主要内容 ： main contents 混合差分解法 Mixed PD Method

1. 第四章 跨音速定常小扰动势流混合差分方法及隐式近似因式分解法chapter 4 The Mixed Finite Difference Method(FDM) for Velocity Potential Function of Steady Small Perturbation and Implicit Approximate Factor Decomposition Methods 主要内容： main contents 混合差分解法 Mixed PD Method 小扰动方程及小扰动激波差分式Small perturbation equation and small perturbation relationship for shock flow 小扰动速势差分方程 The finite differential equation of small perturbation potential function

2. 边界条件及边界条件的嵌入The initial condition and boundary condition • 线松弛迭代解法Linear relaxation iteration method • 升力翼型的跨音速小扰动势流差分方法FD method of velocity potential function for small perturbation • 隐式近似因子分解法Approximate factor decomposition method • AF1方法 AF1 method • AF2方法 AF2 method

3. 方法比较 Comparison of the method 重点： Focus • 混合差分方法Mixed FD Method 难点： Difficulty • 隐式近似因子分解法Implicit Approximate factory decomposition

4. 第四章 跨音速定常小扰动势流混合差分方法及隐式近似因式分解法chapter 4 The Mixed Finite Difference Method for Velocity Potential Function of Steady Small Perturbation and Implicit Approximate Factor Decomposition Methods • 跨音速流：局部超音区与亚音速同时存在的流场 Transonic flow ：Local supersonic flow and supersonic flow exists meantime • 偏微分方程：混合型方程 The PDE：Mixed type equation

5. 混合差分方法：用不同的差分方程求解跨声速流场混合差分方法：用不同的差分方程求解跨声速流场 Mixed Finite difference method is to solve transonic flow with different FDMs • 混合型方程及流场：采用迭代方法求解，求解之前不知道方程的类型 Mixed Equation and flow field, the iterative method is used because the type of the equation is unknown before it was solved • 小扰动方程：小马赫（0.6~1.4）流过薄而微变的叶片（机翼或叶栅）时全速势方程可简化为小扰动方程 Small perturbation equation(SPE): when mach number is small (ie 0.6~1.4)the full velocity potential equation can be simplified to SPE

6. 混合差分:用混合差分格式求解小扰动方程 Mixed FDM :To solve equation using MFDM • 混合差分和松弛迭代法求解全速势方程 Mixed FDM and Relaxation iteration : To solve full velocity potential equation. • 优缺点： Advantage/disadvantage 跨音速松弛法---速度快，有效 Transonic relaxation method faster efficient 时间推进法：适用范围广 Time matching methods, widely usage 近似因子分解法：快速 Approximate factor decomposition：faster 多层网格法：收敛性好 Multi-grid technique：good convergence

7. 4.1 跨声速小扰动速度势方程 Equation of transonic small perturbation velocity potential function 跨声速气流绕过薄翼的情况 For the case of transonic flow pass a thin airfoil • 二维平面速势方程 2D velocity potential equation

8. 气流绕过薄翼

9. 适用范围：亚、跨、超音速无旋流动 • Suitablecase ：subsonic， transonic， supersonicirr-rotationalflow. • 将流动分解为两部分：未经扰动的流动、扰动流动Todecompose the flow into unperturbed flow andperturbflow • 未经扰动的流动就是无穷远前方来流Flowat unperturbed fields is far field flow • 扰动运动速度势可以用 表示。速度可以用 表示 Potential function of perturbation flow is ，perturbation velocity components

10. 两部分的合速度势 The total velocity potential function

11. 代入速势方程可得小扰动速度 应满足的方程 • Substitute the equation and then the small perturbation eq. • 求得速度场之后，可以得到压强及压强系数为 • The pressure and pressure coefficient can be obtained from the following equations.

12. 再用等熵流动的关系式可得到其他参数 • Then introducetheisentropyrelation to get other parameters 比热比 绝热指数

13. 小扰动条件下，扰动速度远小于自由来流速度onsmall perturbation condition, the perturbation velocity lessthanfreestream • 补充条件： • Supplementconditions • 来流不能接近音速 incomingflowvelocity doesnotapproachsonic • 来流非高超声速 incomingflowvelocity doesnotapproachhypersonic • 为进一步简化扰动方程，忽略扰动速度一次项，可得到下列关系：Simplifiedequation

14. 最后得到： • Finalequation • 应用范围: 亚、超声速Suitableforsubsonicandsupersonic • 不适用于跨声速区域: 对于跨声速 ≈1，必须取消补充假设条件，即取消来流不能接近音速的假设，这时速势方程首项的系数一次项不能忽略Fortransonicflowfield(M ≈1)， thesupplementcondition， thefirstitemofthepotentialfunctionequationcannotbeneglected.

15. 跨声速小扰动方程应为：The smallperturbationequationofvelocityfunction • 可以证明：当M∞→1时， It’s proved ,when M∞→1,

16. 因此跨声速条件下，小扰动方程可以写成 So that the small perturbation equation at transonic flow can be written as • 此方程的类型取决于： Type of the equation depends on =B2-4AC=4(M2-1) • 当M<1时, <0,不存在实特征根，没有特征线，为椭圆型 When M<1,no real eigenvalue exists, that is no character line, the equ. is elliptic. • 当M>1时, >0,存在两个特征根，有两条特征线，为双曲型 When M>1,there are two eigenvalue, two character lines, the equ. is hyperbolic eq. • 当M=1时, =0,存在一个特征根，有一条特征线，为抛物型When M=1,there is one eigenvalue, one characteristic line ,the equ. is parabolic

17. 特征线(当M>1时)：斜率The slope of characteristic line 是马赫角 is so call Mach angle • 特征线与x轴夹角为局部马赫角，对称于x轴。 Local Mach angle is the angle between velocity vector and the characteristic line y r’ q p 依赖区 影响区 q’ r o x

18. 影响区：P点下游由两条特征线所夹的区域 Influence zone:upwind zone between characteristic lines • 依赖区： P点上游由两条特征线所夹的区域 Depend zone downstream zone between the characteristic lines • 扰动下的压强系数公式The pressure coefficient on small perturbation condition

19. §4-2小扰动激波关系式 The shock relations of small perturbation . 等熵激波小扰动激波的熵增是三阶小量 For small perturbation shock, entropy increase is third order， so it is isentropy shock。 激波的精确速度关系式：Accurate velocity relation of shock

20. 激波前后的速度关系式（几何关系） Velocity relations in front/rear-shock 即

21. 对于直角坐标系 At Cartesian coordinates 因此 so that

22. 由能量方程可得 From energy equation 由此得到 M∞→1时的方程（跨声速中）From where , the equation when M∞→1,(transonic flow) 超声速中 At supersonic flow • 适用范围：激波前后小扰动方程，适用于等熵波Above eqs. are available for small perturbation flow in front/behind of the shock, i.e. , iso-entropy flow

23. §4-3 跨声速小扰动速势差分方程Small perturbation equation for transonic flow 混合性方程，在同一流场中不同点所用的差分方程 不同。 Mixed equation, different FDE is used for the scheme 一、中心差分格式 Centeral FDE scheme flow field 对速度势 For velocity potential function

24. 一阶导数的差分格式 First order difference equation is obtained as • 二阶导数的差分格式 Plus two equations, and get 2ed order PD 二阶精度 2nd order

25. 二、一侧差分格式 One side FDE of the derivatives • 在超音速流中，气流参数只受上扰动游影响与下游扰动无关。At supersonic flow, the parameters of flow are dependent on upwind perturbation and independent on down flow perturbation • 需建立迎风一侧差分格式 The upwind one side FD scheme is needed to built • 取上游一侧的点构成差分格式 Take the upwind point to construct FD scheme • 一阶精度迎风格式 1st order upwind scheme • 二阶精度迎风格式 2nd order upwind scheme

26. 三、亚音速点的差分方程At subsonic flow equation 取网格点如图：正交等间距网格The space nodes are shown as

27. 中心差分格式构成的差分方程 即

28. 受周围四点的影响，这是亚声速流动的特点 is effect by around four points , this is subsonic feature

29. 四、超声速点的差分方程FDE for supersonic flow • 当计算点为超音速（M大于1）时，方程为双曲线型When local supersonic flow appear ,the equation ishyperbolic • 存在依赖区（上游马赫锥内部）The dependence zone exists ,(up mach core) • 对y的差分可以用中心格式The centurial difference is used for the derivative with sped to y

30. 对x的差分要用迎风格式Upwind scheme is used for X-direction • 显示格式： 差分式取 ，而不用 线法Explicit scheme • 每次都用i网格线上的已知值，可以从左到右逐点计算The known value is used to calculate the value at every node sequently

31. 隐式格式：利用当前网格线上的值构筑差分方程Implicit scheme : using present value to construct FDE 具有三个未知量（在网格线i上） Where there are 3 unknown points • 显式比隐式方便Explicitly scheme is more convenient than implicit scheme

32. 显式格式稳定区域小The stability zone of explicit is smaller than that of implicitly • 稳定性和收敛性Stability and convergence • 收敛性：当步长趋于零时，差分方程解趋于微分方程解Convergence: when step length tends to zero, the solution of the PDF tends to the solution of PDE • 稳定性：差分误差在传播过程中有界且逐渐减小Stability :the error is limited or decreased

33. 对波动方程（双曲型）：稳定性条件是差分方程依赖区不小于微分方程的依赖区For viberation Eq ,the stability condition is that the dependent zone of PDE less than that of PDE • 对超声速势函数 For potential velocity fuction 差分方程依赖区半顶角 The half conical angle The dependent zone of the FDE

34. 微分方程的半顶角the angle of the dependent zone • 差分方程稳定条件为 • 对于跨声速势流，不满足稳定条件，因为For transonic flow, the stability condition is not satisfied

35. 跨声速势流不能用显示格式so transonic potential function can not solve with explicit method • 隐式格式的依赖范围大于微分方程的依赖范围The dependent zone of implicit scheme is great than that of PED J+1 J J-1

36. 双曲方程差分采用一侧隐式格式For hyperbolic equation ,one side implicitly scheme is used • 五、音速点的差分方程The finite diffence at sonic points • 当M=1时，方程为抛物性，存在一族特征线When M=1,the equation is parabolic, there exist a series of characterist line • 速度势方程化为potential equation become

37. 采用差分方程可以写成Using FDE • 六、速度判别式Velocity critical condition • 四种情况: • Four cases • 亚声速sub 亚声速sub • 超声速supe 超声速super • 亚声速sub 超声速super • 超声速 super 亚声速sub super Subsonic supersonic airfoil

38. Ⅰ Ⅱ Ⅲ：过渡连续 continually changes Ⅳ：出现激波 参数不连续 the shock appears, parameters are discontinous Ⅲ：有音速线存在There exists sonic points 逐点判别：根据 系数进行判别 Judge according to the coefficient of

39. 中心差分 • 一侧差分

40. 差分方程形式 • PDE form

41. 七.跨声速小扰动激波的差分方程 PDE for transonic small perturbation shock flow 激波处：速度由超声速过渡到亚声速 At shock, the flow transfer from supersonic to subsonic 激波前流场均匀（近似） In front of the shock ,the flow is uniform supersonic flow shock i i+1 i-1 i-1 j i+1,j i,j j+1 i+1,j+1 i-1 i,j+1

42. 激波后流场均匀（近似） After the shock ，the flow is also uniform 差分方程（跨声速小扰动方程的差分形式） FDE (Transonic small perturbation flow)

43. 对无旋流动（无旋条件） Condition of irrotational flow 其差分形式 Its FD form

44. 考虑了无旋条件的扰动速度差分方程 After considering the irrotatational condition the small perturbation equation becomes 讨论：discussion： 跨声速区小扰动激波差分方程与小扰动激波关系相同

45. 八、超音速点差分方程的人工粘性 • artificialviscousforsupersonic FDE • 速势方法假设了流场均为等熵流 Thevelocity potential method assume that the flow is iso-entropy • 导致流场间断解不唯一（可由亚-超，也可由超-亚） It leads tonon-uniquesolution • 如果采用迎风格式 （单侧差分），则只适合压缩突跃（由超-亚），不可能出现膨胀解。 Continuoussolution，ifthe upwind scheme is used, the solution only suitable for compressible sharp increase (shock), not suitable for sharp decrease.

46. 原因： • 采用1阶迎风格式 1storderupwindscheme • 超声速点差分方程（迎风格式） FDEofthe potential equation at supersonic flow

47. 应用当地M数改成相对应的微分方程 • Using local Mach number M to rewrite the PDE then • 其中 类似于跨音速小扰动粘性流方程中的粘 性项。称为人工粘性Whereis similar as the viscous form of small pertubation equation, so called it artificial viscous • 差分方程的解只含压缩突跃，即激波（是熵增过程） PDE only includes compressed shape change(where the entropy creases ) • 不可能产生膨胀突跃（即熵减过程） Not suitable for expanding shape change(where entropy decreases)

48. 4.4 边界条件及其嵌入 Embeding of Boundary conditions 一、边界条件(Boundary Condition) 1.物面： 无粘，无穿透条件 on wall no normal velocity 对于翼型（叶栅），设物面方程为， 则定常流动边界条件

49. 即： 若翼型上下表面可表示为 则 速度分量可写成

50. 上表面的边界条件为 BC on up surface is 其中， , 为扰动速度 Where , is the perturbation velocity components