3-D NUMERICAL SIMULATIONS OF FLOW LOSS IN HELICAL CHANNEL*

ZHAO Ling-zhi

Institute of Electrical Engineering, Chinese Academy of Sciences, Beijing 100190, China

College of Computing and Communication Engineering, University of Chinese Academy of Sciences, Beijing 100049, China, E-mail: zlz@mail.iee.ac.cn

PENG Yan

Institute of Electrical Engineering, Chinese Academy of Sciences, Beijing 100190, China

LU Fang

China Ship Scientific Research Center, Wuxi 214082, China

LI Jian, LI Ran, LIU Bao-lin

Institute of Electrical Engineering, Chinese Academy of Sciences, Beijing 100190, China

(Received March 27, 2012, Revised July 3, 2012)

3-D NUMERICAL SIMULATIONS OF FLOW LOSS IN HELICAL CHANNEL*

ZHAO Ling-zhi

Institute of Electrical Engineering, Chinese Academy of Sciences, Beijing 100190, China

College of Computing and Communication Engineering, University of Chinese Academy of Sciences, Beijing 100049, China, E-mail: zlz@mail.iee.ac.cn

PENG Yan

Institute of Electrical Engineering, Chinese Academy of Sciences, Beijing 100190, China

LU Fang

China Ship Scientific Research Center, Wuxi 214082, China

LI Jian, LI Ran, LIU Bao-lin

Institute of Electrical Engineering, Chinese Academy of Sciences, Beijing 100190, China

(Received March 27, 2012, Revised July 3, 2012)

The flow loss of a helical channel Magnetohydrodynamic (MHD) thruster without MHD effect was numerically studied with 3-D simulations, and a flow loss coefficient ζ was defined to quantify the flow loss and its influencing factors were studied. The results show that ζ decreases in a first-order exponential manner with the pitch of a helical wall and the Reynolds number, and it declines slowly when t/T＞0.2 and Re>105, a flow guide makes the flow more smooth and uniform, especially in the flow guide and helical wall sub-regions and thus reduces the flow loss greatly, by about 30% with the averaged value ofζ from 0.0385 to 0.027, a rectifier weakens the helical flow and strengthens the axial one in the rectifier and outlet sub-regions, thus reduces the rotational kinetic pressure with the averaged value of ζ declining about 4% from 0.0385 to 0.037, and ζ decreases with a rectifier’s axial length when Re>105.

helical channel, Magnetohydrodynamic (MHD) propulsion, rectifier, flow guide, flow loss

Introduction

With the same bore diameter, the superconducting solenoids allow for lower weight and higher maximum field strength than the dipole magnets. Recent Magnetohydrodynamic (MHD) thruster developments were concentrated therefore on the use of solenoids[1-9]. A superconducting solenoid magnet produces an axial magnetic field and thus requires a special helical channel with a complicated arrangement of electrodes, helical wall, flow guide and rectifier. The function of the flow guide and the rectifier is to convert smoothly the axial flow into the helical one,or vice versa. However, the increase of the electrical efficiency due to the higher magnetic field strength is accompanied by an increase of hydrodynamic losses in the helical channel MHD thruster introduced by the complicated flow guide, helical wall and rectifier. The hydrodynamic loss has a great bearing on the relative efficiency or channel efficiency of a MHD thruster. And it is surely related with the structure of the helical channel. So, the optimization designs of the flow guide, the helical wall and the rectifier, such as the blade shape, the attack angle, the pitch, the profile lines, and the surface finish can reduce the hydrodynamic loss, thereby enhance the channel efficiency of the helical channel MHD thruster.

Compared with the helical wall, the structure of the flow guide and the rectifier is more irregular and complicated and their 3-D physical models cannot be simplified to 1-D or 2-D ones. Thus, the studies of the hydrodynamic loss of a helical channel MHD thrustermainly focused on the helical wall. Peng et al. studied the effect of the helical wall on the performance of a helical channel MHD thruster with a simple 1-D analysis model and found that there was a best loop number to achieve the peak efficiency[10]. Hu et al.[11]analyzed the hydrodynamic characteristics of a helical channel MHD thruster with different pitch ratios and numbers of helical wall using 3-D numerical simulations and found that the MHD thruster would have the best performance with a pitch ratio of about 1. Takeda experimentally studied the impact of the pitch and the diameter of the helical wall on the flow loss and found that the pressure drops in the vicinity of the entrance and the exit of the helical wall were the main causes of the disagreement of the flow loss between the experimental and computed values[12,13]. Based on the hydrodynamic tests, Peng[14]defined a total local pressure coefficient to represent the hydrodynamic loss of the flow guide and the rectifier and found that the total local pressure coefficient depends on the vane number, the length and the attack angle of the flow guide and the rectifier and varies a little with the Reynolds number. In 2010, a superconducting helical channel MHD thruster with a 7 T conduction-cooled solenoid superconducting magnet and a Φ0.178m helical channel was designed and constructed in theInstitute of Electrical Engineering, ChineseAcademy of Sciences[15]. The helical channel, especially,the flow guide and the rectifier were optimized with 3-D simulations and the channel efficiency reached 74% in the test.

F ig.1(a) Inside of the manufactured helical channel

Fig.1(b) Calculating model

In a MHD thruster, there are electromagnetic and fluid flow fields interacting with each other and the hydrodynamic loss is very difficult to estimate. Usually, it is estimated by calculating or measuring the frictional loss and the local loss of the helical channel without MHD effect (the magnetic field strength or the current density is assumed to be equal to 0). In this paper, the inner turbulent flow of the above mentioned demonstration helical channel without MHD effect was numerically studied with 3-D simulations first, then a flow loss coefficient was defined to quantify the flow loss and its influencing factors were investigated. The numerical simulations were validated experimentally. Some guidances to decrease the flow loss were obtained, which provide a theoretical basis for the optimization of a helical channel.

1. Mathematical model

1.1 Physical model

Figure 1(a) shows the inside of a manufactured helical channel with main parameters as shown in Table 1.Andthe attack angleis givenwith the radius for which the average currentdensity is calculated[14]. Fig.1(b) is the corresponding physical model. Besides the flow guide, the helical wall and the rectifier, there are an inlet section and an outlet section with the total length in Z direction of about 2.19 m. The origin of the Cartesian coordinate is on the center of the inlet.

1.2 Governing equations

The flow in the helicalchannel is unsymmetrical andthus 3-D simulations of the inner fluid flow were carried out. The Reynolds number (Re=U0L0/ν, L0=(Do-Di)/2 and U0is the averaged velocity withinthe helical wall region,νis the kineticviscosity) is very high (105-106), sothat the instabilities are likely to develop and sustain to generate turbulence. The Renormalization-Group (RNG) k-ε model was used because of its inclusion of the effectof swirl on the turbulence, which would thus enhance the accuracy for swirling flows. The working liquid is incompressible seawater. The helical channel is horizontal and the gravity is ignored. The equations governing the steady incompressible flow without MHD effect are as follows

where ρis the density, uiis the mean velocity component, p is the mean pressure, andis the fluctuating velocity component. Additional terms now appear that represent the effects of turbulenceand the RNG k-εmodel is used to close the equations.

Table 1 Basic structural parameters

where δijis the Kronecker’s symbol,νtis the turbulent viscosity,Cμis a constant, k is the turbulence kinetic energy and ε is the turbulence dissipation rate. μeff=μ+μl, μl=ρμ(k2/ε), and Gkrepresents the generation of the turbulence kinetic energy due to the mean velocity gradients, C1ε=1.42, C2ε=1.68, akand aεare the turbulent Prandtl numbers for k and ε, respectively.

2. Computational method

2.1 Mesh

The complicated solid model of a helical channel was built by Solidworks and then imported into Gambit toobtain the fluid flow domain as shown in Fig.1(b). According to the components, the fluid flow domain was decomposed into five sub-domains as the inlet sub-domain C1, the flow guide sub-domain C2, the helical wall sub-domain C3, the rectifier sub-domain C4 and the outlet sub-domain C5. The axial length (in Z direction) of C1 is 0.22 m and 1 m for C5.

A proper grid distribution is vital for the accurate solution of the wall-bounded turbulence. The structural hex meshing schemes were adopted. In the nearwall regions, the nodes were clustered so as to correctly resolve the flow field there. The clustering was specified by setting the distance from the wall to the node closest to the wall, the number of nodes to be distributed, and using a successive ratio for the cell sizes. The size of the first cell at the wall was estimated by using a given Reynolds number and the desired y+value. For Re=105and y+=50 the distance in the wall-normal direction of the cell center closest to the wall was 0.0005 m. The adaptive meshing scheme was also adopted during the calculation process to control y+within 400.

Table 2 Static pressure difference with diffe rent number of cells (Re=2.1×105)

Table 2 shows the calculated results against different numbers of cells.It can be seen that with the same meshing scheme and increasing thenumber of nodes, the cell number is increased from 952 000 to 1 674 000, the calculated static pressure difference changes from 19.38 kPa to 17.29 kPa getting more andmore closer to the tested value of 17.30 kPa. Continuing to increase the number of the cells to 2 020 000, the calculated value only varies a little. With due consideration of the calculating time and the precision, the grid with about 1 674 000 cells, as shown in Fig.2, was adopted.

Fig.2 Grid

2.2 Boundary condition and solver

The inlet boundary condition was set witha constant mean velocity normal to the inlet. The outlet boundary condition was specified by the outflow. The reference pressure location was set to the origin of the Cartesian coordinate with a value of 101 325 Pa.

The working fluid was seawater with the densityof 1 025 kg/m3and the kin etic viscosity of 0.001003 kg/ms.

The software Fluent which uses the Finite Volume Method (FVM) to solve the partial differential equations was used to analyze the behavior of the seawater flow in the helical-channel. The secondorder upwind scheme was used in the discretization of the governing equations, and the flow problem was solved in a pressure-velocity coupling manner. The standard wall functions were used in the near-wall treatment and onthe walls, no-slipboundary condition was adopted.Theresiduals of10–5and the surface monitors were used to determine the convergence of the solution.

3. Results and discussions

3.1 Verification

In a MHD thruster, the flow rate and the static pressure differ ence are important fluid parameters and can be obtained through the 3-D numerical simulation. For a helical channel MHD thruster, the seawater still flows somewhat helically inthe outlet sub-domain C5 with a high rotational kinetic energy to be consumed as final heat without contribution to the flow rate. So in this paper, we define the flow loss as

where P1, A1and P2, A2are the area-averaged static pressure and the section area at the helical channel’s inlet and outlet, respectively, and Q is the flow rate.

Fig.3Tested and calculated flow loss of the demonstrationhelical channel

In order to test the validity of the numerical simulation, the flow loss with different flow rates in the demonstration helical channel MHD thruster were calculated and compared with the corresponding tested values as shown in Fig.3. In the hydrodynamic test, the magnetic field was zero and the flow was driven by an au xiliary pump. The flow rate and the inlet andoutlet’s static pressures in the helical channel were measured. It can be seen that the calculated values agree well with the measurements, which shows that the numerical method of this paper is reliable.

3.2 Flow loss coefficient

The flow loss is closely related with the structure of the helical channel. In order to quantify the influence of the structure on the flow loss and to provide a reference parameter, we define a dimensionless flow loss coefficient as

where UMHD=2Q/[(t-δ）（Do-Di)] is the average velocity in the helical wall sub-domain C3 andδ, t are the thickness and the pitch of the helical wall, respectively. The smaller the flow loss coefficient is, the smaller the flowloss is and the higher the channel efficiency is.

3.3 Infl uencing factors

The helical wall, the flow guide and the rectifier all have effects on the fluid flow and in turn on the flow loss. In the study, we consider a single factor at a time, namely, only one structural parameter varies and others are kept unchanged. And in the following figures exceptfor Fig.5(c), the discrete points are calculated values and continuous lines are fitting curves.

Fig.4 Flow loss coefficient varying with the pitch

Figure 4 shows the flow loss coefficient ζ varying with the pitch of the helical wall t. In this case, there is no flow guide or rectifier and the abscissa is the ratio of the pitch t to the axial length of the helical wall L. The flow path decreases with t increasing when L is constant, so ζ decreaseswith t/ L. From Fig.4, it can be seen thatζ decreases slowly in a first-order exponential manner and varieslittle with the Reynolds number when t/L>0.2.

Fig.5(a) Flow loss coefficient varying withflow guide andrectifier

Fig.5(b1) Pathline colored by the static pressure (Lr=Lf= 0 m)

Fig.5(b2) Pathline colored by the static pre ssure (Lr=0.12m, Lf=0m)

Fig.5(b3) Pathline colored by the static pressure (Lr=0m, Lf=0.12m)

Figure 5(a) shows the influence of the flow guide and the rectifier on the flow loss with t/L=0.186 and L=0.5m. The influences of theflow guide and the rectifier are considered separately. The square signsand the corresponding fitting curve represent the case without the flow guide and the rectifier. The triangle and circle signs and the corresponding fitting curve represent the case wit h a rectifier and without a flowguide. The diamond and cross signs and the corresponding fitting curve represent thecase with a flow guide and without a rectifier. It can be seen that ζ varies slowly with the Reynolds number in a firstorder exponential manner. A flow guide can reduce the flow loss by about 30% with the averaged value of ζ from 0.0385 to 0.027. With a rectifier, the averaged value of ζ decreases about 4% from 0.0385 to 0.037. From Fig.5(a), it can also be seen that ζ is almost the same for a flow guide/rectifier with a constant axial length whether the number of vanes n is 3 or 4. Figures 5(b1), 5(b2), 5(b3) and Fig.5(c) are the corresponding pathline and distribution of the pressure along the axial direction with 4 vanes. In Fig.5(c), the lines with solid symbols correspond to the right Y axis, and C3 cove rs from z=0.34m to z=0.84m. It can be seen that with a flow guide, the inlet axial flow is converted into the helical one smoothly, and the static pressure deceases linearly and the dynamic pressure varies not so much as those without a flow guide in C3, namely, a flow guide makes the flow more smooth and uniform especially in C2 and C3 and thus reduces the flow loss. A rectifier weakens the helical flow and strengthensthe axial one in C4and C5, and thus reduces the rotational kinetic pressure. So, a rectifier mainly makes the flow in C4 and C5 smooth and reduces the flow loss there.

Fig.5(c) Distribution of the pressure along axial direction (Re=1.51×105)

Figure 6 shows the flow loss coefficientζ varying with the axial length of the rectifier. It can be se en that with Lrincreasing,ζ decreases when Re＞105.

4. Conclusions

The flow loss of ahelical channel MHD thruster without MHD effect was numerically studied with 3-D numerical simulations. A flow loss coefficientwas defined to quantify the flow loss and its influencing factors were studied. The results show that:

(1)ζdecreases in a first-order exponential manner with t he pitch of a helical wall and declines slowly when t/L>0.2.

(2) ζ decreases in a first-order exponential manner with the Reynolds number and declines slowly when Re＞105.

(3) A flow guide makes the flow more smooth and uniform especially in the flow guide and helica l wallsub-reg ions and thus reduces the flow loss greatly, about 30% with the averaged value of ζ from 0.0385 to 0.027.

(4) A rectifier weakens the helical flow and strengthens the axial one in the rectifier and outlet sub-regions, and thus reduces the rotational kinetic pressure with the averaged value of ζ declining about 4% from 0.0385 to 0.037, and ζ decreases with the axial length of a rectifier when Re＞105.

Fig.6 Flow loss coefficient varying with the axial length ofthe rectifier (Lf=0.12m, t/T=0.186)

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10.1016/S1001-6058(11)60313-2

* Biography: ZHAO Ling-zhi (1977-), Female, Manchu, Ph. D. Candidate, Senior Engineer

PENG Yan,

E-mail: pengyan@mail.iee.ac.cn