Mass and Heat Exchange in Rotating Compressor Cavities With Variable Cob Separation Open Access

Blade tip clearances and thermal stresses affect the efficiency and operating life of gas turbine compressors. Engine thermo-mechanical design and analysis requires accurate prediction of compressor disk temperatures, which are determined by the flow structure and heat transfer within the cavities formed by corotating disks and the rotor shroud. Figure 1 shows a typical aero-engine arrangement, with an axial throughflow of cool air in the bore annulus at low radius. In practice, most compressor cavities are open with thick cobs at the inner radii of the disks where hoop stresses are at a maximum [1]. The cavity shroud is heated by the mainstream annulus and temperature differences lead to buoyancy-induced flow. Such flows feature unsteady and three-dimensional vortex structures, ingress of cool fluid from the axial throughflow, enthalpy, and momentum exchange, and egress of hot fluid from the cavity.

where $Ω$ is the disk rotational speed, b is the cavity outer radius, W is the average throughflow velocity, $Tsh$ is the temperature of the shroud, and $Tf$ is the inlet throughflow temperature. Other symbols are defined in the nomenclature. Note that the Grashof number (⁠$Gr=Reϕ2βΔT$⁠) is often used as a governing parameter. In aero-engines, Gr is $O(1013)$ creating a challenge for computational fluid dynamics to achieve accurate solutions.

Closed Cavity ($η=0$⁠):

The closed cavity is a canonical case that reduces the complexity of the system, isolating the buoyancy-induced phenomena from the interaction with the axial throughflow. The Aachen group measured shroud heat transfer correlations in closed cavities [2], with adiabatic disks, providing evidence of laminar free convection. One such closed cavity (⁠$a/b=0.521$ and $s/b=0.5$⁠) has been the subject of several computational studies. Pitz et al. [3] conducted large-eddy simulation (LES) of flow in this geometry, finding counter-rotating vortices consistent with Rayleigh-Bénard convection and disk boundary layer profiles similar to that of laminar Ekman layers, consistent with the assumptions used by Owen and Tang [4]. Saini and Sandberg [5,6] used LES computations to show the number of structure pairs varied with time, and that compressibility effects at high rotational Reynolds numbers suppressed the formation of these structures and the shroud heat transfer, in agreement with the theoretical findings by Tang and Owen [7].

Flow structures, as well as disk and shroud heat transfer, were measured in a closed cavity with $a/b=0.454$ and $s/b=0.167$ using the Bath Compressor Cavity Rig [8]. Unsteady pressure measurements showed 3-4 vortex pairs and structure slip-to-rotation ratios of less than 1%. A plume model was developed for the prediction of disk and core temperatures by Tang and Owen [9], assuming convective heat transfer via hot and cold plumes between the vortex pairs and conductive disk heat transfer via laminar Ekman layers. The predictions were validated by Lock et al. [10] where a correlation for shroud heat transfer was established using heat flux gauge measurements and experimentally derived local core temperatures. The plume model was further developed for transient operating conditions by Nicholas et al. [11]. Unsteady Reynolds-Average Simulations in a conjugate heat transfer model from Parry et al. [12] were in good agreement with both the experiments and the model.

Open Cavity ($η=1$⁠):

For cases with $η=1$⁠, the disk cobs are removed and the cavity is fully open to interaction with the axial throughflow. Farthing et al. [13] conducted an experimental study of flow structure in such cavities with $a/b≈0.1$ and $0.124<s/b<0.532$ using laser Doppler anemometry. They showed that the average velocity of the buoyancy-driven structures could be up to 10% slower than that of the disks and that there was a peak in this structure slip for a given Ro. They also demonstrated the existence of a toroidal vortex that significantly affected the flow structure. Bohn et al. [14] measured flow structure in a fully open cavity with $a/b=0.3$ and $s/b=0.2$⁠, showing one pair of vortices rotating at 88–90% of the disk speed. Disk heat transfer was measured by both Bohn et al. [14] and Farthing et al. [15]; however, no consistent trends on the effects of Ro and $s/b$ were observed. LES results from Pitz et al. [16] with a fully open cavity of $a/b=0.521$ and $s/b=0.5$ and showed through flow penetration into the cavity increased with Re$z$⁠. The disk boundary layer thickness was again consistent with that of laminar Ekman layers.

Open Cavity (⁠$0<η<1$⁠):

In practice, aero-engine cavities feature $0<η<1$ due to thick cobs at the inner radii of the disks (however, some industrial gas turbines incorporate virtually closed cavities (⁠$η→0$⁠) with small clearances for leakage flow). Long et al. [17] and Long and Childs [18] measured velocities and shroud heat transfer in an open cavity with $a/b=0.318$⁠, $s/b=0.195$⁠, and $η=0.601$⁠. The velocity measurements showed two flow regions in the cavity: at low radii the flow was strongly affected by the radial jet induced by axial throughflow; at high radii the flow was governed by buoyancy effects. The frequency spectrum of the tangential velocity showed two pairs of structures. Atkins and Kanjirakkad [19] presented the effects of operating parameters on disk temperature measurements in the same cavity, from which the calculated disk heat transfer by Tang et al. [20] showed suppressed buoyancy effects at high $Reϕ$ due to compressibility.

Heat transfer and flow structure in an open cavity with $a/b=0.454$⁠, $s/b=0.167$⁠, and $η=0.650$ were investigated in the Bath Compressor Cavity Rig [21–23]. A single pair of counter-rotating structures was observed slipping at 15% in the open cavity, much higher than the equivalent slip of 1% per structure in the closed cavity. This open cavity slip was a function of Ro in the open cavity, and the strength of buoyancy, $βΔT$⁠, in both cavity geometries. At low Ro there was experimental evidence of egress of cavity flow to the axial throughflow in both the upstream and downstream direction. Such reversal flow recirculates heated air from the cavity and reduces the overall heat transfer in the cavity. Pernak et al. [24] have defined subcritical and supercritical flow regimes: in the former, reversal flow is prominent; in the latter the toroidal vortex in the cob region is dominant. A wall-modeled large-eddy simulation (WMLES) study from Gao and Chew [25] using Bath data and geometry for Ro $>0.4$ showed that axial throughflow was entrained radially into the cavity via the cold plumes, and hot air was expelled into the throughflow via the hot plumes. They also calculated flow reversal, which decreased as Ro increased with a corresponding reduction in cavity-throughflow exchange. The plume model for disk temperature prediction was applied to this open cavity by Nicholas et al. [26], determining experimentally-derived correlations for reversal flow and the mass exchange.

A recent study by Fischer and Puttock-Brown [27] gave further evidence for the toroidal vortex in the Sussex rig for $0.11<Ro<3.24$⁠, with time-averaged radial velocity measurements that were the first of their kind. The authors showed the vortex developed primarily for $Ro>0.2$ and that its size and strength increased with Ro.

Various experimental, theoretical and computational studies conducted by different groups have implemented both open and closed cavities with varying cavity dimensions. However, there is a gap in the literature investigating the effect of varying the cob geometry on the shroud heat transfer, flow structure, core and disk temperatures. This paper is the first to study the effect of $η$ on mass and heat exchange for compressor cavities. A combined experimental and modeling approach has been used to capture this effect using cavities with different cob separations. Section 2 introduces the Bath Compressor Cavity Rig, the cavity geometries (⁠$η=0,0.325,0.650$⁠), and the methods of data analysis. A plume model is derived in Sec. 3 to predict the effect of $η$ on disk and core temperatures. In Secs. 4 and 5, experimental measurements of the flow structures, shroud heat transfer, and disk temperatures over a range of Ro, Gr, and Re$ϕ$ are presented, and compared with the model. Section 6 gives practical design implications and Sec. 7 summarizes the key conclusions.

The Bath Compressor Cavity Rig is shown in Fig. 2. Details are available in Luberti et al. [8]. The test section consists of four titanium disks, with a central cavity instrumented with thermocouples across the disk radius, a shroud heat flux gauge and two unsteady pressure sensors at $r/b=0.85$⁠. This data is collected in the rotating frame of reference via a telemetry system. The external surfaces of the disks in the central cavity disks are insulated with 5 mm of low-conductivity Rohacell foam, forming a quasi-adiabatic boundary condition. The axial throughflow is supplied from ambient air in the laboratory and circular heaters at the shroud (combined with a range of rotational disk speeds) provide a range of Gr and Ro. Aluminum ring attachments to the cobs allow the rig to simulate both open and closed cavity configurations. These were bolted to the inner cob surfaces to rotate with the disks.

Figure 2 shows the three geometries, denoted as Cavity A, B, and C and characterized by $η=0,0.325$ and 0.650, respectively. The inner and outer radii of the disks, the axial width in the outer region, and the bore flow dimensions are consistent in all configurations. Detailed geometrical information and operating parameters are listed in Table 1. Thirteen cases are presented in this paper but a total of 189 steady-state cases were collected using the three geometries. The detailed operational parameters of the cases with temperature distributions presented in this paper are given in Table 2.

Not shown in Fig. 2 are thermocouple rakes that measure the radial distribution of temperature in the throughflow annulus (⁠$rs<r<a$—see Fig. 1), as shown in Jackson et al. [29]. These are at the inlet to the annulus, upstream and downstream of the cavity. Such measurements identify whether flow reversal (associated with cavity ingress and egress) is significant. From Ref. [29], the flow may then be separated into subcritical (Ro < 0.4) and supercritical flow regimes (Ro > 0.4): in the former, reversal flow is prominent; in the latter the toroidal vortex in the cob region is dominant. Typical sub- and supercritical throughflow temperature profiles are shown in  Appendix, Fig. 16, for both open cavity geometries B and C.

where $Δp$ is the pressure difference between the centers of the anticyclonic and cyclonic vortices.

where $kd$ and $td$ are the thermal conductivity and thickness of the disk.

Theoretical models have been developed separately for closed and open cavities [11,26]. This section presents an integrated model to capture the effect of cob separation ratio $η$ on the mass exchange with the axial throughflow and consequences for buoyancy-induced flow and heat transfer.

Figure 3 illustrates the flow structure in the cavities: laminar Ekman layers with radial outflow, shroud and hub free convection layers, radial movement of hot and cold plumes, and a toroidal vortex at low radius. These illustrations are schematic as the flow is unsteady, unstable and three-dimensional. The cavity can be divided into an inner region dominated by flow exchange between the cavity and axial flows, and an outer region driven by buoyancy-induced cyclonic and anticyclonic vortices. Plumes of hot and cold fluid flow radially between these rotating vortex pairs. These are shown more clearly in Fig. 4, where the varying cob geometries have a significant effect on this flow structure. Shown in more detail in Sec. 4, increasing $η$ reduces the number of vortex pairs and increases the significance of the exchange and reversal mass flows, $ψex$ and $ψr$⁠, respectively.

For all cavities, the flows on the disks and shroud surfaces are governed by laminar Ekman layers and a free convection layer, respectively. The source flow of the cold plumes characterizes the interface between the inner and outer regions. For closed cavities (⁠$η=0$⁠), the recirculated flow in the outer region is cooled by the cobs (the hub) and is the source of the fluid in the cold plumes. For open cavities (⁠$η>0$⁠), a fraction of the cold plume flow is provided by entrainment (or ingress-egress exchange) of cold fluid from the axial throughflow. The plume mass flowrate is an upper limit for the entrainment/exchange flowrate.

At high Ro, where the axial throughflow impingement is strong even for cases with $η=1$⁠, a part of the cold plume flow in the outer region is supplied by recirculation. This is illustrated in Fig. 3. At low Ro (⁠$<0.4$⁠), the toroidal vortex is suppressed and a greater proportion of the plume flow is provided by entrainment.

For the closed cavity (⁠$η=0$⁠), $ψex=0$⁠. For the open cavities, the empirical exchange-flow coefficients $D1$⁠, $D2$⁠, and $D3$ are a function of $η$⁠. The values for the two geometries B and C (⁠$η=0.325$ and 0.650) were determined from experiments and presented in Sec. 5. The constant $D1$ will increase from cavity B to cavity C, as $η$ increases.

Pr is the Prandtl number and other symbols are defined in the nomenclature.

where $Φ$ is the angle (as a function of radius) between the upwards radial direction and the elemental disk surface area, which is zero except for the hub fillet region.

Thus, when the cavity is fully-open, the hub area $Ahb$ is reduced to 0 and the central right-hand side term in Eq. (29) reduces to zero. This term calculates contribution of convective heat transfer from the hub to the cold plume temperature in the outer region. The right most term describes the contribution of disk heat transfer and compressibilty to the outer cold plume temperature, as a function of both the plume and exchange mass flows. If the cavity is fully closed (cavity A, $η=0$⁠) then the inner region is replaced by cobs and the hub free convection layer, so this equation reduces to just the hub heat flux term.

where $Ash$ is the shroud area and $τsh$ is the nondimensional shroud heat flux. In Eq. (34), $θp̂,b$ is the cold plume temperature at $r=b$⁠, and the right most term is the temperature difference between the plumes caused by convective heat transfer from the shroud.

where $θl$ is the nondimensional temperature of the core in the cavities upstream and downstream of the test section. As discussed in Sec. 2, these adjacent cavities were isolated from the axial throughflow (Fig. 2) and thermal-insulating foam created a quasi-adiabatic boundary condition on the back surfaces of the disks. Temperature boundary conditions were applied at the outer radius of the disk diaphragm and the inner radius of the cobs. Heat flux boundary conditions were applied on the disk surfaces using the characteristics of laminar Ekman layers, [11,26]. Note that due to the conjugate nature of the problem, the temperatures of the core and disk were solved iteratively. The iterative process was repeated until the change of the disk temperature $Δθd<10−5$⁠. Depending on the initial guess, this typically only took 5-6 iterations until convergence. The calculation of the disk and core temperatures for one experimental case takes approximately two seconds using a standard laptop.

The calculation of $θd$ was relatively insensitive to the air temperatures in the upstream and downstream cavities, $θl$⁠, which were isolated from the axial throughflow - see Fig. 2. A $10%$ change in $θl$ produced $<1%$ change in $θd$⁠. Note that the disk temperatures at $r=b$ and $r=a$ were fixed with measured values. Ideally, a correlation for the cooling on the inner cob surface can be used to predict the temperature at $r=a$⁠, and the temperature at $r=b$ can be determined by the heat transfer on the outer shroud surface and adjacent disks.

As discussed in Sec. 2.2, cross-correlation of the two pressure signals provides the slipping speed (⁠$Ωs$⁠) of the counter-rotating structures. Figure 5 shows the variation of slipping speed with different Ro for all three cavities at $βΔT=0.25$⁠. Note that for the closed cavity (Cavity A), the interaction between the cavity and throughflow is prevented, hence there is no direct effect of Ro on the cavity flow structure.

Consistent with the findings in Jackson et al. [21] and Nicholas et al. [26], there is a critical Ro for which there is a peak in structure slip, showing maximum momentum exchange between the open cavity and throughflow. This peak is a combined result of the reversal flow in the axial throughflow and the toroidal vortex generated by the throughflow in the inner region. As $η$ is halved from Cavity C to B, the magnitude of $Ωs$ decreased from $10−15%$ to $2−4%$⁠, indicating an nonlinear reduction in the exchange mass flowrate. In the closed cavity with no momentum exchange, $Ωs<1%$ for all cases and no distinct peak with Ro is observed (as there is no exchange mass flowrate for closed cavities, the parameter Ro for Cavity A is merely an indication of the cooling on the inner cob surface).

Figure 6 shows the fast Fourier transform (FFT) for all three geometries for $Ro=0.2$⁠, $βΔT=0.25$ and $Reϕ=8×105$⁠. The magnitude of $Cp$ denotes the approximate strength of the rotating structures, and this is maximum for Cavity C; here the open area at the cobs is largest and momentum exchange with the axial throughflow is greatest. The frequency of the peak in $Cp$ is the passing frequency of all vortex pairs, $fs,1$⁠. Equation (7) reveals a single pair for Cavity C, with the second peak, at $fs,2=2×fs,1$⁠, corresponding to the asymmetry of the two vortices, as detailed in Ref. [23]. There are three, virtually-symmetric vortex pairs for Cavity A. While these rotating vortices are inherently unsteady, cavities A and C exhibit a regular set of pairs. The intermediate cavity (B) displays an aperiodic behavior. The FFT shows two equal peaks between $0.062<f/fd<0.074$⁠, corresponding to either two or three vortex pairs, as illustrated by the spectrogram in Fig. 7.

As shown in Fig. 7, the closed cavity features a peak frequency at $f/fd=0.022$⁠, with some noise at lower frequencies and three pairs of structures. Cavity C (that with the largest opening at the cobs) shows a constant peak frequency of $f/fd=0.13$ with some unsteadiness and a higher level of overall noise caused by the throughflow interaction. Here there is a single pair of structures with the second peak (⁠$fs,2/fd=0.26$⁠) resulting from vortex asymmetry. Cavity B features two distinct bands of frequency peaks at $f/fd≈$ 0.05 and 0.07, and no clear presence of vortex asymmetry. Cross-correlation of a variety of samples from this case shows that the number of structure pairs are changing frequently in an aperiodic nature, along with a slight change in slip per structure. The upper band corresponds to three pairs of structures and with slip $Ωs/Ωd=2.3%$ per structure, and the lower band to two pairs of structures with slip $Ωs/Ωd=2.5%$⁠.

A summary of the number of vortex pairs (given by the ratio peak frequency, $fs,1$⁠, to structure slip, $Ωs$⁠) for all cavities with varying Re$ϕ$ is given in Fig. 8. For Cavity C there is one pair of structures found from cross-correlation and FFTs of the dual pressure signals. For the closed cavity A, there is a transition from three to four pairs for Re$ϕ≥2.2×106$⁠. For Cavity B, the number of structure pairs is between two and three. The reason for a noninteger value is that the results are averaged across 500 disk revolutions to produce a mean result from the aperiodic structures.

Plume mass flow rates, $ψp$⁠, were derived from the measurements of unsteady pressure using Eq. (9). The shroud Grashof number $Grsh$ (Eq. (13)) was calculated using the measured shroud temperature and the experimentally-derived core temperature $Tc,b′$ from the disk temperature measurements in the outer region and the heat transfer coefficients from the conductive Ekman layer. Note there is a distinction between the experimentally-derived core temperatures and the theoretical- predicted core temperatures from the model in Sec. 5. The former involves solving the inverse problem by matching resultant disk temperatures with experimental values. The predicted values come from the correlations for heat transfer and mass flow.

Both the plume mass flowrate and shroud heat transfer correlations are used in the plume model to determine the magnitude and distribution of core and disk temperatures, as discussed in the next section.

The predicted cavity temperatures at $Reϕ=2.3×106$⁠, $βΔT=0.25$ and $Ro=0.4$ are shown in Fig. 12 (though the Rossby number is not the driving parameter for closed cavity heat transfer, it is presented to show consistent bore cooling conditions with the open cavity cases). Symbols denote disk temperature measurements (averaged across both upstream and downstream disks). Solid lines denote disk temperature predictions from the plume model and dashed lines core temperature predictions. These predictions follow the procedure described in Sec. 3.3, using correlations for shroud and hub heat transfer, plume, reversal, and exchange mass flow, and the temperature and heat flux boundary conditions. In Cavity A, the temperature predictions are limited to the diaphragm section of the test geometry (⁠$0.52<x<0.98$⁠, the outer region). For the open cavities, this prediction extends to the inner or cob region; here there is a step change in core temperature as the radial flow is driven by plume mass flow in the outer region and driven by exchange mass flow in the inner region, where $ψex<ψp$⁠. There is an increase core temperature with radius caused by the compressibility effect at large Re$ϕ$⁠. Note that there is consistent agreement between the measured and modeled disk temperatures for all three geometries.

For $η=0$ (Cavity A), the core temperature is lower than the disk temperature (⁠$θc<θd$⁠) at high radii and higher at low radii, and the crossover radius (where $θc=θd$⁠) is near $x=0.82$⁠. This is consistent with results from Lock et al. [10]. The crossover radius is closer to the shroud than the hub due to the larger shroud surface area and heat transfer.

As $η$ increases, the mass and heat transfer between the cavity and throughflow is increased, hence $θd$⁠, $θc$ and the cross-radius decreases. Despite Cavity B being open to interaction with the throughflow, the distributions of disk temperature were similar to those of Cavity A. This was caused by the reduced exchange mass flowrate relative to Cavity C, equivalent to just $5−10%$ of the plume mass flow. There is a step increase of the core temperature from the inner to outer region for cavities B and C. This is caused by the mixing of the cold plume flow with the hot recirculated flow in the outer region.

Figure 13 is similar to Fig. 12, other than reduced $βΔT$⁠. Consider cavity C, with the largest $η$⁠. Equations (41) and (44) show a reduction in $βΔT$ leads to reduction in Grashof number and $ψp$⁠, resulting in a decrease in $ψex$ and, in turn, an increase in the core temperature $θc$⁠. In turn, there is a reduction in the disk temperature gradient in the outer region. However, for cavities A and B, the exchange flowrate is very low or zero. Thus, there is a relatively minor effect from $βΔT$ on the overall distributions of nondimensional temperature, except a reduction in cob temperature. This is due to weakened convection from the shroud as a result of the reduced Grashof number (see Eq. (17)). Relative to Fig. 12 there is an increase in the gradient of core temperature due to the reduced $βΔT$⁠, which causes an increase in the compressibility parameter $χ$ (see Eq. (26)). This is a result of the dimensional core temperature gradient being proportionately larger due to the lower $ΔT=Tsh−Tf$ value.

Figure 14 shows the impact of $Reϕ$ on temperatures for cavity B. There is a reduction in core temperature with increased Re$ϕ$⁠, which is due to the increase in Grashof number, plume mass flow, and mass and heat transfer between the cavity and throughflow. Moreover, increasing $Reϕ$ increases the compressibility parameter, $χ$⁠, and so increases the radial gradient of core temperature, as observed in both other cavity configurations [10,26].

Figure 15 illustrates the impact of changing from subcritical to supercritical Rossby number for cavities B and C. The former is the reversal flow regime where $Ro<0.4$ and the latter the toroidal vortex regime where $Ro>0.4$⁠. As Ro increases to 0.4, there is a decrease in both $θd$ and $θc$ due to the reduction in reversal flow, $ψr$⁠. From Figure 11, $ψex$ is approximately constant for $Ro<0.4$⁠, but $ψr$ decreases to zero, see Eq. (14). For $Ro>0.4$⁠, the toroidal vortex reduces $ψex$⁠, causing a slight increase in $θd$ and $θc$⁠. This trend in $θc$ is consistent with that of structure slip in Fig. 5, showing a peak in mass and heat transfer between the cavity and throughflow, as well as the peak in shroud heat transfer for $Ro=0.4$ shown in Table 2.

This research has illustrated the nonlinear effect of varying the cob separation on the heat transfer and flow structure in aero-engine compressor cavities. The radial variation of disk temperature is influenced significantly by this cob separation, as will be the thermal stresses and expansion of the compressor rotor. These, in turn, will impact tip clearances, engine operating life and efficiency. The model presented here provides expedient, reduced-order solutions to the complex conjugate heat transfer problem. The methodology is specifically intended for incorporation into practical thermo-mechanical engine design codes.

Engine designers have the challenge of balancing component life with a system-level optimization. From one perspective, enhanced heat transfer effectively cools the disks reduces tip clearances; however, this causes a maximum enthalpy exchange with the axial throughflow, reducing its cooling effectiveness in the later stages of the engine and increases thermal stresses. A compromise governed by the cavity-bore flow interaction is required. The model (validated here by experimental data) provides a quantitative prediction of both the disk temperatures and throughflow enthalpy exchange, better informing iterative engine design decisions.

This paper provides the first study of the effect of cob separation on mass and heat exchange in rotating compressor cavities. A new model has been presented that is able to predict the interaction between the cavity and the axial throughflow, and consequently the radial distribution of disk and fluid-core temperatures. The model was validated using detailed experimental measurements from the Bath Compressor Cavity Rig, which has the unique capability of quantifying disk temperature and shroud heat flux alongside unsteady pressure signals in the rotating cavity. A range of disk-cob spacings was investigated over a range of engine representative conditions.

Measurements of pressure from fast-response sensors on the rotating disks captured unsteady cyclonic and anticyclonic vortex pairs typical of buoyancy-induced flow. The number of structures and their slip relative to the disk depended on the cob separation, with different degrees of ingress/egress to/from the cavity. Specific cases featured fundamentally aperiodic (chaotic) features, even at steady-state conditions of Rossby, Grashof and Reynolds numbers.

Correlations for shroud Nusselt number and radial mass flowrate in the cavity were generated across a wide range of Grashof numbers. The convective heat transfer at the shroud was highest for the largest cob separation, with increased cavity-throughflow momentum exchange and ingress of cold fluid. The rate of radial mass flow reduced as the cob separation increased, with larger slip (relative to the disk) of the rotating structures at increased cavity-throughflow interaction. Correlations were generated for mass exchange between the cavity and axial throughflow, showing a nonlinear reduction in fluid entrainment with reduced cob separation.

The model was able to predict the mass and heat transfer to/from the cavities, together with the radial distributions of core and disk temperatures. Excellent agreement with experiments was achieved for 189 cases that spanned a range of Rossby (Ro), Grashof and Reynolds numbers. There is a critical Ro near 0.4 where heat transfer and core slip are maximized, and core and disk temperatures are minimized. This was shown to be a result of two phenomena: flow reversal at low Ro and a toroidal vortex at high Ro.

The physics-based model developed here is able to provide a reduced-order method to support the prediction of thermal stresses and blade-tip clearance in high-pressure compressors. Appropriate for thermo-mechanical design in industry, the model was created to inform the design of next generation high pressure-ratio aeroengines that require ever greater improvements to efficiency and accurate tip-clearance prediction.

The research presented in this paper was supported by the UK Engineering and Physical Sciences Research Council and in collaboration with Rolls-Royce plc and the University of Surrey, under the Grant No. EP/P003702/1. The authors are especially grateful for the support of Jake Williams and the approval from Rolls-Royce to publish the work.

• Engineering and Physical Sciences Research Council (Award No. EP/P003702/1; Funder ID: 10.13039/501100000266).

The authors attest that all data for this study are included in the paper.

A =

surface area (⁠$m2$⁠)

a =

inner radius of cavity (cob) (m)

$a′$ =

inner diaphragm radius (m)

$b′$ =

outer diaphragm radius (m)

b =

outer radius of cavity (shroud) (m)

C =

plume mass flow correlation coefficient

$cp$ =

specific heat capacity (J/(kgK))

$dh$ =

hydraulic diameter (⁠$=2(a−rs)$⁠) (m)

$D1,D2,D3$ =

exchange flow coefficients

E =

reversal flow correlation coefficient

F =

shroud/hub heat transfer correlation coefficient

$fs$ =

rotational frequency of the flow structure (Hz)

G =

cavity aspect ratio (⁠$=s/b$⁠)

h =

heat transfer coefficient (⁠$W/(m2K)$⁠)

k =

thermal conductivity of air (W/(mK))

$kd$ =

thermal conductivity of disk (W/(mK))

l =

empirical exchange flow exponent

$m˙$ =

mass flow rate (kg/s)

m =

plume mass flow coefficient exponent

n =

number of vortex pairs

N =

rotational speed of disks (RPM)

p =

static pressure (Pa)

$p¯$ =

mean static pressure (Pa)

q =

heat flux (⁠$W/m2$⁠)

r =

radius (m)

R =

specific gas constant (J/(kgK))

s =

cavity width at diaphragm (m)

$s′$ =

cob separation (m)

t =

thickness (m)

$tα$ =

time lag between pressure sensors (s)

T =

temperature (K)

W =

throughflow velocity (m/s)

x =

non-dimensional radial location

y =

non-dimensional axial distance

z =

axial distance (m)

$α$ =

circumferential angle between two pressure sensors (rad)

$β$ =

volume expansion coefficient (⁠$K−1$⁠)

$γ$ =

ratio of specific heats

$δ$ =

Ekman layer thickness (m)

$η$ =

cob separation ratio (⁠$=s′/s$⁠)

$θ$ =

non-dimensional temperature $(=(T−Tf)/(Tsh−Tf))$

$μ$ =

dynamic viscosity (⁠$kg/(ms)$⁠)

$ρ$ =

density (⁠$kg/m3$⁠)

$τ$ =

non-dimensional heat flux $(=b2q/(kdtd(Tsh−Tf)))$

$χ$ =

compressibility parameter (⁠$=Ma2/βΔT$⁠)

$ψ$ =

non-dimensional mass flow rate (⁠$=m˙/(μs)$⁠)

$Φ$ =

angle between disk surface and radial direction (rad)

$Ω$ =

angular velocity (rad/s)

$Cp$ =

coefficient of pressure for FFT $(=(p−p¯)/(2ρfΩd2b2))$

$CΔp$ =

coefficient of pressure difference $(=Δp/(ρfΩd2b2))$

Gr =

Grashof number (⁠$=Reϕ2βΔT$⁠)

Ma =

rotational Mach number $(=Ωb/γR(Ta+Tb))$

Nu =

Nusselt number (⁠$=hb/k$⁠)

Pr =

Prandtl number (⁠$=Cpμ/k$⁠)

Ra =

Rayleigh number (⁠$=PrGr$⁠)

Ro =

Rossby number (⁠$=W/(Ωa$⁠)

$Reϕ$ =

Rotational Reynolds number $(ρΩb2)/(μ))$

$Rez$ =

Axial Reynolds number $(ρWdh/(μ))$

$βΔT$ =

Buoyancy parameter (⁠$=(Tsh+Tf)/Tf$⁠)