﻿ 双层垂直轴水轮机性能计算的当量密实度法
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 哈尔滨工程大学学报  2017, Vol. 38 Issue (8): 1197-1202  DOI: 10.11990/j.issn.1006-7043.201606034 0

### 引用本文

ZHANG Zhiyang, SUN Ke, ZHANG Liang, et al. Equivalent-solidity method for performance analysis of double-H type vertical-axis tidal current turbine[J]. Journal of Harbin Engineering University, 2017, 38(8), 1197-1202. DOI: 10.11990/j.issn.1006-7043.201606034.

### 文章历史

Equivalent-solidity method for performance analysis of double-H type vertical-axis tidal current turbine
ZHANG Zhiyang, SUN Ke, ZHANG Liang, MA Qingwei, LI Bingqiang
College of Shipbuilding Engineering, Harbin Engineering University, Harbin 150001, China
Abstract: To solve the numerical divergence of the stream tube model in the performance calculation of double-H type vertical-axis tidal turbines, an equivalent-solidity method, which is a correction method for modifying the stream tube, was proposed. The equivalent-solidity method made an analogy between single-H type and double-H type turbines and converted a double-H type turbine into a single-H type turbine. This approach decreased the number of disks of the stream tube and avoided numerical divergence. The experimental and CFD calculation data from a single vertical axis turbine validated our numerical programs. Then, problems of the numerical divergence of the performance calculation of the double-H type turbine were analyzed. Therefore, the basic theory of the equivalent-solidity method was proposed; this theory validated the results of this new program based on the CFD calculation results. Comparison results show that the equivalent-solidity method agrees well with the CFD method and can be used to estimate the performance of double-H type vertical-axis tidal turbine.
Key words: tidal current energy    equivalent solidity method    vertical axis tidal turbine    stream tube model    CFD    hydrodynamic performance    double H type vertical axis tidal turbines

1 流管模型数学模型的建立 1.1 流管模型

 图 1 流管模型 Fig.1 Stream tube model

 ${{f}_{x}}=\frac{1}{2}\rho {{W}^{2}}C\Delta h\left( -{{C}_{t}}\sin \theta -{{C}_{n}}\cos \theta \right)$ (1)

 ${{{\bar{f}}}_{x}}=\frac{Z\Delta \theta }{\rm{ }\!\!\pi\!\!\rm{ }}{{f}_{x}}$ (2)

 $\Delta I=2\rho AU_{\infty }^{2}a\left( 1-a \right)$ (3)

 $A=r\Delta \theta \left| \cos \theta \right|\cdot \Delta h$ (4)

 \begin{align} &\quad \quad \quad \quad 2\rho AU_{\infty }^{2}a\left( 1-a \right)= \\ &\frac{Z\Delta \theta }{\rm{ }\!\!\pi\!\!\rm{ }}\cdot \frac{1}{2}\rho {{W}^{2}}C\Delta h\left( -{{C}_{t}}\sin \theta -{{C}_{n}}\cos \theta \right) \\ \end{align} (5)

 \begin{align} &\quad \quad \quad f\left( a \right)=a\left( 1-a \right)= \\ &\frac{ZC{{{\bar{W}}}^{2}}}{4\text{ }\!\!\pi\!\!\text{ }r\left| \cos \theta \right|}\cdot \left( -{{C}_{t}}\sin \theta -{{C}_{n}}\cos \theta \right)= \\ &\quad \frac{\sigma {{{\bar{W}}}^{2}}}{2\left| \cos \theta \right|}\cdot \left( -{{C}_{t}}\sin \theta -{{C}_{n}}\cos \theta \right) \\ \end{align} (6)

1.2 双盘面多流管模型——单层叶轮

 图 2 双盘面多流管模型 Fig.2 Double disk multiple stream tube model

 $\tan \ {{\alpha }_{u}}=\frac{{{U}_{u}}\cos \theta }{\omega r+{{U}_{u}}\sin \theta }$ (7)
 $\tan \ {{\alpha }_{d}}=\frac{{{U}_{d}}\cos \theta }{\omega r+{{U}_{d}}\sin \theta }$ (8)
 $W_{u}^{2}={{\left( r\omega +{{U}_{u}}\sin \theta \right)}^{2}}+U_{u}^{2}{{\cos }^{2}}\theta$ (9)
 $W_{d}^{2}={{\left( r\omega +{{U}_{d}}\sin \theta \right)}^{2}}+U_{d}^{2}{{\cos }^{2}}\theta$ (10)

 $\tilde{W}_{k}^{2}={{\left[ \frac{r\omega }{{{U}_{k}}}+\sin \theta \right]}^{2}}+{{\cos }^{2}}\theta$ (11)
 $\begin{array}{l} \quad \quad \quad \quad f\left( {{a_u}} \right) = {a_u}\left( {1 - {a_u}} \right) = \\ \frac{{ZC}}{{8{\rm{ \mathsf{ π} }}r\left| {\cos \theta } \right|}} \cdot {\left( {\frac{{{W_u}}}{{{U_\infty }}}} \right)^2}\left( { - {C_{tu}}\sin \theta - {C_{nu}}\cos \theta } \right) \end{array}$ (12)
 $\begin{array}{l} \quad \quad \quad \quad f\left( {{a_d}} \right) = {a_d}\left( {1 - {a_d}} \right) = \\ \frac{{ZC}}{{8{\rm{ \mathsf{ π} }}r\left| {\cos \theta } \right|}} \cdot {\left( {\frac{{{W_d}}}{{{U_e}}}} \right)^2}\left( { - {C_{td}}\sin \theta - {C_{nd}}\cos \theta } \right) \end{array}$ (13)

1.3 双盘面多流管模型——双层叶轮

 图 3 双层叶轮示意图 Fig.3 Diagram of double-H type turbine

 图 4 双层叶轮流管示意图 Fig.4 Stream tube of double-H type turbine

 $a\left( 1-a \right)=f\left( \theta ,\sigma ,U,{{C}_{n}},{{C}_{t}} \right)$ (14)

a∈(0, 1) 时，等式左侧的取值范围是(0, 1/4]，当等式右侧的取值范围超出这个范围时，迭代公式无解。

1.4 当量密实度法

 ${{\sigma }_{\text{new}}}={{\sigma }_{\text{out}}}\left[ 1+{{\left( \frac{{{C}_{\text{in}}}}{{{C}_{\text{out}}}} \right)}^{a}}{{\left( \frac{{{R}_{\text{in}}}}{{{R}_{\text{out}}}} \right)}^{b}} \right]$ (15)

 ${C_{{\rm{new}}}} = 2{\rm{ \mathsf{ π} }}{\sigma _{{\rm{new}}}}{{\rm{R}}_{{\rm{out}}}}{\rm{/Z}}$ (16)

2 流管模型程序验证

2.1 固定偏角垂直轴轮机模型实验

 图 5 实验平台 Fig.5 Experiment platform
 图 6 轮机模型 Fig.6 Turbine model
2.2 CFD计算

 图 7 流场域网格 Fig.7 Meshes of liquid field
 图 8 叶片表面网格 Fig.8 Meshes on the surface of blade

 图 9 边界条件 Fig.9 Boundary conditions
2.3 流管模型程序计算

2.4 计算结果对比分析 2.4.1 叶轮总体性能对比

 图 10 能量利用率系数-速比曲线(单层叶轮) Fig.10 Curves of the coefficient of power and the tip-speed ratio(single-H type turbine)
2.4.2 受力情况的对比

 图 11 切向力系数-位置角曲线 Fig.11 Curves of the coefficient of shear force and position angle
 图 12 法向力系数-位置角曲线 Fig.12 Curves of the coefficient of normal force and position angle

3 当量密实度法的验证和分析

 图 13 能量利用率系数-速比曲线(双层叶轮) Fig.13 Curve of the coefficient of power and the tip-speed ratio(double-H type turbine)

 图 14 能量利用率系数-位置角曲线(速比=2.5) Fig.14 Curves of the coffcient of power and position angle (TSR=2.5)
 图 15 能量利用率系数-位置角曲线(速比=3.5) Fig.15 Curves of the cofficient of power and the position angle (TSR=3.5)
4 结论

1) 实验结果验证了当量密实度法和CFD方法的有效性，两种方法都可以用于垂直轴水轮机性能的计算。

2) 当量密实度法修正后的流管模型可以有效解决数值发散的问题，可以高效地计算双层垂直轴水轮机的总体性能。当量密实度法是双层垂直轴水轮机性能计算的有效方法。

3) 本文计算的垂直轴水轮机密实度较小(0.1左右)，对于密实度较大(大于0.1或更高)的垂直轴水轮机，当量密实度法是否有效，还需进一步地论证和分析。

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