High-speed Flow Past WingsThe analytical solution to the transonic small perturbation equation which describes steady compressible flow past finite wings at subsonic speeds can be expressed as a nonlinear integral equation with the perturbation velocity potential as the unknown function. This known formulation is substituted by a system of nonlinear algebraic equations to which various methods are applicable for its solution. Due to the presence of mathematical discontinuities in the flow solutions, however, a main computational difficulty was to ensure uniqueness of the solutions when local velocities on the wing exceeded the speed of sound. For continuous solutions this was achieved by embedding the algebraic system in an one-parameter operator homotopy in order to apply the method of parametric differentiation. The solution to the initial system of equations appears then as a solution to a Cauchy problem where the initial condition is related to the accompanying incompressible flow solution. In using this technique, however, a continuous dependence of the solution development on the initial data is lost when the solution reaches the minimum bifurcation point. A steepest descent iteration technique was therefore, added to the computational scheme for the calculation of discontinuous flow solutions. Results for purely subsonic flows and supersonic flows with and without compression shocks are given and compared with other available theoretical solutions. |
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1.0 x/CHORD FIGURE airfoil algebraic system analytical ANGLE OF ATTACK arbitrary aspect ratio wing boundary conditions calculation scheme chordwise COMP compressible flow compression DATA defined DELTX e.g. reference END C SUBROUTINE equations 23 evaluation flow solutions FORMAT 1H freestream Mach number GO TO 50 harmonic solution IHLF influence coefficients influence function input integral equation ISECT Jacobian Kutta condition linearized solution method of parametric method of steepest NACA NAMELIST Newton's method nonlinear NOT.TWING NUMERICAL ANALYSIS numerical solutions NWING obtained parameter parametric differentiation past a lifting past a non-lifting perturbation velocity PRESENT RESULT PRESSURE DISTRIBUTION PRMT PROFILE rectangular wing SECTIN solution of system SPANWISE STATION steepest descent Subcritical flow Supercritical flow Supercritical flow past supersonic surface swept wing system 23 system 31 Thickness ratio three-dimensional transonic flow two-dimensional U₁ velocity potential wing planform WING RECTANGULAR PLANFORM wings at subsonic WRITE X=X+H YPRIME=-YPRIME כם