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Abstract

Pump-turbines were always running at partial condition with the power grid changing. Flow separations and stall phenomena were obvious in the pump-turbine. Most of the RANS turbulence models solved the shear stress by linear difference scheme and isotropic models, so they could not capture all kinds of vortexes in the pump-turbine well. At present, partially-averaged Navier-Stokes (PANS) model has been found to be better than LES in simulating flow regions especially those with less discretized grid. In this paper, a new nonlinear PANS turbulence model was proposed, which was modified from RNG k-ε turbulence model and the shear stresses were solved by Ehrhardt's nonlinear methods. The nonlinear PANS model was used to study the instability of “S” region of a model pump-turbine with misaligned guide vanes (MGV). The opening of preopened guide vanes had great influence on the “S” characteristics. The optimal relative opening of the preopened guide vanes was 50% for the improvement of the “S” characteristics. Pressure fluctuations in the vaneless space were analyzed. It is found that the dominant frequency at the vaneless space was twice the blade passing frequency, while the second dominant frequency decreased as the preopening increased.

1. Introduction

Pump-turbine, which was one of the key components in a pumped storage power station, played an important role in the optimization of power network. The starting period and stopping period of a pump-turbine ran at a high frequency, leading to a lot of instability problems due to the existence of “S” characteristics. When a pump-turbine ran in the “S” region, pressure fluctuation was very dangerous, even leading to some serious accidents. At present, misaligned guide vanes (MGV) were used in some pumped storage power stations to improve the “S” characteristics, but the influence of the relative opening of preopened guide vanes on the “S” characteristics was unclear.

Pump-turbines were always running at partial condition with the power grid changing, leading to flow separations and stall phenomena in the pump-turbine. The study of instability of a pump-turbine should overcome the difficulties in the simulation of strong vortex and separation flow. Large-scale coherent structures play a crucial role in the internal instabilities. For predicting the fluctuation of the dominant scale of vortexes, the traditional Reynolds-averaged Navier-Stokes (RANS) method suffers from inherent physical limitations [1, 2], which could not capture the flow with all kinds of scales. Large eddy simulation (LES) may not be computationally viable due to the large computational cost. Instead, the mixed computational methods, which combined the desirable aspects of RANS and LES, were used in engineering computation [3, 4]. Speziale [5] proposed a new turbulence model that combines the advantages of RANS method with those of LES [6]. Girimaji et al. [7] developed a bridging method inspired by the modeling paradigm proposed by Speziale [5]. The method was given the name partially averaged Navier-Stokes (PANS) model and was purported for any filter width [8].

At present, partially averaged Navier-Stokes (PANS) model has been found to be better than that of the LES in flow regions where simulations suffered from poor near-wall resolution [9]. Ji et al. [10, 11] studied the unsteady cavitating turbulence flow around a highly skewed model marine propeller based on the PANS method. Ma et al. [12] and Huang and Wang [13] used PANS model to investigate the cavitating flow around a hydrofoil. The PANS models mentioned above were all modified from standard k-ε turbulence model, while the standard k-ε turbulence model was poor in the simulation of strong swirling flows [14]. Most of the RANS turbulence models solved the shear stress by linear difference scheme and they were isotropic models [15], so they could not capture all kinds of vortexes in the pump-turbine well.

In this paper, a new nonlinear PANS turbulence model was proposed. The results based on the new nonlinear PANS turbulence model were compared with experimental results. Then the instability of a pump-turbine was studied by the nonlinear PANS model.

2. Nonlinear PANS Model

For incompressible flow, the continuity equation and Reynolds averaged Navier-Stokes equations are as follows:

\begin{matrix} \frac{\partial U_{i}}{\partial t} + U_{j} \frac{\partial U_{i}}{\partial x_{j}} + \frac{\partial τ (V_{i}, V_{j})}{\partial x_{j}} = - \frac{\partial 〈 p 〉}{\partial x_{i}} + ν \frac{\partial^{2} U_{i}}{\partial x_{j} \partial x_{i}}, \\ - \frac{\partial^{2} 〈 p 〉}{\partial x_{i} \partial x_{i}} = - \frac{\partial U_{i}}{\partial x_{j}} \cdot \frac{\partial U_{j}}{\partial x_{i}} + \frac{\partial τ (V_{i}, V_{j})}{\partial x_{i} \partial x_{j}} . \end{matrix}

(1)

V_i is partitioned into resolved and unresolved parts in the instantaneous velocity field, using an arbitrary homogeneous filter:

V_{i} = U_{i} + u_{i},

(2)

where U_i is the resolved velocity field and u_i is the unresolved field. It is used by (3) instead of the resolved field

U_{i} = 〈 V_{i} 〉 .

(3)

The additional nonlinear term τ(V_i, V_j), which is the generalized central second moment, is defined as:

τ (V_{i}, V_{j}) = (〈 V_{i} V_{j} 〉 - 〈 V_{i} 〉 〈 V_{j} 〉) .

(4)

In RNG k-ε turbulence mode, the equations for turbulent kinetic energy (k) and turbulent dissipation (ε) were shown as follows:

\begin{matrix} \frac{\partial (ρ k)}{\partial t} + \frac{\partial (ρ \overset{—}{U_{j}} k)}{\partial x_{j}} = \frac{\partial}{\partial x_{j}} [α_{k} (μ + μ_{t}) \frac{\partial k}{\partial x_{j}}] + P_{k} - ρ ∊, \\ \frac{\partial (ρ ∊)}{\partial t} + \frac{\partial (ρ \overset{—}{U_{j}} ∊)}{\partial x_{j}} = \frac{\partial}{\partial x_{j}} [α_{∊} (μ + μ_{t}) \frac{\partial ∊}{\partial x_{j}}] + C_{∊ 1}^{*} P_{k} \frac{∊}{k} - C_{∊ 2} ρ \frac{∊^{2}}{k}, \end{matrix}

(5)

where

\begin{matrix} μ_{t} = ρ C_{μ} \frac{k^{2}}{∊}, \\ C_{∊ 1}^{*} = C_{∊ 1} - \frac{η (1 - η / η_{0})}{1 + β η^{3}}, \\ S_{i j} = \frac{1}{2} (\frac{\partial \overset{—}{U_{i}}}{\partial x_{j}} + \frac{\partial \overset{—}{U_{j}}}{\partial x_{i}}), \end{matrix}

(6)

η = {(2 S_{i j} \cdot S_{i j})}^{1 / 2} \frac{k}{∊},

(7)

where C_μ = 0.0845, α_k = α_ε = 1.39, C_ε1 = 1.42, C_ε2 = 1.68, η₀ = 4.377, β = 0.012, and P_k denotes the production terms of turbulence kinetic energy.

For PANS methods,

\begin{matrix} f_{k} = \frac{k_{u}}{k}, \\ f_{∊} = \frac{∊_{u}}{∊} . \end{matrix}

(8)

Filter control parameters are f_k and f_ε. These two parameters are the partial average-quantification. The natural to quantify the extent of PANS filtering using f_k and f_ε. Considering large scales turbulence flows contain most of the kinetic energy, while the smallest scales flows produce much of the dissipation. The parameters dictated that 0 ≤ f_k ≤ f_ε ≤ 1. The smaller the f_k, the finer the filter: f_k = 1 represents RANS and f_k = 0 indicates DNS. Unity value of f_ε implies that the RANS and PANS unresolved small scales are identical. Smaller values of f_ε would require the resolution of dissipative scales of motion.

Equating the source terms, P_k in RNG k-ε turbulence mode has a relationship with P_ku in PANS model shown in (7):

P_{k} = \frac{1}{f_{k}} (P_{k u} - ∊_{u}) + \frac{∊_{u}}{f_{∊}} .

(9)

Then we can obtain PANS model by the modification of RNG k-ε turbulence model, and the equations are

\begin{matrix} \frac{\partial (ρ k_{u})}{\partial t} + \frac{\partial (ρ \overset{—}{U_{j}} k_{u})}{\partial x_{j}} = \frac{\partial}{\partial x_{j}} [α_{k} (μ + \frac{μ_{u}}{σ_{u}}) \frac{\partial k_{u}}{\partial x_{j}}] + P_{k u} - ρ ∊_{u}, \\ \frac{\partial (ρ ∊_{u})}{\partial t} + \frac{\partial (ρ \overset{—}{U_{j}} ∊_{u})}{\partial x_{j}} = \frac{\partial}{\partial x_{j}} [α_{k} (μ + \frac{μ_{u}}{σ_{u}}) \frac{\partial ∊_{u}}{\partial x_{j}}] + C_{∊ 1}^{*} P_{k u} \frac{∊_{u}}{k_{u}} - C_{∊ 2}^{*} ρ \frac{∊_{u}^{2}}{k_{u}} . \end{matrix}

(10)

Thus,

\begin{matrix} μ_{u} = ρ C_{μ} \frac{k_{u}^{2}}{∊_{u}}, \\ σ_{k u} = \frac{f_{k}^{2}}{f_{∊}}, σ_{∊ u} = \frac{f_{k}^{2}}{f_{∊}}, \\ C_{∊ 2}^{*} = C_{∊ 1}^{*} + \frac{f_{k}}{f_{∊}} (C_{∊ 2} - C_{∊ 1}^{*}) . \end{matrix}

(11)

Considering the nonlinear turbulence flow in the pump-turbine, the shear stress was solved by nonlinear turbulence model which was proposed by Liu et al. [15]:

P_{k} = - ρ \overset{—}{U_{i}^{'} U_{j}^{'}} \frac{\partial \overset{—}{U_{j}}}{\partial x_{j}},

(12)

\overset{—}{U_{i}^{'} U_{j}^{'}} = \frac{2}{3} k δ_{i j} - 2 C_{μ P} υ^{2} T S_{i j} + C_{1} C_{μ P} υ^{2} T^{2} (S_{i k} S_{k j} - \frac{1}{3} S_{k l} S_{k l} δ_{i j}) + C_{2} C_{μ k e} υ^{2} T^{2} (Ω_{i k} S_{k j} - Ω_{j k} S_{k i}) + C_{3} C_{μ P} υ^{2} T^{2} (Ω_{i k} Ω_{j k} - \frac{1}{3} Ω_{l k} Ω_{l k} δ_{i j}) + C_{4} C_{μ k e} υ^{2} T^{3} (S_{k i} Ω_{l j} - S_{k j} Ω_{l i}) S_{k l} + C_{5} C_{μ P} υ^{2} T^{3} S_{i j} S_{k l} S_{k l} + C_{6} C_{μ P} υ^{2} T^{3} S_{i j} Ω_{k l} Ω_{k l},

(13)

C_{μ P} = \min (\frac{1}{0.9 S^{1.4} + 0.4 Ω^{1.4} + 3.5}, 0.15),

(14)

Ω_{i j} = \frac{1}{2} (\frac{\partial \overset{—}{U_{i}}}{\partial x_{j}} + \frac{\partial \overset{—}{U_{j}}}{\partial x_{i}}),

(15)

S = \frac{k}{∊} \sqrt{2 S_{i j} S_{i j}},

(16)

Ω = \frac{k}{∊} \sqrt{2 Ω_{i j} Ω_{i j}},

(17)

where $C_{μ k e} = β (k^{2} / ε)$ , C₁ = – 0.2, C₂ = 0.4, $C_{3} = 2.0 - \exp (- {(S - Ω)}^{2})$ , C₄ = – 32.0C_μP², C₅ = – 16.0C_μP², C₆ = 16.0C_μP², T is turbulence time scale, and v is the turbulence velocity scale.

3. Pump-Turbine Geometry

Parameters of the model pump-turbine are shown in Table 1. D₁ denotes the runner inlet diameter in pump mode; Z_S, Z_G, and Z are the numbers of stay vanes, guide vanes, and runner blades, respectively; H_d denotes the rated head; n denotes the rotational speed of the runner; Q_d denotes the rated discharge.

Table 1:

Parameters of the model pump-turbine.

H_d (m)	52.4
Q_d (m³/s)	0.45
n (rpm)	1500
D₁ (m)	0.3
Z	9
Z _S	20
Z _G	20

The pump-turbine's structure is shown in Figure 1. The structure of MGV is shown in Figure 2.

Figure 1:

Profile of pump-turbine.

Figure 2:

Structure of MGV.

4. Simulation Conditions and Verification of Grid Independence

The model's grids, which were composed of an unstructured hexahedron and tetrahedron, were developed using ICEM, which is a commercial software package used for CFD discretization. Hexahedral grids were used for the runner and draft tube, and mixed grids were used for the other components. The mesh of the runner is shown in Figure 3.

Figure 3:

Mesh of different hydraulic regions.

The commercial CFD code FLUENT was used to perform the simulations. The boundary conditions of the pump-turbine were different in the reverse pump mode and in the turbine mode. The velocity at the spiral casing inlet was given for the turbine mode, whereas the pressure was given for the reverse pump mode. The runner's hydraulic region was set as a moving mesh model. The SIMPLEC algorithm was used to enforce mass conservation. For the quad/hex grids and complex flows in turbine mode, a second-order centered difference was used for the pressure interpolation to obtain better results. Taking the higher pressure gradient into consideration for reversed pump mode, PRESTO! (FLUENT Inc., 2005) discretization was used in the pressure interpolation. For present unsteady flow calculation, the time step was 0.00019 s. The nonlinear PANS model was used to calculate the “S” characteristics. For all calculations, simulations were run until convergence, which was determined by a reduction in the residual error to less than 0.0001.

The nonlinear PANS method was performed by the commercial software FLUENT by user defined function (UDF). Couple method was used to correct the pressure during the calculation. Second-order upwind scheme was used in resolving the Navies-Stocks equations. The rate of unsolved unresolved-to-total kinetic energy f_k = 0.2 and the rate of unsolved unresolved-to-total kinetic energy dissipation f_ε = 1 were used in the calculation.

To verify the calculation accuracy in terms of the grid size, several grid scales were selected. The cell number varied from 4 million to 10 million. The efficiency of the pump-turbine with different cell numbers when the pump-turbine was run in turbine mode at a no-load opening is shown in Figure 4. The pressure distribution on the pressure side of the runner blade is shown in Figure 5. The pressure distribution performed by 8.2 million is almost the same as the result performed by 9.7 million. It can be seen that the degree of computational precision is satisfactory when the element number is more than 7 million. A mesh with about 9 million cells in total was chosen for the simulations. The number of nodes and elements in each part is shown in Table 2.

Table 2:

Mesh of different hydraulic regions.

Hydraulic region	Runner	Guide vanes	Stay vanes	Draft tube	Casing	Total

Cells	2,734,700	1,734,700	1,455,324	1,173,888	1,840,506	8,939,118
Nodes	2,842,386	1,904,000	1,585,040	1,196,517	1,952,039	9,479,982

Figure 4:

Efficiency of the pump-turbine with different mesh scales.

Figure 5:

Pressure distribution on the pressure side of the runner blade.

5. Results and Discussion

5.1. “S” Characteristics with MGV

“S” characteristics of the pump-turbine with MGV at no-load opening are shown in Figure 6. The simulation results of the “S” characteristics with synchronized guide vane (SyV) based on PANS model agree well with experimental data. The result of “S” curve based on RNG k-ε model has large error compared with experimental result. The RNG k-ε model could not capture the detailed flow phenomena in the pump-turbine with small opening of guide vanes, causing the error of external characteristics. The capability comparisons of RNG k-ε model and PANS model are shown in the Appendix. MGV can improve the “S” curve, without changing the unit speed and unit flow rate at the runaway point. The flow rate of the pump-turbine with MGV is larger than the result of SyV in turbine mode, while it is smaller when it runs at reverse pump mode.

Figure 6:

“S” characteristics at no-load opening.

The head and flow rate curve is shown in Figure 7 and the torque of the runner and the flow rate is shown in Figure 8. MGV increases the discharge capacity of the pump-turbine within the same head. Flow rates of the two cases are very close at the runaway point. When the pump-turbine runs at reverse pump mode, the flow rate of the pump-turbine with SyV is larger than the unit with MGV. Downward displacement of the torque curve can be found in Figure 8 by the use of MGV.

Figure 7:

H-Q curve.

Figure 8:

Torque curve of the runner.

5.2. Flow in the Pump-Turbine with MGV

During the starting period, the “S” region with Q₁₁ > 0 determines the instability of the pump-turbine, and the relative opening of preopened guide vanes has a relationship with the “S” characteristics, so the “S” characteristics with Q₁₁ > 0 in different relative openings are calculated. The results of the “S” characteristics with different relative openings are shown in Figure 9. The openings of preopened guide vanes are 10°, 15°, and 21°, and their corresponding relative openings are 33%, 50%, and 70%, respectively. The existence of the “S” characteristic can be seen when the relative opening is less than 50%, but the increase of relative opening of guide vane improves the “S” region. The “S” characteristic disappears when the relative opening is 50%, and that is to say there is a one-to-one relationship between the Q₁₁ and n₁₁. The increase of relative opening of preopened guide vane will not make further improvement on the “S” region when the relative opening is larger than 50%. The increase of relative opening will cause flow instability in the pump-turbine, even leading to serious pressure fluctuation. The chosen of optimal relative opening should consider both the “S” characteristics and the flow instability.

Figure 9:

“S” region with different relative openings.

5.3. Flow in the Pump-Turbine with MGV

The runaway point is the point for starting load during the start-up process; it is very important for the instability of a pump-turbine. Streamline in the runner on blade-to-blade surface at no-load condition is shown in Figure 10. Flow in the runner with SyV has periodic distribution. MGV changes the flow in the runner. Different flow rates in the 20 passages of guide vane lead to reverse flow in the runner. The flow rate in the passage of the runner against the preopened guide vanes is larger than others, which may be the reason for the change of the performance of the pump-turbine.

Figure 10:

Streamlines in the runner.

Flow in the casing and the region of guide vanes is shown in Figure 11. No reverse flow can be seen with SyV. Because the relative opening of guide vanes is small, the resistance is large enough for the fluid flowing through the space between two adjacent guide vanes so that a water ring swirling at a high speed is formed in the vaneless space. The use of MGV destroys the water ring, as can be found in Figure 10 (b).

Figure 11:

Streamlines in the casing.

Vortex rope in the draft tube at the runaway point is shown in Figure 12. The vortex rope is calculated by the pressure contour. Vortex rope in the draft tube has large volume with SyV compared with MGV. The use of MGV changes the shape of vortex rope into foam-like structure.

Figure 12:

Vortex rope in the draft tube.

5.4. Pressure Fluctuation in the Vaneless Space

Pressure fluctuations in the vaneless space at runaway point are shown in Figure 13. The fluctuation of the pressure in the pump-turbine with SyV is consistent, and the pressure fluctuation only contains high frequency component. The use of MGV increases the amplitude of the pressure fluctuation. With the increase of the relative opening of preopened guide vane, the low frequency of pressure fluctuation is more and more obvious. Low frequency is very dangerous to the operation of the pump-turbine.

Figure 13:

Pressure fluctuation at time domain.

The pressure fluctuations at frequency domain are calculated by fast Fourier transform (FFT), which are shown in Figure 14. The spectrum characteristics of the pressure fluctuation are shown in Table 3. Results of pressure fluctuation in the vaneless space agree well with the experimental data. The amplitude of pressure fluctuation increases as the opening of preopened guide vane increases. The amplitude is twice the result with SyV when the pre-opening is 15°. At the runaway point, the dominant frequency (DF) of pressure fluctuation is the blade passing frequency. The second dominant frequency (SDF) of pressure fluctuation is two times the blade passing frequency when the preopening is small. The increase of the preopening makes the second dominant frequency lower and lower.

Table 3:

Comparison of pressure fluctuation at the vaneless space.

	6° SyV	10° MGV	15° MGV	21° MGV

ΔH/H (%)
Exp.	7.03	—	13.53	—
Cal.	7.94	8.81	14.45	16.25
DFf₁ (Hz)	9f_n	9f_n	9f_n	9f_n
SDFf₂ (Hz)	18f_n	18f_n	1.6f_n	0.7f_n

Figure 14:

Pressure fluctuation at frequency domain.

6. Conclusions

A PANS model was proposed based on RNG k-ε turbulence model. The shear stress was resolved by nonlinear model. The nonlinear PANS model was used to simulate the flow field in a model pump-turbine and was proved to be accurate in the simulation of “S” characteristics. The “S” characteristics of a pump-turbine with MGV were investigated and compared with the results of SyV. The MGV improved the “S” characteristics by the changing of the flow field in the pump-turbine. It destroyed the water ring in the vaneless space and changed the periodic flow in the runner. The use of MGV aggravated the pressure fluctuation in the vaneless space. The amplitude of pressure fluctuation increased with the preopening increase. The dominant frequency of pressure fluctuation in the vaneless space was the blade passing frequency, but the second dominant frequency gradually decreased as the preopening increased.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Footnotes

Appendix

To check the capability of the nonlinear PANS model for the prediction of complex flow in turbo machineries, a closed-loop test rig of Pedersen et al. [16] shown in Figure 15 was investigated. Fluid was cycling in the closed-loop test rig. Pedersen et al. [16] investigated the flow field of the centrifugal pump by particle image velocimetry (PIV) method. The velocity distributions on the measuring surface were calculated and compared with experimental results.

When the pump runs at small flow rate condition q_v/q_d = 0.25, velocities at different diameters in the impeller on the measuring surface at the small flow rate condition are shown in Figure 16. q_d is the rated flow rate. q_v is the flow rate of the pump. Streamlines on the measuring surface in the impeller performed by three models are shown in Figure 17. Calculation result of the nonlinear PASN model shows that flow in the impeller has a periodic distribution. No periodic phenomenon can be seen from the result of RNG k-ε turbulence model. The velocity distributions performed by the nonlinear PANS model agree well with the experimental result. The nonlinear PANS model can capture the large-scale vortex in the impeller accurately. The velocity at the impeller inlet is larger than the result of the nonlinear PANS model. The result of RNG k-03B5 turbulence model varies greatly from the experimental result. The closed-loop test rig has a periodic distribution in the hydraulic region. The flow in the cycling system should have periodic distribution, so that the result of the nonlinear PANS model is more realistic.

Acknowledgments

The authors would like to thank projects 51076077, 51176168, and 51306018 supported by National Natural Science Foundation of China and project supported by National Science and Technology Ministry (ID: 2011BAF03B01).

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Instability Analysis of a Model Pump-Turbine with MGV Based on Nonlinear Partially Averaged Navier-Stokes Methods

Abstract

1. Introduction

2. Nonlinear PANS Model

3. Pump-Turbine Geometry

4. Simulation Conditions and Verification of Grid Independence

5. Results and Discussion

5.1. “S” Characteristics with MGV

5.2. Flow in the Pump-Turbine with MGV

5.3. Flow in the Pump-Turbine with MGV

5.4. Pressure Fluctuation in the Vaneless Space

6. Conclusions

Conflict of Interests

Footnotes

Appendix

Acknowledgments

References