Flow and Heat Transfer in an Axial Throughflow Rotating Disk Cavity with Dual Inlets Under Variable Conditions

Abstract

The flow and heat transfer in a rotating disk cavity with dual axial inlets are investigated under a range of operating conditions. A full 360° computational fluid dynamics model is employed, with 40 simulation cases varying the rotational Reynolds number (Re_ω= 1.9 × 10⁶–3.1 × 10⁶) and axial throughflow Reynolds number (Re_z = 7.3 × 10⁵–1.2 × 10⁶). The results show that elevated rotation intensifies turbulent mixing and significantly enhances convective cooling on the upstream disk, whereas increasing throughflow improves heat transfer on the downstream disk by promoting deeper coolant penetration. However, an excessive axial flow rate can induce local thermal stratification near the upstream disk, which offsets its heat transfer gains, and strong rotation diminishes the marginal benefits of higher throughflow on downstream cooling. Overall, the study reveals distinct cooling behaviors on the upstream and downstream disk surfaces governed by the interplay between rotation and throughflow. These findings provide insight into optimizing dual-inlet cavity designs and underscore the importance of balancing rotational speed and coolant flow distribution for effective thermal management in gas turbine disk cavities.

1. Introduction

With the continuous advancement of modern aero-engine technologies, turbine inlet temperatures have increased substantially [1]. To ensure the reliable and durable operation of critical hot-section components—such as turbine rotors and blades—under extreme conditions of high temperature, pressure, and rotational speed, it is essential to analyze and optimize the key components of the secondary air system. Among these, the turbine disk cavity plays a central role, often exhibiting complex and unsteady flow behavior due to its rotational nature and intricate geometry.

In modern high-performance aero-engines, turbine disk cavities have evolved to adopt multiple inlet configurations, diverging notably from traditional single-inlet designs. Within these rotating cavities, secondary airflows from different sources interact under the combined action of centrifugal and Coriolis forces, and curved geometric boundaries, producing highly three-dimensional and unsteady internal flow structures. Therefore, investigating the flow organization and heat transfer behavior in rotating cavities with dual axial throughflow inlets and a single outlet is of great significance for optimizing secondary air system design and improving thermodynamic performance.

1.1. Literature Review

Over the past several decades, a rich body of work has characterized the flow and heat transfer in classical cavities with simple inlet/outlet configurations. Seminal studies established baseline behaviors for a single inlet/single outlet cavity, identifying how rotational forces and throughflow interact to produce characteristic flow regimes. Hide et al. [2] divided the cavity flow clearly into a source region, a sink region, and a core region, establishing a fundamental theoretical model for laminar flow in cavities. Owen and Rogers [3] conducted a detailed theoretical analysis of flow characteristics within rotating disk cavities, established numerous theoretical models and empirical correlations, and provided extensive experimental data for model validation. Numerical simulation methods have seen extensive application in studies of rotating disk cavities. Farthing et al. [4] further investigated the radial flow characteristics, explicitly highlighting the influence of rotation on the radial flow patterns. Long et al. [5] analyzed the effects of inlet flow distribution on heat transfer performance.

As high-accuracy algorithms have matured, Sun et al. [6] were the first to use LES at a low Rossby number to resolve cyclonic/anticyclonic vortex pairs and natural-convection-like cells, validating a new Nusselt-number scaling and providing a benchmark for centrifugal-buoyancy physics and turbulence-model calibration. Hickling et al. [7] introduced conjugate LES (LES–CHT) for rotating cavities, simultaneously capturing turbulence and the solid-disk thermal response; their predictions of wall temperature and low-frequency thermal fluctuations clearly outperformed RANS, delivering one of the first LES–CHT validations in this context. Wang et al. [8] used DNS as a yardstick to benchmark WMLES and hybrid approaches with a high-order open-source solver, quantifying the accuracy–cost trade-off and demonstrating credibility for both rotor–stator and co-rotating configurations. Under extreme rotation, Saini et al. [9] revealed a core dominated by a single toroidal vortex with Ekman-layer in/out-flow paths, showed that shroud-side heat transfer follows pure-buoyancy correlations, and clarified long-standing uncertainties in the high-Ro vortex structure.

As for advanced experimental methods, Jackson et al. [10] built an engine-representative co-rotating compressor cavity rig at high Re/Gr, producing high-fidelity wall-temperature and local Nusselt maps that now serve as an authoritative dataset for CFD validation and engineering relevance. Using two-component LDA in a heated multi-disk cavity, Fazeli et al. [11] non-intrusively measured axial and tangential velocities, directly identified coherent vortex pairs and slip frequencies, and cross-validated these features with URANS. In a multistage cavity with throughflow.

Leveraging new tools, Puttock-Brown et al. [12] used a physics-informed neural network (PINN) to infer disk surface heat flux from limited temperature measurements—offering faster, more noise-robust inverse analysis for rapid thermal data assimilation. Zhang et al. [13] trained a neural-network surrogate on a CFD database for a co-rotating cavity with a finned vortex reducer, enabling near-instant predictions (~97% accuracy) of pressure loss and Nusselt distribution and thereby accelerating design optimization. Guo et al. [14] combined a neural-network surrogate with a genetic algorithm to optimize a twin-web cavity/pre-swirl system, achieving marked rim-temperature reduction and improved Nusselt levels and uniformity, demonstrating the potential of AI-enabled multi-objective optimization for complex rotating cavities.

Despite the extensive knowledge gained for single-inlet rotating cavities, little attention has been paid to configurations with dual axial inlets. Almost all the classical experiments and simulations assume a cavity fed by coolant from one side only. In practical situations, however, there are cases where cooling air may be introduced from both sides of a rotor disk cavity. Such a dual axial inlet arrangement could potentially provide a more uniform cooling and symmetric flow structure, but it also introduces additional complexity: interacting vortical flows from each side, potential instabilities, and a different ingestion/sealing behavior. To date, systematic research on the flow and thermal organization in dual-inlet rotating cavities is lacking. A few recent studies have begun to hint at the possibilities—for instance, Wu et al. [15] proposed a computational approach to accurately determine the adiabatic wall temperature, addressing the classical three-temperature problem encountered in multi-inlet cavities. Guo et al. [16] investigated the effect of unequal flow split between two inlets in a novel twin-web turbine disk, finding that the flow rate distribution can strongly influence heat transfer patterns. These studies, however, are still preliminary and leave many open questions. There is a clear gap in understanding regarding how dual inlet jets interact within the cavity, how the classical single-inlet correlations or turbulence models might need adjustment, and what the net cooling benefit of a dual-inlet design may be.

In this context, the present study provides a comprehensive investigation of a rotating disk cavity with axial coolant inlets. The goal is to elucidate the flow structure and heat transfer characteristics under dual-inlet conditions, thereby addressing the noted research gap. The novelty of this work lies in examining the twin-inlet cavity in a systematic manner—using combined numerical simulation and analysis—to reveal the resulting flow patterns and thermal distributions. Key contributions include evaluation of turbulence model performance in this dual-inlet cavity; identification of how two opposing axial inflows interact and whether they achieve a more uniform cooling of the disk surfaces; and development of several perspectives to investigate the interaction between flow and heat transfer. The findings of this study are expected to advance the fundamental knowledge of rotating cavity flows and provide practical guidance for designing dual-inlet disk cooling systems in next-generation gas turbines. In summary, by exploring a previously under-studied cavity configuration, this work contributes both innovative methodology and impactful results toward more efficient and reliable turbine cooling designs.

1.2. Brief Theoretical Background

Coriolis deflection. In rotating cavities, the dominant rotational effect is the Coriolis acceleration, which acts perpendicular to the local velocity. It turns the inlet jet toward the disks and drives near-wall crossflow; this is most noticeable on the upstream disk.

Ekman layer and pumping. Rotation and viscosity form a thin Ekman boundary layer on the disks. As rotation becomes stronger, this layer is thinner and the near-wall sweeping is more effective, which helps remove heat and reorganize the core recirculation.

This study focuses on time-averaged convection in a rotating cavity; solid conduction and thermal inertia are outside the present scope and are noted as limitations.

2. Numerical Methods for Rotating Cavity

2.1. Computation Model

In order to accurately capture the asymmetry of flow characteristics, the rotating disk cavity was modeled as a full 360-degree annular geometry. This rotating disk cavity features two annular axial inlets, referred to as Inlet 1 and Inlet 2, which are positioned at distinct radial locations. Specifically, Inlet 1 is located near the inner radius r_i, while Inlet 2 is situated at a larger radial position r₃. The temperature of the gas entering through Inlet 2 is set to be 20 K higher than that entering through Inlet 1, thereby inducing thermal stratification within the cavity. Finally, the gas within the cavity exits axially through an outlet located at the same radial position as Inlet 1. A schematic illustration of the two-dimensional dimensions of the rotating disk cavity is presented in Figure 1a, and the three-dimensional geometry of the computational model is depicted in Figure 1b–d.

The axial direction is defined as the z-axis. The geometric parameters of the cavity include the inner and outer radius r_i and r₀, the radial positions of inlets r₂ and r₃, the inlet lengths L₁ and L₂, outlet length L₃, high-radius inlet width c, and cavity gap width s. Detailed values of these parameters are listed in

Table 1. Geometric parameters of the rotating disk cavity model.

Parameter	Value (mm)	Parameter	Value (mm)
r0	500	s	100
r2	172.5	r3	398.5
ri	150	L1, L2	90
c	3	L3	90

.

2.2. Parameter Definitions

In the current study, several key physical parameters were employed to characterize the flow and heat transfer characteristics comprehensively, including the rotational Reynolds number Re_ω (1), the axial Reynolds number Re_z (2), the non-dimensional radial position R* (3), the local convective heat transfer coefficient h (4), the local Nusselt number Nu (5), and Rossby number (6) defined as the ratio of the axial Reynolds number to a rotational Reynolds number. The definitions of these parameters are presented as follows:

$$R{e}_{\omega }={\displaystyle \frac{\rho {r_0}^2\omega }{\mu }}$$

(1)

$$R{e}_z={\displaystyle \frac{2\rho v{r}_{\mathrm{i}}}{\mu }}$$

(2)

$$R^{*}={\displaystyle \frac{r}{r_0}}$$

(3)

$$h={\displaystyle \frac{q}{T_{\mathrm{w}}-T_{\mathrm{ref} }}} [\mathrm{W}/{\mathrm{m}}^2·\mathrm{K}]$$

(4)

$$Nu={\displaystyle \frac{hr}{\lambda }}$$

(5)

$$Ro={\displaystyle \frac{R{e}_z}{R{e}_{\omega }}}$$

(6)

where μ is the dynamic viscosity, ρ is the gas density, r₀ is the radius of the rotating disk, ω is the angular velocity of the disk, v is the inlet velocity, r is the local radial position, and λ is the thermal conductivity of air.

In this study, third-kind boundary conditions were applied, where T_w represents the wall temperature under a prescribed heat flux. Another key parameter is T_ref, which is defined as the wall temperature under zero heat flux conditions (q = 0), and both are obtained through calculations. The adiabatic wall temperature is taken as the reference temperature T_ref.

2.3. Mesh Generation and Grid Independence Verification

Structured hexahedral meshes were generated using ANSYS ICEM 2022R1, and numerical simulations were conducted using ANSYS FLUENT 2022R1 in this study. The overall mesh topology and a detailed view of the computational domain are presented in Figure 2a. To verify mesh independence, five mesh configurations consisting of approximately 8 million, 18 million, 23 million, 28 million, and 33 million cells were evaluated. Under identical boundary conditions, the average convective heat transfer coefficient on the upstream disk surface was compared for each mesh, as shown in Figure 2b. Results indicated a significant increase in the average convective heat transfer coefficient as the mesh size was refined from 8 million to 23 million cells. However, further refinement to 28 million and 33 million cells resulted in negligible differences, suggesting that additional mesh refinement beyond 28 million cells would not substantially improve computational accuracy. Therefore, considering both computational cost and solution accuracy, a mesh consisting of approximately 28 million cells was selected for all subsequent simulations in this study.

2.4. Mesh y+ and Numerical Method Validation

To ensure the accuracy of near-wall heat transfer predictions, the distribution of wall y+ values throughout the computational domain was carefully examined. As shown in Figure 3a, the distribution of wall y+ values indicates that most of the near-wall cells satisfy the criterion y+ < 1, especially in regions characterized by high heat transfer rates and steep velocity gradients.

To further verify the accuracy and reliability of the numerical approach, validation studies were conducted by comparing the present numerical simulations with previously published experimental data. Specifically, Test Case 2 reported by Bohn et al. [17] was selected as a reference to assess the predictive performance of both the SST-kω and RNG-kε turbulence models. Experimental conditions corresponding to this test case are summarized in

Table 2. Experimental conditions and geometry of Bohn et al. [17].

Parameter	Value	Parameter	Value
Reω	8 × 105	Rez	2 × 104
ω	56.67 rad/s	rin	0.12 m
a	0.138 m	b	0.4 m
s	0.08 m	Ro	0.025

.

Figure 3b presents the comparison results, demonstrating that predictions obtained using the SST-kω turbulence model exhibit better agreement with experimental data than those obtained with the RNG model, particularly at the mid-radius and high-radius regions on the disk surface. Based on this comparative analysis, the SST-kω turbulence model was considered more suitable and the average error is less than 10%. Therefore, SST-kω was selected for the numerical simulations of the rotating disk cavity with dual axial inlets in the current study.

Because the operating range in this study includes high rotational Reynolds numbers, we complemented the classical low-Re check against Bohn with a validation against the high-Re experiment of Jackson et al. [10]. Figure 4 compares the radial wall-temperature distribution at a representative condition (Re_ω ≈ 3 × 10⁶). Using the same mesh and boundary conditions, we tested both SST–kω and RNG–kε closures. SST–kω follows the measurements closely over the mid–high-radius region, with a mean deviation below 10% (the largest differences are confined near the rim), whereas RNG–kε tends to over-predict temperatures around the mid-radius. Based on this evidence, SST–kω is adopted as the baseline model for the subsequent parametric study.

To quantify potential errors associated with the steady RANS approach, two representative operating points were recomputed using URANS while keeping the mesh and turbulence model (SST–kω) unchanged. URANS was advanced with a second-order scheme and a time step ensuring Courant numbers of order unity in the jet/core region; time averages were formed after discarding transients. As shown in Figure 5, the disk surface Nusselt distributions predicted by URANS and steady RANS agree closely—differences are typically within 10% and limited mainly to a narrow near-rim region. This agreement supports using steady RANS as the baseline for the wide parametric sweep reported in this study.

2.5. Boundary Conditions and Detailed Simulated Cases

To investigate the effects of different operating conditions on flow and heat transfer characteristics, this study adopts multiple sets of boundary conditions for varying operating conditions. The specific boundary conditions are shown in

Table 3. Boundary conditions.

Parameter	Value	Parameter	Value
Mass Flow Rate at Inlet1 (kg/s)	0.513/0.617/0.729/0.831	Temperature at Inlet1 (K)	300
Mass Flow Rate at Inlet2 (kg/s)	0.170/0.206/0.241/0.277	Temperature at Inlet2 (K)	320
Rotational Speed of Cavity (rpm)	1150/1325/1500/1650/1850	Heat Flux Density (W/m2)	0/2000

.

In this study, a total of 20 operating conditions were designed by varying the rotational Reynolds number and axial Reynolds number, as listed in

Table 4. Summary of operating conditions and corresponding case numbers for numerical simulations, defined by rotational Reynolds number Re_ω and axial Reynolds number Re_z.

	Re	7.27 × 105	8.80 × 105	1.03 × 106	1.18 × 106
Reω
1.92 × 106		Case1	Case6	Case11	Case16
2.21 × 106		Case2	Case7	Case12	Case17
2.50 × 106		Case3	Case8	Case13	Case18
2.79 × 106		Case4	Case9	Case14	Case19
3.08 × 106		Case5	Case10	Case15	Case20

. To determine the wall heat transfer coefficient, two separate cases—one with adiabatic disk surfaces and another with constant heat flux disk surfaces—were calculated for each operating condition. Thus, a total of 40 simulation cases were conducted.

3. Disk Surface Heat Characteristics Under Variable Operating Conditions

This section consists of two parts, presenting an analysis of the heat transfer characteristics on the rotating disk surfaces. The primary focus is placed on examining the temperature distribution variations on both the upstream and downstream disks under different operating conditions, as well as characterizing the local Nusselt number distribution, which indicates the heat transfer performance of the disk surfaces. By comparing these temperature metrics, the overall trends in disk surface heating under varying operating conditions can be clearly observed.

3.1. Temperature Distribution Characteristics on Disk Surfaces

Figure 6 shows typical temperature contours on the upstream and downstream disk surfaces of a rotating cavity. A noticeable difference in temperature distributions between the two disks can be observed. On the upstream disk, the presence of the high-radius inlet divides the disk surface into two distinct thermal regions: a higher-temperature zone between the high-radius inlet and the disk rim, and a lower-temperature convection zone located between the high-radius and low-radius inlets, primarily due to the substantial influx of cooler air from the low-radius inlet. In contrast, the downstream disk exhibits significant temperature variation primarily in the region influenced by the cooler airflow entering at the low-radius inlet.

The temperature distributions on the upstream and downstream disks exhibit distinct patterns under varying operating conditions. To visualize these differences, Figure 7 illustrates the variations in average temperatures of the upstream and downstream disks across different operating conditions. It is evident that the average temperature of the upstream disk exhibits a decreasing trend with increasing rotational Reynolds number, whereas the downstream disk demonstrates a decreasing average temperature with increasing axial Reynolds number. Additionally, the average temperature of the downstream disk is consistently higher compared to that of the upstream disk.

To further investigate the detailed temperature distribution on the disk surfaces, two representative datasets were selected to plot the radial temperature distribution curves for both upstream and downstream disks, as illustrated in Figure 8. Consistent with the previous observations, the upstream disk surface is divided into two distinct regions. This phenomenon is clearly reflected in the temperature profiles: upstream disk temperature initially increases and then gradually decreases radially outward up to the high-radius inlet, after which it rapidly rises and subsequently decreases. Conversely, the downstream disk temperature profile exhibits a stable increase from the lower radius toward the higher radius, followed by a temperature drop near the disk rim. It is worth noting that at the lowest Re_ω (black curve), the upstream-rim peak at R* ≈ 0.9 arises from wall–jet deceleration and a weak rim recirculation that thicken the local thermal boundary layer; with increasing Re_ω, stronger rotation-induced crossflow flushes the rim and the peak diminishes.

3.2. Nusselt Number Distribution Characteristics on Disk Surfaces

To quantify and further understand the spatial variations in heat transfer across the rotating disk surfaces, the radial distributions of local Nusselt number were carefully examined under varying operating conditions. The Nusselt number, which is a dimensionless representation of convective heat transfer, provides clear insights into the local heat transfer effectiveness at the disk–fluid interface.

Figure 9 illustrates the radial distributions of the local Nusselt number on the upstream disk surface at four fixed axial Reynolds numbers (Re_z = 7.27 × 10⁵, 8.80 × 10⁵, 1.03 × 10⁶, and 1.18 × 10⁶), each with varying rotational Reynolds numbers Re_ω. As clearly depicted, the upstream disk Nusselt number distribution exhibits a pronounced convex shape, with distinct peaks occurring approximately at the radial location corresponding to the high-radius inlet (around R* ≈ 0.75). These pronounced peaks result primarily from intensified convective heat transfer induced by enhanced local fluid mixing and strong turbulence generated due to the interaction between the incoming axial jets and the rotating cavity flow.

It can be clearly observed from Figure 9 that, at each fixed axial Reynolds number, the local Nusselt number significantly increases with increasing rotational Reynolds number. Notably, the peak Nusselt number values near the high-radius inlet region are approximately 3–5 times greater than those found at lower-radius regions. This highlights the substantial local heat transfer enhancement driven by higher rotational velocities, which strengthen turbulence and local shear near the disk surface, thereby promoting convective heat exchange efficiency.

In contrast to the upstream disk, the downstream disk displays distinctly different heat transfer characteristics, as presented in Figure 10. At four fixed rotational Reynolds numbers (Re_ω = 1.92 × 10⁶, 2.21 × 10⁶, 2.50 × 10⁶, and 2.79 × 10⁶), variations in axial Reynolds number significantly affect the radial distribution of the Nusselt number on the downstream disk surface. Compared with the upstream disk, the downstream disk demonstrates notably lower peak Nusselt numbers, indicating generally weaker convective heat transfer performance. Additionally, the downstream disk’s Nusselt number distributions clearly exhibit a concave profile, characterized by relatively higher values at the disk rim region (R* ≈ 0.9−1.0) and notably lower values around the mid-radius region (R* ≈ 0.5−0.7). This concave profile is indicative of intensified convective heat transfer at the disk periphery, likely attributable to increased radial convection and turbulence-driven flow disturbances at the disk edge. By “radial convection” we mean forced convection caused by radial advection within the rotating boundary layer, not buoyancy-driven effects. The effect is strongest near the rim because the tangential speed grows with radius (V = Ω × r), increasing wall shear and turbulence production, and the high-radius inlet reinforces the near-wall crossflow. These factors enhance mixing and raise the local Nusselt number at the edge.

Further examination of Figure 10 reveals that, for each fixed rotational Reynolds number, increasing the axial Reynolds number generally enhances the overall Nusselt number across the disk surface. Particularly pronounced improvements are observed at both the lower-radius inlet region (R* ≈ 0.3−0.4) and near the disk rim region at higher axial Reynolds numbers. Distinct local peaks form near the lower-radius inlet region under high axial Reynolds number conditions, indicating significant local heat transfer enhancement driven by stronger axial jet interactions. Conversely, the mid-radius region shows only a moderate increase in Nusselt number with increasing axial Reynolds number, reflecting its comparatively weaker sensitivity to axial jet momentum.

The significant differences in local Nusselt number distributions between the upstream and downstream disks, as well as the distinct responses to varying rotational and axial Reynolds numbers, underscore the complex interplay of fluid mechanics and heat transfer phenomena within the rotating disk cavity. The markedly varied local heat transfer behaviors observed in this chapter emphasize the necessity of exploring internal flow structures, turbulence intensities, and axial jet penetration characteristics within the cavity to gain deeper insights into the underlying physical mechanisms.

3.3. Chapter Summary

This chapter provided a detailed analysis of disk surface heat transfer characteristics under variable operating conditions, focusing specifically on temperature distributions and local Nusselt number variations on both upstream and downstream disks. It was demonstrated that disk surface temperature distributions exhibit clear spatial non-uniformities, influenced distinctly by rotational and axial inflow conditions. Moreover, the Nusselt number distributions highlighted significant radial variations, characterized by pronounced peaks on the upstream disk and distinctive concave profiles on the downstream disk.

The observations made in this chapter clearly reveal that both rotational and axial Reynolds numbers profoundly influence the convective heat transfer performance of the disk surfaces. Specifically, increased rotational Reynolds number markedly enhances local Nusselt numbers on the upstream disk, whereas elevated axial Reynolds number significantly improves convective heat transfer across the downstream disk. However, the underlying physical mechanisms driving these heat transfer variations remain intricately linked to complex internal flow and turbulence characteristics.

Therefore, to thoroughly elucidate these mechanisms, the next chapter systematically examines the internal flow dynamics within the rotating disk cavity. By investigating flow structures, turbulence distributions, vorticity patterns, and jet penetration behaviors, the upcoming analysis aims to provide critical insights required to interpret the heat transfer phenomena observed in the current chapter comprehensively.

4. Flow Characteristics Inside the Disk Cavity Under Variable Operating Conditions

The previous chapter primarily investigated the heat transfer characteristics on the disk surfaces under various operating conditions. In this chapter, attention shifts towards the flow characteristics inside the rotating disk cavity, providing critical insight into the internal fluid dynamics, which fundamentally influence heat transfer performance within the system. The analysis presented here aims to clarify the complex interplay between rotational effects and axial jet flows, highlighting the flow structures formed under different rotational and axial Reynolds numbers.

Figure 11 illustrates the locations of representative cross-sections selected for detailed analysis in this and subsequent chapters. Specifically, cross-sections parallel to the disk surfaces (C1, C2 near the upstream disk; C3 at mid-cavity; C5, C6 near the downstream disk) and one axial cross-section (C4, y = 0, aligned with the jet direction) are introduced. Among these, the mid-cavity cross-section (C3, z = 0.05) and the axial cross-section (C4, y = 0) will be frequently utilized throughout Section 4 and Section 5 to visualize and quantify internal flow patterns, turbulent kinetic energy distributions, jet penetration behavior, and associated heat transfer phenomena. These cross-sections are strategically chosen to facilitate a comprehensive analysis of flow behaviors and clearly demonstrate how changes in operating parameters alter the internal cavity dynamics.

While the current chapter primarily focuses on mid-cavity cross-section (C3) and axial cross-section (C4), additional cross-sections closer to the disk surfaces (C1, C2 near the upstream disk and C5, C6 near the downstream disk) will be utilized in Section 5 to further explore internal heat transfer characteristics.

By systematically employing these representative cross-sections, this chapter provides in-depth insights into how rotational and axial inflow conditions influence the internal flow structures, turbulence intensity, and jet dynamics within the rotating disk cavity. The findings established herein will form an essential foundation for understanding the heat transfer mechanisms discussed thoroughly in the subsequent chapter.

To comprehensively characterize the complex flow behavior within rotating disk cavities, appropriate physical parameters must be carefully selected. Among these, vorticity magnitude is a crucial parameter representing the intensity of local fluid rotation. By examining the distribution and strength of vortical structures within the cavity, we can effectively illustrate how the internal flow responds to variations in rotational and axial Reynolds numbers, thereby clearly highlighting regions of enhanced momentum exchange and potential heat transfer augmentation.

Furthermore, the flow in the rotating cavity is inherently turbulent, and thus, turbulent characteristics significantly influence mixing and heat transfer. Therefore, we selected turbulent kinetic energy (TKE) to quantify the intensity of velocity fluctuations. Regions with higher TKE values typically indicate strong turbulent mixing and consequently enhanced convective heat transfer capabilities. By investigating TKE distributions alongside axial jet penetration behavior, we can explicitly identify how variations in operating conditions alter turbulence intensity, jet penetration depth, and mixing within the cavity.

Thus, the combination of vorticity and turbulent kinetic energy analyses provides a robust framework to fully reveal the internal flow dynamics and associated heat transfer phenomena within the rotating disk cavity.

4.1. Vorticity Distribution and Flow Structures at Mid-Cavity Cross-Section (C3)

To gain deeper insight into the internal flow structures and the corresponding variations under different operating conditions, vorticity magnitude contours along with streamline patterns at the representative mid-cavity cross-section (C3, z = 0.05) were systematically analyzed. Figure 12 illustrates the flow characteristics at this cross-section under varying rotational and axial Reynolds numbers.

Figure 12a–c presents the vorticity magnitude contours and streamline patterns as the rotational Reynolds number increases from 1.92 × 10⁶ to 3.08 × 10⁶, with a constant axial Reynolds number of 8.8 × 10⁵. At a relatively low rotational Reynolds number (Figure 12a), the internal vortical structures appear diffuse and exhibit moderate vorticity intensities, predominantly characterized by lighter blue colors, indicative of weaker fluid rotation. Streamlines reveal relatively simple flow patterns, suggesting limited interaction and mixing within the cavity.

As the rotational Reynolds number increases (Figure 12b), significant changes in the flow structures emerge. The vorticity magnitude contours display notably deeper color shades, transitioning from lighter blue toward green and red, clearly signifying a substantial intensification of fluid rotation. Particularly near the high-radius inlet region, pronounced vortex clusters develop, accompanied by more intricate and stronger secondary flow patterns, as evidenced by increasingly complex streamline configurations. Such enhanced vortex formation results from stronger centrifugal and Coriolis forces generated by the increased rotational velocity.

At the highest rotational Reynolds number examined (Figure 12c), vortical structures are further intensified and become more coherent, with extensive regions of distinctly elevated vorticity magnitudes depicted by deep red and yellow colors. This progressive deepening of color clearly highlights the amplified local fluid rotation within the cavity. Correspondingly, streamline patterns become increasingly complex, with clearly defined multiple vortices indicating vigorous momentum exchange and effective fluid mixing. The enhanced internal vortices, due to intensified rotation, could profoundly influence local convective heat transfer, a phenomenon investigated thoroughly in subsequent analyses.

Figure 12d–f illustrates the evolution of vortical structures and streamline patterns as the axial Reynolds number increases from 7.27 × 10⁵ to 1.03 × 10⁶, at a fixed rotational Reynolds number of 1.92 × 10⁶. Under relatively low axial Reynolds number conditions (Figure 12d), the flow exhibits clear yet less intense vortical structures, characterized by moderate vorticity levels and relatively simple, symmetric streamline patterns, indicating a stable internal flow regime with limited turbulence intensity.

Increasing axial Reynolds number to 8.8 × 10⁵ (Figure 12e) significantly alters the flow dynamics within the cavity. The axial jet momentum strengthens substantially, disrupting previously stable vortices and resulting in fragmented, complex flow structures. The vorticity contours indicate deeper color shades around the jet inlet and cavity mid-region, signaling increased local rotation intensity due to enhanced interactions between axial inflow and rotating cavity fluid. Streamline patterns further reveal pronounced asymmetry and complexity, suggesting enhanced internal turbulence and mixing resulting from stronger axial jet interactions.

With increasing axial throughflow, the coherent vortex cores undergo breakdown and partial merging due to the stronger axial momentum inflow, which reduces their coherence and lowers the number of dominant cores in the cavity. The vorticity magnitude contours vividly depict highly intensified vortical structures, with extensive zones exhibiting deeper colors such as red and orange. This heightened vorticity intensity clearly demonstrates the vigorous fluid rotation triggered by the high axial jet momentum, significantly disturbing and restructuring the flow field within the cavity. Corresponding streamline patterns show highly asymmetric and turbulent flow features, implying substantial enhancement of fluid mixing and turbulence intensity.

Overall, the vorticity distribution and streamline analyses at the mid-cavity cross-section clearly demonstrate distinctive trends resulting from variations in rotational and axial Reynolds numbers. Increased rotation strongly enhances vortex formation and coherence within the cavity, leading to more stable yet significantly intensified flow structures. Conversely, increased axial Reynolds number markedly increases flow complexity and turbulence, creating highly fragmented and dynamic internal vortical structures. These insights clearly illustrate how local fluid rotation intensity and flow mixing characteristics are profoundly altered under varying operational conditions, which are anticipated to significantly affect the heat transfer processes analyzed in subsequent chapters.

4.2. Turbulent Kinetic Energy and Jet Penetration Analysis on Axial Cross-Section (C4)

To further elucidate the internal turbulent characteristics and axial jet penetration behavior within the rotating disk cavity, turbulent kinetic energy (TKE) distributions and corresponding flow streamlines were examined on the axial cross-section (C4, y = 0), aligned with the high-radius jet direction. Figure 13 and Figure 14 present the TKE contours, streamline patterns, and associated variations in jet penetration length under different rotational and axial Reynolds numbers, respectively.

In this work, the jet penetration length denotes the maximum wall-normal distance from the disk surface to the jet core inside the cavity. Practically, a meridional plane through the inlet midline is extracted, the time-averaged field is used, and the jet centerline on this plane is traced as the locus of local maxima of the axial velocity u_z (the TKE ridge yields an equivalent path). The penetration length is the largest normal distance from the disk to this centerline.

Figure 13a–c illustrates TKE contours and jet streamlines under increasing rotational Reynolds number conditions (Re_ω = 1.92 × 10⁶ to 3.08 × 10⁶), at a constant axial Reynolds number (Re_z = 8.8 × 10⁵). At a lower rotational Reynolds number (Figure 13a), the axial jet exhibits a relatively deep penetration length (15.48 mm), accompanied by substantial turbulence intensity within the cavity, as reflected by the pronounced turbulent kinetic energy distribution downstream from the jet inlet. Streamlines indicate significant axial flow penetration and robust interaction between the jet and the cavity interior, facilitating effective turbulent mixing.

However, as the rotational Reynolds number increases (Figure 13b,c), notable changes in flow patterns occur. Specifically, the jet penetration length significantly shortens from 15.48 mm to 6.52 mm, indicative of a pronounced suppression of axial jet penetration caused by increased centrifugal and Coriolis forces. Despite the reduced jet penetration depth, local turbulence intensity near the high-radius inlet markedly intensifies, as clearly evidenced by deeper coloration in the TKE contours. This local enhancement in turbulence arises from more vigorous shear interactions between the incoming axial jet and the strengthened circumferential rotation of fluid near the inlet. Yet, further downstream, deeper inside the cavity, turbulence intensity substantially diminishes due to rapid radial and circumferential redistribution of axial momentum.

The quantitative trend of jet penetration length versus rotational Reynolds number is summarized explicitly in Figure 13d, confirming that increased rotation notably limits axial jet penetration, thus confining enhanced turbulence primarily to regions close to the inlet.

Figure 14a–c presents the variations in turbulent kinetic energy distributions and jet penetration length as the axial Reynolds number Re_z increases from 7.27 × 10⁵ to 1.03 × 10⁶, maintaining a fixed rotational Reynolds number (Re_ω = 1.92 × 10⁶). At the lowest axial Reynolds number considered (Figure 14a), the axial jet penetration length is relatively limited (10.51 mm), with moderate turbulence intensity characterized by comparatively lighter color shades in TKE contours, indicating lower levels of turbulent mixing within the cavity. Streamlines reflect a relatively shallow jet penetration, suggesting limited axial momentum and reduced interaction with downstream regions.

Upon increasing the axial Reynolds number to 8.8 × 10⁵ (Figure 14b), the jet penetration length significantly extends (15.48 mm), accompanied by a marked increase in turbulence intensity, particularly in regions further downstream from the inlet. Higher axial momentum clearly intensifies turbulent mixing, with deeper and more extensive regions of elevated TKE. Flow streamlines confirm enhanced axial penetration, leading to more vigorous interactions within the cavity.

At the highest axial Reynolds number of 1.03 × 10⁶ (Figure 14c), an even more dramatic increase in jet penetration length occurs, reaching up to 41.8 mm. Concurrently, turbulent kinetic energy intensities reach notably higher levels, with prominent regions of intensified turbulence extending deep into the cavity interior. This substantial increase in turbulence intensity and jet penetration highlights the profound influence of axial jet momentum on the internal fluid dynamics, generating highly turbulent conditions throughout the cavity. Figure 14d explicitly summarizes this trend, clearly illustrating the nonlinear increase in jet penetration length with increasing axial Reynolds number.

It should be noted that regions exhibiting elevated turbulent kinetic energy (TKE) are indicative of enhanced local mixing, which is anticipated to significantly influence the convective heat transfer processes analyzed in detail in the subsequent chapter. Furthermore, the variations observed in axial jet penetration depth are particularly important, as they influence local fluid temperature distributions and thus have direct implications for heat transfer efficiency within the cavity, which will be elaborated in Section 5.

4.3. Chapter Summary

This chapter systematically investigated the internal flow characteristics of the rotating disk cavity, focusing on the effects of rotational and axial Reynolds numbers on vorticity distributions, flow structures, turbulent kinetic energy distributions, and axial jet penetration behaviors. It was observed that increasing the rotational Reynolds number significantly enhances vortical structure intensity and local turbulence near the cavity inlet, while simultaneously limiting axial jet penetration. Conversely, increasing the axial Reynolds number substantially extends the jet penetration depth and notably amplifies turbulence intensity throughout the cavity.

These distinctive flow behaviors highlighted in this chapter directly influence the convective heat transfer characteristics within the rotating disk cavity. To fully elucidate this relationship, the next chapter will comprehensively analyze internal heat transfer characteristics by integrating temperature distributions and turbulence-induced heat exchange phenomena observed at various representative cross-sections.

5. Heat Transfer Characteristics Inside the Disk Cavity Under Variable Operating Conditions

In previous chapters, the effects of rotational Reynolds number and axial Reynolds number on disk surface heat transfer and internal flow structures have been systematically discussed. In this chapter, the focus shifts towards comprehensively analyzing the internal heat transfer characteristics within the rotating disk cavity. Specifically, the influences of turbulence intensity, temperature distributions, and sectional temperature gradients on local heat transfer performance are examined under various operating conditions.

5.1. Turbulent Kinetic Energy (TKE) Distributions and Their Influence on Heat Transfer

Figure 15 illustrates the variations in average turbulent kinetic energy (TKE) across five typical cross-sections parallel to the disks (C1, C2, C3, C5, and C6) under different Reynolds numbers conditions. The results reveal a clear correlation between the changes in Reynolds numbers and turbulence intensification.

As shown in Figure 15a, under fixed rotational Reynolds number, increasing the axial Reynolds number significantly enhances the TKE across all considered cross-sections. This enhancement is particularly pronounced at cross-sections closer to the upstream disk surface (e.g., C1 and C2). Similarly, as depicted in Figure 15b, at a fixed axial Reynolds number, an increase in the rotational Reynolds number also significantly elevates TKE values for all cross-sections, with the most substantial increase again occurring near the upstream disk (C1).

Higher turbulent kinetic energy values indicate intensified fluid mixing and stronger local turbulent convection within the cavity, thus greatly promoting convective heat transfer. Therefore, elevated turbulence levels induced by higher rotational and axial Reynolds numbers are expected to substantially enhance the internal heat transfer efficiency.

5.2. Temperature Distribution and Flow Structure Characteristics on Axial Cross-Section (C4)

To elucidate the influence of flow structure on internal thermal fields, Figure 16 presents temperature contours on the axial cross-section (C4) under various operating conditions. Figure 16a–c correspond to cases with a fixed axial Reynolds number (Re_z = 8.80 × 10⁵), and progressively increased rotational Reynolds numbers (Re_ω = 1.92 × 10⁶, 2.50 × 10⁶, and 3.08 × 10⁶, respectively). Conversely, Figure 16d–f show temperature contours at a constant rotational Reynolds number (Re_ω = 1.92 × 10⁶) but with increased axial Reynolds numbers (Re_z = 7.27 × 10⁵, 8.80 × 10⁵, and 1.03 × 10⁶, respectively).

From Figure 16a–c, it is clearly observed that, at fixed axial Reynolds number, an increase in rotational Reynolds number markedly shortens the axial penetration length of the high-radius inlet jet, causing the jet to closely interact with the upstream disk surface. This phenomenon results in enhanced local mixing and consequently improved heat transfer near the upstream disk surface. However, the downstream disk region experiences relatively limited improvement due to diminished jet penetration.

Figure 16d–f demonstrate that at a constant rotational Reynolds number, increasing the axial Reynolds number significantly enhances axial jet penetration, thereby improving the heat transfer effectiveness around the downstream disk surface. Temperature gradients near the downstream disk surface are considerably reduced, indicating a more uniform thermal field within the cavity due to the strengthened axial jet penetration.

These observations are consistent with the internal flow characteristics described in Section 4, further highlighting the critical role of axial jet structure and turbulence intensity in governing the internal heat transfer mechanisms.

5.3. Evaluation of Local Heat Transfer Efficiency Based on Inter-Sectional Temperature Differences

To provide a simplified yet intuitive indicator for local heat transfer evaluation near the disk surfaces, sectional temperature differences between adjacent cross-sections were defined and analyzed. ΔT is defined as the difference in the area-averaged static temperatures on the two axial sections, ΔT = ⟨T_Ci − T_Cj⟩, where ⟨⋅⟩ denotes an area-weighted average over the native mesh; no field remapping is performed. Specifically, the temperature difference between sections C2 and C1 (ΔT_C1→C2) was employed as an indicator near the upstream disk, while the difference between sections C5 and C6 (ΔT_C6→C5) was used near the downstream disk. The closer the temperature difference between the two surfaces is to zero, the more effective the heat transfer between them. Figure 17 presents these temperature difference curves under various operational parameters.

As seen in Figure 17a, under fixed axial Reynolds numbers, increasing rotational Reynolds number clearly reduces the temperature difference near the upstream disk, indicating enhanced heat transfer efficiency in that region. This can be attributed to the increased turbulent mixing and stronger fluid–wall interactions associated with higher rotational velocities.

Figure 17b indicates that, for the downstream disk region, the temperature differences remain relatively stable across various rotational Reynolds numbers, except under conditions with lower axial Reynolds numbers. This observation implies that the rotational Reynolds number exerts limited influence on heat transfer enhancement near the downstream disk surface.

Figure 17c reveals that under fixed rotational Reynolds numbers, increasing axial Reynolds numbers surprisingly results in larger temperature differences near the upstream disk. This phenomenon indicates that stronger axial jet penetration may cause local thermal stratification near the upstream disk surface, negatively impacting heat transfer effectiveness in that region.

Conversely, Figure 17d demonstrates a clear reduction in the temperature difference near the downstream disk with increasing axial Reynolds number under fixed rotational conditions. Thus, enhanced axial jet momentum significantly improves heat transfer performance near the downstream disk surface. However, this improvement becomes less pronounced at higher rotational Reynolds numbers, suggesting that strong rotational effects may mitigate the axial jet’s capability to enhance heat transfer.

In summary, these inter-sectional temperature difference analyses further clarify the beneficial impacts of increasing rotational Reynolds numbers on upstream disk heat transfer and axial Reynolds numbers on downstream disk heat transfer. Nevertheless, the limitations of this simplified evaluation method must be acknowledged, and it is suggested that future studies incorporate detailed local Nusselt number analyses for more comprehensive assessment.

5.4. Chapter Summary

This chapter systematically investigated the internal heat transfer characteristics within the rotating disk cavity under variable operating conditions. The key findings can be summarized as follows:

Increasing rotational and axial Reynolds numbers significantly elevates turbulent kinetic energy, thereby enhancing turbulent mixing and convective heat transfer within the cavity.

Enhanced rotational Reynolds numbers notably improve heat transfer near the upstream disk by shortening axial jet penetration and intensifying local turbulence, but have limited influence on downstream disk heat transfer.

Increasing axial Reynolds numbers markedly improves heat transfer near the downstream disk surface due to deeper jet penetration, but may adversely affect heat transfer efficiency near the upstream disk surface by inducing local thermal stratification.

The simplified approach of evaluating local heat transfer efficiency using inter-sectional temperature differences provides intuitive insights into the trends of heat transfer changes, though it has inherent limitations in precision and scope. Future studies are recommended to combine these insights with detailed local Nusselt number distributions for comprehensive heat transfer evaluation.

The subsequent chapter will integrate all findings presented in this study, summarize major conclusions, and offer recommendations for practical engineering applications and future research directions.

6. Conclusions

In this study, a systematically validated computational approach was employed to investigate the detailed flow structure and heat transfer characteristics in a rotating disk cavity with dual axial inlets under variable rotational and axial Reynolds numbers. The analyses spanned rotational Reynolds numbers from approximately 1.9 × 10⁶ to 3.1 × 10⁶ and axial Reynolds numbers from 7.3 × 10⁵ to 1.2 × 10⁶. Results underscore the complex interplay between rotational-induced turbulence and axial jet penetration, significantly influencing the local heat transfer distribution on both upstream and downstream disk surfaces. The key conclusions are summarized as follows:

• The rotational Reynolds number significantly governs heat transfer on the upstream disk surface. With increased rotation, the local turbulence near the high-radius inlet intensifies markedly, leading to enhanced convective heat transfer characterized by pronounced peaks in the Nusselt number distribution. However, the rotation simultaneously reduces axial jet penetration, limiting effective cooling benefits for the downstream disk region.
• The axial Reynolds number primarily influences downstream disk cooling efficiency. Increased axial flow enhances coolant penetration depth within the cavity, substantially improving convective cooling on the downstream disk surface by reducing local temperature gradients. Nevertheless, excessive axial throughflow may cause localized thermal stratification near the upstream disk, slightly offsetting heat transfer gains previously achieved by increased rotational speed.
• Integrated consideration of rotational and axial flow effects reveals the necessity for balanced operating conditions. An optimal configuration involves sufficiently high rotational speed to enhance upstream disk cooling, combined with appropriate axial coolant flow rates to ensure effective downstream disk cooling. Misalignment between rotation and axial throughflow conditions can lead to non-uniform cooling performance and reduced overall heat transfer efficiency.

This work provides essential new insights into the governing mechanisms of flow and heat transfer within dual axial inlet rotating disk cavities, laying a foundation for the informed optimization of secondary air systems in modern aero-engines. Future studies incorporating conjugate heat transfer analyses and high-fidelity simulations will further enhance our understanding and guide the design of advanced cooling strategies.