Flow around permeable and thick airfoils and numerical solution of singular integral equations
Flow around permeable and thick airfoils and numerical solution of singular integral equations
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透水厚翼型绕流及奇异积分方程数值解
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发表时间：
1992
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通讯作者：
I. M. Molyakov
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作者：
I. Lifanov;A. Matveev;I. M. Molyakov
将可透型和厚翼型绕流问题简化为分段上奇异的第二类变系数积分方程的解或此类方程的方程组。构造了求解该问题的数值方法。该方法基于用拉格朗日多项式逼近光滑解分量。计算了特定气动问题的解。许多理论和应用问题在被简化为积分方程时，自然会导致线性一维奇异积分方程或这种方程在开曲线或闭曲线上的方程组。例如，在空气动力学中，这些问题出现在模拟机翼周围涡度层的流动中，在电磁波衍射中，它们出现在通过感应电流模拟反射表面时，在弹性理论中，它们出现在通过分布力模拟物体边界时，等等。这种方程的理论出现在上世纪末D.休伯特和A.庞加莱的著作中。它的基础是F. Neter和T. Karleman的研究奠定的。在本世纪上半叶，这一理论得到了苏联数学家的大力发展。在著名的N.I. Muskhelashvili和f.d. Gakhov的专著中，这个理论有了一个比较完整的形式。此外，这类方程的理论在许多方向上分支，比如对求解的类别（所谓的L理论）施加不那么严格的条件，对方程的系数等等。泛函分析和代数方法在奇异积分方程的现代研究中得到了广泛的应用。在奇异积分方程的许多应用中，需要将其解提高到数值结果，因此，应该开发有效的计算机辅助解的近似方法。建立奇异积分方程数值解法的先驱工作可以追溯到20世纪30年代[13,20,37]。这些著作开创了s.g. Mikhlin、a.n. Kalandii、v.v. Ivanov、i.c. Gokhberg、b.g. Gabdulakhaev、i.k. Lifanov、Z. Presdorf、D. Elliott、B. Zilberman及其追随者的著作中各种近似方法对奇异积分方程适用性的系统研究。目前有许多研究奇异积分方程近似解的著作。按照惯例，大多数工作可以被分类为：(1)理论性质的工作，涉及计算方案的构建及其收敛性和稳定性分析；(2)应用中遇到的特殊奇异积分方程的数值解。第一组作品的作者通常远离具体应用问题的数值解。他们建议适当地解决一些问题（LK.Lifanov等人），但并不总是必要的事情。第二组作品的作者解决了一些必要的问题，但并不总是正确的。然而，由于应用问题变得越来越复杂，并且由于计算机工程的进步，迫切需要由“纯粹”和“应用”数学家组成的科学团体，旨在构建和证明有效的计算方案，以解决特定的应用问题，并将其解决方案提升到数值计算机实现。本文作者属于茹科夫斯基空军工程院成立的用离散涡法求解气动问题的研究小组之一。最近，在离散涡方法中，我们一直在发展（见[42,43]）以下方法的数值解决的问题，一个理想的不可压缩流体周围的厚翼型，特别是那些具有尖锐尾缘（所谓的茹科夫斯基型）。选择一条平均线，在翼型表面上的不透水边界条件被转移到这条线上。结果，在这条平均值线之外，我们得到了拉普拉斯方程的边值问题，其边界条件包含所求函数的法向导数和正切导数的组合，在曲线的不同边的组合是不同的。通过在这条线上放置涡度层和源层，并满足不渗透边界条件，得到了这条线上的奇异积分方程组。该系统的具体形式在第2节中给出。本文采用涡旋法计算了透水与不透水、薄厚翼型表面的气动特性，使所研究的气动问题简化为奇异积分实系数方程的求解。对于弱变形不透水无限薄翼（a型）的绕流问题，将其简化为带柯西核的一维奇异第一类线性积分方程。对于可渗透薄翼型周围的流动，许多问题（例如，降落伞周围的流动）导致奇异第二类积分变系数方程（在这种情况下，奇异积分的系数是一个多项式）。如果考虑厚翼型绕流问题，我们得到了更一般的奇异第二类积分方程或奇异积分方程组。对于可渗透的薄有限翼，气动问题可简化为二维二阶积分方程和Hadamard有限值积分（见第1节）。在第三节中，我们考虑了复变平面上任意分段光滑曲线上奇异第二类积分变效率方程的一般理论。第1节和第2节中描述的奇异积分方程是此类方程的特殊情况。在本节中，我们将介绍奇异积分算子，它可以方便地开发求解所研究的奇异积分方程的数值方法。讨论了用柯西型多重积分发展奇异积分方程理论的可能性。将奇异积分方程的解问题表示为一个算子方程，并给出了近似解的格式。第3节中考虑的奇异积分方程近似解的方法在第4节和第5节中通过第1节和第2节中描述的空气动力学问题中出现的一个实际奇异积分方程的例子来说明。在这方面，我们涉及到这类方程的解的问题的表述的正确性。详细介绍了可透厚翼型绕流的计算方案。本文提出的具有柯西核的奇异积分方程数值解的方法已在计算机上实现。在第6节中，我们给出了一些特定气动问题的数值计算结果。结果与精确解和其他格式的计算结果进行了比较。1. 一个可渗透的翼型周围流动让一个薄均匀可渗透的翼型部分b[41]被一个理想的不可压缩流体，它有一个速度UQ在无限飞行。让我们绑一个坐标系氧的翼型如图1所示，并表示弦的翼型ba。这个问题在数学上表述如下。除了翼型表面S外，流在任何地方都是势。扰动速度势φ在任意点M£S服从拉普拉斯方程：
Problems of flow around permeable and thick airfoils are reduced to solution of singular second-kind variable-coefficient integral equations on a segment or to the systems of such equations. A numerical method for solving the problems is constructed. The method is based on approximating a smooth solution component by the Lagrangian polynomials. Solutions to particular aerodynamic problems are calculated. When being reduced to integral equations, many theoretical and applied problems naturally lead to linear one-dimensional singular integral equations or to the systems of such equations on open or closed curves. For example, in aerodynamics such problems arise in modelling the flow of a vorticity layer around an airfoil, in diffraction of electromagnetic waves they arise in modelling a reflecting surface by induced currents, in the theory of elasticity they arise in modelling the boundary of a body by distributed forces, etc. A theory of such equations appeared at the end of the last century in the works of D. Hubert and A. Poincare. Its foundations were laid by the studies of F. Neter and T. Karleman. In the first half of this century the theory was significantly developed by Soviet mathematicians. A somewhat complete form was given to the theory in wide-known monographs of N.I. Muskhelashvili and F. D. Gakhov. Further the theory of such equations branched in many directions such as imposing less strict conditions on the class of the sought-for solutions (the so-called L -theory), on the coefficients of the equations, etc. The methods of functional analysis and algebra are extensively used in modern studies devoted to singular integral equations. In numerous applications of singular integral equations bringing their solutions up to a numerical result is required and, consequently, efficient approximate methods of their computer-aided solution should be developed. Pioneer works on constructing numerical methods of solution of singular integral equations date back to the 1930s [13,20,37]. These works initiated a systematic study of the applicability of various approximate methods to singular integral equations in the works of S. G. Mikhlin, A. N. Kalandii, V. V. Ivanov, I. C. Gokhberg, B. G. Gabdulakhaev, I. K. Lifanov, Z. Presdorf, D. Elliott, B. Zilberman and their followers. At present there are many works that deal with approximate solution of singular integral equations. By convention, most of the works may be classified as (1) the works of theoretical character dealing with construction of computational schemes and analysis of their convergence and stability; (2) the works dealing with numerical solution of particular singular integral equations encountered in applications. The authors of the works of the first group are as a rule far from numerical solution of particular applied problems. They recommend to solve something properly 110 LK.Lifanov et al. but not always something that is necessary. The authors of the works of the second group solve something that is necessary but not always properly. However, due to applied problems becoming more and more complex and due to a progress in computer engineering there is an urgent demand for scientific groups that would consist of both 'pure' and 'applied' mathematicians and aim at constructing and justifying efficient computational schemes for solving particular applied problems and bringing their solutions up to the numerical computer implementation. The authors of this paper belong to one of such groups formed in N. E. Zhukovsky Airforce Engineering Academy for solving aerodynamic problems by a discrete vortex method. Lately in the discrete vortex method we have been developing (see [42,43]) the following approach to numerical solution of of the problem of the flow of an ideal incompressible fluid around thick airfoils, especially those with sharp trailing edge (the so-called Zhukovsky profiles). A mean line is chosen and impermeability boundary conditions on the airfoil surface are carried over to this line. As a result, outside this mean line we arrive at a boundary-value problem for the Laplace equation with the boundary conditions that contain a combination of normal and tangent derivatives of the sought-for function, the combination being different on different sides of the curve. By placing the vorticity and source layers on this line and satisfying the impermeability boundary conditions we obtain a system of singular integral equations on this line. The particular form of the system is given in Section 2. In this paper aerodynamic characteristics of permeable and impermeable, thin and thick airfoil surfaces are calculated using the vortex method which allows us to reduce the aerodynamic problems under study to solution of singular integral real-coefficient equations. For the flow around a weakly-deformable impermeable thin infinite wing (a profile) the problem is reduced to a linear one-dimensional singular first-kind integral equation with the Cauchy kernel. For the flow around a permeable thin airfoil a number of problems (for instance, the flow around a parachute) lead to singular second-kind integral variable-coefficient equations (in this case the coefficient of a singular integral is a polynomial). If the problem of flow around a thick airfoil is considered, we obtain a more general singular second-kind integral equation or the system of singular integral equations. For a permeable thin finite wing the aerodynamic problem is reducible to a two-dimensional second-kind integral equation with the Hadamard finite value integral (see Section 1). In Section 3 we consider a general theory of singular second-kind integral variablecoefficient equations on an arbitrary piecewise smooth curve in the complex-variable plane. Singular integral equations described in Sections 1 and 2 are special cases of such equations. In this section we introduce singular integral operators that are convenient in developing numerical methods for solving singular integral equations under study. We also discuss the possibility of developing the theory of singular integral equations with the Cauchy-type multiple integrals. The problem of solution of singular integral equations is represented as an operator equation for which a scheme of approximate solution is given. An approach to the approximate solution of singular integral equations considered in Section 3 is illustrated in Section 4 and 5 by an example of a real singular integral equation that appears in the aerodynamic problems described in Sections 1 and 2. In this connection we touch upon the correctness of formulation of the problem of solution of such equations. A detailed description of the computational scheme is Flow around permeable and thick airfoils 111 presented. The proposed method of numerical solution of singular integral equations with the Cauchy kernel has been implemented on a computer. In Section 6 we give some results of numerical calculations of particular aerodynamic problems. The results are compared with exact solutions and the calculations obtained with other schemes. 1. FLOW AROUND A PERMEABLE AIRFOIL Let a thin uniformly permeable airfoil section [41] be flown around by an ideal incompressible fluid which has a velocity UQ at infinity. Let us tie a coordinate system Oxy to the airfoil as shown in Fig. 1 and denote the chord of the airfoil as ba. Mathematically the problem is stated as follows. The flow is potential everywhere except for a surface S of the airfoil. The perturbed velocity potential φ obeys the Laplace equation at any point M£S: