the linear velocity of
the surface of the outer cylinder was two times higher than the linear velocity of
the inner cylinder. One of the intermediate rotation variants of the outer cylinder