工程科学与技术   2022, Vol. 54 Issue (2): 141-149

1. 四川大学 水利水电学院，四川 成都 610065;
2. 四川大学 水力学与山区河流保护国家重点实验室，四川 成都 610065

Analytical Solution to Flow Field Around an Axisymmetric Filled Drainage Hole
ZHANG Shuai1,2, YE Fei1,2, HU Chao1,2, ZHENG Shuang1,2, FU Wenxi1,2
1. College of Water Resource & Hydropower, Sichuan Univ., Chengdu 610065, China;
2. State Key Lab. of Hydraulics and Mountain River Eng., Sichuan Univ., Chengdu 610065, China
Abstract: Drainage holes are usually arranged to reduce the seepage pressure inside rock mass. The partial blockage of the drainage hole will limit the drainage capacity and threaten the safety of an engineering project. At present, the analysis of the filled drainage hole only considers the fluid movement in the unfilled area, and there are few theoretical studies that consider the flow in both the filled and unfilled regions. In view of this, a clear-seepage flow coupled theoretical model was presented to study the velocity distribution of the drainage hole with axisymmetric filled medium in the rock mass. In the model, the Navier–Stokes equation was adopted to describe the motion of the clear water in the hole, and the Darcy–Brinkman equation was employed to describe the movement of the seepage water in the filled medium. The analytical solutions of velocity distribution and discharge were derived under the shear stress jumping interface boundary condition, and the results of analytical solution for velocity shown good agreement with that of the numerical solution obtained by the fourth order Runge–Kutta method. Parameter sensitivity analyses shown that the flow velocity had a positive relation with the Darcy number (Da) and the relative thickness of the open region, and a negative relation with the ratio of viscosity and the stress jump coefficient. The flow velocity was more sensitive to the variation of the typical parameters and the influence of viscosity ratio on the velocity distribution could not be neglected when the stress jump coefficient is less than 0. Furthermore, with the increase of the relative thickness of the open region, the interface velocity gradually decreases when Da=1, while it firstly increases and then decreases when Da=0.1, and it gradually increases when Da<0.1. The research results of this paper can provide a theoretical basis for the calculation of the drainage capacity of the partially filled drainage hole. It is of great significance for developing the clear-seepage flow coupled theory of drainage holes and can provide a reference for the follow-up drainage hole dredging.
Key words: axisymmetric filled drainage hole    clear-seepage coupled model    flow field analytical solution    stress jump coefficient

1 理论描述

 图1 岩体内部轴对称填充排水孔模型 Fig. 1 Model of axisymmetric filled drainage holes in the rock mass

1.1 排水孔纯水流控制方程

 $\dfrac{{\partial {v_{{\text{f}}x}}}}{{\partial x}} + \frac{{\partial {v_{{\text{f}}y}}}}{{\partial y}} + \frac{{\partial {v_{{\text{f}}\textit{z}}}}}{{\partial \textit{z}}} = 0$ (1)

 \begin{aligned}[b] {f_x} - \frac{{\text{1}}}{\rho }&\frac{{\partial p}}{{\partial x}} + \upsilon \left( {\frac{{{\partial ^2}{v_{{\text{f}}x}}}}{{\partial {x^2}}} + \frac{{{\partial ^2}{v_{{\text{f}}x}}}}{{\partial {y^2}}} + \frac{{{\partial ^2}{v_{{\text{f}}x}}}}{{\partial {\textit{z}^2}}}} \right) = \hfill \\& \frac{{\partial {v_{{\text{f}}x}}}}{{\partial t}} + \left( {{v_{{\rm{f}}x}}\frac{{\partial {v_{{\text{f}}x}}}}{{\partial x}} + {v_{{\rm{f}}y}}\frac{{\partial {v_{{\text{f}}x}}}}{{\partial y}} + {v_{{\rm{f}}\textit{z}}}\frac{{\partial {v_{{\text{f}}x}}}}{{\partial \textit{z}}}} \right) \hfill \\[-2pt] \end{aligned} (2)

 $\frac{{\partial p}}{{\partial x}}{\text{ = }}{\mu _{\text{f}}}\left( {\frac{{{\partial ^2}{v_{{\text{f}}x}}}}{{\partial {y^2}}} + \frac{{{\partial ^2}{v_{{\text{f}}x}}}}{{\partial {\textit{z}^2}}}} \right)$ (3)

 $\frac{{{\partial ^2}{v_{{\text{f}}x}}}}{{\partial {y^2}}} + \frac{{{\partial ^2}{v_{{\text{f}}x}}}}{{\partial {\textit{z}^2}}} = \frac{{{\partial ^2}{v_{{\text{f}}x}}}}{{\partial {r^2}}} + \frac{1}{r}\frac{{\partial {v_{{\text{f}}x}}}}{{\partial r}}$ (4)

 ${\mu _{\text{f}}}\left(\frac{{{{\text{d}}^2}{v_{{\text{f}}x}}}}{{{\text{d}}{r^2}}} + \frac{1}{r}\frac{{{\text{d}}{v_{{\text{f}}x}}}}{{{\text{d}}r}}\right) - \frac{{{\text{d}}p}}{{{\text{d}}x}} = 0,{\text{ }}r \in (0\sim{R_0})$ (5)
1.2 填充介质渗流控制方程

 $\frac{{\partial {v_{{\text{p}}x}}}}{{\partial x}} + \frac{{\partial {v_{{\text{p}}y}}}}{{\partial y}} + \frac{{\partial {v_{{\text{p}}\textit{z}}}}}{{\partial \textit{z}}} = 0$ (6)
 \begin{aligned}[b] n\rho {f_x} - &n\frac{{{\mu _{\text{f}}}}}{K}{v_{{\text{p}}x}} - n\frac{{\partial p}}{{\partial x}} + {\mu _{\text{f}}}\left( {\frac{{{\partial ^2}{v_{{\text{p}}x}}}}{{\partial {x^2}}} + \frac{{{\partial ^2}{v_{{\text{p}}x}}}}{{\partial {y^2}}} + \frac{{{\partial ^2}{v_{{\text{p}}x}}}}{{\partial {\textit{z}^2}}}} \right) = \hfill \\& \rho \frac{{\partial {v_{{\text{p}}x}}}}{{\partial t}} + \rho \left( {{v_{{\text{p}}x}}\frac{{\partial {v_{{\text{p}}x}}}}{{\partial x}} + {v_{{\text{p}}y}}\frac{{\partial {v_{{\text{p}}x}}}}{{\partial y}} + {v_{{\text{p}}\textit{z}}}\frac{{\partial {v_{{\text{p}}x}}}}{{\partial \textit{z}}}} \right) \hfill \\[-10pt] \end{aligned} (7)

 ${\mu _{\text{f}}}\left(\frac{{{{\text{d}}^2}{v_{{\text{p}}x}}}}{{{\text{d}}{r^2}}} + \frac{1}{r}\frac{{{\text{d}}{v_{{\text{p}}x}}}}{{{\text{d}}r}}\right) - n\frac{{{\text{d}}p}}{{{\text{d}}x}} - n\frac{{{\mu _{\text{f}}}}}{K}{v_{{\text{p}}x}} = 0,{\text{ }}r \in ({R_0}\sim R)$ (8)
1.3 流场分布解析解

 \begin{aligned}[b]& \xi =r/R,M={\mu }_{\text{eff}}/{\mu }_{\text{f}}=1/n,Da=K/{R}^{2},\\& S=1/\sqrt{MDa},U=u{\mu }_{\text{f}}/\left(G{R}^{2}\right) \end{aligned} (9)

 $\frac{{{{\text{d}}^2}{U_{\text{f}}}}}{{{\text{d}}{\xi ^2}}} + \frac{1}{\xi }\frac{{{\text{d}}{U_{\text{f}}}}}{{{\text{d}}\xi }} + 1 = 0,{\text{ }}\xi \in (0\sim \gamma )$ (10)
 $\frac{{{{\text{d}}^2}{U_{\text{p}}}}}{{{\text{d}}{\xi ^2}}} + \frac{1}{\xi }\frac{{{\text{d}}{U_{\text{p}}}}}{{{\text{d}}\xi }} + \frac{1}{M} - {S^2}{U_{\text{p}}} = 0,{\text{ }}\xi \in (\gamma \sim 1)$ (11)

 ${U_{\text{f}}} = - \dfrac{1}{4}{\xi ^2} + {A_1}\ln \xi + {A_2},{\text{ }}\xi \in (0\sim \gamma )$ (12)
 ${U_{\text{p}}} = \frac{1}{{M{S^2}}} + {A_3}{I_0}(S\xi ) + {A_4}{K_0}(S\xi ),{\text{ }}\xi \in (\gamma \sim 1)$ (13)

1）在孔中心（ξ=ξc=0），纯水流流速达到最大值：

 $\frac{{{\text{d}}{U_{\text{f}}}}}{{{\text{d}}\xi }} = 0$ (14)

2）在填充介质和纯水交界面（ξ=ξi=R0/R），满足流速连续剪应力跳跃边界条件[17-18]

 ${U_{\text{f}}} = {U_{\text{p}}}$ (15)
 $M\frac{{{\text{d}}{U_{\text{p}}}}}{{{\rm{d}}\xi }} - \frac{{{\text{d}}{U_{\text{f}}}}}{{{\rm{d}}\xi }} = \beta \frac{{{U_{\text{p}}}}}{{\sqrt {Da} }}$ (16)

3）在填充介质和不透水岩石界面（ξ=ξr=1），流速等于0：

 ${U_{\text{p}}} = 0$ (17)

 ${A_1} = 0$ (18)
 \begin{aligned}[b] {A_2} =& \dfrac{1}{4}\frac{{\gamma (2D{a^{0.5}} + \beta \gamma )\left[ {{I_0}(S){K_0}(S\gamma ) - {I_0}(S\gamma ){K_0}(S)} \right]}}{{D{a^{0.5}}MS{\varphi _1} - 4\beta {\varphi _2}}} + \hfill \\& {\text{ }}\dfrac{1}{4}\frac{{D{a^{0.5}}{K_1}(S\gamma )\left[ { - 4{I_0}(S\gamma ) + (4 + M{S^2}{\gamma ^2}){I_0}(S)} \right]}}{{D{a^{0.5}}M{S^2}{\varphi _1} - 4S\beta {\varphi _2}}} + \hfill \\& {\text{ }}\dfrac{1}{4}\frac{{D{a^{0.5}}{I_1}(S\gamma )\left[ {(4 + M{S^2}{\gamma ^2}){K_0}(S) - 4{K_0}(S\gamma )} \right]}}{{D{a^{0.5}}M{S^2}{\varphi _1} - 4S\beta {\varphi _2}}} \hfill \\ \end{aligned} (19)
 \begin{aligned}[b] {A_3} =& - \frac{{2\beta \left[ {{K_0}(S\gamma ) - {K_0}(S)} \right]}}{{2D{a^{0.5}}{M^2}{S^3}{\varphi _1} - 2M{S^2}\beta {\varphi _2}}} - \hfill \\& {\text{ }}\frac{{D{a^{0.5}}MS\left[ {S\gamma {K_0}(S) + 2{K_1}(S\gamma )} \right]}}{{2D{a^{0.5}}{M^2}{S^3}{\varphi _1} - 2M{S^2}\beta {\varphi _2}}} \hfill \\[-9pt] \end{aligned} (20)
 \begin{aligned}[b] {A_4} =& \frac{{2\beta \left[ {{I_0}(S\gamma ) - {I_0}(S)} \right]}}{{2D{a^{0.5}}{M^2}{S^3}{\varphi _1} - 2M{S^2}\beta {\varphi _2}}}{\text{ + }} \hfill \\& {\text{ }}\frac{{D{a^{0.5}}MS\left[ {S\gamma {I_0}(S) - 2{I_1}(S\gamma )} \right]}}{{2D{a^{0.5}}{M^2}{S^3}{\varphi _1} - 2M{S^2}\beta {\varphi _2}}} \hfill \\[-9pt] \end{aligned} (21)

 ${\varphi _1}{\text{ = }}{I_1}(S\gamma ){K_0}(S) + {K_1}(S\gamma ){I_0}(S)$ (22)
 ${\varphi _2}{\text{ = }}{I_0}(S\gamma ){K_0}(S) - {K_0}(S\gamma ){I_0}(S)$ (23)

 $\begin{gathered} {Q_{\text{f}}} = \int {{U_{\text{f}}}{\text{d}}{A_{\text{f}}}} {\text{ }} = \int_0^\gamma {{U_{\text{f}}}} {\text{d}}\text{π} {\xi ^2} {\text{ }} = \text{π} ({ - }1{\text{/}}8{\gamma ^4}{\text{ + }}{A_2}{\gamma ^2}) \hfill \\ \end{gathered}$ (24)
 \begin{aligned}[b] {Q_{\text{p}}}{\text{ = }}&\int {{U_{\text{p}}}{\text{d}}{A_{\text{p}}}} {\text{ }} = \int_\gamma ^1 {{U_{\text{p}}}} {\text{d}}\text{π} {\xi ^2} {\text{ }} =\hfill \frac{{\text{π} (1 - {\gamma ^2})}}{{M{S^2}}} +\\& \frac{{2\text{π} }}{S}\left( \begin{gathered} {A_3}\left[ {{I_1}(S) - \gamma {I_1}(S\gamma )} \right] - {A_4}\left[ {{K_1}(S) - \gamma {K_1}(S\gamma )} \right] \hfill \\ \end{gathered} \right) \hfill \\ \end{aligned} (25)

 $\tau {\text{ = }} - {\mu _{\text{f}}}\frac{{{\text{d}}u}}{{{\text{d}}r}} = \frac{1}{2}G{R_0}$ (26)

2 有效性检验

 图2 流速分布数值解和解析解对比 Fig. 2 Comparison of velocity distribution between analytical solution and numerical solution

 图3 界面应力跳跃 Fig. 3 Stress jump on the interface

 图4 Da变化情况下的流速分布 Fig. 4 Velocity distribution for representative values of Da

 图5 ${\boldsymbol{\gamma}}$ 变化情况下的流速分布 Fig. 5 Velocity distribution for representative values of ${\boldsymbol{\gamma}}$

3 分析与讨论

3.1 黏度比M对流速分布U的影响

Givler和Altobelli[24]通过试验发现M的范围在1.0～7.5。Al–Azmi[25]通过分析Vafai和Kim[26]提出的模型，发现在速度和应力连续的界面边界条件下，M的变化对流场流速变化影响很小。

 图6 M变化对流速分布的影响( $\;{\boldsymbol{\beta}}$ =1.5) Fig. 6 Efect of M on velocity distribution( $\;{\boldsymbol{\beta}}$ =1.5)

 图7 M变化对流速分布的影响( $\;{\boldsymbol{\beta}}$ =–0.8) Fig. 7 Effect of M on velocity distribution( $\;{\boldsymbol{\beta}}$ =–0.8)

 图8 M变化对界面流速的影响 Fig. 8 Effect of M on the interface velocity

3.2 应力跳跃系数β对流速分布U的影响

 图9 $\;{\boldsymbol{\beta}}$ 变化对流速分布的影响 Fig. 9 Effect of $\;{\boldsymbol{\beta}}$ on velocity distribution

3.3 达西数Da对流速分布U的影响

Da对流速分布的影响见图10，流场流速随着Da的增加而增加。这是因为Da与填充介质的渗透率成正比，Da的增加会使颗粒对流体的阻力减小，能量损失减小。当Da<10–5时，填充介质中的渗流流速非常小，填充介质几乎可视为不透水，此时Darcy−Brinkman公式不再适用，排水孔中纯水流满足泊肃叶定律；当Da=1时，填充介质可视为全透水，整个模型中流体可全视为纯水流。

 图10 Da变化对流速分布的影响 Fig. 10 Effect of Da on velocity distribution

 图11 Da变化对界面流速的影响 Fig. 11 Effect of Da on the interface velocity

3.4 相对开度γ对流速分布U的影响

 图12 ${\boldsymbol{\gamma}}$ 变化对流速分布的影响 Fig. 12 Effect of ${\boldsymbol{\gamma}}$ on velocity distribution

 图13 $\;{\boldsymbol{\beta}}$ 变化对界面流速的影响 Fig. 13 Effect of $\;{\boldsymbol{\beta}}$ on the interface velocity

 图14 Da变化对流速分布的影响 Fig. 14 Effect of Da on the interface velocity

4 本文模型与传统泊肃叶定律对比

 图15 本模型与传统泊肃叶定律对比 Fig. 15 Comparison between present model and Poiseuille’s law

5 结　论

1）流场流速与黏度比（M）呈负相关。当应力跳跃系数β>0时，M对0流速分布的影响很小；然而当β<0时，M对流速分布的影响十分显著，且不可忽略。

2）流场流速与应力跳跃系数（β）呈负相关。β增加，填充介质对流体的阻力增加，这会导致流场流速减小。当β<0，典型参数（MDaγ）的变化对流速分布的影响更显著。

3）流场流速与达西数（Da）呈正相关。当Da<10–5时，可认为填充介质区域不透水，Darcy−Brinkman方程不再适用。当Da=1时，填充介质对流体的阻力可忽略不计，通道内的速度分布与泊肃叶流相似。

4）流场流速与空隙相对开度（γ）呈正相关。然而，在不同的Daβ值下，界面速度并不总是与γ呈正相关。随着γ的逐渐增加，当Da=1时，界面速度逐渐减小；当Da=0.1时，界面速度先增大后减小；当Da<0.1时，界面速度逐渐增大。

5）本模型同时考虑孔洞纯水流和填充介质中渗流，比传统泊肃叶定律更合理。模型理论解可为分析部分填充排水孔泄流能力提供参考。

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