气冷涡轮气热弹耦合有限差分算法研究
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摘要
航空工业的迅速发展对航空发动机提出了更高的要求,即高的推重比和热效率,导致涡轮的进口温度不断提高,目前该温度已远远高于涡轮材料的耐热极限,这就需要采用复杂的冷却技术来保证涡轮叶片安全的工作和较长的工作寿命。准确预测叶片内热传导以及热应力的分布是提高涡轮冷却效率、保证叶片工作寿命的关键性问题,气热弹数值仿真技术是解决该问题的有效方法和工具之一。本文主要对应用有限差分方法进行气冷涡轮气热弹数值仿真的关键性问题进行研究,并对气冷涡轮内气热弹耦合现象进行初步探讨。
首先通过深入分析气冷涡轮内复杂的多场耦合现象,研究了气热弹多场耦合的物理模型,给出了以柱坐标系下参数为不变量的任意曲线坐标系下的无量纲时间平均雷诺N-S方程,给出了直角坐标系中考虑热传导、热辐射以及热应变对能量耗散影响的温度场方程并将其在任意曲线坐标系中展开;推导了直角坐标系和任意曲线系中考虑热变形影响以及不考虑热变形影响的以位移为求解量的弹性固体应力场平衡微分方程的表达式;讨论了气热、气弹、热弹、气热弹等四类多场耦合边界条件的给定方式,从而建立了多场耦合的计算模型。由于采用了多块结构化网格离散气冷涡轮气热弹耦合的计算域,又给出了四类不同耦合计算网格块之间的数值传递方法,实现了多场耦合网格之间的数据传递。
然后研究了流场、温度场、弹性固体应力场求解的差分格式及其数值方法,这是应用有限差分法进行涡轮气热弹多场耦合计算中的重要问题。三维粘性流场控制方程采用具有Godunov性质的三阶TVD格式离散,并采用AF方法求解。温度场采用了稳态和非稳态的两种求解方式:稳态温度场采用隐式ADI交替方向方法求解;对于非稳态温度场,构造紧致格式离散一次和二次交叉偏微分项,并应用高阶格式的ADI法求解离散后的方程,该方法具有时间方向一阶精度、空间方向四阶精度,经验证表明,该方法较显格式和C-N格式具有更高的求解精度。引入位移法求解二维弹性应力方程,为了验证该方法的可行性和可靠性,对具有解析解的悬臂梁进行受力分析,结果表明:第二类边界条件的求解精度对整个计算的结果有重大的影响,平衡方程二阶精度的计算值与解析解吻合程度好,而一阶精度的计算值偏离解析解较大。然后对三维弹性固体应力场控制方程进行了无量纲化。在Possion模型方程的基础上推导了交替方向求解方式,分别构造了空间方向三点(二阶精度)及五点(四阶精度)的差分格式,在求解中,有变量分离式以及耦合式两种方法供选择。计算表明:高阶格式具有较高的求解精度;变量分离式求解运算量小,收敛速度快,耦合式求解具有较好的稳定性,但计算耗时长。然后对应力场求解的稳定性问题进行了研究。计算表明,平衡方程求解的稳定性决定着整个应力场求解的稳定性。边界上的平衡方程和内部节点的平衡微分方程的耦合求解具有较好的稳定性,而两类方程分离计算的算法和编译程序相对容易实现,多块网格之间的数据传递也较易实现。为了提高迭代计算的稳定性,对微分方程离散后形成的线性方程组矩阵主对角线元素进行了修改,需要指出的是,这会导致迭代速度有所降低。经解析解算例对比表明:采用上述方法的数值结果与解析结果吻合较好,其中内部节点值较边界节点值吻合得更好。
接着建立了应用有限差分方法的气冷涡轮气热弹耦合数值仿真平台,并对MARKII型叶片进行气热弹耦合计算分析。由CFX10多个湍流模型、转捩模型结果得知在MARKII型叶片整个前缘边界层内流动为层流状态,而在吸力面的中后部因为激波分离诱发转捩变为湍流,在靠近压力面的尾缘区域,边界层流动逐渐变为全湍流。使用HIT-3D (哈尔滨工业大学自行研发的三维流场计算程序)耦合求解器的气热耦合计算表明,采用各种湍流、转捩模型都能够得到与真实流动吻合较好的边界层外流场,但其对边界层内部的流动和传热过程的模拟能力差别很大,B-L代数模型将整个流场当作湍流来处理,不能辨认出来转捩,求解的壁面温度和换热系数均高于实验值;低雷诺数的q ?ω模型可以部分地模拟转捩的效果,预测的叶片温度以及热传导系数优于B-L代数模型;而考虑了转捩影响的B-L&AGS模型对壁面的温度预测较其它两种模型预测更为准确,但由于没有考虑间歇因子的沿壁面法向的输运效应,在局部预测的对流换热系数低于实验值。
最后在气热耦合计算结果的基础上对MARKII型叶片进行气弹和热弹分析,结果表明:相对于温度载荷对叶片的作用力产生的形变及应力,气动力载荷对叶片的作用力产生的形变及应力在数量级相差较大,是一个小量;而叶片内的热变形、热应力与温度场、温度梯度、叶片形状以及叶片的约束有关,温度越高叶片形变越大,温度梯度越大叶片应力值越大。同时采用不同湍流、转捩模型的结果对比发现:以B-L湍流模型计算的温度场为载荷的叶片热应力要高于以B-L&AGS转捩模型以及q ?ω模型的相应结果,其中B-L&AGS模型的叶片表面温度值低、叶片内部温度梯度小,具有较小的热应力值。为了验证本文开发的有限差分程序的热弹分析能力,又采用了有限元法的ANSYS程序进行了相同条件下的热弹计算,对比表明,在同条件下这两个求解器的结果很接近,这就初步表明本程序已经具备了一定的热弹分析的能力。
To improve the thrust-weight ratio and thermal cycle efficiency, the gas temperature at turbine inlet has been increased, and it has greatly exceeded the yielding limit of the metal material. Thus an effect cooling system is required to maintain the blade operation. It is a key problem for improving the cooling efficiency that the thermal load and the thermal stress in the blade should be accurately predicted. And the thermal-flow-elastic coupling technique has shown its great potential in solving the problem mentioned above. The purpose of the paper is to investigate the key problems of applying finite difference method to thermal-flow-elastic coupling simulations, and to study the thermal-flow-elastic problems in the air-cooled turbines.
Firstly the physical model of the thermal-flow-elastic coupling problems is studied, and the controlling equations are deduced. Such equations consist of three parts: (1) the dimensionless time-averaged N-S equation systems in both the cylindrical coordinates and body-fitted coordinates, (2) the thermal transport controlling equation taking account of the effects of thermal conduction, thermal radiation and thermal deformation on the thermal dissipation in the Cartesian coordinates and its expanded form in the arbitrary curvilinear coordinates, (3) elastic stress equilibrium differential equations with displacement the solving variable both considering and not considering the effects of thermal deformation in the Cartesian coordinates and their expansion in the arbitrary curvilinear coordinates. Then the posing methods of coupling boundary conditions, including coupled heat transfer, flow-elastic, thermal-elastic, and thermal-flow-elastic coupling conditions, are analyzed. The numerical models for the multi-field coupling problems are constructed based on the controlling equations and the coupling boundary conditions. Since the multi-block structured grids are employed to discretize the computational field, the data transfer methods for such four kinds of coupling simulations are also provided.
Secondly the discretizing schemes and numerical methods for solving the flow, thermal and elastic controlling equations are studied because of quite the importance of such problem for the thermal-flow-elastic simulations employing finite difference method. For the 3-D viscous flow controlling equations the third-order accurate TVD difference scheme with Godunov characteristic is employed. For the thermal field controlling equation there are two kinds of numerical methods, viz. an implicit ADI method for the steady thermal controlling equation and another kind of ADI method combining a kind of high-order accurate scheme for the unsteady one. The latter method uses a compact scheme to discretize the first- and the second-order cross partial differential terms, and the method is with first-order accuracy in time and forth-order accuracy in space, which is higher than those of the explicit method and the C-N method. For the 2-D elastic stress controlling equations the displacement method is utilized, since the Dirichlet problem is easier constructed with the displacement method than that with the stress method. And projecting beam is selected as the validation case. The comparison between the predicted stress distribution and the analytic one show that the accuracy of the Neumann problems greatly affects the final numerical results, and that there are slight deviation between the numerical result and the analytic one with a second-order scheme for the equilibrium equation but large deviation with a first-order scheme. The dimensionless 3-D solid elastic stress controlling equations are deduced, and the orders of magnitude of the variables and their coefficients are obtained. For the sake of convenience in constructing solving method, the elliptic elastostatics equations are selected as the controlling equations. And ADI method is deduced on the basis of Possion modeling equation, and two kinds of difference scheme, a three-node one and a five-node one, are constructed. The variable-separating solving method and the coupling solving method are utilized in the simulations separately, and the results show that the former one is with less computational load and converging speed than the latter one, but the latter one is with quite nice stability. For the equilibrium equation containing Neumann problem, its computational stability is crucial to that of the whole iteration. For the equilibrium equation at the boundary nodes and the equilibrium differential equation at the inner nodes, the coupling method is with fairly good computational stability, but the separating method is with simpler algorithm. In addition the latter one could be easily programmed, and it is also with simple formula for the data transfer between multi-block grids. To improve the stability of computation, the principle diagonal elements of discretized linear equations system matrixes are modified, which unfortunately reduces the computational speed. The simulation with the methods mentioned above is carried out. And the numerical results, especially those at the inner nodes, agree rather well with the analytic ones.
Thirdly the thermal-flow-elastic coupling solver employing finite difference method is developed, and the thermal-flow-elastic coupling simulations by the solver are carried out, with the test case as MARKII guide vane. The simulations by use of CFX10 with several turbulence models andγ?θtransition model show that the laminar flow exists at the whole leading edge of the vane, and that because of the strong shock wave at the suction midst and the strengthened instable flow at the pressure trailing edge the turbulent flow occurs at the aft suction side and the pressure trailing edge. Coupled heat transfer simulation utilizing HIT-3D (a CHT solver developed by Harbin Institute of Technology) with B-L and q ?ωturbulence models and B-L&AGS transition model are accomplished. The predicted flow fields in the main flow field agree well with the measured one, otherwise the predicted boundary layer flows and vane thermal loads differ from the employed models. The B-L algebraic turbulence model, not able to model transition process, over-predicts the temperature and heat transfer coefficient (HTC). q ?ωlow-Re turbulence model, able to model the effects of transition on the flow and heat transfer, predicts temperature and HTC distributions with less deviations from the measured ones than the B-L model does. And the B-L&AGS transition model, able to model the transition process, predicts thermal load agreeing best to the measured one, otherwise it under-predicts the HTC at several part of the vane surface since such model neglects the transport of the intermittency along the outer normal direction of the wall.
Finally the flow-elastic coupling and thermal-elastic coupling simulations of the MARKII vane, on the basis of CHT results, are carried out. Compared to the strain and stress caused by thermal load induced acting force, those caused by aerodynamic load induced acting force is rather small. The thermal-elastic results reveal that the thermal deformation and thermal stress of the vane are influenced by the thermal field, temperature gradient, vane geometry and the constraint on the vane, high temperature and temperature gradient leading to large vane deformation and vane stress separately. The results also reveal that the thermal stress cause by the thermal load predicted by different turbulence and transition models differ from the model selected. That with B-L model is higher than those with the other models, and that with B-L&AGS transition model is with the smallest value. For the sake of comparison, there is thermal-elastic simulation utilizing ANSYS, a finite element solver. And the differences of the predicted thermal-elastic results by the different solvers are quite slight, which proves the ability of the developed solver in thermal-elastic analysis.
首先通过深入分析气冷涡轮内复杂的多场耦合现象,研究了气热弹多场耦合的物理模型,给出了以柱坐标系下参数为不变量的任意曲线坐标系下的无量纲时间平均雷诺N-S方程,给出了直角坐标系中考虑热传导、热辐射以及热应变对能量耗散影响的温度场方程并将其在任意曲线坐标系中展开;推导了直角坐标系和任意曲线系中考虑热变形影响以及不考虑热变形影响的以位移为求解量的弹性固体应力场平衡微分方程的表达式;讨论了气热、气弹、热弹、气热弹等四类多场耦合边界条件的给定方式,从而建立了多场耦合的计算模型。由于采用了多块结构化网格离散气冷涡轮气热弹耦合的计算域,又给出了四类不同耦合计算网格块之间的数值传递方法,实现了多场耦合网格之间的数据传递。
然后研究了流场、温度场、弹性固体应力场求解的差分格式及其数值方法,这是应用有限差分法进行涡轮气热弹多场耦合计算中的重要问题。三维粘性流场控制方程采用具有Godunov性质的三阶TVD格式离散,并采用AF方法求解。温度场采用了稳态和非稳态的两种求解方式:稳态温度场采用隐式ADI交替方向方法求解;对于非稳态温度场,构造紧致格式离散一次和二次交叉偏微分项,并应用高阶格式的ADI法求解离散后的方程,该方法具有时间方向一阶精度、空间方向四阶精度,经验证表明,该方法较显格式和C-N格式具有更高的求解精度。引入位移法求解二维弹性应力方程,为了验证该方法的可行性和可靠性,对具有解析解的悬臂梁进行受力分析,结果表明:第二类边界条件的求解精度对整个计算的结果有重大的影响,平衡方程二阶精度的计算值与解析解吻合程度好,而一阶精度的计算值偏离解析解较大。然后对三维弹性固体应力场控制方程进行了无量纲化。在Possion模型方程的基础上推导了交替方向求解方式,分别构造了空间方向三点(二阶精度)及五点(四阶精度)的差分格式,在求解中,有变量分离式以及耦合式两种方法供选择。计算表明:高阶格式具有较高的求解精度;变量分离式求解运算量小,收敛速度快,耦合式求解具有较好的稳定性,但计算耗时长。然后对应力场求解的稳定性问题进行了研究。计算表明,平衡方程求解的稳定性决定着整个应力场求解的稳定性。边界上的平衡方程和内部节点的平衡微分方程的耦合求解具有较好的稳定性,而两类方程分离计算的算法和编译程序相对容易实现,多块网格之间的数据传递也较易实现。为了提高迭代计算的稳定性,对微分方程离散后形成的线性方程组矩阵主对角线元素进行了修改,需要指出的是,这会导致迭代速度有所降低。经解析解算例对比表明:采用上述方法的数值结果与解析结果吻合较好,其中内部节点值较边界节点值吻合得更好。
接着建立了应用有限差分方法的气冷涡轮气热弹耦合数值仿真平台,并对MARKII型叶片进行气热弹耦合计算分析。由CFX10多个湍流模型、转捩模型结果得知在MARKII型叶片整个前缘边界层内流动为层流状态,而在吸力面的中后部因为激波分离诱发转捩变为湍流,在靠近压力面的尾缘区域,边界层流动逐渐变为全湍流。使用HIT-3D (哈尔滨工业大学自行研发的三维流场计算程序)耦合求解器的气热耦合计算表明,采用各种湍流、转捩模型都能够得到与真实流动吻合较好的边界层外流场,但其对边界层内部的流动和传热过程的模拟能力差别很大,B-L代数模型将整个流场当作湍流来处理,不能辨认出来转捩,求解的壁面温度和换热系数均高于实验值;低雷诺数的q ?ω模型可以部分地模拟转捩的效果,预测的叶片温度以及热传导系数优于B-L代数模型;而考虑了转捩影响的B-L&AGS模型对壁面的温度预测较其它两种模型预测更为准确,但由于没有考虑间歇因子的沿壁面法向的输运效应,在局部预测的对流换热系数低于实验值。
最后在气热耦合计算结果的基础上对MARKII型叶片进行气弹和热弹分析,结果表明:相对于温度载荷对叶片的作用力产生的形变及应力,气动力载荷对叶片的作用力产生的形变及应力在数量级相差较大,是一个小量;而叶片内的热变形、热应力与温度场、温度梯度、叶片形状以及叶片的约束有关,温度越高叶片形变越大,温度梯度越大叶片应力值越大。同时采用不同湍流、转捩模型的结果对比发现:以B-L湍流模型计算的温度场为载荷的叶片热应力要高于以B-L&AGS转捩模型以及q ?ω模型的相应结果,其中B-L&AGS模型的叶片表面温度值低、叶片内部温度梯度小,具有较小的热应力值。为了验证本文开发的有限差分程序的热弹分析能力,又采用了有限元法的ANSYS程序进行了相同条件下的热弹计算,对比表明,在同条件下这两个求解器的结果很接近,这就初步表明本程序已经具备了一定的热弹分析的能力。
To improve the thrust-weight ratio and thermal cycle efficiency, the gas temperature at turbine inlet has been increased, and it has greatly exceeded the yielding limit of the metal material. Thus an effect cooling system is required to maintain the blade operation. It is a key problem for improving the cooling efficiency that the thermal load and the thermal stress in the blade should be accurately predicted. And the thermal-flow-elastic coupling technique has shown its great potential in solving the problem mentioned above. The purpose of the paper is to investigate the key problems of applying finite difference method to thermal-flow-elastic coupling simulations, and to study the thermal-flow-elastic problems in the air-cooled turbines.
Firstly the physical model of the thermal-flow-elastic coupling problems is studied, and the controlling equations are deduced. Such equations consist of three parts: (1) the dimensionless time-averaged N-S equation systems in both the cylindrical coordinates and body-fitted coordinates, (2) the thermal transport controlling equation taking account of the effects of thermal conduction, thermal radiation and thermal deformation on the thermal dissipation in the Cartesian coordinates and its expanded form in the arbitrary curvilinear coordinates, (3) elastic stress equilibrium differential equations with displacement the solving variable both considering and not considering the effects of thermal deformation in the Cartesian coordinates and their expansion in the arbitrary curvilinear coordinates. Then the posing methods of coupling boundary conditions, including coupled heat transfer, flow-elastic, thermal-elastic, and thermal-flow-elastic coupling conditions, are analyzed. The numerical models for the multi-field coupling problems are constructed based on the controlling equations and the coupling boundary conditions. Since the multi-block structured grids are employed to discretize the computational field, the data transfer methods for such four kinds of coupling simulations are also provided.
Secondly the discretizing schemes and numerical methods for solving the flow, thermal and elastic controlling equations are studied because of quite the importance of such problem for the thermal-flow-elastic simulations employing finite difference method. For the 3-D viscous flow controlling equations the third-order accurate TVD difference scheme with Godunov characteristic is employed. For the thermal field controlling equation there are two kinds of numerical methods, viz. an implicit ADI method for the steady thermal controlling equation and another kind of ADI method combining a kind of high-order accurate scheme for the unsteady one. The latter method uses a compact scheme to discretize the first- and the second-order cross partial differential terms, and the method is with first-order accuracy in time and forth-order accuracy in space, which is higher than those of the explicit method and the C-N method. For the 2-D elastic stress controlling equations the displacement method is utilized, since the Dirichlet problem is easier constructed with the displacement method than that with the stress method. And projecting beam is selected as the validation case. The comparison between the predicted stress distribution and the analytic one show that the accuracy of the Neumann problems greatly affects the final numerical results, and that there are slight deviation between the numerical result and the analytic one with a second-order scheme for the equilibrium equation but large deviation with a first-order scheme. The dimensionless 3-D solid elastic stress controlling equations are deduced, and the orders of magnitude of the variables and their coefficients are obtained. For the sake of convenience in constructing solving method, the elliptic elastostatics equations are selected as the controlling equations. And ADI method is deduced on the basis of Possion modeling equation, and two kinds of difference scheme, a three-node one and a five-node one, are constructed. The variable-separating solving method and the coupling solving method are utilized in the simulations separately, and the results show that the former one is with less computational load and converging speed than the latter one, but the latter one is with quite nice stability. For the equilibrium equation containing Neumann problem, its computational stability is crucial to that of the whole iteration. For the equilibrium equation at the boundary nodes and the equilibrium differential equation at the inner nodes, the coupling method is with fairly good computational stability, but the separating method is with simpler algorithm. In addition the latter one could be easily programmed, and it is also with simple formula for the data transfer between multi-block grids. To improve the stability of computation, the principle diagonal elements of discretized linear equations system matrixes are modified, which unfortunately reduces the computational speed. The simulation with the methods mentioned above is carried out. And the numerical results, especially those at the inner nodes, agree rather well with the analytic ones.
Thirdly the thermal-flow-elastic coupling solver employing finite difference method is developed, and the thermal-flow-elastic coupling simulations by the solver are carried out, with the test case as MARKII guide vane. The simulations by use of CFX10 with several turbulence models andγ?θtransition model show that the laminar flow exists at the whole leading edge of the vane, and that because of the strong shock wave at the suction midst and the strengthened instable flow at the pressure trailing edge the turbulent flow occurs at the aft suction side and the pressure trailing edge. Coupled heat transfer simulation utilizing HIT-3D (a CHT solver developed by Harbin Institute of Technology) with B-L and q ?ωturbulence models and B-L&AGS transition model are accomplished. The predicted flow fields in the main flow field agree well with the measured one, otherwise the predicted boundary layer flows and vane thermal loads differ from the employed models. The B-L algebraic turbulence model, not able to model transition process, over-predicts the temperature and heat transfer coefficient (HTC). q ?ωlow-Re turbulence model, able to model the effects of transition on the flow and heat transfer, predicts temperature and HTC distributions with less deviations from the measured ones than the B-L model does. And the B-L&AGS transition model, able to model the transition process, predicts thermal load agreeing best to the measured one, otherwise it under-predicts the HTC at several part of the vane surface since such model neglects the transport of the intermittency along the outer normal direction of the wall.
Finally the flow-elastic coupling and thermal-elastic coupling simulations of the MARKII vane, on the basis of CHT results, are carried out. Compared to the strain and stress caused by thermal load induced acting force, those caused by aerodynamic load induced acting force is rather small. The thermal-elastic results reveal that the thermal deformation and thermal stress of the vane are influenced by the thermal field, temperature gradient, vane geometry and the constraint on the vane, high temperature and temperature gradient leading to large vane deformation and vane stress separately. The results also reveal that the thermal stress cause by the thermal load predicted by different turbulence and transition models differ from the model selected. That with B-L model is higher than those with the other models, and that with B-L&AGS transition model is with the smallest value. For the sake of comparison, there is thermal-elastic simulation utilizing ANSYS, a finite element solver. And the differences of the predicted thermal-elastic results by the different solvers are quite slight, which proves the ability of the developed solver in thermal-elastic analysis.
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