Study on the influence of different structural parameters on the performance of Tesla valve

In this paper, the fluid flow of Tesla valve after the change of three structural parameters, such as diverting Angle, bend outlet width and stages, is analyzed, and their effects on the performance of Tesla valve single channel are studied. The results show that the Di value of the Tesla valve presents an upward trend with the increase of the three parameters, that is, the performance of the single pilot pass is getting better and better, but the improvement of its performance will be limited to a certain extent. When the diverting Angle increases, the pressure drop and energy loss caused by the reverse flow of the fluid will be larger, but if the diverting Angle is too large, the forward flow of the fluid in the Tesla valve will be affected. When the width of the bend outlet increases to a certain size, the fluid flowing into the bend pipe is close to the upper limit, and the single pilot performance of the Tesla valve will not be greatly improved. The increase of multiple Tesla valve stages has an effect on both forward and reverse flow of fluid.


Introduction
The Tesla valve is a non-moving parts valve with a fixed geometry, Nikola Tesla filed the patent in 1920.Its special structure makes it easier for the fluid to pass through the forward direction than the reverse direction, so the Tesla valve has the title of fluid secondary pipe.Because the Tesla valve is simpler than the traditional valve structure, and there are no moving parts, it is easy to mass produce, and the service life is longer.Tesla valve has potential application value in the fields of single pilot and fluid pressure reduction, so it has great research significance.
So far, many researchers have explored and studied the Tesla valve, for example, Gamboa.R. et al. calculated the GMF Tesla valve unit structure [1]; A. Y. Nobakht et al studied the influence of different types of Tesla microvalves on the single pilot connectivity [2]; Hai-yang Liu et al. studied the single pilot connectivity of Tesla valve [3]; S.zhang et al. proposed seven structural parameters that may affect the efficiency of Tesla valve [4]; Liu Zhe et al have proposed a symmetrical Tesla valve and studied the scaling law of its unidirectional flow characteristics [5];Quynh M. Nguyen et al. revealed that the connection between the duality, early turbulence, and pulsation of the Tesla valve can provide information for fluid mixing and pumping applications [6].
In addition, many researchers have also studied the practical application and feasibility of Tesla valves.S.M. Thompson proposed a plate oscillating heat pipe with a Tesla valve type check valve, which preliminarily verified the feasibility of this optimization [7]; Wen-ming Li et al. developed a new micro-channel flow boiling heat sink with Tesla valve [8]; Jin-yuan Qian et al. analyzed the pressure drop rule of the countercurrent Tesla hydrogen reduction valve at a large inlet speed range, providing a reference for further application of the Tesla valve and other related academic research on hydrogen fuel cell electric vehicles [9][10].Derakhshan S proposed a piezoelectric micropump with Tesla nozzle-diffuser microvalve and improved its performance [11].Yong-ming Yao et al. developed a valveless piezoelectric pump with reverse diversion channel to reduce backflow and improve output performance [12].Tabish Wahidi proposed that the application of Tesla valve in supercritical fluid natural circulation loop (NCL) can better stabilize all supercritical pressures and heat inputs considered in the study [13][14].
However, these studies did not carefully analyze the flow state of the fluid in the Tesla valve, the change of energy and the change amplitude of the Di value of the Tesla valve after the structural parameters were changed.In view of this, this study changes the structure of Tesla valve by adjusting the diverting Angle, the width of the exit of the bend and the series of the valve within a certain range, and then uses Fluent software to calculate and compare the changed structure model, so as to explore the influence of these parameters on the single pilot connectivity of Tesla valve after changes.

Working principle and performance of Tesla valve
As shown in Figure 1, the basic structure of the Tesla valve consists of an inlet pipe, an outlet pipe, and a distributary pipe, and the distributary pipe is composed of a straight pipe and a curved pipe.On the whole, due to the particularity of the structure of the Tesla valve, the loss of the reverse flow Tesla valve is more than the energy loss of the forward flow, and the pressure drop generated by the reverse flow is much larger than the pressure drop generated by the forward flow, so the Tesla valve has a single guide.
The single pilot connectivity effect of Tesla valve can be measured by Diodicity, that is, Di value [15], which is the ratio of pressure difference at the inlet and outlet of reverse flow to pressure drop difference at the inlet and outlet of forward flow, and its expression is Di is Diodicity; ∆ is the pressure difference between the outlet and the inlet of the Tesla valve after the reverse flow of fluid; ∆ is the pressure difference between the outlet and the inlet of the forward flow Tesla valve.The greater the Di value, the greater the pressure drop of the fluid flow back through the Tesla valve compared to the forward flow, the more difficult the reverse flow, and the stronger the single pilot connectivity of the Tesla valve.

Geometric model and construction
The Tesla valve used in this study is a miniature single T45-R Tesla valve [16].The T45-R Tesla valve is composed of three parts: inlet pipe, distributary pipe and outlet pipe.When the fluid flow direction is different, the inlet pipe and outlet pipe are exchanged, and the distributary pipe is composed of a straight pipe and a curved pipe.When the fluid enters the Tesla valve through the inlet pipe that is collinear with the straight of the diverting pipe, it is a forward flow.When the fluid enters through the inlet pipe at the other end, it is a reverse flow.In this study, the two-dimensional model of T45-R Tesla valve is adopted as the research object.The entrance cross section is a square with a side length of 100μm, and the entrance with a pipe width of 100μm is displayed in the two-dimensional model.In order to eliminate the inlet effect, the inlet pipe and outlet pipe length L1 and L2 are 1200μm, which is 12 times the equivalent length of the pipe.The radius R of the circular arc in the bend of the diverting pipe is 228μm; The length of the line segment L3 tangent to the arc is 235μm; The exit width of the bend w is 100μm; The diverting Angle θ of the diverting pipe is 45°.
Since the diverting angle, bend outlet width and series are the main structural parameters that affect the performance of Tesla valve, therefore, the diverting Angle, bend outlet width and series of T45-R Tesla valve are changed respectively to obtain different structure of Tesla valve [17].The adjusted diverting angles θ were 30°, 45°and 60°respectively.The exit width of the bend w is 80μm, 100μm and 120μm, respectively [18].In order to highlight the impact of the series of Tesla valves on performance, this study will not use single-stage Tesla valves for comparison, but adjust the series n of Tesla valves to 4, 5 and 6 stages for comparison.The model in Table 1 is obtained by combining the three parameters separately.
Table 1.The model after changing the structural parameters.
Each model was meshed using Gambit software.Due to the change of structural parameters, and the mesh density of each model is roughly the same, the minimum mesh number of single-stage Tesla valve is 50,000, and the maximum mesh number of multi-stage Tesla valve is 160,000.The mesh is composed of triangular mesh and quadrilateral mesh, so as to ensure that the arc can be smoother under limited computing power and reduce calculation errors.Finally, the Fluent software is used for calculation.

CFD numerical calculation
Calculations were performed using the SST k-ω model; The fluid is liquid water, and its density is set to 1000kg/m³ and its viscosity is set to 0.001kg/(m•s);The inlet boundary is set as the flow rate of 10m/s, the outlet boundary as the static pressure condition, and the reference pressure is 0Pa; When the residual is less than 10 -5 and the maximum velocity of the monitoring outlet is stable, the calculation result is judged to be convergence.
Firstly, the inlet and outlet are set according to the direction of the forward flow, and the ∆ can be obtained by calculation; Then switch the inlet and outlet, that is, the direction of the reverse flow, and calculate the ∆ according to this; By substituting the calculated results into equation ( 1), the single pilot pass parameter Di can be solved numerically.
By simulating the simulation and optimization experiment of Tesla valve by T-Q.Truong et al [19], the error between the simulation results and the experimental results is between 2.29% and 2.99%.The difference between the experimental results and the simulation results is relatively small, which verifies the validity of the numerical model.As shown in Figure 3, the numerical model is validated and can be used to optimize the design of the Tesla valve.

Influence of diverting Angle on performance of single-stage Tesla valve
Fix the exit width of the bend, change the diverting Angle to 30°, 45°and 60°and simulate the singlestage Tesla valve.Figure 4 shows the variation of Di value of single-stage Tesla valve with diverting Angle under the condition that exit widths of bend are fixed at 80μm, 100μm, and 120μm respectively.As can be seen from the figure, the Di value of the Tesla valve increases with the increase of the diverting Angle.When the fluid flows in the reverse direction in the Tesla valve, when the diverting Angle increases, more fluid will flow into the curved pipe, and the pressure drop and energy loss caused by the fluid in the curved pipe impacting the pipe wall will be larger.Moreover, the Angle between the fluid flowing through the curved pipe and the confluence direction of the fluid flowing through the straight pipe will also be larger after flowing out of the curved pipe outlet, resulting in greater pressure drop and energy loss.In addition, with the increase of the diverging Angle, the return area in the diverging pipe, the confluence of the diverging pipe and the outlet pipe due to the secondary flow and separation flow will also become larger, which will also affect the flow of the fluid, as shown in Figure 5. Therefore, it is more difficult for fluid to reverse flow in a Tesla valve with a larger diverting Angle, and the stronger the single pilot connectivity of the Tesla valve, the better its performance.In the two processes in which the diverting Angle increases in Figure 4, when the diverting Angle increases from 30°to 45°, the Di value of Tesla valve with the exit width of the bend 80μm, 100μm and 120μm increases by 22.32%, 15.68% and 16.81%, respectively; When the diverting Angle increases from 45°to 60°, the Di value of the Tesla valve with the exit width of the bend 80μm, 100μm, and 120μm increases by 7.03%, 7.85% and 8.95%, respectively.When the diverting Angle of the Tesla valve increases from 30°to 45°, the Di value changes greatly, and the performance is greatly improved; When the diverting Angle increases from 45°to 60°, the change of Di value is small, and the change of performance is not as big as the former.This is because the increase of diverting Angle makes the reverse flow more difficult, and its forward flow will also be affected.When the fluid flows forward in the Tesla valve, the larger the diverting Angle, the larger the Angle between the straight pipe and the outlet pipe, and the larger the pressure drop and energy loss caused by the collision between the fluid flowing into the outlet pipe from the straight pipe and the pipe wall.Therefore, the larger the Angle, the more difficult the forward flow will be.
To sum up, the performance of the Tesla valve is not the greater the diverting Angle of the diverting pipe, but also the resistance of the forward flow should be considered.The Di value of the Tesla valve with a diverting Angle of 30°is very low, and the single guiding effect is poor; When the diverting Angle increases from 45°to 60°, the Di value increases too slowly, indicating that the fluid flow is greatly affected by the positive resistance, and the diverting Angle should not be too large.The optimal diverting Angle of the model adopted in this study should be between 45°and 50°.In practical application, the diverting Angle should be reasonably selected to achieve the optimal single channel effect under the premise of ensuring the smooth forward flow.

Influence of exit width of the bend on Tesla valve performance
Fix the diverting Angle, the exit width of the bend was changed to 80μm, 100μm and 120μm respectively to simulate the single-stage Tesla valve.Figure 5 shows the variation of Di value with the exit width of the bend under the condition that the shunt Angle is fixed at 30°, 45°and 60°r espectively.As can be seen from the figure, the overall change of Di value is much smaller than that when the diverting Angle is changed, but it also has a certain impact on the single guiding effect of the Tesla valve.When the exit width of the bend increases from 80μm to 100μm, the Di value of the Tesla valve with the diverting Angle of 30°, 45°and 60°increases by 9.71%, 3.76% and 4.55%, respectively.On the one hand, with the increase of the exit width of the bend, more fluid will flow into the curved pipe and cause energy loss, as shown in Figure 7(a) and 7(b).And the return area of the fluid in the straight pipeline due to the secondary flow and separation flow will be greatly increased,as shown in Figure 8(b), 8(d) and 8(f).On the other hand, the larger the exit width of the bend, the more obvious the diffusion of fluid molecules here, the stronger the collision between the fluid flowing out of the bend and the fluid flowing through the straight pipe, and the greater the pressure drop and energy loss.When the exit width of the bend increased from 100μm to 120μm, the Di value of the Tesla valve with the diverting Angle of 30°, 45°and 60°only increased by 0.23%, 1.21%, and 2.24%.This is because after the bend outlet width is increased from 100μm to 120μm, the increase in the amount of fluid flowing into the bend pipe is greatly reduced compared with the increase from 80μm to 100μm and there is not much energy loss, as shown in Figure 7(c).However, for a Tesla valve with a diverting Angle of 30°, there is still more fluid increase, but its Di value changes the least, because its performance is poor, even if the bend outlet width is changed, its single pilot connectivity will not be significantly improved.In summary, the influence of the exit width of the bend on the performance of the Tesla valve is limited.When the reverse flow of fluid increases with the width of the bend outlet, no more fluid can flow into the bend pipe, resulting in energy loss, the enhancement of Tesla valve single pilot connectivity will gradually enter a saturation state.The optimal size of the model in this study should be slightly more than 80μm but less than 100μm.

Influence of stage on Tesla valve performance
Fix the diverting Angle and bend outlet width, The diverting Angle is fixed to 30°, 45°and 60°r espectively, and the exit width of the bend is fixed to 100μm.Change the Tesla valve stage 4, 5, 6 stages respectively to simulate.Figure 9 shows the change of Di value with the width of the bend outlet when the diverting Angle is fixed at 30°, 45°and 60°and the bend outlet width is fixed at 100μm.In Figure 9, when the number of Tesla valve stages increases from 4 to 5, the Di value of Tesla valve with a diverting Angle of 30°, 45°and 60°increases by 9.46%, 7.63% and 6.04% respectively; When the stage is increased from 5 to 6, the Di value of the Tesla valve with the diverting Angle of 30°, 45°and 60°increases by 6.34%, 5.32% and 4.31%, respectively.
It can be seen that the increase of the series of Tesla valve can increase the Di value of Tesla valve to a certain extent, that is, the single pilot effect is better.But as the series increases, the effect on the forward flow of the fluid will be too large, correspondingly, the increase of Di value of Tesla valve also decreases, and the Tesla valve stages suitable for the fluid system should be selected according to the specific flow situation and needs of the fluid system in practical application.

Influence of diverting Angle on performance of multistage Tesla valve
In order to explore the effect of Angle change on the performance of multistage Tesla valve, the above conditions are changed to: the stages are fixed as 4, 5 and 6 levels respectively, and the exit width of the bend is fixed as 100μm.Change the diverting Angle Change the shunt Angle of 30°, 45°and 60°r espectively and analyze the obtained data, as shown in Figure 10.In Figure 10, in the two processes in which the diverting Angle increases, when the diverting Angle increases from 30°to 45°, the Di value of the Tesla valve of stage 4, stage 5 and stage 6 increases by 28.55%, 26.40% and 25.19%, respectively; When the diverting Angle increases from 45°to 60°, the Di value of the Tesla valve of stage 4, stage 5 and stage 6 increases by 6.39%, 4.83% and 3.82%, respectively.Compare the above data with the data on the influence of diverting Angle on the performance of single-stage Tesla valves.When the diverting Angle increases from 30°to 45°, the Di value of the multistage Tesla valve increases more than that of the single-stage Tesla valve, in this process, the influence of diverting Angle on the performance of multistage Tesla valve is greater than that of single-stage Tesla valve; When the diverting Angle increases from 45°to 60°, the Di value of the multistage Tesla valve increases less than that of the single-stage Tesla valve, this indicates that the influence of diverting Angle change on the performance of multistage Tesla valve is smaller than that of single-stage Tesla valve.This is due to the fact that when the Tesla valve series was increased from 5 to 6, the Di value decreases greatly with the increase of the series.

Conclusion
In this paper, the three structural parameters of the diverting Angle, the exit width of the bend and the series of the Tesla valve are changed respectively to establish the corresponding model, and each model is calculated by CFD technology.According to the calculation results, the flow of fluid inside the valve was analyzed, and the change rule of Di value of the Tesla valve was compared.The influence of these three structures on the performance of the Tesla valve was further explored, and the following conclusions were drawn:  The performance of Tesla valve increases with the increase of diverting Angle.When the diverting Angle is too small, the flow of the reverse flow into the curved pipe is less, the pressure drop and energy loss of the fluid in the curved pipe are less, and the single flow effect is poor; When the diverting Angle is too large, the pressure drop and energy loss caused by the forward flow of fluid through the pipe joint will also increase, thus affecting the performance. The effect of diverting Angle on the performance of multistage Tesla valve is greater than that of single-stage Tesla valve.Compared with the single-stage Tesla valve, the Di value curve of the multistage Tesla valve has a larger increase in the growth stage and a faster gradual leveling stage.This is because the effect of the diverting Angle on the performance of the single-stage Tesla valve will have a superimposed effect as the number of stages increases. The single-pass effect of the Tesla valve becomes better with the increase of the width of the exit of the bend.However, after the width of the bend outlet increases beyond a certain range, there will be no more fluid flowing into the bend pipe, and the amount of fluid in the bend pipe will reach the upper limit, and the improvement of the single pilot performance of the Tesla valve will also reach a gradually stable trend. The increase of Tesla valve series in the process of reverse flow of fluid can make the single pilot effect of Tesla valve better to a certain extent.However, with the increase of the series, the forward flow of the Tesla valve is also greatly affected, and the effect of the series on the single pilot performance of the Tesla valve is becoming less and less.

Figure 1 .
Figure 1.Tesla valve basic structure.The main structure of the Tesla valve consists of a straight pipe and a curved pipe, which relies on the asymmetry of its own structure and takes advantage of the inertia of the fluid, resulting in a large difference in the pressure drop generated by the fluid flowing into and through the Tesla valve from the two entrances respectively.When the fluid flows into the Tesla valve from the forward inlet, most of the fluid flows into and through the straight pipe, and only a small part of the fluid flows into the curved pipe.Because the straight pipe has less resistance to the fluid, the pressure drop of the fluid is smaller and the total energy loss is less when most of the fluid flows through the straight pipe.When the fluid enters the Tesla valve from the reverse inlet, contrary to the forward flow, most of the fluid will flow into and through the curved pipe, while a small part of the fluid will flow into the straight pipe.Due to inertia, the fluid will collide with the wall of the curved pipe, resulting in more pressure drop and energy loss.In addition, the fluid flowing out of the curved pipe will have a secondary collision with the fluid flowing through the straight pipe.After the collision of the two parts of the fluid, part of the energy will be lost again.The comparison of the forward flow field and the reverse flow field is shown in Figure 2.

Figure 2 .
Figure 2. (a) The flow field of fluid flowing forward in the Tesla valve; (b) The flow field in which the fluid flows in reverse within the Tesla valve.

Figure 3 .
Figure 3.Comparison of experimental data with simulated data.

Figure 4 .Figure 5 .
Figure 4. Di value of single stage Tesla valve under different diverting Angle conditions.

Figure 6 .
Figure 6.Di values of Tesla valves under different exit widths of bend.

Figure 9 .
Figure 9. Di value of Tesla valve under different stage conditions.

Figure 10 .
Figure 10.Di values of multistage Tesla valves under different diverting Angle conditions.