Effect of Air Content on the Internal Fluid Characteristics of Liquid Floated Gyro

With the continuous development of inertial technology, new gyroscopes represented by laser gyroscope, fiber optic gyroscope, and atomic gyroscope are gradually showing up in the civil field. However, compared with the new gyroscopes, the traditional mechanical gyro represented by liquid floated gyro (LFG) has the advantages of high precision, long life, and good reliability [1], which still occupies an important position in aviation, aerospace, and other military high-tech fields [2]. As a core instrument of the inertial guidance and navigation system for the long-range intercontinental strategic missile, LFG is used to realize the angular motion information measurement of the carrier relative to the inertial space, and its accuracy level is the most direct and important index to measure the performance of the inertial system. According to the projection, for each order of magnitude decrease in the drift rate of the gyroscope, the circular probability error of the missile can be reduced by a factor of 10, which is equivalent to increasing the explosive yield of the missile warhead by nearly 100 times. The performance of LFG directly affects navigation positioning, hit or orbit accuracy, and the instrumentation accuracy determines more than 70% of the guidance accuracy of the entire navigation system [3,4]. Therefore, researchers have taken error reduction and accuracy improvement as the direct goal to improve the performance of inertial instruments.

In order to balance the gravity of the gyro floater (a core sensitive part of LFG), the special floating oil with large density and high damping is filled around the floater inside the gyro instrument. In addition, the high damping characteristics of the oil ensure its good physical stability, enhancing the output shaft's interference resistance. When filling the floating oil into the instrument, a certain amount of air will be mixed into the oil by the process level, environmental conditions, and other factors, forming tiny bubbles suspended in the oil. The tiny bubbles dispersed in the ordinary liquid are generally in a non-equilibrium state, and these bubbles gradually merge into a continuous gas phase through slow motion. For floating oil, which are high specific gravity viscous liquids, they are difficult to form a continuous phase in a short period of time [5], and the presence of bubbles may affect the temperature field, flow field, and viscous drag of the oil, thus affecting the drift rate of the instrument. At present, there is no specific report on the effect of bubbles on gyro oil, but some scholars have conducted more studies on the rheological characteristics of bubbling oil–gas two-phase flow. In the 1990s, An et al. of Zhejiang University systematically studied the calibration of air content, rheological properties of oil and gas two-phase flows, and their effects on friction lubrication [6–10]. Wen and Che concluded that flow direction, velocity magnitude, flow pattern, and bubble size are the main factors affecting the distribution of air content in vesicular flow [11]. Zhang et al. conducted an experimental study using turbine oil and nitrogen as materials and found that the viscosity of the oil–gas two-phase flow increased with the increase of air content [12]. Yang et al. concluded that the effect of air content on the viscosity of oil–gas two-phase flows is shear rate-dependent, and the viscosity increases at lower shear rates but decreases at higher shear rates as the air content increases [13]. With the in-depth study of oil–gas two-phase flow, the effect of air content on practical engineering objects such as angle seat valves, mixing pumps, and bearings has been paid more and more attention [14–19]. At the same time, the development of computational fluid dynamics (CFD) simulation technology also provides a powerful tool for the study of similar problems, so that complex heat and mass transfer phenomena can be visualized in the scientific research [20–22]. In summary, the air content has a certain influence on the rheological properties of oil and the working performance of the application object, however, there is a lack of related research on gyro oil. Therefore, this paper adopts the CFD method to establish a fluid–solid conjugate heat transfer model of LFG, based on which the influence law of air content on the dynamic characteristics of gyro oil is analyzed to provide theoretical reference for the manufacture of LFG.

The original assembly model of the liquid floating gyroscope (Fig. 1(a)) was established by modeling software pro/e, following the principle of not changing the basic and important dimensions of the gyro, and ignoring the unimportant features such as chamfers, holes, and slots of the parts, and on this basis, the gyro model was reasonably simplified (Fig. 1(b)). The simplified model was imported into the finite element analysis software cfx, and the mesh module was used to discretize the geometry, and a mesh model with a computational scale of 40 million nodes was obtained (Fig. 1(c)) with an average mesh quality of 0.76.

Material intrinsic parameters and boundary conditions are necessary to obtain a fixed solution for the model field quantities. The main material of the instrument parts is metal special material with high thermal conductivity, good mechanical strength, and stable structure. The gyro floating oil is organic synthetic oil with high viscosity and high density. The physical parameters of the main materials are shown in Table 1. The heat source of the three-floating gyroscope includes the internal gyro motor and the surface insulation resistor with thermal power of 4 W and 1.5 mW, respectively. In addition, the convective heat transfer parameters between the outer surface of the instrument and the ambient air are: ambient temperature of 55 °C and heat transfer coefficient of 20 W/(m² K). The CFD model with complete boundary conditions is shown in Fig. 1(d).

1. Continuity equation

   $∂ρ∂t+∇⋅(ρV)=0$

   (1)

   where ρ is the fluid density, and V is the fluid velocity vector [23,24].
2. Momentum conservation equation

   $DVDt=f−1ρ∇p+μρ∇2V+13μρ∇(∇⋅V)$

   (2)

   where f is the unit volume force, p is the pressure, and μ is the fluid viscosity [23,24].
3. Energy conservation equation

   $ρd(e+V2/2)dt=ρf⋅V+∇⋅(V⋅τij)+∇⋅(λ∇T)+ρc$

   (3)

   where e is the internal energy, c the is external heat source, and τ the is surface stress [23,24].

4. Heat exchange equation

Considering the solution scale and convergence accuracy comprehensively, the equation iteration residual (root-mean-square (RMS)) is selected as the convergence standard, the convergence criterion is RMS ≤ 2 × 10⁻⁵, and the convergence process of the calculation model with air volume fraction of 1% is shown in Fig. 2. As the floating oil flows slowly, the iteration process of the fluid energy equation is relatively long, and it reaches the convergence standard after about 90 iteration steps.

Figure 3 reveals the temperature distribution of the gyro oil at different air content levels, from which it can be seen that: the overall temperature distribution is relatively uniform, the high-temperature area is located on the left side near the sensor and the right side near the torquer, and the low-temperature area is located at the end caps on the left and right sides, which is mainly because the structure of the left and right sides is relatively complex and the heat transfer shows differentiation. On the whole, the air content has no significant effect on the temperature field distribution of the gyro oil under the calculation conditions.

To quantitatively evaluate the effect of air content on the temperature distribution characteristics of the gyro oil, as shown in Fig. 4, the temperature gradients of the oil in the left, middle, and right regions are extracted at different air content levels, which are marked as left ΔT_L, middle ΔT_M, and right ΔT_R, respectively, and the temperature gradient of the whole oil domain is labeled as ΔT_W. Figure 4 shows that the temperature gradient of the oil in each region has a small increase when the air content goes from none to yes, and thereafter there is no significant change as the air content increases, which is consistent with the phenomenon reflected by the temperature cloud map, indicating that the air content has no significant effect on the temperature field of the gyro oil in the smaller range studied in this paper.

According to the fluid rheology model shown in Eq. (6) of Sec. 2.2, the air content has a direct effect on the viscosity of the gyro oil, so the air content will affect the flow field inside the gyro. Figure 5 shows the flow field of the gyro oil at different air content levels, from which it can be seen that the flow field distribution of the gyro oil is basically the same under different air content levels, two vortices are formed in the middle section of the oil, the overall flow velocity is small in magnitude, and the region with relatively large flow velocity is located at the left-right end, which is consistent with the temperature distribution.

To quantitatively evaluate the effect of air content on the flow field distribution of the gyro oil, as shown in Fig. 6, velocity vectors in X, Y, and Z directions are extracted at different air content levels, which are labeled as VX, VY, and VZ, respectively, and the average velocity is labeled as AV. Figure 6 shows that VX, VY, VZ, and AV all decrease significantly with the increase of air content, which is because the oil viscosity increases with the increase of air content by the tension of the internal bubbles and the oil flow need to overcome a greater internal friction. When the air volume fraction increased from 0% to 3%, the average velocity of the oil flow decreased by about 4.15%.

Figure 7 shows the force distribution on the outer surface of gyro oil at different air content levels. From the cloud diagram, the air content has no obvious effect on the surface force. However, these surface forces will form an axial viscous drag torque, which will affect the output accuracy of the gyro.

To quantitatively evaluate the effect of air content on the viscous drag torque, as shown in Fig. 8, the drag torques introduced by oil flow in the left, middle, and right regions are extracted at different air content levels, which are labeled as M_L, M_M, and M_R, respectively, and the total drag torque of the whole oil domain is labeled as M_W. Figure 8 shows that due to the small magnitude of the viscous force, there is no obvious variation of the drag torque with the air content in a single region. The variation curve of the total torque M_W shows that its response to the change of air content is nonlinear. M_W increases significantly when the air content goes from none to yes; when the air volume fraction β ≤ 1%, M_W changes insignificantly; when β increases further, M_W shows an approximately linear increasing trend. It can be seen that the effect of air content on the viscous drag torque of gyro oil is complex, and although it does not produce a cross-order of magnitude effect, it will lead to a slow angular velocity drift of the gyro. The increase of air volume fraction β from 0% to 3% lifts M_W by about 0.02%, which will affect the accuracy stability of the gyro instrument. Therefore, air bubbles should be avoided as much as possible during the oil filling process, or necessary measures should be taken to remove them after they are generated.

In this paper, the fluid–solid conjugate heat transfer model of liquid floated gyro is established by the computational fluid dynamics method, and the effect of air content on the temperature field, flow field, and viscous drag torque of the gyro oil is calculated and analyzed, which can provide theoretical reference for the development of gyros. Under the calculated conditions, the following conclusions are obtained in this paper:

1. The air content has no significant effect on the temperature field distribution pattern of the floating oil inside the gyro.

2. The flow velocity of gyro oil decreases significantly with the increase of air content, and the average velocity decreases by about 4.15% when the air volume fraction β increases from 0% to 3%.

3. There is a nonlinear effect of air content on the viscous drag torque introduced by oil flow, when the air volume fraction β increases from 0% to 3%, the torque increases by about 0.02%, which reduces the instrument accuracy stability to a certain extent.

This work was funded by Anhui University of Technology (Grant Nos. RZ2400001421 and RZ2400002569).

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.