Design, Multi-Point Optimization, and Analysis of Diffusive Stator Vanes to Enable Turbine Integration Into Rotating Detonation Engines

Due to the combustion chamber's total pressure gain, rotating detonation engines may offer less fuel consumption than state-of-the-art, deflagration-based gas turbine engines. The performance benefit of the prospective thermodynamic cycle has been identified for almost two decades, with potential increases of up to 20 percentage points in efficiency [1]. Some of these studies assume idealized engine components, not considering the effect that the new combustor may have on the rest of the engine, mainly on the turbine efficiency. Sousa et al. [2] developed an engine model in which the conventional combustor is replaced by a rotating detonation combustor (RDC). By using NASA's T-MATS library, the impact of the new combustor on the entire engine was precisely assessed to quantify the effective benefits of rotating detonation engines. The efficiency gain over the Joule–Brayton cycle rose to 5% at low pressure ratios but vanished at high-pressure ratios.

However, the combustion process is sustained by one or more rotating detonation waves [3], creating an unsteady flow behavior that propagates downstream toward the exhaust. Consequently, unsteady Reynolds-averaged Navier–Stokes (URANS) simulations have been used in the past years for the numerical analysis of RDCs. Braun et al. [3] evaluated a premixed H₂–air mixture at different inlet pressures, evaluating the differences in the transient flow field, whereas Meng et al. [4] focused on the detonation wave characteristics in H₂–air non-premixed combustors. On the other hand, reduced order models have also been proposed for the assessment of RDCs. Sousa et al. [5] modeled the flow field in a Rotating Detonation Engine (RDE) combining a 1D chemical kinetics solver with the 2D method of characteristics, being able to extract the radius-to-detonation height that provides the lowest level of irreversibilities. Kaemming et al. [6] developed a reduced order model including additional phenomena, such as deflagrative burning, providing a good agreement in the precombustion conditions. On the experimental side, Bach et al. [7] equipped an RDC with a series of nozzle guide vanes, showing the pressure gain is not altered by the vane presence. However, the preferred direction of the detonation wave is proved to be affected by the vane inclination, emphasizing the need for proper combustor-turbine integration. Bach et al. [8] also measured the total pressure gain in RDCs for different geometries and mass fluxes, using the data to develop an empirical model and to identify potential design corridors to attain positive pressure gain.

The transient nature of these novel combustors may introduce high-amplitude oscillations in flow angle at the turbine inlet. Braun et al. [9] studied the flow conditions downstream of a detonation combustor for different accelerating and diffusing passages and showed that the outlet flow angle variations have an amplitude of up to 100 deg, ranging between −50 deg and +50 deg. This will induce large inlet flow incidences in the vane row, significantly higher than those found in high- and low-pressure turbines in traditional gas turbines, which generally range from −10 deg to +10 deg [10]. Xisto et al. [11] used URANS to investigate numerically the integration of a transonic axial turbine stage with a pulsed detonation combustor. The complicated shock structure was responsible for the generation of local separation bubbles and unwanted rotor vibrations. The time-dependent rotor inlet conditions introduced sharp periodic variations of the incidence angle, which drastically diminished the turbine efficiency. Therefore, new design rules and guidelines may be needed to make turbine airfoils more resilient to high incidence levels.

Additionally, the Mach number will significantly differ from conventional subsonic turbines. In this case, the inlet profile may present fluctuations ranging from the high subsonic to the supersonic regime, based on the total-to-static pressure ratio of the combustor [3]. To guarantee safe operation, the vane passage must remain unchoked for subsonic and unstarted for supersonic inlet conditions. Liu et al. [12] suggested increasing the channel height from inlet to outlet using a smooth, contoured endwall profile to avoid choking the passage. He also attained considerable turbine stage efficiency improvements by optimizing the endwall, both at steady and unsteady conditions. However, only the fluctuations in the inlet total pressure were reported, which are expected to have a much lower detrimental effect on the row performance, compared to inlet flow angle variations. Mushtaq et al. [13] discussed in more detail the impact of the endwall design in supersonic turbines for RDEs, finding that the endwall losses play a major role in the overall loss. Thanks to a multi-step optimization approach, both the efficiency and the work extraction were considerably enhanced.

This paper aims to identify the geometric features that enhance performance under significant flow incidences for high-subsonic inflows, assess their impact on the flow field around the vane, and present design recommendations. A 3D RANS multi-point, multi-objective optimization with three different inlet flow angles is conducted, including the endwall and vane design. Pressure loss and rotor forcing are the objective functions, with significant reductions in all operating conditions. Ultimately, a transient analysis of the baseline and optimized designs confirms the adequacy of the multi-point, multi-objective optimization methodology.

Turbine vane passages generally have a pronounced area contraction from the inlet up to the throat to turn the flow and increase the outlet tangential velocity and the turbine specific work. However, as the throat area is reduced, the maximum inlet Mach number that the vane passage can ingest is diminished. If this limitation is not considered, it can lead to an unsuccessful combustor-turbine integration, where the turbine is not able to operate at the combustor design condition, effectively changing the combustor operating point. This can be particularly concerning for RDCs, due to their inherently unsteady flow field and their susceptibility to changes in their operating conditions, which can induce vastly different operating modes and the number of waves [14].

Equation (1) defines the isentropic area ratio or starting limit, establishing a relation between the maximum inlet Mach number and the throat-to-inlet passage area ratio [15]. In the subsonic regime, as the Mach number grows, the minimum allowable area ratio increases. This restricts the contraction of the passage, which effectively restricts flow turning and power extraction. To be able to ingest high-subsonic flow without reducing the vane turning, endwall contouring may be applied, which consists of increasing the channel height from vane inlet to outlet. As a result, the reduction in area due to flow turning is counteracted by endwall diffusion.

Two approaches are proposed to optimize the vane geometry for changing inlet flow angles. The first alternative is to run unsteady simulations (URANS), imposing an inlet profile for the flow angle. In this case, the entire wave period is evaluated and the unsteady phenomena, such as the damping of the fluctuations, are also captured. However, the computational cost is remarkably higher than in steady RANS. This reduces the number of geometries that can be evaluated, making it more challenging to extract valid design recommendations. The second approach is to conduct a multi-point steady optimization (RANS), where every individual is evaluated at several conditions, which may be different points along the inlet profile. These simulations are computationally less expensive, offering the speed that the unsteady method lacks. A three-point multi-point optimization can be completed approximately in a 120 h timeframe, using 384 cores, each with 2 GB of memory and running six processes in parallel. The entire inlet period is not assessed but if the design is optimized for several points along the inlet profile, the overall performance will also be improved. Thus, this alternative was selected for the optimization.

The optimization has two objectives, the first one is to minimize pressure loss, using the pressure loss coefficient $(YP=(P01−P02)/(P02−P2))$⁠. Since it is not practical to have a function for each simulation, some type of averaging needs to be applied to include the results of the three cases into one objective. The pressure loss coefficient will be averaged considering the different outlet dynamic pressures, as expressed in Eq. (3). This puts more weight on the cases with higher dynamic pressures, which implies larger net losses.

The second objective is to avoid structural vibrations in the rotor by reducing the forcing induced by the stator [16]. The rotor blade moves circumferentially along the pressure field provided by the preceding vane row. In the rotor's frame of reference, the non-uniformities are seen as an inlet boundary condition dependent on time [17]. Consequently, Puente et al. [17] suggested using only steady computations, considering the non-homogeneity of the pressure field as the main source of rotor forcing, preventing the use of fluid-structural simulations. Following the results from Ref. [18], the standard deviation (STD) along the pitch will be employed to measure the non-uniformity of the pressure field (Eq. (4)). To obtain a representative metric over the entire downstream plane, an area average is done. The standard deviation is calculated at ten different spans, from 5% to 95%, and each value is considered valid over an area of 10% span. This way, an area-averaged value is computed, using Eq. (5). To obtain an average between the three simulations, a simple average is employed, as shown in Eq. (6). Even though both objectives may have been included in a single objective function, the aim of the optimization was to develop clear design guidelines including the trade-off between pressure loss and rotor forcing. Thus, a multi-objective approach was preferred.

Additionally, the optimized vane should be able to operate under fluctuating high-subsonic inlet Mach numbers, higher than 0.6 and ideally up to Mach 1. Therefore, as explained in Sec. 2.1, the throat-to-inlet area ratio should be constrained to make sure the vanes can ingest inflows in that range of Mach numbers. However, the optimization software that has been employed only allows either design parameters or fluid-dynamic metrics to be used as constraints in the optimization; thus, the throat-to-inlet area ratio could not be directly constrained. Nevertheless, a similar result may be obtained using the outlet Mach number as a constraint. For a given inlet Mach number, the lower the throat-to-inlet area ratio, the higher the outlet Mach number (until it gets choked) and vice versa.

The multi-objective differential evolution optimization code is developed by numeca (FineDesign3D), relying on the strength Pareto evolutionary algorithm [20]. The mapping of the design space is performed using a Latin hypercube sampling approach [21], with 200 individuals. This value is chosen following numeca's recommendations [19], with slightly more than four times the number of design parameters (47). Regarding the type of surrogate model, Kriging is selected [19]. The optimization is run on 384 cores, with six simultaneous processes. Both objective functions are given the same relative weights.

The parametric model determines the discretization of the geometry into a set of parameters, being a crucial step in the optimization. The nature of the generated geometries depends on the model's details. Ideally, each parameter should have a remarkable effect on the geometry, easing the analysis of the optimal parameters [18]. However, as the number of parameters (design vector) is reduced, the ability to introduce slight design changes is also diminished. Additionally, the design space is defined by the ranges of all parameters. In this case, a balance must be found, so that the optimal design can be achieved while avoiding many illogical or non-physical profiles.

The strategy consists of three vane-to-vane sections, defined at three radial locations (0%, 50%, and 100% span). The 2D sections are built with a camber line, a suction side (SS) and a pressure side (PS) curve, using Bezier polynomials for the three of them. The camber line uses the inlet metal angle, the outlet metal angle, and the stagger angle. The intersection of the inlet and outlet angle lines defines the intermediate control point of the polygon, as depicted in Fig. 1. The suction and pressure sides employ four and three intermediate control points, defined perpendicularly to the camber line. The spacing along the camber line increases by 1.2 from point to point (stretching factor). The leading edge (LE) radius and trailing edge radii and wedge angle close the airfoil geometry (Fig. 1). Due to structural and manufacturing constraints, the trailing edge radius is kept constant at 0.75 mm.

The 3D vane geometry is created by radially stacking the 2D airfoil sections. The stacking law specifies the circumferential position of the trailing edge along the radial direction. It is defined using a spline curve with three control points uniformly distributed along the span and with the trailing edge position at the hub remaining fixed, as shown in Fig. 2. Including a stacking law enables higher control over the outlet flow topology, being able to define different lean configurations [17]. Moreover, the axial chord can vary along the radial direction. The law is defined with a spline curve along the trailing edge, with the axial position at the hub kept fixed and two control points at 50% and 100% span. By imposing that the radial evolution along the leading edge is symmetric to the trailing edge, the entire law can be modeled with only three parameters, the axial chord of each airfoil section.

The flow in a turbine row is three dimensional and to accommodate for that the endwall geometry may be non-axisymmetric and with asymmetrical hub and shroud contours. However, to simplify the design and to reduce the number of parameters, the endwall is treated as axisymmetric and with the hub and shroud profiles symmetric to the mean radius. This way, the contour is modeled with a spline with four equidistant points along the axial direction, which can only move radially, as depicted in Fig. 3. This way the endwall is uniquely defined with only four parameters since the inlet and outlet coordinates are fixed. As a consequence of the endwall shape and outlet-to-inlet height ratio, the radial extension of the vane varies along the axial coordinate. Finally, the three 2D sections, together with the stacking, meridional, and endwall profiles, and the vane count add up to a total of 47 parameters.

The vane baseline geometry is inspired by a high-pressure transonic turbine [22]. The most important geometric features are shown in Table 1. The endwall contour has an outlet-to-inlet height ratio of 3, allowing for a very high level of area ratio and a considerable amount of turning.

FineTurbo, developed by numeca, was chosen as the solver for the optimization. Turbulence effects are accounted for with the two-equation k–w shear stress transport model, assuming the boundary layer to be fully turbulent. To guarantee a precise solution of the 3D RANS equations, the solver is compared with CFD++, which has been previously validated [23,24]. The steady-optimized geometry presented in Ref. [12] is run with the same mesh and boundary conditions. The outlet mass-flow-averaged total pressure and flow angle are computed, with relative errors of 0.11% and 0.18%, respectively, showing a good agreement between solvers.

Autogrid5 from numeca is employed as a structured mesher. The y+ is kept below 1 in all the domains, resolving the viscous sublayer. A grid independence analysis is performed, evaluating three different meshes, coarse (1.62M cells), medium (2.59M), and fine (4.07M), with an approximate relative increase of 20% in the number of cells in each direction. The percentual differences in the outlet mass-flow-averaged total pressure and flow angle are below 0.07% and 0.17%, respectively, a negligible variation from one mesh to another. The independence is also quantified following the methodology of Roache [25] (Table 2). The results indicate that the solutions are clearly within the asymptotic range of convergence.

Nevertheless, it is also relevant to check the radial distribution of one of these metrics to verify that there are no significant local differences in the flow field. Figure 4 represents the outlet total pressure distribution as a function of the span for the three grids. To quantify the level of discrepancy, the standard deviation of the difference in absolute value between the fine grid and the medium and coarse grids are computed, being 0.011 bar and 0.032 bar, respectively. This shows that the deviation to the outlet total pressure profile of the fine mesh is almost three times larger for the coarse than for the medium grid. Thus, the latter one is selected for the optimization.

The boundary conditions of the three operating points are included in Table 3. Uniform total pressure, total temperature, and flow angle are specified at the inlet, with mass flow imposed at the outlet. When the flow is not axial, the mass flow is reduced accordingly, to keep a mass-flow-averaged inlet Mach number of 0.6. The fluid is air, treated as a calorically perfect gas, and all walls are adiabatic. The axial flow condition has been included in the optimization to guarantee that an enhanced performance at high inlet flow angles is not achieved at the expense of poor performance for a zero inlet angle.

Overall, 84 populations are generated with six individuals per population (504 geometries in total) with an estimated computational time of 125 h. As mentioned in Sec. 2.2, a constraint on the outlet Mach number was employed; however, selecting the appropriate penalty weight and threshold is not a straightforward process [19]. To guarantee that the vanes will be able to ingest flow above Mach 0.6 and potentially even higher, the throat-to-inlet area ratio should be around 1. Based on the results from the design of experiments, the outlet Mach number for the designs with that area ratio was around 0.55, depending on the exact loss levels. Thus, this value was chosen as the threshold for the constraint. The selection of the penalty weight was slightly more arbitrary but based on initial estimates of the potential reduction in both objective functions, a value of 100 was set (relative to the weight on the objective functions).

These settings were used for the first 34 populations; however, the success rate was excessively low, around 30%, which meant that many of the generated geometries crashed at some point during the optimization. Since the constraint penalty was not strong enough, the optimizer suggested designs with very low area ratios, which could reduce both objective functions but that could not operate at an inlet Mach number of 0.6, causing the solver to crash or providing unphysical results. To solve this, the penalty weight was increased to 150 for the last 50 populations and the success rate rose to approximately 75%. The difference between the two sets of populations can be observed in Fig. 6, which plots both objective functions against each other. In the first set, a better performance was obtained but at the expense of a throat-to-inlet area ratio below what was desired, which considerably shortens the inlet Mach number operating range. The opposed result was obtained in the second set, not so optimal performance but with a larger area ratio. Additionally, due to the higher success rate, enough individuals were generated and the optimization converged to a Pareto front.

Figure 7 presents a zoomed-in view of Fig. 6 to visualize better the results of the optimization. The improvement compared to the baseline has been remarkable, with a reduction of up to 37% and 77% for the averaged pressure loss coefficient and averaged pressure distortion, respectively. However, for geometries with an area ratio above 1, the level of enhancement in the averaged pressure loss coefficient is reduced by up to 21%. Additionally, despite the use of the constraint, many geometries have a throat-to-inlet area ratio below 1. This suggests that the impact of the area ratio on the vane performance is strong and that the reduction in the objective functions can counteract the effect of the constraint penalty for designs with outlet Mach numbers slightly higher than the constraint threshold. To confirm the dependency, Fig. 8 plots the averaged pressure loss coefficient as a function of the throat-to-inlet area ratio, which increases by more than ten percentage points when the area ratio rises from 0.9 to 1.1. Thus, minor differences in area ratio cause significant changes in performance.

As shown in Table 3, when the inflow is not axial, the angle at the domain inlet is set to be ±30 deg. Nevertheless, the flow angle at the vane's leading edge is higher than that. Figure 9 represents the evolution of the mass-flow-averaged flow angle as a function of the axial coordinate for the baseline design for the three different operating conditions. For the cases with non-axial inflow, the angle grows along the diffusing region upstream of the vane from the initial value of ±30 deg up to ±42 deg. The reason is the reduction in the axial velocity component along this region, whereas the tangential velocity remains unchanged. Thus, the angle increases from the inlet up to the LE (positive becomes more positive and negative more negative). This effect may be detrimental to the vane performance, as it increases the incidence under which it is operating. Therefore, lower diffusion upstream of the vane can have an additional benefit due to the reduction in the flow angle growth. Nonetheless, applying most of the radial diffusion along the vane may lead to extensive flow separation and larger pressure losses; thus, a balance must be found on how the radial diffusion is distributed.

It is also relevant to assess the behavior of the pressure loss coefficient with vane incidence. Figure 10 depicts the pressure loss coefficient of the 20 individuals with the lowest average pressure loss. Due to the difference in diffusion upstream of the vane between geometries, the leading edge flow angle varies slightly for the non-axial inflow cases. The solid line is a quadratic polynomial fit of the averaged values for every case and serves as a decent estimate of the overall trend, with an R² coefficient of 0.919. The trend agrees qualitatively with other references in the turbine field [27,28], with a more pronounced growth of the loss for positive than for negative incidence. However, the exact shape of the curve will depend on each specific vane design and how tolerant it is to incidence. Table 4 includes the averaged values, the relative increase to the on-design condition, and the standard deviation.

Similarly, the impact of incidence on the downstream pressure distortion must also be assessed. Figure 11 presents the pressure distortion as a function of the leading edge flow angle for the same 20 individuals. Negative incidence diminishes the pressure difference between the pressure and suction sides, with more acceleration on the PS and less on the SS. Thus, the pressure field downstream of the vane is more uniform. The opposite may happen for positive incidence, where a larger pressure difference may induce a higher pressure distortion. However, the growth in rotor forcing for positive incidence cannot be accurately estimated due to the high dispersion in the results. In this case, the solid line is a rough estimation (R² = 0.726) and cannot be used to describe the full trend of the pressure distortion. The averaged values and standard deviation are included in Table 4.

Two geometries are selected to be compared with the baseline. Their main performance parameters are included in Table 5 and their location within the optimization can be found in Figs. 7 and 8. IND 238 is selected among the group of individuals that have a lower area ratio but also the greatest reduction in pressure loss. Within that group, it has the lowest pressure loss coefficient, the largest area ratio, and the lowest vane count. On the other hand, IND 483 is chosen from those geometries that have an area ratio similar to the baseline. The purpose is to illustrate the effect of different area ratios on the improvement with respect to the baseline and also its impact on the main performance metrics. The optimized designs offer significant enhancements in both objective functions. The reduction in pressure loss for a similar area ratio is around 19%; however, if the operating range can be less extensive and the area ratio can be diminished, then the loss reduction can be doubled, underlining the important effect of area ratio on loss. Regarding the vane count, both designs include one additional vane.

The main effect of the inlet flow angle on the flow field can be visualized in Fig. 12, which includes the isentropic Mach number distributions at mid-span for IND 238 for the three flow angles, together with the Mach number contours for positive and negative incidences. As previously mentioned, a positive inlet angle creates a larger pressure difference by accelerating the flow more on the SS and less on the PS. The figure shows that the stagnation point is moved toward the PS, which induces a massive acceleration in the SS as the flow surrounds the leading edge. The vast adverse pressure gradient that appears after the suction peak creates a separation line that extends along the entire suction side.

Moving to the negative angle case, the stagnation point is shifted toward the SS, causing larger acceleration on the pressure side and lower on the suction side, drastically reducing the vane loading. In this case, the adverse pressure gradient is now located on the pressure side; however, no separation is observed. The outlet Mach number is lower for the two cases with non-axial inflow, compared to the axial one. This is simply due to a lower mass flow, as noted in Table 3.

The inlet angle also impacts the development of the secondary flows and the loss mechanisms. A larger pressure difference between PS and SS creates a stronger pressure gradient in the tangential direction, which pushes the PS leg of the horse-shoe vortex and the passage vortex toward the SS, impinging on the SS more upstream than for the on-design case. This is perfectly observed in Fig. 13 with the entropy contours at 10% span for IND 238 (compound lean). For a positive inlet angle, as soon as the secondary flows start to grow, they are immediately pushed toward the SS. Contrarily, for a negative angle, the passage vortex remains in the middle of the passage all the way until the outlet, due to the reduction of the pressure gradient from PS to SS. Furthermore, a separation bubble appears owing to the adverse pressure gradient on the PS, creating additional pressure losses.

The determination of the location where the secondary flows hit on the suction side is essential because a 3D separation line starts precisely at this point [29,30], inducing the radial growth of these vortical structures toward the center of the channel. This is confirmed in Fig. 14, which depicts the pitch-wise averaged entropy contours for the same three cases. It is observed that the point of radial separation is pushed upstream for positive incidence, compared to the axial inflow case. This introduces larger pressure losses, and the outflow is now much more dominated by these vortexes, perturbing the mid-span flow completely and covering almost the entirety of the span. On the other hand, for negative incidence, since the passage vortex does not impinge on the SS, these flow structures do not grow radially and remain near the endwalls. This abates the influence of the secondary flows on the outflow, especially at mid-span, and provides lower pressure losses, compared to positive incidence.

The impact of the inlet flow angle on the vane design can be evaluated by comparing the optimized geometries from the multi-point optimization with the best designs from a previous single-point optimization at zero inlet angle. Figure 15 shows the curvature distribution along the SS as a function of the curve distance for the multi-point and single-point optimized geometries, at 10% and 50% span. To include the leading edge region in the plot, the initial point has been shifted toward the PS, as indicated in the blade-to-blade views on the right. At 10% span, the curvature peak in the leading edge is considerably reduced in the multi-point geometry. This is caused by a larger LE radius, which helps create airfoils more resilient to incidence, minimizing the velocity peaks around the leading edge [31]. The averaged LE radius in the single-point optimized design is 1.47 mm, whereas in the multi-point optimized case it is 1.72 mm, 17% more.

The SS curvature distribution has a smoother evolution in the multi-point design, at both spans. This implies that the turning is applied more gradually along the curve, instead of concentrating it almost entirely along the first part of the vane. A lower curvature and curvature derivative prevent local velocity peaks and their corresponding adverse pressure gradients, reducing the chances of flow detachment [31]. Therefore, it can be a valuable strategy to improve the vane performance at high positive incidence. However, shifting the turning toward the aft (generally called “aft loading”) is more likely to cause flow separation in the rear suction side, owing to a thicker boundary layer and higher flow velocities in that region. Nevertheless, due to the superior level of positive incidence, it is preferable to have separation further downstream rather than right after the leading edge.

The endwall contour also presents notable differences. Figure 16 includes the hub endwall control points for the 20 individuals with the lowest pressure loss in the single-point and multi-point optimizations. Since a spline curve is used to build the contour, the control points are part of the profile. The averaged values are highlighted in squares, using them to compute an estimate of the angle at each section. Furthermore, the average location of the vane's leading edge for each optimization is included in the figure. In the single-point case, the radial diffusion upstream of the vane is limited and most of it happens between 40% and 80% of the total length, which corresponds with approximately the first half of the vane axial chord, whereas in the final 20%, the endwall profile is almost straight.

On the other hand, the geometries optimized for positive and negative inlet angles have a more conical endwall design, with more divergence upstream of the vane and also in the final 20% of the length, but significantly less along the first half of the vane axial chord. As observed in Fig. 14, for positive incidence the passage vortex moves immediately toward the suction side, triggering the separation of the secondary flows. Since adverse pressure gradients favor the radial growth of these vortexes, reducing the radial diffusion near the leading edge can be helpful to minimize this effect and abate pressure losses. In the case of negative incidence, less diffusion is also beneficial to shorten the length of the separation bubbles, by minimizing the adverse pressure gradient. Nonetheless, more diffusion upstream of the vane could induce flow detachment.

Finally, the multi-point optimized vanes appear to have longer chords compared to those optimized only for an axial inflow since the average location of the leading edge is more upstream. This implies that the diffusing region is shortened, which may be positive for minimizing flow detachment caused by the larger endwall angles. Additionally, a longer axial chord may help in reducing pressure losses by reducing the pitch-chord ratio, especially for cases with high inlet flow angles.

Regarding the outlet metal angle, Table 6 includes the mean values and standard deviation for the 20 individuals with the lowest averaged pressure loss in the multi-point optimization and the difference with the mean values from the single-point optimization with axial inflow. First, it is observed that the designs with low pressure loss have a considerable amount of turning, taking into account that the outlet metal angle is allowed to vary from 53 deg to 68 deg. This is opposed to what was initially expected, as in conventional turbine design more turning generally implies larger pressure losses. Nonetheless, since the vane designs are highly diffusive, more convergent vane passage geometries can be beneficial to avoid massive diffusion and flow separation, and this may be done by increasing the vane turning.

However, as shown in Table 6, these values have been reduced in the multi-point optimization, especially near the endwalls. In axial inflow cases, a high turning increases the velocity toward the outlet, minimizing the risk of flow detachment, but for high inlet flow angles, it may become detrimental beyond a specific value, inducing larger separation and penalizing the performance. Since excessive turning increases the vane loading and the chances of secondary flow detachment, the metal angles at the hub and tip are lower than at mid-span.

After the optimization is completed, a transient analysis is performed on the baseline and the selected optimized geometries shown in Table 5. The aim is to confirm that the proposed methodology is effective in reducing pressure loss and pressure distortion under transient inlet flow angles. To do so, the performance benefits at pulsating conditions will be compared to the steady, multi-point optimization results.

To ensure a fair analysis, the transient and steady boundary conditions must match. In the steady multi-point optimization, the outlet mass flow is imposed and varied in each case to have an inlet Mach number of 0.6, as indicated in Table 3. In the unsteady case, both constant and time-varying outlet boundary conditions can be defined. Setting a time-changing outlet value may not be the right approach since the user is pre-defining the shape and the frequency at the domain outlet, which a priori are unknown. Hence, a constant outlet is preferred for the transient analysis. Static pressure is set for the outlet boundary condition, since it is reported to be more robust than mass flow [19], enhancing convergence. The downstream static pressure is selected as the average of the outlet pressures in the three steady simulations, as indicated in Table 7. As a result, the transient operating condition should be similar to the three steady cases.

Anyway, to minimize any sources of discrepancy and to have comparable results, the baseline and optimized geometries have been re-evaluated, using the same inlet steady conditions included in Table 3, but with the averaged outlet static pressure reported in Table 7, which will be also used in the transient simulations. Table 8 presents the averaged pressure loss coefficient and averaged pressure distortion for the steady cases with the new downstream pressure boundary condition. The percentual variation of the two optimized geometries versus the baseline is included too. Compared with the previous steady results shown in Table 3, except for the pressure distortion in the baseline design, all the metrics experience very slight deviations, highlighting that the newly selected operating point is extremely similar to the one employed in the multi-point optimization.

A phase-lagged approach is employed to resolve the unsteady flow field around the turbine vane. A sinusoidal flow angle profile is defined at the domain inlet, fluctuating from 30 deg to −30 deg. The outflow of an RDC may have a different flow angle profile shape; nevertheless, a sinusoidal wave has been chosen because it covers the entire range of flow angles with a similar time exposure to all values. Furthermore, due to the absence of strong spikes or discontinuities, it can help to achieve convergence.

The solver employs a second-order upwind spatial discretization scheme with a Courant–Friedrichs–Lewy number of 1 and a dual time stepping technique. To achieve convergence, the number of inner iterations at every time-step is set to 100, per numeca's recommendation [19]. A time-step of 35 μs is selected, providing 38 time-steps per inlet flow angle period, well above numeca's suggestion of at least 20 [19]. The adequacy of this value depends on the expected frequency of the flow phenomena and will be verified with a subsequent frequency analysis. Finally, to accelerate convergence, the computation is initialized from the steady solution with a zero inlet flow angle.

The simulation is run for 20 periods on 64 cores, with a convergence time of 24 h. Axial thrust, torque, pressure loss, and inlet and outlet mass flows were monitored to verify unsteady convergence, applying the method outlined by Clark and Grover [33]. Figure 17 illustrates the behavior of the torque and inlet mass flow of IND 238 for the last four periods. The frequency content of the last two periods is also included, proving convergence has been attained. It can be observed that no significant frequency content is present above 5 kHz. This confirms the adequacy of the selected time-step, which results in a maximum sampling frequency above 14 kHz.

Table 9 presents the main unsteady performance metrics for the selected designs. Both optimized geometries outperform the baseline in pressure loss and pressure distortion. Analyzing the steady results shown in Table 8, the percentual enhancement of the optimized designs compared to the baseline agrees very well between steady and unsteady cases, except for the pressure distortion for IND 483. This confirms that the proposed steady, multi-point optimization strategy is effective in reducing pressure loss and pressure distortion under pulsating inlet flow angles, providing percentual improvements similar to those obtained with unsteady computations.

The steady and unsteady metrics presented in Tables 8 and 9, respectively, are included in Table 10 for a direct comparison between the RANS and URANS performance. In all cases, unsteady pressure loss and pressure distortion levels are considerably lower than in steady conditions. The difference could be due to unsteady phenomena that depend on the frequency and shape of the inlet fluctuations and that cannot be captured with steady simulations. However, another source for discrepancy lies in how the RANS metrics are averaged. In this case, two of three steady conditions are highly detrimental in terms of performance (−30 deg and 30 deg) compared to the rest of the sinusoidal inlet angle profile, which could be the reason for the worse steady performance. In the proposed optimization methodology, only three inlet flow angle conditions (0 deg, −30 deg, and 30 deg) are resolved. To have a closer comparison with URANS, more cases may need to be run to obtain a higher resolution on the inlet flow angle profile. Nonetheless, it would come at the expense of increased computational cost and a longer optimization time.

This paper presents a design analysis of diffusive stator vanes, optimized to operate in the high-subsonic regime with significant inlet flow incidences, potentially enabling turbine integration into rotating detonation engines. Initially, the requirements to ingest flow with high Mach numbers are identified and implemented in the vane geometry using endwall contouring, allowing a high level of turning with an extended operating range. To assess the effect of the vane incidence on the flow field and performance and to extract design recommendations, a multi-point optimization with three different inlet angles is conducted, employing 3D steady RANS simulations. Compared to the baseline case, the optimized designs offer enhancements in averaged pressure loss and averaged pressure distortion of up to 37% and 50%, respectively. However, if the operating range cannot be shortened, then the improvement is reduced to 19% and 41%, respectively, due to the strong effect of the throat-to-inlet area ratio on the vane performance. Compared with geometries optimized for axial inflows, a larger leading edge radius and an aft-loaded design with a more progressive SS curvature distribution are beneficial for vane geometries operating under high incidences. Additionally, a more conical endwall profile with less radial diffusion along the first half of the vane chord and lower turning near the endwalls are demonstrated to help minimize pressure losses.

To conclude, a transient evaluation of the baseline and optimized vane geometries is performed, demonstrating the effectiveness of the presented optimization approach in minimizing pressure loss and pressure distortion for unsteady inlet flow angles. The rationale for the performance differences between steady and unsteady computations is also outlined.

This material is based upon work supported by the U.S. Department of Energy under Award Number DE-FE0032075. The authors would like to acknowledge the U.S. Department of Energy for the part-time faculty appointment of Professor Paniagua to the Faculty Research Participation Program at the National Energy Technology Laboratory. The authors would like to thank Logan Tuite for his help with the optimization setup and troubleshooting.

There are no conflicts of interest.

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.

f =

frequency (Hz)

f̅ =

reduced frequency

h =

channel height (mm)

$m˙$ =

mass flowrate (kg/s)

r =

radius (mm)

x =

axial coordinate (mm)

A =

area (mm²)

C =

chord length (mm)

L =

total axial length (mm)

M =

Mach number

P =

static pressure (bar)

Q =

constraint parameter

R =

specific gas constant (J/kg K)

S =

curve distance (mm)

T =

static temperature (K)

V =

absolute flow velocity (m/s)

W =

penalty weight

M_is =

isentropic Mach number

P₀ =

total pressure (bar)

T₀ =

total temperature (K)

Y_P =

pressure loss coefficient

y+ =

non-dimensional wall distance

α =

flow angle (deg)

α_1m =

vane inlet metal angle (deg)

  α_2m =

vane outlet metal angle (deg)

γ =

specific heat ratio

δ =

boundary layer thickness (mm)

δ_TE =

trailing edge wedge angle (deg)

θ =

azimuthal coordinate (deg)

λ =

stagger angle (deg)

ν =

kinematic viscosity (m²/s)

σ =

pressure standard deviation (bar)

1 =

passage inlet

2 =

passage outlet

av =

averaged

ax =

axial direction

h =

hub

in =

inlet

LE =

leading edge

m =

mid-span

out =

outlet

s =

isentropic

t =

tip

TE =

trailing edge

th =

throat

 thres =

threshold

DoE =

design of experiments

GCI =

grid convergence index

HPT =

high-pressure turbine

IND =

individual

NGV =

nozzle guide vane

PS =

pressure side

RANS =

Reynolds-averaged Navier–Stokes

RDC =

rotating detonation combustor

SPEA =

strength Pareto evolutionary algorithm

SS =

suction side

SST =

shear stress transport

STD =

standard deviation

URANS =

unsteady Reynolds-averaged Navier–Stokes