《时间推进法》PPT课件.ppt

第五章 时间推进法,内容守恒形式欧拉方程非定常欧拉方程的特征线非定常欧拉方程显式差分多维流的时间分裂法非定常欧拉方程有限体积法无粘流计算的人工粘性加速收敛的方法及算例重点多维流的时间分裂法非定常欧拉方程有限体积法,5-1 守恒形式的非定常欧拉方程,一、引言 激波存在时，流场有旋，不存在势函数，不能用速势方法。不记粘性时，可以用欧拉方程描述流场。非定常二维可压缩欧拉方程,方程的性质方程是双曲型（对时间）跨音速区包含激波时间推进分法可以克服跨音速计算困难基本思路：把定常问题化为非定常问题的渐进解（稳态）全场统一用一种数值方法可以使用有限体积方法二、积分形式的守恒型非定常方程组只有写成守恒形式的方程才能代表物理守恒律和间断面上的物理守恒律。,连续方程：动量方程：能量方程：令 绝势流动能量方程为：,三、微分形式的守恒非定常流欧拉方程（3D）或引入总焓，则,根据连续方程改写为,四、守恒的欧拉方程组的缩写,其中，U,F,G,H是列向量,通用形式,可写成是向量矩阵形式,则,积分型的矢量矩阵表达式,五、气体状态方程,其中，,引入完全气体状态方程 方程组封闭可解 例:一维流欧拉方程具体表达式,令,则,F是复合函数,令,方程可写为,同理可写出二维欧拉方程的通用表达式,其中,5-2非定常欧拉方程的特征线（自学）,5-3 非定长欧拉方程的显式格式一、简单线性波动方程,其解析解存在,沿特征线上,二、一阶精度显示差分,t,x,x=at+c,0,截断误差,差分依赖区边界上,（微分依赖区与差分依赖区重合）,精确平移条件：特征线 上u不变,一阶显示差分格式将不稳定，不能用,i-1,i,特征线,三、二阶精度的显示格式 利用Taylor级数可构造二阶精度显示差分格式,差分方程稳定性：（差分方程依赖区不小于微分方程依赖区）,令,则有,当CFL=1时，差分方程的依赖区与微分方程依赖区重合，得到的结果与精确解相同,CFL(Courant-Friedrichs-Lowy)数,四、二阶精度显示两步差分,校正：,即,具有二阶精度,预估：,二步格式的构造,向后差分,给出中间结果,校正：用中间结果构造向前差分,可以反过来，先向前再向后差分，即,，具有一阶精度,得到二阶精度,预估：,五、一维流欧拉方程组差分格式 方程通用格式 V、F表达式同前预估式 V具有一阶精度校正式 V具有二阶精度 与其等价的微分方程为,稳定性条件：差分方程依赖区不小于微分方程依赖区。V其稳定性条件 即,或 CFL！没有经过严格证明的结论六、二维流欧拉方程组 方程通用形式 其中U,F,G同前 两步法格式:预估校正,以差分算子Lxy表示，则 MacCormark二阶精度差分格式分“七点式”“五点式”稳定性条件：或,5-4 多维流的时间分裂法 Time deposition method of Multi-dimension flow 维数增加，稳定性所允许的最大时间步长减小。Number of dimensions increase leads the stability time step decrease显示格式的计算率降低 Efficiency of explicit scheme decrease用两步时间分裂的差分格式将多维差分方程分解为多个一维差分格式 Two step time decomposition method is to decompose computation into two step,或记为,依赖于x,y平面内的九个点，先对y求解，再对x求解，为消除x,y顺序影响，第二个时间步可先对x求解再对y求解。It depends on 9 points in x y plane,firstly to solve it for x then for y in order to eliminated the effect on sequence,second step is for x first and then for y.,在各个方向都按各自的稳定性限制条件来确定推进时间步长 To determine time step individual for x and y各方面均选取最大允许的值。On both direction,the time step can be maximum value.举例：三角形翼型的流动。契形顶角 Example：triangle airfoil AOA 10,Angle of leading edge,Take=const y方向分三区：近场、中场、远场 Divide 3 zones in y direction,near，middle，far field,估算x和y方向时间步长Calculate the time steps in x and y direction.,时间步长：time step:中间场：middle 近 场：near远 场：far,最大时步长,各区的运算可规定为 The computation regular for every zone,中间,Max time step,近场,near,middle,远场,四步推时,的运算可规定为,)computation,中,Middle,near,far,4 steps match(,近,near,近,中,middle,远 1232网格 1次 far,远,far,可提高效率 Improve efficiency,近场 432网格 4次 near,中 832网格 2次 middle,近场，中场，远场均执行2次，共1536次 Near middle far perform 2 times,1536,推进4，执行的运算次数（时间）,Total computational time for 4 in total matching,若三区网格数相同，全部时间为,允许最大时间步Max time step,If the mesh number are same for three zones（2432）,时间分裂格式的相对数值效率为The numerical efficiency of time matching scheme,其中Tst代表单位推进需要的计算机时Where Tst denotes time required for every step,结果见p117中图,非定常欧拉方程组中，用总焓方程代替非定常能量方程也能求得定常解 In unsteady Euler Eqs.The energy equation can be replace by equation of total,当,时，方程趋于定常，整个流场总焓不变,When the equation becomes steady form,5-5非定常欧拉方程有限体积法 The finite volume method for Euler equations,限体积法：用基本方程积分，以空间体积元素为对象离散化方程 Finite volume method:to use integral form of basic equations,and express discrete equation in form of volume,其中（对二维问题）where(for 2d problem),为控制面的法向量 Where is normal vector of control surface,总焓均匀且不随时间变化的Euler流The Euler flow in which the total enthalpy is uniform and does not change with time,一、Maccormark 时间分裂有限体积法 Time decomposition method of Maccormark 二阶精度显示两步法格式 2nd order explicit FD with two steps matching,y,x,o,网格单元面积（三维问题则为体积）the area of mesh,单元边界长度矢量（面积矢量）the vector of boundary edges,差分格式的积分表形式 the integrated form of FD,其中 代表网格中心点的值 where donates the value of center of the mesh,引入算子表达式 introduce FD calculator,稳定条件 stability condition,(二)非正交曲线坐标网格Non-orthogonal grids 有限体积格式不仅可用于正交网格，也可用于非正交网格 FVM can be apply not only in orthogonal grids but also in non-orthogonal grids,当 为常数时，格式是有二阶精度Where are constant,the scheme is of 2nd precision,1,2,3,4,对非正交网格For non-orthogonal grids,1,2,3,4,体积（面积）Volume（area）,以连续方程为例，写出差分方程有限体积格式Take continuity equation as an example，the FD scheme for FVM can be written as,例：叶栅通道,Maccormack格式用于叶栅通道拟流线为直线/曲线前后缘设置尖劈,S2,S1,S3,S4,i,j,A,B,C,D,E,F,G,H,S,P,二、Denton方法Denton methodABCD网格单元，由拟流线组成Mesh is constructed with quasi-streamlines,计算点位于拟流线上且在单元的中央 Computational nodes are on quasi-streamline and the center of the meshDenlon 改进格式,以f表示通量（）则可简化为 To press the flux with f,then FD can be simplified as following其中，Cf和Cp是通量和压强修正量 Where Cf and Cp are flux and pressure flux,Ff 是通量插值函数，由（i，j），（i-1，j），（i-2,j）三个 Ff is the interpolation function,it can be obtained from 计算点的通量内插得到 Three points(i,j),(i-1,j),(i-2,j)Fp是压强插值函数，由（i-1,j），（i,j），（i+1,j）三点内插 Fp is the interpolation function obtained from points（i-1,j），（i,j），（i+1,j）,是松弛因子 is the relaxation factor同理可写出 和 的表达式 Based the same principle,and can be obtained注意：上述格式中，速度分量用旧速度压强用新速度和旧速度组成差分格式先求解密度和压强，再求解动量方程求新速度场,三、边界条件,进/出口边界条件 Generally three types of BC,inletoutlet BC 周期性 Periodic BC物面边界条件 Wall BC远场边界条件 Far field BC,一般有四种：,对于叶栅通道内流动，有四种：For a cascade flow channel,three are four BC进口边界（AH）Inlet boundary(AH),周期性边界（AB CD HG FE）Periodical Boundary（AB CD HG FE）出口边界（ED）Outlet Boundary(ED)进口（AH）当 时，需三个条件：进气角总温总压 Inlet(AH),when,three BC are required,angle of velocity,total temperature,total pressure,边界上值受内通道影响effected by inner flow,当 时，边界值不受内通道影响，可以给定速度,When,boundary values are not influenced by inner flow,出口处（ED）：Outlet(ED)当（亚音速）需一个条件，一般给压强 when(subsonic),pressure as one BC is needed 当 时，边界值可以外插，无需条件,叶片表面上（BSC或GPF）On the surface of cascade(BSC,or GPF)速度矢量与表面相切 The velocity parallels the surface,周期性边界条件（AB和HG，BC和FE）On periodic BC(AB,HG,BC,FE):边界上对应点参数相同.the parameters on corresponding points are same 可向上、下各延伸一点（i,0）和（i,N+1）the grid are extended up and down one point respectively(i,0)(i,N+1),(i,N),(i,1),有限体积法中物面通量为0，只需要计算物面压强 For FVM,the flux on surface are zero,only the pressure on boundary is needed,物面法向动量方程：The equation of holmium in the normal of wall,对平面流动 For plane(2D)flow其中R是曲率半径 Where R is radius of curvative其差分格式 its FD scheme is其中 是i 点距物面的距离 Where is the distance to the wall,可以用外插法，由内点外得到物面上的压强 Extrapolation method can be used also计算精度受曲率计算精度影响比较大,5-6 无粘流计算的人工粘性The artificial viscous of inviscous flow computation欧拉方程二阶精度显式差分方程截断误差为：The trancation error of 2nd explicit FDE for Euler Eqs is不含粘性It does not includes viscousity在激波附近会出现压强和速度的波动和过高峰值The pressure and velocity will fluctuate near the shockwave须加入适当人工粘性 The suitable artificial viscousity must be introduced 过高人工粘性会影响求解精度Over high artificial viscosity will influence the precise,对二维Euler流动，人工粘性一般取：For 2D Euler flow，the artificial viscosity is generally其中Cx，Cy是人工粘性系数，取00.5Where Cx，Cy are coefficients of artificial viscosity,given as value 00.5,人工粘性相当于给方程增加了两项：The artificial viscosity adds two terms to PDE 对应的方程与粘性流N-S方程相比Compared with the corresponding N-S,其中 为第二粘性系数 Where is second viscous coefficient 法向粘性应力项The normal viscosity term is相应的x方向动量方程粘性The corresponding momentum equation in x diraction与人工粘性具有同样的表达式和含义,代表粘性影响（人工粘性）McCormack人工粘性Denotes the artificial viscousity of McCormack scheme光滑变化区域，人工粘性是四阶小量，不影响差分格式精度In the smooth flow field，the artificial viscosity is 4th order，no influence on the precision of the FDE当出现激波，二阶系数很大，该项会产生明显的粘性作用，适当选取Cx可以很好地模拟激波But when shock appears，the 2nd order partition different becomes larger，it may present significant in viscous effect,例：一维收扩喷管过度膨胀流场Example:1 D Converge-Diverge Nozzle over Expanded,激波前：Ma数光滑过渡In front of Shock,Ma distribute smoothly.激波后：稍有波动Behind the shock,there exist fluctuation激波位置在三个网格之间Shock located in between three grids较好的抑制了波动 restraining the fluctuation perfectly.,简单的人工粘性：利用加权平均方式引进数值阻尼Simple artifical viscosity,to introduce artifical viscosity using weighted average method 其中 是阻尼系数。When is the damping coefficient.人工阻尼方法相当于在微分方程中引入修正项Artificial damping method is equivalent to,当 时，不会影响二阶格式的精度When it does Not influence the precision of 2nd FD scheme.阻尼系数 取值原则：The principle for receiving value of 激波区有较强光滑作用，使激波保持在2-4个网格之间It has smooth effect in shock zone.在激波区之外，则希望没有光滑作用，因此应取不同的值。Out of the shock zone,it has no smooth effects.激波捕获法：在差分方程中添加高阶项，取激波间断展宽，但仅为连续的薄层Capture of shock,introduce high order FD to widen the shock and keep it continuous in a thin layer.,5-7 加速收敛的方法及算例Example of Computation Acceleration一、NACA转折角为800 亚声速叶栅 NACA cascade with 800 turning angle Denton 方法（1）Denton method（1）,McCormack时间分裂有限体积法（2）McCormack time decomposition FUH(2)压强分布Pressure distribution,叶背前缘：方法2较方法1有改善Expanded surface method 2 is better.叶盆：比实验值高Compressed surface,results is higher than that of experimentationMa数分布比较：,二、NACA转折角95具有激波的叶栅 Cascade with 95 deg of turning angle 方法2：用人工粘性，可以较好捕获激波 Method 2，using artificial viscosity can capture the shock well,三、加速收敛的方法Acceleration method to iterate computation 定常问题Steady problem收敛结果与初值和计算过程无关Converged results has nothing to do with initial flow field and computation proc.可以使用不同的时间步长Diff.time step can be used当地时间步长法，取当地允许的最长时间步长Local time step method，max.allowed time step can be given当地时间步长取决于当地网格Local time step depends on local space step and flow para.收敛之前的流场没有物理意义，只有收敛后才有The flow field before convergence has no meaning原理是加快扰动传播速度The principal of acceleration is to speed up perturbation,网格逐步加密方法Method of grid reframe稀网格计算快，但精度低Coarse mesh can speed up computation but its precision is lower密网格精度高但计算速度慢Fine grid can give higher precision but convergence is worst疏密结合，先疏后密，可提高计算速度与精度The best way is to combine coarse-fine grid表5-2例,多维流的时间分裂法The time decomposition method for multi-dimensional flow,小结,Summary,主要内容:Main contents,守恒Euler方程Conservational Euler equation,非定常流的特征显示格式Explicit FDS of steady Euler flow,非定常Euler流 Unsteady Euler flow,有限体积法 Finite volume method of unsteady Euler flow,无粘流人工粘性 Artificial viscosity for inviscous flow加速收敛的方法Acceleration of Methods,重 点Importance,加速收敛法Acceleration of Methods,多步显式格式Multi-step explicit scheme,多维流时间分法Multi-dimensional time decomposition method,有限体积法Finite volume method,有限体积法Finite volume method,难 点Difficulty,时间分裂法time decomposition Method,