粘性/粘弹性流体流动和热迁移问题的微分求积法
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摘要
一般讲，在流体力学中由于控制方程是非常复杂的非线性方程组，所以不可能得到问题的精确解。因此为了得到非线性方程组的解，提出了各种数值计算方法，其中，有限差分法(FD)和有限元(FE)是两类常用的方法。实际上，在许多场合中我们只需要在少数点上求得适当精度的解就够了。但是在采用FD和FE时，为了得到在少数点上适当精确的解往往需要使用大量的网格点。因此当使用这些方法时需要很大的工作量和存储量。但是，如果采用1970年Bellman提出的微分求积法(DQ)则只需要少量的节点就能得到较高精度的解。此外，由于DQ还具有使用方便、节点间距选取任意等优点，因此近年来吸引了许多研究者的注意。
传统的DQ只适用于正规区域的问题，并且缺少迎风机制来处理流体流动的对流性质。为了使DQ能适用于求解不规则区域中的流体流动问题，本文中提出了一种具有迎风机制的局部化的DQ(称为ULDQ)。利用ULDQ对一些不可压与热迁移耦合的粘性和粘弹性流体的二维流动问题求得了满意的数值结果。本文的主要成果如下：
1．因为在传统的DQM中缺乏迎风机制来刻画流体流动中的对流特性，所以当雷诺数Re较大时，流体流动的数值试验常常失败。在本文中提出了一种预估校正法，它在每一时间迭代步中，对对流项先用传统的迎风差分格式进行了预估，然后用DQ对全部空间变量进行校正，这种方法称为混合微分求积法。利用这种方法对二维不可压的Navier-Stokes方程和热迁移的耦合问题进行了数值模拟。结果表明这种混合微分求积法对较大雷诺数时的流体的计算仍是适用的，和传统的有限差分方法相比，混合微分求积法具有较好的收敛性、较高的精度和较少工作量等优点。
2．虽然微分求积法已被用来求解许多流体力学中的问题，但它们仅局限于正规区域的解，同时缺乏迎风机制来刻画流体流动的对流性质。本文中提出了一种方法，它直接把迎风机制引入到DQ中，同时使用一种局部化的技巧来处理不规则区域的流动问题。我们称经过上述改进的DQ为迎风的局部微分求积法。利用这种方
Shanghai University DOctoral Dissertation
法对一个在不规则区域中的二维不可压的流体和热迁移的祸合问题进行了数值模
拟。和差分法相比较，这种方法是更为精确的，并且具有较少的工作盘。
3.本文讨论了在两个平行板中的不可压的二阶粘弹性流体的二维稳态流体问
题。利用对方程中的小参数进行展开的方法，求得了零阶和一阶近似方程。利用
DQM和本文提出的一种迭代方法成功地得到了问题的数值解。数值结果表明在入
口处，流体的弹性对流动有明显的影响:而在离入口较远处，流体弹性的影响是很
微弱。
4.进一步讨论了在两个平行板中的不可压的二阶粘弹性流体和热迁移藕合的
二维稳态流动问题，其中，在能量方程中包含了一个粘性的耗散函数。当流体的弹
性性质微弱时，利用摄动方法得到了零阶和一阶近似方程.为了在管壁附近得到较
高精度的数值解，半区域被分成两个小区域，其一是靠近板壁的薄区域，称为内部
区域;另一是厚的区域称为外部区域.在内部区域和外部区域中的控制方程分别用
DQ方法进行离散;并对得到的离散方程用分界面处的匹配条件连接起来(这种技
巧称为区域分裂技巧)。本文给出了一种迭代方法来求解所得到的离散方程组。利
用这种区域分裂技巧和DQ方法成功地完成了两个平行平板之间的二维不可压二阶
粘弹性流体的热迁移祸合的稳态流动的数值模拟。所得数值结果和已有结果比较在
定性上是非常吻合的。根据数值结果以看到，在中心线附近弹性的影响是微弱的，
在交界面和平板附近，弹性的影响是很强地。同时数值结果也表明，使用区域分裂
技巧的DQ是一种非常有效的离散方法，它具有精度高、收敛性好以及工作量少等
优点。
In general, it is impossible to obtain the exact solutions of problems in fluid mechanics, due to their governing equations are a very complicated system of nonlinear equations. Hence various numerical methods are presented for solving these ones. The finite difference (FD) method and finite element (FE) method are two kinds of usual methods. In many cases it is enough to obtain moderately accurate solutions at a few points. But one has to use a large quantity of grid points in order to obtain moderately accurate solutions at a few points as the FD method and FE method are adopted. Thus, a large quantity of the computational workloads and storages are required when these methods are used. However, for the differential quadrature (DQ) method proposed by R. Bellman in 1970, it only needs applying a few grid points in order to get high-precise solutions. Furthermore the DQ method possesses advantages including easy to use and arbitrary to choose the grid spacing. Hence the DQ method has attracted many researche
rs attention in recent years.
But the traditional DQ method is only suitable for solving problems with regular domains and there is lack of upwind mechanism to treat with the convection in the fluid flow. In order to make the DQ method to be appropriate for solving problems in the fluid mechanics with irregular domains, a localized DQ method having upwind mechanism (ULDQ) is proposed in this dissertation. By using the ULDQ method, some numerical results for incompressible two-dimensional flow problems of viscous or viscoelastic fluid coupled with heat transfer are obtained. The main results contain as follows:
1. Because there is a lack of upwind mechanism to characterize the convection of the fluid flow in the traditional differential quadrature method, so the numerical experiments for fluid flow are usually became defeat as the Reynolds number are larger. In the Chapter 3, at the every temporal iterative step, a prediction-adjustment method in which at first the traditional upwind difference scheme is applied to predict for convective terms, then the DQ method are used to adjust all terms in spatial variables. This method is called the mixed differential quadrature method. By using this mixed method the numerical experiments for the coupled problems of two-dimensional incompressible Navier-Stokes
equations with heat equation are made successfully. The results obtained show that the mixed differential quadrature method is appropriate to solve the fluid flow with higher Reynolds numbers. These results also point out that the mixed differential quadrature method owns advantages including the good convergence, high accuracy and less workloads comparing with the conventional differential quadrature method.
2 . Although the differential quadrature method has been applied to successfully solve various problems in fluid mechanics, it is limited with regular domains and an absence upwind mechanism to characterize the convection of the fluid flow. The upwind mechanism is directly introduced into the traditional DQ method and a localization technique is applied to deal with the irregularity of flow regions in Chapter 4. The DQ method improved above is called the upwind local differential quadrature method. By using this method, the numerical simulations for the coupled problem of incompressible laminar flow with heat transfer in an irregular region are made successfully. Comparing with the low-order finite difference method, the upwind local differential quadrature method is more accurate and it only requires less computational workload.
3. A problem of two-dimensional steady flow for an incompressible second-order viscoelastic fluid between two parallel plates is discussed by using the perturbation technique in Chapter 5. By expanding the governing equations with respect to a small parameter, the zero and first order approximation equations are obtained. By using the differential quadrature method and an iterative technique presented in the thesis the numerical solution is successfully obtained.
The numerical r
传统的DQ只适用于正规区域的问题，并且缺少迎风机制来处理流体流动的对流性质。为了使DQ能适用于求解不规则区域中的流体流动问题，本文中提出了一种具有迎风机制的局部化的DQ(称为ULDQ)。利用ULDQ对一些不可压与热迁移耦合的粘性和粘弹性流体的二维流动问题求得了满意的数值结果。本文的主要成果如下：
1．因为在传统的DQM中缺乏迎风机制来刻画流体流动中的对流特性，所以当雷诺数Re较大时，流体流动的数值试验常常失败。在本文中提出了一种预估校正法，它在每一时间迭代步中，对对流项先用传统的迎风差分格式进行了预估，然后用DQ对全部空间变量进行校正，这种方法称为混合微分求积法。利用这种方法对二维不可压的Navier-Stokes方程和热迁移的耦合问题进行了数值模拟。结果表明这种混合微分求积法对较大雷诺数时的流体的计算仍是适用的，和传统的有限差分方法相比，混合微分求积法具有较好的收敛性、较高的精度和较少工作量等优点。
2．虽然微分求积法已被用来求解许多流体力学中的问题，但它们仅局限于正规区域的解，同时缺乏迎风机制来刻画流体流动的对流性质。本文中提出了一种方法，它直接把迎风机制引入到DQ中，同时使用一种局部化的技巧来处理不规则区域的流动问题。我们称经过上述改进的DQ为迎风的局部微分求积法。利用这种方
Shanghai University DOctoral Dissertation
法对一个在不规则区域中的二维不可压的流体和热迁移的祸合问题进行了数值模
拟。和差分法相比较，这种方法是更为精确的，并且具有较少的工作盘。
3.本文讨论了在两个平行板中的不可压的二阶粘弹性流体的二维稳态流体问
题。利用对方程中的小参数进行展开的方法，求得了零阶和一阶近似方程。利用
DQM和本文提出的一种迭代方法成功地得到了问题的数值解。数值结果表明在入
口处，流体的弹性对流动有明显的影响:而在离入口较远处，流体弹性的影响是很
微弱。
4.进一步讨论了在两个平行板中的不可压的二阶粘弹性流体和热迁移藕合的
二维稳态流动问题，其中，在能量方程中包含了一个粘性的耗散函数。当流体的弹
性性质微弱时，利用摄动方法得到了零阶和一阶近似方程.为了在管壁附近得到较
高精度的数值解，半区域被分成两个小区域，其一是靠近板壁的薄区域，称为内部
区域;另一是厚的区域称为外部区域.在内部区域和外部区域中的控制方程分别用
DQ方法进行离散;并对得到的离散方程用分界面处的匹配条件连接起来(这种技
巧称为区域分裂技巧)。本文给出了一种迭代方法来求解所得到的离散方程组。利
用这种区域分裂技巧和DQ方法成功地完成了两个平行平板之间的二维不可压二阶
粘弹性流体的热迁移祸合的稳态流动的数值模拟。所得数值结果和已有结果比较在
定性上是非常吻合的。根据数值结果以看到，在中心线附近弹性的影响是微弱的，
在交界面和平板附近，弹性的影响是很强地。同时数值结果也表明，使用区域分裂
技巧的DQ是一种非常有效的离散方法，它具有精度高、收敛性好以及工作量少等
优点。
In general, it is impossible to obtain the exact solutions of problems in fluid mechanics, due to their governing equations are a very complicated system of nonlinear equations. Hence various numerical methods are presented for solving these ones. The finite difference (FD) method and finite element (FE) method are two kinds of usual methods. In many cases it is enough to obtain moderately accurate solutions at a few points. But one has to use a large quantity of grid points in order to obtain moderately accurate solutions at a few points as the FD method and FE method are adopted. Thus, a large quantity of the computational workloads and storages are required when these methods are used. However, for the differential quadrature (DQ) method proposed by R. Bellman in 1970, it only needs applying a few grid points in order to get high-precise solutions. Furthermore the DQ method possesses advantages including easy to use and arbitrary to choose the grid spacing. Hence the DQ method has attracted many researche
rs attention in recent years.
But the traditional DQ method is only suitable for solving problems with regular domains and there is lack of upwind mechanism to treat with the convection in the fluid flow. In order to make the DQ method to be appropriate for solving problems in the fluid mechanics with irregular domains, a localized DQ method having upwind mechanism (ULDQ) is proposed in this dissertation. By using the ULDQ method, some numerical results for incompressible two-dimensional flow problems of viscous or viscoelastic fluid coupled with heat transfer are obtained. The main results contain as follows:
1. Because there is a lack of upwind mechanism to characterize the convection of the fluid flow in the traditional differential quadrature method, so the numerical experiments for fluid flow are usually became defeat as the Reynolds number are larger. In the Chapter 3, at the every temporal iterative step, a prediction-adjustment method in which at first the traditional upwind difference scheme is applied to predict for convective terms, then the DQ method are used to adjust all terms in spatial variables. This method is called the mixed differential quadrature method. By using this mixed method the numerical experiments for the coupled problems of two-dimensional incompressible Navier-Stokes
equations with heat equation are made successfully. The results obtained show that the mixed differential quadrature method is appropriate to solve the fluid flow with higher Reynolds numbers. These results also point out that the mixed differential quadrature method owns advantages including the good convergence, high accuracy and less workloads comparing with the conventional differential quadrature method.
2 . Although the differential quadrature method has been applied to successfully solve various problems in fluid mechanics, it is limited with regular domains and an absence upwind mechanism to characterize the convection of the fluid flow. The upwind mechanism is directly introduced into the traditional DQ method and a localization technique is applied to deal with the irregularity of flow regions in Chapter 4. The DQ method improved above is called the upwind local differential quadrature method. By using this method, the numerical simulations for the coupled problem of incompressible laminar flow with heat transfer in an irregular region are made successfully. Comparing with the low-order finite difference method, the upwind local differential quadrature method is more accurate and it only requires less computational workload.
3. A problem of two-dimensional steady flow for an incompressible second-order viscoelastic fluid between two parallel plates is discussed by using the perturbation technique in Chapter 5. By expanding the governing equations with respect to a small parameter, the zero and first order approximation equations are obtained. By using the differential quadrature method and an iterative technique presented in the thesis the numerical solution is successfully obtained.
The numerical r
引文
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[111] Wang X., Wang Y. and Zhou Y. Application of a new vibrational analysis of beams and frame structures. J. of Sound and Vibration, 2004, 269:1133-1141.
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[113] Zhu Y.D., Shu C., Qiu J. and Tani J. Numerical simulation of natural convection between two elliptical cylinders using DQ method. Int. J. Heat Mass Transfer, 2004, 47(4):797-808.