## Abstract

Supersonic inlet flow is usually considered to be detrimental to the performance of turbine systems. A new class of bladeless turbines was developed, which allows for power extraction from supersonic axial inflows without swirl with minimal maintenance cost. This is achieved through a wavy hub surface that promotes shocks and expansion fans and hence generates torque. In a first step, a baseline bladeless turbine is designed and the power extraction is analyzed. The bladeless turbine surface is parametrized in matlab and subsequently imported into Hexpress (Numeca) to mesh an unstructured grid. The first layer thickness is kept below one for all the simulations. The unstructured mesh is loaded into cfd++ (Metacomp) to solve the three dimensional steady Reynolds-averaged Navier–Stokes (RANS) equations. Second, the work extraction principle is broken down from a three-dimensional unsteady problem into a two-dimensional steady phenomenon. An experimental campaign is outlined and test details are discussed. Finally, after the experimental characterization, the operational envelope and scaling of the bladeless turbine are described for several reduced mass flows, reduced speeds, and geometrical features of the turbine (amplitude of the wavy surface, helix angle, hub radius, and length of the axial turbine).

## 1 Introduction

Commercial supersonic flight is constrained by noise and emissions issues [1]. In 2018, the International Council on Clean Transportation [2] reports that depending on the engine configuration, a supersonic aircraft requires three to nine times more fuel per passenger than subsonic aircraft equipped medium to high bypass fan ratio engines [3]. Hence, in high-speed propulsion, small-scale and high temperature compliant turbines are vital to minimize the relative size of the engine bay and overall weight. Turbine-based combined cycles include combustors delivering supersonic axial outlet flows [4], alas conventional turbines are designed to operate efficiently at inlet Mach numbers of around 0.1 and require advanced cooling and special materials. Paniagua et al. [5] showed that a conventional turbine subjected to an inlet Mach number of 3.5 has pressure losses over 80% and is unstarted. Unstarting is a phenomenon in which the isentropic limit (for subsonic flows) and the Kantrowitz limit [6] (for supersonic flows) is not satisfied and not all mass flow can be ingested by the turbine. Consequently, a pressure or shock wave builds up in front of the turbine and travels upstream toward the combustor. Hence, for these high-speed inflows, new classes of expansion devices able to efficiently handle high velocity flow have been investigated in the past years. Efficiencies beyond 80% can be achieved for conventional turbines using ad hoc diffusers integrated with the turbine [7]; however, these are prone to separation at off-design operation [8].

For supersonic inlet conditions, supersonic axial turbines were developed in which the axial Mach number remains supersonic throughout the passage [9]. The supersonic turbine is characterized by slender blade shapes to limit shock losses with a low amount of turning. This turning (and consequent power extraction) is constrained by the Kantrowitz limit that sets the maximum flow angle. Through an engine analysis, it was proven that for supersonic flights, a low-pressure ratio turbojet equipped with a supersonic turbine could be up to 5% more fuel efficient than conventional turbojet engines [10].

A third class of fluid machine is the bladeless turbine. Vinha et al. [11] proposed an axial bladeless turbine (MBDA patent EP 2868864 [12]) operating in the steady supersonic flow regime (flows with Mach number around 3.5 and an inlet relative swirl angle of 16 deg). In this type of smooth/flat axial turbine, the power is extracted by only using the viscous forces. Additionally, a preswirler or nozzle guide vanes are needed to generate power (absolute inlet flow angle of 30 deg [11]). The power extraction was explained by means of velocity triangles and the effect of the viscous forces were studied via steady Reynolds averaged Navier–Stokes (RANS) simulations. Bladeless power extraction provides significant benefits for flows exposed to high speeds. Nikola Tesla introduced this concept for radial bladeless turbines in 1913, driven only by the viscous forces of the flow impinging on the rotating disks [13]. Li et al. [14] recently carried out experimental and numerical work for a Tesla turbine exposed to an incompressible fluid. Neckel and Godinho [15] designed convergent-divergent nozzles for Tesla turbines using air as working fluid. Schmidt reported the use of radial bladeless turbines for power production from biomass combustion and achieved efficiencies from 11% to 13% [16]. For an axial turbine, a power extraction up to 0.4 MW was achieved for a bladeless turbine with a length of 0.7 m and with a hub radius of 0.345 m and an inlet flow angle of 30 deg [11]. Braun et al. [17] studied the influence of wavy axial bladeless turbines for unsteady moving shocks for one single operational point and found a significant effect of the pressure forces acting on the hub.

Unlike bladed turbines, the current bladeless turbine concept can handle a wide range of supersonic operating points, without experiencing unstarting, with simpler end wall cooling, and easier maintenance costs. Bladeless turbines may be additionally used as topping or bottoming cycle, to drive the auxiliary power system, at the expense of limited pressure losses documented in this paper.

In this work, an axial bladeless turbine able to deliver power from axial supersonic flows without swirl is investigated in detail. First, the design and operation is discussed followed by an experimental validation. Finally, the operational envelope of the bladeless turbine is discussed, as well as scaling procedures to allow for the practical implementation of this concept in supersonic applications.

## 2 The Wavy Bladeless Turbine

### 2.1 Definition of the Turbine Geometry.

Figure 1 illustrates the principle of operation of the bladeless turbine investigated in this work. The radius of the hub is 0.094 m and height of the channel is 0.02 m. The shroud is assumed to be fixed and not rotating. Figure 1(a) depicts the bladeless turbine with a flat design with a working operation as detailed in Ref. [11] in which a swirl angle is crucial for power generation from the viscous forces acting on the hub. Figure 1(b) shows a bladeless turbine with a wavy surface. This wavy surface generates additional compression waves, shocks, and expansion fans that can lead to increased power extraction.

Figure 2 depicts the parametrization of the unrolled hub endwall contoured with a wavy surface. This wavy surface consists of multiple periodic wavy contours that each have a rise that results in a “hill” and subsequent descent that ends in a “valley.” Figure 2(a) sketches an unrolled view of the bladeless turbine. The inlet of the hub is flat which after a distance *d* evolves into a wavy surface pattern with an amplitude *A* and *W* denotes the number of waves. The waves have an inclination of *α*_{helix} relative to the axis of the turbine, the flow has a certain flow angle (denoted as *α*_{flow angle}). The sense of the angles is defined by the rotation of the hub. Figure 2(b) displays the parametrization to smoothly transition from a flat entrance to a wavy surface. The parametrization consists of a Bezier curve where the first two and last two control points are fixed in height and space and the middle point has two degrees-of-freedom (horizontal and vertical direction).

### 2.2 Numerical Methodology.

This geometry was subsequently meshed through Hexpress (Numeca). A parasolid file was generated from a matlab script and loaded into igg (Numeca). Both at the hub and casing endwalls, the first layer thickness (*y*^{+}) was below 1 to ensure adequate solving of the viscous sublayer. Figure 3(a) highlights the entire mesh, which was around 8 × 10^{6} cells for this geometry. Figure 3(b) shows a close up on the wavy surface in which a refinement was added to ensure detailed capturing of the subsequent compression and acceleration of the supersonic flow over the hub.

The numerical domain is illustrated in Fig. 4. The shroud of the bladeless turbine was modeled as a no-slip wall, the hub as no-slip wall in the relative frame of reference (set by the rotational speed of the shaft). The outlet was a mixed supersonic–subsonic boundary condition with a certain back pressure. At the inlet a static pressure, temperature and velocity components were imposed to reach supersonic conditions and in this study the inlet Mach number varied from 1.25 to Mach 2.4. Steady Reynolds Averaged Navier–Stokes equations were solved with the cfd++ solver of Metacomp [18], the turbulence closure was provided by the *k*–*ω* shear stress transport (SST) turbulence model. This turbulence model was selected based on its ability to capture the turbulence effects in the near-wall region as well as in the freestream. The fluid was solved in the relative frame of reference of the hub. The momentum that was exercised on the hub was automatically saved, both the viscous and the inviscid contribution. The boundary conditions and main assumptions for the computational fluid dynamics setup are summarized in Table 1.

Boundary | Boundary conditions |
---|---|

Hub | Moving wall |

Shroud | No-slip wall adiabatic wall |

Outlet | Mixed supersonic-subsonic |

Inlet | Velocity, static pressure and temperature |

Periods 1 and 2 | Periodicity (90 deg) |

Computational time | ∼9.2 h |

# CPU’s | 16 (Purdue RICE cluster) |

Turbulence model | k–ω SST |

Gas constant | 287 |

γ | 1.4 |

Boundary | Boundary conditions |
---|---|

Hub | Moving wall |

Shroud | No-slip wall adiabatic wall |

Outlet | Mixed supersonic-subsonic |

Inlet | Velocity, static pressure and temperature |

Periods 1 and 2 | Periodicity (90 deg) |

Computational time | ∼9.2 h |

# CPU’s | 16 (Purdue RICE cluster) |

Turbulence model | k–ω SST |

Gas constant | 287 |

γ | 1.4 |

To simplify the analysis and interpret the trends, a calorically perfect gas with specific heat ratio of 1.4 and gas constant of 287 was used. The influence of the temperature on the specific heat ratio was verified with a thermally perfect gas (air) and a deviation of 7% in torque and 4% of difference in total pressure loss was retrieved. Periodicity of 90 deg was assumed for the simulations. However, a half wheel bladeless turbine was run to justify this choice and the difference in torque was below 0.3%.

The designed inlet flow speed can reach Mach numbers as low as 1.25, which would unstart the turbine and induce a traveling pressure wave upstream toward the combustor if the channel were straight (constant inlet-to-outlet area). Hence, the turbine passage was opened by 50% as seen in Fig. 4.

Figure 5(a) plots the convergence of torque for a typical simulation. After 1000 iterations, the change in viscous and inviscid torque on the hub is less than 0.01%, hence 2000 iterations were selected as convergence criterion. Figure 5(b) shows the mesh convergence: four meshes were investigated (ranging from 2.9 until 11.9 × 10^{6} cells), and the torque is visualized. The *y*^{+} for each mesh was kept constant while the mesh was refined by a factor 1.2 in radial, tangential, and axial direction. Figure 5(b) depicts an asymptotic behavior for the mesh convergence and hence a mesh of 8 × 10^{6} was selected for this study. The relative difference between the finest (11.9 M) and the finer (8 M) mesh was 1% with respect to torque on the hub and 1% of difference in total pressure loss (inlet-to-outlet), as shown in Table 2.

### 2.3 Performance Results.

A baseline bladeless turbine that consists of a wavy hub with a helix angle of 45 deg, an amplitude-to-channel-height of 10% (2 mm), and a total number of 20 tangential waves was selected. In a previous study for unsteady flows, [17], positive power extraction was observed in bladeless turbines where the helix angle had the same sense as rotation of the hub (Fig. 2). The geometrical features that describe the baseline case are tabulated in Table 3.

Boundary | Boundary conditions |
---|---|

R (m) | 0.094 |

L (m) | 0.15 |

d (m) | 0.02 |

A (m) | 0.002 |

h (m) | 0.02 |

α_{helix} (deg) | 45 |

α_{flow angle} (deg) | 0 |

W (-) (full wheel) | 20 |

ω (rad/s) | 1800 |

Boundary | Boundary conditions |
---|---|

R (m) | 0.094 |

L (m) | 0.15 |

d (m) | 0.02 |

A (m) | 0.002 |

h (m) | 0.02 |

α_{helix} (deg) | 45 |

α_{flow angle} (deg) | 0 |

W (-) (full wheel) | 20 |

ω (rad/s) | 1800 |

In Fig. 6, the different flow structures that contribute to the power extraction in bladeless turbines with a wavy surface become apparent. Due to the helix angle inclination and the rise and descent of the wavy contour, shock waves and expansion waves arise. These are accompanied by an increase in static pressure along the rise of the wavy contour (deceleration of the flow) and a decrease of pressure along the descent of the wavy contour (acceleration of the flow). Due to the pressure difference and inclination of the wavy contour, a tangential force on the hub is created which results in torque on the shaft (hub). In contrast to flat/smooth bladeless turbines, torque can be generated from pressure forces from axial inflow (flow with no tangential component, *α*_{flow angle} = 0 deg). The hub (Fig. 6) is colored by pressure and the inlet and section at constant tangential location is marked by the Mach number. Zone “A” lies within a rise of the wavy contour and a drastic pressure increase is observed, zones “B” and “C” show locations at the descent of the wavy contour in which the flow accelerates. Close to the hub, the flow is, however, not axial (marked by low Mach number regions) and is detached from the axial core flow. This effect lowers the power extraction.

Figure 7(a) highlights the pressure contour on the hub in function of the axial distance and R coordinate. The transition from flat inlet to the first hill of the wavy contour is accompanied by a large pressure increase (0 till 0.02 m downstream of the inlet), due to the generation of compression and shock waves. Downstream of the first “hill,” the pressure decreases due to the expansion fans beyond which a periodic interaction of increase and decrease of pressure is observed. Figure 7(b) depicts the tangential shear stress on the hub with locally negative values downstream of the valleys which marks flow that does not follow the axial core flow and hence does not participate in pressure decrease and limits power extraction (observed at 0.04 m from the inlet). The contribution at each axial location of the pressure and viscous forces is integrated along the tangential plane and is depicted in Fig. 7(c). The contribution of the pressure to the power (inviscid contribution ) at each axial location is always positive and highest during the initial rise (0 till 0.02 m downstream of the inlet). The viscous contribution (due to the shear stress) is always negative as a result of the axial inflow, which in the frame of reference of the rotating hub generates negative forces and hence a negative power contribution. Local regions of lower power/length are due to detachment in the flow.

The ratio between the viscous to the total power is 28%. The reduction and enhancement of power extraction along the axial chord is function of the shock interactions as well as local separation downstream of the wavy contours and is highly dependent on the Mach number at the inlet. Figure 7(d)) displays the cumulative power in function of the axial coordinate. Power extraction is highest at the inlet, followed by a linear decay along the axis of the machine, at an approximate rate equal to 0.4 MW/m, which could be used to perform a quick length assessment of bladeless turbines. In contrast to bladeless designs with similar dimensions and without wavy surface [11], where no power can be extracted in this situation, 63 kW of power was extracted. The total pressure loss from inlet to outlet was 7.6% and constitutes an increase of 3.3% points compared to a nozzle with the same area ratio and axial length. Total temperature ratio (*T*_{01}/*T*_{02}) was 1.0012 (Δ*T*_{0} ∼ 3 K).

## 3 Experimental Characterization

### 3.1 Design of the Test Model.

To study the flow features within the bladeless turbine, several axial cuts of the Mach number are plotted in Fig. 8. The contours highlight the interaction of the shock waves due to the rise of the wavy contour as well as local detachment from the main core flow (identified by low Mach numbers). Figure 8(b) depicts the static pressure contour of a tangential cut in a bladeless turbine exposed to a steady inlet of Mach 2.1. On the other hand, Fig. 8(c) illustrates the pressure contour of the same hub and shroud geometry solved in a two-dimensional frame with a steady inlet Mach number of 2.1.

Several similarities exist between the two-dimensional and three-dimensional visualization: a strong compression shock compresses the flow, followed by an expansion and a recompression at the onset of the second wavy surface. Similar flow features are seen for the third wavy surface. From this analysis, it is concluded that the main flow features that determine the power to the turbine (the static pressure on the hub) are preserved for the first two consecutive waves in the two-dimensional representation.

The two-dimensional simulation provides additional insights into the physical mechanism causing the power/torque generation. A zoom of the two-dimensional RANS simulation (Fig. 9) reveals several zones of interest. A first set of compression waves (1) compresses the flow (2), followed by an acceleration zone in which the pressure decreases (3), followed by a detachment shock (4) and ultimately into a separation zone (5). This mechanism is repeated for the second and third wavy contour and results in a positive pressure force (in the *z* direction). The same shock pattern is observed for a bladeless turbine exposed to a steady inlet (Fig. 9(b)). The inclination of the helix angle for the bladeless turbine provides a net force in the tangential direction of the wave and generates torque to rotate the hub. Similar flow structures were detected by Sun et al. [19] who performed direct Navier–Stokes simulations of a wavy wall structure to study the curvature and Mach number effects on the turbulent flow. Tyson and Sandham [20] investigated the wavy wall structures for several amplitude-to-wavelength ratios. Gorle et al. [21] performed a RANS/direct Navier–Stokes comparison and discussed the uncertainty quantification for the RANS simulations.

To experimentally test the shock structures that drive the power extraction, a suitable test section was built able to fit into a linear wind tunnel at Purdue [22]. The test section design is based on several RANS simulations. Figure 10(a) represents a two-dimensional RANS simulation of the convergent divergent section and wavy surface in which the compression shock (at the inception of the wavy contour) and separation zones (in the valleys of the wavy contour) are observed. To design the convergent and divergent section, Bezier curves are used which guarantee curvature continuity. Figure 10(b) depicts the control points of the Bezier curve of the convergent part (in red) and the control points of the divergent section (in green). The first derivative of the divergent curve was ensured to be continuous to guarantee a smooth flow. At the inlet of the wavy hub surface, the Mach number was around 2. The inlet height was scaled to represent the bladeless turbine at representative conditions.

One of the criteria for the design was modularity: the rendering of the test section (Fig. 10(c)) reveals that the test article consists of four pieces that can be mounted and dismounted via bolts. The first piece (1) is the convergent part, the second piece is the convergent-divergent nozzle (2), the third piece is the wavy hub article, (3) and the fourth piece is the baseplate (4) and creates the correct area ratio between the upper wall and the test article. Figure 10(d) depicts the actual manufactured piece out of aluminum.

The final CAD rendering of the supersonic convergent-divergent-wavy wall test section is portrayed in Fig. 11(a) in the linear wind tunnel and the test section has an axial length of 0.5 m, a height of 0.23 m and depth of 0.17 m. The four individual pieces are first assembled outside of the test section and then mounted into the linear wind tunnel. The mounted test article is seen on Fig. 11(b) within the actual wind tunnel. The two quartz windows are then mounted and allows for complete optical access for Schlieren and shadowgraph measurements [22].

The Schlieren system is of a *Z*-type configuration and the camera is a Basler AG camera (Ahrensburg, Germany) with a frame rate up to 50 frames/s at maximum resolution.

The convergent–divergent and wavy surface contained 50 static pressure taps with a hole diameter of 0.8 mm, which is small enough to guarantee measurement accuracy and large enough to prevent clogging. The holes were connected via tubes of 1.6 mm diameter to Scanivalve modules. The uncertainty on the pressure measurements was ±230 Pa. Kiel probes were used to measure the total pressure.

### 3.2 Experimental Results.

Figure 12 visualizes the total pressure during the run. When the valve opens, a peak is seen in the total pressure and this is due to the operation of the facility. To reach the right mass flow at the correct temperature, a second line—the purge line—is used and when the desired test conditions are met, the valve toward the test section is opened and afterward the purge line is closed. The red line depicts the total pressure upstream in the test section and shows a constant pressure during the supersonic regime, the blue line (downstream) however fluctuates due to unsteady shock interactions. This test duration was around 100 s. At the end of the experiment, the test section valve is again opened, and the pressure equalizes to the atmospheric pressure.

Figure 13(a) depicts a snapshot in time of a typical Schlieren image during a test. At the onset of the first hill, a first compression zone and consequent shock is formed, followed by a separation shock and a shear layer in the valley, with a similar pattern in the second wave. From the Schlieren imaging, unsteadiness is observed in the shear layer as well as the separation and downstream compression zone. This can be explained by the interaction of the shear layer with the consequent compression shock, as described by Leger et al. [23] and Deshpande and Poggie [24]. The numerical Schlieren (Fig. 13(b)) from the RANS simulations captures the different flow structures well. The shock angle at the first compression ramp matches with the experiments; however, the shape of the shear layer is slightly off as well as the angle of the second separation shock.

Figure 14 shows the repeatability of the static pressure measurements on the hub surface in function of normalized pressure. During several months, multiple tests were performed and the max-to-min discrepancy between several tests was below 2% in the convergent part and at a location within the wavy part (*z* = 0.39 m).

Figure 15 plots the static pressure distribution along the hub and different areas of interest are marked. Region A depicts the subsonic-supersonic section with the first compression ramp in which the discrepancy ($pexperiment\u2212pRANS/pexperiment$) between the RANS and the experiment remains below a maximum of 4.9% (at a location of *z* = 0.355 m). Area B is the region where separation occurs and discrepancy raises to 13%. Afterward, in zone C, the flow attaches again with discrepancy below 4%. Region D is characterized by a second separation zone in which the discrepancy is between 10% and 15%. Finally, the flow attaches again in region E and discrepancy drops below 14%. In the separated region, the pressure predicted by the RANS results is lower, which can be explained due to a slight mismatch between the point of separation and attachment in the computational fluid dynamics and the experiment.

(Unsteady) Reynolds-averaged Navier–Stokes are usually adequate tools to predict the onset of the separation bubble; however, they over-predict the extent of the separation bubble, which justifies the need for experimental validation. Spanwise variation of static pressure at the inlet of the convergent section was below 1%. At the throat, spanwise variation was maximum and had a measured value of 3% and decreased to 1.5% at the start of the wavy hub surface.

## 4 Bladeless Turbine Operational Envelope

The isentropic state was defined by the total pressure quantities upstream and downstream of the turbine in the absolute frame of reference. A similar optimal efficiency was also reached at around 30,000 RPM. This optimum point of efficiency is related to the rotation of the hub: the shock strength decreases as RPM increases and consequently entropy losses decrease throughout the channel. The maximum thermodynamic efficiency for this reduced mass flow lies around 8%. At optimal RPM, the absolute flow angle at the outlet was 0.7 deg and due to the high speeds, this change in flow angle produced a power extraction of ∼70 kW. The specific Euler work was around 3.9 kJ/kg and is around 50–100 times smaller than typical values for traditional subsonic turbines. At the optimal rotational speed of around 3000 rad/s, the relative inlet flow angle was 14 deg, and the power extraction corresponds to a temperature drop of approximately 4 K (*T*_{01}/*T*_{02} = 1.0015).

Figure 17(a) plots the influence of the number of waves on the reduced torque. A maximum reduced torque and efficiency is achieved for 28 waves. The difference of reduced torque between all of the geometries is less than 20% when the number of waves is more than doubled, owing to a small sensitivity of torque to the number of waves for a given inlet condition. In Fig. 17(b), the influence of the helix angle on the performance of the bladeless turbines is analyzed. A peak reduced torque was observed near 53 deg and a similar trend is observed when the hub speed was zero. The efficiency, however, peaks at lower helix angles for this specified rotational speed of 1800 rad/s. Peak efficiencies for higher helix angles are achieved for higher rotational speed.

When the hub has no rotation, the relative difference in torque is less than 20% for the different helix angle and hence the sensitivity of the helix angle is small for a wide range of helix angles. The ratio of the viscous torque (which has a negative effect) over the total torque exerted on the hub varied between 30% for lower helix angles and 15% for high helix angles.

Around 40 simulations were run with different inlet Mach numbers, total pressure, total temperatures, and RPMs for one single geometry. Figure 18(a) depicts the reduced torque in function of $m\u0303\u2009\xb7\u2009n\u0303$. All points with the same inlet Mach number form iso-reduced-mass lines, typical for supersonic flows in which the mass flow is only dependent on the Mach number and upstream total pressure and total temperature. Simulations were performed over a range of Mach 1.25 to Mach 2.4, the higher Mach number calculations result into lower reduced mass flows. The reduced torque is dependent on both reduced mass as well as reduced speed. The reduced torque is highest for low reduced speeds, as seen on Fig. 18(a). Zone A (for which the inlet Mach number is 1.25) and Zone B (for which the inlet Mach number is 2.1) depict bladeless turbine conditions in which static pressure increases and total temperature is maintained constant. Although all the data at Mach 1.25 appear to result in a single reduced torque, we observe a larger variation at Mach 2.1, which is associated with the variation in the Reynolds number, which drives the extent of the separation bubble. Highest reduced torque is achieved for highest Mach number and this is due to the fact that the outlet flow angle is increased for higher Mach number and is attributed to the difference in shock structures. Figure 18 depicts the mechanical efficiency in terms of $m\u0303\u2009\xb7\u2009n\u0303$, and the maximum for this geometry lies above 10%. Iso-reduced mass flow lines are also represented and higher reduced mass yields higher efficiency.

Figure 19 elaborates on this effect. The reduced torque is plotted in function of *ρ***u*_{axial, inlet} based on the inlet conditions (which can be related to the Reynolds number based on the diameter of the channel and viscosity of the fluid). For Mach 1.25, the lines at several static pressures form a line with a low slope, whereas for Mach 2.1, this slope is increased and tends to an asymptotic behavior at higher Reynolds numbers. Additionally, the pressure on the hub and at a constant tangential slice is shown in which a strong pressure wave is observed near the midaxial location.

The increased separation for high Mach number cases is evidenced in Fig. 20(a) in which the tangential shear stress is shown at the hub. An increase of negative shear stress indicates that a higher amount of flow does not follow the axial direction. This also penalizes the local power density (Fig. 20(b)) with negative contributions at 0.06 m downstream of the inlet due to the local flow detachment.

The maximum power density at 0.02 m downstream of the inlet is slightly lower than the Mach 1.25 case (Fig. 7). The effect of the Mach number is investigated in Fig. 21 where the power is depicted in function of the inlet Mach number (for a constant speed line). An increase of reduced torque is observed, which is mainly due to a higher increase of torque compared to the decrease of static pressure at the inlet. The power extraction is maximal for a Mach number of around 1.4. This optimum Mach number could be dependent on the geometry because the inclination and strength of the compression and reflections of the hub and the shroud surface affect the local flow separation that penalizes power extraction. For this reduced speed line, maximum efficiency was found in the low Mach number region.

Finally, the influence of the amplitude of the wavy surface on the performance of the bladeless turbine is investigated at a Mach number of 2.1. Three different amplitudes of the wavy contour of the bladeless turbine were performed (10% of amplitude-to-channel-height which was the baseline, 15% and 20%). Figure 22 shows the amplitude in function of power and efficiency. Due to the stronger deceleration, higher local pressures and higher work extraction can be achieved for higher amplitudes. The increase of power was 210% when the amplitude was doubled. The viscous-to-total contribution decreased from 21% (baseline case) to 14% for the highest amplitude case. Additionally total pressure loss increased from 13% (baseline case) to 21% for the largest amplitude case. The reduced torque has a parabolic trend with an *R*^{2} value of 0.99998 as well as the trend of the efficiency with an *R*^{2} value of 0.995, hence these slope coefficients can be used in engine models and eventually be used to extrapolate results for higher amplitudes of the wavy surface. Increase of amplitude-to-channel-height to 30% yielded efficiencies above 25% at optimal reduced speed (*T*_{01}/*T*_{02} ∼ 1.02).

Figure 22(b) plots the amplitude of the wave for two different hub radii against the reduced torque and it is demonstrated that the nondimensional number “reduced torque” can be used to predict the power for several radii and that the discrepancy between the points is less than 7%.

## 5 Conclusions

In this paper, the principle of power extraction in bladeless turbines with a wavy hub surface is explained. A baseline turbine was designed with a sinusoidal wavy surface that had an amplitude-to-channel-height of 10% and a helix angle of 45 deg. Around 75 kW could be extracted from axial inflow at Mach 1.25. Due to the rise and descent of the wavy contour and its inclination (helix angle), subsequent shock waves and expansion fans were generated, which triggered favorable power extraction. The viscous forces acted against the hub and the viscous-to-total torque ratio was around 28% for the baseline case at Mach 1.25. The total pressure loss was around 7% and this only constituted a 3%-point increase compared to a nozzle without wavy surface and same inlet-to-outlet area ratio.

Second, an experimental validation was performed based on an axial slice of the bladeless turbine. This two-dimensional representation relates the power extraction principle to a compression wave, followed by an expansion fan, detachment shock, and separation zone. The wavy surface was tested at relevant conditions to study the pressure and shock structures and the RANS results compared well with the experimental data.

Finally, the operational envelope of bladeless turbines for supersonic axial inflow is discussed. The ideal RPM for maximum efficiency and power was around 30,000. Reduced torque was defined as nondimensional parameter for the bladeless turbine and plotted against reduced mass flow and reduced speed as well as the efficiency. A wide range of helix angles were modeled and showed sensitivity below 20% in terms of torque. Similarly, the effect of the number of waves on the reduced torque and the efficiency was studied. The impact on the amplitude of the wavy contour was considerable, with a 210% increase in power production when the amplitude was doubled. Several hub radii were investigated and the reduced torque proved to be a good nondimensional parameter to scale the performance of the bladeless turbine for a wide range of geometries.

## Acknowledgment

The authors thank Valeria Andreoli and Lukas Inhestern for the technical discussions and Zhe Liu for revising the paper.

## Nomenclature

*c*_{p}=heat capacity (kJ/kg K)

*d*=axial hub distance to reach from “flat” to “wavy” (m)

*L*=axial length of the turbine (m)

*M*=Mach number

- $m\u02d9$ =
mass flow (kg/s)

- $m\u0303$ =
reduced mass flow

- $n\u0303$ =
reduced speed

*p*=pressure (Pa)

*P*=power (kW)

*R*=radius (m)

- RANS =
Reynolds averaged Navier–Stokes

- RPM =
rotational speed (rpm)

*s*_{tangential}=tangential shear stress (kPa)

*T*=temperature (K)

*W*=number of waves

*y*=tangential coordinate (m)

*y*^{+}=nondimensional wall distance

*z*=axial coordinate (m)

*α*_{flow}=flow angle (deg)

*α*_{helix}=helix angle (deg)

- $\tau $ =
torque (N·m)

- $\tau \u0303$ =
reduced torque

*θ*=tangential angle (rad)

*ω*=rotational speed (rad/s)

### Subscripts

### Appendix

The experimental validation of the simplified bladeless turbine passage is performed at the Purdue Experimental Turbine Aerothermal lab.

As depicted in Fig. 23(a), the Purdue Experimental Turbine Aerothermal lab has two wind tunnels: a linear wind tunnel and an annular wind tunnel. The high pressure side of the wind tunnel is connected to the high pressure tanks in Zucrow that contain air pressure at 14 bar with a volume of 56 m^{3} [22]. The exhaust of the wind tunnel is connected to a vacuum tank with a volume of 280 m^{3}. This vacuum pump is driven by a Dekker vacuum pump able to reach pressures below 30 mbar. In this work, the linear wind tunnel is employed (Fig. 23(b)). The linear test section is completely transparent for visible spectra through two lateral Quartz windows and is aimed at technical readiness level 1 and 2. Through the vacuum pump and high pressure line, a wide variety of pressure ratios can be set, allowing for testing from low subsonic to Mach 6 through a wide range of Reynolds numbers (Fig. 23(c)). The linear wind tunnel is rated of providing air mass flows up to 30 kg/s, air temperatures up to 700 K, and pressures up to 8 bar in the settling chamber. The facility is uniquely suited for extended duration tests, in some cases, even over an hour as well as short transients for heat flux measurements.