3.3. Comparison with the Previous Method
As mentioned in
Section 1 and
Section 3.2, in the previous research into the fluid lubricated screw-nut pairs, the flow fields were solved based on the simplified Reynolds equation in MATLAB. When the Reynolds-averaged Navier–Stokes equation was deduced from the Navier–Stokes equation, the Reynolds stress was neglected, for convenience. Meanwhile, the inertia term and the viscous force term were ignored or greatly simplified. Furthermore, the momentum equation in the direction of the gap thickness were also neglected, due to the narrow size of the clearance.
In addition, the solving pattern of the flow equations in the previous method is opposite to that in the new method. The field was solved based on the predefined motions according to the kinetic equilibrium theory. Meanwhile, the influences of the screw and nut on the field were not considered.
A qualitative comparison between the method proposed in this paper and the previous method was made in
Section 1 and
Section 3.2. In order to show the drawbacks in the previous method more intuitively, a quantitative contrastive analysis was carried out in this section.
In line with the simplification operations described above, the simplified flow equations of the system in this paper are given in Equations (17) and (18).
Under different loading conditions, the fluid domain built in
Section 2 was solved based on Equations (17) and (18) in MATLAB. The angular velocity of the screw
was set as
, and the oil supply pressure
was
.
As shown in
Table 3 and
Table 4, the simulation results using the previous method are compared with those based on the new method under different loading conditions. Method “a” and “b” represent the newly proposed method in this paper and the previous method based on the simplified Reynolds equation in MATLAB, respectively.
is the external load,
is the variation of the oil film thickness.
and
are the upper and lower recess pressures, respectively.
In
Table 3,
is nearly
in both Method “a” and “b”, when a non-loaded system is solved. In Method “a”,
fluctuates dynamically around
, and the recess pressure also fluctuates around a value, rather differently from
due to the restriction from the restrictor. However, while using Method “b”, an absolutely non-loaded state is obtained based on
, and the recess pressures are very close to
, resulting in the fact that the restriction from the restrictors are almost not reflected. The simplification of the flow equations in Method “b” is responsible for such a phenomenon, and the flux out of the recess obtained using Method “b” is far less than that in the actual condition. Therefore, a higher recess pressure is obviously brought about by less flux, when the field is still solved according to the flow conservation. Correspondingly, the pressures of the whole field are also larger than those in the actual condition.
It is noteworthy that in Method “a” the approximately equal signs before the pressures and do not mean that the values are constant, but represent that the fact that the pressures and fluctuate around those values. However, in Method “b”, the approximately equal signs before the pressures mean the rounding of the constant values.
The simulation results using the two methods under loading conditions are shown in
Table 4. Considering the difference of the solution methods, the simulation results at
under Case “1” were selected to be compared with the eventual results under Case “2” and “3” in this section. The meanings of the approximately equal signs in the two methods are similar to those in
Table 3.
As shown in
Table 4,
,
, and
under Case “1” fluctuate around some values eventually, when an external load of
is applied to the system for approximately
using Method “a”. However, for the same
, there is only a very small external load applied to the system, according to the results obtained using Method “b” under Case “2”. And the external load obtained using Method “b” is a constant. As mentioned above, the approximately equal sign before the load value means that the calculation result has been rounded. Furthermore, the difference between
and
obtained is very tiny, and they are still very close to
. That is to say, a much larger external load is needed in Method “a” to obtain the same
as Method “b”, which can be accounted for by the minor restriction from the restrictors in Method “b”. As mentioned in
Table 3, a lesser role is played in Method “b” by the restrictors, owing to the significantly reduced flux, which is a result of the simplification of the flow equations. Correspondingly, in Method “b”, the upper and lower oil film pressures obtained based on a given
are both larger than the actual pressures, hence the variations of the pressures are smaller, especially when
is very tiny. Therefore, in
Table 4, compared with Case “1”, there is a lesser difference between the upper and lower oil film pressures under Case “2”.
Judging from
Table 4, when an external load of approximately
is applied to the system, the variations of the recess pressures obtained using Method “a” under Case “1” are much larger than those under Case “3”. However, a larger
is produced using Method “b” under Case “3”. As mentioned above, the restrictor has a minor effect on the flow in Method “b”, leading to small variations of the oil film pressures, hence the immunity of the system against the external load is lower than that in Method “a”.
In order to show the differences between the two methods more intuitively, the pressure fields under Case “1”−“3” were compared in the circumferential direction and the direction of the oil film thickness. The pressure contours of the lower surface of the screw under Case “1”−“3” are displayed in
Figure 9. The pressure field obtained in FLUENT using Method “a” is shown in (a), and the pressure distributions obtained in MATLAB using Method “b” under Case “2” and “3” are shown in (b) and (c), respectively.
The differences of pressure distributions and pressure values between
Figure 9b,c is too slight to be distinguished in the figures, the reason for which has been explained in
Table 4. The overall trend of the pressure distribution in
Figure 9a is similar to those in
Figure 9b,c. Apart from the difference of pressure value, it can be found that the pressure distribution on the inner side of the recess in
Figure 9a is slightly different from that on the outside, unlike the symmetric distribution around the recess in
Figure 9b,c. Although the diffusions to the edges are promoted by the pressure gradients, the influence of the central force on the diffusion on the outside of the recess is opposite to that on the inner side. Hence, the pressures of the outside of the recess decrease faster than those of the inner side in
Figure 9a. However, the difference is not reflected in the results obtained using Method “b”.
As mentioned above, the pressure and velocity gradients in the -axis direction were neglected, due to the narrow gap while using Method “b”. In order to display the shortcomings brought about by such an operation more clearly, the oil film pressures obtained using the two methods were compared in the direction of the oil film thickness.
The fluid particles, which had the same angular and radial coordinates at some time, were selected as the research objects. That is to say, they were on the same normal line of the screw surface. As the length scale in the direction of the oil film thickness was very small, the differences among the pressures of different particles were also very small when compared with the pressure values. In order to show the variation of the pressures in this direction more obviously, a normal line going through the lower recess was selected, owing to its larger range of the flow field covered than those in the sealing surfaces. The line on which the fluid particle had a position of “”, was taken as an example. The oil film pressures acting on the particles were then compared between the two methods.
Owing to the neglect of the pressure gradient in the direction of the oil film thickness, the pressures were equal in this direction while using Method “b”. That is to say, the pressures of all the particles on the selected line were equal. Moreover, the pressures of all the particles in the recess were also the same in Method “b”. As mentioned in
Table 4, the recess pressures under Case “2” and “3” were constants close to the oil supply pressure, and it was found in the simulation that the differences between the upper and lower recess pressures were very tiny, due to the poor influence of the restrictors in Method “b”. However, while using Method “a”, the pressures acting on the particles on the line were different and variable. Therefore, the relation curve between the pressure and the position of the fluid particle on the line is shown in
Figure 10, in order to show the variation of the pressures in the direction of the oil film thickness in Method “a”.
Taking the lower surface of the screw as a reference, the distance from any particle with a position of “
” in the gap to the surface was set as the independent variable. And the way in which the pressures acting on the particles vary with distance is displayed in
Figure 10. An irregular variation is shown by the pressures. And it was found in the simulations that the curve varied irregularly over time due to the turbulent fluctuation. The density of data points is related to the distribution of the grid nodes; therefore, the data points are more densely distributed near the narrow gap than the recess. Compared with the constant pressures in the
-axis direction in Method “b”, the variation of the pressures in Method “a” can reflect the flow field more accurately.
Through the above analyses, the method proposed in this paper was obviously superior to the previous method in illustrating the field.