Flow Characteristics and Mass Flow Distribution Mechanism Within Multi-Inlet and Multi-Outlet Corotating Disc Cavities

Abstract

This study reveals the governing mechanism of mass flow distribution within a multi-inlet, multi-outlet corotating disc cavity, providing critical insights for designing advanced gas turbine secondary air systems. An experimentally verified numerical investigation is conducted across a range of rotational Reynolds numbers $R{e}_{\varphi }=5\times {10}^6~2\times {10}^7$ and axial Reynolds numbers $R{e}_z=2\times {10}^5~5\times {10}^5$, corresponding to Rossby numbers $Ro$ from 0.01 to 0.10. Results highlight that $Ro$ governs the internal flow and outlet mass flow distribution through two distinct regimes. In the rotation-dominated regime at low $Ro$, the radial outlet mass flow ratio $MR$ decreases sharply, and a stable, dual-zone vortical structure forms. As $Ro$ increases, growing inflow inertia disrupts this structure, causing vortices to merge, which enhances swirl uniformity and slows the rate of $MR$ decrease. This transition dictates outlet performance: the radial outlet discharge steadily improves with $Ro$, while the axial outlet performance increases abruptly around $Ro=0.02$ before saturating.

1. Introduction

Secondary air systems (SAS) remain integral to modern gas turbine designs, ensuring the delivery of cooling flows to thermally critical components in turbine sections. SAS designs are commonly a labyrinthine network composed of seals, orifices, and rotating cavities [1]. This configuration creates corotating disc cavities characterized by multiple inlets and outlets, which collectively govern the supply and distribution of SASs, cooling airflow to targeted regions. The arrangement of these components directly dictates the thermal management requirements for high-thermally-stressed turbine components. Typical SAS designs feature the lower radial multi-inlet-outlet rotating cavities that this study is interested in. Traditionally, SAS airflow design has employed generous redundancy in flowrate to guarantee adequate cooling air delivery to critical components. However, driven by the escalating demands for improved thermal efficiency in next-generation gas turbines, optimizing the utilization of limited SAS airflow has become imperative. This necessitates balancing the maximization of overall engine efficiency with the maintenance of component reliability under extreme thermal operating conditions. Consequently, engine designers require a detailed understanding of flow behavior and the ability to predict mass flow distribution within complex corotating cavities.

The fundamental understanding of flow within corotating disc cavities is built upon decades of theoretical, experimental, and numerical research. Early theoretical and analytical work by pioneers such as Hide [2] and Owen et al. [3] categorized the flow into distinct source, sink, and core regions bounded by Ekman boundary layers. These foundational concepts were later validated and expanded upon by numerical investigations such as by Chew et al. [4], which successfully replicated key flow structures and explored more complex phenomena like vortex breakdown in cavities with axial throughflow. Parallel investigations into cavities with axial throughflow, notably by Farthing et al. [5], identified more complex phenomena, including the formation of toroidal vortices and the occurrence of axisymmetric and non-axisymmetric vortex breakdown, which were shown to be dependent on geometric parameters and the Rossby number.

Based on these prior works, Owen and Roger [6] synthesized these concepts into a more comprehensive model for superposed flow, in which they developed a categorized flow structure model for a superposed radial outflow corotating disc cavity with axial inflow, as shown in Figure 1. According to this model, the axial inflow jet impinges on the downstream disc and transitions into a radially outward wall jet. This jet separates into two different flow paths: one portion becomes entrained by the rotating downstream disc, while the remainder diverges to form the source region at the lower radius before ultimately being entrained by the upstream disc. As the entrained flows develop along the corotating discs, they form Ekman-type boundary layers that converge at the shroud into a unified sink layer, which ultimately exits the cavity as radial outflow. An interior core region lies between the Ekman-type layers and the source/sink regions. This is characterized by rotating fluid velocities lower than the local disc rotational velocity, in which this core exhibits minimal shear, which is a stark contrast to the high-velocity gradients observed in the adjacent boundary layers.

While experimental and analytical methods dominated early fundamental studies of corotating cavities, numerical approaches have increasingly complemented these conventional methodologies in recent years. Much of this computational work has focused on configurations relevant to modern gas turbines, particularly the high-pressure (HP) compressor cavities. For instance, Sun et al. [7] employed both Reynolds-averaged Navier–Stokes (RANS) and Large-Eddy Simulation (LES), successfully replicating key flow structures like the rotating core and Ekman layers in a cavity with radial inflow and axial throughflow. Beyond validating fundamental structures, numerical methods have been employed to explore the influence of specific operating conditions. Xu et al. [8] investigated the impact of varying inflow swirl ratios on corotating disc flow structures using their findings to identify distinct flow patterns and refine pressure drop prediction models. Furthermore, the practical implications of these flows, such as windage and thermal effects, have been studied. Zhang et al. [9] combined RANS simulations with a novel static testing technique to quantify windage torque in a conical corotating cavity, directly linking the flow physics to the windage-induced temperature rise in compressor discs. Corotating cavity research has also progressed towards engineering novel designs to actively control these complex flow structures; for instance, Shen et al. [10] designed and validated a hybrid vortex reducer that uses de-swirl shroud orifices and fins to suppress Ekman layer development and significantly reduce overall pressure loss.

While existing research has extensively explored various aspects of corotating cavities, a notable focus has been on configurations with single inlets and outlets. Guo et al. [11] investigated the flow characteristics in a twin-web turbine disc cavity, which inherently features a multi-outlet design. The work highlights how different coolant inlet strategies, such as central versus pre-swirl modes, lead to distinct flow distributions among the outlets. Similarly, Lin et al. [12] performed a comprehensive evaluation of a multi-inlet and multi-outlet pre-swirl rotor-stator system, revealing that the secondary seal outflow significantly decreases the main pre-swirl supply flow rate that reaches the turbine blades. Liu et al. [13] conducted experimental and theoretical evaluation of another rotor-stator pre-swirl system with similar inlet-outlet configuration, whose focus was on quantifying the loss in swirling velocity due to the secondary sealing flows. However, these studies are largely focused on rotor-stator pre-swirl systems rather than corotating disc cavities. In addition, the work that focuses on the mass flow distribution within multi-inlet and multi-outlet configuration such as Lin et al. [12] do so with fixed secondary mass flow rates, leaving the distribution mechanism between two outlets unexplored.

A critical factor underpinning the accuracy of these numerical investigations is the choice of turbulence model, which must capably resolve the complex phenomena inherent to rotating cavities. Regarding turbulence modeling, an earlier numerical investigation by Morse [14] utilized an in-house elliptical solver with a low-Reynolds-number k-ϵ model to predict source–sink flow structures in 2D asymmetrical cavities. Although the numerical results exhibited strong agreement with experimental data for radial inflow, the study highlighted challenges in resolving turbulent Ekman layer development during radial outflow, especially at reduced flow rates and rotational speeds. Subsequent comparative analyses by Kumar et al. [15] evaluated the Spalart–Allmaras (SA), k-ϵ, and Reynolds stress models (RSM) against experimental data and momentum-integral methods [16]. Their findings indicated that the RSM provided more accurate predictions than the SA and k-ϵ models, which exhibited excessive turbulent diffusion in core regions. However, recent work by Xu et al. [8] demonstrated superior predictive accuracy of the SST k-ω model over both RSM and k-ϵ formulations. This was attributed to the model’s hybrid formulation, combining the near-wall precision of the k-ω approach with k-ϵ-like behavior in free-stream regions, thereby enabling improved resolution of steep gradients and swirling vortex core structures characteristic of corotating cavities. Additionally, Onori et al. [17], explored large eddy simulation (LES) applications for corotating cavities, yielding flow predictions consistent with SST k-ω results and experimental measurements.

The review of the existing relevant literature reveals a significant gap, where the vast majority of research has focused on simplified cavity models characterized by a single primary inlet and a single primary outlet. This is in sharp contrast to the practical configurations in modern gas turbines, which often feature complex secondary air system networks with multiple inlet and outlet paths. While a few studies have explored multi-inlet and multi-outlet systems, these have predominantly centered on rotor-stator pre-swirl cavities. Consequently, the interaction between inflow jets and their subsequent division among various outlets in corotating disc cavities, which is a mechanism that cannot be fully captured by models developed for simpler geometries, still remains a critical and underexplored area. As advancing next-generation gas turbine efficiency hinges on maximizing cooling performance under stricter flowrate constraints, investigation into the fluid dynamics of multi-inlet and multi-outlet corotating disc cavities is essential to bridge the gap between fundamental research and the predictive capabilities required for advanced SAS design.

To address this critical knowledge gap, the present study employs numerical simulations to systematically investigate the flow characteristics and mass flow distribution within a multi-inlet and multi-outlet corotating disc cavity. The investigated flow condition for rotational Reynolds number $R{e}_{\varphi }$ ranges from $5\times {10}^6$ to $2\times {10}^7$ and for axial Reynolds numbers $R{e}_z$ from $2\times {10}^5$ to $5\times {10}^5$, which corresponds to Rossby number $Ro$ ranging from 0.01 to 0.10. The aim is to characterize the dominant flow structures and reveal the underlying mass flow distribution mechanisms, providing foundational insights for the design and analysis of complex secondary air systems.

2. Numerical Methodology

2.1. CFD Model and Boundary Conditions

The schematic diagrams of the investigated model are presented in Figure 2. Cooling airflow is supplied to the cavity via two inlets, one at the bottom of the cavity and one in the middle radial region at the upstream-side rotor, which will be referred as low inlet (LI) and high inlet (HI), respectively. The main flow region known as the corotating disc cavity is comprised of the encasing rotor walls and an upper shroud at the disc rim. While the low inlet and axial outlet are connected via a straight annular passage, the radial outlet is situated at the upper center shroud as a narrow circumferential gap. The low inlet and high inlet are both assigned as a mass flow inlet. The mass flow rate of the low inlet is set to values corresponding to the inflow axial velocity reflecting the target axial Reynolds number $R{e}_z$, expressed as follows:

$$R{e}_z=\rho V_{z,in}{r}_{LI}/\mu ,$$

(1)

Meanwhile, the mass flow rate of the high inlet is assigned with an identical value as the low inlet. A fully developed swirl assumption was given to both inflows, in which the tangential components are set to match the local tangential speed of the rotor disc. All wall surfaces are given a rotating wall boundary condition which matches the target rotational Reynolds number $R{e}_{\varphi }$, given as follows:

$$R{e}_{\varphi }=\rho \mathsf{\Omega }{r}_b^2/\mu ,$$

(2)

The investigated $R{e}_z$ and $R{e}_{\varphi }$ flow conditions in this study range across $2\times {10}^5<R{e}_z<5\times {10}^5$ and $5\times {10}^6<R{e}_{\varphi }<2\times {10}^7$, respectively. Temperature-wise, the inlet mass flows are assigned at 300 K, and all walls are set to adiabatic. For the radial and axial outlets, they were assigned a static pressure boundary condition of 0 Pa, assuming discharge to an ambient environment. As shown in Figure 2a, the circumferentially periodic configuration permits a 30° sector domain simplification, with the side walls assigned with periodic boundary conditions.

As displayed in Figure 2b, the geometrical parameters of the cavity are chosen from a simplification and abstraction of the lower SAS cavities. The overall outer radius of the entire cavity is denoted with $r_b$. While the outer and inner radii of the axial channel are $r_a$ and $r_c$, respectively, the median radius of the low inlet is denoted as $r_{LI}$. For the high inlet, the median radius is located at $r_{HI}$. The annular gap width of the high inlet $g_{HI}$ is determined by matching the area of the high inlet to be identical to the area of the low inlet. The gap width between the two corotating disc is $g_s$. At the radial outlet, the gap width $g_{RO}$ is determined by an outlet area ratio $S=A_{RO}/A_{AO}$, which is the area of the radial outlet $A_{RO}$ relative to the area of the axial outlet $A_{AO}$. The details of the geometrical parameters are summarized in

Table 1. Cavity geometrical parameters.

Parameters	Values
$g_s/r_b$	$0.4$
$r_{LI}/r_b$	$0.25$
$r_c/r_b$	$0.1$
$r_{HI}/r_b$	$0.75$
$A_{RO}/A_{AO}$	$1$

.

2.2. Grid and Mesh Independence Study

The computational domain of the corotating disc cavity model is discretized using a tetrahedral mesh generated via ANSYS Fluent Meshing from ANSYS Fluent 2022 R1, as illustrated in Figure 3. The meshing generation setting sets the maximum face size and maximum cell length as equal. Local grid refinement is implemented in the narrow radial outlet gap regions to resolve steep velocity gradients, which is achieved by halving the maximum face size and cell length. To accurately capture near-wall viscous effects, a boundary layer mesh with thin prismatic elements is applied to wall-adjacent regions. The first layer height is set to 0.001 times the radial thickness of the axial channel, ensuring compliance with wall function requirements by maintaining dimensionless wall distances $y^{+}$ below 5 throughout the cavity. A total of 10 thin boundary layers are generated at the walls, with a progressive expansion ratio of 1.2 to ensure smooth transition to the bulk tetrahedral mesh.

A grid independence study is performed to determine the optimal mesh resolution, with computational grids spanning element counts from 400,000 to 4,000,000. The grids are prepared by reducing the maximum face size and maximum cell length simultaneously. The study was conducted using boundary conditions of an axial Reynolds number of $R{e}_z=2\times {10}^5$ and a rotational Reynolds number of $R{e}_{\varphi }=5\times {10}^6$. As shown in Figure 4, the variation of axial-inlet to radial-outlet pressure difference plateaus as the mesh density increases, indicating diminishing sensitivity to further refinement. Relative to the finest grid, deviations in pressure difference fall below 1% for grids exceeding 2.838 million elements. To balance numerical precision with computational efficiency, the mesh configuration satisfying this convergence threshold is selected for subsequent simulations. This approach ensures solution accuracy while maintaining tractable computational resource requirements.

2.3. Numerical Method and Validation

The numerical method employed in this study utilizes the commercial computational fluid dynamics (CFD) code ANSYS Fluent 2022 R1 to solve steady-state RANS equations for the corotating disc cavity domain, utilizing a pressure-based solver. The coupled scheme is adopted for pressure–velocity coupling. Gradient discretization is implemented using the Least Squares Cell-Based scheme, while spatial discretization for pressure employs a second-order scheme. All other physical quantities are discretized with a second-order upwind discretization scheme.

The fluid in the flow domain is modeled as ideal compressible air. Under the adiabatic wall assumption adopted in this investigation, the following fluid properties are prescribed as constant: specific heat of air $c_p=1006.43 \mathrm{J}/(\mathrm{k}\mathrm{g} \mathrm{K})$, thermal conductivity $k=0.0242 \mathrm{W}/(\mathrm{m} \mathrm{K})$, and dynamic viscosity $\mu =1.7894e-05 \mathrm{k}\mathrm{g}/(\mathrm{m} \mathrm{s})$.

According to the literature review conducted on previous numerical investigations, the selection of an optimal turbulence model for corotating disc cavity flows remains unresolved within the research community. Consequently, this study undertakes a comparative analysis of three turbulence models identified as promising in prior literature: the Reynolds Stress Model (RSM), realizable k-ϵ, and SST k-ω. Numerical predictions were validated against experimental data from Owen et al. [3] and a theoretical reference solution. The experimental dataset comprises radial swirl ratio distributions $\beta$, defined as the fluid tangential velocity $V_{\varphi }$ nondimensionalized by the local rotating wall velocity $\Omega r$, measured in the cavity core region at $R{e}_{\varphi }$ ranges of ${10}^5$ to ${10}^6$. For validation at $R{e}_{\varphi }={10}^7$, the swirl distribution in the core region of a radial outflow cavity was derived using linear Ekman layer equations from established theoretical work [18]. The equation is expressed as follows:

$${\beta }_{core}=1-2.22C_w^{5/8}R{e}_{\varphi }^{0.5}{\left(r/r_b\right)}^{13/8},$$

(3)

where $C_w$ is the dimensionless mass flow rate ${\dot{m}}_{in}/\mu b$.

For all validation cases, $C_w$ was selected to correspond with the experimental conditions, specifically at $C_w=2500$. However, it is important to note that this validation was performed against the experimental data from Owen et al. [3], which describes a simplified single-inlet, single-outlet source-sink cavity. This differs fundamentally from the multi-inlet, multi-outlet configuration analyzed in the present study, which introduces significantly more complex flow interactions from competing inlet jets and outlet paths. As illustrated in Figure 5, all turbulence models generated numerical predictions that generally conform to experimental and theoretical trends. However, detailed comparative analyses indicate notable discrepancies. The Reynolds Stress Model (RSM), depicted by the lightest dotted line, underpredicts the swirl distribution across all $R{e}_{\varphi }$ values. Conversely, the k-ϵ model (represented by the dash-dot line) exhibits good agreement with validation data at higher $R{e}_{\varphi }$ cases; however, its numerical results substantially overpredict the swirl ratios in lower radial regions. In contrast, the numerical results obtained using the SST k-ω model demonstrate consistently high accuracy across the entire range of $R{e}_{\varphi }$. The average errors $\overline{err}$, summarized in

Table 2. Numerical methods errors [19].

Case	$\mathit{R}{\mathit{e}}_{\mathit{\varphi }}$	${\overline{\mathit{e}\mathit{r}\mathit{r}}}_{\mathit{k}-\mathit{ϵ}}$	${\overline{\mathit{e}\mathit{r}\mathit{r}}}_{\mathit{S}\mathit{S}\mathit{T}\mathit{k}-\mathit{\omega }}$	${\overline{\mathit{e}\mathit{r}\mathit{r}}}_{\mathit{R}\mathit{S}\mathit{M}}$
Exp1	$5.47\times {10}^5$	20.34%	9.21%	10.14%
Exp2	$8.17\times {10}^5$	11.17%	7.99%	8.85%
Exp3	$1.1\times {10}^6$	6.02%	8.74%	11.00%
Theory	$1\times {10}^7$	2.90%	4.19%	5.91%

[19], further corroborate the aforementioned analysis, indicating that the SST k-ω model maintains errors below 10%. Despite the inherent limitations of validating against a geometrically dissimilar case, this comparative analysis is crucial. It demonstrates that the SST k-ω model, with its hybrid formulation suited for resolving both near-wall boundary layers and free-stream vortex structures, is the most robust choice for capturing the physics of the corotating disc cavity under investigation. Hence, the validated numerical methodology incorporating the SST k-ω model was selected for this investigation.

3. Flow Similarity Under Rossby Number Characterization

The investigation conducted in this study reveals that flow characteristics within the multi-inlet and multi-outlet corotating disc cavity are governed by the interaction between the axial inflow inertia and the rotational effects of the discs. To characterize the balance between inertial and rotational forces, the dimensionless Rossby number $Ro=R{e}_z/R{e}_{\varphi }$ is introduced as a critical parameter quantifying the ratio of axial inflow inertia to rotational effects. This parameter enables clear distinction between flow regimes: rotational dominance occurs at low $Ro$ (corresponding to high $R{e}_{\varphi }$ or low $R{e}_z$ conditions), whereas axial inertial forces prevail at high $Ro$ (high $R{e}_z$ or low $R{e}_{\varphi }$ conditions).

Figure 6 illustrates the streamlined diagram overlayed on the corresponding swirl ratio contours of the cavity radial–axial midplane. The figure presents two pairs of cases where the Rossby number is held constant while the constituent Reynolds numbers $R{e}_z$ and $R{e}_{\varphi }$ are varied. Despite the different individual Reynolds numbers, the resulting flow fields exhibit striking similarity. According to Figure 6a,b, both $Ro=0.02$ cases show a large vortical structure in the lower half of the cavity and a group of smaller vortices adjacent to the high inlet. The swirl ratio contours are nearly identical, indicating that the tangential velocity distribution is governed by the same underlying physics. Similarly, according to Figure 6c,d, the flow patterns for a higher $Ro=0.04$ case are again remarkably consistent with each other but are distinctly different from the $Ro=0.02$ cases. The size and interaction of the vortical structures have changed, yet the flow similarity between the two cases with the same Rossby number is evident. This demonstration of flow similarity confirms that the Rossby number is a valid parameter for characterizing the flow regime, even in corotating cavities with complex multi-inlet and multi-outlet configurations. Therefore, the following analysis is framed against an increasing Rossby number. This approach allows for a systematic investigation of the flow development from a rotation-dominated regime to a mixed-flow regime where the influence of inflow inertia becomes increasingly significant.

A primary focus of this investigation is the distribution of mass outflow between the radial and axial outlets. The distribution is quantified by the mass ratio, $MR$, which represents the fraction of the total inflow mass rate (${\dot{m}}_{LI}+{\dot{m}}_{HI}$) that exits through the radial outlet (${\dot{m}}_{RO}$), while the remaining portion ($1-MR$) exits through the axial outlet. Figure 7 illustrates the relationship between mass flow ratio $MR$ and Rossby number $Ro$ across different $R{e}_z$ conditions of $2\times {10}^5$, $3\times {10}^5$, $4\times {10}^5$, and $5\times {10}^5.$ The graph demonstrates that $Ro$ is capable of collapsing the inertial and rotational effects into a single governing parameter, consolidating different $R{e}_z$ conditions within a multi-inlet and multi-outlet rotating cavity onto a unified similarity curve. According to Figure 7, the $MR$ plot reveals two distinct regimes of flow behavior, demarcated by an approximate $Ro$ of 0.02, denoted as Regime I and Regime II in the figure. Under low $Ro$ conditions within the range of Regime I, where the rotational effects dominate the flow, the $MR$ is high and is observed to decrease swiftly as $Ro$ increases. In contrast, under high Ro conditions within the range of Regime II, where the inertial effects begin to take hold over rotational effects, the $MR$ is noticeably lower than in Regime I. While the $MR$ exhibited a linear reduction trend in Regime I, a gradual power-law decay $MR$ trend is exhibited in Regime II. Compared to Regime I, the rate of MR reduction diminishes significantly within Regime II. In addition, further enhancement of axial inertial effects results in only marginal decreases in MR, demonstrating a reduced sensitivity of the cavity system to inertial forces at elevated $Ro$ levels.

4. Radially-Stratified Vortical Flow Structure

The streamline diagrams of the cavity, captured at the radial–axial midplane, are displayed in Figure 8. Subfigures (a)–(h) depict the flow structures under an increasing $Ro$ condition of 0.01 to 0.10. In general, the streamline diagrams provided in Figure 8 shows that the flow within the cavity is characterized by a complex, radially stratified, dual-zone vortical structure that evolves significantly with an increasing Rossby number. This dual-zone formation is a direct consequence of the interaction between the two distinct inlet jets, creating a more complex flow field than the single, dominant vortex typically observed in the foundational single-inlet, single-outlet cavity studies. The two primary vortical zones are located in the lower half of the cavity, driven by the low inlet, and the upper half, which is influenced by the high inlet.

At the lowest Rossby number of 0.01, Figure 8a shows that rotational effects dominated the cavity flow. The inflow jet from the low inlet is immediately deflected upwards and entrained by the upstream disc as it enters the cavity. A small counter-clockwise vortex occupies the space between the low inflow jet and the upstream disc. As it flows radially upwards, it merges with the inflow supplied by the high inlet. An axially elongated clockwise vortex is situated in the upper radial region, directly under the high-inlet jet. As $Ro$ increases to 0.02 in Figure 8b, the flow structure changes significantly. The low inlet jet now forms an axial throughflow moving towards the outlet, though some of this flow impinges on the downstream disc, creating a clockwise vortex in the low radial region. Meanwhile, the high-inlet jet penetrates a short axial distance before being deflected upwards by rotational forces. The jet generates a series of counter-rotating vortices across the high radial region. Further increasing the Rossby number to 0.03 and 0.04, as illustrated in Figure 8c,d, results in minimal alteration to the recirculation zone in the lower radial region. The primary flow structure changes are observed in the upper recirculation zone. With rising $Ro$, the high-inlet jet achieves greater axial penetration before being deflected radially upward in a smooth trajectory. Consequently, the downstream vortices are compressed against the disc, while the upstream vortex expands, filling the upper corner.

As Ro progresses to 0.05, the main change observed in Figure 8e is that the downstream-side vortices within the upper recirculation zone merge into a single vortex. The upper recirculation zone is now characterized by a pair of counter-rotating vortices, with the high-inlet jet flowing between them directly toward the radial outlet. Meanwhile, the large vortex in the lower recirculation zone experiences a reduction in its radial span. According to Figure 8f–h, further increases in Ro to 0.10 saw the lower radial recirculation zone evolve into a structure of multiple radially stacked vortices. The initial primary vortex, formed by the impingement of the low-inlet jet on the downstream disc, continues to contract in its radial extent. Meanwhile, in the upper recirculation zone, the trajectory of the high-inlet jet remains stable. However, the greater inflow momentum of the high inlet associated with the higher Rossby number causes a stronger impingement of the deflected jet on the shroud. The resulting flow divergence leads to the expansion of the clockwise vortex in the upper recirculation zone, causing it to extend downwards past the radial position of the high inlet. This overflowing mass from the upper vortex subsequently spills into the lower recirculation zone, feeding the complex, radially stacked structure and severely disrupting the local flow field.

As introduced in the preceding section, the swirl ratio $\beta =V_{\varphi }/\mathsf{\Omega }r$ quantifies the nondimensionalized tangential velocity distribution within the cavity relative to the local tangential velocity of the rotor surface. Figure 9 presents swirl ratio contours on the radial-axial midplane of the cavity under increasing Rossby number, similar to the streamline diagrams discussed earlier. Overall, the contours reveal distinct radially stratified zones that develops due to the interaction between the two inlet jets and the rotating cavity walls. With increasing radius, the swirl ratio shows an initial decrease. This trend is interrupted by the high inlet, which introduces high tangential momentum and locally increases the swirl ratio before it begins to diminish again.

At the lowest Ro condition of 0.01, Figure 9a shows that the supplied inflow jets of both low and high inlets are highly swirled, and these high-swirl regions remain localized near the inlets and surface of the upstream disc. As the inflow jet travels radially upwards, the swirl ratio of the jet trajectory is shown to decrease. In addition, the swirl ratio contour reaches its lowest point where the jets from the low and high inlets combine. As $Ro$ progresses onward to 0.05 between Figure 9b–e, the momentum from the stronger low inlet jet compresses the low-swirl zone at the central radial region of the cavity. Simultaneously, the increased momentum of the high inlet jet expands the high-swirl region around it, causing it to penetrate deeper into the cavity core. The core of the lowest swirl ratio shifts to the downstream disc, coinciding with the upper part of the primary vortex in the lower recirculation zone. As Ro increases beyond 0.06, this low-swirl region diminishes and is pushed further against the downstream disc, as shown in Figure 9f–h. Concurrently, the high-swirl area tied to the upper recirculation zone overflows, extending below the high inlet’s radial position. These observations suggest that higher inflow rates increase the tangential momentum of the overall cavity flow and enhances mixing that distributes this momentum, resulting in a more uniform rotating flow field.

In order to further analyze the cavity swirl ratio in detail, the axial distributions are probed at several radial locations of $r/r_b=0.3, 0.5, 0.7, 0.9$ from the axial location of the upstream rotor surface ($z=-0.1 \mathrm{m}$) to the axial location of the downstream rotor surface ($z=0.1 \mathrm{m}$), as presented in Figure 10. Subfigures (a) to (d) present the swirl ratio axial distributions under varying Rossby numbers of Ro = 0.01, 0.02, 0.04, 0.08, respectively.

Several general characteristics of the axial swirl ratio $\beta$ distributions are apparent according to Figure 10. Firstly, the swirl ratio is consistently approaching unity near the upstream and downstream rotor surfaces, indicating the expected behavior of fluid corotating with the discs. Secondly, between the two rotor surfaces, the profiles exhibit steep gradients near the walls and a relatively flat profile in the central core region. These features are characteristic of rotating cavity flows, representing the thin Ekman boundary layers where viscous effects from the discs are dominant, and an axially central core where the flow is largely inviscid. Thirdly, a distinct radial stratification of the swirl ratio is evident, with the innermost radial location ($r/r_b=0.3$) generally exhibiting the highest swirl ratios across the axial span and often shows slight swirl fluctuations near the downstream disc. In contrast, the central cavity radial locations ($r/r_b=0.5$ and $0.7$) generally display lower swirl ratios compared to the $r/r_b=0.3$ location. The outermost probed radius at $r/r_b=0.9$ often presents a swirl ratio level higher than the central cavity radial locations.

Figure 10 reveals that increasing $Ro$ noticeably enhances the swirl ratio $\beta$ magnitude across all probed locations. At the lowest $Ro$ condition of 0.01, the profile in Figure 10a shows a clear high-swirl region near the upstream disc, particularly at the low-radial region ($r/r_b=0.3$), which corresponds to the immediate radially upwards deflection shown previously in Figure 8a. The weak penetration of the inflow jet under highly rotational-dominated flow conditions is reflected by the axial swirl profile of $r/r_b=0.3$, extending a quarter axial span of the cavity and immediately falling flat at axial locations $z\approx -0.05 \mathrm{m}$. As Ro increases to 0.02, the swirl ratio at $r/r_b=0.3$ in Figure 10b initially fluctuates around $\beta =1$ near the upstream disc ($z<0 \mathrm{m}$) and subsequently spans a range of approximately $\beta =0.9$ near the downstream side ($z>0 \mathrm{m}$). In Figure 10c,d, the fluctuation is further amplified as $Ro$ is further increased to 0.04 and 0.08, which is able to elevate the fluctuation peak to approximately 1.2 and 1.4, respectively. This fluctuation near the upstream side at $r/r_b=0.3$ is attributed to the complex tangential shear interaction occurring at $r_a$, the interface between the low axial channel inflow and the opposing recirculating fluid within the corotating disc cavity.

For the axial swirl ratio distributions at $r/r_b=0.7$, a location just below the high inlet ($r_{HI}/r_b=0.75$), the swirl ratio profile changes the most dramatically across Figure 10c,d as $Ro$ increases beyond 0.04, with the mode swirl ratio escalating from approximately 0.6 to over 1.0. This sharp increase corroborates the contour plot analysis, confirming that the influence of the strongly swirled high-inlet jet extends radially inward, energizing the tangential flow in the lower portions of the cavity. Concurrently, the pronounced dip in the swirl ratio profiles at $r/r_b=0.5$ clearly marks the axial location of the primary vortex core in the lower recirculation zone. As the Rossby number increases, this vortex core dip migrates from the upstream to the downstream side of the cavity. This feature is most prominent at $Ro=0.08$, where the swirl ratio plunges from over 1.2 near the upstream disc to a minimum of approximately 0.5. As seen in the Figure 9 contour plots, this demonstrates that despite the strong downward overflow of high-swirl fluid from the upper zone, the dynamics in the compressed lower region remain dominated by the impingement of the low-inlet jet.

To complement the axial profiles, Figure 11 presents the radial distribution of the circumferentially averaged swirl ratio examined at the cavity midplane ($z=0 \mathrm{m}$), which illustrates the effect of varying $Ro$ conditions. The vertical dotted line at $r_{HI}/r_b$ marks the radial position of the high inlet, providing a reference for the flow interactions occurring in the outer region of the cavity.

A dominant trend is that an increase in the Rossby number consistently results in a higher swirl ratio across the entire radial span. At lower Rossby numbers of $Ro=0.01$ to 0.04, where the lower and upper recirculation zones are distinctly separated, the profile exhibits a characteristic U-shape. This is defined by a significant dip in the mid-radial region ($r/r_b\approx 0.5-0.7$) and a subsequent peak beyond $r/r_b=0.75$, which corresponds to the influence of the high-inlet jet. As Ro increases, this low-swirl region is progressively compressed. For instance, the radial span of the U-shaped dip contracts from approximately 0.9 wide at $Ro=0.01$ to 0.5 wide at $Ro=0.04$. This quantitative evidence supports the earlier observation that higher inflow rates enhance the transport of tangential momentum into the cavity, promoting better mixing and resulting in a more uniform, high-swirl rotating flow field. In stark contrast, the swirl profile at $Ro=0.08$ is fundamentally different, exhibiting a distinct M-shaped distribution. This shape is indicative of the more complex flow field, highlighting the presence of the multiple, radially stacked vortex structures that form at higher Rossby numbers. The multiple inflection points in the profile directly reflect the severe disruption caused by the overflowing, high-swirl mass from the upper recirculation zone interacting with the lower zone.

This evolution from a simple U-shaped profile to a complex M-shaped distribution is a defining feature of the studied multi-inlet and multi-outlet corotating disc cavity system. It reflects the competing influences of the two inlets, which is a phenomenon entirely absent in single-inlet models, which tend to produce simpler swirl profiles without such abrupt variations in the mid-radial region.

In summary, the flow within the multi-inlet, multi-outlet corotating cavity is characterized by a radially stratified vortical structure that is highly sensitive to the Rossby number. At low $Ro$ conditions, the rotational forces dominate, leading to distinct upper and lower recirculation zones with minimal interaction. As $Ro$ increases, the growing influence of inflow inertia alters this structure significantly. The high-inlet jet penetrates deeper, compressing and eventually merging the vortices in the upper zone, while the lower zone contracts and develops a more complex multi-vortex system. This development directly impacts the swirl ratio, with higher Ro numbers enhancing cavity swirl and a more uniform swirl distribution.

5. Influence of Rossby Number on Multi-Outlet Flow Characteristics

As the governing vortex flow structure has been established in the preceding section, this section now examines the consequences on the multi-outlet flow characteristics. The mass flow distribution between both axial and radial outlets is fundamentally driven by the pressure generated within the cavity. Therefore, the analysis begins with a detailed investigation of the cavity pressure distribution. Building on this, the investigation then introduces discharge coefficients to quantitatively evaluate the performance of each outlet, framing the results against the Rossby number to unify the findings across the different operating conditions.

5.1. Cavity Pressure Distributions

To analyze the pressure field across various operating conditions, the static pressure is normalized into a pressure coefficient, defined as $C_P=(P-P_{LI})/\left(0.5\rho {\Omega }^2{r}_b^2\right)$, with the low-inlet static pressure $P_{LI}$ serving as the reference. Figure 12 presents the resulting $C_P$ contours for Rossby numbers from 0.01 to 0.10. A fundamental characteristic observed across all cases is the strong radial pressure gradient. The pressure consistently increases from the inner radius towards the outer shroud. Consequently, the highest-pressure coefficient is consistently located at the outermost radius of the cavity near the upper shroud.

According to Figure 12, it is evident that an increase in $Ro$ introduces significant local variations to this general pressure field. At the lowest inflow condition of $Ro=0.01$ shown in Figure 12a, the pressure distribution is comparatively smooth, exhibiting a gentle radial gradient. As the inflow from both inlets increases, Figure 12b presents a distinct low-pressure pocket, which begins to form in the high recirculation region near the upstream disc. As $Ro$ further increases to beyond 0.03 in Figure 12c–h, this low-pressure pocket becomes more pronounced. This low-pressure region, seen in the upper upstream corner of the cavity, corresponds directly to the core of the expanding upstream vortex in the upper recirculation zone, as identified in Figure 8. The increased momentum of the high-inlet jet allows it to penetrate deeper into the cavity, enhancing local velocity within this upstream vortex. Concurrently, the radially upwards deflection of the high-inlet jet impinges on the shroud, causing stagnation pressure to build up at the outer radial regions of the cavity. It is noted that the pressure coefficient adjacent to the shroud progressively increases with higher $Ro$.

Furthermore, a localized region of higher pressure can be observed in Figure 12b, starting at Ro = 0.02 onwards, on the downstream disc where the low-inlet jet impinges, marking a distinct stagnation point at the base of the lower recirculation zone. The increase in $Ro$ is observed to intensify the localized region of high pressure due to low-inlet jet impingement. However, the pressure in this low radial region remains significantly lower than the high-pressure regions near the shroud.

To further investigate the radial pressure variations, Figure 13 presents the circumferentially averaged radial distribution of the pressure coefficient $C_P$ at the cavity midplane ($z=0 \mathrm{m}$), which illustrates the effect of varying $Ro=0.01$ to $0.08$. The vertical dotted line at $r_{HI}/r_b$ marks the radial position of the high inlet, providing a reference for the flow interactions occurring in the outer region of the cavity.

According to Figure 13, increasing $Ro$ from $0.01$ to $0.08$ causes an overall increase in the pressure coefficient $C_P$ across the majority of the cavity radius, confirming the radial gradient observations in Figure 12. For the lower $Ro$ conditions, the profiles show a relatively smooth increase. The $Ro=0.01$ condition exhibits a gentle, nearly linear profile, peaking at a $C_P$ of approximately 0.2. As $Ro$ increases to 0.02 and 0.04, the pressure gradient becomes progressively steeper, with the peak $C_P$ near the shroud increasing to approximately 0.4 and 0.7, respectively. In stark contrast, the pressure profile for Ro = 0.08 deviates significantly from a smooth curve. Instead, it exhibits large fluctuations, particularly beyond the high-inlet radial location $r_{HI}/r_b$. These pressure fluctuations are a direct consequence of the large, low-pressure pocket associated with the strong vortex in the upper recirculation zone, as observed in the pressure coefficient contours.

The cavity pressure distribution is strongly dictated by the Rossby number. While a fundamental radial pressure gradient exists due to the rotational effects of the corotating disc cavity, the overall magnitude of the pressure field is determined by the strength of the inflows from the two inlets. A stronger inflow, associated with a higher Rossby number, results in a more forceful impingement on the downstream disc and the upper shroud. This intensified impingement creates a high-pressure stagnation zone, significantly raising the pressure near the shroud and the axial outlet.

5.2. Discharge Coefficient of the Different Outlets

In order to examine the performance of the two outlets of the cavity, a discharge coefficient for the radial and axial outlets are to be defined. The discharge coefficient, $C_d$, which is defined as the ratio of the actual measured mass flow rate of the outlet ${\dot{m}}_{actual}$ to the theoretical mass flow rate ${\dot{m}}_{theoretical}$ that corresponding outlet would experience under ideal conditions [20], is shown as follows:

$$C_d={\dot{m}}_{actual}/{\dot{m}}_{theoretical},$$

(4)

An inviscid ideal flow assumption is made for the theoretical mass flow rate for the radial outlet ${\dot{m}}_{theoretical,RO}$, where the pressure difference across the radial outlet is entirely transformed into kinetic energy. This ideal assumption allows for the utilization of Bernoulli’s equation to determine the exit velocity under ideal conditions, consequently the mass flow rate can be calculated from the pressure difference:

$${\dot{m}}_{theoretical,RO}=A_{RO}\sqrt{2\rho \mathsf{\Delta }{P}_{RO}},$$

(5)

The pressure difference $\mathsf{\Delta }{P}_{RO}$ driving the radial outlet mass flow is assumed to be the difference between the pressure at the shroud $P_b$ and the radial outlet back pressure $P_{RO}$. Based on the observations made in the previous section regarding cavity pressure distributions, the shroud pressure is the result of two superimposed mechanisms: the global pressure rise from centrifugal forces and the localized stagnation effect from the high inlet jet’s impingement. For this calculation, $P_{T,HI}$ is taken as the mass-weighted average of the total pressure at the high inlet boundary, assuming negligible loss before impingement. This shroud pressure $P_b$ can be formulated as given:

$$P_b={P_{T,HI}+P}_{LI}+{\displaystyle \frac{1}{2}}\rho {\Omega }^2(r_b^2-r_{LI}^2),$$

(6)

A similar ideal-flow approach is used for the axial outlet. However, the determination of the relevant pressure differential $\mathsf{\Delta }{P}_{AO}$ is similarly complex. As observed in the pressure coefficient distributions (Figure 12), the axial throughflow due to the low inlet creates a high-pressure stagnation region on the downstream rotor surface near the axial outlet. This stagnation pressure is considered the effective driving potential for the axial outlet. Here, the low inlet total pressure $P_{T,LI}$ is taken as the mass-weighted average of the total pressure at the axial inlet boundary, assuming negligible loss before impingement. Consequently, $\mathsf{\Delta }{P}_{AO}$ is defined as the difference between the low inlet total pressure $P_{T,LI}$ and the axial outlet back pressure $P_{AO}$ as the driving pressure behind the theoretical axial outlet mass flow rate ${\dot{m}}_{theoretical,AO}$:

$${\dot{m}}_{theoretical,AO}=A_{AO}\sqrt{2\rho (P_{T,LI}-P_{AO})},$$

(7)

The discharge coefficients for the radial outlet $C_{d,RO}$ and axial outlet $C_{d,AO}$ are plotted against the Rossby number $Ro$, as shown in Figure 14a,b, respectively. For both outlets, the data points corresponding to different axial Reynolds numbers $R{e}_z$ collapsed onto a single, well-defined curve. This further confirms that the $Ro$ is the dominant dimensionless parameter, successfully unifying the individual effects of inflow inertia and cavity rotation to govern the performance of each outlet. While both coefficients generally increase with $Ro$, their distinct characteristics reveal the competing mechanisms that dictate the overall mass flow distribution between the two outlets.

According to Figure 14a, $C_{d,RO}$ exhibits a positive correlation with $Ro$, and a closer inspection reveals a subtle change in gradient beyond $Ro=0.04$. Under low $Ro$ conditions, the $C_{d,RO}$ trend exhibits a steeper increasing gradient compared to high $Ro$ conditions beyond $Ro=0.04$. This is due to the vortex structure in the upper recirculation zone, which effectively obstructs the outlet path, as shown in Figure 8a–c. As $Ro$ increases towards the transition point, this obstructing vortex gradually diminishes, allowing the outlet’s performance to improve progressively. Under higher $Ro$ conditions, this growth in $C_{d,RO}$ continues; however, its gradient decreases. Beyond Ro = 0.04, the continued but more gradual rise in $C_{d,RO}$ is attributed to a different mechanism. As the streamline diagrams in Figure 8 indicate, the flow trajectory toward the radial outlet is already largely unobstructed. Consequently, the slighter increase in the discharge coefficient is driven by pressure effects rather than further optimization of the flow path. Specifically, the increasing momentum of the high-inlet jet intensifies its impingement on the shroud, creating a localized stagnation pressure that supplements the centrifugal pressure head and continues to improve the radial outlet’s discharge performance.

In contrast, $C_{d,AO}$ exhibits a distinctly different characteristic, as shown in Figure 14b. Its performance is highly sensitive to the flow transition. Unlike the gradual improvement of the radial outlet, $C_{d,AO}$ increases sharply once inflow inertia is sufficient to overcome rotational suppression at around $Ro=0.02$. This is evident between Figure 8a,b, where the abrupt establishment of a direct throughflow path from the low inlet is observed when $Ro$ increases from 0.01 to 0.02. Once this path is fully formed, the performance of the axial outlet effectively saturates, causing the rate of growth for $C_{d,AO}$ to diminish and plateau at a value of approximately 0.85 for $Ro$ greater than 0.04.

In summary, the performance of the different outlets shows a clear quantitative dependence on the Rossby number. Both outlets exhibit distinct trends in their performance and influencing mechanisms. The discharge coefficient of the radial outlet $C_{d,RO}$ is influenced by a combination of factors: the overall rotational effects within the cavity, the obstruction or clearing of the flow path by the upper vortex structure, and the localized stagnation pressure from the high-inlet jet impingement on the shroud. In contrast, the discharge coefficient of the axial outlet $C_{d,AO}$ is primarily dictated by the establishment of a direct flow path from the low inlet, which is highly sensitive to rotational suppression at low Rossby numbers and plateaus once the path is formed.

6. Conclusions

In this study, an experimentally validated numerical method is employed to analyze the flow characteristics and mass flow distribution within a corotating disc cavity with a multi-inlet and multi-outlet configuration. The current investigation systematically examines the flow behavior of the cavity across rotational Reynolds numbers $R{e}_{\varphi }$ spanning $0.5\times {10}^7$ to $2.0\times {10}^7$ and axial Reynolds numbers $R{e}_z$ ranging from $2\times {10}^5$ to $5\times {10}^5$, corresponding to Rossby numbers $Ro$ ranging from 0.01 to 0.10. The presented results underscore the competing influence between the rotational effects of the corotating discs and the inertial effect of the two streams of inflow jets on the flow characteristics, which in turn drives the mass flow distribution between the multiple outlets. The key conclusions are summarized as follows:

1.

The relationship between the mass flow ratio $MR$ and the Rossby number $Ro$ is characterized by two distinct regimes: in the rotation-dominated regime at low $Ro$ values (Ro < 0.02), the high $MR$ decreases sharply, while in the mixed rotational-inertial regime at higher $Ro$ values, the $MR$ is lower, and its rate of decrease becomes significantly more gradual.

2.

The flow within the cavity is characterized by a radially stratified, dual-zone vortical structure, the evolution of which is highly sensitive to the Rossby number $Ro$. At low $Ro$ conditions, where rotation dominates, two distinct upper and lower recirculation zones are formed with minimal interaction, a complexity not present in simpler single-inlet cavities. As $Ro$ increases, the growing influence of inflow inertia causes the upper zone vortices to compress and eventually merge, leading to an overflow of high-swirl flow that disrupts the lower zone and creates a multi-vortex structure. This structural evolution directly enhances the cavity tangential momentum and leads to a more uniform swirl ratio distribution at higher $Ro$.

3.

The Rossby number $Ro$ dictates the mass flow distribution by governing the performance of each outlet through distinct mechanisms. The radial outlet discharge coefficient $C_{d,RO}$ steadily increases with $Ro$, a trend driven first by the clearing of an obstructing vortex at low Ro and then by rising shroud stagnation pressure at high $Ro$. In contrast, the axial outlet discharge coefficient $C_{d,AO}$ increases abruptly around $Ro=0.02$ as inflow inertia overcomes rotational suppression to establish a direct throughflow path, after which its performance saturates and plateaus.

This study reveals new insights into the governing factors of flow characteristics within corotating disc cavities featuring complex multi-inlet and multi-outlet configurations. From an engineering perspective, these findings establish a clear framework for predicting cooling flow distribution. This insight sets a clear path for designing a predictive model for engine designers. By calculating the Rossby number based on the operating inflow rates and rotational speeds, designers can anticipate whether the system will operate in a rotation-dominated or a mixed-flow regime, which in turn allows for the targeted optimization of the cooling supply. While this paper provides valuable initial insight, future work is required to investigate a broader range of operating conditions on cavities with different geometrical parameters, such as varying inlet/outlet area ratios and locations. Future work should investigate the influence of heat transfer from the discs, as temperature gradients may alter fluid properties and influence the stability of the vortical structures. Expanding this research will allow for the development of a comprehensive and robust predictive model, enabling designers to fine tune secondary air systems for next-generation gas turbines with even greater precision and efficiency.