﻿ 通风及内热源参数的方腔内混合对流模拟
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 哈尔滨工程大学学报  2021, Vol. 42 Issue (2): 273-279  DOI: 10.11990/jheu.201905105 0

### 引用本文

SUN Mengnan, SONG Guiqiu, ZHOU Shihua, et al. In-cavity mixed convection simulation based on ventilation and internal heat source parameters[J]. Journal of Harbin Engineering University, 2021, 42(2): 273-279. DOI: 10.11990/jheu.201905105.

### 文章历史

1. 东北大学 机械工程与自动化学院, 辽宁 沈阳 110819;
2. 沈阳工业大学 机械工程学院, 辽宁 沈阳 110870

In-cavity mixed convection simulation based on ventilation and internal heat source parameters
SUN Mengnan 1, SONG Guiqiu 1, ZHOU Shihua 1, DONG Zhixu 2
1. School of Mechanical Engineering&Automation, Northeastern University, Shenyang 110819, China;
2. School of Mechanical Engineering, Shenyang University of Technology, Shenyang 110870, China
Abstract: To consider the influence of heat source shape and ventilation condition on fluid flow in a cavity and its heat transfer performance, mixed convection in the ventilated square cavity with a central heat-generating element was studied. The coupled algorithm of the multi-relaxation time model lattice Boltzmann method (MRT-LBM) and finite difference method (FDM) were used. Under different positions of inlet and outlet and different shapes of heat source, the flow, temperature, and heat transfer characteristics of mixed convection were analyzed, and the flow line distributions, isotherms, and Nusselt number of the heat-generating surface were given. Simulation results indicated that, when the locations of the inlet and outlet are fixed, the shape of the heat source determines the local convection heat transfer intensity but has little influence on the temperature field; the average Nusselt number increases with increasing radius R, and reaches its maximum value when the inlet is located at the middle and the outlet at the top.
Keywords: heat transfer    mixed convection    MRT-LBM    finite difference method    coupled algorithm    square cavity    shape of heat-generating element    positions of inlet and outlet

1 方腔内混合对流模型的建立 1.1 问题描述

1.2 计算模型

 $\frac{{\partial U}}{{\partial X}} + \frac{{\partial V}}{{\partial Y}} = 0$ (1)
 $\frac{{\partial U}}{{\partial \tau }} + U\frac{{\partial U}}{{\partial X}} + V\frac{{\partial U}}{{\partial Y}} = - \frac{{\partial P}}{{\partial X}} + \frac{1}{{Re}}\left( {\frac{{{\partial ^2}U}}{{\partial {X^2}}} + \frac{{{\partial ^2}U}}{{\partial {Y^2}}}} \right)$ (2)
 $\frac{{\partial V}}{{\partial \tau }} + U\frac{{\partial V}}{{\partial X}} + V\frac{{\partial V}}{{\partial Y}} = - \frac{{\partial P}}{{\partial Y}} + \frac{1}{{Re}}\left( {\frac{{{\partial ^2}V}}{{\partial {X^2}}} + \frac{{{\partial ^2}V}}{{\partial {Y^2}}}} \right) + Ri\theta$ (3)
 $\frac{{\partial \theta }}{{\partial \tau }} + U\frac{{\partial \theta }}{{\partial X}} + V\frac{{\partial \theta }}{{\partial Y}} = \frac{1}{{RePr}}\left( {\frac{{{\partial ^2}\theta }}{{\partial {X^2}}} + \frac{{{\partial ^2}\theta }}{{\partial {Y^2}}}} \right)$ (4)

 $Nu = - \frac{{\partial \theta }}{{\partial \mathit{\boldsymbol{n}}}}$ (5)
 $N{u_{{\rm{av}}}} = \frac{1}{A}\int_A {Nu} {\rm{d}}A$ (6)

 $\begin{array}{l} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \mathit{\boldsymbol{f}}(r + {c_i}\delta t,t + \delta t) - \mathit{\boldsymbol{f}}(r,t) = \\ - {\mathit{\boldsymbol{M}}^{ - 1}}\mathit{\boldsymbol{S}}(\mathit{\boldsymbol{m}}(\mathit{\boldsymbol{r}},t) - {\mathit{\boldsymbol{m}}^{{\rm{(eq)}}}}(\mathit{\boldsymbol{r}},t)) + \mathit{\boldsymbol{F}} \end{array}$ (7)

 ${c_i} = \left\{ {\begin{array}{*{20}{l}} {(0,0),}& \ \ \ \ {i{\rm{ = 0}}}\\ {c(\cos[(i - 1){\rm{ \mathsf{ π} }}/2],\sin[(i - 1){\rm{ \mathsf{ π} }}/2]),}&{{\rm{1}} \le {\rm{ }}i \le 4}\\ {\sqrt 2 c(\cos\frac{{(2i - 9){\rm{ \mathsf{ π} }}}}{4},\cos\frac{{(2i - 9){\rm{ \mathsf{ π} }}}}{4}),}&{{\rm{5}} \le {\rm{ }}i \le 8} \end{array}} \right.$ (8)
 $\omega = \left( {\frac{{\rm{4}}}{{\rm{9}}}{\rm{, }}\frac{{\rm{1}}}{{\rm{9}}}{\rm{, }}\frac{{\rm{1}}}{{\rm{9}}}{\rm{, }}\frac{{\rm{1}}}{{\rm{9}}}{\rm{, }}\frac{{\rm{1}}}{{\rm{9}}}{\rm{, }}\frac{{\rm{1}}}{{{\rm{36}}}}{\rm{, }}\frac{{\rm{1}}}{{{\rm{36}}}}{\rm{, }}\frac{{\rm{1}}}{{{\rm{36}}}}{\rm{, }}\frac{{\rm{1}}}{{{\rm{36}}}}} \right)$ (9)
 $\begin{array}{*{20}{c}} {{F_i} = 3{\omega _i}{\rho _0}\alpha (T - {T_0})g\frac{{{\mathit{\boldsymbol{c}}_i}\mathit{\boldsymbol{\bar y}}}}{{{c^2}}}}&{i{\rm{ = 0}},{\rm{ 1}},{\rm{ }} \cdots ,{\rm{ 8}}} \end{array}$ (10)

 $\rho = \sum\limits_{i = 0}^8 {{f_i}}$ (11)
 $\rho \mathit{\boldsymbol{u}} = \sum\limits_{i = 0}^8 {{f_i}} {c_i} + \delta t\mathit{\boldsymbol{F}}$ (12)

 ${f_1}({x_w},t + \delta t) = (1 - q){f_1}({x_f},t + \delta t) + q{f_1}({x_s},t + \delta t)$ (13)
 Download: 图 3 曲面插值边界条件 Fig. 3 Interpolation boundary condition for curved wall

 $\begin{array}{l} {f_3}({x_s},t) = \frac{1}{{1 + q}}[q{f_3}({x_f},t) + q{f_1}({x_f},t) + \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} (1 - q){f_1}({x_{ff}},t)] \end{array}$ (14)

 $\frac{{\partial \theta }}{{\partial t}} = {\theta ^{n + 1}}(m,n) - {\theta ^n}(m,n)$ (15)
 $\left\{ \begin{array}{l} \frac{{\partial \theta }}{{\partial x}} = \frac{{\theta (m + 1,n) - \theta (m - 1,n)}}{2}\\ \frac{{\partial \theta }}{{\partial y}} = \frac{{\theta (m,n + 1) - \theta (m,n - 1)}}{2} \end{array} \right.$ (16)
 $\left\{ \begin{array}{l} \frac{{{\partial ^2}\theta }}{{\partial {x^2}}} = \theta (m + 1,n) - 2\theta (m,n) + \theta (m - 1,n)\\ \frac{{{\partial ^2}\theta }}{{\partial {y^2}}} = \theta (m,n + 1) - 2\theta (m,n) + \theta (m,n - 1) \end{array} \right.$ (17)

1.3 模型验证

 Download: 图 4 本文和文献[9]的流场和温度场的算例验证 Fig. 4 Comparison of streamlines and isotherms between the present work and that of ref.[9]

2 对流换热的计算结果与分析

 Download: 图 5 进流口位于底部时，R和出流口位置对流函数线和等温线的影响 Fig. 5 Effects of R and outlet location on the streamlines and isotherms for cavity with bottom inlet
 Download: 图 6 进流口位于中部时，R和出流口位置对流函数线和等温线的影响 Fig. 6 Effects of R and outlet location on the streamlines and isotherms for cavity with middle inlet
 Download: 图 7 进流口位于顶部时，R和出流口位置对流函数线和等温线的影响 Fig. 7 Effects of R and outlet location on the streamlines and isotherms for cavity with top inlet

 Download: 图 8 不同腔体结构下，R对局部努赛尔数Nu的影响 Fig. 8 Effect of R on local Nusselt number in different cavity structures

 Download: 图 9 不同腔体结构下，R对平均努赛尔数Nuav的影响 Fig. 9 Effect of R on average Nusselt number in different cavity structures
3 结论

1) 流场和温度场分布与热源形状关系不大，而主要受进、出流口位置影响。当进流口位于腔体底部或顶部时，随着出流口位置与进流口位置之间距离的增大，热源周围的环流被挤压，并逐渐被外部来流所取代；而当进流口位于中部时，外部来流始终流经热源的上、下表面，出流口位置不改变腔内流体的流动模式。在温度场中，由于受到外部强制对流的影响，等温线会向进流口附近的热源表面以及出流口位置聚集。

2) 当热源截面为圆形时(R=0.5d)，热源表面的局部努赛尔数Nu曲线具有平缓变化的单峰值；随着R的减小，Nu曲线上会逐渐出现多个尖锐峰值。Nu的最大值出现在热源表面靠近进流口位置的一侧，而出流口位置对Nu的影响不大。当进、出流口分别位于顶部和底部时，可在正方形热源截面(R=0)的左上角获得最佳局部对流换热位置。

3) 当进、出流口位置固定时，热源表面的平均努赛尔数Nuav随着R的增大而增大。当R不改变时，Nuav的最大值在方腔具有中部进流口和顶部出流口时出现。因此，在进、出流口分别位于中部和顶部时，采用圆形截面的热源(R=0.5d)可以使热源整体表面的对流换热最强。

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