Advantages of the aerodynamic performance of micro‐Tesla turbines

Tesla turbines, featuring bladeless turbomachinery and energy transformation using the viscosity of the working medium, have wide application prospects in renewable energy utilization as micropower equipment and portable power units and show better aerodynamic performance with a smaller turbine size. In this paper, a miniaturization method of Tesla turbines, including the stator and rotor, was proposed based on its sensitivity analysis results and flow similitude law. A typical Tesla turbine with a disc outer diameter of 100 mm was miniaturized to 50, 20, and 10 mm, respectively. In addition, the miniaturized Tesla turbines, including both the simplified single‐channel turbine model and the multichannel turbine model used in practical applications, were simulated numerically. The results show that the miniaturization method that maintains the two high‐impact dimensionless parameters at their optimal values is simple, reasonable, and effective. The isentropic efficiency of the single‐channel Tesla turbine decreases slightly with its scaling down. However, for the multichannel Tesla turbine in practical applications, the isentropic efficiency increases significantly with a decrease in turbine size, due to a decrease in the impact of the outermost disc channel on the flow fields of the inner disc channels (called the casing wall effect). This is embodied by the phenomenon that part of the working medium in the inner disc channels flows into the outermost disc channel through the nozzle‐rotor chamber, and the proportion of this part of the working medium decreases significantly with a decrease in turbine size. In conclusion, the Tesla turbine with a smaller turbine size exhibits better aerodynamic performance and has great potential in the field of microturbomachinery.


| INTRODUCTION
A Tesla turbine, a type of unconventional bladeless turbomachinery, was invented and patented by the famous inventor and scientist Nikola Tesla in 1913, 1 and has a high potential for the utilization of renewable energy (e.g., biomass energy waste heat recovery, 2 geothermal power generation, 3 and solar-thermal power plant 4 ), power supply and power plant for microequipment, 5 portable power unit, 5 and the small scale organic Rankine cycle (ORC) system. 6,7 The Tesla turbine consists of a stator and rotor, similar to a conventional bladed turbine, as shown in Figure 1. The rotor comprises several parallel and concentric rotating disks that are closely spaced, and the distance between adjacent disks is on a hundred-micron scale. Several holes were located around the axis to discharge the working medium from the rotor. A volute 8 or several nozzles serve as the stator and surround the rotor. Both the stator and rotor are encapsulated in the case.
The viscous frictional force is the main source of energy loss in a bladed turbine. However, in a Tesla turbine, this represents the major force for rotating the rotor and highlights the essential differences between these two types of turbines. 9 In the Tesla turbine, the working medium expands and accelerates in the stator, reaching its maximum velocity at the stator outlet and then injecting into the disc channels formed by adjacent disks in an approximately tangential direction. It then flows through the rotor spirally and transforms the thermal energy into mechanical energy, making use of the viscous stress between the working medium and disc walls. Finally, it flows out of the rotor axially via the disc holes.
Due to its structure and working principle, the isentropic efficiency of the Tesla turbine increases with decreasing turbine size. 10 This solves the major problem of a bladed conventional turbine when it scales down and is the most attractive advantage of the Tesla turbine. In addition, the turbine is easy to manufacture and maintain, low-cost, highly adaptive of working medium, and self-cleaning owing to the centrifugal force. 11 As a result, in recent years, research has increasingly focused on the research of the Tesla turbine in an attempt to elucidate its internal flow mechanism, propose an optimization design method, and expedite its commercial application.
In the last century, researchers have primarily investigated the overall aerodynamic performance and internal flow characteristics using theoretical analysis and experimental methods. The most representative study was conducted by Rice's research team, which includes the theoretical analysis of compressible flow and incompressible flow using differential and integral methods, respectively, and includes an experimental analysis of the overall aerodynamic performance. [12][13][14] However, its detailed flow status has not been studied due to the limitations of current research methods and experimental techniques.
Since the beginning of the 21th century, the rapid development of computer technology and computational fluid dynamics has facilitated the numerical calculation of internal flow fields in Tesla turbines, providing insights into its internal flow mechanism and the variation rules of aerodynamic performance. Guha's research team established a one-dimensional theoretical analysis model to predict aerodynamic performance, 15 and also researched the effects of discrete inflows, disc thickness, and radial clearance on the fluid dynamics and performance of Tesla turbine numerically. 16 The results show that small disc thickness, blunt disc tip, and proper nozzle-rotor (N-R) radial clearance all lead to a highly efficient Tesla turbine. An experimental study on the aerodynamic performance of a Tesla turbine was also conducted by Guha's research team, and a new simple method, the angular acceleration method, for measuring output torque and power was developed. 17 Moreover, a new nozzle utilizing a plenum chamber was adopted in this experiment to improve its flow efficiency. 18 Turbine models with uniform and nonuniform inlets were numerically calculated, and the effects of inlet conditions and flow coefficient on the aerodynamic performance of the Tesla turbine were analyzed. 19 Studies show that the flow coefficient is a significant factor affecting turbine efficiency, and inlet nonuniformity reduces turbine efficiency. Galindo et al. studied the influence of disc spacing distance and pressure on the aerodynamic performance of the Tesla turbine numerically and experimentally and found that the optimal values of disc spacing distance and pressure were 0.9 mm and 103.89 kPa, respectively. 20 The energy loss in the Tesla turbine was divided into four categories, namely stator losses, N-R peripheral viscous losses, end wall ventilation losses, and leakage losses, and their individual impacts on the overall performance were investigated experimentally. The results show that stator losses affect overall efficiency most significantly, and should be addressed appropriately. 21 In another study, a Tesla turbine was applied to a supercritical carbon dioxide power generation system as an expander, producing an expansion power of 1.25 MW and a predicted isentropic efficiency of 59.3%. 22 In addition, with the development of modern flow testing technology, the flow fields, especially the velocity distributions in narrow disc channels, were experimented using highly resolved optical PIV. In this context, a fully automated, nonintrusive technique using reflections on the rotor walls of a traversable constant wave laser was developed to overcome the problem of the limited measurement volume. 23 Based on the above references, two kinds of Tesla turbine models are usually applied in their research, the single-channel and multichannel Tesla turbine models. In practical applications, a Tesla turbine must be a structure with multidiscs due to better aerodynamic performance. The single-channel turbine model is simplified based on the assumption that the flow field in each channel is the same, which ignores the influence of disc thickness and the casing wall effect. Compared to the multichannel turbine model, the geometry of the single-channel turbine model is much simpler, but the prediction accuracy is lower due to the geometrical simplification. However, in the theoretical analysis, numerical simulation, and experimental research of flow fields, the single-channel turbine model is usually applied due to its special advantages, such as higher solvability in the theoretical analysis, less computing time and computing resources in the numerical analysis, and stronger operability in the flow field experiments.
In these studies, the researchers mainly focused on the aerodynamic performance and flow characteristics of Tesla turbines using numerical and experimental methods, providing many valuable results and conclusions. The most significant advantage of a Tesla turbine is widely considered to be its advanced aerodynamic performance with a smaller turbine size, which can solve the problem that the isentropic efficiency of a conventional bladed turbine decreases significantly when it scales down and makes a Tesla turbine high applications in microturbomachinery. However, little attention has been paid to Tesla turbine miniaturization. Based on open literature, only Guha conducted relevant research. 24 A systematic dimensional analysis and similitude study for the three-dimensional rotating flow in the narrow disc channels of a Tesla turbine was investigated using the Buckingham Pi theorem, however, the flow in the stator was not considered, and the aerodynamic performance of micro-Tesla turbines in practical applications was not researched.
In the present study, a miniaturization method for Tesla turbines, including the stator and rotor, was proposed, and a typical Tesla turbine with a disc outer diameter of 100 mm was miniaturized into three different turbine models. In addition, the simplified singlechannel Tesla turbine model and the multichannel Tesla turbine model in practical applications were simulated numerically to determine the variation rules of aerodynamic performance and flow characteristics to verify the advantages of a micro-Tesla turbine in terms of aerodynamic performance. This study provides a strong theoretical basis and technical support for the research and application of micro-Tesla turbines and provides a reference for the miniaturization research of similar fluid machinery.

| MINIATURIZATION METHOD
Based on the design method of Tesla turbines and the flow similitude theory, a miniaturization method of Tesla turbines was proposed herein, including two steps, the miniaturization of a single-channel turbine model and the miniaturization of a multichannel turbine model. Based on this miniaturization method, a Tesla turbine with multiple disks in practical applications can be scaled down to a specified size simply and reliably.
In the step of the miniaturization of a single-channel turbine model, the thermodynamic parameters of the working medium at the turbine inlet and the static pressure at the turbine outlet should be kept constant. In this step, similar flow fields and isentropic efficiency of the turbine models with different turbine sizes are expected to obtain. In the step of the miniaturization of a multichannel turbine model, besides the above constraints, a new constraint of constant mass flow rate is added. In this step, the mass flow rate remains unchanged by increasing the disc number, and thus the turbines with the same flow capacity but different turbine sizes are obtained.
Previous studies on the sensitivity analysis of isentropic efficiency and numerical research for Tesla turbines reported that the dimensionless relative tangential velocity at the rotor inlet Ŵ o,d , and Ekman number E k are the two most sensitive parameters of isentropic efficiency. 25 Each of them has an optimal value that maximizes the turbine isentropic efficiency. Therefore, in the miniaturization method of Tesla turbines, their optimal values (or at least in their optimal ranges) should be maintained. The specific implementation methods are as follows.
First, the geometrical parameters of the model turbine are calculated based on the geometrical parameters of the prototype turbine and the prescribed miniaturization factor S, including the disc outer diameter, disc inner diameter, the radius at the nozzle inlet, and width of the nozzle throat: where the subscripts m and p represent the model and prototype turbines, respectively. Clearly, the geometrical parameters of the model turbine are in proportion to the miniaturization factor. As mentioned previously, the dimensionless relative tangential velocity at the rotor inlet should be kept constant when the turbine is miniaturized. Thus, if the absolute tangential velocity at the rotor inlet v θ,o,d is known, the tangential velocity at the rotor inlet and the rotating speed can be obtained according to the following equation: In the miniaturization method, the tangential velocity at the rotor inlet is constant. Thus, the rotating speed is calculated as follows: which shows that the rotating speed is inversely proportional to the disc radius and the miniaturization factor.
Although most geometrical parameters and rotating speed have been computed, the disc spacing distance has not been determined. According to the definition of the Ekman number, E b ν ω = 0.5 / / k , the ratio of half of the disc spacing distance to the Ekman boundary layer thickness, ν ω / , the disc spacing distance of the model turbine can be calculated as follows: where the angular speed has been previously calculated. The kinematic viscosity coefficient mainly depends on the thermodynamic parameters of the working medium, and that of the model turbine has a slight variation from the prototype turbine, which has little influence on the miniaturization design. Thus, the disc spacing distance of the model turbine is inversely proportional to the square root of the angular speed. The N-R radial clearance can be obtained according to the disc spacing distance considering the installation problems caused by too little stator-rotor gap, and the calculating formula is, To be noted, Tesla turbines exhibit better aerodynamic performance with a smaller disc thickness, and from this aspect, the disc thickness should be as small as possible under the condition that the material allowable stress is satisfied. However, from the aspect of processing technology, the processing difficulty increases, and the disc flatness decreases with a decrease in disc thickness. Therefore, the disc thickness should be determined combined all the above constraints. In detail, the stress analysis of the rotor should be conducted to determine the minimum disc thickness after the other geometrical parameters are determined, including disc installation details, and then the processing technology should be considered to determine a proper disc thickness. In this research, for simplicity, the disc thickness is calculated as the formula, t t b b = / m p m p . In addition, the nozzle outlet geometrical angle remains unchanged when a Tesla turbine scales down. At this point, the geometrical parameters of the turbine have been determined. The mass flow rate is calculated based on the continuity equation, and the isentropic efficiency of the model turbine slightly decreases due to the variation in the disc spacing distance not changing proportionally with the miniaturization factor. Thus, the aerodynamic performance parameters of the model turbine have been obtained. Owing to the lack of relevant empirical values, the QI ET AL.
| 1737 miniaturization design may be repeated twice to obtain the final design. Table 1 shows the geometrical and operational parameters of the prototype turbine and the model turbines scaled down based on this miniaturization method. The isentropic efficiency of the model turbine is obtained according to reasonable experiential assumptions, and the power is calculated according to the isentropic efficiency and operating parameters of the turbine.
The miniaturization method is performed in a singlechannel turbine model. As for the miniaturization of a multichannel Tesla turbine, it is scaled down by varying the disc number on the basis of the miniaturized singlechannel turbine while ensuring that the mass flow rates of the different turbine models are unvaried. In this case, the disc number of the multichannel Tesla turbine increases from 5 to 16 and 65, when the disc outer diameter decreases from 100 to 50 and 20 mm, respectively, as shown in Table 1.

| Physical model
Two types of turbine models, single-channel and multichannel models, with different disc outer diameters, were simulated numerically to verify the aerodynamic performance advantage of micro-Tesla turbines. The working medium was compressed air. The boundary condition of the total pressure of 345 kPa and total temperature of 373 K was applied to the simulated domain inlet. The static pressure of 101 kPa was given as the condition of the domain outlet. The frozen rotor method was used to transfer data between the stationary and rotating parts.
The geometrical and operational parameters of the single-channel turbine model are listed in Table 1. The geometrical parameters of the multichannel Tesla turbine are the same as those of the single-channel turbine except for the disc number. The detailed T A B L E 1 Geometrical and aerodynamic parameters of miniaturized Tesla turbines.

Parameters Symbols Units
Prototype  Table 2.
The rotor volume denotes the volume of the cylinder with the disc side as the bottom and the rotor axial length as the height. Obviously, the rotor volume decreases, despite the rotor axial length increasing with a decrease in disc outer diameter, which indicates that the turbine size decreases when the turbine scales down.
One-to-one turbine and one-to-many turbine are two common types of multichannel turbines with different nozzle geometries, as shown in Figure 2. A previous study shows that for the two kinds of Tesla turbines, the variation rules of the aerodynamic performance with influencing factors are the same, and the one-to-one turbine performs better than the one-to-many turbine. Thus, in the present study, only the one-to-one Tesla turbine was numerically calculated. To improve computing efficiency, the simulation model was reduced to a quarter of the whole model according to rotational periodicity circumferentially and the symmetry of the geometry and operating condition axially, denoted by the pink region in Figure 2. The calculation domain model is illustrated in Figure 3.

| Numerical solver and mesh sensitivity
In this study, the flow fields were simulated numerically by the computational fluid dynamics software ANSYS CFX. The shear stress transport (SST) turbulence model was used to close the flow governing equations of the Reynolds-averaged Navier-Stokes method, which was also used by other researchers. 10,23,26,27 The overall accuracy results of the second order were obtained by discretizing the space term with a second-order central difference scheme and discretizing the time term with a second-order backward Euler scheme. The structural mesh was generated for the whole calculation domain using the software ANSYS ICEM CFD. The Y-type mesh was used in the nozzle beveled section, and O-type mesh was applied to the rotor and the N-R chamber to improve local mesh quality, as shown in Figure 4. In addition, for both the single-channel and multichannel turbine models, the first grid height of 0.002 mm on the solid walls was used to satisfy the requirement of the SST turbulence model and obtain credible results.
To obtain accurate results, mesh independence was conducted for both the single-channel and multichannel Tesla turbines. The results are shown in Tables 3 and 4. The relative variations in the tables were calculated based on the results with the maximum mesh number.
The actual shaft power divided by the ideal isentropic power is defined as the isentropic efficiency of the Tesla turbine, which is a total-static efficiency, as follows: where P and m represent the actual shaft power and mass flow rate, respectively. h Δ isen is the isentropic enthalpy drop. p nt and T nt represent the total pressure and total temperature at the turbine inlet, and p i,d stands for the static pressure at the turbine outlet.
As shown in Tables 3 and 4, the mesh in Case 3 for the single-channel Tesla turbine and the mesh in Case 2 for the multichannel Tesla turbine met the requirements of the mesh independence, thus the mesh with 1.70 million and the mesh with 3.27 million were used for numerical analysis. For other turbine models with different disc outer diameters, the nodes were changed based on geometrical variations.

| Aerodynamic performance analysis
The internal flow fields of four single-channel Tesla turbine models with different disc outer diameters in Table 1 were numerically simulated at different nondimensional tangential velocity differences at the rotor inlet (tangential velocity difference) by varying the rotating speed. Figure 5 illustrates the variation curves of the aerodynamic performance of four single-channel Tesla turbines versus the tangential velocity difference. With decreasing disc outer diameter, the isentropic efficiency, torque coefficient and specific power all decrease slightly, while the flow coefficient increases.
The flow coefficient C m , torque coefficient C T , and specific power C P are defined as follows:   (5) where u o,d and v o,d are the average tangential velocity and radial velocity at the outer edge of the disc, respectively.
The results show that the isentropic efficiency of the single-channel turbine model does not increase when it scales down. The turbine models with different disc outer diameters all have optimal values of the tangential velocity difference, which increases slightly with decreasing disc outer diameter. In addition, the design value of the tangential velocity difference of 0.31 is in the optimal range of isentropic efficiency, which indicates that the miniaturization design is rational and feasible. Table 5 compares the theoretical and numerical results of the aerodynamic performance of Tesla turbines with different disc outer diameters at their design rotating speeds. It can be seen that for each aerodynamic performance parameter, a small difference exists between the theoretical value and the CFD result, which results from the estimation of the isentropic efficiency. With a decrease in disc outer diameter, the mass flow rate and power decrease significantly, and the isentropic efficiency goes down slightly. This is due to the fact that as the turbine size decreases, the flow area decreases significantly, and therefore the mass flow rate and power decrease.
The main reason for the decrease in isentropic efficiency is due to the fact that as the Tesla turbine scales down, the ratio of the N-R radial clearance to the disc outer diameter (relative radial clearance) increases gradually, and as the disc outer diameter decreases from 100 to 50, 20, and 10 mm, the relative radial clearance increases from 0.005 to 0.007, 0.0109, and 0.0154, respectively. The air flows out of the nozzle through the N-R chamber before entering the disc spacing channel, during which the high-velocity air diffuses in the N-R chamber, the air velocity decreases, and the flow angle increases (relative to the tangential direction). As the disc outer diameter decreases, the relative radial clearance increases, and the working medium diffuses more significantly, leading to a greater energy loss in the N-R chamber and a lower isentropic efficiency. In the following section, the influence of the disc outer diameter on the performance is analyzed based on the flow status and flow mechanism. Table 6 shows the aerodynamic performance of the Tesla turbines with different disc outer diameters at optimal rotating speeds. Comparing Tables 5 and 6, it can be found that the optimal rotating speed of the miniaturized single-channel Tesla turbine is less than that of the design rotating speed (the maximum relative error is 7.7%). The dimensionless tangential velocity difference corresponding to the optimal rotating speed increases with decreasing disc outer diameter. The last column in Table 6 shows the dimensionless tangential velocity difference range corresponding to a turbine isentropic efficiency greater than 99% of the maximum efficiency. For all miniaturized turbines, the tangential velocity difference at the design speed of 0.31 is within this range, which demonstrates that this miniaturization method is reasonable and feasible. As the disc outer diameter decreases, the tangential velocity difference increases, which provides practical theoretical guidance for miniaturization design methods.

| Flow characteristics
The flow characteristics of the single-channel Tesla turbines were studied based on the detailed flow fields to analyze the variation laws of the aerodynamic performance of single-channel Tesla turbines.    Figure 6 shows the pressure ratio contours on the middle sections of the Tesla turbines with different disc outer diameters at their design rotating speeds. It can be seen that the distribution of the pressure ratio is almost unchanged. The working medium expands and accelerates in the nozzle, and the pressure decreases. Additionally, it changes significantly at the throat of the nozzle. The air continues to expand and accelerate, entering the rotor and transforming air energy into mechanical energy using air viscosity. In addition, with a decrease in disc outer diameter, the pressure at the nozzle outlet and rotor inlet increases, the pressure drop in the nozzle goes down, and the pressure drop in the rotor goes up. Figure 7 shows the Mach number contours and streamlines on the middle sections of the Tesla turbines with different disc outer diameters at design rotating speeds. Obviously, with decreasing disc outer diameter, the air velocity at the nozzle outlet increases, and the pressure drop in the nozzle decreases, as shown in Figure 6. As the disc outer diameter decreases, the rotating speed increases sharply, creating a higher flowing resistance in the rotor, which leads to a higher pressure drop in the rotor and, thus, a lower pressure drop in the nozzle. In addition, according to the miniaturization method, the ratio of the nozzle length to the nozzle hydraulic diameter decreases, resulting in a decrease in nozzle frictional loss. For this reason, the Mach number and air velocity at the nozzle outlet increase, although the pressure drop in the nozzle decreases.
In addition, the relative rotor radial clearance increases with a decrease in disc outer diameter. There is a region with a small Mach number outside the rotor, which represents the N-R chamber. This Mach number saltation is primarily caused by the influence of the rotor rotating. As the disc outer diameter decreases, the air angle flowing from the nozzle outlet into the rotor decreases, and the flow path lines become longer, creating an increase in the energy loss in the rotor. Owing to the increase in relative radial clearance, the working medium with a high velocity at the nozzle outlet diffuses more seriously in the N-R chamber, leading to a lower velocity in the outer region of the rotor.
As the disc outer diameter decreases, the Mach number at the rotor outlet increases, indicating the increasing leaving velocity loss. This suggests that more unnecessary working medium flows through a smaller rotor, resulting in a higher energy loss in the rotor. Therefore, in conclusion, the nozzle loss decreases, the rotor loss increases, and the leaving velocity loss increases with decreasing disc outer diameter. Figure 8 presents the Mach number contours on different axial sections with circumference angles of 0°and 90°of the rotor for the Tesla turbines with different disc outer diameters. The circumference angle is shown in Figure 1, and the circumference angles of 0°and 180°are where the two nozzles are installed. The contours were stretched to five times their size in the axial direction for clearer display. It can be seen that the Mach number distributions of the turbines with different disc outer diameters on each axial section are similar. Specifically, the Mach number on the axial section of 0°first decreases and then increases from the rotor inlet to the outlet, which is mainly due to the combined action of energy transformation and the pressure drop in the rotor. The turning point of the Mach number is found to | 1745 occur earlier for the turbine with a smaller disc diameter, resulting from a larger pressure drop. The Mach number on the axial section of 90°increases from the rotor inlet to the outlet, and the growth rate goes up with decreasing disc diameter. In addition, the Mach number at the rotor outlet increases as disc diameter decreases, indicative of an increase in the leaving-velocity loss. Figure 9 shows the entropy increase of each component of the Tesla turbine at the design rotating speed. The entropy increase represents the amount of energy loss, where the greater the entropy increase, the more energy loss. As the disc outer diameter decreases, the entropy increase in the nozzle (green line in Figure 9) decreases greatly, whereas the loss in the N-R chamber (blue line) increases. The sum of the two, the entropy increase in the stator decreases (gray line), and the decreasing speed turns down.
The entropy increase in the rotor (red line) increases with a decrease in disc outer diameter, which is mainly caused by the longer streamlines in the smaller rotor and more unnecessary working medium flowing through the smaller rotor (as shown in Figure 7). The total entropy increase in the turbine, including the stator and rotor (orange line), first decreases and then increases. Moreover, little variation in the entropy increase is observed in the turbine as the disc outer diameter changes. These findings indicate that with a decrease in disc outer diameter, the stator loss decreases, rotor loss increases, and the leaving velocity loss also increases. Owing to the actions of these losses, the isentropic efficiency of the single-channel Tesla turbine ultimately decreases with the disc outer diameter.

| Aerodynamic performance analysis
The results presented above indicate that the mass flow rate and power of the simplified single-channel Tesla turbine reduce significantly after miniaturization, while the isentropic efficiency decreases slightly. This section will analyze the influence of miniaturization on multichannel Tesla turbines for use in practical applications.
Based on a previous study, the optimal rotating speeds of single-channel and multichannel turbines with the same structural and operating parameters are different. For the turbines with a disc outer diameter of 100 mm, the optimal rotating speeds are 43,500 and 30,000 r·min −1 , respectively. In this section, according to the same rotating speed relationship, the flow fields of the multichannel Tesla turbines with disc outer diameters of 50 and 20 mm were numerically simulated at their optimal rotating speeds of 58,620 and 141,380 r·min −1 , respectively. Table 7 lists the overall aerodynamic performance of the miniaturized multichannel Tesla turbines. With a decrease in disc outer diameter and rotor volume, the mass flow rate of the multichannel Tesla turbine stays approximately the same, with a difference less than 1.1% (compared to the multichannel Tesla turbine with a disc outer diameter of 100 mm), while the isentropic efficiency and power increase greatly. To sum up, the aerodynamic performance of the Tesla turbine improves significantly as it scales down.

| Flow characteristics
Previous research shows that the main factors responsible for the lower isentropic efficiency of multichannel Tesla turbines compared with single-channel Tesla turbines are the disc thickness and casing wall effect. Specifically, disc thickness makes it more difficult for the working medium to flow into the disc channel from the nozzle, resulting in a greater energy loss in the N-R chamber and the outer area of the rotor. The casing wall effect causes part of the working medium in the disc channels of the rotor middle (relative to the casing side wall, called the inner disc channels) to flow into the outermost disc channel through the N-R chamber, resulting in most of the working medium not being fully used to transform energy. To analyze the difference in flow fields in Tesla turbines with different turbine sizes, the percentages of mass flow rate and torque in each disc channel are evaluated in detail. Table 8 shows the mass flow rate percentage and torque percentage in each disc channel of the multichannel Tesla turbines. The outermost disc channel is the disc channel in the rotor closest to the casing side, which contains one rotating disc side and one stationary casing side. The inner disc channels refer to all disc channels but the outermost one, which consists of two rotating disc sides. This percentage refers to the percentage of the mass flow rate or torque of each disc channel in the total mass flow rate or total torque of the turbine model. The percentages of the inner disc channels are not constant, but the variation range is small, as shown in Table 8. The data indicates that the closer to the outermost disc channel, the smaller the mass flow rate percentage, indicating that the more working medium from this disc channel flows into the outermost disc channel. Compared with the inner disc channels, much more working medium flows through the outermost disc channel, however, its torque percentage is the smallest, which is much lower than its theoretical average value. Therefore, the low flow efficiency of the outermost disc channel reduces the efficiency of the whole turbine.
As shown in Table 8, with decreasing disc outer diameter, the disc channel number increases greatly, and the mass flow rate percentage of each inner disc channel is closer to its theoretical average value. This indicates that the influence of the outermost disc channel becomes less, leading to a decrease in the corresponding energy loss. It is worth noting that the torque percentages of some inner disc channels are less than their corresponding theoretical values, and they are all close to the outermost disc channel. Although the mass flow rate percentages in these disc channels are less than the corresponding theoretical value, part of the working medium in the inner disc channels flows into them and worsens the flow status, thereby reducing the flow efficiency. Figure 10 shows the streamlines and contours of the Mach number of the miniaturized multichannel Tesla turbines and the corresponding single-channel Tesla turbine on the middle sections of the different disc channels. The figures on the left, middle, and right represent the innermost disc channel, the outermost disc channel of the multichannel Tesla turbine, and the disc channel in the single-channel Tesla turbine, respectively. The flow fields of the single-channel turbines are those at optimal rotating speeds.
Comparing the turbines with the same disc outer diameter, the Mach number at the nozzle outlet in the multichannel turbine is higher than that in the singlechannel turbine. This is attributed to the fact that the rotating speed of the multichannel turbine is lower than that of the single-channel turbine, leading to a decrease in the pressure at the nozzle outlet. In addition, a sudden decrease exists in the flow area between the N-R chamber and the rotor, which also leads to a slight decrease in the pressure at the nozzle outlet. Compared with the single-channel turbine, the air velocity of the multichannel turbine in the inner disc channels is higher because of the higher velocity at the nozzle outlet. Thus, the flow angle in the inner disc channels is smaller, and the streamlines is longer. The flow field in the outermost disc channel is significantly different from that in the single-channel turbine owing to the air flowing from the inner disc channels through the N-R chamber into the disc channel. In addition, in the single-channel T A B L E 7 Aerodynamic performance of the miniaturized multichannel Tesla turbines. turbine, when the working medium flows into the disc channel from the nozzle outlet, low-velocity zones form at the rotor inlet on both sides of the nozzle, generating vortexes. However, in the multichannel turbine, the vortex zone disappears. Although the working medium in a multichannel turbine has a low speed in this region, it flows into the outermost disc channel through the N-R chamber and does not generate a vortex in this area. In the single-channel turbine, the low-velocity working medium in this area has no other region to flow, ultimately generating a vortex in this area. A comprehensive comparison of the subdiagrams shows that, for all Tesla turbines with different disc outer diameters, the flow field in the outermost disc channel is markedly different from that in the innermost disc channel. Part of the working medium in the innermost disc channel flows out of the rotor spirally, while the other part flows out of the N-R chamber. The working medium in the outermost disc channel consists of a flow from the nozzle and a flow from the chamber. As the disc outer diameter decreases, the ratio of the working medium flowing into the N-R chamber in the innermost disc channel to all the working medium flowing out of the nozzle channel decreases, as shown in Figure 10, which is consistent with the results in Table 8. Figure 11 shows the 3D streamlines of the multichannel Tesla turbines with different disc outer diameters. Part of the working medium from the inner nozzle channels flows through the N-R chamber and finally to the outermost disc channel. As the disc outer diameter decreases, the disc number increases to maintain the mass flow rate unchanged. For the turbine with a disc outer diameter of 100 mm, most of the working medium from the inner nozzle channels flows into the outmost disc channel. For those of 50 and 20 mm, the working medium in most inner nozzle channels flows into the corresponding disc channels, and only that in several nozzle channels close to the casing wall flows into the outermost disc channel. All these flow phenomena support the detailed data in Table 8 and the corresponding analysis. To summarize, these findings indicate that the increase in the isentropic efficiency of the multichannel Tesla turbine after its miniaturization is mainly due to the increase in disc number. As the disc number increases, the flow in the inner channels of a multichannel Tesla turbine F I G U R E 11 Three-dimensional streamlines of the multichannel Tesla turbines with different disc outer diameters: (A) 100 mm, (B) 50 mm, and (C) 20 mm. is less affected by the outermost disc channel. From this, it can be inferred that the optimal rotating speed of a multichannel Tesla turbine with a larger disc number is closer to that of a single-channel turbine. Although the single-channel Tesla turbine will not be applied in practice, there is a need to research it, since it can be considered as the flow status in the inner channels of the multichannel Tesla turbine with relatively more disks.
Based on the discussions above, it can be concluded that the variation rules of the isentropic efficiency for these two Tesla turbine models with disc outer diameters are different. It decreases a little for single-channel Tesla turbines, which indicates that based on the miniaturization method proposed above, single-channel Tesla turbines can be scaled down to a specified size with approximately constant flow efficiency, and the miniaturization method is feasible. The isentropic efficiency increases significantly for multichannel turbines, which manifests that Tesla turbines, in practical applications, perform better with a smaller turbine size and have high application potential in the field of microturbomachinery.

| CONCLUSIONS
In this study, a feasible and simple miniaturization method of the whole Tesla turbine (including the stator and rotor) was proposed, and three micro-Tesla turbines with different disc outer diameters were miniaturized based on a typical Tesla turbine. The flow fields were numerically investigated to verify whether the isentropic efficiency of the Tesla turbine in practical applications increases with a decrease in turbine size. This work offers strong theoretical support for the study and application of Tesla turbines, as well as approaches for the miniaturization of similar rotating machinery. The main conclusions are as follows: (1) Based on the design method of Tesla turbines and the flow similitude theory, a miniaturization method of Tesla turbines, including the stator and rotor, was proposed. The two most important influencing parameters, the dimensionless tangential velocity difference at the rotor inlet and the Ekman number, should be maintained during the miniaturization process. A typical turbine with a disc outer diameter of 100 mm was miniaturized to 50, 20, and 10 mm. The design rotating speeds are 88,000, 220,000, and 446,000 r·min −1 , respectively. (2) The flow fields of the four single-channel turbines were numerically studied. With a decrease in turbine size, the mass flow rate and power of the turbine decrease significantly, and the isentropic efficiency decreases slightly. The optimal value of the dimensionless tangential velocity difference at the rotor inlet increases slightly with a decrease in turbine size, and its design value is within its optimal range, which verifies the rationality of the miniaturization method proposed in this paper. Thus, in the subsequent miniaturization, this parameter of the miniaturized turbine should be slightly higher than that of the prototype turbine. (3) The overall aerodynamic performance and internal flow characteristics of the micro-multichannel Tesla turbines in practical applications were numerically studied. The mass flow rate of each turbine model was identical under the design conditions. As disc outer diameter decreases from 100 to 50 and 20 mm, the isentropic efficiency of the miniaturized turbines increases significantly from 21.46% to 30.78% and 32.21%, respectively. In addition, the flow status of the inner disc channels is more similar to that of the single-channel turbine, and the influence of the outermost disc channel on the flow status in the inner disc channels is markedly reduced, leading to an increase in the isentropic efficiency of the multichannel Tesla turbine. This highlights the advantages of the isentropic efficiency of Tesla turbines, which promotes the commercial application of Tesla turbines in the field of micropower equipment.