4.0 Introduction
The critical load-bearing structural components of the compressor and turbine in a gas turbine engine are essentially airfoils (vanes and blades) and disks, which rotate at high angular velocity with the blades mounted on them; the vanes are mounted on the static structure. Although the blades directly participate in energy conversion in the primary flow paths of compressors and turbines, the disks are exposed to internal cooling and sealing flows. The failure of a rotor disk with its extremely high rotational kinetic energy is considered catastrophic for the entire engine. In Chapter 3, we discussed an arbitrary duct and orifice being the two core flow elements of an internal flow system and presented their 1-D flow modeling as needed to assemble a flow network model, including its robust numerical solution method. In this chapter, we will expound on some novel concepts and flow features associated with gas turbine internal flows over rotor disks and in cavities formed between a rotor disk and either another rotor disk or a static structure. As in Chapter 3, we continue here our emphasis on the 1-D modeling of disk pumping flow, swirl and windage distributions, and centrifugally-driven buoyant convection in compressor rotor cavities with or without a bore cooling flow.
Most of the concepts presented in this chapter with good physical insight are by and large outside the mainstream of thermofluids education at the senior undergraduate and graduate levels in most universities around the world. Nevertheless, these concepts are critically important for the design and analysis of gas turbine internal flow systems, for example, to design rim seals to minimize, or to prevent, hot gas ingestion; to develop an optimum preswirl system for the turbine blade cooling air; and to accurately compute rotor axial thrust for sizing the thrust bearing.
Although the references list a number of leading references on various topics covered in this chapter, readers may refer to Owen and Rogers (1989, 1995) and Childs (2011) for a comprehensive bibliography, particularly related to free disk, rotor-stator, and rotor-rotor systems.
4.1 Rotor Disk
In a gas turbine, compressor and turbine blades are mounted on rotor disks. These disks must have acceptable temperature distributions to ensure their structural integrity during both steady and transient engine operations. Unless the coolant flow over the disk co-rotates at the same angular velocity as the disk, it gets pumped in the disk boundary layer; radially outward if the flow rotates slower than the disk and radially inward if it rotates faster than the disk, as physically explained later in this chapter. In the following sections, we discuss two disk pumping situations. In one, the free disk pumping, the free-stream air next to the rotating disk is stagnant. In the second, the disk pumping beneath a forced vortex, the flow is co-rotating at a constant angular velocity, which is a fraction of the disk angular velocity.
4.1.1 Free Disk Pumping
Figure 4.1 depicts the boundary layers of radial and tangential velocities for a disk rotating in a quiescent fluid far away from the disk. The growth of the radial velocity boundary layer from r = 0r=0 to r = Rr=R occurs through fluid entrainment via the axial velocity. Note that the free rotating disk flow features zero radial pressure gradient imposed from the adjacent stagnant fluid outside the boundary layers. In the same way as the flat pate, the boundary layers in this case become turbulent for Re(r) = ρr2Ω/μ>3 × 105Rer=ρr2Ω/μ>3×105 – the local rotational Reynolds number.
Figure 4.1 Free disk pumping.
Schlichting (1979) presents von Karman momentum integral boundary layer solutions for both laminar and turbulent boundary layers in a free rotating disk. Here we present the key results only from the turbulent boundary solutions. For the one-seventh power law profile in the boundary layer, assuming turbulent boundary layer right from r = 0r=0, the fee disk pumping mass flow rate is given by
(4.1)which may be alternatively written as
(4.2)where Re = ρR2Ω/μRe=ρR2Ω/μ is the disk rotational Reynolds number based on the disk radius RR. Although a free rotating disk is not found in a gas turbine design, Equations 4.1 or 4.2 estimates the upper limit for the disk pumping flow. Note that, for a laminar boundary layer, the disk pumping flow rate varies as r2r2, see for example Schlichting (1979). Because the disk area also varies as r2r2, the axial velocity of entrainment remains uniform over the disk. For the turbulent boundary layer, however, the disk pumping flow rate according to Equation 4.2 varies as r2.6r2.6, as a result, the axial velocity of the entrained flow increases with radius for an incompressible flow.
Another design parameter of interest is the torque produced by the tangential wall shear stress distribution on the rotor disk. This torque for one side of the disk wetted by the fluid is given by
(4.3)which from the solution, expressed in terms of the moment coefficient, yields
(4.4)Because the disk is rotating with angular velocity ΩΩ, the total disk torque computed from Equation 4.3 will impart windage equal to ΓΩΓΩ to the boundary layer flow (pumping flow).
4.1.2 Disk Pumping Beneath a Forced Vortex
Figure 4.2a shows the boundary layer flows on a rotating disk when the fluid outside the boundary layer itself rotates as a forced vortex at a fraction of the disk angular velocity. The ratio of fluid core angular velocity and the disk angular velocity is represented by the swirl factor SfSf. At Sf = 0Sf=0, the flow field of Figure 4.2a reverts to that of the free rotating disk shown in Figure 4.1 and yields the maximum pumping flow rate in the boundary layer. When Sf = 1Sf=1, the fluid gets into solid-body rotation with the disk with no pumping flow.
Figure 4.2 (a) Disk pumping beneath a forced vortex and (b) fraction of free disk pumping mass flow rate versus swirl factor.
Newman (1983) extends the momentum integral method of von Karman, presented in Schlichting (1979) for a free rotating disk, to cases for which the outer flow is rotating at a constant angular velocity. Boundary layers for both radial and tangential velocities are assumed turbulent right from r = 0r=0. From the solution results presented by Newman (1983), we obtain the following formula to compute disk pumping mass flow rate for the one-seventh power law velocity profile assumed in the boundary layer:
(4.5)where ζζ, given by Equation 4.6, is the fraction of the free disk pumping mass flow rate computed by Equation 4.1.
As shown in Figure 4.2b, ζζ depends strongly on the swirl factor SfSf, yielding the maximum value of the disk pumping mass flow rate for Sf = 0Sf=0 and no pumping for Sf = 1Sf=1. Equation 4.5 is a useful design equation to estimate pumping flow rate on a rotor surface between two radii.
4.1.3 Rotor Disk in an Enclosed Cavity
Figure 4.3 shows a disk rotating in a stationary housing with zero inflow and outflow. As a result of no-slip boundary conditions, the fluid assumes local velocity of the rotor and stator in contact. The rotor disk acts like a bladeless compressor or a pump (incompressible flow) and pumps the fluid radially outward. Because the disk pumping flow increases radially outward, it is continuously being fed (axial flow from the stator boundary layer to the rotor boundary layer) from the radially inward flow along the stator. The presence of the stationary shroud makes the stator torque somewhat higher than the rotor torque. As a result, the fluid core is expected to rotate at less than half of the disk angular velocity (Sf < 0.5Sf<0.5). As the gap between the stator and the rotor decreases, the fluid core will tend to rotate at Sf = 0.5Sf=0.5. The radial static pressure gradient in the enclosure will be established corresponding to the forced vortex with Sf = 0.5Sf=0.5.
Figure 4.3 Rotor disk in an enclosed cavity.
Although the torque on the stator surface does not do any work, there is continuous work transfer into the fluid from the rotor disk. As a result, the fluid temperature within a perfectly insulated enclosure will rise continuously.
4.2 Cavity
The rotor-rotor and rotor-stator cavities are the most dominant and ubiquitous features of internal flow systems of gas turbines. Assuming a turbulent cooling and sealing flow in these systems, the interplay of flow behavior on a rotor surface, a stator surface, and the mass flow rate associated with radially outward or inward flow is responsible for a variety of flow features found in theses cavities. A good understanding of these flow features is the key to their one-dimensional modeling for the flow network simulation of these internal flow systems.
A rotor surface tends to pump the flow radially outward and acts like a bladeless compressor if the adjacent fluid core rotates at a fraction of the disk angular velocity and like a bladeless turbine if the fluid core co-rotates faster than the disk. In the first case, the energy transfer occurs from the rotor disk to the fluid and in the second case from the fluid to the disk. On the disk itself, tangential velocity varies linearly with radius. Stator torque acts to reduce the angular momentum of the flow regardless of the flow direction and rotation. The stator does not partake directly in the energy transfer to or from the fluid. For a small flow influenced by rotor and stator torques, the core behaves like a forced vortex rotating at a fraction (around 0.5) of the rotor disk angular velocity. For a large flow, which is not influenced by rotor and stator torques, the flow behaves more like a free vortex, keeping a nearly constant angular momentum. In this case, the angular velocity of a radially outward flow decreases downstream and for a radially inward flow increases in the flow direction, at times exceeding the rotor angular velocity.
In general, a fluid flow seeks the path of least resistance. In a rotating flow, the difference between the angular velocity of the flow and that of the wetted wall determine the torque. If the wall rotates faster than the fluid, it will increase the flow angular momentum. If the wall angular velocity is less than that of the fluid, the torque produced will decrease the flow angular momentum. Accordingly, the stator torque always reduces the angular momentum of the adjacent fluid flow.
4.2.1 Rotor-Stator Cavity with Radial Outflow
Figure 4.4 shows a rotor-stator cavity with a superimposed radial outflow. For a small outflow rate, shown in Figure 4.4a, the flow streamlines, fully meeting the demand of the induced pumping flow, are along the rotor surface. At a radius where the disk pumping flow rate exceeds the superimposed flow rate, the fluid is entrained from the radially inward flow induced on the stator surface to make up for the difference, as shown in the figure. For a large superimposed radially outflow, exceeding the disk pumping flow, no radially inward flow on the stator surface occurs, as shown in Figure 4.4b.
Figure 4.4 Schematic of a rotor-stator cavity with superimposed radial outflow: (a) small outflow rate and (b) large outflow rate.
4.2.2 Rotor-Stator Cavity with Radial Inflow
Figure 4.5 shows a rotor-stator cavity with a superimposed radial inflow. For a small inflow rate with Sf < 0.5Sf<0.5, shown in Figure 4.5a, the flow enters the cavity along the stator surface. Some of this flow is peeled off by the rotor to satisfy its pumping flow requirement, featuring a flow reversal over a part of the cavity near the rotor surface. At a lower radius, the flow starts swirling faster like a free vortex and preferably migrates to descend down the rotor surface so as to minimize the overall wall shear force opposing it. This part of the rotor disk, where the fluid is flowing radially inward, acts like a bladeless turbine.
Figure 4.5 Schematic of a rotor-stator cavity with superimposed radial inflow: (a) small inflow rate and (b) large inflow rate.
In case of a large radial inflow, shown in Figure 4.5b, the flow behaves more like a free vortex and preferentially flows down the rotor surface so as to minimize the overall shear force. The entire rotor disk in this case behaves like a bladeless radial turbine with work transfer from fluid to the rotor.
4.2.3 Rotating Cavity with Radial Outflow
Figure 4.6 shows the complex shear flow streamlines of a radial outflow in the cavity between two rotating disks. The axial flow entering the cavity through the upstream disk undergoes a sudden geometric expansion. The growth of the outer shear layer of the annular jet occurs through entrainment of the pressure-gradient-driven backflow from the downstream stagnation region. This creates the primary recirculation region shown in the figure. The size and strength of this recirculation region are found to depend mainly on the flow rate and rotational speed as discussed in Sultanian and Nealy (1987). The entering axial flow turns 90 degrees over the concave corner and flows radially outward, aided in part by frictional pumping over the downstream disk induced by its rotation. A part of the flow (almost half in this case!) turns back toward the upstream disk and moves radially outward as a result of similar pumping action over that disk.
Figure 4.6 Schematic of a rotating cavity with superimposed radial outflow.
4.2.4 Rotating Cavity with Radial Inflow
In the rotating cavity shown in Figure 4.7, we have a radial inflow. Like the case of radial outflow, the rotating cavity features essentially four regions. Both the source region at the inlet and sink region at the outlet are complex shear flows. The core in the mid-section of the cavity features nearly zero axial and radial velocities, and it is bounded by Ekman boundary layers on both disks. These boundary layers are essentially nonentraining. In essence, the flow entering the rotating cavity splits almost in half and flows down radially on each disk with no intermediate entrainment. Because we normally associate disk pumping with a radially outward flow in the disk boundary layer, the flow features shown in Figure 4.7 may appear somewhat counterintuitive to some.
Figure 4.7 Schematic of a rotating cavity with superimposed radial inflow.
4.3 Windage and Swirl Modeling in a General Cavity
A major task in the design of gas turbines is to compute windage and swirl distributions throughout the path of an internal flow system. These distributions are needed to determine the thermal boundary conditions for structural heat transfer analysis and for establishing static pressure distributions for axial load calculations. Figure 4.8 shows the schematic of a general gas turbine cavity and its key features. This cavity includes multiple axisymmetric surfaces, which may be rotating, co-rotating, counter-rotating, or stationary. Each disk surface may comprise of radial, conical, and horizontal surfaces; for example, shown for surface 2 in the figure, and may feature three-dimensional protrusions, called bolts, which tend to destroy the overall symmetry of the cavity about the axis of rotation. Additionally, the cavity may have multiple inflows and outflows with different swirl, pressure, and temperature conditions.
Figure 4.8 Schematic of a general gas turbine cavity and its key features.
Because the flow field in the general cavity shown in Figure 4.8 is highly complex and three dimensional, a 3-D CFD appears to be the only viable analytical method for its analysis and predictions. Such an analysis may not, however, support the shrinking design cycle time and realizing a robust design requiring multiple runs to account of statistical variations in boundary conditions. In the following sections, we present a 1-D flow modeling methodology based on the large control volume analysis for a general gas turbine cavity encountered in design. Because the methodology uses some of the published correlations to compute torque of stator and rotor surfaces, it behooves the readers (designers) to modify them in their design applications based on their design validation studies.
Daily and Nece (1960) studied, both experimentally and theoretically, the fundamental fluid mechanics associated with the rotation of a smooth plane disk enclosed within a right-cylindrical chamber, as shown in Figure 4.9. In this investigation, the torque data were obtained over a range of disk Reynolds numbers from Re = 103Re=103 to Re = 107Re=107 for axial clearance to disk radius ratios from G = 0.0127G=0.0127 to G = 0.217G=0.217 for a constant small radial tip clearance; the velocity and pressure data were obtained for both laminar and turbulent flows. The tangential and radial velocity profiles are schematically shown in Figure 4.9a for the case of merged boundary layers and in Figure 4.9b for the case of separate boundary layers.
Figure 4.9 Rotor disk in an enclosed cavity: (a) merged boundary layers (Regimes I and III) and (b) separate boundary layers (Regimes II and IV).
The study of Daily and Nece (1960) identifies the existence of the following four basic flow regimes, which are delineated in Figure 4.10 for various combinations of ReRe and GG. The rotor disk moment coefficient in each regime is summarized as follows:
Regime I: Laminar flow with merged boundary layers (small clearance)
Regime II: Laminar flow with separate boundary layers (large clearance)
Regime III: Turbulent flow with merged boundary layers (small clearance)
Regime IV: Turbulent flow with separate boundary layers (large clearance)
Figure 4.10 Delineation of four flow regimes in the flow of a disk rotating in an enclosed cavity (Daily and Nece, 1960).
Regime IV is generally considered relevant for gas turbine design applications. For our 1-D modeling of a general cavity, we make use of the rotor and stator moment coefficients proposed by Haaser, Jack, and McGreehan (1988) and extend them for partial disks, which may be co-rotating and counter-rotating with arbitrary angular velocities.
Based on the actual gas turbine test experience and the experimental data of Daily and Nece (1960); Haaser, Jack, and McGreehan (1988) proposed the empirical correlation for the shear coefficient on one side of the rotor disk as
and that on one side of the stator disk as
where the full disk Reynolds number (Re=ρΩRo2/μ
) has been modified to
(4.13)in order to use the correlations for a partial disk (Ri>0Ri>0). Note that Equations 4.11 and 4.12 have been deduced from the following moment coefficient correlations assumed for the full disk by assuming a uniform radial distribution of the tangential shear stress. Under this assumption, the shear coefficient and moment coefficient are related as follows:
(4.14)Using Equation 4.14, we obtain the moment coefficient equation from Equation 4.11 for the partial rotor disk as
(4.15)for the full rotor disk as
for the partial stator disk as
(4.17)and for the full stator disk as
Equations 4.15 and 4.16 for the rotor disk or Equations 4.17 and 4.18 for the stator disk of outer radius RoRo yield the following relation for the ratio of the torque for a partial disk with Ri>0Ri>0 to that for a full disk with Ri = 0Ri=0:
(4.19)The plot of Equation 4.19 in Figure 4.11 shows that for Ri/Ro ≤ 0.5Ri/Ro≤0.5, the equation yields Γpartial>ΓfullΓpartial>Γfull, which is physically unacceptable. To mitigate this problem, we make the assumption that, instead of a constant tangential shear stress over the disk, as assumed in Haaser, Jack, and McGreehan (1988), the local shear coefficient of the tangential shear stress is constant, giving
(4.20)
Figure 4.11 Variation of disk torque ratio (Γpartial/ΓfullΓpartial/Γfull) with radius ratio (Ri/RoRi/Ro).
Thus, from Equations 4.16 and 4.20, we obtain for the rotor disk
(4.21)Similarly, from Equations 4.18 and 4.20, we obtain for the stator disk
(4.22)Based on Equations 4.21 and 4.22, we obtain for rotor and stator disks the following relation
(4.23)which is plotted in Figure 4.11. The figure shows that the anomaly associated with Equation 4.19 is absent from Equation 4.23.
In using Equations 4.21 and 4.22 for calculating local tangential shear stress on the rotor surface and stator surface, respectively, one is expected to use the dynamic pressure 0.5ρΩ2r20.5ρΩ2r2. For a general 1-D modeling of cavities, it is more appropriate to use the fluid tangential velocity relative to the surface to compute the dynamic pressure, which for the rotor becomes 0.5ρ(1 − Sf)2Ω2r20.5ρ1−Sf2Ω2r2 and for the stator 0.5ρSf2Ω2r2
. Accordingly, Equations 4.21 and 4.22 are recast as follows:
where Re=ρRo2Ω/μ
.
Let us now consider the 1-D steady adiabatic flow modeling in a simple rotor-stator cavity shown in Figure 4.12a. With ṁin=ṁout=ṁ
, the steady continuity equation in the cavity is satisfied. Because the flow is assumed adiabatic, the change in fluid total temperature occurs entirely as a result of work transfer from the rotor. The stator torque participates in the torque-angular momentum balance only but not directly in the energy transfer with the fluid. To capture accurate variations of flow properties in the cavity, we divide it into a number of control volumes. For the control volume k whose inlet surface is designated by j and the outlet surface by j+1, we write the following angular momentum equation:
(4.26)
Figure 4.12 Cavity with throughflow: (a) rotor-stator cavity, (b) cavity of co-rotating disks, and (c) cavity of counter-rotating disks.
Assuming a forced vortex core with swirl factor SfkSfk such that Sfj + 1 = SfkSfj+1=Sfk and substituting
and
in Equation 4.26, we obtain
(4.27)which is a transcendental equation in the unknown SfkSfk.
Note that in the marching solution from the cavity inlet to outlet we have Sfj = Sfk − 1Sfj=Sfk−1, which is obtained from the solution for the upstream control volume. One can use the regula falsi method, presented, for example, in Carnahan, Luther, and Wilkes (1969), as a robust and fast iterative solution technique for obtaining SfkSfk from Equation 4.27; see also Appendix D.
Knowing SfkSfk in the control volume, the static pressure change from inlet to outlet can be obtained using the radial equilibrium equation
(4.28)Using an average value of density ρ¯=0.5ρj+ρj+1
for the control value, Equation 4.28 can be integrated to yield
Thus, the change in fluid total temperature as a result of windage in the control value can be obtained using the equation
In the foregoing derivations, we have tacitly assumed that Sf < 1Sf<1 in which case the work transfer occurs from the rotor disk to the fluid. For Sf>1Sf>1, however, the fluid does work on the rotor. In the following section, we will account for this possibility in the modeling of a general cavity with arbitrary inflow conditions.
Figures 4.12b and 4.12c depict a cavity with two rotor disks, which are either co-rotating or counter-rotating. If we set Ωref = 0Ωref=0, these cavities revert to that of Figure 4.12a. Considering the rotor with the highest angular velocity as the reference rotor in a multirotor cavity and using its angular velocity (ΩrefΩref) to normalize other rotor and fluid angular velocity, we can easily extend Equation 4.24 to express the local shear coefficient for any rotor in the cavity as
where
Note that the local dynamic pressure to be used in conjunction with Equation 4.29 equals 0.5ρβ−Sf2Ωref2r2
. Further note that Equation 4.29 is applicable to all rotors in the cavity, including the reference rotor with β = 1β=1.
For the stator surface, we re-write Equation 4.25 as
4.3.1 Arbitrary Cavity Surface Orientation: Conical and Cylindrical Surfaces
A cavity may have a rotor or stator disk comprising conical and cylindrical surfaces. For the conical part of a rotor disk we use the shear stress coefficient correlation given by Equation 4.29, and if the conical surface is a part of a stator disk, we use Equation 4.30 to compute its local shear coefficient. For the conical surface segment of the rotor disk, shown in Figure 4.13a, we express its torque as
(4.31)where
Figure 4.13 (a) Disk with a conical surface and (b) disk with a cylindrical surface.
For the corresponding conical surface segment of a stator disk, we express its torque as
(4.32)where
For the cylindrical rotor surface segment shown in Figure 4.13b, we adopt the shear stress coefficient correlation proposed by Haaser, Jack, and McGreehan (1988), extended for a rotor in addition to the primary rotor
where
Using Equation 4.33, the torque of the cylindrical surface segment of the rotor can be obtained as follows:
(4.34)For the corresponding cylindrical surface segment of a stator disk, we extend Equation 4.12 for the shear coefficient to the generalized form
which yields the corresponding torque as
(4.36)
4.3.2 Bolts on Stator and Rotor Surfaces
Bolts are three-dimensional protrusions on rotor and stator surfaces. They significantly influence both the windage generation and swirl distribution in the cavity. The bolt-to-bolt spacing has a profound effect of its drag force. As the one bolt falls in wake of its upstream bolt, relative to the tangential flow velocity, its drag contribution decreases. Figure 4.14a shows bolts on a disk with small bolt-to-bolt interference, while the increased number of bolts shown in Figure 4.14b result in higher bolt-to-bolt interference.
Figure 4.14 (a) Disk with bolts with small interference and (b) disk with bolts with large interference.
Following the approach of Haaser, Jack, and McGreehan (1988), the torque as a result of bolts in an axisymmetric cavity is computed as follows:
Bolts on rotor surface:
(4.37)Bolts on stator surface:
(4.38)where
Nb≡Nb≡ Number of bolts
h ≡ Bolt height from the disk surface
b ≡ Bolt width along the radial direction
CDb≡CDb≡ Baseline drag coefficient of each bolt (≈ 0.6)
Rb≡Rb≡ Bolts pitch circle radius
Ib≡Ib≡ Bolts interference factor (a function of s/ds/d, see Figure 4.14a)
Based on the empirical data presented in Hoerner (1965), the following approximate correlations are obtained to compute IbIb as a function of s/ds/d:
For 1≤sd<2
(4.39)For 2≤sd<7
(4.40)These correlations yield Ib = 0.25Ib=0.25 for s/d = 1s/d=1 and Ib = 1.0Ib=1.0 for s/d>6s/d>6.
4.4 Compressor Rotor Cavity
Axial compressors of gas turbines for aircraft propulsion and power generation feature multiple stages to achieve high compression ratio (20–40) needed in today’s high-performance engines. Compressed air temperatures in aft stages of the compressor tend to be very high. As a result of increasing air density along the compressor flow path, its annulus area also decreases with smaller vanes and blades. The thermal growth of compressor rotor disks relative to their outer casing play an important role in determining the blade-tip clearances, which impact compressor aerodynamic performance. During transients of engine acceleration and deceleration, rim-to-bore temperature gradients in each disk determine its low cycle fatigue (LCF) life. This calls for a transient conduction heat transfer analysis of each rotor disk with convection boundary conditions from the air in the cavity (called rotor cavity) formed with the adjacent disk. These rotor cavities are found in all shapes and sizes, some of them are completely closed, while others are cooled with an axial throughflow, called bore flow. Completely closed compressor rotor cavities are generally found in industrial gas turbines. In aircraft engines, except for initial stages of a high-pressure compressor (HPC), the rotor cavities feature a bore flow, which is designed to influence the rotor-tip clearances for better compressor aerodynamic performance and higher LCF life of each rotor disk.
Figure 4.15 schematically shows a compressor rotor cavity with bore flow. Without the bore flow, the rotor cavity may be considered a closed cavity. Because the static pressure at the vane exit is higher than that at its inlet, a reverse flow through each inter-stage seal in the main flow path occurs. Note that the windage generated in these inter-stage seals of an axial-flow compressor significantly change the thermal boundary conditions on rotor surfaces exposed to the main flow path.
Figure 4.15 Schematic of an axial compressor rotor cavity.
4.4.1 Flow and Heat Transfer Physics
In order to better understand the unsteady and three-dimensional nature of the centrifugally-driven buoyant convection (CDBC) in a typical compressor rotor cavity, schematically shown in Figure 4.15, let us first look at the spin-up and spin-down flow behavior in a closed adiabatic rotor cavity shown in Figure 4.16. When we impulsively rotate a closed cavity to the steady angular velocity ΩΩ, shown in Figure 4.16a, the fluid initially at rest inside the cavity eventually reaches a solid-body rotation with the cavity walls. During the transient, the angular momentum from the walls is transferred to the fluid through the action of viscosity and no-slip boundary condition at the walls. On the cylindrical surface parallel to the axis of rotation, we have negligible axial pressure gradients and the viscous boundary layer (called Stewartson boundary layer) grows radially inward. Each disk surface, which is normal to the axis of rotation, behaves like a free disk rotating adjacent to a nonrotating fluid, discussed earlier in this chapter, and results in the radially-outward pumping secondary flow in the Ekman boundary layer, continuously entraining fluid from the core.
Figure 4.16 (a) Fluid spin-up from rest in a rotating closed cavity and (b) spin-down of fluid under solid-body rotation to rest in a nonrotating closed cavity.
Let us try to physically understand how the radially-outward secondary flow is initiated during spin-up. Being stationary, the core features uniform zero pressure gradient, which is also imposed on the boundary layers on the cavity surfaces. Within the disk boundary layer, the fluid attains angular momentum and causes centrifugal body force, which is balanced by an adverse radial pressure gradient (dPs/dr=ρrΩf2
). To ensure zero net pressure gradient within the boundary layer, as imposed from the core fluid, a favorable radial pressure gradient is created, which causes increase in radial momentum flow while also overcoming the opposing viscous force from the radial wall shear stresses. This radial disk pumping flow grows by entraining fluid from the core. Thus, to satisfy continuity, circulating flows develop in the meridional plane, as shown in Figure 4.16a.These circulating flows speed up the transient process and the whole system reaches the state of solid-body rotation by an order of magnitude faster than through viscous diffusion alone. If the disks were frictionless, the secondary flow would seize to exist, and the steady state would take place through the growth of the Stewartson boundary layer alone.
During spin-down from solid-body rotation, when the rotor cavity walls are suddenly stopped, as shown in Figure 4.16b, secondary flows in each meridional plane within the Ekman and Stewartson boundary layers circulate in the directions opposite to those in the spin-up process. The occurrence of a radially inward flow in the Ekman boundary layer on each disk can be physically explained as follows. The fluid core initially in solid-body rotation has a radial pressure gradient (dPs/dr = ρrΩ2dPs/dr=ρrΩ2), which is imposed on each disk boundary layer. Because the fluid in the boundary layer has lower angular velocity than that in the core at the same radius, it experiences a net radially-inward pressure force causing it to flow toward the axis of rotation. Thus, circulating secondary flows develop in the cavity to satisfy continuity.
The spin-down process with its interesting feature of radially-inward flow in the Ekman boundary layer can be easily demonstrated in a simple home experiment. Let us take a cup of water and drop in it a few mustard seeds. With a straw, let us now vigorously stir the water and bring it to a state of solid-body rotation. When we stop stirring and pull the straw out, the water spins down to rest within a minute or so. During the spin-down process, we notice that the mustard seeds, which were mostly rotating away from the axis of rotation, collect at the bottom of the cup, not at the periphery but near the center (around the axis of rotation). The mustard seeds are carried there by the radially-inward flow at the bottom surface.
It may be noted here that, during the spin-up process, the work transfer (windage) from the adiabatic cavity walls will somewhat raise the total temperature of the cavity fluid. In the steady state of solid-body rotation, the fluid features a radial pressure gradient and becomes isothermal. This is true even for a compressible fluid like air. For an adiabatic spin-down, no work transfer occurs, and the fluid total temperature remains constant.
When the annular bore flow passes over the open compressor rotor cavity, shown in Figure 4.15, it undergoes a sudden geometric expansion and may impinge on the web region of the downstream disk. The jet expansion over the cavity results in a toroidal vortex, which recirculates in the axial direction and moves in solid-body rotation with the outer core flow in the cavity under adiabatic conditions. Unlike a driven cavity flow, the radial extent of the toroidal vortex is confined to the web region. For the case of disks temperature being different from that of the bore cooling air flow, CDBC drives a very complex flow structure in the rotor cavity. In their comprehensive review of buoyancy-induced flow in rotor cavities, Owen and Long (2015) note that, for the situation when the temperature of the disks and shroud is higher than that of the air in the cavity, the unstable and unsteady three-dimensional cavity flow features sources (high-pressure regions) and sinks (low-pressure regions), which in the rotating coordinate system (disks rotating counterclockwise, aft looking forward) appear as anti-cyclonic (rotating clockwise) and cyclonic (rotating counterclockwise) flows, respectively. In addition, a part of the axial bore flow enters the cavity through radial arms and return via Ekman layers on the disks. There is little hope of micro-modeling this cavity flow as a means to perform heat transfer calculations for design applications. In the next section, we present a practical approach to 1-D heat transfer modeling of compressor rotor cavities under CDBC.
4.4.2 Heat Transfer Modeling with Bore Flow
In free convection, both the flow and heat transfer are intricately coupled, that is, one depends on the other. Our daily experience involves free convection largely driven by the gravitational force field with a nearly constant acceleration (g = 9.81 m/s2g=9.81m/s2). In the chimney attached to the fireplace in our homes, we know that the hot air rises and the cold air sinks to satisfy continuity, generating a free convection flow, which we call gravitationally-driven buoyant convection (GDBC). As we know, water has its maximum density at 4oC4oC. In winter, when a lake starts to freeze, GDBC plays the role in ensuring that the water stays in the liquid phase to save marine life below the frozen top layer, which also acts as a good thermal insulator. When we have cold air next to a hot vertical wall, the free convection flow goes upward in the wall boundary layer. When the vertical wall is colder than the adjacent air, the free-convection flow comes downward in the wall boundary layer. For a downward facing horizontal wall, however, GDBC happens only when the wall is colder than the air; when this wall is hotter than the air, the flow is stably stratified with no GDBC.
The acceleration associated with the centrifugal force in a rotating system is given by gc = rΩ2gc=rΩ2, which varies directly as the radius of rotation and as the square of the angular velocity. At r = 0.5 mr=0.5m, for a cavity rotating at 3000 rpm, we have gc/g≈5000gc/g≈5000, which clearly shows the strength of CDBC in a compressor rotor activity vis-à-vis GDBC for similar temperature differences between cavity surface and the surrounding air. In the case of CDBC, cold fluid moves radially outward and warm fluid moves radially inward. Thus, when the wall is hotter than the fluid, the fluid flows radially inward in the disk boundary layer, and, when the wall is colder than the fluid, it flows radially outward in the boundary layer.
For 1-D heat transfer modeling of a compressor rotor cavity with a bore flow, shown in Figure 4.17, we divide the cavity in multiple control volumes, compute the convective heat transfer in each control volume and find the total convective heat transfer rate (Q̇c
), which becomes the boundary condition for the steady-flow energy equation over the bore CV to determine the air total temperature change from inlet to outlet. Accordingly, we can write
(4.41)
Figure 4.17 1-D Heat transfer modeling of compressor rotor cavity with bore flow.
For calculating the convective heat flux q̇c=hTw−Taw
from each wall of the cavity CV, we need three quantities: wall temperature (TwTw); adiabatic wall temperature (TawTaw), which acts as the fluid reference temperature; and heat transfer coefficient (hh), which is obtained from a specified empirical correlation. In this multitemperature problem, it is not obvious which temperature we should use as the appropriate adiabatic wall temperature. In the present approach, we make use of the fact that, under adiabatic conditions, the rothalpy will remain constant for any excursion of the fluid in solid-body rotation within the rotor cavity. Thus, we can write
(4.42)where rr is the central radius of a cavity control volume and rborerbore corresponds to the mean radius of the bore flow. Owen and Tang (2015) also suggest the use of Equation 4.42 for the fluid reference temperature to compute the convective heat transfer between the cavity wall and air in CDBC. They have, however, arrived at this equation using the compressibility effects and invoking the isentropic relationship between the pressure ratio and temperature ratio at two points.
As to the empirical heat transfer correlations for CDBC on the disk and cylindrical rim surfaces, we present here a set of standard correlations for GDBC from McAdams (1954). These correlations conform to the common equation form
The coefficient C and the exponent m for the Rayleigh number RaRa used in Equation 4.43 for various situations are tabulated in Table 4.1. In view of the geometric complexity of a compressor rotor cavity with arbitrary temperature distribution on its bounding surfaces, the empirical correlations suggested by Equation 4.43 are to be treated as nominal correlations for initial modeling purposes only. Such correlations must be later refined and established for a consistent design practice from extensive thermal surveys for various designs. These empirical correlations then become proprietary to the original equipment manufacturer and do not belong to a textbook.
Table 4.1 Constants C and m used in Equation 4.43
| Physical situation | Ra range | C | m |
|---|---|---|---|
| Vertical surface | 104 − 109104−109 | 0.59 | 14 |
| Vertical surface | 109 − 1012109−1012 | 0.13 | 13 |
| Horizontal surface (Tw>TawTw>Taw, CDBC) | 105 − 2 × 107105−2×107 | 0.54 | 14 |
| Horizontal surface (Tw>TawTw>Taw, CDBC) | 2 × 107 − 3 × 10102×107−3×1010 | 0.14 | 13 |
| Horizontal surface (Tw < TawTw<Taw, CDBC) | 3 × 105 − 3 × 10103×105−3×1010 | 0.27 | 14 |