## Abstract

New compact engine architectures such as pressure gain combustion require ad hoc turbomachinery to ensure an adequate range of operation with high performance. A critical factor for supersonic turbines is to ensure the starting of the flow passages, which limits the flow turning and airfoil thickness. Radial outflow turbines inherently increase the cross section along the flow path, which holds great potential for high turning of supersonic flow with a low stage number and guarantees a compact design. First, the preliminary design space is described. Afterward a differential evolution multi-objective optimization with 12 geometrical design parameters is deducted. With the design tool autoblade 10.1, 768 geometries were generated and hub, shroud, and blade camber line were designed by means of Bezier curves. Outlet radius, passage height, and axial location of the outlet were design variables as well. Structured meshes with around 3.7 × 10^{6} cells per passage were generated. Steady three-dimensional (3D) Reynolds-averaged Navier–Stokes (RANS) simulations, enclosed by the k-omega shear stress transport turbulence model were solved by the commercial solver CFD++. The geometry was optimized toward low entropy and high-power output. To prove the functionality of the new turbine concept and optimization, a full wheel unsteady RANS simulation of the optimized geometry exposed to a nozzled rotating detonation combustor (RDC) has been performed and the advantageous flow patterns of the optimization were also observed during transient operation.

## Introduction

In traditional engines that follow the Joule/Brayton cycle, conventional deflagration combustors deliver low Mach numbers at the inlet of the turbine, with typical values around Mach 0.1 [1]. At such low subsonic inlet speeds, high turning angles can be achieved throughout the turbine nozzle guide vane, while guaranteeing that the flow remains unchoked satisfying the isentropic limit. Similar to the design of subsonic stators, supersonic passages should comply with the isentropic limit, and hence constrains the maximum turning angle. In contrast to the subsonic case, higher supersonic inlet Mach numbers allow for higher turning in the supersonic vane [2]. Higher Mach numbers, however, are accompanied by higher pressure loss; consequently, the designer should compare power output versus pressure loss. To increase the cycle efficiency, new cycles such as detonation (pressure gain) cycles can achieve higher stagnation pressure due to a near constant combustion [3]. This near constant volume combustion is accompanied by a higher cycle efficiency. For the same turbine inlet conditions, the detonation-based cycle results in a significant lower production of entropy, hence delivering a higher cycle efficiency. Several detonation engine architectures exist. A first one is a pulse detonation combustor in which a cyclic refill occurs in a tube [4]. Another architecture is a rotating detonation combustor (RDC) in which a rotating shock continuously burns the fuel. In RDCs, the Mach number at the outlet of the combustor is around one with fluctuations below and above the speed of sound [5]. These type of combustors have been experimentally [6–8] and numerically [9,10] investigated since the 1960s. Finally, reduced models to get fast predictions of the flow field have been presented via method of characteristics solvers [2,11].

Paniagua et al. [12] found that there will be a loss of 80% when diffusing from Mach 3.5 to subsonic (Fig. 1). Hence, to efficiently cope with these high Mach number flow and pressure fluctuations, new turbine classes need to be designed and several architectures for efficient downstream power extraction are possible. A first choice is to integrate the turbine with a diffuser to significantly reduce the turbine entry Mach number, which then would allow to use conventional axial turbines; however, this requires a rather long diffuser, or an integration of the diffuser with the turbine nozzle guide vane [13]. The optimization of the nozzle guide vane end walls can result in an efficiency of about 80%. The increase of efficiency compared to a baseline turbine without contouring was above 30% points. Another solution is to design nozzles to further boost the Mach number, damp the flow angle fluctuations, and design axial supersonic turbines. This class of turbine was developed by Sousa et al. [2] and Paniagua et al. [12] and is characterized by axial supersonic flow throughout the passage, slender airfoils with limited turning to remain below the Kantrowitz limit [14], and guarantee starting of the turbine. Recently, Sousa et al. [15] performed an engine analysis with a reduced model of the rotating detonation combustor and the supersonic turbine. They found an increase of several percentage points at lower compression ratios for a turbo jet engine configuration. Finally, Huff et al. [16] experimentally investigated radial inflow turbines coupled to a radial rotating detonation engine enabling a compact design with power output of 70 kW. Higashi et al. [17] also tested a centrifugal compressor, RDC, and single stage radial flow turbine. Akbari and Polanka tested pressure gain combustor devices with radial turbine designs as well [18].

This paper describes the design procedure for compact radial outflow turbines for supersonic flows. First, the design method and space is described followed by an optimization, and finally, a full unsteady investigation of rotating detonation waves exposed to optimized radial outflow turbines.

## Design and Optimization

### Baseline Design.

The Kantrowitz line plays an important role to design efficient and compact supersonic turbines. The area ratio of the throat *A*_{throat} to inlet area *A*_{inlet} needs to be above the Kantrowitz line to allow a self-starting of the turbine and thus, enabling supersonic flow through the entire passage as visualized in Fig. 2.

In conventional supersonic turbines the criterion of a large enough *A*_{throat} restricts the turning of the flow and thus, the wheel torque. Hence, radial outflow designs with inherently increasing cross section are favorable to obtain high flow turning and compact machine designs. To be able to combine an axial outflow RDC with the designed turbine, an axial inflow into the turbine passage must be realized. Figure 3 shows the chosen passage geometry for the baseline design of the novel axial inflow radial outflow turbine. A circular shroud curvature, as can be found in the radial outflow compressor design by McKain and Holbrook [19], and a constant passage height was defined. The passage shroud turning radius to passage height ratio (1.5) has the order of commonly found radial turbomachinery. The passage height was kept constant to guarantee an expansion of the flow cross section as shown in Fig. 2. From these design constraints an outlet diameter of 0.22 m was retrieved. Hence, the ratio of orthogonal passage area to inlet duct area is monotonously increasing, so that the smallest area *A*_{throat} can be found at the inlet of the turbine passage giving freedom to high flow turning. However, considering the thickness of the blades at the leading edge (LE), the cross-sectional area at the inlet can significantly be reduced and endanger the self-starting of the passage. Thus, the leading edge had to be cut back toward higher axial positions, where *A*_{throat}/*A*_{inlet} has a higher value, until the passage was started.

The blades are designed with a blade thickness of 1.6 mm at the hub and of 0.8 mm at the tip. A tip gap of 0.4 mm height was assumed. The camberline was designed by means of a Bezier curve defined by six control points (CPs) constantly distributed over the blade as shown in Fig. 3(b). The same distribution of rel. chord length to *θ* was defined at hub and shroud. The first point was set to obtain zero incidence at the inlet. The design speed at the outlet tip was set to 650 ms^{−1}. McKain and Holbrook [19] state stress stability for even higher centrifugal forces acting on blades with comparable blade thickness. The resulting wheel geometry was modeled with BladeGen (NUMECA) and is shown in Fig. 3(c). A relatively low blade number of ten was chosen to reduce the number of leading edge shocks entering the passage and increase the self-starting probability. The inlet bulb was designed as a cone to reduce shock losses in the inlet.

During a preliminary design phase, numerous geometries were evaluated to maximize the outlet blade angle. An increase in the *β*-angle at low chord length prevents flow separation at the shroud due to the guidance from axial to radial flow direction. However, a large increase causes flow separation on the pressure side (PS) close to the leading edge on the blade, which restricts the design space. Although a higher outlet blade turning reduces the passage cross section, reducing the exit flow velocity, a maximum flow turning was identified to be most beneficial for torque production. The maximum outlet bade angle was restricted by flow separation close to the outlet, where centrifugal forces and blade angle are high. Thus, an outlet blade angle of 42 deg at the shroud and of only 14 deg at the hub was realized. While the restraint on the outflow angle limits the power extraction, high shock losses in the inflow section and at the leading edge potentially reduces efficiency of the expansion device.

### Computational Fluid Dynamics Setup.

For the computational domain of the optimization an 80 mm long inlet duct was modeled upstream of the tip of the inlet bulb. As shown in Fig. 3(a), the outlet of the domain was set 60 mm behind the trailing edge (TE) of the blade. To reduce the numerical expenses of the optimization, steady three-dimensional (3D) Reynolds-averaged Navier–Stokes (RANS) simulations, enclosed by the *k*-omega shear stress transport turbulence model, were solved for only one blade passage by the commercial solver CFD++ [20]. The solver was validated with a Mach 5 ramp case to demonstrate the shock boundary layer interaction was fully resolved [21]. The solver was further validated with experimental results of a transonic vane with supersonic outlet flow [22]. In the contour plot of the Mach number at midspan (Fig. 4(a)), the trailing edge shock impinging on the neighboring vane suction side (SS) is visualized. The comparison of the computational fluid dynamics (CFD) and experiments for the isentropic Mach number on the vane (Fig. 4(b)) shows good agreement and the error at the location of the shock impingement is around 0.3%.

While the inlet boundary conditions were set by imposing the $Min,\u2009Tin$, and *p*_{in} according to the values in Table 1, the outlet boundary was set to the supersonic outlet condition. By means of NUMECAs autogrid5, three different structured grids with an HOH block structure were generated to demonstrate mesh convergence of the two optimization objectives (shaft power and mass flow averaged entropy 30 mm behind the trailing edge). As can be seen in Table 2, the relative derivation from the finest mesh results is small. Due to the small difference (below 0.25%) between “medium” and “fine” mesh, the middle cell density, which is depicted in Fig. 5, was considered for this study. However, since the geometry changes and might increase the passage length during the optimization, more cells are required for the optimization. Additionally, a strict compliance of a *y*^{+}-value below one is desired. Thus, the final mesh with around 3.79 × 10^{6} cells per passage and a maximum *y*^{+}-value of 0.84 was selected. The tip gap was meshed with 41 cells between upper blade surface and shroud.

M_{in} | 2 |

T_{in} | 1380 K |

P_{in} | 0.5000 bar |

T_{t}_{,in} | 2283 K |

P_{t}_{,in} | 4.386 bar |

$m\u02d9$ (one passage) | 0.1481 kg s^{−1} |

$m\u02d9red.$ (one passage) | 1.614 kg s^{−1} bar^{−1} K^{0.5} |

M_{in} | 2 |

T_{in} | 1380 K |

P_{in} | 0.5000 bar |

T_{t}_{,in} | 2283 K |

P_{t}_{,in} | 4.386 bar |

$m\u02d9$ (one passage) | 0.1481 kg s^{−1} |

$m\u02d9red.$ (one passage) | 1.614 kg s^{−1} bar^{−1} K^{0.5} |

Coarse | Medium | Fine | |
---|---|---|---|

$(sout\u2212sout,Finesout,Fine)$ | 0.0359 % | 0.0077 % | 0.0000 % |

$(W\u02d9\u2212W\u02d9FineW\u02d9Fine)$ | −0.296 % | 0.032 % | 0.000 % |

$(\eta tt\u2212\eta tt,Fine\eta tt,Fine)$ | −1.75 % | −0.22 % | 0.00 % |

Maximum y^{+} | 4.45 | 2.23 | 0.84 |

Cells | 1,933,824 | 2,972,224 | 3,792,960 |

Coarse | Medium | Fine | |
---|---|---|---|

$(sout\u2212sout,Finesout,Fine)$ | 0.0359 % | 0.0077 % | 0.0000 % |

$(W\u02d9\u2212W\u02d9FineW\u02d9Fine)$ | −0.296 % | 0.032 % | 0.000 % |

$(\eta tt\u2212\eta tt,Fine\eta tt,Fine)$ | −1.75 % | −0.22 % | 0.00 % |

Maximum y^{+} | 4.45 | 2.23 | 0.84 |

Cells | 1,933,824 | 2,972,224 | 3,792,960 |

### Parametrization.

*Z*-coordinate and the variability of the outlet CPs was limited by the inlet wall

*R*-coordinate as visible in Fig. 6. Furthermore, freedom was given to the

*Z*-coordinate of the shroud at the passage outlet and to the passage height. Since flow toward higher radii results in a reduction of the power

*w*according to Eq. (1) but at the same time allows a further acceleration and higher flow turning, the blade outlet radius was set as optimization parameter as well.

*c*

_{u}_{,2}with no swirled inflow (Eq. 2).

Furthermore, the first control point of the Bezier curve of the camberline was adjusted according to the set speed to obtain zero incidence at the tip of the blade. All other control points of the camberline were set as free optimization parameters. An important parameter to secure a high number of started turbine passages (“individuals”) was the definition of the *Z*-coordinate of the leading edge. For this parameter, the location was imposed to obtain the same area ratio *A*_{throat}/*A*_{inlet} as in the baseline design depending on generated hub and shroud geometries.

### Optimization Procedure.

## Supersonic Optimization Results

Although a higher outlet radius is inherently associated with a power penalty (demonstrated by Eq. (1)), here it allows to provide a power gain from a larger wheel outlet area with consequent further flow acceleration and the benefit of a longer passage allowing smoother flow turning. Since the outlet tip speed was fixed for all geometries, centrifugal forces are reduced in the outlet region. Furthermore, longer blades and thus, smoother curvatures can be realized due to an extended flow channel (hub and shroud lines) and prevents flow separation at the outlet section, while guaranteeing high-blade turning angle. The optimal configuration expands the working fluid over a total pressure ratio of 4.86, a total temperature ratio of 1.243, and the power delivery is 831 kW. This constitutes a rise of 728 kW compared to the baseline design, and with 65.4% efficiency nearly 45% points more than the baseline design. This is primarily achieved by ensuring the turbine passage is started, preventing both the appearance of a standing normal shock and the abatement of the separated region. In turbochargers, radial turbines exposed to transients into off-design operation exhibit reductions in efficiency of nearly 40% [28] (in stationary internal combustion engine condition). Conventional axial turbine exposed high inlet flows deliver an efficiency of barely 50%; however, the optimization of the endwall can yield over 30% of efficiency improvement [29].

By comparing the optimized passage in Fig. 8(a) with the baseline passage in Fig. 6, aforementioned conclusions can be confirmed. Table 3 summarizes the geometrical details of the optimized geometry compared to the baseline case. The optimized geometry has a higher outlet diameter of 295 mm combined with a higher passage height. Like in the baseline design, the highest change of the blade curvature appears at relatively low chord-length as shown in Fig. 8(b). The outlet blade turning was increased significantly to 72 deg at the shroud. Figure 8(d) shows that the blade normal distance to the neighbor blade remains rather constant, which decreases flow separation. Hence, the outlet passage height is the main driver behind the increase in cross-sectional area and hence flow acceleration. The outlet diameter in combination with the constant outlet speed gives the reduced blade tip speed of 937 rpm/K^{0.5} for the optimized geometry. The optimized geometry was filed as provisional patent [30].

Baseline | Optimized | |
---|---|---|

D_{shroud,in} | 100 mm | 100 mm |

D_{hub,in}$(Dhub,inDshroud,in)$ | 20 mm (0.2) | 20 mm (0.2) |

z_{LE}$(zLEDshroud,in)$ | 15 mm (0.15) | 53 mm (0.15) |

h_{LE}$(hLEDshroud,in)$ | 40 mm (0.4) | 30 mm (0.3) |

h_{TE}$(hTEDshroud,in)$ | 40 mm (0.40) | 59 mm (0.59) |

$\u03f5LE$$(\u03f5LEhLE)$ | 0.4 mm (0.01) | 0.4 mm (0.013) |

$\u03f5LE$$(\u03f5TEhTE)$ | 0.4 mm (0.01) | 0.4 mm (0.0068) |

D_{out}$(DoutDshroud,in)$ | 220 mm (2.2) | 295 mm (2.95) |

β_{out,hub} | 14 deg | 55 deg |

β_{out,shroud} | 42 deg | 72 deg |

Baseline | Optimized | |
---|---|---|

D_{shroud,in} | 100 mm | 100 mm |

D_{hub,in}$(Dhub,inDshroud,in)$ | 20 mm (0.2) | 20 mm (0.2) |

z_{LE}$(zLEDshroud,in)$ | 15 mm (0.15) | 53 mm (0.15) |

h_{LE}$(hLEDshroud,in)$ | 40 mm (0.4) | 30 mm (0.3) |

h_{TE}$(hTEDshroud,in)$ | 40 mm (0.40) | 59 mm (0.59) |

$\u03f5LE$$(\u03f5LEhLE)$ | 0.4 mm (0.01) | 0.4 mm (0.013) |

$\u03f5LE$$(\u03f5TEhTE)$ | 0.4 mm (0.01) | 0.4 mm (0.0068) |

D_{out}$(DoutDshroud,in)$ | 220 mm (2.2) | 295 mm (2.95) |

β_{out,hub} | 14 deg | 55 deg |

β_{out,shroud} | 42 deg | 72 deg |

## Optimized Geometry Aerodynamics

As observed in Figs. 8(a) and 8(c), the leading edge of the blade is relatively far downstream from the inlet bulb. The shock structure at the inlet of the turbine wheel in Fig. 9 shows that a far downstream position is profitable for the shock losses. The flow enters the passage axially with a Mach number of 2 and is decelerated due to the diagonal shocks emanating from the inlet cone and the compression waves of the hub. Additionally, the cone shock reflects at the shroud causing a local separation, which reattaches before entering the blade passage. Figure 9 shows a “started” turbine that complies with the Kantrowitz limit (Fig. 2), in which the turbine passage is mostly supersonic with oblique shocks formed at both endwalls. Thus, the leading edge shock, which appears to be relatively orthogonal due to the low Mach number for all simulated geometries, has lower intensity. Although the absolute and relative Mach number is locally reduced below 1, the turbine is started as seen at 10% span in Fig. 10(a). At higher radial positions like 50% span in Fig. 10(b) and 75% span in Fig. 10(c), the relative inlet Mach number is reduced to values of around 1.6 due to the shock structure at the inlet and the leading edge shock crosses the passage diagonally. The leading edge shock crosses the passage diagonally and remains supersonic for the entire passage. Figure 11 depicts the static surface pressure and the shocks impingement due to the inlet cone is observed. These shock impingements are dominant on the pressure side and leads to further blade loading and power. The static pressure distribution at 75% span shows that the leading edge shock first hits the suction side, which can be explained by the incoming flow angle. Afterward, the shock is reflected and hits the pressure side of the next blade. Additionally, the shock that first arrives the pressure side is clearly reflected to the suction side. However, the reflection angle is much sharper due to the convex curvature of the pressure side.

### Subsonic Inflow.

When the flow enters the blade passage, the flow experiences a reduction of the cross-sectional area due to the thickness of the blades (Fig. 12). The optimized geometry and most generated radial turbine geometries have a monotonously growing passage cross section. While the curvature of the hub introduces compression waves, the convex shape of the shroud introduces expansion waves over a significantly larger circumference. Hence, the turbine passage is characterized by favorable conditions to transition from subsonic to supersonic flow throughout the passage. Thus, the performance of the fluid machine for subsonic inflow was assessed.

To ensure a fair comparison of the performance parameters, the inlet mass flow was maintained to the one of the supersonic inlet condition. Furthermore, the total inlet temperature was kept constant to simulate the same reduced rotational speed. With these subsonic boundary conditions, an inlet Mach number of around 0.6 stabilized in the inlet duct. As observed in Fig. 13, the relative Mach number reaches one close the throat (LE) of the passage at all spans. From the LE the flow continuously accelerates, as depicted in Fig. 14. However, due to the concave surfaces of hub and blade pressure side, the flow takes longer for a complete transition to fully developed supersonic flow close to this surface. The flow is accelerated until it reaches high relative outlet Mach numbers of around three and absolute Mach numbers of around two in the stationary frame of reference.

Due to the subsonic inflow no strong leading edge shock develops, which significantly reduces the overall losses of the turbine. Also, further shock reflections and following loss generation mechanisms are diminished as visualized in Fig. 14 and in Fig. 15. Due to the subsonic inflow, the surface pressure level is high near the inlet section. The surface pressure distributions appears smoother and does not show shock impingement patterns. However, at 30% chord length a local flow separation occurs on the suction side. In general, the flow quality was improved significantly by using subsonic instead of supersonic inflow. Although the turbine power is slightly reduced to 807 kW, the total-to-total efficiency increases to 79.1% (nearly 14%points increase). Lower losses reduce the total pressure ratio to 3.35, while the total temperature ratio nearly maintains at 1.236. The reduction of shaft power can be explained by the missing shock impingement patterns on the pressure side. In conclusion, the optimized geometry is able to handle supersonic and subsonic inflow, which allows to use the turbine in numerous application without the need of a (Laval) nozzle to generate supersonic inflow.

### Off-Design Analysis.

First, only the inlet total pressure was varied and the inlet total temperature and static values of temperature and pressure were maintained constant at the levels listed in Table 1. Table 4 lists all the relevant nondimensional parameters for the evaluated operating points. The total pressure ratio as well as the reduced mass flow is maintained nearly constant for all supersonic and subsonic inlet conditions. The power output increases linearly with the total pressure, because of the linear dependency to the massflow.

Supersonic inflow | ||||
---|---|---|---|---|

$m\u02d9$ (kg s^{−1}) | 1.111 | 1.481^{a} | 1.852 | 2.222 |

$m\u02d9red.$ (kg s^{−1} bar^{−1} K^{0.5}) | 16.14 | 16.14^{a} | 16.14 | 16.14 |

$W\u02d9$ (kW) | 619 | 832^{a} | 1044 | 1253 |

η_{tt} (%) | 64.6 | 65.4^{a} | 65.7 | 65.4 |

π_{tt} | 4.905 | 4.859^{a} | 4.849 | 4.891 |

$Tt,inTt,out$ | 1.241 | 1.243^{a} | 1.245 | 1.242 |

$pt,in$ (bar) | 3.290 | 4.386^{a} | 5.483 | 6.579 |

Subsonic inflow | ||||

$m\u02d9$ (kg s^{−1}) | 1.111 | 1.481 | 1.852 | 2.222 |

$m\u02d9red.$ (kg s^{−1} bar^{−1} K^{0.5}) | 25.47 | 25.51 | 25.54 | 25.57 |

$W\u02d9$ (kW) | 602 | 807 | 1012 | 1218 |

η_{tt} (%) | 77.8 | 79.1 | 80.0 | 80.4 |

π_{tt} | 3.406 | 3.352 | 3.314 | 3.307 |

$Tt,inTt,out$ | 1.234 | 1.236 | 1.236 | 1.237 |

p_{t}_{,in} (bar) | 2.081 | 2.770 | 3.457 | 4.146 |

Supersonic inflow | ||||
---|---|---|---|---|

$m\u02d9$ (kg s^{−1}) | 1.111 | 1.481^{a} | 1.852 | 2.222 |

$m\u02d9red.$ (kg s^{−1} bar^{−1} K^{0.5}) | 16.14 | 16.14^{a} | 16.14 | 16.14 |

$W\u02d9$ (kW) | 619 | 832^{a} | 1044 | 1253 |

η_{tt} (%) | 64.6 | 65.4^{a} | 65.7 | 65.4 |

π_{tt} | 4.905 | 4.859^{a} | 4.849 | 4.891 |

$Tt,inTt,out$ | 1.241 | 1.243^{a} | 1.245 | 1.242 |

$pt,in$ (bar) | 3.290 | 4.386^{a} | 5.483 | 6.579 |

Subsonic inflow | ||||

$m\u02d9$ (kg s^{−1}) | 1.111 | 1.481 | 1.852 | 2.222 |

$m\u02d9red.$ (kg s^{−1} bar^{−1} K^{0.5}) | 25.47 | 25.51 | 25.54 | 25.57 |

$W\u02d9$ (kW) | 602 | 807 | 1012 | 1218 |

η_{tt} (%) | 77.8 | 79.1 | 80.0 | 80.4 |

π_{tt} | 3.406 | 3.352 | 3.314 | 3.307 |

$Tt,inTt,out$ | 1.234 | 1.236 | 1.236 | 1.237 |

p_{t}_{,in} (bar) | 2.081 | 2.770 | 3.457 | 4.146 |

Corresponds to the design condition of the optimization.

Second, the inlet total pressure and total temperature were maintained constant with the values mentioned in Table 1 and inlet Mach number plus rotational speed were varied independently. Figure 16 displays the efficiency and the power output as a function of the reduced massflow. Both efficiency and power output increase with increasing rotational speed, and lowering the inlet Mach number, all related to weaker leading edge shocks and higher massflow toward the critical massflow. Note that when the inlet Mach number was reduced below 1.8, the turbine did not start anymore. Thus, a zone spans from 21 to 25 kg s^{−1 }bar^{−1} K^{0.5}, where the turbine cannot operate. When the Mach number is increased up to Mach numbers around 2.5 the flow separates at the shroud endwall and the passage chokes. The highest efficiency in supersonic inlet condition of 71.8% was achieved with an inlet Mach number of 1.8, which is close to the starting limit. For the subsonic inflow case, the inflow Mach number is a result of the throat geometry and rotational speed and thus cannot be influenced by the boundary conditions. Hence, only variations of the rotational speed were simulated for this type of inflow. The variation of speed with subsonic inflow shows a maximum at design speed. An increase of the speed causes growing incidence losses, so that the initial trend of growing efficiencies is inverted.

### Unsteady Rotating Detonation Combustor Inflow

#### Inlet Condition.

For this study, the designed turbine was assessed with the flow of a hollow RDC such as described by Stoddard et al. [31] and Anand et al. [32,33]. In the chosen setup, the exhaust flow of the RDC was further accelerated by means of a divergent nozzle to finally achieve a mass flow averaged Mach number of 2. The resulting flow field 20 mm upstream of the turbine wheel is shown in Fig. 17(b). Due to the short circumference, only one moving oblique shock was modeled to rotate in positive *θ*-direction and the flow field is characterized by high unsteadiness. Static pressure fluctuations of 192%, total temperature fluctuations of 78%, and Mach number fluctuations of 67% appear in the inlet flow as presented in Fig. 18. The core flow was modeled with steady boundary conditions at Mach 2. To avoid a breakdown of the inlet flow, the static pressure had to be set to the mean value of the surrounding RDC pressure and the Mach number was set to 2.

#### Unsteady Numerical Settings.

Finally, the optimized geometry (under steady flow conditions) was assessed with the aforementioned highly transient flow of a hollow RDC. Unsteady RANS simulations with a time-step of 5.7 × 10^{−7} s and 25 internal iteration steps were performed by means of the CFD solver CFD++. Thus, one entire rotation of the oblique shock was resolved by 300 time steps. During this period the rotor rotates by 45.6 deg. The entire rotor with ten blades was modeled with 37 M cells.

#### Unsteady Flow Results.

As plotted in Fig. 19, the rotating oblique shock mixes with the core flow before reaching the passage of the rotor and the peak in total temperature is spread from 10% to 90% span of the flow passage. Thus, high total temperatures can be found over a circumference of 170 deg close to the hub. The radial and circumferential mixing leads to an increase of radially averaged total temperature at the passage inlet (visible in Fig. 20). Due to the mixing with the core flow and the inflow shock pattern, the radially mass flow averaged total pressure spike appears reduced and the peak is divided into two total pressure peaks. Inflow shocks and the overall inflow geometry lead to a considerable deceleration of the flow up to transonic Mach numbers. However, due to the capability to start with subsonic inflow, the turbine remains started in all turbine passages so that outlet flow Mach numbers are oscillating around 3. Thus, the reduced Mach number allows to have less leading edge shock losses. In general, total pressure, total temperature relative flow angle variations are damped noticeably through the rotor passage. Fluctuations in the outlet Mach number appear due to the mixing of the wake effect. The relative flow angle is reduced up to 20 deg before entering the rotor passage. This reduction occurs due to the flow deceleration in the inlet section, while the inlet flow has a mass flow averaged flow angle of around 0 deg (Fig. 18). Highest values of the inlet flow angle are situated tangentially downstream and upstream of the oblique shock. The relative flow angle at the outlet is reaches values up to −55 deg and is damped with fluctuations below 8.2 deg.

Figures 21 and 22 show the relative Mach number and relative total pressure at mid span over one entire cycle. It can be seen how the oblique shock rotates in the same direction as the rotor. Just upstream of the LE, the flow behind the oblique shock becomes subsonic, allowing lower leading edge shock losses when then pressure peak enters the passage. Furthermore, over the entire circumference upstream of the LE, a complex system of weak shocks can be identified. This shock pattern comes from the inlet cone at the hub and from the shock reflection at the shroud (Fig. 9). Upstream of the LE, a complex Mach number distribution is visible. At the LE weak leading edge shocks appear due to the locally low supersonic flow and before the oblique shock hits the blade, compression waves generate flow speeds with a relative Mach number below one at the pressure side of the inducer. At midchord, the flow of the oblique shock creates spots of extreme acceleration on the SS, which leaves the blade passage flow to expand and to accelerate up to absolute Mach numbers of around 3.5.

Figure 22 plots the system of weak shocks at the inlet which barely create total pressure loss and a similar trend is observed for the LE shock. Figure 23 represents the surface static pressure curves for the same time instances as represented in Figs. 21 and 22. Due to the rotating shock, static pressure maxima on the PS are significantly higher than with steady flow. Interestingly, the fluctuations of static wall pressure on the SS are of much lower magnitude. While static surface pressure fluctuations become less on the PS toward lower spans, fluctuations on the SS are slightly increasing. Hence, highest blade loading and highest amplitudes of blade loading fluctuations can be expected in the inducer at higher spans. Downstream of that location, the blade loading decreases continuously.

An over one entire cycle time averaged power of 1.41 MW was obtained with the transient boundary condition of the hollow RDC. In comparison to a separately executed steady-state simulation with averaged static inlet conditions and axial velocity (same mass flow and inlet Mach number) this corresponds to a drop in power of 18%. The reason might be the lower core total temperature of the hollow RDC and efficiency penalties due to transient flow conditions.

## Conclusions

A Mach 2 supersonic axial inlet and radial outflow turbine was designed to produce a baseline configuration. The resulting turbine geometry was subsequently optimized using a genetic multi-objective optimization procedure with twelve design parameters to define the blade surface, hub, and shroud. The optimization resulted in 768 converged simulations of individuals with started supersonic inflow. From the optimization the following design criteria can be drawn to achieve high efficient axial inflow radial outflow supersonic turbines for a given inlet diameter that depends on the upstream combustor and/or nozzle:

Cutting back of leading edge to achieve self-starting ability.

Extending hub inlet bulb for advantageous shock system.

Aiming for continuously growing passage cross section and high blade turning.

Reaching high levels of blade turning at low chord length avoids separation at the shroud at the inlet of the turbine passage.

Increasing passage height for cross section expansion.

Keeping a rather constant blade normal distance which results from high blade turning.

Comparing geometries with constant outlet speed with higher outlet radius reduce centrifugal forces and thus flow separation in exducer and allows for higher turning angles. Furthermore, a higher outlet radius gives more area potential which can be invested in flow turning or flow acceleration.

Extending hub and shroud lines allow for longer blades and thus, smother curvatures. Hence, flow separation and flow turning can be improved.

Compared to the baseline design with an efficiency of 20%, the performance was enhanced by 728 kW of shaft power, delivering a total-to-total efficiency of 65.4%. The specific shape of the optimized axial inflow radial outflow turbine enables ingestion of subsonic flow and converting it to supersonic outflow. This reduces LE shock losses significantly, so that total-to-total efficiencies of 79.1% were achieved. The analysis at off-design conditions revealed unstarted turbine passages for Mach numbers below 1.8, close to this value the turbine experiences its peak efficiency. The optimized turbine was modeled with highly transient supersonic inflow of a hollow rotating detonation combustor.

A future experimental demonstration of the present turbine will require the design of a cooling system to ensure long creep life. Furthermore, stress and dynamic analysis should be performed, as well as evaluations of the blade lifetime. During this process it is likely the design of the turbine may need to be revised to ensure that the lifetime of the blade remains within the required number of cycles.

## Acknowledgment

Lukas Benjamin Inhestern was partially supported by FEDER and the Spanish Ministry of Economy and Competitiveness through grant number TRA2016-79185-R. His research stay as visiting scholar at Pudue University was additionally funded by Universitat Politécnica de Valencia. The authors would also 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. Special thanks to Zhe Liu for his assistance in the validation of the CFD solver.

## Funding Data

National Energy Technology Laboratory (Faculty Research Participation Program) (Funder ID: 10.13039/100013165).

Spanish Ministry of Economy and Competitiveness (Grant No. TRA2016-7918-R).

Universitat Politecnica de Valencia (Travel Grant).

U.S. Department of Energy (Part-Time Faculty Appointment, Funder ID: 10.13039/100000015).

## Nomenclature

### Latin Letters

*A*=area

*c*=absolute velocity

*c*=_{p}isobaric heat capacity

- CFD =
computational fluid dynamics

- CP =
control point

- LE =
leading edge

- $m\u02d9$ =
massflow

*M*=Mach number

*p*=pressure

- PS =
pressure side

*R*=radius

- RDC =
rotating detonation combustor

- SS =
suction side

*T*=temperature

- TE =
trailing edge

*v*=relative velocity

*w*=specific power

- $W\u02d9$ =
power

*y*^{+}=nondimensional wall distance

*Z*=axial coordinate