On the Flow in the Gap between Corotating Disks of Tesla Turbine with Different Supply Configurations: A Numerical Study

Abstract

Momentum diffusion and kinetic energy transfer in turbomachinery have always been significant issues, with a considerable impact on the performance of the bladeless Tesla turbine. This radial turbine shows high potential for various energy applications, such as Organic Rankine Cycle or combined heat and power systems. Analyzing the flow inside the gap between the corotating disks of the Tesla turbine presents challenges due to several factors, including submillimeter length scales, variations in flow cross-section, interactions of body forces arising from rotation with turbulence, interactions between the turbine’s inlet nozzles and rotor, and moving walls. General design parameters, e.g., number of nozzles, also pose a challenge in order to achieve the full potential of this turbine. In this research, two different variants of the supply system are considered with six and forty nozzles. To minimize computational expenses, a portion of the entire domain is considered. The flow in each domain, consisting of one inlet nozzle and a segment of one gap between the disks, is examined to reveal the complexity of flow structures and their impact on the Tesla turbine performance. Large Eddy Simulation (LES) with the Smagorinsky subgrid-scale model is used to verify the results of the k-ω Shear-Stress Transport (SST) turbulence model in the first case study with six nozzles. Analyzing the results indicates that the k-ω SST model provides valuable insights with appropriate accuracy. The second case study, with forty nozzles, is simulated using the k-ω SST turbulence model. The research compares flow structure, flow parameters, and their impact on the system’s performance. From the comparison between the k-ω SST turbulence model and LES simulation, it was observed that although the k-ω SST model slightly overestimates the general parameters and damps fluctuations, it still provides valuable insights for assessing flow structures. Additionally, the mesh strategy is described, as the LES requirements make this simulation computationally expensive and time-consuming. The overall benefits of this method are discussed.

1. Introduction

In recent years, there has been a rising fascination with mini expanders, which are increasingly utilized in diverse industries, such as combined heat and power, or systems based on the Organic Rankine Cycle (ORC) [1,2,3]. These devices are characterized by their notable attributes, including reliable operation under various conditions and cost-effective production [4]. Among these mini expanders is the Tesla turbine, classified within the realm of friction turbomachinery [5]. The efficiency of friction turbomachines hinges on the transfer of momentum occurring between the operational flow and rotating disks through momentum diffusion [6,7]. Walls play a crucial role as the primary source of turbulence and the establishment of boundary layers. In the context of friction turbomachinery, these walls function as kinetic energy exchangers, exerting a noteworthy influence on the overall performance of such systems [8,9].

To assess the flow behavior in the near-wall region, the theoretical wall functions have been formulated [10,11,12,13]. Nevertheless, in scenarios involving small flow cross-sections, it becomes imperative to consider other influential parameters and phenomena, such as the interaction of boundary layers. This consideration may result in outcomes that deviate from the theoretical predictions [14,15,16].

Despite being invented more than a century ago, the Tesla turbine received minimal attention until the past decade. The increasing demand for harnessing low-energy sources has sparked a renewed interest in adopting expanders with characteristics akin to the Tesla turbine, known for their simplicity, affordability, and adaptable construction.

Diverging from conventional bladed turbines, the Tesla turbine is a bladeless machine for fluid flow, converting fluid enthalpy into mechanical energy through viscous forces [17,18]. This transformation occurs within the rotor, which consists of multiple disks spaced closely together. By leveraging fluid viscosity, momentum diffusion from the fluid to the disks transpires. The working medium enters the rotor at the outer edge of the disks, and the initial fluid layer adheres to the disk surface due to adhesive forces and, in some cases, the Coanda effect. Collisions between particles from faster and slower fluid layers facilitate momentum exchange, transferring momentum to the first fluid layer in contact with the disk, thereby initiating rotor rotation.

Reducing the area of the flow cross-section has a notable impact on flow characteristics [19,20,21]. In the context of submillimeter dimensions, the development and interaction of boundary layers, coupled with flow fluctuations, lead to a transient behavior that is challenging to predict. Much of the numerical research addressing this issue diverges from experimental data due to the limited accuracy of the applied models [22]. Simulating such phenomena using Computational Fluid Dynamics (CFD) poses challenges, and considering the mesh requirements of various turbulence models, the k-ω SST model emerges as the most suitable option for simulating flow in such a small domain [6,23]. The investigation of flow between corotating disks, with a uniform inlet flow at the outer diameter, resembling a simplified version of the corotating disks in the Tesla turbine, has revealed the crucial importance of understanding flow behavior in such domains in greater detail [7]. For a more realistic representation of the Tesla turbine operation, where nozzles deliver fluid at given points, challenges arise in characterizing flow features, particularly in the vicinity of the jet. In this zone, the interaction of the inlet jet with developing boundary layers from the corotating disks intensifies the transient behavior, necessitating a more meticulous analysis. Rusin et al. [24] provided an analysis and evaluation of a Tesla turbine model, highlighting the influence of turbulence models on predicting the turbine’s operational conditions. They examined various turbulence models using different time and space discretization approaches. Additionally, they determined the distribution of power units across the disks and compared the turbine’s power predictions from numerical analysis with preliminary experimental findings. They observed an overestimation across all utilized turbulence models compared to experimental data.

In one of their earlier works, Rusin et al. [25] constructed a Tesla-type turbine featuring five corotating disks and a plenum chamber equipped with 4 nozzles. They conducted an optimization based on efficiency to identify optimal geometrical and operational parameters. Throughout their investigations, they aimed to minimize radial tip clearance to prevent rotor displacement from causing disk rubbing against the casing due to rotational speed and manufacturing inaccuracies. They focused on efficiency-based numerical optimization of the Tesla turbine, optimizing parameters such as inlet nozzle height, inter-disk gap, nozzle angle, pressure, and rotational velocity.

Building upon recent literature, the current study focuses on a turbine with optimized parameters to further investigate flow characteristics within the gap between corotating disks. A high-fidelity simulation method like LES proves beneficial in providing a more accurate estimation of the transient behavior of the system [26,27,28].

The lack of experimental data in flow analysis inside the gap between corotating disks of a Tesla turbine poses a challenge in accepting the results of turbulent models and highlights the need to prove their accuracy and reliability. Moreover, the literature suggests that the turbulence models, e.g., k-ω SST, tend to dampen fluctuations, particularly in areas where the flow is turbulent but not fully developed, resulting in smoother flow behavior in the computational domain [29,30]. Moreover, the body forces arising from the rotation may contribute to the overestimation of eddy viscosity in the case of RANS models. This study focuses on considering only a fraction of the Tesla turbine rotor, namely 1/6 and 1/40 of its perimeter, employing a supply system with 6, and 40 nozzles. The first case with a lower number of nozzles was simulated to verify the results obtained from the k-ω SST model against a high-fidelity model. Another aspect investigated is the effect of the number of nozzles on flow structures. Considering the derived results from the first case, the k-ω SST model is employed to simulate the case with the higher number of nozzles. The objective is to precisely analyze the flow between the corotating disks of the turbine and to assess the reliability of k-ω SST model in this study case.

2. Governing Equations

The inquiry detailed in the paper involved conducting numerical simulations using both the k-ω SST turbulence model and LES.

The governing equations involve continuity, momentum, and energy conservation in their respective forms.

$${\displaystyle \frac{\partial \rho }{\partial t}}+{\displaystyle \frac{\partial }{\partial x_j}}\left(\rho U_j\right)=0$$

(1)

$${\displaystyle \frac{\partial \left(\rho U_j\right)}{\partial t}}+{\displaystyle \frac{\partial }{\partial x_j}}\left(\rho U_i{U}_j\right)=-{\displaystyle \frac{\partial p}{\partial x_i}}+{\displaystyle \frac{\partial {\tau }_{ij}}{\partial x_j}}+\rho f_i$$

(2)

$${\displaystyle \frac{\partial \left(\rho \left({e+\frac{1}{2}U}_i{U}_i\right)\right)}{\partial t}}+{\displaystyle \frac{\partial }{\partial x_j}}\left(\rho U_j\left({e+{\displaystyle \frac{1}{2}}U}_i{U}_i\right)\right)=-{\displaystyle \frac{\partial }{\partial x_j}}\left(p{U}_j\right)+{\displaystyle \frac{\partial }{\partial x_j}}\left({\tau }_{ij}{U}_i\right)-{\displaystyle \frac{\partial }{\partial x_j}}\left(q_j\right)+\rho f_i{U}_i$$

(3)

The $k-\omega SST$ model constituted the turbulence closure, in which the transport equations of turbulence kinetic energy k and specific dissipation rate ω are solved:

$${\displaystyle \frac{\partial \left(\rho k\right)}{\partial t}}+{\displaystyle \frac{\partial }{\partial x_j}}\left(\rho U_{j}k\right)={\displaystyle \frac{\partial }{\partial x_j}}\left[\left(\mu +{\displaystyle \frac{{\mu }_t}{{\sigma }_k}}\right){\displaystyle \frac{\partial k}{\partial x_j}}\right]+P_k-\rho k\omega$$

(4)

$${\displaystyle \frac{\partial \left(\rho \omega \right)}{\partial t}}+{\displaystyle \frac{\partial }{\partial x_j}}\left(\rho U_j\omega \right)={\displaystyle \frac{\partial }{\partial x_j}}\left[\left(\mu +{\displaystyle \frac{{\mu }_t}{{\sigma }_{\omega }}}\right){\displaystyle \frac{\partial \omega }{\partial x_j}}\right]+\alpha {\displaystyle \frac{\omega }{k}}{P}_k-\beta \rho {\omega }^2$$

(5)

In the LES, closing the Navier–Stokes equations is accomplished through the implementation of the wall-modeled large eddy simulation (WMLES) model, which calculates the subgrid-scale (SGS) eddy viscosity. The foundational algebraic WMLES formulation was initially introduced in the research of Shur et al. [31]. This model integrates a mixing length model with a modified Smagorinsky model [32] and incorporates the wall-damping function proposed by Piomelli et al. [33].

$$\nu =min\left[{\left(\kappa d_w\right)}^2, {\left(C_{Smag}\mathsf{\Delta }\right)}^2\right].S.\left\{1-exp\left[-{\left({\displaystyle \frac{y^{+}}{25}}\right)}^3\right]\right\}$$

(6)

$$\mathsf{\Delta }=min\left(max\left(C_w.d_w;C_w.h_{max},h_{wn}\right);h_{max}\right)$$

(7)

where $\kappa =0.4187$, $C_{Smag}=0.2$, and $C_w=0.15$ are constants.

3. Model Definition

3.1. Calculation Domain and Boundary Conditions

The numerical investigations were conducted using Ansys Fluent (2024 R1), a commercially available software based on the finite volume method. Two Tesla turbine concepts were the focus of the study: one with six nozzles (referred to as N6) and the other with forty nozzles (referred to as N40). Air was selected as the operating fluid, and the simulations followed ideal gas behavior. In each case, the geometry of the investigation exhibited symmetrical behavior every 60 and 9 degrees. To optimize computational efficiency, the simulation domain represented only a portion of the entire turbine. Simulating the entire apparatus would require a large number of cells, leading to a substantial increase in both simulation time and costs while not providing a significant improvement in the data quality.

In simulations, the absolute coordinate system was employed. The computational domain consisted of three separate sections. The first section is a plenum chamber and acts as the supply system. The second section is a converging nozzle with a throat size of 0.7 mm, oriented at an 8-degree angle with respect to a tangential direction. The final part represents 1/6 and 1/40 of the gap between two corotating disks (60 and 9 degrees), with outer and inner diameters of 160 mm and 80 mm, respectively. The considered interdisk gap size is 0.75 mm, and the radial tip clearance between the casing and the disks is set at 0.25 mm, which is minimized based on the constraints of real Tesla turbine design and manufacturing limitations. Periodic boundary conditions were applied on both sides of the disc sector in N6 with a central angle of 60 degrees, and in N40 with an angle of 9 degrees, as shown in Figure 1. The simulation considered the thickness of the plenum chamber to be equal to the gap size, and symmetric boundary conditions were applied on both sides of the chamber. The flow in the computational domain was simulated in an absolute frame, and the walls of corotating disks were set to rotate. To simplify the computational model, the thickness of the domain was equated to the size of the gap. This configuration establishes a one-to-one design where each gap is linked to its corresponding nozzle jet [34]. This setup facilitates the direct passage of the nozzle jet through the gap, thereby ignoring losses from the interaction between the inlet jet and the disk tips. The rotational speed, which was set at 17,500 RPM, the geometry of the turbine, and selected operating and boundary conditions were determined based on our previous study of a Tesla turbine, which resulted in maximum efficiency [16]. Additionally, the inlet conditions include a gauge total pressure of 2 bars, a total temperature of 300 K, and a turbulence intensity of 5%. Ambient pressure is applied at the outlet.

The numerical solution of the conservation equations Involves a weighted average of second-order upwind and central interpolations of variables. Time discretization is implemented using a second-order implicit transient formulation.

Figure 1 depicts a schematic of the studied cases, including different parts of the computational domains and the employed boundary condition in each case. Moreover, the utilized mesh in each case is illustrated. The meshing strategy is discussed in the next section.

3.2. Mesh for k-Omega SST Model

In this section, the mesh for N6, which was simulated using the uRANS method and k-ω SST turbulence model, is presented. The mesh independence study was conducted by varying the number of nodes in radial, circumferential, and axial directions within the range of 60–412, 27–156, and 15–43, respectively. In all cases, the hexahedral mesh was generated with the first layer thickness of 0.5 μm to ensure that $y^{+}<1$.

In the mesh independence check, the focus was on the gap between corotating disks. In the gap, the node distribution in the radial direction was denser in the rotor tip region, with the minimum and maximum node distances of 0.1 mm and 0.5 mm, respectively. In the circumferential direction, the higher resolution close to the nozzle jet was applied with minimum and maximum spacing of 0.1 mm and 0.3 mm, respectively. In the plenum chamber, the hexahedral mesh is generated with an edge size in the range of 0.05–0.2 $\mathrm{m}\mathrm{m}$ and a total node number of 600 k. Throughout all cases, a constant number of nodes in the plenum chamber was considered to ensure the same conditions at the inlet into the gap.

In each case, the total torque $T$ and power generated $N$ considering the full gap between two corotating disks were calculated by:

$$T=\int r\times \tau dA$$

(8)

$$N={\omega }_a\int r\times \tau dA$$

(9)

Torque generation, as a global parameter, is used to verify the mesh independence study. This parameter directly or indirectly accounts for many flow and thermodynamic quantities, such as velocity gradients, pressure gradients, and changes in viscosity due to temperature, all of which contribute to wall shear stresses that form the basis of torque. It is worth mentioning that torque is quite sensitive to mesh quality and refinement, making it a strong candidate for mesh assessment.

Table 1. Node distribution, total number of nodes, and torque generation for the mesh independence study.

Nodes Distribution				Torque (Nm)
Radial Direction	Circumferential Direction	Spanwise	Total
156	412	43	2,763,696	0.04157
150	400	40	2,400,000	0.04157
122	362	30	1,324,920	0.0415
65	154	24	240,240	0.04145
27	60	15	24,300	0.04101

presents the generated torque for all the studied meshes in this regard. The generated power in every mesh setup is depicted in Figure 2. As it is shown, changes in power generation for the mesh size in the gap higher than 1.3 million are lower than 0.25%. Furthermore, for mesh sizes higher than 2.4 million, the generated power is almost constant. Then, the chosen number of nodes for further steps of study is 2.4 million nodes in the gap between corotating disks, utilizing 400 nodes in the circumferential direction, 150 nodes in the radial direction, and 40 nodes along the gap with a growth ratio of 1.38. The total number of nodes in this case reached 3 million, including the plenum chamber. Since N40 was a subset of N6, the mesh generation for this case involved a distribution of 60 nodes in the circumferential direction, 150 nodes in the radial direction, and 40 nodes in the spanwise direction, with an equal number of nodes in the plenum chamber.

The selected time step of 0.001 s resulted in smooth convergence. To assess the convergence of the solution, the mass balance, torque generation, and fluctuation of static pressure and velocity magnitude at selected points with the crucial nozzle effect were measured. Additionally, after convergence, the residuals were observed to be between 10⁻⁴ and 10⁻⁵. The study examined the relationship between parameter fluctuations and the size of time steps used in simulations. Transient flow simulations were conducted with testing of time steps ranging from 10⁻³ s to 10⁻⁶ s. Starting with a time step of 10⁻⁶ s, which corresponds to the Courant number of 1 in the whole computational domain, we tested progressively larger time steps in each phase to evaluate the parameters’ sensitivity. No parameter fluctuations were detected during this stage of the study. Therefore, a time step of 0.001 s was selected for other calculations.

3.3. Mesh for LES

The required mesh for LES simulation is only generated in the gap between the corotating disks of N6. The supply chamber is simulated by employing the same mesh as the k-ω SST. For the initial phase of the LES simulation, 5 million cubic cells were employed, initialized with the converged k-ω SST solution obtained after 10,000 iterations. The success of an LES simulation requires the resolution of at least 80% of the turbulent kinetic energy. To achieve this, it is necessary to resolve eddies whose sizes are larger than approximately half the size of the integral length scale ($l_0$). The integral length scale is a local quantity that can be evaluated using $k$ and $\omega$ values obtained from the Reynolds–Averaged Navier–Stokes (uRANS) simulation:

$$l_0={\displaystyle \frac{k^{0.5}}{\left(C_{\mu }\omega \right)}}$$

(10)

where $C_{\mu }=0.09$.

Based on Kolmogorov’s energy spectrum to resolve 80% of the eddies we have:

$${\displaystyle \frac{l_0}{\mathsf{\Delta }}}\ge 4.8$$

(11)

where $\mathsf{\Delta }=\sqrt[3]{cell volume}$.

To adhere to the mesh requirements stipulated by the LES, adjustments were made to the cubic mesh. The presence of rotating walls induced the generation of small eddies in the adjacent wall area. To meet the LES criteria in at least 90% of the domain, the mesh underwent two rounds of modification, achieving a total number of cells of 50 million. The mesh modification process was conducted to ensure that the ratio of the integral length scale to cell size ($l_0$/Δ) remained higher than 4.8. This entailed two rounds of modification to the first two layers of the source mesh near the walls, accompanied by a switch from the k-ω SST model to LES. The thickness of the first mesh layer adjacent to the wall, after two modifications, was adjusted to 13 μm, corresponding to a $y^{+}$ value of approximately 1.6.

The simulation achieved convergence after 72,000 iterations, with the initial time step set at ${10}^{-8}$ and gradually increased to ${10}^{-6}$. The Courant number was checked at each step. The maximum Courant number in the case of the time step equal to ${10}^{-7}$ was equal to one. A schematic representation of the computation domain, boundary conditions, and the modified mesh is depicted in Figure 1.

4. Results and Discussion

4.1. Comparison k-ω SST and LES Simulations for N6

The findings from both k-ω SST and LES simulations indicate that the mass flow rate considering the full gap between corotating disks was measured at 0.00221 kg/s and 0.0023 kg/s, respectively. At the nozzle’s inlet and outlet, the average gauge pressures were determined to be 2 bar and 0.3 bar, respectively. Additionally, the average total temperature at the nozzle’s inlet was 300 K, and the average static temperature at the nozzle’s outlet was 238 K. This difference implies a nozzle efficiency of 96%.

The primary goal of this investigation is to thoroughly assess the flow characteristics within the gap between corotating disks. Consequently, the average parameters at the inlet of the gap are of considerable significance. The mass average radial velocity for k-ω SST and LES were −28.19 m/s, and −33 m/s, respectively. Moreover, the circumferential velocity of k-ω SST was 302.44 m/s, and this value for LES simulation was equal to 284.18 m/s. The primary component of velocity in this context is the circumferential one. In the case of LES, the high-fidelity simulation resolves the generated vorticities at the outer diameter of the disks, which introduces additional resistance to the flow in this region. At the same time, the literature suggests that the k-ω SST model tends to dampen fluctuations (due to overestimation of eddy viscosity), particularly in areas where the flow is turbulent but not fully developed [29,30]. Due to the undeveloped boundary layer at the outer diameter of the disks, this damping effect is most pronounced in this area. Moreover, the angle of attack can slightly change due to the aforementioned phenomena, which impacts the velocity components. Consequently, a higher level of vorticities presented by LES and damping of fluctuations by k-ω SST causes the LES to present lower circumferential velocity compared to the k-ω SST at the entrance to the gap. Furthermore, the average temperature of mass flow was calculated to be 250 K and 257 K for k-ω SST and LES simulation, respectively, while the area-averaged gauge pressure was recorded as 22,637 Pa and 24,775 Pa.

The crucial factor In assessing the turbine’s performance involves analyzing the torque generated on the walls. In the initial stage, this parameter is scrutinized using both the k-ω SST turbulence model and the LES simulation. The torque values obtained from the k-ω SST and LES simulations were 0.0417 Nm and 0.0400 Nm, respectively.

The system’s efficiency is determined based on the total-to-static enthalpy drop by the following formula:

$$\eta ={\displaystyle \frac{N}{\dot{m}{c}_p{T}_{in}\left(1-{\left(\frac{p_{out}}{p_{in}}\right)}^{\frac{\gamma -1}{\gamma }}\right)}}$$

(12)

Considering Equation (9), the power derived from k-ω SST and LES simulations were 76.4 W and 73.3 W, respectively. Furthermore, from Equation (12), the system efficiency in the k-ω SST and LES simulations was determined to be 43.0% and 39.21%, respectively.

The 4.2% overestimation of torque by the k-ω SST model prompts further consideration, particularly considering the distribution of wall shear stress on the rotating walls. To explore this further, it is possible to visualize the circumferential and radial wall shear stress, represented as a contour on a rotating wall and as distribution along the radial lines. These lines are presented in Figure 3, where they are labeled, as A, B, C, and D. Lines A and D are located at 15 and 45 degrees in mid-gap. Additionally, lines B and C are positioned at a 30-degree angle on the surfaces of both rotating disks.

The wall shear stress represents the energy transfer between the operating flow and the rotating disks, with only the circumferential component generating torque around the rotational axis. Figure 3 illustrates the distribution of both components of wall shear stress on the rotating disk.

As depicted in Figure 3, the maximum values of both components of wall shear stress are derived at the outer diameter of the disks. The maximum radial wall shear stress occurs particularly in the vicinity of the inlet jet. The zone near the outer diameter of the disks holds a significant influence on torque generation due to its greater distance from the rotational axis. As illustrated in Figure 3, the k-ω SST simulation anticipates a larger region with maximum circumferential wall shear stress near the outer diameter. In both simulations, it was observed that the primary area contributing to torque generation is confined to the outer 12.5% of the rotating wall surface. As the flow progresses from the outer to the inner diameter, there is a noticeable reduction in the generated wall shear stress. Upon closer examination of the provided plot, it becomes evident that the most substantial decline in generated wall shear stress occurs in this region, accounting for nearly 90% of the energy loss in the operating flow. In the inner sections, changes in this parameter are insignificant. The domain most affected by the inlet jet is shown in detail in the zoomed area of Figure 3. The k-ω SST dampened the fluctuations and presented a smoother drop on both components of wall shear stress. The k-ω SST model predicts a larger area directly influenced by the jet shown in the circumferential wall shear stress contour. In the vicinity of the jet’s outlet, the presence of an underdeveloped boundary layer results in elevated circumferential wall shear stress. In this region, the boundary layer is still in the process of development, leading to a high-velocity gradient near the wall surface.

To determine the distribution of wall shear stress on the disk’s surfaces, components of this parameter are plotted along lines B and C in Figure 4. The area with the higher fluctuations is zoomed in on part of the plot. The k-ω SST model displays significant symmetry along lines B and C, in contrast to LES. The k-ω SST model predicts higher wall shear stress in radial and circumferential directions, especially in the final 20% near the nozzle jet. Moreover, the fluctuation of wall shear stress in the LES simulation reveals a transient behavior of the flow, which is different from the k-ω SST model that shows smoother distributions.

Figure 5 represents the pressure and velocity distributions along lines A and D, defined in Figure 3, in the middle of the gap at angles of 15 and 45 degrees relative to the horizontal line. Lines A and B are selected to be just before and after the area with the maximum effect of the nozzle jet, respectively.

Upon analyzing Figure 3 and Figure 5, it is evident that there is a noticeable effect of the nozzle, characterized by a drop in static pressure and an increase in velocity when moving from line A to line B. This effect is primarily observed at the outer edge of the corotating disks. However, as one moves from the outer edge to the inner edge, both lines exhibit similar values for pressure and velocity. Additionally, Figure 5 effectively illustrates the velocity fluctuation obtained from the LES throughout the entire studied domain. This fluctuation is particularly pronounced in line D, which is more influenced by the inlet jet.

Figure 6 depicts the dimensional radial velocity profile of the flow within the gap at three specified locations: 1, 2, and 3 illustrated in Figure 3. Moving from the outer to the inner diameter of the disks, the radial velocity increases. This increase in the radial component of the velocity is evident along all studied lines, reaching its maximum in the area close to the wall at line 3.

The shape of the velocity profile in the radial direction is influenced by three factors: centripetal force, pressure force, and viscose forces. The magnitude of their effects may vary with the rotational speed. The combined impact of these forces results in a “W” shape of the velocity profile, with the minimum absolute velocity occurring in the mid-gap. Figure 6 illustrates that in the vicinity of the jet, there is a reverse radial flow at mid-gap in the radial direction. This phenomenon can be attributed to the high rotational speed of the system, affecting the force balance on the operating flow. Examining the three locations presented in Figure 6, it is noteworthy that reverse flow is observed in both k-ω SST and LES simulations at location 1. However, at location 2, only the LES simulation predicts reverse flow.

Figure 7 depicts the dimensionless relative velocity profile of the flow inside the gap obtained from parallel corotating disks at three specified locations: 1, 4, and 5 for the $y^{+}$ between 1 and 200.

The symbols are $U^{+}={\displaystyle \frac{U}{u_{\tau }}}$, $y^{+}={\displaystyle \frac{y{u}_{\tau }}{\nu }}$, and $u_{\tau }=\sqrt{{\displaystyle \frac{\tau }{\rho }}}$. In the k-ω SST simulation, the velocity profiles taken from both surfaces converge at the center. However, the disparity in the profiles in the case of LES simulation indicates unequal wall shear stress generated on the disks and highlights the transient behavior of flow characteristics. At locations 4 and 5, the dimensionless relative velocity profile is nearly identical, but there is a noticeable difference between location 1, and two others. The higher observable $U^{+}$ at dimensionless relative velocity profile at location 1 can be attributed to an undeveloped velocity profile, influenced by the interaction between the inlet flow and the evolving boundary layers on the corotating walls. As one progresses from location 1 to 5, the boundary layers become well-developed.

Examining locations 4 and 5, where boundary layers are well-developed and wall shear stress is consistent, the velocity profiles appear nearly identical, particularly in the LES simulation.

Upon scrutinizing both LES and k-ω SST simulations at all specified locations, it becomes apparent that k-ω SST predicts higher wall shear stress in both radial and circumferential directions. As a result, the velocity profiles derived from the k-ω SST model exhibit a downward shift.

Compared to the circumferential component, the limited portion of the radial component from the total velocity causes the W-shape of this component to not be significantly influential on the overall formation of the velocity profile at the studied locations. The noteworthy phenomenon observed is the interaction of developing boundary layers with the inlet jet.

The fluctuations in parameters during the transient simulation of the flow using the k-ω SST turbulence model are neglectable. This consistent behavior was insensitive to the time step size, which varied between 10⁻³ and 10⁻⁶ s. Throughout the entire domain, the smallest tested time step produced a Courant number lower than one, yet no significant parameter fluctuations were observed. In the converged solution, the parameter fluctuations nearly vanished. In contrast, in the LES simulation, the visible fluctuation was observable. Even post-convergence in the LES simulation, persistent parameter fluctuations highlight the ongoing transient behavior of the system.

Vorticity is another important parameter that is discussed to compare the performance of k-ω SST and LES simulation of N6. To evaluate this parameter, two surfaces are defined at the outer edge of the gap and mid-radii. The surface closer to the outer diameter is positioned in an area where the nozzle jet has the highest influence, resulting in maximum wall shear stress in both radial and circumferential directions, reverse flow, and vorticity. The surface at mid-radii is specified to assess vorticity in the area with a lower impact from the inlet jet. Four distinct lines on the edges of these surfaces are drawn to examine vorticity variation near the wall surface and mid-gap region more precisely. Figure 8 demonstrates the vorticity contours on two surfaces indicated at the outer edge of the disks and mid-radii. The surfaces are generated from the revolution of lines along the gap, between 20° and 30° located at the outer edge of the disks and in mid-radii, and it is presented in Figure 8. On each surface, two lines are defined on the edges at 20° and 30°. The lines are chosen in the described locations to better demonstrate the effect of the jet and the development of boundary layers on fluctuations of parameters. Line 1 is chosen as a location with the maximum influence of the inlet jet, and line 2 is 10° away from line 1 to show how the development of boundary layers will dampen the fluctuations. The two other lines are also chosen in the mid-radii location to determine the vorticity level in a region with fewer transient effects.

The nearly symmetrical vorticity pattern observed in the k-ω SST simulation underscores the impact of its time step being 10–1000 times larger compared to the LES simulation.

In a confined region near the nozzle, a notable variation in vorticities by approaching the disk’s surfaces is observable. In this specific region, the jet flow, affected by developing boundary layers and the interaction of the inlet jet with the evolving boundary layers, induces fluctuation and additional vorticities in the mid-gap. As the boundary layers progress and the distance from the nozzle increases, the vorticities near the wall escalate. This phenomenon is more pronounced in the LES simulation, which exhibits more fluctuation and represents more transient behavior.

Figure 8 highlights the effect of the inlet jet on the rising level of vorticities observed in both LES and k-ω SST simulations. As depicted in the mid-radii contour, the fluctuation of parameters, even in LES simulation, diminishes, and the maximum magnitude of vorticities, which occurs near the walls, is in a lower range in this area. Figure 9 depicts vorticities on the lines defined in Figure 8. With the advantage of a high-resolution mesh and the capability to resolve at least 80% of the eddies in the computational domain, the LES demonstrates well-resolved vorticity close to the rotating walls. Due to the appropriate time step, an asymmetrical pattern of vorticity distribution is observable in the examined lines. Considering the utilized mesh for k-ω SST and LES simulations, the k-ω SST can resolve the vorticities closer to the disk’s surfaces. As mentioned earlier, to satisfy the mesh requirements of the k-ω SST simulation, a fine mesh with the $y^{+}<1$ was utilized. The very small thickness of the first layer enables modeling vorticities close to the disk’s surface. However, in adhering to the LES mesh requirements, the mesh consists of cubes. Consequently, although the total number of nodes in the LES simulation is almost 17 times higher than that in the k-ω SST simulation, considering the dimensions of cells in the employed mesh in each model, the k-ω SST yields a higher magnitude of vorticities in the near-wall area, but in the rest of the domain, the LES simulation shows a higher value of resolved vorticities. In the vicinity of the jet, the k-ω SST model shows a higher level of vorticity in the mid-gap area and as mentioned earlier, damps the fluctuations caused by the inlet jet. LES, on the other hand, resolves lower levels of vorticity in this area, leading to higher fluctuations. Additionally, as the distance from the jet increases, LES predicts higher levels of vorticity in the mid-gap, which reduces the velocity.

From the k-ω SST simulation, it is apparent that vorticity tends to increase in the near-wall area. As we move further away from the jet, a more developed boundary layer exhibits more vorticities in this region. However, owing to the larger parameter fluctuations in the LES simulation, the obtained results do not align with this trend.

The rotating walls produce eddies in the nearby area, leading to an increase in vorticities in this region. As we approach the mid-gap, the range of vorticities decreases. Throughout the entire gap, the transient behavior of the system and parameter fluctuations are observable in the case of LES simulation, underscoring the importance of high-fidelity simulation for such a phenomenon.

To further illustrate the significance of simulating the transient behavior of the system, the Fast Fourier Transform (FFT) of the pressure fluctuation derived from LES simulation at a specific point defined on line 1 is presented. The pressure values are extracted from the corresponding position along Line 1 where Z = 0.

FFT is used to analyze the modes of fluctuations. Using the FFT, the fluctuation signal is converted from the time domain to the frequency domain. In this way, one can get an overview of the entire signal and see how the parameter is distributed across the frequency spectrum. In this respect, the FFT analysis of pressure fluctuations is presented in Figure 10.

From Figure 10, many fluctuations of parameters with microamplitude are observable in higher frequencies, but the main peak of amplitude in both studied time steps is happening in lower values of frequencies. Figure 10b depicts a wider spectrum of frequencies observed from smaller time steps. As previously discussed, the k-ω SST simulation dampens all parameter fluctuations, making this model incapable of predicting the transient behavior of the system and any associated effects.

4.2. Comparison between N6 and N40

Analyzing the results obtained from LES and k-ω SST simulation of N6 indicated that, although k-ω SST is damping all the fluctuations, it still provides a valuable insight into the flow structures and distribution of parameters. Consequently, considering the required computational expenses for LES simulation, k-ω SST model is an appropriate alternative numerical solution. In this part, the effect of the number of nozzles on the flow between the disks is discussed. N40 was simulated by the k-ω SST model. The parameters derived from the simulation of N6 and N40 are presented in

Table 2. Comparison between obtained parameters from the simulation of N6, and N40.

Parameter	N6 (LES)	N6 (k-ω SST)	N40 (k-ω SST)
The angle of fraction [deg]	60	60	9
Mass flow rate [kg/s]	0.0023	0.0022	0.0147
Mass-averaged radial velocity [m/s]	−33.0	−28.2	−25.0
Mass-averaged circumferential velocity [m/s]	284.8	302.4	273.7
Mass-averaged static temperature [K]	257	250	261
Area-averaged gauge pressure [Pa]	24,775	22,637	71,287
Torque [Nm]	0.0400	0.0416	0.1750
Specific Power [Ws/kg]	31,882.6	34,570.1	21,836.7
Efficiency	39.2	43.0	26.8

.

A comparison of N40 and N6 shows a 16.18% drop in system efficiency by increasing the number of nozzles. Although the generated power in N40 is higher than in N6, the considerably higher mass flow rate in N40 caused the drop in efficiency.

The circumferential and radial wall shear stress are visually depicted both as contours on a rotating wall and along line A. As the derived results from the k-ω SST simulation represented quite symmetrical behavior, the wall shear stress distribution on both corotating disks is the same. A single line is defined on the surface of one of the rotating disks, positioned at a 15-degree angle compared to the horizontal line, as illustrated in Figure 11. To enhance the representation of areas with higher circumferential and radial wall shear stress, the contours and legends in the figures are presented in logarithmic scale.

Figure 11 illustrates higher values of wall shear stress in both circumferential and radial directions in N40. The increased number of jets in N40 leads to a higher mass flow rate, resulting in a wider range of shear stress in both studied directions.

Upon closer examination of the zoomed area, it is evident that in N6, the lower interaction of the jets results in a wider area with the maximum effect of the jet. However, this phenomenon is primarily observed at the outer edge of the disk.

Line A is positioned just before the area affected by the inlet jet. In Figure 12, circumferential and radial wall shear stress on this line is represented for both N6 and N40. The results derived from N40 show significantly higher values, with most radii exhibiting values nearly four times larger than those observed in N6.

In N40, the effect of jet interactions causes the fluctuation of both components of wall shear stress in the outer edge of the disk, which is evident in Figure 12. In N6, as one moves from the outer to the inner edge of the disks, there is a rapid drop in both components of wall shear stress near the outer edge. However, beyond this drop, the wall shear stress remains relatively constant throughout the rest of the domain. In contrast, in N40, while there is also a drop in wall shear stress near the outer edge, it increases as one approaches the outlet.

In Figure 11, line B is defined in mid-gap, positioned at a 15-degree angle compared to the horizontal line. Figure 13 represents the velocity and pressure distribution along line B obtained from both N6 and N40. It is shown that the fluctuation of velocity near the outer edge of the disk indicates the impact of jet interaction in N40. In N6, as one moves from the outer to the inner edge of the disks, the kinetic energy of the operating flow decreases, indicating the conversion of flow energy to rotation of the disks. However, in N40, this parameter fluctuates between 240 m/s and 260 m/s, with a noticeable increase in the vicinity of the outlet area.

In both cases, moving from the outer to the inner edge of the disk, there is a decrease in pressure. However, there is a significant disparity in static pressure between N6 and N40. The nearly threefold higher pressure observed in N40, particularly in the vicinity of the outer edge of the disks, indicates a considerably higher mass flow rate in this case. A comparison of Figure 12 and Figure 13 shows that moving from the outer to the inner edge of the disks, the drop in kinetic energy of the operating flow in N40 is neglectable. The higher number of jets in N40 causes a more than sixfold higher mass flow rate.

5. Conclusions

The k-ω SST simulation overestimates the torque of 4.25% and wall shear stress with the representation of the steady-state behavior of the system, though the LES represents a more accurate estimation of all parameters with a demonstration of the transient behavior of the system, revealing the need for high fidelity simulation to observe the fluctuation of such a phenomenon.

Although the k-ω SST model dampens parameter fluctuations, it still provides a valuable solution for the flow. Considering the computational costs of LES simulations, this highlights the k-ω SST model as a fast approach with appropriate accuracy.

In the LES simulation of N6, the parameters exhibit the maximum range of fluctuation in the vicinity of the outer edge of the disks. In the rest of the domain, the transient behavior appears to be smoother.

Assessment of the results reveals that interactions between the inlet jet and developing boundary layers from parallel corotating disks establish vorticities approaching the disk’s surface. Due to the extremely transient behavior of the flow in that region, it does not follow a specific trend.

The outer edge of the corotating disks plays the most important role in power generation, as this area has the maximum distance from the rotational axis. Consequently, in an optimized design of a Tesla turbine, the level of kinetic energy of operating flow should drop in this area.

In N6 and considering both components of wall shear stress, the maximum total wall shear stress occurs in the outer 20% of the disk surface, and in the rest of the domain represents an almost constant value. Considering the high number of nozzles studied in N40, this parameter depicted an increase in the vicinity of the outlet.

Using a high number of nozzle jets results in a noticeable interaction of the nozzle jet, manifesting as fluctuations in parameters near the outer edge of the disks. Moreover, employing a high number of nozzles significantly increases the mass flow rate, resulting in nearly four times more power generation. However, this also causes the efficiency of the system to drop by almost 16%.