# Numerical Simulation and Uncertainty Analysis of an Axial-Flow Waterjet Pump

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. CFD Uncertainty Analysis Procedure

#### 2.1. Verification

_{S}is composed of the numerical error δ

_{SN}and the modeling error δ

_{SM}, viz, δ

_{S}= S − T = δ

_{SN}+ δ

_{SM}. When a numerical solution falls in the asymptotic range, the δ

_{SN}can be estimated together with the error contained in that estimate via the corrected approach; otherwise, only the numerical uncertainty U

_{SN}can be estimated to provide a boundary for the δ

_{SN}via the uncorrected approach. In this work, only the uncorrected approach is used; details about the corrected approach can be found in [10].

_{G}and U

_{T}in (1) are replaced by a combined discretization uncertainty U

_{GT}that is evaluated by refining the grids and time-step simultaneously at uniform refinement ratios. The U

_{SN}is alternatively expressed as

_{P}, is not investigated and will be neglected in the present analysis.

_{I}is defined as

_{1}, S

_{2}and S

_{3}denote the solutions yielded from fine, medium, and coarse grids and time steps, respectively. The convergence is monotonic when 0 < R < 1 but oscillatory when −1 < R < 0. If |R| > 1 the solutions diverge and the uncertainty cannot be estimated. For monotonic convergence, the RE approach is used to estimate the U

_{GT}. For oscillatory convergence, the U

_{GT}is estimated by

_{U}and S

_{L}denote, respectively, the maximum and minimum values in the simulation results. Note that the U

_{GT}yielded from (5) is just a rough estimate for the boundary of δ

_{SN}.

_{est}is an estimate for the limiting order of accuracy of the first term as the grid and time-step sizes go to zero and the asymptotic range is reached, i.e., C→1. The uncorrected approach is used to estimate the U

_{SN}only. The U

_{GT}is estimated by the following formulae that are proposed by Wilson et al. [21] and recommended in [10],

#### 2.2. Validation

_{SM}by using benchmark experimental data and, when conditions permit, estimating the modeling error δ

_{SM}itself. This process investigates whether the correct equations are solved [20]. The comparison error E is defined as the difference between experimental data D and simulation result S, i.e.,

_{V}is defined as

_{D}denotes experimental uncertainty. To determine whether the validation has been achieved, the comparison error E is compared to the validation uncertainty U

_{V}and the programmatic validation requirement U

_{reqd}. If the three variables are unequal to each other, one of the following cases must be true:

- (1)
- |E| < U
_{V}< U_{reqd} - (2)
- |E| < U
_{reqd}< U_{V} - (3)
- U
_{reqd}< |E| < U_{V} - (4)
- U
_{V}< |E| < U_{reqd} - (5)
- U
_{V}< U_{reqd}< |E| - (6)
- U
_{reqd}< U_{V}< |E|

_{V}level, but the modeling error cannot be estimated as the comparison error is below the uncertainty (noise) level. Particularly, in the first case the validation is successfully achieved at a level below U

_{reqd}.

_{V}, the modeling error dominates and is approximately equal to E. In case (4) the validation is successful at the |E| level.

## 3. Unsteady RANS Simulation of an Axial-Flow Pump

_{G}= Δh

_{2}/Δh

_{1}= Δh

_{3}/Δh

_{2}= $\sqrt{2}$, where h

_{3}, h

_{2}and h

_{1}denote, respectively, the sizes of coarse grid G

_{3}, medium grid G

_{2}, and fine grid G

_{1}. Figure 5 shows the surface grids for rotor and stator blades. The key parameters of the computational grids are listed in Table 1.

_{a}at three streamwise stations when Q = 0.42 m

^{3}/s. It seems that the boundary layer becomes fully developed from 1.7 m downstream of the inlet (0.45 m to the nose of shaft cap), and the inlet velocity profile prescribed by the software would have little influence on the modeling uncertainty.

^{+}~ 1) or wall-function type meshes (y

^{+}> 30). In the layer next to the wall, functions of the y

^{+}are used to specify the turbulent dissipation rate ε and the turbulent viscosity μ

_{t}. Otherwise, the transport equation for ε is solved. The equation for k is solved across the entire flow domain. The realizable k-ε model was proposed by Shih et al. [23], which consists of a new model equation for the turbulent dissipation rate and a new eddy viscosity formulation. The model was validated for different flow types, including rotating homogeneous shear flows, and found to perform much better than the standard k-ε model.

_{T}= Δt

_{2}/Δt

_{1}= Δt

_{3}/Δt

_{2}= 2, are used in the unsteady simulations, where Δt

_{3}, Δt

_{2}, and Δt

_{1}denote coarse, medium, and fine time-step sizes, respectively. Note that the grids and time steps must be coarsened or refined simultaneously. To expedite convergence, steady-flow simulations are used to initialize unsteady simulations. In each time step, 20 iterations are performed to reduce the residuals to an acceptable level.

## 4. Numerical Uncertainty Analysis for the Axial-Flow Pump

^{3}/s–0.471 m

^{3}/s) are considered which cover a wide range of the pump’s operating conditions. The uncertainties in the simulated head and power are evaluated.

_{d}and p

_{u}denote shroud-surface pressures averaged over the circumferences at 2d downstream and 2d upstream of the rotor, respectively; V

_{d}and V

_{u}denote the axial velocities averaged over the shroud cross sections at 2d downstream and 2d upstream of the rotor, respectively. The locations where the pressures and axial velocities are numerically evaluated are the same as those in the experiments for the pump model considered here. In (12) the gravitational acceleration and the density of water are denoted by g and ρ, respectively; the torque and the rotational speed (r/min) of the rotor are denoted by M and N, respectively.

#### 4.1. Verification

_{tip}is 2.3 × 10

^{6}, where R

_{tip}is the tip radius of the rotor. The coarse time step size Δt

_{3}is set to 1.1494 × 10

^{−4}s, which corresponds to a blade angular displacement of 1° per time step. The medium and fine time step sizes, Δt

_{2}and Δt

_{1}, correspond to 0.5° and 0.25° blade angular displacement per time step, respectively. To resolve the flow in viscous sub-layer, the near-wall grid layers need to satisfy y

^{+}~ 1. Corresponding to the grid sizes as listed in Table 1, the ranges of surface-averaged y

^{+}are given in Table 2 for the flow rates simulated. Figure 7 shows the limiting streamlines on rotor and stator blade surfaces when Q = 0.42 m

^{3}/s. On rotor blade surfaces, flow separation occurs mainly on the suction side, in inner radii and close to the trailing edge. On stator blade surfaces, the flow is converging on the suction side but diverging on the pressure side; flow separation occurs on the suction side only, in close proximity to the trailing edge, and from about 40% span to the tip. It seems that the stator blade geometry may need to be improved. The flow patterns simulated with the coarse grid G

_{3}and the fine grid G

_{1}are quite similar to each other.

^{−5}–10

^{−6}, those in velocity components to 10

^{−5}–10

^{−7}, and those in k and ε to 10

^{−6}–10

^{−8}when the solution converges. The speed of convergence is slow at low flow rates.

^{3}/s as an example. The heads differ by less than 0.05% in the last two revolutions for the three solutions, but speed of convergence is much lower than that of the power.

_{R}= mZ

_{S}, k and m are positive integers; Z

_{R}and Z

_{S}are blade numbers of the rotor and the stator, respectively; N is the rotational speed (shaft frequency) of the rotor. For the axial-flow pump considered here, Z

_{R}= 7, Z

_{S}= 9, the lowest frequency of the interaction components is 63 times the shaft frequency. To remove the interaction components, Fourier analyses are performed for the simulated unsteady head and power first; then, the amplitudes at the interaction frequencies are set to zero and the time series are reconstructed. Figure 10 shows a comparison of the time series before and after removing the interaction components. For the combination of rotor and stator blade numbers considered here, the interaction components have little influence on the simulated total fluctuations.

_{GT}, the uncertainties due to grid and time-step sizes. The following can be observed:

- (1)
- The convergence ratio is 0 < R < 1, which indicates the simulation results converge monotonically when the discretization in time and space is refined simultaneously and consistently. This result justifies the use of the generalized Richardson extrapolation (RE) for evaluating the observed order of accuracy p and the estimated error ${\delta}_{RE}^{*}$.
- (2)
- The estimated limiting order of accuracy p
_{est}is set to 2, since the governing equations are discretized with second-order schemes in space, and ${r}_{T}={r}_{G}^{2}$ although a first-order scheme is used for the discretization in time. - (3)
- The correction factor C is sufficiently far from 1 in most cases. Therefore, only the uncertainty U
_{GT}is evaluated to give a boundary of the simulation error. - (4)
- The iteration uncertainties as shown in Table 4 are negligibly smaller than U
_{GT}, hence U_{SN}≈ U_{GT}. - (5)
- The numerical uncertainties are less than 4.3%; and the uncertainties in simulated head are higher than those in simulated power, especially at the low are high flow rates.

#### 4.2. Validation

_{1}and fine time step size Δt

_{1}. The validation uncertainty U

_{V}is calculated according to (11). Due to the lack of experimental uncertainty data, the experimental uncertainty U

_{D}is assumed to be 0.8% by summing up the system accuracies in measuring the flow rate Q, the pressure difference p

_{d}− p

_{u}, and the torque M, etc. The results in Table 6 indicate that

- (1)
- The validation is successfully achieved at the U
_{V}level of 1–4%, except for the power P at Q = 0.35 m^{3}/s and Q = 0.471 m^{3}/s. - (2)
- For the power P at Q = 0.471 m
^{3}/s, the validation is successful at the |E| level of 1%, although the comparison error is larger than the validation uncertainty. - (3)
- For the power P at Q = 0.35 m
^{3}/s, the comparison error is much larger than the validation uncertainty, which indicates that the modeling error is large, and the validation is not achieved. - (4)
- In most cases investigated here, the principal source of error is unidentifiable since the comparison errors are quite close to the validation uncertainties.

## 5. Simulated Flow Features

_{1}at the flow rate Q = 0.42 m

^{3}/s.

#### 5.1. Tip Clearance Flow

#### 5.2. Interactions between Rotor and Stator Blades

_{tip}and 0.95R

_{tip}. The pressures on a rotor blade oscillate nine periods when the rotor completes a revolution, because the rotor blade sweeps across all the nine blades of the stator. For the same reason, the pressures on a stator blade oscillate seven periods since the rotor is seven-bladed. The oscillation amplitudes at the two radii shown in Figure 17 are quite close to each other, for both the rotor and the stator.

## 6. Concluding Remarks

_{G}and time step uncertainty U

_{T}are replaced by U

_{GT}, the numerical uncertainty when grid and time-step sizes are refined simultaneously and consistently. For complex three-dimensional flow problems, the parameter refinement ratios need to be chosen appropriately so that the number of grids is not excessively large, and the time-step size is not too small. However, by doing so, it is almost impossible for the solutions to reach the asymptotic range. The analysis results indicate that the numerical uncertainties in present simulations are less than 4.3%. The validation is successfully achieved in most cases, except for the power at the lowest flow rate considered due to large modeling errors. For the flow rates considered, it is impossible to identify the principal source of error because the comparison errors are quite close to the validation uncertainties. So far as the head and power are concerned, it seems that the present simulation method based on block-structured grids works well from a practical point of view. However, from an uncertainty point of view, the grid and time-step sizes are still not small enough although further refinement would be very challenging computationally and even impractical.

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**The geometric model of the seven-bladed rotor and the nine-bladed pre-swirl stator. The stator blades are connected to the shroud surface without clearance. The shroud surface is omitted.

**Figure 2.**The computational domain for the axial-flow pump. The domain is a circular cylinder bounded by the shroud surface, 16 times the shroud diameter in length. The rotor is located at the longitudinal center of the domain, while the stator is located upstream of the rotor.

**Figure 4.**The grid structure in the tip clearance of a rotor. The sections perpendicular to the nose-tail chord of the tip are shown at 10% (

**top**), 50% (

**middle**), and 90% (

**bottom**) chord length.

**Figure 6.**Simulated axial velocity profiles in shroud-surface boundary layer, where h is the normal distance from the shroud surface, X

_{0}is the distance measured from the inlet, and X

_{S}is the distance to the nose of shaft cap.

**Figure 7.**Limiting streamlines on rotor (

**top row**) and stator (

**bottom row**) blade surfaces, Q = 0.42 m

^{3}/s.

**Figure 9.**The convergence history of the averaged head H (

**left**) and power P (

**right**) in the last five revolutions before convergence, Q = 0.42 m

^{3}/s. The abscissa, n, denotes the number of revolutions relative to the last revolution (n = 0). The ordinates H

_{n}and P

_{n}denote the head and power of the nth revolution, respectively.

**Figure 10.**Comparison of the unsteady head H (

**top row**) and power P (

**bottom row**) before (black lines) and after (red lines) removing the components arising from stator/rotor interaction. The abscissa, θ

_{R}, denotes the angular position of a rotor blade. The head and power are expressed respectively in percents relative to H

_{m}and P

_{m}, the averages over a complete revolution. Q = 0.42 m

^{3}/s.

**Figure 12.**Pressure contours on the tip surface and streamlines in the tip clearance. Q = 0.42 m

^{3}/s.

**Figure 13.**The streamlines (colored by static pressures) in sections across the tip surface, where x

_{c}is the chordwise distance from the leading edge in fractions of the chord length. Q = 0.42 m

^{3}/s.

**Figure 14.**In-plane absolute velocity profiles in the tip clearance at 10% (

**left**) and 50% (

**right**) chord length. The velocity profiles shown at five instantaneous positions of the rotor θ

_{R}cover the angular spacing between adjacent stator blades. From suction side to pressure side, three sections are taken at 10%, 50% and 90% of the local thickness t, respectively. The velocity magnitude is zero at the vertical straight lines. The shroud is stationary. The rotor tip rotates from left to right. Q = 0.42 m

^{3}/s.

**Figure 15.**The pressure (

**top row**) and vorticity (

**bottom row**) around stator and rotor blade sections at 0.7R

_{tip}. The rotor blade angle θ

_{R}= 0°, 10°, 20°, 30° (from

**left**to

**right**). Q = 0.42 m

^{3}/s.

**Figure 16.**The pressure (

**top row**) and vorticity (

**bottom row**) around stator and rotor blade sections at 0.95R

_{tip}. The rotor blade angle θ

_{R}= 0°, 10°, 20°, 30° (from

**left**to

**right**). Q = 0.42 m

^{3}/s.

**Figure 17.**Blade-surface pressure oscillations in a complete revolution of the rotor at 0.7R

_{tip}(

**top row**) and 0.95R

_{tip}(

**bottom row**). x

_{c}denotes the chordwise location from section nose in fractions of the chord length. θ

_{R}denotes the angular position of a rotor blade. Q = 0.42 m

^{3}/s.

**Figure 18.**Oscillations of the torques on a rotor blade (

**left**) and a stator blade (

**right**). θ

_{R}denotes the angular position of the rotor blade. Q = 0.42 m

^{3}/s.

Grid ID | Maximum Cell Size on Blade Surface (mm) | First-Layer Cell Height from Blade Surface (mm) | Number of Cells in Tip-Clearance | Total Number of Cells (Million) | |||
---|---|---|---|---|---|---|---|

Stator | Rotor | Stator | Rotor | Radial | Circumferential | ||

G_{3} | 5 | 4 | 0.02 | 0.01 | 10 | 16 | 1.79 |

G_{2} | 3.54 | 2.83 | 0.014 | 0.007 | 15 | 22 | 4.87 |

G_{1} | 2.5 | 2 | 0.01 | 0.005 | 20 | 30 | 13.17 |

Grid ID | Stator | Rotor | Tip Clearance |
---|---|---|---|

G_{3} | 3.2–4.2 | 1.9–2.3 | 5.7–5.8 |

G_{2} | 2.3–3.0 | 1.4–1.6 | 4.7–4.8 |

G_{1} | 1.7–2.3 | 1.0–1.3 | 3.3–3.4 |

**Table 3.**Comparison of simulation results and experimental data [25] for the axial-flow pump.

Q (m^{3}/s) | Grid, Time-Step Size | Simulation Result S | Experimental Data D | Comparison Error E (%D) | |||
---|---|---|---|---|---|---|---|

H (m) | P(kW) | H (m) | P (kW) | H | P | ||

0.35 | G_{3}, Δt_{3} | 7.320 | 29.473 | 7.450 | 31.341 | −1.47 | −5.96 |

G_{2}, Δt_{2} | 7.373 | 29.541 | −1.04 | −5.74 | |||

G_{1}, Δt_{1} | 7.406 | 29.584 | −0.59 | −5.61 | |||

0.42 | G_{3}, Δt_{3} | 5.221 | 25.343 | 5.400 | 25.957 | −3.31 | −2.36 |

G_{2}, Δt_{2} | 5.247 | 25.414 | −2.83 | −2.09 | |||

G_{1}, Δt_{1} | 5.267 | 25.473 | −2.45 | −1.87 | |||

0.471 | G_{3}, Δt_{3} | 3.476 | 20.341 | 3.600 | 20.305 | −3.44 | 0.18 |

G_{2}, Δt_{2} | 3.521 | 20.456 | −2.18 | 0.75 | |||

G_{1}, Δt_{1} | 3.546 | 20.513 | −1.49 | 1.02 |

Q (m^{3}/s) | G_{3}, Δt_{3} (%S) | G_{2}, Δt_{2} (%S) | G_{1}, Δt_{1} (%S) | |
---|---|---|---|---|

0.35 | H | 0.03 | 0.02 | 0.02 |

0.42 | 0.03 | 0.03 | 0.01 | |

0.471 | 0.04 | 0.04 | 0.01 | |

0.35 | P | 0.03 | 0.02 | 0.01 |

0.42 | 0.03 | 0.03 | 0.01 | |

0.471 | 0.04 | 0.04 | 0.01 |

Q (m^{3}/s) | ε_{21} | ε_{32} | R | p | ${\delta}_{RE}^{*}$ | C | U_{GT} | U_{SN} (%S) | |
---|---|---|---|---|---|---|---|---|---|

0.35 | H | −0.033 | −0.053 | 0.642 | 1.280 | −0.060 | 0.558 | 0.113 | 1.52 |

0.42 | −0.021 | −0.025 | 0.814 | 0.593 | −0.091 | 0.228 | 0.231 | 4.27 | |

0.471 | −0.025 | −0.045 | 0.564 | 1.651 | −0.033 | 0.772 | 0.048 | 1.33 | |

0.35 | P | −0.043 | −0.068 | 0.625 | 1.355 | −0.071 | 0.599 | 0.129 | 0.41 |

0.42 | −0.059 | −0.071 | 0.836 | 0.516 | −0.302 | 0.196 | 0.787 | 3.03 | |

0.471 | −0.057 | −0.115 | 0.496 | 2.025 | −0.056 | 1.017 | 0.062 | 0.30 |

Q (m^{3}/s) | |E| (%D) | U_{V} (%D) | ||
---|---|---|---|---|

H | P | H | P | |

0.35 | 0.59 | 5.61 | 1.72 | 0.90 |

0.42 | 2.46 | 1.86 | 4.35 | 3.14 |

0.471 | 1.50 | 1.02 | 1.55 | 0.86 |

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## Share and Cite

**MDPI and ACS Style**

Qiu, J.-T.; Yang, C.-J.; Dong, X.-Q.; Wang, Z.-L.; Li, W.; Noblesse, F. Numerical Simulation and Uncertainty Analysis of an Axial-Flow Waterjet Pump. *J. Mar. Sci. Eng.* **2018**, *6*, 71.
https://doi.org/10.3390/jmse6020071

**AMA Style**

Qiu J-T, Yang C-J, Dong X-Q, Wang Z-L, Li W, Noblesse F. Numerical Simulation and Uncertainty Analysis of an Axial-Flow Waterjet Pump. *Journal of Marine Science and Engineering*. 2018; 6(2):71.
https://doi.org/10.3390/jmse6020071

**Chicago/Turabian Style**

Qiu, Ji-Tao, Chen-Jun Yang, Xiao-Qian Dong, Zong-Long Wang, Wei Li, and Francis Noblesse. 2018. "Numerical Simulation and Uncertainty Analysis of an Axial-Flow Waterjet Pump" *Journal of Marine Science and Engineering* 6, no. 2: 71.
https://doi.org/10.3390/jmse6020071