第三章 特征线法 Chapter3 Characteristic Line Method 内容 特征线理论 特征线定义 特征线方程 相容性方程 特征线数值方法 重点 特征线理论 特征线方程 相容性方程 特征线数值方法 3-1 引言 Introduction 求解超音速流动中无旋/有旋二维及轴对称流场 To solve the supersonic irrotional/rotational 2D/Axisymmetric flow. 适用于双曲型偏微分方程 It is a method for hyperbolic PDE 3-2 特征线法理论 The theory of characteristic line method 一、速度势方程 Equation of velocity potential function 二维轴对称流动的速度势方程 The equation of velocity potential function for 2D and Axis-symmetric flow. (1 a 2 x 2 ) xx 2 x y a 2 y 2 (1 a 2 ) yy y y 0 对二维平面流,取 0 0 for 2D flow 对轴对称流动,取 for Axisymmetric flow 1 1 对于气体 For gas a a 2 *2 k 1 V 2 2 或 a a 2 *2 r 1 V 2 2 方程是二阶非线性偏微分方程 The potential flow equation is 2nd PDE 由于其最高阶导数是线性的,故也称拟线性 It is so called quasi-linear equation because the coefficient of its highest order PD is linear 可改写为标准形式: rewrite as a general standard expression A xx 2 B xy C yy D 方程的类型 The type of the equation are B AC 0 双曲型 Hyperbolic 抛物型 Parabolic 椭圆型 Elliptic 2 B AC 0 2 B AC 0 2 对于超音速流动(V>a),因为: For supersonic flow (V>a), since B AC ( 2 x y a 2 x 2 ) (1 2 a )( 1 2 y 可见M>1时是双曲型的 Evidently, when M>1, it is hyperbolic PDE,. B AC 0 2 2 a ) 2 v a 2 a 2 2 Ma 2 1 0 双曲型方程可以用特征线方法求解 Hyperbolic PDE can be solved using characteristic line method 特征线法的基本思路:将二阶偏微分方程问题转化为 沿特征线方向的一阶常微分方程组问题。 The idea of characteristic line method: To transfer the 2nd order PDE into a set of the 1st order ordinary equations. 二、特征线的定义 Definition of characteristic line 超音速流中特征线在物理上相当于马赫线,代表空 间不连续的间断面(线),是流场中的弱间断线。 In physics, the Mach line of supersonic flow corresponds to Mach line on which the flow parameters are not continuous 在此线上速度是连续的,而速度导数可能不连续。 The velocity on this line is continuous ,but the derivatives of the velocity is not continuous. 满足方程速度 the general equation of velocity potential function xx dx xy dy d x xy dx yy dy d y A 2 B C 0 xx xy yy 由上面的方程组可得 xy The solution of above linear equation for xy A dx 0 0 dx d y C 0 dy A dx 0 2B dy dx Adyd x Ddydx Cdxd y Ady 2 2 Bdxdy Cdx C 0 dy (A) 2 当分母为零时二阶导数将不确定。对应这样的曲线称为特征 线,特征线在(x-y)平面上的投影叫做物理面特征线 When the denominator is zero ,the 2nd order mix derivatives will be undetermined ,the corresponding curves are so called characteristic line, its projection on xoy plane is called physics plane characteristic line . 三、特征线方程 The equation of characteristic line 分母为零: The denominator is zero: A( dy ) 2B( 2 dx dy )C 0 dx 其解为物理面特征方程 Its solution denotes the characteristic equations in physic plane dy dx B B 2 A AC 特征线方程规定了物理面特征线斜率的变化规律 The characteristic line equation restricts the slope of characteristic lines 只有当B2-AC>0有意义,即只有对超音速流动才有意义。 Only when B2-AC>0, the root will be real number , and the corresponding solution will be meaningful only for supersonic. 当B2-AC>0有两个解,即代表两条特征线,称为第一特族征 线和第族二特征线。 When B2-AC>0, the characteristic line equation have two roots, which denote the two family of characteristic lines. ( dy dx ) B B AC 2 A ( dy dx ) B B AC 2 A x , y 沿特征线的变化规律 物理面特征线确定 The characteristic lines determine the change regulation of the velocity components 四、相容性方程 compatibility equation 在特征线上,(A)式分母为零,要求其数学上有意义 ( xy 存在),则必须使分子也为零,即: On the characteristics line, the denominator is aero, so the numerator must be zero to insure the solution is meaningful Adyd x Ddydx Cdxd y 0 d y A dy D dy 即( ) ( )c ( )c d x C dx C dx B B AC 2 A D C ( dy d x )c 特征线的相容方程规定了沿特征线dVX和dVy必须满足的关 系。 The compatibility equations of characteristic line restricts the difference of Vx and Vy 五. 二维平面和轴对称流的特征线方程和相 容方程 The characteristic line equation and compatibility equations for 2D and axis- symmetry flow 由速度势方程可得知 Using the velocity potential function x 2 A 1 1 2 a B 2 a x C 1 D a 2 2 y a 2 y Vx y 2 V xV y a 2 1 Vy a y 2 Vy y 2 因此特征线的斜率为 The slope of the characteristics line dy dx c V xV y a 2 Vx V y 2 2 a 2 1 2 1 Vx a 2 利用速度三角形关系及马赫角定义 Using the velocity triangle relations and the mach angle of supersonic flow V x V cos V y V sin sin 1 Ma 整理得 rewritten dy tg ( ) c dx c dy tg ( ) c dx c c 和 c 表示特征线的斜率。 c and the slopes of denote c the characteristic curves 或写为 or dy tg dx 任意一条特征线与速度矢量的夹角为马赫角 denotes the angle between velocity and the characteristic line 速度矢量与 X 轴的夹角为倾角 The angle between velocity vector and X axis 由此得相容性方程: The consult equation V 2 x a dV x 2V xV y V a dV y 2 2 x 2 2 a Vy y dX 0 讨论: Discussion 1.利用特征线方法可以把求解速度势的二阶偏微分方程问 题化为求解速度分量的一阶常微分方程 The 2nd order PDE was transferred to 1st order ordinary different equations. 2.利用特征线方法求出 V x , V y 之后,相当于求解 此不用直接求解 x , y To solve Vand x using characteristic line method instead of solving V y x , y,因 directly 3-3 特征线数值解法 过1点的右行特征线C 与过2点的左行特征线 C 相交于4 点 Two characteristic curves 1-4 and 2-4 intersect at point 4 对交点4应用相容性方程 For point 4,the characteristics equation and compatibility equations are dy tg dx Q dV x R dV y S dV 0 其中 where Q Vx a 2 2 R 2V x V y (V x a ) 2 a Vy 2 S y 2 差分形式方程为 The limited difference forms are follows y x 特征线方程characteristics Q V x R V y S x 0相容性方程compatibility equations 一.流场内点计算过程 The computational procedure for inner points 给定两个初始点1和2的坐标速度(大小,方向) give out two initial points 1 and 2 根据特征线方程和相容性方程写出4点与1,2两点之间的几 何关系和流场速度关系 Found the relations according two characteristics equations and the compatablity equations y4 y2 ( x4 x2 ) y 4 y1 ( x 4 x1 ) y4 x4 y2 x2 y 4 x 4 y1 x1 由上式可得4点的坐标( x 4 , y 4) To get the coordinates of point 4 x4 1 1 y4 x2 y 1 x1 1 1 y4 x2 y 1 x1 y4 1 1 其中 可以用下列分式计算 Where can be expressed as following t g ( tg 1 s in ( V y ) ) Vx 1 ( 1 M ) 点4的速度可以用相容性方程求解 The velocity at point 4 can be solve using compatibility equations . Q (V x 4 V x 2 ) R (V y 4 V y 2 ) S ( x 4 x 2 ) 0 Q (V x 4 V x 1 ) R (V y 4 V y 1 ) S ( x 4 x1 ) 0 or Q V x 4 R V y 4 S ( x 4 x 2 ) Q V x 2 R V y 2 Q V x 4 R V y 4 S ( x 4 x1 ) Q V x 1 R V y 1 其中, wher e, Q 2 (V x 2 a 2 ) 2 2 R 2 (2V x 2 V y 2 Q 2 2 ) Q 1 (V x1 a x1 ) 2 R1 (2V x1 V y1 Q 1 1 ) 2 2 S 2 a 2 V y 2 y2 2 S 1 a V y1 y1 采用欧拉预估-校正方法求解,则 Using Euler’s prediction-correction method 预估 predict V x2 V x2 , V y 2 V y 2 , y 2 y 2 V x1 V x1 , V y1 V y1 , y1 y1 求解线性方程组(相容性方程的差分等式)得 To solve the linear equations composed using the compatibility equation. T2 T1 V x4 Q2 Q1 V x4 R4 R1 T2 T1 Q2 Q1 Q2 Q1 R2 R1 R2 R1 校正步计算中 Correct calculation V x2 V x1 Vx2 Vx4 , V y2 2 V x1 V x 4 , V y1 2 Vy2 Vy4 , y2 2 V y1 V y 4 , y1 2 其中应用的值计算采用 Where the corresponding value a 2 a 2 s in r 1 2 1 ( a V ) 2 V y2 y4 2 y1 y 4 2 流程图(Flow chart) 对称面(轴)的处理(可以简化计算过程) Deal with the Axis 当点4处于对称面(轴)上时,则2是1的镜像点,且 When point 4 locate on symmetry boundary, the point 2 is the mirror image of point 1, and y4 0 Vy4 0 4 0 这时只需要解特征线方程中一个(点1的右伸特征线)就 可以计算出 x4 Then only the characteristic equation of 1-4 is needed to get y 4 x 4 y1 x y4 0 1 y 4 , x4 relation x 4 x1 y1 只解1,4点的相容方程就可以得到V x 4 x4 Also only the compatibility equation of 1-4 is necessary Q V x 4 R V y 4 T V x 4 T Q 二、固壁点的计算过程 Computational procedure for points on boundary 设固壁的曲线方程为 Assume equation of the boundary curve is following y a bx cx dy dx 2 tg b 2 cx 由于壁面为流线,其切线与速度方向一致,故 Due to the boundary curve is streamline, the slope of its tangent is tg Vy b 2 cx Vx 特征线方向和相容性方程可写成 The characteristics equation and compatibility equations can be written as y a bx cx 2 y 4 2 x 由 y 4 a b x 4 cx 4 V y4 tg b 2 x 4 V y4 得出 x4 x4 y2 x2 Q V x 4 R V y 4 T 其中 tg ( ) T S ( x 4 x 2 ) Q V x 2 R V y 2 Q (V x a ) 2 2 R ( 2V x V y Q ) S a V y / y 2 可以采用预估-校正方法求解边界点的 x 4、 y 4、 v x 4、 v y 4 The corrected value on boundary can be also determined using prediction-correction method 直接法:可能导致边界上点过于稀疏 Direct method , the disadvantage is that the points on boundary might be too few 逆方法:先在边界上预设多个点然后利用特征线方法求预设点上的坐 标值及流场参数(速度值) Inverse method, to preset more points on the boundary first, and then using characteristics method to predict the flow parameters. 步骤:1)先给出 x 4 , y 4 Steps: 1). Give out x 4 , y4 v x4 v x4 ,vy4 ,vy4 2 利用特征线方程和相容性方程计算2点的坐标和速度 Using compatibility equations and the characteristics equations to calculate the coordinates and velocity components at point 4 4 3 2 3) 校正2点的值和位置 To correct the coordinates and velocity of point 2 4) 校正4点的值和位置 To correct the coordinates and velocity of point 4 三、 自由边界点的计算过程 Computational procedure on free boundary 确定自由边界的位置和流动参数的方程 To determine the position and flow parameters on free boundary 自由边界点的流动参数 P4 4 T 4V 4 一般情况都是已知的, Usually the flow parameters on boundary are known 气动函数关系式 Aerodynamics function V4 其中, p * * , T , Pa V x 4 V y 4 f ( p , T , Pa ) 2 2 * 是总压,总温,边界压强 * 4 3 Where p * , T * , P are total pressure a Total temperature and static pressure on boundary 2 由特征线2-4的相容性 According to the compatibility equation of characteristic line 2-4 Q x V x 4 R V y 4 T Q T R V 4 Q R 2 2 Q R 2 Vx4 1 V x 4 V 4 V x 4 2 2 2 2 2 T 1 2 2 确定4点的位置 To determine the position of point 4 自由边界是一条流线,速度方向与切线一致,故 The velocity propeller to the sorrface on the free stream boundary due to it’s a stream line. dy dx vy vx 0 积分得integrated it and obtained y 4 0 x 4 y 3 0 x3 其中 where 0 3 tg 3 利用特征线2-4方程得到 The compatibility equations on the characteristic lines are y 4 x 4 y 2 x3 以上两式联立可得 Coupling above two equations then 1 y 3 0 x3 1 x y 3 x 2 4 1 0 1 y 3 0 x3 0 x y 3 2 y4 1 0 1 校正方法 用 ( x2 x4 ) 2 代替 x 2,用 Use ( x x ) 2 to substitute xand 2 recalculate all above parameters. 2 4 ( x3 x 4 ) 2 代替 x 3 ,重复面的计算 ( x 3 xto ) 4 substitute 2 , and x 3 then 作业2:用特征线方法设计轴对称收-扩喷管的扩张段型面,保证 在壁面速度方向与切线方向一致,馆内没有膨胀波和激波。 喉道音速线 总结 内容 特征线理论 特征线定义 特征线方程 相容性方程 特征线数值方法 重点 特征线理论 特征线方程 相容性方程 特征线数值方法 难点 特征线数值方法