Influence of Various Stator Parameters on the Open-Water Performance of Pump-Jet Propulsion

Abstract

In order to improve the hydrodynamic performance of pump-jet propulsion (PJP) when matching stator with the rotor, the RANS method with SST k-ω turbulence model is employed to study the influence of six kinds of stator parameters, which are classified into three groups, i.e., stator solidity, stator angles and rotor–stator spacing (S). Results show that the stator solidity involves the blade number (Ns) and chord length (L), has an obvious acceleration effect at and after stator, and produces a higher thrust and torque with a slight efficiency change. Further comparing Ns and L results, we find greater distinctions between the two cases when stator solidity is greatly adjusted. Three stator angles, i.e., stagger angle (α), lean angle (γ), and sweep angle (β), are studied. The α has the biggest effect on the thrust, torque, and efficiency; meanwhile, it shifts the advance number that corresponds to maximum efficiency. The effect of γ is similar to α, but its influence is far less than α. However, there is little difference between various β cases except for off-design conditions, where the efficiency drops dramatically as β increases. The S has a slight effect on PJP performance. Even though S decreases 34% relative to the original PJP, the rotor thrust and torque increase by less than 1%. In addition, we compare torque balance locations under various parameters, and each component force is analyzed in detail to explain the reason for performance variation. The present work is conducive to future optimization in PJP design.

1. Introduction

The pump-jet propulsion (PJP), which consists of a stator (a stationary blade row), a rotor (a rotating blade row), and a duct, is a special type of underwater thruster. PJP has been widely used for various underwater vehicles. According to the position of the stator, there are two types of PJP: pre-swirl PJP and post-swirl PJP. The submarines and some offshore vessels generally adopt the former for noise reduction [1]. In contrast, underwater vehicles such as torpedoes typically employ post-swirl PJP, for the post-stator can recover part of the wake energy that is lost with conventional propellers. Hence, the stator generates more thrust and improves the open-water efficiency of PJP.

In earlier years, PJP research mainly focused on the design method. Mccormick et al. [2] are the first academics who discussed the design of a post-stator PJP. They performed the numerical evaluation of the efficiency and cavitation of PJP. The shroud and propeller were replaced by an equivalent system of ring source and sinks, ring vortices, and vortex filaments. Henderson et al. [3] introduced a method for the design of PJP. A quasi-one-dimensional method of blade design was described that used compressor data. Meanwhile, the arrangement of shroud and vanes is discussed. Furuya et al. [4] presented a report describing the blade-to-blade design with corrections due to the three-dimensional approach.

There is some literature about hydrodynamic research on PJP by experimental and numerical methods. In the experimental study, Zierke et al. [5] are earlier researchers who performed various test methods in the underwater turbomachine with stator. They conducted experimental measurements of a high Reynolds number axial pump in the Applied Research Laboratory (ARL) at Penn State. Flow visualization and velocity measurement by laser doppler velocimeter (LDV) were carried out. In addition, both slow-response and fast-response pressure probes were used to test pressure fluctuation. Suryanarayana et al. [6,7,8] conducted a series of experiments of PJP on an axisymmetric underwater body. The investigation of PJP was tested at a wind tunnel, which focuses on the axial velocity and tangential velocity, the self-propulsion point, thrust magnitude, and torque balance. In addition, experiments at the cavitation tunnel showed that the inception position took place at the rotor tip at a high advance ratio and the leading edge of the suction side at a low advance ratio. However, the duct and stator were free from cavitation. Shirazi et al. [9] also investigated a post-swirl PJP on a full-scale underwater vehicle experimentally and numerically. The experiments include the bollard pull, self-propulsion, and bare hull resistances were conducted to validate the design procedure and verify the numerical results. Further conditions in which PJP works were numerically studied.

In numerical simulation, many academics utilized the RANS method with various turbulence to study PJP; Lee et al. [10] investigated an IGV/rotor propulsion pump using a standard k-ε turbulence model with a low-Reynolds-number approach near the wall. Results such as blade loadings and wake vortex patterns are compared with Zierke’s [5] experimental results. The authors of [11,12,13] also used the k-ε model to investigate the hydrodynamic performance of PJP in torpedo propulsion. Park et al. [14,15] presented a numerical simulation of the waterjet propulsion system and single-stage ducted marine propeller, respectively; both of which solved RANS equations with the k-ε model. Huyer [16] also employed the k-ε model to study the lateral force of a post-swirl propulsor under a controllable pitch stator; both ducted configuration and open case were researched. Ahn [17] studied a pump-jet with and without a ring by solving the Spalart–Allmaras equation. The ring cases showed lower vortex strength at the tip. The tip clearance is a key factor for the shrouded rotor and has been a research hotspot for PJP. The authors of [18,19,20,21] numerically investigated the influence of tip clearance on post-stator PJP. Results showed that efficiency is closely related to the size of clearance due to the energy loss generated by tip leakage vortex (TLV). The larger the tip clearance size, the lower efficiency of PJP, while the efficiency remains unchanged when it increases to a certain value. The tip vortices, including tip separation vortex (TSV) caused by separation at the leading edge of the rotor and TLV, are also influenced by clearance. As the tip clearance size increases, the TSV spreads toward the suction side. Decreasing the advance ratio (increasing rotation speed or decreasing the inlet velocity) will reinforce the vortices and make the cavitation phenomenon more possible to happen.

Reviewing the previous literature, most researchers focus on the hydrodynamics of the rotor and flow details around the rotor, especially the tip clearance between the rotor and duct inner surface. Little research has been conducted on pre-swirl stator parameters except for [22], which mainly investigated the rotor fluctuation but ignored some important results of PJP components, such as the stator force, stator torque, and duct force. From Yu’s work, it is obvious that the pre-whirl stator has a significant impact on PJP global performance. The fundamental reason is that the hydrofoil performance [23] of rotor blade is influenced by the inflow condition caused by various stator parameters. The comprehensive study of stator parameters is worthwhile for future PJP design. In this paper, six kinds of stator parameters, organized into three groups, are thoroughly studied based on the RANS method. Apart from the hydrodynamics of all components of PJP, we emphasize the flow field between stator and rotor. The layout is as follows:

Section 2 presents the governing equations and the details of numerical discretization method. In Section 3, the geometry of prototype PJP and the definition of stator parameters are given. The computational domain and numerical setup are described in detail. Then, we perform the validation of the numerical method. Section 4 presents the detailed hydrodynamic results of PJP with various parameters. Finally, Section 5 gives the summary of key results.

2. Numerical Simulation Methods

PJP works in a uniform flow, and the fluid is water at a constant temperature, and thus heat transfer effect is neglected. Due to an extremely low Mach number, the water is treated as an incompressible fluid. In the present work, the steady simulation is adopted to investigate the open-water performance of PJP under the different stator parameters. Reynolds Averaged Navier–Stokes (RANS) equations include mass conservation equation and momentum conservation equation are:

$$\frac{\partial U_i}{\partial x_i} = 0$$

(1)

$$\rho {\overline{u}}_j\frac{\partial {\overline{u}}_i}{\partial x_j} = \rho {\overline{F}}_i - \frac{\partial \overline{p}}{\partial x_i} + \frac{\partial }{\partial x_j}\left(\mu \frac{\partial {\overline{u}}_i}{\partial x_j} - \rho \overline{u_i^{\prime }{u}_j^{\prime }}\right)$$

(2)

In the equations, ρ is water density, x_i and x_j are cartesian coordinate components (i = 1, 2, 3; j = 1, 2, 3), and F represents the mass force of an element, such as gravity. For the present work, the F equals to zero. μ and p are the dynamic viscosity and pressure, respectively. u_i and u_j are the absolute velocity component. $\rho \overline{u_i^{\prime }{u}_j^{\prime }}$ is the Reynold stress. ${\overline{u}}_i$ and ${\overline{u}}_j$ represent the time-averaged item, the turbulence model is needed for the closure of Equation (2).

The SST k-ω turbulence model is a two-equation eddy-viscosity model proposed by Menter [24], which has become widely used in flow simulation of underwater vehicles and PJP [25,26,27,28,29,30,31]. The shear stress transport (SST) formulation combines the best of k-ω model and k-ε model, is able to investigate a model of low Reynolds number without using the wall function method. In the free-stream far away from the wall, the SST can freely switch to the k-ε formulation and thus avoids the common problem exists in k-ω formulation that the model is too sensitive to the inlet free-stream turbulence conditions. Meanwhile, the SST has a better behavior in adverse pressure gradients and separating flow. Qin [19] carried out numerical simulations of PJP with three different turbulence models, namely standard k-ε, realizable k-ε, and SST k-ω turbulence models. The results showed that the SST configuration is much more accurate. Therefore, the SST k-ω formulation is chosen to study the performance and flow characteristics of PJP. The turbulent kinetic energy k and specific dissipation rate ω formulations can be written as follows:

$$U_j\frac{\partial k}{\partial x_j} = P_k - {\beta }^{*}k\omega + \frac{\partial }{\partial x_j}\left[\left(\nu + {\sigma }_k{\nu }_T\right)\frac{\partial k}{\partial x_j}\right]$$

(3)

$$U_j\frac{\partial \omega }{\partial x_j} = \alpha {S_0}^2 - \beta {\omega }^2 + \frac{\partial }{\partial x_j}\left[\left(\nu + {\sigma }_{\omega }{\nu }_T\right)\frac{\partial \omega }{\partial x_j}\right] + 2\left(1 - F_1\right){\sigma }_{\omega 2}\frac{1}{\omega }\frac{\partial k}{\partial x_i}\frac{\partial \omega }{\partial x_i}$$

(4)

where the P_k is the production rate of turbulence used to prevent the built-up of stagnation regions:

$$P_k = min\left({\tau }_{ij}\frac{\partial U_i}{\partial x_j}, 10{\beta }^{*}k\omega \right) with {\tau }_{ij} = {\nu }_T\left(\frac{\partial U_i}{\partial x_j} + \frac{\partial U_j}{\partial x_i}\right)$$

To properly predict the onset and amount of flow separation from the surface, the eddy-viscosity ν_T should be limited, where $v_T = a_{1}k/max\left(a_1\omega , S{F}_2\right)$. Notice that $v_T = {\mu }_T/\rho$. S₀ is an invariant measure of strain rate and F₂ is a blending function similar to F₁. Specific formulations and constant values can be found in reference [32].

In the present work, the Ansys/CFX solver is employed to numerically investigate the global performance of PJP. The solver uses the element-based finite volume method. The governing equations are reorganized in the volume integral and surface integral forms and then discretized with each element. The advection scheme for four equations from (1) to (4) all adopt a high-resolution scheme to ensure a more accurate result and good robustness.

3. Numerical Simulation of PJP

3.1. Model Geometry

Figure 1 shows the typical model-scaled PJP. The pre-swirl stator with eight blades is fixed on the conical hub, which is the extension of the submarine tail and has the same conical degree. The six-post rotor blades are installed on a cylindrical hub, with a tip clearance equal to 1 mm. In this paper, the rotor has a diameter of D_r = 146 mm. The area ratio and hub diameter ratio of the rotor are 0.8 and 0.3, respectively. Both the profile curves of stator hub and shroud are not parallel to the flow direction, thus resulting in unequal diameters of stator at upstream and downstream, which are D_sin = 101.8 mm and D_sout = 91.6 mm, respectively. The duct length is Ld = 177.8 mm, and the diameters of inlet and outlet are D_in = 210.4 mm and D_out = 146.0 mm, respectively.

At present, we classify six stator parameters into three groups, i.e., stator solidity, stator angle, and rotor stator spacing (S). The first includes two parameters, the blade numbers (N_s) and chord length (L). The second contains three angles, as shown in Figure 2.

In Figure 2 view1, i.e., the circumferential view, the original foil and the modified foil at 0.7 span are presented. We adjust the stagger angle (α) of the original model with a step of 4°. However, the blade height (h) increases to h₁ when decreasing α. In order to exclude the influence of blade height on performance of PJP, the chord length (C) is scaled in a proper ratio. The scaled ratio in Figure 2 should be h₁/h.

In Figure 2 view2, i.e., the meridional view, a swept blade with a sweep angle β = 4° in dashed line is shown.

In Figure 2 view3, i.e., the axial view, the solid line, and dashed line are the leading edge of the stator. The lean angle (γ) with a positive value means that the blade leans to the pressure side. Accordingly, negative γ denotes blade leans to the suction side.

The third parameter S is the axial distance between the trailing edge of the stator and the leading edge of the rotor at the root of the blade. It will be modified by translating the stator blade along the axial direction with a step 5 mm and treated as a dimensionless value normalized by D_r. The S for the original PJP is 0.35 D_r.

In the present work, five sets of cases for each parameter will be studied. The computational cases are listed in

Table 1. Computational cases for each stator parameters.

Case ID	Stator Parameters
	Solidity		Angle			Rotor–Stator Spacing
	Ns	L	α	β	γ	S
1	6	6/8C	−8°	−8°	−4°	0.35Dr (origin)
2	7	7/8C	−4°	−4°	0° (origin)	0.32Dr
3	8 (origin)	1.0C (origin)	0° (origin)	0° (origin)	4°	0.29Dr
4	9	9/8C	4°	4°	8°	0.26Dr
5	10	10/8C	8°	8°	12°	0.23Dr

.

3.2. Mesh and Numerical Setup

The entire computational domain is divided into three subdomains: external domain, stator domain, and rotor domain. The external domain is shown in Figure 3. PJP works in a cylindrical tunnel with a length of 15D_r, and a radius of 5D_r. The inlet is located 5D_r upstream of the rotor, and the outlet is located 10D_r downstream of the rotor. Both the stator and rotor domains are generated by a single periodic domain that contains one blade.

Figure 4 shows the coarse grid, the medium grid, and the fine grid of the single domain of rotor and stator.

Table 2. Mesh number.

Mesh Groups	ID	Stator Domain (M)	Rotor Domain (M)	External Domain (M)	Total (M)
fine	1	5.13	6.07	3.73	14.92
medium	2	1.35	1.45	2.69	5.49
coarse	3	0.52	0.61	0.87	2.01

lists the number of meshes. The relation between y⁺ and the width of the first cell to the wall y is driven by the Blasius equation, expressed as follows:

$$y = 6{\left(\frac{V_{ref}}{\nu }\right)}^{-\frac{7}{8}}{\left(\frac{L_{ref}}{2}\right)}^{\frac{1}{8}}{y}^{+}$$

(5)

where the V_ref is reference velocity, defined as $\sqrt{V_A^2 + {\left(\pi n{D}_r\right)}^2}$ (V_A denotes the inflow velocity, n denotes the rotation speed). The L_ref is reference length, corresponding to the chord length of rotor blade at r/R = 0.75. According to Equation (5), the height of first layers, under the typical operating condition of V_A = 2.6 m/s and n = 20 r/s, is 2 × 10⁻⁶ m for all solid surface except the tunnel wall.

The boundary conditions can be seen in Figure 3. A uniform inflow velocity V_A is added at the inlet boundary, and the turbulence intensity is set to 1%. For outlet boundary, an average static pressure of zero is imposed to simulate the far-field. The solid surface of all PJP components is consider to be the no-slip wall. The outer tunnel wall is set to a free-slip wall. In addition, three subdomains need to be connected with the interface boundary. There are three types of interfaces provided by CFX for frame change models, the mixing plane, frozen rotor, and transient rotor–stator methods. Among them, the frozen rotor is the most useful way to produce a steady-state solution for the multi-frame problem, and is adopted in this work. The external domain and stator domain are set to stationary. The rotor domain is set to rotate around the negative z-axis with a constant speed of 20 r/s. The alternate rotation model, in which the absolute frame velocity is advected instead of relative frame velocity, is chosen for a significant reduction in numerical error.

3.3. Verification of Grid

In order to assess the performance of PJP, the dimensionless hydrodynamic coefficients are defined as follows.

$$J = \frac{V_A}{n{D}_r}, K_{Tr} = \frac{T_r}{\rho n^2{D}_r^4}, K_{Ts} = \frac{T_s}{\rho n^2{D}_r^4}, K_{Td} = \frac{T_d}{\rho n^2{D}_r^4}, K_T = \frac{T_r+T_s + T_d}{\rho n^2{D}_r^4} = K_{Tr} + K_{Ts} + K_{Td}$$

$$K_{Qr} = \frac{Q_r}{\rho n^2{D}_r^5}, K_{Qs} = \frac{Q_s}{\rho n^2{D}_r^5}, \eta = \frac{T{V}_A}{2\pi n{Q}_r} = \frac{\left(T_r + T_s + T_d\right)V_A}{2\pi n{Q}_r} = \frac{J}{2\pi }\frac{K_T}{K_{Q_r}}$$

(6)

where J is the advance ratio. T is the thrust, with the subscript of r, s, d denote stator, rotor and duct. Q represents torque, Q_r and Q_s are the torque of rotor and stator, respectively. K_Tr, K_Ts and K_Td are the thrust coefficient of the rotor, stator, and duct, respectively. K_Qr and K_Qs are the torque coefficient of the rotor and stator, respectively. η is the open-water efficiency. During calculation, the rotor speed is fixed at 20 r/s. By adjusting the V_A, we can acquire the PJP’s performance at various J.

This section mainly discusses the results of the K_T and K_Qr to verify the grid. The formulations will be illustrated by taking the K_T as example. The convergency ratio of thrust coefficient R(K_T) is:

$$R(K_T) = \frac{S_2(K_T) - S_1(K_T)}{S_3(K_T) - S_2(K_T)}$$

(7)

where the ‘S’ with subscript ‘1’, ‘2’, ‘3’ represent three different configurations with fine, medium, and coarse meshes. According to the results of R(K_T), three conditions are obtained as follows:

(1)

0 < R(K_T) < 1: Monotonic convergence.

(2)

−1 < R(K_T) < 0: Oscillatory convergence.

(3)

|R(K_T)| > 1: Divergence.

The uncertainty can be calculated by a correction of factor [33,34] or factor of safety [35]. The present work uses the factor of safety, i.e., value of 1.25 to calculate the uncertainty, which has an alternate definition: the Grid Convergency Index (GCI). The GCI for a fine mesh is defined as:

$$GC{I}_{Fine}^{21}(K_T) = \frac{1.25{\delta }_{21}^a(K_T)}{r^{p_l(K_T)}-1} = U(K_T)$$

(8)

The following formulations are used to calculated the $GC{I}_{Fine}^{21}(K_T)$,

$$\begin{array}{l}p(K_T) = \frac{1}{ln\left(r_i\right)}\left[ln\left(\frac{S_3(K_T) - S_2(K_T)}{S_2(K_T) - S_1(K_T)}\right) + q(p(K_T))\right]\\ q(p(K_T)) = ln\left(\frac{r_{21}^{p_l(K_T)} - s(K_T)}{r_{32}^{p_l(K_T)} - s(K_T)}\right)\\ s(K_T) = 1*\mathrm{sign}\left(\frac{S_3(K_T)-S_2(K_T)}{S_2(K_T) - S_1(K_T)}\right)\\ {\delta }_{21}^a = \left|\frac{S_2(K_T) - S_1(K_T)}{S_1(K_T)}\right|\end{array}$$

(9)

where the order of accuracy p(K_T) is generally calculated by the fixed-point iteration method, but the refinement ratio r_i is set to a constant value of 1.4 in this work, which means the q(p(K_T)) is zero. Meanwhile, we can use the K_T and r_i to solve p(K_T). Following the above steps, we can also obtain the CGI of K_Qr.

Table 3. Uncertainty analysis under different advance coefficients.

	J = 0.2		J = 0.4		J = 0.6
	ξ = KT	ξ = 10KQr	ξ = KT	ξ = 10KQr	ξ = KT	ξ = 10KQr
S1(ξ)	0.6306	0.9818	0.6306	0.9818	0.6306	0.9818
S2(ξ)	0.6284	0.9655	0.6284	0.9655	0.6284	0.9655
S3(ξ)	0.6218	0.9403	0.5975	0.9152	0.6218	0.9403
R(ξ)	0.3447	0.6445	0.6579	0.8444	0.3493	0.7900
p(ξ)	3.1656	1.3054	1.2443	0.5025	3.1259	0.7005
${\delta }_{21}^a\left(\xi \right)$	0.0036	0.0166	0.0058	0.0258	0.0013	0.0182
$GC{I}_{Fine}^{21}\left(\xi \right)$	0.0024	0.0376	0.0140	0.1751	0.0009	0.0857
	J = 0.8		J = 1.0		J = 1.2
	ξ= KT	ξ= 10KQR	ξ= KT	ξ= 10KQR	ξ= KT	ξ = 10KQR
S1(ξ)	0.5274	0.8669	0.4638	0.8062	0.3755	0.6867
S2(ξ)	0.5263	0.8513	0.4632	0.7867	0.3751	0.6723
S3(ξ)	0.5246	0.8288	0.4625	0.7622	0.3745	0.6512
R(ξ)	0.6683	0.6906	0.7390	0.8008	0.5705	0.6818
p(ξ)	1.1976	1.1003	0.8990	0.6603	1.6678	1.1386
${\delta }_{21}^a\left(\xi \right)$	0.0021	0.0180	0.0012	0.0243	0.0010	0.0209
$GC{I}_{Fine}^{21}\left(\xi \right)$	0.0054	0.0501	0.0041	0.1220	0.0017	0.0561

lists the verification results under different J. The convergency ratio of K_T and K_Qr are all located between 0 to 1, illustrating that the mesh groups are monotonic convergence. Although the GCI of K_T and K_Qr are discrepant at different J, all of them are below 1%, demonstrating that the refinement of grid does not cause a significant change of hydrodynamic coefficients. In order to get a more detailed flow field, the fine mesh is adopted for a further study of the effect of the stator on the PJP’s performance.

Figure 5 presents PJP performance curves with fine mesh. The curves of K_T, K_Tr and η are plotted in solid line. The dash lines are applied to other thrust coefficients and torque coefficients of PJP components. In addition, the experiment data of K_Tr and K_Qr [36] are given in the style of the symbol. The results show that the changing trend of K_T, K_Qr, and η is similar to conventional propellers. The difference is that η for PJP has a relatively large value in a wide range of J. The maximum η location is approximately 0.9 of J, where the η reach up to approximately 0.6. Furthermore, the duct generates additional effective thrust at low J, but produces resistance at high J; the turning point of J after which the duct never produce effective thrust is approximately 0.7. In contrast, the stator always provides drag force.

Comparing the K_Tr and K_Qr obtained by numerical simulation with the experimental data, it is evident that the numerical results at the whole range of J are relatively smaller than experiment data, especially at large J. The maximum error of K_TR and K_QR appears at J = 1.2 is 8%; other errors range 4%~7%. We think this relatively high error should be attributed to inherent numerical analysis, plus an unconformity with experimental environments, such as attachment used for fixing PJP components or relatively small flow tunnel may have an undesirable influence. However, the trend of numerical results corresponds well to experimental data. The numerical method is acceptable to get a further study on PJP. In addition, the torque directions of the stator and rotor are opposite to each other. The larger difference in rotor torque and stator torque may generate a high overturning moment for the submarine, affecting its maneuverability. Therefore, we define the torque difference ΔK_Q as (K_QR − K_QS)/K_QS × 100 for a quantitative evaluation. Figure 5b exhibits the ΔK_Q at different J. It is clear that the absolute value of ΔK_Q reduces first and then increase with the increase in J. The optimum condition is where the ΔK_Q equals zero, which means that the submarine does not need to produce extra force to balance the overturning moment. The balance condition of torque for the original PJP locates at slightly higher than 0.9 of J.

Figure 6 shows the contour of wall y⁺ at J = 0.8. The rotor exhibits a high y⁺ compared with the stator and duct. Meanwhile, the y⁺ at the leading edges of the stator, rotor, and duct is relatively higher than at other regions. The mean y⁺ at the stator and duct are 0.3 but 0.8 at the rotor, satisfying the turbulence model demand and is appropriate for high Reynolds number flow.

4. Flow Results of Different Stator Parameters

4.1. Influence of Stator Solidity

The stator solidity, determined by N_s and L, is generally one of the first key factors in the design procedure. The stator solidity increases proportionally to the increase in N_s or L. In order to compare the effect of these two parameters on the PJP’s performance, the range and interval of stator solidity of N_s cases and L cases are kept the same.

A.

Stator number

The hydrodynamic performance of PJP at J = 0.2~1.2 with five sets of stator numbers is presented in Figure 7, where the case of N_s = 8 represents the original case. As shown in Figure 7a, the thrust and torque of PJP increase linearly with the increase in N_s. The difference in total thrust and torque at low J between two adjacent N_s cases ranges 4% ~ 5%. The efficiency of PJP slightly changes or even presents a negligible change with the increase in N_s, and the maximum efficiency locates at J = 0.9. However, after the 0.9 of J, the efficiency reduces dramatically with the decrease in N_s, especially at J = 1.2. For instance, the efficiency of N_s = 6 at J = 1.2 is reduced by 16% relative to the original case. The thrust coefficients of PJP, which consist of K_Tr, K_Ts, K_Td, are plotted in histogram form in Figure 7b. It is clearly seen that the larger the N_s, the more resistance the stator creates. For each additional stator blade, the resistance increases by 2.5 N at J = 0.8, or causes an increase in K_Ts by 15%~20%. For K_Tr and K_Td, there is an approximate 5% and 2% increase, respectively. Figure 7c exhibits torque difference at the whole operating conditions. Generally, the change rules of torque and thrust with J are similar. That is, both K_Qr and K_Qs increase with the increase in N_s. However, the different increased rate of K_Qr and K_Qs with N_s eventually shifts the torque balance position. As N_s increase, the balance condition moves toward lower J. Specifically, the torque balance positions for N_s = 6~10 are located at 1.1, 1.0, 0.9, 0.8, and 0.7, respectively.

Figure 8 presents pressure contour at the plane of x = 0 and velocity contour at four different planes under J = 0.8 (principal analysis condition in later). The four planes are separately located upstream of the stator, in the middle of the stator, downstream of the stator, and upstream of the rotor. Note that the plane of A₄ located in the rotor domain, the variable ‘velocity in stn frame’, which denotes the absolute velocity, is plotted to avoid all-one-color at A₄ or other planes due to excessive velocity difference between stator domain and rotor domain. In addition, both the pressure and velocity are dimensionless quantities normalized by 0.5ρV_A² and V_A, respectively. The contour of C_P shows some local-low pressure regions which correspond to free vortices in the wake of the rotor. Results indicate that the local low-pressure position and magnitude are essentially unchanged as N_s varies. The velocity contours at A₃ and A₄ show some low-speed zones, and the shape of which get narrow from the hub to the shroud. The radial-going trail of the low-speed zone at A₃ resembles the form of the trailing edge of the stator.

Comparing the velocity contour at the surface of A₁~A₄, there is a significant acceleration effect for flow velocity. It is worth noting that flow velocity gradually becomes more uniform between two adjacent stator blades with N_s increase. Meanwhile, the velocity magnitude at A₃ is gradually increased from V_A to 1.2 V_A as N_s increases from 6 to 10. Since the larger N_s, the narrower flow passage between two adjacent stator blades and the more obvious acceleration effect it performs. The increased velocity of incidence flow for the rotor eventually causes an increase in rotor thrust. Since the increased amplitude of rotor thrust is greater than the resistance of stator, the total thrust of PJP keeps growing even though more drag is produced by added stator blade.

B.

Chord length

It is necessary to investigate the effect of chord length on the PJP’s performance before studying other paraments. Because when changing other paraments, such as stagger angle or sweep angle, the chord at each stator profile will vary inevitably. In this section, the chord is modified on the basis of N_s studied before. The chord of the original PJP is taken as a reference to other cases with extended foil or shrunken foil, namely L = 6/8C, 7/8C, 9/8C, and 10/8C. Figure 9 presents the performance of PJP with different chord length. In order to get a distinct contrast with the N_s effect, the ordinate scales are the same as that of Figure 7. According to the figures, the trend of all coefficients with L and with N_s is similar. The increase in K_T and K_QR range 3%~5% as L increase. The efficiency basically stays the same value at lower J but drops quickly at high J. Further, both K_Ts and K_Tr grow with the increase in L as rapidly as with the increase in N_s, and the K_Td also become larger as L increases when the duct produces thrust but become smaller when it generates drag. As for the torque difference, there are distinct differences of balance position between N_s configurations and L configurations under the same stator solidity. For L = 6/8C and 10/8C, the balance condition of J is slightly smaller than 1.1 and slightly higher than 0.7, respectively. However, for L = 7/8C and 9/8C, the balance conditions are the same as with the corresponding N_s cases (N_s = 7 and N_s = 9), i.e., J = 0.8 and J = 1.0, respectively.

To further figure out the difference in the effect of N_s and L on the PJP’s open-water performance, each hydro-coefficient difference between two sets under the same stator solidity is plotted in Figure 10. We define the difference as $\Delta \phi = 100\cdot ({\phi }_1-{\phi }_2)/{\phi }_2$, among which the φ represents hydrodynamic coefficients such as K_Tr or K_Ts, with the subscript ‘1′ and ‘2′ separately denote the coefficients of L and N_s configuration. The labels from ‘conf.1′ to ‘conf.4′ at the horizontal axis represent the cases of N_s = 6, 7, 9, 10, and each of them corresponds to L = 6/8C, 7/8C, 9/8C, 10/8C. Note that the conf.2 and conf.3 are reduced and increased stator solidity, respectively, based on the original PJP cases with a same magnitude, so are the conf.1 and conf.4.

As is shown in the figure, the difference in all coefficients under conf.1 is distinctly larger than conf.2. Take the K_Tr as an example, the Δφ for K_Tr is 3.16% under conf.1, but reduces to 1.03% under conf.2; similarly, the Δφ under the conf.4 is larger than that under the conf.3 except for the K_Qs, which illustrates the higher extent of stator solidity modified, the more distinct difference between N_s effect and L effect on the PJP’s performance even under the same stator solidity. Furthermore, all the signs of coefficients for increased stator solidity are opposite to those for reduced except for K_Qs. For the increased stator solidity cases, i.e., the conf.3 and conf.4, the Δφ are positive except for K_Td, which indicates the effect of L is more significant than N_s for increasing the PJP’s thrust or efficiency, and the higher of stator solidity, the more obvious this effect. However, for the reduced stator solidity cases, i.e., the conf.1 and conf.2, the Δφ is negative except for K_Td and K_Qs, which indicates the reduction in the N_s is more likely to deteriorate the PJP’s performance compared with the decrease in L. Based on the above analyses, we can draw a conclusion that it is more obvious to modify L for the improvement of PJP’s performance rather than adjust N_s.

To further investigate the reason for the L effect on the PJP’s performance, it is essential to study the flow field at and downstream of the stator. Figure 11 presents the C_P contour at the suction side and pressure side of the stator blade. The limiting streamlines are also exhibited. The label ‘H’ at ordinate represents the maximum span length in x-axis.

The contour of the pressure side presents a decrease in pressure from top left to bottom right, with streamlines parallel to the conic surface of the hub and shroud. It seems that the chord length L has little impact on streamline direction. The corner between the trailing edge and hub exhibits a low-pressure region that gradually enlarged with the increase in L. The results of the suction side surface show a local high-pressure region (axial direction: 0~0.3L, radial direction: 0.8~1H) in the corner between the shroud and leading-edge, which may be influenced by high stagnation pressure of duct inlet. The separation of the boundary layer occurs near the trailing edge. As L increases, the separation position moves downstream, namely from 0.8L to 0.9L. At the same time, the streamlines on the root of the suction side present radial-going flow and concentrate near the trailing edge. Furthermore, a small vortex is obviously presented close to the shroud and trailing edge. Due to the enormous pressure difference between the pressure side and suction side, together with a low-energy at the stator boundary layer, a secondary crossflow occurs, and the flow direction points to the hub.

Figure 12 presents the velocity distribution at the slice of A₃ and A₄. The figure shows narrow low-velocity regions along circumferential direction periodically. The periodicity of the low-speed zones at A₃ along circumference direction is consistent with N_s, i.e., eight; this is because of the occurrence of separation at stator trailing edge and its prolonged effect on downstream. However, the influence on location at A₄ by stator wake is not as strong as at A₃. Thus, the periodicity turns into six (the number of rotor blades), with some small strip-shaped and round-shaped low-speed regions existing due to the slight impact of stator wake. The results also show that a slight grow of velocity magnitude with the increase in L.

In order to investigate the flow details at A₃ and A₄, the velocity V and velocity components: axial velocity V_z, circumferential velocity V_c, radial velocity V_r along the circumferential direction at three different spans are presented in Figure 13. The circumferential angle at the horizontal axis is described in Figure 12. For the V_r, the direction from hub to shroud denotes positive; the opposite direction is negative.

The velocity components change periodically except at 0.3span of A₄. The periodicity at A₃ is eight but six at A₄ except for V_r. However, the V_r at A₄ has the same periodicity as at A₃, which indicates V_r is deeply influenced by the stator. The stator effect is also observed at 0.3span of A₄, since there are induced peaks by stator blade for V_z and V_c, thus causing an uncertain periodicity. The proportions of the three velocity components vary greatly. Take the 0.3span of A₃ as an example, the V_z curves are especially close to V curves, meaning that the highest velocity proportion is V_z, the maximum value is above 0.9V_A. Followed by the V_c, the maximum value is approximately 0.4 V_A.

Comparing the velocity components with various L, the large L seems to accelerate flow. Both V_z and V_c get larger as L increases. That explains why the larger L, the closer the boundary layer separation position to the trailing edge of the stator. In contrast, the L has little effect on V_r. The V_r curves at A₃ and A₄ under for various L cases almost coincide with each other. The resultant velocity finally has a 1.5%~2% increase with the increase in chord length.

4.2. Influence of Stator Angles

The stator angles can be summarized as three parameters in cylindrical coordinate views as described previously: the stagger angle for the circumferential view, the lean angle for the axial view, and the sweep angle for the meridional view. Each parameter is modified with a same step (4°) to investigate and compare their effect.

A.

Stagger angle

The stator stagger angle α is an essential parameter for PJP performance, since it greatly influences the effective incidence angle of rotor, while the blade height will be changed when adjusting the angle directly. Therefore, the chord length of hydrofoil at each section needs to be scaled to maintain a constant blade height.

Table 4. The scaled factor of various stator blade.

Cases	#1	#2	#3	#4	#5	#6	#7	#8	#9	#10
α = −4°	0.9922	0.9903	0.9866	0.9833	0.9814	0.9804	0.9801	0.9808	0.9825	0.985
α = −8°	0.9894	0.9856	0.9782	0.9719	0.968	0.9661	0.9655	0.967	0.9702	0.9751
α = 4°	1.0128	1.0149	1.0188	1.0223	1.0245	1.0255	1.0259	1.025	1.0232	1.0205
α = 8°	1.0311	1.0353	1.0436	1.0508	1.0554	1.0577	1.0584	1.0566	1.0528	1.0471

lists the specific scaled factor of various angle cases of each stator profile.

Figure 14 shows the PJP’s performance with different α. In Figure 14a, as α increases, the K_T and K_Qr increase considerably. The α has a greatly obvious effect on the PJP’s performance compared with the previous parameters. The growth rate of K_T from −8° to −4° is 13.96% relative to the former. The K_T increases by 13.78% from −8° to 0°, followed by 12.68% and 12.96% for 0° to 4° and 4° to 8°, respectively. The growth rates of K_Qr are 11.58%, 12.16%, 12.09%, and 13.7%, respectively, for −8° to −4°, …, and 4° to 8°. The different growth rates of K_T and K_Qr eventually lead to a slight efficiency difference at J = 0.8, which is a turning point of the efficiency difference between various α cases. For J < 0.8, the η stays almost the same, but when J is greater or equal than 0.8, the η increase considerably as α increase. Further, the large α will cause the highest efficiency to move slightly toward high J.

As shown in Figure 14b, the K_Tr and K_Ts positively correlate with α. However, for the K_Td, it increases with the increase in α when the duct provides thrust but decreases when the duct generates drag. In general, K_Ts and K_Tr are the most affected coefficients with the rise of α. The average growth rates for KTs and K_Tr are approximately 50% and 13% at J = 0.8.

Figure 14c exhibits a relatively larger torque difference value compared with the configurations of the previous two parameters. The maximum ΔKQ locates at J = 0.2 exceeds 100% when α = −8°. As α increases, the torque balance position moves toward low J. For α = −8°~4°. The balance points locate separately at J = 1.2, 1.05, 0.9, 0.7, 0.4.

Figure 15 shows the C_p contour for various cases at three different spanwise surfaces. Each subfigure exhibits only half of the computational domain for saving space. Obviously, the α has a dramatic influence on the pressure distribution in both stator and rotor domains. The pressure at the suction side of the stator and rotor reduces with the increase in α, but there is little change at the pressure side. It can be inferred that the rise of α will produce a larger stagnation pressure at the leading edge of the stator and cause the separation on the suction side of the stator to occur in advance. Meanwhile, the velocity flow direction is changed along the stator section as α increases and eventually leads to a larger attack angle for hydrofoil at each rotor blade section, which is similar to the function of increasing the pitch angle of the rotor. The large pressure-reduced area near the suction side of the rotor not only produces thrust and torque but may affect the contiguous region around the stator. The drag force caused by the stator is much higher at the same time when α increases, but the increased resistance of the stator is insufficient to offset the increased thrust of the rotor. Further, the increased rate of stator torque is also far less than the rotor. Consequently, the large α contributes to the increase in PJP total thrust and causes a higher torque difference at usual working conditions, further making torque balance point move to higher J. In addition, a strong change of pressure at 0.95 span near the tip of the rotor blade is observed, which results in tip leakage flow and rotor wake.

B.

Lean angle

In this section, the lean angle γ, as one of the secondary parameters in PJP design, is modified with the same step of 4° as α does. A series of PJP performances with various γ are presented in Figure 16. Similar to other previous parameters, as γ increases, K_T, K_Qr, and η all increase in varying degrees. The increased rate of K_T and K_Qr between two adjacent γ cases distributes in the range of 2%~4% at J = 0.8. In contrast, the η has a slight change but no more than 1%. For the component thrust, the K_Tr and K_Ts also increase linearly at 3% and 11% with the increase in γ, respectively. However, for K_Td, the relative variation between two adjacent configurations does not exceed 3%. As for torque difference, the change in ΔK_Q with γ is opposite to K_T or K_Qr. The maximum variation of ΔK_Q is located at J = 0.2 for γ = −8°, and it is smaller than that for previous parameters that have been analyzed. Consequently, the torque balance condition of J for γ = −8° turns small, i.e., J = 1.0; For γ = 8°, it locates at J = 0.8. However, in other cases, the balance positions distribute between J = 0.8 and 1.0.

Further analysis on velocity distribution, as shown in Figure 17 between stator and rotor, is needed to better understand the reason for performance variation. As has been shown in Figure 12, there are six radial-going low-velocity-belts distributed evenly at constant Z plane due to stator wakes, with high-velocity between two adjacent belts. When γ increases, the velocity at the high-velocity region becomes slightly larger. The same phenomenon is obviously seen at 0.5 span, especially at the stator wake region. In Figure 18, the flow angle φ, defined as arctan (V_z/V_c), is presented at 0.5 span. It is found that φ decreases slightly with the increase in γ, especially at the region between stator and rotor. However, the decreased rate of flow angle is far less than the increased rate of velocity, thus causing an increase in K_T and K_Qr.

C.

Sweep angle

In previous parameters, the stator always maintains a constant value of blade height to avoid the possible influence on the PJP’s performance. In this section, we will investigate the global performance of PJP with various sweep angle β. Considering the mate dimensions of duct and stator, the lower bound of β is −4°, with a step length and upper bound are 4° and 12°, respectively. The open-water performance and thrust coefficient of all components are plotted in Figure 19a. The results show that the β has little effect on the performance of the rotor, with no more than a 1% difference in K_T, K_Qr, η between two different β situations without considering the off-design condition. The thrust of all components, including the stator, rotor, and duct, are plotted in Figure 19b. The K_Tr changes little. However, there are slight changes in K_Ts and K_Td. Take J = 0.8 as an instance, the K_Ts and K_Td separately decrease and increase with an average rate of 2% as a 4° step increase in β. Since both of K_Ts and K_Td are in the same order of magnitude, the resistance decreased on stator just offset the increased drag of duct, hence the thrust coefficient of PJP is essentially unchanged. For torque difference, a slight difference yields at low and high J compared with the original PJP, whereas the ΔK_Q stays almost the same at middle J, which results in a fixed torque balance position.

Figure 20 presents the C_P at a constant span of 0.5. There are intense pressure changes at the leading edge and trailing edge. However, the curves of various β situations coincide with each other well. The same phenomena exist at other profile foils, thus demonstrating that changing β has essentially no effect on the pressure of the rotor.

The pressure coefficient of the stator blade is presented in Figure 21. The local high pressure in the upstream shroud-corner region on the suction surface for β = −4° is much higher than for other cases. With the increase in β, namely the increase in the distance from the duct inlet to the high-span leading edge of the stator, the local high pressure gradually fades away. Finally, it forms a low-pressure belt at three sides, enclosing a much lower pressure region, which is obvious for high β. Thus, we can deduce that the pressure at the upstream shroud corner is deeply affected by the distance from the duct inlet to the leading edge of the stator. Compared with the negative β or lower β case, the higher β may benefit to structural performance due to a more uniform pressure distribution on the surface rather than a high local pressure that may result in stress concentration. Another noticeable phenomenon is that the concentrated point of streamlines near the hub and the trailing edge shifts slightly toward upstream when β increases. The vortex at the shroud corner of downstream is observed when β equals 8° and 12°, which will give rise to a large shear action on fluid tightly close to this area. Similarly, the CP of pressure surface at the upstream shroud-corner position is higher in the β equals −4° case than others, with the increase in β, the high-pressure area gradually reduced, thus causing the decrease in stator resistance.

Figure 22 shows the velocity components Vi obtained from 0.5 span at the cut plane of A₃ and A₄. The influence of β on Vi at A₃ is more obvious than at A₄. A tiny distinction between various β cases is observed, while at A₄, the curves almost coincide. As for the flow angle, the maximum value is generally smaller at A₃ than at A₄. Specifically, the value of the flat part does not exceed 68° at A₃, but it is larger than 70° when the flow enters the rotor domain. Since the absolute increase in V_z is larger than V_c, the flow gradually deflects towards the axis direction after passing through the stator. It is also found that φ curves are much more identical at A₄ than at A₃ except for some special flat parts at A₄, thus explaining the identical pressure distribution of the rotor blade section for various β cases, as shown in Figure 20.

4.3. Influence of Rotor–Stator Spacing

The stator-rotor spacing S is an important parament in the design of PJP. Since the rotor–stator interaction become intense when the S becomes small and causes a disastrous hydro-oscillation force on the rotor. In contrast, a big S will cause hydraulic loss that may have an adverse impact on the performance of PJP. According to some design manuals, the values of S typically shall not be less than 0.05D_r in axial-flow pump design, and the smallest S modified in this paper far satisfy the devising standard. In this section, the interface stays the same location. Only the stator blades are moved toward the rotor. Numerical simulation settings remain the same except that the number of nodes in the stator domain is properly increased and decreased before and after the stator.

Figure 23 presents the performance of different S situations, the K_Qr and η of PJP are slightly affected by S, while the K_T gives the same value for all cases. By further studying the force of each component, it is found that the variation of force for duct and stator between all cases is highly varied than that of rotor force. For example, when J = 0.8, although the increased rates of K_Ts and K_Td are separately approximately 2.5% and 1.5%, the K_Tr increases less than 1%. Eventually, the K_T is almost unchanged because of one magnitude order larger than K_Ts and K_Td that K_Tr owns. For torque difference, as J reduces, the variation of ΔK_Q gradually becomes larger. The maximum difference between the case of maximum S and minimum S (i.e., 0.35Dr and 0.23Dr) is less than 10%. The balance conditions for all S cases regularly distribute at approximately J = 0.9.

Due to the slight change of K_Tr, the pressure curves at 0.5 span rather than pressure contour are presented in Figure 24 to investigate the main reason for slight change of the performance coefficients. The figure shows that the C_p at the pressure side has no change with variation of S. However, the C_p at the suction side decreases slightly with the reduction in S, especially within the range of 0~0.5 chord of the rotor. Thus, it causes a slight increase in KTR when S becomes smaller.

Figure 25 presents the velocity component at 0.5 span. Due to the frozen interface between stator and rotor, the flow is interrupted when entering into the rotor domain, thus causing the length of wake to decrease accordingly with the decrease in S, which is non-physical. The rotor acceleration is obviously seen in the figure of V_z contour. The V_z increases to approximately 1.4 V_A at the channel and wake of the rotor. However, the V_c, plotted in the absolute coordinate reference system, is close to zero on the pressure side, so are the regions after the rotor. However, the high V_c exists on the suction side of the rotor and stator. The difference is that the high V_c locates at the trailing edge of the stator, but it locates at the leading edge of the rotor. As for V_r, there are negative values in a wide area, especially near the pressure side of the stator, which demonstrates that the flow has a trend towards the hub. Additionally, the velocities distribution becomes more complicated as S decreases due to the more intense interaction between stator and rotor. Further transient simulation is worthwhile for the study of the fluctuant pressure and the PJP’s oscillation force.

More specific velocity components along the circumferential direction of 0.5 span at A₃ and A₄ are plotted in Figure 26. It is evident that the lower S leads to a higher velocity at A₃. The maximum value of V_c increases tremendously with the decrease in S compared with V_z and V_r; however, the effect of S on the velocity at A₄ seems not as obvious as at A₃. The flow angle varies erratically at A₃, yet in most regions, it presents an increasing trend with the decrease in S; whereas at A₄, the change in flow angle becomes regular with S, i.e., increase slightly with S decreases, thus causing a slight increase in K_Tr.

5. Conclusions

We have numerically investigated the hydro-performance of PJP with different stator parameters and classified these parameters into three groups:

(A)

The blade number Ns, chord length L;

(B)

The stagger angle α, lean angle γ, sweep angle β;

(C)

The rotor–stator spacing S.

The SST k-ω turbulence model and frozen interface technique are employed to predict performance at various conditions. Based on RANS results, the verifications of mesh groups are performed to confirm appropriate mesh for further study. The results of the selected mesh give good accuracy with experiment results, which demonstrates the computation model is reasonable to predict the PJP’s performance. To better understand why PJP global performance varies, we mainly analyze the global performance and flow results under J = 0.8, where the efficiency of the original PJP is a little lower than maximum efficiency. The numerical results are summarized as follows:

The parameters related to stator solidity, i.e., Ns and L, can accelerate flow when increasing them, yield higher incidence velocity for the rotor, and increase thrust and torque. At the same time, the efficiencies at design conditions stay almost the same. In addition, there are some distinctions between the two parameters for effect on PJP performance. The higher degree of stator solidity varies from the original PJP, the greater variation of results for two parameters.

The stator angles defined in various views are modified with the same step value (4°) to investigate their influence. The results indicate that KT, KQr, and η are positively associated with α and γ. However, no significant change occurs when adjusting β except for the off-design condition. The α has a greatly higher influence on the PJP’s performance than the other two angles. The operating condition of J corresponding to maximum efficiency is shifted to a higher value when α increases. However, the γ only slightly affects the maximum efficiency, which increases with the increase in γ. In addition, we noted that a higher γ case gives a larger velocity and lower flow angle upstream of the rotor and eventually results in a rise of rotor thrust owing to a greatly higher change rate of velocity than flow angle. From the steady simulations, increasing α produces a large area of low-pressure regions on the suction side of the stator or rotor, which may lead to a higher fluctuation pressure and deteriorate the noise performance. In addition, the higher β, the lower chance of the local high-pressure region at shroud corner close to the leading edge of the stator, which can relieve the stress concentration without changing the PJP’s performance and merits a further assessment.

The rotor–stator spacing has only a slight or limited influence on PJP performance. Even the stator has been translated towards the rotor 20 mm, or S decreases 34% relative to the original PJP, the rotor thrust at J = 0.8 rise only 1%, so does the K_Qr or η. Further investigation on the pressure coefficient of profile foil illustrates that Cp within 50% chord length from leading-edge becomes lower as S decrease.

Overall, among these stator parameters, the α mostly affects the PJP’s global performance, followed by N_s and L, then is γ, the last two are S and β. In general, the torque and thrust for the blade (including stator and rotor) increase or decrease synchronously. The share of rotor thrust is extremely higher than the stator at design conditions. The increased K_Ts, caused by the increased α, stator solidity, or γ, does not significantly change K_T. In comparison, the duct provides thrust at heavy-loading conditions, which occupies 25% and 14% of the total thrust at J = 0.2 and J = 0.4 for the original PJP. A significant change of K_Td hence leads to an obvious change of K_T. For normal operating conditions, i.e., J = 0.8, the proportion of K_Td is as small as that of K_Ts and less than 10%, which is also not enough to change the total thrust. The ΔK_Q mainly is related to the change of K_Qr. The change rates of K_Qr for studied stator parameters are higher than K_Qs. The higher of K_Qr, the ΔK_Q becomes lower. Consequently, the J corresponding to torque balance location becomes lower.