Numerical analysis of flow instability with vortex cavitation in venturi tube

The swirling flow with cavitation can cause pressure pulsations in a draft tube of a hydro turbine at off-design operating conditions. Especially at the full load condition, the cavitation volume may fluctuate greatly, causing pressure pulsations with flow rate fluctuations known as cavitation surge throughout the hydraulic system. The cases that predicted the phenomenon qualitatively using the CFD analysis have been reported previously, whereas the methods to predict the cavitation volume and the fluctuation frequency in the cavitation surge condition quantitatively have not yet been established. Therefore, in this study, unsteady CFD analysis was conducted for the experimental apparatus that simulates the flow in the draft tube downstream of a runner using the stationary guide vane and the venturi tube. In CFD analysis, the influence of the turbulence models by using LES and SST and the influence of the number of meshes in LES simulation were investigated. It was founded that LES simulation captured the vortex generation finely and improved the prediction accuracy of the cavitation volume than SST simulation. On the other hand, it was founded that too fine mesh has a small effect on the prediction accuracy of the cavitation volume, although the resolution of the vortex is improved.


Introduction
Recently, the power load adjustment is important for the hydraulic power plant to cope with fluctuating loads due to the renewable energy.Thus, it is necessary to develop the hydro turbine that can operate stably over a wide operating range.But, at off-design operating conditions, the swirling flow with cavitation can cause pressure pulsation in a draft tube of a hydro turbine, and it disturbs the stable operation.Especially at the full load condition, the cavitation volume may fluctuate greatly, causing pressure pulsations with flow fluctuations known as cavitation surge throughout the hydraulic system.Although previous studies [1] ~ [4] reported the investigations about the mechanism and the evaluating the stability of this phenomenon, the detailed mechanism elucidation and the high accurate CFD analysis method have not yet been established.In particular, the cases that predicted the phenomenon qualitatively using the CFD analysis have been reported, whereas the methods to predict the cavitation volume and the fluctuation frequency in the cavitation surge condition quantitatively have not yet been established.
Therefore, in this study, unsteady CFD analysis was conducted for Waseda University experimental apparatus [4] that simulates the flow in the draft tube downstream of a runner using the stationary guide vane and the venturi tube, and the prediction accuracy was evaluated.This report presents the influence of the turbulence model by using LES model and SST model and the influence of the number of meshes in LES simulation.

Experimental apparatus
The experimental apparatus in Waseda University is shown in figure 1.It is the closed loop, and the water is circulated by the pump.Furthermore, the shape of the venturi tube is shown in figure 2. The pressure were measured on the venturi tube wall, and the two pressure measurement points are axially identical and circumferentially offset by 120 degrees.The venturi tube is made of acrylic to visualize the internal flow.The cavitation behavior in the venturi tube was recorded using a high-speed camera in the experiment, and the cavitation volume was calculated by image processing of the recorded movies.The calculation method of the cavitation volume assumes that the cavitation shape is cylindrical in the depth direction of the recorded images and the void fraction in the cavitation region is 1.0, so the method tends to overestimate the volume.The void fraction in the cavitation region is about from 0.6 to 0.8 in CFD results using LES model, so it is seemed that the experimental calculation method overestimates the actual cavitation volume by 20 % to 40 %.

Numerical settings
In CFD analysis, the numerical model including the upstream and downstream tanks and the venturi tube between the tanks was used (shown in figure 3).The pressure measurement points were identical with the experimental points.The list of numerical settings is shown in table 1.The homogeneous model was used as the cavitation model.The inlet boundary condition was constant mass flow rate, and the outlet boundary condition was constant static pressure.As the turbulence model, SST model and LES model were used to evaluate the influence of the difference of the turbulence model.The time step for each simulation was set to the respective value for stability of the simulation.In addition, the numerical condition was set to flow rate Q = 650 L/min, swirl number at the inlet of venturi tube S w = 1.1, and the cavitation coefficient σ = 2.0, and it was the condition under which the cavitation surge occurred clearly in the experiment.swirl number S w is shown in equation ( 1), and cavitation coefficient σ is shown in equation ( 2).
where   is the axial velocity,   is the circumferential velocity, and R is the radius.
where   is the static pressure on the gas phase in the downstream tank,  is the water density,  is the acceleration of gravity, h is the height between the water level in the downstream tank and the axial center of the venturi tube,   is the saturated vapor pressure of the water, and u is the mean axial velocity at the throat of the venturi tube.

Numerical mesh
The comparison was made between the results of the simulation using SST model and LES model.The numerical meshes have a slightly difference between each turbulence models.Each mesh nodes number is shown in table 2. The mesh nodes number used in LES simulation is higher than that in SST simulation, because the swirler mesh is difference between LES and SST simulation.The swirler meshes are shown in figure 4. The swirler mesh in SST simulation was made by tetrahedral mesh, while that in LES simulation was made by hexahedral mesh to stable the LES simulation.Other than the mesh, the analysis settings are identical.

Prediction results of cavitation volume
Figure 5 shows the results of the vortex visualization in the venturi tube for both SST and LES simulation.The typical snapshots (t1 ~ t4) in figure 5 are also illustrated in figure 7, correspondingly.The only rough vortices were captured in SST simulation, while fine vortices were able to be captured in LES simulation, and that a large vortex was formed by the collection of the fine vortices in LES simulation.
In addition, figure 6 shows the results of cavitation visualization by using an isosurface with the void fraction of 0.1 for each simulation.The cavitation shape was captured more complex in LES simulation than that in SST simulation.It is guessed that LES simulation was able to reproduce a more complex cavitation shape by capturing the finer vortices.
The comparison of the time variation of the cavitation volume is shown in figure 7. SST simulation underestimated the time average of the cavitation volume compared to the experimental result, while the amplitude was overestimated.In contrast, the prediction accuracy of the cavitation volume was improved in LES simulation.These results clearly indicate that capturing the vortex generation finely in LES simulation improves the prediction accuracy of the cavitation volume.Note that the cavitation volume measured in the experiment tends to overestimate the actual volume by 20 % to 40 %, so the value plotted in figure 7 is 30% smaller than the measurement result.Furthermore, the reason for the irregular fluctuation of cavitation volume in the experiment may be due to the error caused by the measurement method in the experiment.On the other hand, as shown in Table 3, the frequency of cavitation surge was predicted lower in LES simulation than that in SST simulation, and it was also underestimated to the experimental result by almost half.The main reason why the frequency does not match the experimental result is that the CFD analysis assumes a constant flow rate at the inlet boundary.Figure 8 shows the time variation of the flow rate at the inlet and outlet in LES simulation.In this simulation, only the outlet flow rate varies because the inlet flow rate is assumed to be constant.On the other hand, the inlet flow rate also fluctuated in the experiment.This difference in boundary conditions may be the reason why the frequency does not match that of the experiment.

Prediction results of pressure and velocity distribution
Figure 9 shows the pressure on the venturi tube wall.The pressure fluctuation component is out of phase with P1 and P2, which means that there is a phase difference in the circumferential direction.Thus, it indicates that this pressure fluctuation component is caused by the vortex behavior.In addition, the experimental and LES simulation results also show a frequency component higher than the frequency due to vortex behavior, but it can be not confirmed in SST simulation.This is because SST simulation does not resolve fine vortices, and the high frequency components are assumed to be random noise generated by fine vortices.The frequency of the pressure fluctuation caused by the vortex behavior is shown in table 4. The frequency in SST simulation was higher than the experimental results, while good guess was obtained in LES simulation.
Figure 10 shows the velocity distributions and the void fraction distributions of both SST and LES simulation at the venturi tube throat shown in figure 11.The cavitation generation region differs between the SST and LES simulation, which also results in different flow velocity distributions.In addition, swirl number in SST simulation calculated by this velocity distribution is higher than that in LES simulation by about 4 %.Therefore, the frequency of SST simulation due to the vortex behavior is higher than that of LES simulation.

Estimation of minimum vortex scale
The minimum vortex scale in the boundary layer was estimated with reference to the method shown by Kato [5] to evaluate the influence of the mesh number in LES simulation.In this study, the minimum vortex scale refers the scale of the predominant influence on the generation of turbulence.The wall shear velocity   is shown in equation (3).Generally, the wall shear stress coefficient   does not change much as Reynolds number changes, and wall shear velocity   is about 4 % of the mainstream velocity  ∞ as a first approximation.Furthermore, the diameter of the vortex  is about 30 at the dimensionless distance from the wall  + . + is shown in equation ( 4).Thus, the diameter of the vortex  is shown in equation ( 5) using the viscous scale    ⁄ .Additionally, the circumferential spacing of vortices  and the streamwise length  are estimated from the diameter and are shown in equation (5).Based on the above, the calculated vortex scales are shown in table 5.
where   is the wall shear stress,   is the wall shear stress coefficient, and  ∞ is the mainstream velocity.
where  is the distance from the wall, and  is kinematic viscosity coefficient.

Comparison of computational mesh
The numerical model for this investigation is shown in figure 3, and the mesh of the venturi tube and the swirler were changed mainly to evaluate the influence of the mesh difference.Two types of mesh shown in figure 12 and table 6 were created by using the vortex scale shown in table 5 as a reference.Mesh A is identical to the mesh used in LES simulation in the previous chapter, and the radial size of Mesh A in the mainstream is about four times larger than diameter of the minimum vortex scale.On the other hand, Mesh B has the same radial size near the wall as Mesh A and is the finer mesh in the mainstream than Mesh A.

Numerical results
The results of the vortex visualization in the venturi tube are shown in figure 13.It was confirmed that Mesh B has a much finer vortex resolution than Mesh A. On the other hand, the results of cavitation visualization shown in figure 14 indicate that there is no significant difference between the cavitation shapes of Mesh A and Mesh B. Note that the timing (t1) of snapshots in figure 13 and figure 14 are also illustrated in figure 15, correspondingly.
In addition, the time variation of the cavitation volume is shown in figure 15, and the frequency of the cavitation surge is shown in table 7.Although the frequency differs slightly between Mesh A and Mesh B, the effect on the accuracy of the cavitation volume prediction was found to be negligible.Therefore, it is founded that the finer mesh than Mesh A, whose radial size in the mainstream is about four times the diameter of the minimum vortex scale, has a small effect on the prediction accuracy of the cavitation volume, although the resolution of the vortex is improved.

Conclusion
In this study, the influence of the difference between turbulence models and the influence of the mesh number in LES simulation were investigated.It was founded that LES simulation captured the vortex generation finely and improved the prediction accuracy of the cavitation volume than SST simulation.On the other hand, it was founded that the finer mesh than the mesh whose radial size in the mainstream is about four times the minimum vortex scale, has a small effect on the prediction accuracy of the cavitation volume, although the resolution of the vortex is improved.

Figure 2 .
Figure 2. Venturi tube and pressure measurement point.

Figure 3 .
Figure 3. Numerical region and pressure measurement point.

Figure 9 .
Figure 9.Comparison of pressure on venturi tube wall.

Table 4 .
Frequency of pressure fluctuation caused by vortex behavior.

Figure 10 .
Figure 10.Comparison of velocity and void fraction distribution.

Figure 11 .
Figure 11.Measurement position of velocity distribution.

Figure 12 .
Figure 12.Comparison of mesh of venturi tube.

Table 3 .
Frequency of cavitation surge.

Table 5 .
Estimation of vortex scale.

Table 6 .
Mesh sizes of venturi tube.

Table 7 .
Frequency of cavitation surge.