Temperature Field Study on the of Natural Convection and Its Effect on the Critical Speeds in a High Pressure Rotor System

The coupling fluid flow-thermal model of natural convection heat in a High-Pressure Rotor (HPR) of the aero engine was established with Computational Fluid Dynamics (CFD) method. The temperature distributions and flow fields of the HPR with different shutdown time were studied. Then, the temperature distributions of the HPR during the hot start-up with different shutdown time were calculated. Moreover, the effects of hot start-up and shutdown temperature fields on the critical speeds of the HPR were carried out. The results show that the flow rule in the cavity of HPR is different from that in a quadrate cavity and presents diversity and complexity of flow. During 40-90 min of shutdown time, the temperature difference of the HPR is more than 20°C. When the shutdown time reaches to 60 min, the first critical speed of the HPR is decreased by 22.3% from 2256 to 1768 rpm.


INTRODUCTION
Aero-engine rotor system usually works in the conditions of high rotational speeds, non-symmetric and unsteady temperature effect.When the engine is in the cooled state after shutdown, the heat transfer coefficient of natural convection on the upper part of the High Pressure Rotor (HPR) is less than that on the lower part.So the HPR comes into being a strongly non-symmetrical and non-steady temperature field.It results in a larger thermal bending of the HPR.When the aero-engine starts again (hot-start), the HPR forms big vibration amplitudes induced by the nonsymmetrical temperature field.In particular, some accidents of the aero-engine often happen because of the big thermal bending during hot startup.For example, the HPR of Olympus 593# aero-engine had a more serious thermal bending at 1.5 h after shutdown.If the aero-engine is forced to hot start up, a serious rubbing fault between the rotor and stator will be formed in the HPR systems (Zhang, 2008).At present, for the aero-engine rotor system, the problem of hot starting up with thermal bending induced by nonsymmetrical temperature field has been into "the program of the turbine engine structural integrity" (Zhu et al., 2007) and it must be considered in design and test of the aero-engine.To understand thermal vibration response characteristics of the HPR system, we must understand the temperature distributions of the HPR in the condition of natural convection heat transfer.Based on above project background, the natural convective heat transfer inside a certain type of HPR after shutdown is study in this study.
So far, the studies about the natural convection flow and heat transfer focuses on the simply 2D geometric models, such as rectangular enclosures (Girgis, 2000;Sharifa and Liu, 2003;Aounallah et al., 2007;Wang et al., 2007), vertical plates (Kimura et al., 1998;Mendez and Trevino, 2000;Mamun et al., 2008Mamun et al., , 2009) ) and the cavities with porous media (Atmane et al., 2003;Dong and Li, 2004;Basak et al., 2006;Atayılmaz and Teke, 2010;Sankar and Do, 2010).However, these simple models are not fit to study the practical problems, especially for High Pressure Rotor (HPR) of aero-engine.At present, there are not literatures reported the natural convection flow and heat transfer in the real high pressure rotor of aero-engine.
In this study, the CFD model of the HPR is established to study the natural convection flow and heat transfer.The calculated temperature distributions of the HPR at the rotational speed of 1200 rpm are used for the temperature boundary conditions.Based on the Boussinesq hypothesis, the transient velocity and temperature fields in the HPR are firstly studied with different shutdown time under the natural convection.Moreover, the effects of hot start-up and shutdown temperature fields on the critical speeds of the HPR are firstly analyzed.The calculated results provide some theory reference for the thermal deformations and thermal vibration of the HPR.

COUPLING FLOW-THERMAL MODEL OF NATURAL CONVECTION
Mathematical model: In this study, the following assumptions are adopted.The incompressible and Newtonian fluid is employed with the Boussinesq hypothesis.The inertia force is neglected because of high viscosity.Based on above assumptions, the basic equations describing the behavior in the Cartesian frame can be written as follows: The continuity equation can be expressed by: The momentum equations are defined as: The equation of the non-isothermal conservation of energy is given as: Therefore, we use a two-dimensional and axisymmetric model in Fig. 2 to study the natural convection in the HPR due to its geometrical symmetry.
Figure 3 shows the CFD model of the PHR.The mesh is constructed by using the commercial software, Gambit 3.2.It is noted that the grid should be refined in those regions, such as round solid wall and sudden changed region of flow velocity.The Fig. 3 consists of 11,310 cells and 12

In this study, the following
The incompressible viscous uid is employed with the Boussinesq hypothesis.The inertia force is neglected because of high viscosity.Based on above assumptions, the basic equations describing the behavior in the Cartesian continuity equation can be expressed by: (1) The momentum equations are defined as: and y directions, The coefficient of thermal expansion Figure 1 shows the 3D geometric model of the HPR.It can be seen that the cavity of the HPR has very complex and symmetry geometric structure, which is completely different from the simple quadrate cavity in natural convection heat transfer.
dimensional and axisymmetric model in Fig. 2 to study the natural convection in the Figure 3 shows the CFD model of the PHR.The mesh is constructed by using the commercial software, Gambit 3.2.It is noted that the grid should be refined in those regions, such as round solid wall and sudden changed region of flow velocity.The CFD model in 12,442 nodes.The  Temperature at outer surface of top/bottom for the HPR is determined by: where, T f (x, y) are the calculated temperature values at the rotated speeds of 1200 rpm of the outer surface, as shown in Fig. 4.

RESULTS AND DISCUSSION
Velocity and temperature distributions: Figure 5 presents the velocity distributions in the HPR cavity under the natural convection heat transfer.It can be seen that the calculated velocity distributions of the entire computational domain are very complex because of the complex geometry and temperature boundary conditions.Especially, there are many kinds of vortices between adjacent disks in the cavity.This is mainly because the temperatures in the low-level disks are lower than those in the high-level disks and the temperature difference is the thermal source of air flow in the cavity.In addition, the velocities in upper between adjacent disks of the HPR are more complex than those in lower cavity.Velocity vectors in the HPR cavity can clearly show the air flow rules of natural convection.Because of the complexities of the geometrical model and big density of grids, it is necessary to be magnified the local regions to better understand the local velocity vectors.The different local regions in the HPR cavity are shown in Fig. 6.And the velocity vector distributions in those regions are shown in Fig. 7.
Figure 7 shows magnified velocity vectors in the different regions from Region A to F for the HPR cavity.As can be seen from Fig. 7a, the flow characteristics in three adjacent upper disks (such as the fifth, sixth and seventh level disks) are similar.They all form vortices in the three upper disks, but the velocities differ slightly.For example, in the high temperature disk cavity, there are three vortices.But there is only one vortex in the adjacent low temperature zone.Figure 7b shows velocity vector distribution in the cavity between first and secondary level disks.When the air flows in the small flow region, most of air flows in the third and fourth level disks and forms the left and right vortices.And there is only a small amount flowing into the small gap of the high pressure rotor.A typical oval eddy between the upper and lower surfaces of the HPR cavity can be observed in Fig. 7c, in which the flow characteristic is similar to the natural convection in a square cavity (Girgis, 2000).In other region such as region D, E and F, the flow characteristics in those regions are similar to above analysis.In short, the HPR has complex geometry and non-linear temperature boundary conditions changed along the axis.This results in the rules of natural convection transfer heat in the HPR are very complex and a lot of vortex are form, which completely different from the simple square cavity in natural convection (Sharifa and Liu, 2003), showing the diversity and complexity of flow vectors.
Figure 8 shows the transient temperature distributions on top of the HPR with different shutdown time.At the beginning of shutdown in Fig. 8a and b, the temperature isolines are high density and temperature values of the HPR are relatively high.Especially, the third-level and fourth-level disks have large temperature gradient.With increase of shutdown time in Fig. 8c and d, the temperature values and temperature gradients of the HPR gradually decrease.It indicates that the temperature distributions of the HPR are becoming more uniform.By comparison, it is found that the rates of temperature decrease in the third and fourth level disks are relatively small.This is because the velocities and convective heat transfer coefficient in third and fourth level disks are relatively small.The temperature changes of different positions with different shutdown time are illustrates in Fig. 9.It is observed that the temperatures in all nodes gradually decreased when the shutdown time increases.Meanwhile, the first level disk has the slowest cooling speed and sixth level disk has the fastest cooling speed in all of nodes of the HPR.When the shutdown time goes to 90 min, the cooling speed of each node of the HPR becomes slow.But the first and second level disks still have relative big temperature difference because of the initial high temperatures.At last shutdown time, the temperatures in all nodes of the HPR are equal to the room temperature, which illuminates the heat exchange tends to balance.
Figure 10 depicts the temperature difference between upper and lower surfaces of the HPR with different shutdown time.We can see that the temperatures at top and bottom surfaces of the HPR is not simplified increased with the increase of shutdown time, but show the phenomenon of initial increasing and then decreasing.Meanwhile, the maximum temperature difference between the top and bottom surface is about 23°C at shutdown time of 60 min for the HPR and the minimum temperature difference between the top and bottom surface is about 4.8°C on the sixth level disk at about 15 min.In addition, the temperature difference between the top and bottom surface of the first level disk is relative big and more than 20°C during the shutdown time from 40 to 90 min.It shows a good agreement with the actual measurement values (Zhang, 2008) and verifies the correctness of the simulations in this study.So the HPR come being into relatively strong asymmetric temperature field, which will must change the nonlinear vibration of the HPR system.
Figure 11 shows the temperature curves of different nodes during hot-start process after shutdown time of 300 and 1200 sec.As can be seen from Fig. 10a, when the HPR startups after the shutdown time of 300 sec, the temperature values of the HPR decrease and then gradually increased with the increase of time.The minimal temperature values of different nodes also correspond to the different startup time.Meanwhile, the minimal temperature value of first level disk corresponds to the longest startup time of about 35 sec and that of fifth level disk corresponds to the shortest time of 12 sec.When the shutdown time increases to 1200 sec, as shown in Fig. 11b, the temperature values of the first and second level disks decrease firstly and then gradually increased with the increase of startup time.And other nodes are all increased gradually with the increase of startup time.As the shutdown time increases, the temperature change has tended to balance and the temperature values of all nodes for the HPR equal to room temperature.
Effect of temperature field on critical speeds of the HPR: From above analysis, we have known that the HPR has great temperature changes during the frequent starting and shutdown processes.At the same time, the temperature distributions in each axis section of the HPR are not same each other.Because of the uneven temperature distribution of the HPR, the thermal stresses come into being and lead to some thermal distortion, which results in the change of rigidity matrix and influences the vibration characteristics of the HPR.According to D'Alembert's principle, the bending vibration equation of rotor system under the action of temperature field can be described as (Zhu et al., 2010): Critical speeds (rpm) ----------------------------------------------------------------------- where, [M] = Mass matrix of the rotor system [C] = Damping matrix including gyroscopic effect [K] = Stiffness matrix of the rotor system [K 2 ] = The stiffness matrix induced by the thermal bending {P (t)} = The thermal stress vector {U}, {ˡ{ Ӕ and ˡ ӕ = The displacement matrix, velocity matrix and acceleration matrix, respectively Figure 12 shows the three dimensional finite element models of the HPR to calculate the rotor dynamics.The 3D element with 20 nodes is used to mesh the three-dimensional HPR.The FE model of the HPR has 8635 elements and 15874 nodes.In this section, firstly, the steady temperature field of the HPR is calculated based on FEM.Secondly, the thermal stress distributions of the HPR with different shutdown time are studied by using thermal-structure coupled theory.At last, the thermal stress effect reset into the stiffness matrix, the thermal-structure-dynamics coupling equations are employed to study the thermal vibration performance of the HPR.
Table 1 shows the effects of temperature fields with different shutdown time on the critical speeds of the HPR system.During 0~60 min, the critical speeds gradually decrease with the increase of shutdown time due to the increase of temperature difference in the HPR.As the critical speed orders increase, the effects of temperature fields on the critical speeds decrease gradually.When the shutdown time reaches 60 min, the first critical speed decreases by 22.3%.

CONCLUSION
The CFD model of natural convective heat transfer in the HPR is established in the shutdown process.The velocity and temperature field distributions are calculated with different shutdown time.The results show that the HPR has high temperatures at the beginning of shutdown, especially for the third and fourth level disks.With increase of shutdown time, the temperature values in the HPR decrease gradually.When the shutdown time are during 40~90 min, the temperature difference between the top and bottom surface of the HPR is more than 20°C.
The transient temperature distributions in the HPR are studied during hot starting process after different shutdown time.When the shutdown time is relatively short, temperature values in the HPR initially increase and then decrease.The temperature values of different nodes have different minimal temperature values corresponding to the different shutdown time.Meanwhile, the third-level disk has maximal startup time when its temperature value is lowest, while the eight-level disk has minimal startup time.
The temperature fields with different shutdown time have great effects on the critical speeds of the HPR.During 0~60 min, the critical speeds of the HPR gradually decrease with increase of shutdown time.As the critical speed orders increase from first to fifth order critical speed, the effects of temperature fields on the critical speeds decrease gradually.When the shutdown time reaches 60 min, the first critical speed reduces by 22.3%.
Figure1shows the 3D geometric model of the HPR.It can be seen that the cavity of the HPR has very complex and symmetry geometric structure, which is completely different from the simple quadrate cavity in natural convection heat transfer.

Table 1 :
Effects of temperature fields with different shutdown time on the critical speeds