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Procedía Engineering 126 (2015) 680 - 685

Procedía Engineering

www.elsevier.com/locate/procedia

7th International Conference on Fluid Mechanics, ICFM7

Effect of gap flow on the torque for blades in a rim driven thruster

without axial pressure gradient

Q. M. Caoa*, W. F. Zhaoa, D. H. Tanga, F. W. Honga

a National Key Laboratory on Ship Vibration & Noise CSSRC, Wuxi, 214082 China

Abstract

This paper applied RANS solver and angular momentum current analyzing method to study the effect of gap flow in a rim driven thruster on the torque of the blades when the propeller was replaced by a rotating disk. The rim's torque was divided into two parts, outer surface and end faces. The torque of outer surface was resulted by turbulent Taylor vortex flow in the radial gap between inner and outer cylinder. The torque of end faces was resulted by radial outflow and inflow in the axial gap. Based on N-S equations angular momentum current was conserved in radial gap by neglecting of influence of end faces. We employed RANS solver to investigate the effect of the ratio between inner and outer cylinder radius on the torque of the outer surface by keeping the axial gap ratio. As such the effect of axial gap ratio on the torque of end faces was explored by holding radial gap ratio. Whereafter angular momentum current conservation was applied to obtain a predicted formula to calculate the torque of the outer surface. A new predicted method of calculating the torque of end faces was obtained by fitting the CFD calculated results. Finally the developed method was validated by experiments in the large cavitation tunnel of CSSRC. It showed that the torque calculated by the new predicted method agreed well with experiments comparing with the empirical formulas from Bilgen and Boulos, Daily and Nece.

© 2015 The Authors. Published by ElsevierLtd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.Org/licenses/by-nc-nd/4.0/).

Peer-review under responsibility of The Chinese Society of Theoretical and Applied Mechanics (CSTAM) Keywords: rim driven thruster; gap flow; torque; angular momentum current; RANS

1. Introduction

Rim driven thrusters have some big changes in drive and layout that transforms the motor's stator to the duct's interior and the motor's rotor to the tip of the blade via the rim to obtain lower noise and higher efficiency. However these changes bring some different problems comparing with conventional motor-driven thrusters. Most important

* Corresponding author: Qingming Cao, Tel.: +0-0510-85556647; fax: +0-0510-8555-8105. E-mail address: caoqingming@aliyun.com

1877-7058 © 2015 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

Peer-review under responsibility of The Chinese Society of Theoretical and Applied Mechanics (CSTAM) doi: 10. 1016/j .proeng .2015.11.265

Nomenclature

Ri outer radius of rim R2 inner radius of duct hole

Rin radius of propeller Lr axial length of rim

h radial gap clearance or height sin inlet axial gap clearance

sex exit axial gap clearance Rd outer radius of duct

Ld axial length of duct n =R1/R2, inner and outer radius ratio

S =s/Ri, axial clearance ratio ur azimuthal velocity

w angular velocity Rei = a Rh / v, radial Reynolds number

Rer = cvRf / v, axial Reynolds number Ta = a)Rlhyjh/RJv, Taylor number

№ = Ja / Ja , Nusselt number lam G non-dimensional torque of outer surface

CM = M / (0.57rpa>2Kfh), dimensionless torque of outer surface

Cq = 2Miist / (0.5/xo2Rf), non-dimensional torque of end faces

problem is fictional torque of the rim because the gaps flow between the duct's inner surface and the rim surface. This gap flow is very complicated and difficult to simulate for boundary layer flow in the gap going with the gap flow leading by axial pressure difference. The leading influence of the gap flow changes the hydrodynamic performance of the propeller with the rim and the duct.

Recently some researchers investigated the gap flow and the frictional torque of the rim by numerical and empirical method. Oosterveld [1] applied the frictional resistance formula of smooth flat plate to evaluate the rim's effect, and he considered that the rim lead to increase the torque and reduce the thrust of the propeller. Mishkevish [2] employed Taylor-Couette flow to predict the frictional torque of the rim. Michael et al.[3] divided the rim's surface into three parts, outer surface, inner surface and end faces. Different empirical formulas were employed to predict the frictional torque of the three parts correspondingly. Cao et al. [4] investigated the rim's frictional torque for a rim driven thruster and validated by experiment. Yang and Wang etc applied CFD tools to design rim-driven ducted propeller under design and off-design conditions. Some researchers applied CFD tools to investigate the gap flow by choosing appropriate mesh and turbulent model. Wild et al.[5] applied different turbulent model to calculate Taylor vortex flow in a centrifugal rotor at two radius ratios, and compared with experimental results. Batten and Bressloff etc.[6] considered that the turbulence models were more sensitive to the chosen coefficients for the gap flow. In order to overcome this deficiency the Wilcox low Reynolds number k-w model was used to studied turbulent Taylor vortex flow. Combined the research progress home and abroad, many researchers separated the radial and axial gap flow into Taylor-Couette flow and rotating disk flow in housing independently, and applied empirical formulas to predict the rim's frictional torque. This treatment neglected the interaction between the radial and axial gap flow and the flow driven by the pressure difference. Actually, the radial and axial gap flow in rim driven thrusters differed with one of the two flows in flow features.

In order to take the interaction and geometric features (shorter length of the rim, rotating ring flow in housing) into account, this paper employed RANS solver and angular momentum current analyzing method to study the effect of gap flow on the torque of the blades under the circumstance that the aspect ratio was small. The effect of the ratio between inner and outer cylinder radius on the torque of the outer surface was investigate by holding the axial gap ratio. As such the effect of axial gap ratio on the torque of end faces was explored by holding radial gap ratio. Whereafter angular momentum current conservation was applied to obtain a modified empirical formula to calculate the torque of the outer surface. A new predicted formula to calculate the torque of end faces was obtained by fitting the CFD calculated results. Finally the developed formulas were validated by experiments in the large cavitation tunnel. It showed that the torque calculated by the new predicted formulas agreed well with experiment comparing with the empirical formulas by Bilgen and Boulos, Daily and Nece.

2. Models for simulation

The model for calculation is a cylindrical duct with its middle region subtracting a cylinder to fix the rim. The rim is linked with the propeller and has axial and radial gap with the duct. The test models and geometric notions are shown in Fig. 1. The axial length Ld of the duct is 200mm, the inner radius of the duct Rin is 100mm, the outer radius

Rd of the duct is 150mm. The radius R2 at the rim's position is 130mm, and the axial length Lr of the rim is 70mm. When we investigate the effect of the radial gap on the gap flow, the radial clearance is adjusted by changing the outer diameter of the rim and holding the axial clearance of 2.0mm. Likewise, when studying the effect of the axial gap on the gap flow, the axial clearance is adjusted by changing the length of the middle duct and holding the radial clearance 2.0mm and the length of the rim. All the calculate cases are shown in Table 1. The case "Radi-Gap-4" is identical with the case "Axia-Gap-5".

Table 1. Simulation cases (Unit: mm )

Radi-Case Radial gap clearance h Axia-Case Axial gap clearance Sin/Sex

Radi-Gap-1 5.00 Axia-Gap-1 20.00

Radi-Gap-2 4.00 Axia-Gap-2 10.00

Radi-Gap-3 3.00 Axia-Gap-3 6.00

Radi-Gap-4 2.00 Axia-Gap-4 4.00

Radi-Gap-5 1.60 Axia-Gap-5 2.00

Radi-Gap-6 1.18

Fig.1. Geometry defintion of the rim and duct 3. Numerical method and boundary conditions

RANS solver SST k-w turbulence model was employed to calculate the torque of the rim's out surface and end faces under the condition that no axial pressure difference in the gap. The computational domain has the length of nine diameters of the propeller with three diameters of the propeller in the radius. The upstream inflow boundary is setted at four diameters of the propeller. The downstream outflow boundary is setted at five diameters of the propeller. The type of the mesh in the domain is mixed mesh consisting of structured and unstructured mesh, where unstructured mesh is applied in the domain in the propeller and hub, structured mesh in the rest of the domain. The structured mesh in the gap is fined enough to ensure the calculated velocities in the viscous layers. The mesh is shown in Fig. 2. The rotating speed of the propeller and rim is 20r/s, corresponding Reynolds number ran je Re;=104~105

Fig.2. Grid structure and detail with enlarged scale Fig.3. The configurations of test setup

(a-global grid, b-gap grid)

4. Experimental setup and methodology

In order to validate the calculated torque of the inner cylinder and develop new empirical formulas, a series of experiments applying a duct with thin and thick disks were conducted in the large cavitation tunnel of CSSRC. The tunnel is a closed water circuit with a cylinder test section of diameter 0.8m, having a length of 3.2m. The range of the flow speed in the test section is 3 m/s ~20m/s. The test models are three radial gap cases, including the cases "Radi-Gap-2", "Radi-Gap-5" and "Radi-Gap-6". The radial gap clearance is adjusted by changing the diameter of the thick disks. The configuration of hydrodynamic test setup is shown in Fig. 3.

5. Numerical results and analysis 5.1. Numerical validation test

In RANS solver, modified near-wall approach in k-w turbulence model needs more fine mesh to simulate the viscosity-affected region. In order to validate the turbulence model and near-wall mesh resolution in the gap, Radi-Gap-2 case was chosen to simulate. According to grid convergence rule, we chose three types of mesh growth to study, the inner cylinder's wall was about 0.3,3,30 respectively. The angular velocity varying with wall-adjacent cells distance in the radial gap at X=0 plane is shown in Fig. 4. Obviously, there is small difference in the angular

velocity between the wall y+ 0.3 and 3. Taking account of the expense and time, the second mesh type was chosen to simulate the gap flow and calculate the torque of the inner cylinder. Figure 5 shows the comparison of the angular velocity in the radial gap. The empirical formula for the core region in the gap is derived by Busse [7], and in excellent agreement with the experiment of Lewis & Swinney [8]. Except the viscous layers near the walls of the duct and rim, the angular velocity of the cases Axia-Gap-5, Axia-Gap-4 and Axia-Gap-2 agrees well with Busse's results. It implies the mesh configuration and SST k-w turbulence model can simulate the gap flow accurately. 5.2. Results and analysis

• Effects of radial or axial gap clearance on the gap flow

Figures 6 and 7 show the velocity distribution in the gap for axial clearance changing. In the axial gap, caused by centrifugal forces there exists a circular flow, a radial outflow originating from the rim's end faces, when arriving at the inner surface of the duct a radial inflow developing along the side surfaces of the duct for momentum commutation. The position of the circular flow varies with the axial gap clearance by holding the radial clearance. When the axial clearance ratio is small, the boundary layer of the rim's end faces merges with the side surfaces' of the duct. Further increase the axial clearance, the square of the circular flow expands and the centre moves toward the side surfaces of the duct. Figures 8 and 9 show the velocity distribution in the gap for radial clearance changing. When the axial and radial clearances are all small, Taylor vortices appear only in the edge of the rim outer surface. It implies that there is small interference between the axial and radial gap flow. But the number and region of Taylor vortex in the radial gap will rise with the radial clearance ratio.

Fig.4 Effect of grid on angular velocity in radial gap

Fig.6. Velocity in Axia-Gap-4 without pressure difference

Fig.7. Velocity in Axia-Gap-2 without pressure difference

Fig.8. Velocity in Radi-Gap-4 without pressure difference Fig.9. Velocity in Radi-Gap-1 without pressure difference

• Effects of radial or axial gap clearance on the torque

Numerical results show that when the axial clearance ratio is larger than 0.03 the non-dimensional GnW- Re:18 holds constant approximately. The dimensionless torque of the end faces rises with the axial clearance ratio increasing, and agrees with Daily and Nece. However there is a large difference in values between them. The is due to the fact that the experimental results of Daily and Nece is for rotating disk flow in housing, not resemble the axial gap flow in rim driven thrusters. Taking account of analogy, the dimensionless torque of the rim's end faces is expressed relations with the axial gap clearance ratio and Reynolds number by fitting the calculated results. CQ_mmd Re0'25 = f (S) = -0.001634^1003 + 0.2282 ( 1 )

The calculated end faces' torque coefficient varies with the radius ratio. It demonstrates that when the radius ratio n<0.97 the dimensionless torque c Re°2S holds invariable and approach the results of Daily and Nece. When the

radius ratio n>0.97 the difference between empirical and numerical widens with the radius ratio increasing. The cause is likely that the tangential velocity in the tip edge of axial gap decreases and its derivative rises. Given that the interaction between the axial gap flow and radial flow in the edge is very complicated, the difference for the radius ratio n>0.97 is neglected to study the effect of the radial gap flow on the outer surface's torque coefficient. The non-dimensional torque coefficient ^ ■ Re:1-8 of the rim's outer surface changes with the radius ratio, and decreases as power function accompanying with increasing the radial clearance ratio, similar with Bilgen and Boulos. Actually the decrease is caused by Reynolds number Re~L8, the torque coefficient rises with increasing the

radial gap clearance. According to angular momentum current theory, the angular velocity current is defined by using continuity equation and commutations,

Ja= r3((urm)/ 5r) (2)

as a relevant conserved transport quantity, representing the flux of angular velocity from the inner to the outer cylinder. The dimensionless angular velocity current was made by its value j°° in laminar flow case,

Nm = Jm / Jm

J ' J lam

J,1 = -vr3d{w)AJ / dr \= 2virl2 ) / (rl-r°22 The dimensionless torque is defined as G = Q/ (2 Lpv2) = V-2 J" = v2 NVZ Eckhardt et. al. considered that the convective dissipation rate per unit mass and Nusselt number were related with the Reynolds number Rew of the wind profile. The dissipation of the bulk layer is modeled by a term x Re3w, and the dissipation in the boundary layer is proportioned to Re2.5 . When the gap flow is dominated by the bulk or background fluctuations, the dimensionless torque of the inner cylinder is

Grimou, =^2JLN^ (1+#V221 "tfT'Ref0 (6)

When Re > 104 the dimensionless torque of the inner cylinder is defined as follows,

Gnmo:ll x f (rf) Re18 =r? (1 - r()b Rel 8 (7)

A new predicted formula to calculate the outer surface's torque is developed by fitting the calculated results. Grimout = 0.01668^ 1818(1 -rj) 1757 Re18 (8)

• Comparisons between numerical results and experiments

Figures 10 and 11 depict the comparisons between the measured and empirical results at different rotating speed, corresponding to the case "Radi-Gap-5" and "Radi-Gap-2" respectively. The empirical results are computed by two methods, one is the Eqs.(1) and (8) developed by this paper, and the other is by Daily and Nece, Bilgen and Boulos. They show that the difference between the experimental results and developed method is less than 5%, especially for large radial clearance ratio the difference becomes no more than 1%. Nevertheless, it appears very large difference (almost 30%) between the experiment data and the predicted results according to Daily and Nece, Bilgen and Boulos. This is due to the fact that the empirical equations from Daily and Nece, Bilgen and Boulos are just for rotating disk flow in housing and Taylor-Couette flow separately, and the interaction between the axial gap flow and radial gap flow and the geometric features of rim driven thruster are not taken into account. Consequently, the comparisons indicate the new predicted method has a great improvement comparing to the previous empirical formulas, and can be used to calculate the rim's torque for rim driven thruster.

Fig . 10 The torque of end faces and outer surface at different speed Fig . 11 The torque of end faces and outer surface at different speed (Radi-Gap-5 ) (Radi-Gap-2 )

6. Conclusion

In this paper the effect of the axial and radial gap flow on the rim's torque for rim driven thrusters has been analyze d numerically and experimentally. A parametric study is performed to investigate the independently effect of the axial and radial gap flow on the torque of the outer surface and end faces by changing the axial or radial clearance ratio. Based on angular momentum current analyzing method and numerical results, a new predicted method is developed by fitting the results. Finally the developed predicted method is validated by experiments in the large cavitation tunnel of CSSRC. It indicates that the torque calculated by the new method is in good agreement with the measured, and has a great improvement comparing with the empirical formulas by Bilgen and Boulos, Daily and Nece. From this paper some interesting results are obtained as follows,

(1) There exists a circular flow in the axial gap consisting of a radial outflow originated from the rim's end faces, arriving at the inner surface of the duct a radial inflow developing along the side surfaces of the duct for momentum commutation. The position of the circular flow is varied with the axial gap clearance.

(2) Taylor vortices appear only in the edge of the rim outer surface when the axial and radial clearances are all small. The number and region of Taylor vortex in the radial gap rise with the radial clearance ratio.

(3) The dimensionless torque of the end faces rises with the axial clearance ratio increasing.

(4) The dimensionless torque holds invariable when the radius ratio n<0.97, but the difference between empirical and numerical widens with radial gap ratio increasing when the radius ratio n>0.97. The torque coefficient rises with increasing the radial gap clearance.

Actually, there is pressure difference in the axial and radial gap when the propeller rotating. It triggers totally different flow in the gap and a different tendency of the rim's torque varying with the axial and radial gap clearance consequently. The effect of pressure difference in the gap on the torque will be studied further.

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