Analysis of Aerodynamic Characteristics of Rotating Detonation Turbine Based on Proper Orthogonal Decomposition Method

Abstract

The unsteady interactions in rotating detonation turbine engines (RDTE) remain poorly understood. To address this, a 2D numerical model integrating a rotating detonation combustor (RDC) with a first-stage turbine is established to analyze flow structures and aerodynamics under various detonation modes. Proper orthogonal decomposition (POD) reveals intrinsic links between flow features and performance metrics. Results show that while the RDC generates total pressure gain, it induces significant unsteady flow. Guide vanes partially suppress pressure fluctuations but cannot eliminate total pressure losses or circumferential non-uniformity, reducing rotor efficiency. Increasing detonation wave numbers decreases total pressure gain at rotor inlet but improves flow uniformity: the counterclockwise double-wave mode exhibits optimal performance (27.9% work gain, 5.0% instability, 86.4% efficiency), whereas the clockwise single-wave mode shows the poorest (20.9% work gain, 11.8% instability, 84.0% efficiency). POD analysis indicates first-order modes represent time-averaged flow characteristics, while low-order modes capture non-uniform pressure distributions and pairing phenomena, reconstructing wave propagation. The study highlights discrepancies between turbine inlet’s actual unsteady flow and conventional quasi-steady design assumptions, proposing enhancing mean flow characteristics and increasing first-mode energy proportion to improve work extraction. These findings clarify the detonation wave mode–turbine performance correlation, offering insights for RDTE engineering applications.

1. Introduction

In recent years, pressurized combustors based on detonation combustion have gained significant attention due to their high heat release rates, efficiency, and low entropy increase. Among these, the rotating detonation combustor (RDC) has emerged as a research hotspot in new propulsion systems because of its compact design, high-frequency operation, and stable thrust output [1,2]. Compared to traditional constant-pressure combustors, RDCs enhance thermodynamic performance through auto-pressurization and efficient cycles. Rotating detonation turbine engines (RDTE) can reduce compressor stages and lower compression work, thereby improving overall engine efficiency and representing a key direction in turbine engine development [3,4,5,6].

However, the coupling characteristics between RDCs and turbines present a significant research challenge. The complex flow fields generated by RDCs, particularly under multi-detonation wave mode, pose substantial difficulties for turbine design. Extensive research has been conducted both domestically and internationally on RDC combustion characteristics and their interaction with turbine vanes. The US Air Force Institute pioneered relevant experiments by integrating a T63 turboshaft engine [7,8]. Results showed that RDC exit flow fields exhibit high-frequency pressure fluctuations and non-uniform temperature distributions, potentially leading to significant deterioration in turbine aerodynamic performance. Experimental studies by Wu et al. [9,10,11] revealed that when the detonation wave propagation direction is inconsistent with the blade flow angle, the vane pressure drop increases by at least 10%. Zhou et al. [12] experimentally demonstrated that guide vanes could reduce pressure oscillations by 64%, but narrowed the engine’s stable operating range. Numerical studies by Li et al. [13] confirmed that oblique shock-vane interactions increase wake losses, while Ji et al. [14] found that flow field inhomogeneity raises vane loading and exacerbates flow losses. Although these studies have revealed some macroscopic phenomena, most have been limited to observations and preliminary analyses of specific phenomena, failing to systematically uncover the intrinsic physical mechanisms of the coupled flow field between RDC and turbines. Particularly under multi-detonation wave mode, the complexity and nonlinearity of the flow field make traditional research methods insufficient for comprehensively capturing flow details and key features, hindering the establishment of a clear physical relationship between transient flow structures and performance losses, and limiting the ability to guide turbine matching design optimization. To clarify mechanisms behind rotating detonation turbine engine (RDTE) performance, modal decomposition emerges as a key research direction. Dynamic mode decomposition (DMD) may misinterpret strong nonlinear transients like shock reflection and multi-wave coupling as discrete frequency superpositions, while fast Fourier transform (FFT) identifies time-frequency features but fails to resolve spatial modal correlations, limiting spatiotemporal analysis of detonation–turbine interactions.

Proper orthogonal decomposition (POD) offers a new approach to overcoming the aforementioned challenges. Initially proposed by Lumley [15] for turbulent coherent structure analysis, POD has increasingly been applied to complex flow phenomena to uncover hidden characteristics. In combustor research, Antonio et al. [16] used POD to quantify the spatiotemporal features of unsteady flow fields, accurately characterizing different combustion stages. For compressor near-stall conditions, Wang et al. [17] employed POD to capture dominant flow features in stall flow fields. In synthetic double-jet studies, POD revealed the evolution of jet vortex pair structures [18]. Additionally, combining POD with deep neural network (DNN) data-driven models has shown potential for accurate complex flow field feature capture and low-cost simulations [19]. Current POD research trends focus on the spatiotemporal evolution mechanisms of complex flow fields, making it an effective tool for understanding unsteady flow mechanisms. However, studies on the application of POD analysis in the flow fields of rotating detonation turbines remain relatively limited. Given the complex spatiotemporal coupling information inherent in the combined flow field of RDCs and turbines, POD methods can effectively achieve spatiotemporal decoupling of the flow field, thereby clearly analyzing its spatial distribution characteristics and temporal fluctuation properties. This provides a novel perspective for a deeper understanding of the coupling effects between RDCs and turbines.

This study focuses on the aerodynamic characteristics of turbines under different operating modes of rotating detonation combustors (RDCs). A two-dimensional numerical model encompassing an RDC and a single turbine stage was developed to systematically analyze the impact of various detonation inflow conditions on turbine aerodynamic performance. Using proper orthogonal decomposition (POD), the study emphasizes modal feature extraction of turbine stage flow fields, revealing flow modality under different detonation inflow conditions and clarifying the intrinsic relationship between detonation wave mode and turbine aerodynamic characteristics.

2. Materials and Methods

2.1. Two-Dimensional Numerical Model and Boundary Conditions

In this study, numerical simulations of the combined model of RDC and turbine are conducted using the density-based solver on the ANSYS Fluent platform version 16.0. The unsteady Reynolds-averaged Navier–Stokes (URANS) equations are solved to accurately capture the dynamic interaction between the detonation wave and turbine blades, which has demonstrated high computational efficiency and the ability to capture transient features, as widely adopted in previous relevant studies [20,21]. The turbulence model adopts the k-omega SST model, which is advantageous in accurately capturing boundary layer flows and adapting to highly non-uniform flow field characteristics [22,23]. In terms of numerical schemes, the Roe-FDS method is applied for flux decomposition, while the second-order upwind scheme is used to discretize the convective terms. A dual time-stepping implicit iterative strategy is adopted for time advancement, with the physical time step set at $1\times {10}^{-7}$ s following the methodology in reference [24]. The simulation employs premixed hydrogen–air as the fuel, and the chemical reaction mechanism is based on a four-component one-step global reaction for hydrogen–air mixtures. The chemical reactions are calculated using the laminar finite rate model, and the reaction rate constant $k_f$ can be calculated by the Arrhenius formula:

$$k_f=A{T}^b\exp \left(-E/RT\right),$$

(1)

where $A=9.87\times {10}^8$, $b=0$, $E=3.1\times {10}^7\mathrm{J}/(\mathrm{kg}·\mathrm{mol})$. The above Arrhenius coefficients are chosen based on previous references [20,25] and have been widely used in the RDC simulations.

This study adopts the geometry of a turbine based on the first-stage blade of the high-pressure turbine of the GE-E3 gas turbine engine [26]. To simplify the computational model, the annular geometry of the rotating detonation combustion chamber (RDC) is approximated as a two-dimensional rectangular domain by unfolding it along one of its generating lines. This simplification is reasonable because the annular width of the RDC is significantly smaller than its axial length and circumference, allowing the curvature effects to be neglected. The validity of this approach has been demonstrated in previous studies [27,28,29]. Therefore, the two-dimensional computational domain, as shown in Figure 1, consists of the RDC and a single-stage turbine cascade. The RDC has a width W = 540 mm, and a length L1 = 100 mm. The turbine guide vanes are located at a distance L2 = 250 mm downstream of the RDC outlet. The computational domain is discretized using structured meshes, and the mesh density is enhanced in the vicinity of the turbine cascade to achieve a nondimensional wall distance (y⁺ ≈ 1), thereby ensuring accurate resolution of the near-wall regions.

The left end of the computational domain is set as a velocity inlet. Hydrogen–air premixed gas is injected into the RDC through a series of micro Laval nozzles [30]. The injection conditions are set as follows: total pressure $P_{t,0}$ = 0.5 MPa, total temperature $T_{t,0}$ = 300 K, and Laval nozzle area ratio $A_w/A_t$ = 16. Assuming isentropic flow in the tube, two Mach numbers are present at the Laval nozzle exit: ${\mathit{Ma}}_{b1}$ (>1) and ${\mathit{Ma}}_{b2}$ (<1), which correspond to $P_{b1}$ and $P_{b2}$, respectively, as shown in (2):

$${\displaystyle \frac{A_w}{A_t}}={\displaystyle \frac{1}{\mathit{Ma}}}{[{\displaystyle \frac{2}{\gamma +1}}(1+{\displaystyle \frac{\gamma -1}{2}}{Ma}^2\left)\right]}^{{\displaystyle \frac{\gamma +1}{2(\gamma -1)}}},$$

(2)

Next, the inflow velocity of the premixed gas at the inlet will be calculated based on the Laval nozzle theory combined with wall pressure calculations. This will be analyzed under the following scenarios:

(1)

When $P_w$ > $P_{t,0}$, the premixed gas is not injected into the combustor.

(2)

When $P_{b2}$ < $P_w$ < $P_{t,0}$, the premixed gas is injected into the combustor at subsonic speed.

(3)

When $P_{b1}$ < $P_w$ < $P_{b2}$, the throat of the nozzle remains choked, with a normal shock wave downstream, and the premixed gas is injected into the combustor at subsonic speed.

(4)

When $P_w$ < $P_{b1}$, the premixed gas is injected into the combustor at supersonic speed.

The upper and lower boundaries of the computational domain are modeled as periodic boundary conditions, while the outlet is defined as a pressure outlet with a fixed back pressure of 1 atm. The turbine blades are treated as adiabatic, no-slip walls, and the rotor blades move translationally along the negative y-axis at a constant velocity of 300 m/s. To initiate detonation wave propagation, a premixed zone (0 < x < 0.01 m, 0.125 < y < 0.15 m, P = 0.1 MPa, T = 300 K) and a high-temperature, high-pressure ignition zone (0 < x < 0.01 m, 0.115 < y < 0.125 m, P = 2.5 MPa, T = 2500 K) are established at the initial time step. This setup successfully generates a quasi-periodic flow field with stable detonation wave propagation. Additionally, the positive and negative y-axis directions are defined as clockwise (CW) and counterclockwise (CCW) propagations, respectively.

Grid independence tests are conducted with three mainstream mesh sizes (fine mesh: 0.3 mm, medium mesh: 0.4 mm, and coarse mesh: 0.5 mm) under an equivalent ratio of 0.7. For the three mesh sizes, the thickness of the first boundary layers around the turbine is the same, and the total numbers of grids are 1.36 million, 0.92 million, and 0.66 million, respectively. Figure 2a presents the comparison of the azimuthal pressure distribution at the axial position $x=0.002$ m for the three mesh sizes. The pressure distribution curve clearly exhibits good consistency. Since the specific detonation wave structure is not discussed herein, and the medium grid size is sufficient for capturing the typical detonation flow field characteristics, the medium grid size is selected for the following research to balance accuracy against cost.

Figure 2b presents the flow field temperature distribution of a single detonation wave. From the figure, the typical structure of rotary detonation combustion can be clearly observed: A is the rotating detonation wave, B is the oblique shock wave induced by the detonation wave, C is the slip line between freshly detonated products and older products, D is the region with blocked micro nozzles, E is the fresh premixed gas injected from micro nozzles, and F is the mixing region between fresh premixed gas and product gas. The above results are consistent with the detonation wave structure observed in experiments [31], further confirming the accuracy of the numerical method adopted herein.

To further verify the accuracy of the numerical method in solving detonation parameters, this study conducts a numerical simulation check on one-dimensional detonation phenomena in a shock tube. The shock tube is 0.5 m long and filled with hydrogen–air premixed gas at an equivalence ratio of 1, with an internal pressure of 1 atm and a temperature of 300 K. Ignition and explosion are initiated at the initial moment by a high-temperature and high-pressure hot spot (with a pressure of 60 atm and a temperature of 3000 K) on the left side.

Table 1. Comparison of the detonation parameters between numerical results and CJ theoretical values.

	Detonation Pressure/MPa	Detonation Temperature/K	Detonation Wave Velocity/(m/s)
Numerical result	3.15	2539	1664
CJ theoretical value	3.26	2630	1798
Error	3.37%	3.46%	7.45%

compares the errors between the numerical results and the CJ theoretical values. It can be seen that the hydrogen–air one-step chemical reaction model can accurately predict detonation combustion parameters. The relative errors of $P_{CJ}$ and $U_{CJ}$ are small, while the larger error of $T_{CJ}$ could be attributed to the neglect of intermediate reaction products in the one-step global reaction model, leading to an overestimation of the released heat.

2.2. Proper Orthogonal Decomposition Method

The POD method can be applied to the analysis of any flow variable in the flow field. This study focuses on the extraction of pressure oscillation characteristics in the flow. Based on the Reynolds decomposition principle, the transient spatiotemporal pressure field $p(x,t_i)$ can be decomposed into the time-average component $\bar{p}(x,t)$ and the fluctuating component $p^{\prime }(x,t_i)$. Among them, the fluctuating component can be expanded using orthogonal basis functions:

$$p^{\prime }(x,t)={\sum }_{j=1}^{\infty }{a}_j\left(t\right){\varphi }_j\left(x\right),$$

(3)

where the functions ${\varphi }_j\left(x\right)$ are orthogonal basis functions relating to spatial characteristics, while the functions $a_j\left(t\right)$ are time-dependent functions relating to temporal characteristics. Note that functions $a_j\left(t\right)$ correspond only to ${\varphi }_j\left(x\right)$ and are independent of any other $\varphi$ functions.

Thus, all fluctuating pressure components from N snapshots of the flow field obtained at m grid points can be written into an m × N matrix (where N ≪ m):

$${\mathit{p}}^{\prime }=\left[{\mathit{p}}_1,{\mathit{p}}_2,\dots ,{\mathit{p}}_N\right],$$

(4)

By performing singular value decomposition (SVD), the spatiotemporal characteristic mode can be obtained as

$${\mathit{p}}^{\prime T}=\mathit{U}\mathit{\Sigma }{\mathit{V}}^T,$$

(5)

The characteristic function ${\varphi }_j\left(x\right)$ and its corresponding eigenvalue ${\lambda }_j$ are extracted from matrix ${\mathit{V}}^T$ and $\mathit{\Sigma }$, respectively, while the modal coefficient $a_j\left(t\right)$ is computed from $\mathit{U}\mathit{\Sigma }$. Based on the SVD results and (3), a series of POD modes can be constructed to perform low-dimensional approximation processing on the original flow field. For more detailed information about matrix processing in the POD method, please refer to the relevant literature [32].

In the POD analysis of the flow field, different POD modes represent various flow structures in the flow field, and the magnitude of the corresponding eigenvalues indicates the proportion of energy captured. In other words, POD modes with higher eigenvalues correspond to large-scale coherent structures that dominate in the flow field [33]. The spatial structure of each POD mode remains unchanged over time, while the time-varying modal coefficients reflect the fluctuations in modal energy intensity within the instantaneous flow field. Through the POD process, dominant flow structures and their temporal and spatial characteristics can be extracted from the unsteady flow field. The essence of the flow field can be better described and analyzed based on the first few POD modes of the flow field.

3. Results and Discussion

3.1. Turbine Flow Field Analysis

In RDTE, the intake air faced by the turbine components has a high-frequency pulsating pressure and temperature, which is bound to have a significant impact on the stability and reliability of the turbine, and this effect will change with the transformation of RDC operating mode. To better understand the variation of turbine flow structure and working characteristics under the rotating detonation flow, a simulation is performed under the equivalent ratio $\varphi =0.5$, and the aerodynamic characteristics of the turbine under different RDC working modes are analyzed.

Figure 3 illustrates the temperature distribution in the coupled flow field between the RDC and the turbine under various detonation wave propagation modes, and its flow field results are similar to the flow field structures of rotating detonation turbine engines (RDTE) in References [13,25]. Due to the inherent strong unsteadiness of rotating detonation waves, the temperature distribution in the single-wave flow field exhibits significant non-uniformity. In the multi-wave flow field, multiple rotating detonation waves are evenly distributed. The detonation wave structures in different propagation directions are generally consistent, resulting in the downstream turbine being periodically impacted by multiple oblique shock waves. These oblique shock waves, formed downstream of the detonation waves, directly collide with the turbine guide vanes, generating a series of reflected wave systems. A portion of the reflected waves propagates upstream, thereby influencing the operation of the RDC. The remaining reflected waves undergo multiple reflections within the turbine guide vane passage before continuing to propagate downstream. This process subjects the turbine rotor to severe circumferential distortion of the inlet flow, further complicating the aerodynamic environment.

Figure 4 presents the transient circumferential pressure distribution curves at different axial positions in the flow field for two detonation wave propagation directions, along with the variation of relative pressure peaks with flow direction positions. The sections labeled S1 to S4 correspond to axial positions of x = 0 mm (inlet of the combustion chamber), x = 100 mm, x = 230 mm (outlet of the combustion chamber and inlet of the turbine guide vanes), and x = 305 mm (outlet of the turbine guide vanes), respectively. As shown in the figure, the pressure behind the detonation wave exhibits a significant step, and the pressure peak is much higher than the average value of the cross-section. As the flow field develops and mixes downstream, the pressure peak gradually attenuates, but noticeable pressure fluctuations still exist until reaching the turbine guide vane inlet cross-section. This indicates that there is still significant circumferential non-uniformity of the airflow at the outlet of the guide vane.

$${\pi }_{\mathit{RDC}}={\displaystyle \frac{P_{t_{\mathit{MFA}}}^{S_n}}{P_{t_{\mathit{MFA}}}^{S_1}}},$$

(6)

$$\sigma ={\displaystyle \frac{{\int }_{ t_1}^{ t_1+n\times T_{cycle}}\mathit{RMS}\left(P_t\right)\mathit{dt}}{n\times { T}_{cycle}}},$$

(7)

Here, $P_{t_{MFA}}$ denotes the instantaneous mass-weighted average total pressure at the cross-section, $P_{t_{MFA}}^{S_i}$ the time-averaged value over several cycles, $T_{cycle}$ the detonation wave propagation period, and $S_1$, $S_n$ the combustor inlet and outlet cross-sections, respectively. The total pressure non-uniformity $\sigma$ is defined as the time-averaged circumferential relative standard deviation at the outlet.

Figure 5 presents the total pressure gain and exit non-uniformity of the detonation combustor across different propagation modes. In the single-wave mode, counterclockwise propagation offers a much higher total pressure gain than clockwise propagation. As the number of detonation waves increases, the total pressure gain in both propagation directions declines and converges. This results from weakened detonation intensity and increased shock losses due to more oblique shocks. Moreover, the single-wave mode sees relatively high total pressure distribution non-uniformity, while double- and triple-wave modes see significant reductions. The detonation wave propagation direction barely affects the combustor exit total pressure distribution non-uniformity. For a rotating detonation turbine engine, a single-wave mode combustor provides greater total pressure gain for the turbine but causes more intake distortion. Conversely, a multi-wave mode combustor reduces total pressure gain but offers smoother turbine intake, enhancing work stability and efficiency.

The gas fluctuation is suppressed after the gas passes through the TGV channel, and a loss of the total pressure simultaneously occurs. Figure 6 compares the total pressure loss $∆P_t$ of the TGV channel in different propagation modes. It can be seen that with the increase in the number of detonation waves, the total pressure loss of the TGV channel in the clockwise mode shows a nonlinear trend of decreasing first and then increasing. Among them, the loss in the single-wave flow field is the highest (6.43%), while the loss in the double-wave flow field is the lowest (4.30%). The loss of the TGV channel in the counterclockwise mode gradually increases with increasing number of detonation waves, among which the loss in the three-wave flow field is the highest (3.41%), and the loss in the single-wave flow field is the lowest (1.42%). Furthermore, in the flow field with the same wave number, the total pressure loss of the TGV channel under the clockwise condition is higher than that under the counterclockwise condition, which is caused by the different evolution processes of the incident wave in the TGV channel. However, as the number of detonation waves increases, the differences in the loss of the TGV channel between the two propagation directions are gradually reduced, which in the single-wave, double-wave, and three-wave flow fields are 5.1%, 1.98%, and 1.29%, respectively.

The performance parameters of the gas at the outlet of the TGV channel are dramatically different when the propagation direction of the detonation wave changes, thus affecting the efficiency and power stability of the turbine rotor to different extents. Figure 7 shows circumferential distributions of the relative flow angle at the inlet of the rotor under different working conditions, which fluctuate wildly in the positive and negative range. Under clockwise conditions, the maximum/minimum flow angles in the single-wave, double-wave, and three-wave flow fields differ by 178°, 147°, and 94°, respectively, while the values under the counterclockwise condition are 135°, 87°, and 88°, respectively. This signifies that the fluctuation amplitude of relative flow angle at the rotor inlet under the counterclockwise condition is smaller compared with the clockwise condition. Moreover, it can be used to reduce the fluctuation amplitude of flow angles by increasing the number of detonation waves, so that the flow separation of rotor blades occurs less likely during operation, which helps the flow loss decrease.

In a rotating detonation turbine engine, the turbine components benefit from the total pressure gain of rotating detonation combustion, but must also endure the negative impact of the incoming flow’s pulsation. The actual total pressure gain at the rotor inlet is influenced by both the detonation combustion gain and the guide vane losses. To analyze and compare these effects quantitatively, the impact of rotating detonation combustion on the turbine rotor’s work characteristics can be assessed using the turbine rotor’s specific work. Here, the total pressure and temperature at the deflagration combustor exit are consistent with those at the detonation combustor inlet and exit, respectively. The turbine rotor’s specific work $W_T$ can be calculated as

$$W_T=C_{p,4}\times T_{t,4}-C_{p,5}\times T_{t,5},$$

(8)

where $C_p$ and $T_t$ denote the average specific heat capacity and average total temperature of the gas, respectively. The subscripts 4 and 5 indicate the inlet and outlet sections of the turbine rotor, with values relative to the average specific work under deflagration combustion. Rotor work efficiency under different modes can be calculated using Equation (9), where subscripts 4 and 5 represent the rotor’s inlet and outlet sections, and k is the specific heat ratio.

$${\eta }_T={\displaystyle \frac{1-\frac{T_{t,5}}{T_{t,4}}}{1{-\left(\frac{P_{t,5}}{P_{t,4}}\right)}^{\frac{\gamma -1}{\gamma }}}},$$

(9)

Table 2. Specific power gain and instability of the turbine rotor under different working conditions.

	CW			CCW
	1 Wave	2 Wave	3 Wave	1 Wave	2 Wave	3 Wave
Inlet total pressure gain (%)	26.9	25.5	23.6	42.1	28.6	24.3
Rotor efficiency	0.840	0.887	0.890	0.758	0.864	0.875
Specific work gain (%)	20.9	27.1	27.5	29.1	27.9	27.8
Instability (%)	11.8	7.1	6.5	7.8	5.0	5.6

lists the specific work gain and its instability of the turbine rotor for the RDTE in different modes compared to those for a conventional turbine engine. The instability mentioned above is evaluated by the relative standard deviation of the rotor specific work over several cycles. As shown in the table, as the number of detonation waves increases, the rotor efficiency increases continuously, while the change of the total pressure gain at the rotor inlet shows the opposite trend. Due to the comprehensive influence of combined rotor efficiency with total pressure gain, the specific work gain in different propagation directions shows different trends. In the clockwise mode, the specific work gain of the two-wave and three-wave flow fields is basically the same and obviously higher than that of the single-wave flow field, while in the counterclockwise mode it is the highest in the single-wave flow field. In addition, for the stability of the rotor specific work, the multi-wave flow field is better than the single-wave flow field, and the counterclockwise condition is better than the clockwise condition. Nevertheless, the more the number of detonation waves, the smaller the difference in the two directions. Hence, considering the specific work gain and stability under the conditions discussed in this paper, when working in the counterclockwise two-wave mode, the comprehensive performance of the RDTE is better.

3.2. Analysis Based on POD

Since the rotating detonation wave mainly interacts with the TGV, this section focuses on POD analysis of the TGV area.

Figure 8 shows the pressure gradient distribution of the transient flow field of the guide vane in the single-wave mode. There are a series of wave structures in the above TGV flow field, mainly including the incidence of oblique shock waves, the collision and reflection between oblique shock wave and the guide vane, etc. Moreover, these wave structures are seriously affected by the detonation wave propagation direction, especially in the clockwise mode where the wave structures are more complicated. Therefore, POD analysis is applied to capture the key characteristics of the regional flow in turbines, enabling a better understanding of the effect of detonation flow on turbine characteristics.

Based on the POD basic principle, modes are sorted by the energy proportion of each decomposed mode. The energy proportion is defined as the sum proportion of each mode’s eigenvalue to all modes’ eigenvalues. Figure 9 shows the energy proportion distribution and energy accumulation curves of each guide vane flow field mode under different rotational detonation combustion conditions. In the figure, the energy of the first-order mode in flow fields with different detonation wave numbers is at least first-order higher than that of other higher-order modes, indicating a fundamental difference in the flow structure represented by the first-order mode. Compared to the first-order mode, the energy proportions of other adjacent higher-order modes are “paired”, suggesting these “paired” modes may correspond to specific coherent flow structures. Also, when detonation waves propagate in different directions, lower-order mode energy distribution characteristics vary, meaning the main flow structure characteristics in the guide vane channel differ. For example, in single-wave flow fields of clockwise and counterclockwise modes, the first-order mode energy proportion reaches 30.2% and 31.7%, respectively. As the mode order increases, the energy proportion of higher-order modes in the single-wave flow field rapidly declines to near zero, with the 13th mode’s energy proportion being less than 1%. Thus, targeted analysis of low-order modes aids in understanding complex guide vane channel flow characteristics.

The mode coefficient $a_j\left(t\right)$ from POD decomposition reflects each mode’s temporal evolution and energy capture in the instantaneous flow field, with its absolute value indicating the energy intensity variation. Figure 10 presents the mode coefficient variation with time under single-wave conditions. The first-order mode coefficient’s absolute value is one order higher than others, with a small fluctuation range and no obvious regularity over time. Other low-order mode coefficients show evident periodicity, fluctuating greatly between positive and negative values. As the order increases, the mode coefficient amplitude decreases and becomes chaotic over time. Higher-order adjacent modes also exhibit a “pairing” phenomenon in their development trends, where “paired” modes have similar waveforms with a certain phase difference. Via fast Fourier transform (FFT) analysis of each mode coefficient, it is found that the dominant frequencies of the second- and third-order mode coefficients match the operating frequency $f_D$ of the upstream detonation wave under different propagation directions. The dominant frequency of other high-order mode coefficients is a multiple of the detonation wave’s operating frequency, with the correlation between the dominant frequency of higher-order mode coefficients and the detonation wave’s operating frequency weakening gradually.

Based on the distribution of modal energy contributions, Figure 11 further presents the flow fields of each modal order obtained from POD decomposition under the single-wave mode. It can be observed that the first-order modal flow field does not exhibit significant wave structures, indicating its essential representation of time-averaged flow characteristics. Second-order and higher low-order modes all show significant non-uniform pressure distributions and coherent wave structures, with adjacent “paired” modes (e.g., second and third order) demonstrating clear structural similarities. Combining the dominant frequency information of each modal coefficient shown in Figure 11, it can be concluded that the dominant frequencies of energy changes for the second-order and third-order modes are consistent with the upstream detonation wave working frequency. Moreover, the flow characteristics of both the second-order and third-order mode manifest as unidirectional flow in the circumferential and axial directions under high-pressure inflow conditions. Thus, it can be inferred that the superposition of the second-order and third-order mode collectively reflects the dynamic flow of oblique shock waves in the original flow field. The remaining higher-order modes (e.g., 5th to 30th order) have dominant frequencies of energy changes as multiples of the upstream detonation wave working frequency, and their flow characteristics manifest as localized high pressures on the blade surface and within the passage. Therefore, these low-order modes can be considered to reflect the dynamic flow of reflected waves in the original flow field. Higher-order modes (e.g., 100th order) reflect more detailed and complex flow structures in the original flow field. Such flows have lower intensity and contain significantly less energy.

When the operating mode of the rotating detonation combustor changes, the energy distribution and flow structure of each order mode in the flow field are inevitably influenced. As shown in Figure 9, with the increase in the number of detonation waves, the proportion of the first-order mode energy rises markedly. In the clockwise and counterclockwise mode, the proportions of the first-order mode energy in the two-wave and three-wave flow fields reached 43.9%, 51.2%, and 44.9%, 52.1%, respectively. This indicates that the sum of the proportions of other mode energies except the first-order mode is decreasing simultaneously, meaning that the flow characteristics of these modes are suppressed, the overall flow structure is simplified, and the flow stability is enhanced. The proportion of the first-order energy in the clockwise mode is consistently lower than that in the counterclockwise mode, indicating that the flow structure in the clockwise mode is relatively more complex, which is consistent with the original flow process in the flow field. At the same time, it can be observed that, except for the first-order mode, the distribution of energy in other low-order modes also shows obvious differences with the change in the number of detonation waves, while the energy distribution in high-order modes remains basically unchanged. Moreover, the “pairing” phenomenon of adjacent low-order modes also shows different changes; for example, in the three-wave flow field, the fourth- and fifth-order modes do not “pair” like those in the single-wave and two-wave flow fields. From the energy accumulation curve, it can be seen that with the increase in the number of detonation waves, the accumulation rate of low-order mode energy significantly increases. In the clockwise and counterclockwise mode, the proportions of the first 30-order energy increased from 64.1% and 68.6% in the single-wave flow field to 85.4% and 85.6% in the three-wave flow field, respectively. This again demonstrates that in multi-wave flow fields, the complex flow characteristics corresponding to high-order modes are further weakened, the non-stationarity of the flow field decreases, and the flow stability improves.

This suggests that variations in the number of detonation waves can influence the distribution of low-order modal energy, possibly modifying the flow characteristics of other low-order modes beyond the first order. Figure 12 presents the instantaneous flow structures corresponding to the second- and fourth-order modes under different operating conditions. To enhance the comparability of spatial characteristics across conditions, the total energy of the flow fields at each selected transient moment is kept equal. Due to the quasi-steady periodic flow, the flow states at different circumferential positions of the blade channels can be regarded as those of a single blade channel at different times. From the figure, it can be observed that the flow characteristics of the same-order modes vary under different operating conditions. In the second-order mode flow field, the high-pressure region in the single-wave condition affects about five blade channels, while the others are in a pressure recovery state. This means that after being impacted by the oblique shock wave, the duration of high pressure and subsequent recovery in the blade channels is longer. Within a cycle, significant changes occur in the flow state of a single channel, with varying degrees of gas expansion inside and different gas states at the outlet at different times, leading to substantial intake distortion for the rotor and a decrease in its work efficiency. In contrast, under double-wave condition and triple-wave condition, the shorter cycle of incident oblique shock waves reduces the duration of high pressure and subsequent recovery in a single blade channel. The fluctuation degree of the flow state within a cycle of a single blade channel decreases significantly, and the gas fluctuation at the blade channel outlet improves, mitigating the adverse impact on rotor efficiency. Moreover, examining the fourth-order mode flow structures under different operating conditions reveals that as the number of detonation waves increases, the distribution of reflected wave system structures within the blade channels becomes more uniform and regular, and the transient influence on the gas flow state inside the channel diminishes. This similarly helps reduce the pulsation of the gas state at the blade channel outlet and enhances rotor efficiency.

To sum up, in this section, based on the modal energy contribution analysis of different detonation wave propagation modes and the aerodynamic performance analysis in Section 3.1, it can be concluded that a higher proportion of the first-order mode energy, which represents the average flow structure, corresponds to lower proportions of other low-order mode energies that represent the pulsating wave system structure. Under such circumstances, the turbine rotor’s steady-state work efficiency and stability are enhanced. This is attributed to the strengthened average flow characteristics, which simplify the overall flow structure, make the flow more periodic, and reduce the amplitude of flow state fluctuations in the turbine components. As a result, the flow condition aligns more closely with the quasi-steady assumption made during turbine design. For a rotating detonation turbine engine, in addition to mitigating upstream inflow pulsations, effective measures to enhance the turbine components’ average flow characteristics can increase the first-order mode energy proportion, decrease pulsating energy, and thereby boost the turbine’s work efficiency and stability.

4. Conclusions

This study investigates turbine-stage flow structures and aerodynamic characteristics under typical detonation wave propagation modes using a two-dimensional numerical model coupling a rotating detonation combustor with a single-stage turbine. The mechanisms by which rotating detonation wave propagation modes influence turbine operational behavior are systematically analyzed. Through proper orthogonal decomposition (POD) of quasi-steady flow fields, spatial distribution patterns and temporal fluctuation features are extracted, revealing critical dynamic characteristics of the flow. The main conclusions are summarized as follows:

• In the turbine, rotating detonation combustion yields total pressure gain yet causes significant unsteady flow. Guide vanes somewhat suppress detonation-induced pressure fluctuations, but total pressure loss and airflow non-uniformity still exist, lowering rotor efficiency. More detonation waves reduce rotor inlet pressure gain but improve airflow uniformity and rotor efficiency. Using the rotor specific work gain–pulsation instability ratio as the performance metric, the turbine performs best in the double-wave counterclockwise mode (27.9% work gain, 5.0% instability, 86.4% efficiency) and worst in the single-wave clockwise mode (20.9% work gain, 11.8% instability, 84.0% efficiency).
• POD modal analysis on the turbine guide vane region reveals distinct flow structures for each mode. The first-order mode shows time-averaged flow without clear wave systems, while other modes display non-uniform pressure distributions, indicating pulsating waves. Low-order modes beyond the first exhibit “pairing,” with adjacent modes sharing similar energy ratios and spatiotemporal structures but differing in phase. This suggests the guide vane channel’s wave system motion results from paired mode superposition.
• In the rotating detonation turbine guide vanes flow field, wave propagation direction and wave front number influence the energy distribution and stability. As the number of detonation waves increases, the first-order mode energy proportion in the guide vanes flow field enhances, from 30.2% and 31.7% for single-wave to 51.2% and 52.1% for triple-wave. The cumulative proportion of the first 30-order energy also rises from 64.1% and 68.6% for single-wave to 85.4% and 85.6% for triple-wave, indicating decreased flow unsteadiness. The counterclockwise mode exhibits superior time-averaged flow dominance under identical wave numbers, attributed to its channel reflection wave system characteristics.
• In a rotating detonation turbine engine, the detonation chamber’s outflow is highly pulsatile, and the turbine components’ inflow is quasi-periodically unsteady, deviating from the design’s quasi-steady assumption and impacting the turbine’s work capability. Analysis shows that a higher first-order mode energy ratio and lower pulsating wave system mode energy ratio enhance rotor work efficiency and stability. Thus, enhancing the flow field’s average flow characteristics and increasing the first-order mode energy ratio can improve the rotating detonation turbine’s work performance.

These findings advance the understanding of coupled interactions between detonation flow fields and turbine rotors in rotating detonation turbine engines (RDTE), offering theoretical insights to support the development of RDTE systems.