A numerical study on the unsteady flow in the guide vane of a water-jet propulsion mixed-flow pump

Mixing and dissipation of the complex vortices in the water-jet propulsion mixed-flow pump is an important source of flow loss. Furthermore, some vortices have strong unstable feature, which adversely affect vibration and noise. The unsteady flow in the guide vane of a mixed-flow pump is studied by numerical methods. Through the spectral results of hydrodynamic excitation and the instantaneous flow field of the mix-flow pump, the basic unsteady features of the unsteady flow are obtained. Peak frequency of approximately 1.68 rotation frequency is found as the dominant frequency of the excitation in the guide vane. Dynamic mode decomposition (DMD) is then used to extract the key flow modes in the guide vane. The results illustrate the dominant flow structure in the guide vane, including upstream sweeping structures, the separate vortex structures, and large-scale passage vortex structure. The upstream sweeping wake caused the fluctuation in the boundary layer and induced flow separation. Following the separation, passage vortices are induced which has the dominant frequency 1.68 rotation frequency.


Introduction
Water-jet propulsion is a new and efficient propulsion method, which is widely used in high-speed ships [1].It facilitates the reaction force of the high-speed jet sprayed by the propulsion pump.The unsteady flow in the propulsion pump and the solid wall interaction form the hydrodynamic excitation forces, not only cause vibration but also induce noise [2,3] , which have adverse effect on the water-jet propulsion.As the main pump type of the water-jet propulsion, guide vane mixed flow pump has typical unsteady flow phenomena such as rotor-stator interaction, rotating stall and separate flow [4][5][6][7][8].Separate flow consists of many kinds of vortex structures.These vortex structures are importance source of the flow loss.Some of the vortex structures have strong unsteady characteristics.
Compared with rotor-stator interaction and rotating stall, there is relatively little research on the separate flow in the guide vane of the pumps.Many researches on the secondary flow in the blade passage of a gas turbine can be referred to this study [9][10][11][12][13].Representative turbine secondary flow vortex structures include horseshoe vortex, passage vortex, corner vortex, etc.The passage vortex is the largest scale vortex structure, which is affected by the upstream horseshoe vortex.Then, the corner vortex is induced by the passage vortex.Moreover, the rotor-stator interaction flow has important effect on the vortex structures.Upstream wake bends, stretches, and deforms in the blade passage, induced unsteady effect on the boundary flow [14,15].
In the field of pump design, researchers reduce separate flow losses by controlling load distribution and stack of the blade [16,17].Many experiments on the dynamic pressure of an axial-or mixed-flow The 17th Asian International Conference on Fluid Machinery (AICFM 17 2023) Journal of Physics: Conference Series 2707 (2024) 012034 IOP Publishing doi:10.1088/1742-6596/2707/1/012034 2 pump have shown that the effect of rotor-stator interaction becomes weaken in the guide vane area.Some strong low frequency components are found in design condition, which is not induced by rotating stall [18,19].Unsteady secondary flow vortices may dominant this excitation phenomenon, However, there is still a lack of study on the temporal and spatial evolution of these vortices in an axial-or mixedflow pump.
For the complex time-varying flow field, it is difficult to analyze the revolution of key flow structures from the huge amount of the computed datasets.Hence, model decomposition methods, such as proper orthogonal decomposition (POD) [20] and dynamic mode decomposition (DMD) [21], are proposed.A set of eigenmodes of the unsteady flow field can be obtained via these modal decomposition methods.Through these eigenmodes, the specify flow structure of the complex time-varying flows can be analyzed.POD splits the time-dependent flow into orthogonal spatial eigenmodes.DMD decomposes the unsteady flow field into eigenmodes with individual frequency and amplified rate.Hence, one DMD represent the flow structures with the same time-scale and similar phenomena, which makes DMD a useful tool in unsteady flow analysis and stability analysis [22,23].
In this paper, the internal flow in a mixed-flow pump is simulated by CFD.The time-varying hydrodynamic excitation of the pump and the instantaneous flow field are studied.The unsteady flow in the guide vane region is decomposed by DMD.Through the DMD results, the complex unsteady vortex structures with its temporal feature can be analyzed in detail.This paper is organized as follows: In section 2, the numerical method and DMD are introduced briefly.In section 3, the hydrodynamic excitation, instantaneous and the DMD result are illustrated.

Numerical simulation
The computation model is a guide vane mixed-flow propulsion pump designed by MARIC.The impeller and guide vane are illustrated on Figure 1.The blade number of impeller is 6.The blade number of guide vane is 11.The inlet diameter of the pump mode is 270 mm, outlet diameter is 54mm and the tip clearance is 0.3 mm.The design flow rate is 0.41 m3/s and the design head is 16.5 m.The numerical simulation is implemented on commercial CFD software STAR-CCM+.The computational domain and the mesh are shown on Figure 2. The boundary conditions are set as inlet mass flow and the outlet pressure.The computational domain is divided into 4 regions: region of inlet passage, region of impeller, region of guide vane and region of outlet passage.Setting rotation motion of the impeller region to simulate the rotating blade.The shroud surface in impeller region is set as stationary wall.Sliding mesh interface are set as the interface between impeller region and guide vane region to transfer the numerical solution data.All other walls are set as non-slip wall.Prism layer mesh is generated near all solid walls.The height of the first prism layer meets the requirement of  + ~1.The growth rate of the prism layer is set as 1.2 to improve the boundary resolution.The areas outside the boundary layer in the impeller region and guide vane region are filled with polyhedral meshes provided by the CFD software in order to capture the complex swirling flow in the pump.In the inlet passage and outlet passage region, the remaining areas are filled with hexahedron mesh to improve the mesh quality and the numerical efficiency.The K−ω SST model is used to simulate the turbulence flow, which is proved to have good prediction capability of pump flow [25].The time step of the unsteady simulation is corresponding to the duration of 1° rotation of the impeller.In order validate the numerical result, the four computation meshes (mesh1, mesh2, mesh3, mesh4) are compared, which have 5.71 × 10 6 ， 7.27 × 10 6 ， 9.25 × 10 6 ，1.05 × 10 7 cells, respectively.The simulation results of time-average pump efficiency and pump head are compared with experiment results, illustrated as Figure 3.It can be found that there is little change in efficiency and head when the total number of meshes reaches 9.25 × 10 6 (mesh 3).Moreover, the simulation results of mesh 3 is in good agreement with the experimental results.The accuracy of the numerical simulation can be validated.

DMD method
The basic methodology in this paper is based on DMD.Therefore, the DMD algorithm is illustrated.Assume that the instantaneous flow field is approximately related by a linear operator ,  ∈ ℝ × : where the vector   ∈ ℝ × is the sequential snapshots for the flow.The procedure of DMD is to compute the approximate eigenvalues and the eigenvectors of , which begins with the two snapshots matrices   and   ： According to the assumption (1), the relation between   and   is approximate:   ≈   .Due to the advantage of numerical stability, the SVD-based algorithm is widely used in DMD algorithm [24].The snapshot matrix   is decomposed by the singular value decomposition (SVD) into: The operator  ̅ can be further calculated: For the discretization of unsteady flow field, the spatial dimension n usually far larger than the temporal dimension m.It is very inefficient to compute  ̅ directly.Hence, a low-dimensional operator  ̃ ∈ ℝ × is defined in DMD algorithm: The eigendecomposition of  ̃ can be computed as  ̃ ̃=  ̃ ̃.The matrix of the DMD modes  = [    ⋯  − ] are obtained: while the DMD eigenvalue is obtained from It should be noted that the second norm of each DMD mode represents the corresponding modal energy and each DMD eigenvalue contains the frequency of the corresponding DMD mode.The angular frequency of the DMD mode could be computed by: where  is the time interval of the snapshot.Hence, the highest frequency of the DMD mode will not exceed the /().After simulating 6 rotation cycles, the hydrodynamic characteristics of the pump begin, the hydrodynamic characteristics of the pump begin to oscillate periodically.Spectral analysis is performed on the time-varying hydrodynamic characteristics.The spectral result of the pump efficiency is illustrated on Figure 4.The non-dimensional frequency f * of the horizontal ordinate is defined as f * = f/f r (f r is the rotation frequency of the impeller).It can be found that the spectrum has broadband feature with the peak frequency of f * = 1.67.Some line spectrums appear at f * = 11 and its harmonics, which is related to the guide vane (blade number of the guide vane is 11).In order to study the unsteady feature of the pump excitation, set 2 monitors of fluctuate pressure are set in the pump.One is inside the tip clearance tip, the other is at the passage of guide vane, illustrated as Figure 1.The spectral results of the fluctuate pressure is shown on Figure 6.Due to the periodically rotating blades, the frequencies of the dominant components at point 1 are f * = 6 and its harmonics.However, the components of f * = 1.67 can also be found.At point 2, component of f * = 1.67 has the highest amplitude.Hence, the component of f * = 1.67 induced by some vortex structures in guide vane region have great effect on the unsteady forces and hydrodynamic excitation of the guide vane mixedflow pump.The following study captures these vortices and its feature.

Instantaneous flow field
In order to study the flow mechanism in the guide vane region, the local instantaneous flow field should be analyzed in detail.The local planes in the guide vane are illustrated on Figure 7.The red planes are defined as H planes, which approximate the span surface of the guide vane blade.The streamwise flow can be studied in H planes.The blue planes are defined as A planes, which is the axial surface and illustrates the radial flow.Due to the strong adverse pressure gradient in the streamwise direction, the flow separates rapidly after passing through the leading edge of the suction surface.Then, large-scale separate vortices appear.On the H 60% plane, the lateral pressure gradient of the passage is enhanced, while the steamwise adverse pressure gradient is weakened.The flow separates at the middle part of the blade.The separate vortices appear at the area near the trailing edge.On the H 90% plane, there is no obvious flow separation under the condition of strong lateral pressure gradient and weak steamwise adverse pressure gradient.
Figure 9 show the distribution contour of instantaneous pressure and streamlines on different A planes.It can be found that there is also obvious radial pressure gradient from the shroud to the hub of the passage in addition to the lateral pressure gradient.The phenomenon may be due to the centrifugal force of the mixed-flow pump.Affected by the rotating impeller, the flow on the A 5% plane is mainly circumferential (from the suction surface to the pressure surface of the next blade).On the A 25% plane, separation vortices appear near the hub.Then, the vortices develop towards pressure surface of the next blade with the circumferential flow.On the A 50% plane, the separate vortices become larger in scale and more complex in structure.On the A 95% plane, where is near the outlet of the guide vane passage, the scale of separate vortices further increasing.Due to the lateral pressure gradient, the separate vortices are mainly located on side of suction surface.Affected by the separated vortices, the flow on this plane is mainly radial.

DMD results
From the results of unsteady hydrodynamics characteristics and instantaneous flow field, it can be found that the vortex structures in the guide vane have very complex temporal and spatial feature.In order to further study these vortex structures and find the flow structure corresponding to f * = 1.67.DMD is used to analyze the unsteady flow field in the guide vane passage.The vorticity magnitude is selected as the snapshot variable.The spectrum of the fluctuate pressure in the guide vane (Figure 6 (b)) illustrates that the high frequency components are very weaken in the flow field.Hence, one flow field snapshot is extracted every 5 simulation steps (impeller rotates every 5°), which corresponds to the highest mode frequency f * = 36.The total sample duration is 6 rotation cycles, corresponding to 432 overall snapshots.
Snapshots on different H plane are analyzed by DMD, respectively.The decomposition results are shown on Figure 10.The DMD eigenvalues on all H plane are on or in the unit cycle, which illustrate that the flow in the guide vane have the periodical feature.From the modal spectrum, it can be found that the modes with f * = 0 have the highest modal energy.Modes with f * = 6 and f * = 12, which equals to the blade passing frequency and its harmonics are the leading energetic modes.Moreover, the modes with f * = 1.67 also have relatively high modal energy in these H planes, which corresponding to the important unsteady flow structures affecting the unsteady hydrodynamic excitation.The distribution contour of f * = 0 modes on different H planes are shown on Figure 11.The mode values are normalized.f * = 0 illustrates that the flow structures of this mode do not vary with time.Hence, the modes represent the time-average flow field and show the region with high vorticity magnitude.It can be found that the time-average flow structure is distributed uniformly within each passage.On the H 30% plane, most area starting from the front of the passage are affected by vortices.
On the H 60% plane, high vorticity magnitude is located at the mid part of the passage.On the H 90% plane, there is less high vorticity magnitude distributed within the passage.Modes with f * = 1.67, f * = 6 and f * = 12 are illustrated on Figure 12.Modes with f * = 6 all have the second highest modal energy.As the modal frequency is equal to the blade passing frequency, these modes illustrated the flow structures induced by the rotor-stator interaction and they are the dominant unsteady flow structure in all H planes. On all H planes, it can be found that there is an upstream wake sweeping structure with a circumferential wave number of 6 near the inlet of the guide vane.On H 30% plane, the upstream sweeping wake promotes the unsteady boundary layer flow, which separated from the suction surface rapidly after passing the leading edge.Then, strong separated vortex structures are generated.Despite the lateral pressure gradient in the passage, the separated vortices still move to the pressure surface of the next blade (black arrow in the figure) with the circumferential flow.On H 60% plane, the upstream sweeping wake induced periodic boundary flow on the suction surface, and cause the boundary layer separating in the mid part.Different from the H 30% plane, separate vortices are affected by the lateral pressure gradient, moving downstream of the passage.On H 90% plane, the upstream sweeping wake only affect the boundary flow on the suction surface.There are no obvious separate vortex structures.
Modes with f * = 12 are the harmonics of the modes with f * = 6.These modal structures are also related to the rotating impeller.On all H planes, there are upstream wake sweeping structures with circumferential wave number of 12.Moreover, flow structure in f * = 12 modes are smaller than the f * = 6 modes in scale, which are supplementary details on the development of flow structures, such as fluctuate or rotation (solid arrow).
Modes with f * = 1.67 illustrate the key flow structure in this study.On all the H planes, the modal structures are larger in scale than the f * = 6 modes and are located at the large-scale vortex region in Figure 8.On H 30% and H 60% plane, modal structures appear behind the separated vortices caused by the rotor-stator interaction (refer to the black dash line in the figure).Then, the modal structure further development to the downstream.Moreover, the modal structure is not symmetric in different passage, which gradually develop in the opposite direction of the rotating impeller (dash arrow).Hence, these flow structures are also affected by the motion of the upstream flow.In this manuscript, the modal structures with f * = 1.67 are defined as the passage vortex.The generation and the development of the passage vortex can be concluded.The boundary layer flow separated from the suction surface due to the upstream sweeping wake, separated vortices move to the pressure side and downstream.Behind the separated position, low-frequency large-scale passage vortex is induced and development to the downstream.Moreover, horseshoe vortices of the pressure side of the next blade are found to move and inflow the passage vortex on H 30% plane.On the H 90% plane, there is no obvious separated vortices and passage vortex inside the passage.The modal structure only shown the shedding vortices of the trailing edge.It can be found that the passage vortices also develop in the radial direction.In order to study the revolution of the flow structures in the radial plane, the unsteady vorticity magnitude on different A planes are analyzed by DMD.The modal spectrum is illustrated as Figure 13.It can be found that modes with f * = 6 and f * = 12 are the leading energetic modes in the front part of the passage (A 5% plane and A 25% plane).In the end part of the passage (A 95% plane), modes of f * = 6 and f * = 12 are not the dominant flow structure, which illustrates that the effect of rotor-stator interaction is gradually weaken in the passage.Modes of f * = 1.67 are found in DMD results on all A planes.Opposite to f * = 6 modes, modal energy of f * = 1.67 modes increase gradually in the passage and become the dominant unsteady flow structures since A 50% plane.Figure 14 shows the distribution contour of f * = 0 modes.These modes represent the time-average flow filed.Region with high vorticity magnitude is coincidence with the vortex structures in Figure 9 Modes with f * = 6, f * = 12 and f * = 1.67 are illustrated on Figure 15.On A 5% plane, modal structures with f * = 6 are located on the suction surface and the hub.These modes represent the effect of the upstream sweeping wake on boundary layer flow of the suction surface and the hub, which is the circumferential motion (wave number 6) and the radial motion (solid arrow).The radial motion is due to deviation angle between the upstream wake and the blade angle of the guide vane.The hub is firstly affected by the wake, then is the shroud.f * = 12 modes also represent the effect of rotor-stator interaction, which is a supplement to the details of f * = 6 modes.Modal structure of f * = 1.67 is located at the hub end of the suction surface.As the modal energy is low, it is not the dominant unsteady flow structure on this plane.
On A 25% plane, the f * = 6 mode illustrates the separate vortices (dash line) which is induced by the upstream sweeping wake.f * = 6 mode further represents the detailed characteristics of the development of separated vortices.The modal energy of the f * = 1.67 mode is still relatively low.Hence, the mode illustrates the primary form of the passage vortex.It can be found that the passage vortex appears near the hub end of the suction surface and has the feature of circumferential motion.The affect region of the passage vortex is below the separated vortices (dash line).Hence, the low frequency passage vortex is induced by the flow separation.
On A 50% plane, f * = 6 mode also illustrates the separate vortices induced by the upstream wake.The vortices are moving to the pressure side of the next blade.f * = 12 mode shows the swirling and swing of the passage vortices.On this plane, f * = 1.67 is the leading energetic mode, which represents the important unsteady flow structure.It is found that the modal structures also appear below the separate vortices.Moreover, the rotation motion is found in the modal contour.
On A 95% plane, f * = 6 mode also illustrate the development of the separate vortices, which is swirling in the passage (solid arrow).On this plane, f * = 1.67 mode is the dominant unsteady structure.Although the scale of the passage vortices is very large, they are still below the separation position of the boundary layer.Moreover, the swiping motion is found (solid arrow).

Conclusion
The unsteady flow in a water-jet propulsion mixed-flow pump is studied in this paper.CFD is used to obtained the hydrodynamic excitation of the pump and the instantaneous flow field.The snapshots are extracted from the time-varying flow field of vorticity magnitude.DMD is then used to decompose the unsteady flow field.Several conclusions are drawn as follows: 1) Dominant component of f * = 1.67 is found in the spectrum results of time-varying pump efficiency, axial force and the fluctuate pressure in the guide vane region.Lateral and radial pressure gradient are found in the instantaneous flow field of the guide vane passage.Moreover, complex vortex structures appear in the passage.
2) The DMD results illustrates that the dominant unsteady flow structures are the rotor-stator interaction and the passage vortices.The frequencies of rotor-stator interaction are the blade passing frequency and its harmonics.The dominant frequency of the large-scale passage vortices is f * = 1.67.These flow structures have great effect on the hydrodynamic characteristics.
3) The flow structures of rotor-stator interaction are mainly represented as the upstream sweeping wake and the separation vortices.Upstream wake caused the fluctuation in the boundary layer flow and induces flow separation.The separation vortices moving in lateral and axial direction due to the pressure gradient.Behind the separation position, large-scale passage vortices are induced.

Figure 1 .
Figure 1. the impeller and the guide vane of the computation model.

Figure 4 .
Figure 4. spectral analysis of the time-varying pump efficiency.

Figure 5 .
Figure 5. spectral analysis of the time-varying axial force.

Figure 5
Figure5shows the spectral result of the unsteady axial force of the impeller.Similarly with the pump efficiency, components of f * = 1.67 and f * = 11 have relatively high amplitude.In order to study the unsteady feature of the pump excitation, set 2 monitors of fluctuate pressure are set in the pump.One is inside the tip clearance tip, the other is at the passage of guide vane, illustrated as Figure1.The spectral results of the fluctuate pressure is shown on Figure6.Due to the periodically rotating blades, the frequencies of the dominant components at point 1 are f * = 6 and its harmonics.However, the components of f * = 1.67 can also be found.At point 2, component of f * = 1.67 has the highest amplitude.Hence, the component of f * = 1.67 induced by some vortex structures in guide vane region have great effect on the unsteady forces and hydrodynamic excitation of the guide vane mixedflow pump.The following study captures these vortices and its feature.

Figure 6 .
Figure 6.spectral analysis of the fluctuate pressure at 2 monitor points (a.monitor point 1, b. monitor point 2).

Figure 7 .
Figure 7. local planes in the guide vane.