Summary. - Computational fluid dynamics (CFD) simulations are carried out to examine the effect of geometric

parameters of Tesla valve on flow patterns, pressure drop and flow diodicity. In forward flow, the flow is relatively

horizontal) shows that optimal values of these angles exist. For a certain range of angles, diodicity is greater than 2.

The effect of multi-staging of the Tesla valve is studied, and it is found that the flow unsteadiness and overall diodicity

increase with the number of stages.

Keywords: Micro/Nano Fluidics, Tesla Valve, Computational Fluid Dynamics, Pressure drop, Diodicity.

1. Introduction. - A non-return valve (NRV) is a common device in piping systems that allows the fluid to flow in

one direction while restricting its flow in the opposite direction. The ‘one-way’ operation of the valve in these systems

may be necessary for various reasons, such as safe operation of the mechanical equipment, avoiding mixing or

contamination of fluid streams or in general to ensure desired system performance. Many of the NRVs utilize a

ball/poppet with a seat and spring arrangement. The ball moves in a particular direction due to fluid force, allowing

the fluid flow in one direction while blocking it in the opposite direction. Another type of non-return valve is the Tesla

valve, which, because of its geometrical design, offers more resistance when fluid flows in the reverse direction

compared to its forward movement. It does not require any manual or electronic operation, and unlike other NRV

like a diode in electronics, which allows the current to flow only in one direction and blocks the current flow in the

opposite direction. Like a diode, the Tesla valve offers higher flow resistance due to its complex geometry, creating

vortex-type flow structures as it flows in one direction and offers much less resistance when its flow direction is

reversed. This feature of the Tesla valve is defined in terms of its diodicity, which is the ratio of the pressure drop in

the reverse direction to the pressure drop in the forward direction. The pressure drop in forward/reverse direction or

diodicity depends on the geometrical characteristics of the Tesla valve, the fluid properties and the flow velocities. The

resurfacing of the Tesla valve technology in modern-day micro and nano fluidics has been discussed in detail in a

historical review paper by Purwidyantri and Prabowo (2023). A comprehensive literature review shows that several

studies exist that investigate the various parameters of the Tesla valve. Nobakht et al. (2013) carried out CFD

simulations and studied flow behavior in various sections of the Tesla valve. The work showed that significant energy

processes of heat transfer or phase changes in the Tesla valve. For example, Han et al. (2024) examined the two-phase

boiling flow in several microchannel geometries. The results showed that vapor backflow was reduced in the Tesla

valve channel, which improved the flow boiling performance. Huang et al. (2024) performed flow and thermal analysis

that showed the presence of numerous eddies at the entrance of the valve, which greatly influenced the heat transfer

rate. A Tesla valve with a symmetric shape was studied by Liu and co-researchers (2022), and they proved that the

Hagen number was proportional to the square of the Reynolds number at higher velocities. Jiang et al. (2025)

investigated temperature and pressure oscillations to suggest design improvements in a Tesla valve. The study by Du

et al. (2023) predicted fluid flow and heat transfer performance in a Tesla valve using an artificial neural network

technique. The optimal designs from their work increased the thermal and hydraulic efficiencies by 27% and 78%,

respectively. Apart from these papers, several others considered Tesla valve for different applications such as

2.1 Tesla valve geometry and computational domain. - To examine flow behavior in a Tesla valve, a computational

domain was constructed as shown in Figure I. The valve consists of an inlet, an intermediate section consisting of

branched channels for two possible flow paths and an outlet. One branched region is straight, while the other is curved.

The curvature of the curved channel is defined with the help of two angles, α and β, which are the angles made by the

end and beginning sections of the curved branch with the horizontal. The height of the channel is 0.3 mm, and the

outer radius of the curved section is 0.75 mm.

2.2 Boundary conditions and assumptions. - The effect of geometry on flow patterns and pressure drop is studied by

specifying the inlet either to the left edge or to the (inclined) right end. These two cases are termed ‘forward’ and

‘reverse’ flow, respectively. Velocity was specified at the inlet, and a pressure outlet boundary condition was used at

the outlet. The other surfaces were defined as ‘wall’, where the no-slip condition was applied. The effect of body

force/gravity was neglected. The fluid was assumed to be water, which is a Newtonian and an incompressible fluid.

refined in the portions where the variation in velocity was expected to be higher, such as near the walls and in the

regions where the fluids from two branches mix. The grid size was sufficient to obtain reliable pressure drop and

diodicity values. For example, the results of diodicity were compared for valve geometries with α = 45° and 60° at Re

= 2000 for various grid sizes, as shown in Figure II. The difference in diodicity was found to be less than 1% when

cells were greater than 25000 cells, indicating that the results are independent of the grid. The convergence criterion

2.4 Governing equations and discretization. - Since most microfluidic applications involve low velocities, the

problem in most cases turns out to be that of laminar two-dimensional flow. The governing equations were the

continuity and the momentum equations for 2D unsteady flow that were solved using the Ansys Fluent code.

Where dh is hydraulic diameter, which is equal to twice of channel height h and μ is absolute viscosity.

At higher Reynolds numbers, particularly in the case of ‘reverse’ flow, the solution did not converge under steady

conditions, indicating that the flow was not perfectly laminar. The solution in such a situation was obtained in unsteady

mode without addition of any turbulence model. The unsteady/transient simulations were carried out using a second-

order implicit scheme with a time step of 0.002 ms.

2.5 Validation. - To validate the computational findings of this study, comparisons were made with previously

published experimental and numerical results. The diodicity for one of the geometries considered in this paper (α =

45°, β = 75°) was calculated at different Reynolds numbers and compared with the experimental and numerical results

in a similar geometry by Gamboa et al. (2005). This comparison is shown in Figure III, which shows that the difference

between the diodicity values is within 30%. Since the diodicity values are low, this difference was considered

acceptable and confirms the reliability of the results presented in this paper.

straight section, and the diversion of fluid into the curved section/branch is minimal. Flow recirculation is observed at

the point where the curved section begins, while in the remaining portion of this curved loop, the fluid moves with a

low velocity. A small recirculation zone could be seen downstream at the exit of the curved section where the curved

and straight sections of the channel reunite, causing flow separation and recirculation (Figures IVa and IVc). When the

flow direction is reversed (Figures IVb and IVd), the inlet fluid stream divides into two portions, creating prominent

vortex flow in both sections of the valve. As the two flow streams remix at the outlet section of the channel, a much

larger recirculation region could again be observed. Comparison of Figures (IVa with IVc) and (IVb with IVd) shows

the effect of increasing the Reynolds number. As expected, the flow recirculation in both the curved and exit sections

increased and became stronger at a higher Reynolds number value of 2000. The effect of angle α on flow patterns is

shown in Figure V. The shape of the valve appears to change from a blunt shape to a relatively streamlined shape as

repetitive or constant behavior in property variation is nearly obtained. The plots show that the flow is steady in the

forward direction with all the angles considered. However, in the reverse direction, the flow is unsteady, except when

α = 60°. The pressure required in reverse flow is found sufficiently larger than the pressure required for forward flow.

The quantitative analysis based on time-averaged pressure drop shows that the diodicity [the ratio of pressure drop in

reverse flow (ΔpR) to the pressure drop in forward flow (ΔpF)] is maximum when α is 45°.

3.1.2 Effect of β on flow behavior in Tesla valve. - The effect of variation of β, defined as the angle of inclination of

the left portion of the curved section, is also studied when α is kept constant at 45°. As illustrated in Figure VII, for

forward flow, an increase of β results in a slightly increased flow recirculation at the entrance of the curved section. In

other regions, the velocity profiles are almost the same. In the case of reverse flow, at higher values of β (such as 60°

much significant pressure drops for β = 60° or 75°.

3.1.3 Pressure drop, diodicity and mass flow distribution. - A comparison of the time-averaged pressure drop and

diodicity values obtained for the different valve geometries considered in this study is given in Table I. The results

show that when α is 15°, in forward flow, the pressure drop is higher than the other valves. This results in the lowest

diodicity values with this valve at both the Reynolds numbers. Among the four tested valve geometries (while fixing β

= 8.5), the geometry with α equal to 45° leads to maximum diodicity. The variation of β from 30°-75° while keeping α

= 45°, indicates that there is no appreciable change in pressure drop for forward flow. However, for reverse flow, the

pressure drops are higher for large β angles. The comparison provided in Table 1 shows that valve geometry with α =

45° and β = 60° has maximum diodicity and is considered as the most suitable design among the tested valve shapes.

The influence of geometry on mass flow in the two sections of the valve is also determined. The mass flow in the

bottom (straight) and the top (curved) sections is represented by m1 and m2, respectively. The bar plot in Figure IXa

shows that the mass flow rate in the top section is less than 10% of the inlet flow in forward flow. In addition, the mass

flow rate in the top section (m2) decreases with an increase in Reynolds number, indicating that the fluid entry into the

curved section is further reduced at higher flow velocities. The ratio of mass flow also depends on the valve geometry,

sufficient mass of the fluid to move through this section. Thus, for reverse flow, m2/m1 is mostly greater than 1.

3.1.4 Transient flow features. - As observed earlier in Figures VI and VIII in terms of pressure drop, the flow in the

forward direction is steady, while the flow in the reverse direction is usually transient at Re = 2000. The flow patterns

in a Tesla valve are now discussed in further detail with the help of instantaneous velocity contours. The contours

shown in Figure X are for the valve which has maximum diodicity. The results show that in the outlet section, the high

velocity zone not only changes in magnitude but also in size and shape. This zone appears to be of extended size at one

instant (for example, at 0.2 or 0.8 ms) and is found to split into multiple zones at other instants (0.6 and 1.2 ms). The

changes that occur in the high velocity region lead to the movement of low velocity flow recirculation regions formed

near the top and bottom walls of the outlet section.

showing that the flow is steady in the top and bottom sections.

3.2 Performance of multi-stage valves. - To further increase the system’s performance, multiple valves can be

connected in series. Thus, the flow behavior in the valve, which has maximum diodicity (α = 45°, β = 60°), is examined

with two and three stages. It is found from the time-averaged velocity profiles (not shown here) that the locations of

high and low velocity regions are the same and repeat in different stages. The unsteady behavior, however, significantly

changes as the number of stages is increased. A comparison is made in Figure XII in terms of the root mean square

(RMS) values of velocity variation in time for the two and three-stage Tesla valves. The contour plots of Figure XIIa

show that for the forward flow with two stages, the flow is steady. For a three-stage valve, near the exit section of the

third stage, the velocity is found to vary considerably, showing flow unsteadiness. In reverse flow, the flow is unsteady

and curved sections of the valve. The mass flow rate is calculated at different time intervals in the straight and curved

sections of the three-stage valve, and its variation is plotted in Figure XIII. A comparison of mass flow in the straight

sections A, C and E shows that the mass flow rate at A has less variation, showcasing the flow’s quasi-steady nature,

while at location C, the variation increased, but its value remained negative, indicating that the flow is still in the reverse

direction. However, at location E, which lies in the third stage, the mass flow rate turned positive for a portion of the

time interval and remained negative for the rest of the interval. This shows that the flow direction is continuously

reversing in the straight section of the third stage. Since the total flow rate is constant, an increase in flow in the straight

section decreases flow in the curved section. Like the straight section, the flow rate is less fluctuating in the first stage

and more fluctuating in the third stage. The diodicity for two and three-stage valves is calculated and summarized in

Table II. The results show that for both Reynolds numbers, the diodicity increases with the increase in the number of

stages. The number of stages can therefore be adjusted, based on the pressure/mass flow rate requirements in practical

applications.

4. Conclusions. - The numerical work presented in this paper examines the fluid flow, pressure drop, diodicity and

mass flow distribution in Tesla valves for Reynolds numbers up to 2000. When flow is in the forward direction, the

particularly in reverse flow. The effect of variation of angle α shows that for small angles (e.g., α = 15°), the fluid takes

sharp turns, thus resulting in increased pressure drop in reverse as well as forward flow. On the other hand, when α is

large, such as 60°, the streamlined shape of the valve reduces pressure loss in both forward and reverse directions. The

diodicity values are therefore lower when α = 15° and 60°. For β = 8.5°, the valve with α = 45° results in maximum

diodicity. The variation of β indicates that an increase in this angle does not affect the pressure drop in forward flow.

In reverse flow, however, its larger values are found to increase pressure drop and hence diodicity. Multiple stages for

the Tesla valve are considered, and flow patterns are examined in various stages. It is noticed that, though the time-

averaged velocities are the same in different stages, the unsteadiness in flow is increased in the latter stages. The present

work performs fluid flow analysis of the Tesla valve and examines its performance in terms of pressure drop/diodicity.

Future work, in addition to fluid dynamics, will include the study of the Tesla valve for systems involving other

p pressure (Pa)

t time (s)

u x-component of velocity (m/s)

v y-component of velocity (m/s)

x horizontal / x-direction (m)

Karachi for this work.

Data Availability Statement: The paper includes findings from CFD simulations performed on the commercial code

Ansys Fluent. The simulation files are available from the authors. The description of the Tesla valve can be found in

the patent (U.S. Patent 1329559A). The details of the code/software are available in software Ansys Fluent user guide.

[3] Chou, C.Y., Kuo, G.C., & Chueh, C.C. (2024). Numerical analysis of thermal-hydraulic influence of geometric

flow baffles on multistage Tesla valves in printed circuit heat exchangers. Applied Thermal Engineering, 251, Article

[12] Nobakht, A.Y., Shahsavan, M., & Paykani, A. (2013). Numerical Study of Diodicity Mechanism in Different

Tesla-Type Microvalves. Journal of Applied Research and Technology. 11(6), 876–885.

https://doi.org/10.1016/S1665-6423(13)71594-3

[13] Purwidyantri, A., & Prabowo, B.A. (2023). Tesla Valve Microfluidics: The Rise of Forgotten Technology.

https://doi.org/10.3390/ chemosensors11040256)

[15] Zhao, Y., Wang, R., Gao, D., Chen, H., & Zhang, H. (2024). Numerical investigation and optimization of a multi-

stage Tesla-valve channel based photovoltaic/thermal module. Renewable Energy, 228, Article e120573.

https://doi.org/10.1016/j.renene.2024.120573

[16] Zhao, Y.J., Tong, J.B., Zhang, Y.L., Xu X.W., & Tong, L.H. (2024). Hydraulic loss experiment of straight-

through Tesla valve in forward and reverse directions. Science Progress, 107(3) Article e39285767

1. Conception and design of the study

2. Data acquisition

3. Data analysis

4. Discussion of the results

5. Writing of the manuscript

6. Approval of the last version of the manuscript

Hernández Gobertti.