Numerical Solution for Airfoils near Stall in Optimized Boundary-Fitted Curvilinear Coordinates

Abstract

Boundary-fitted curvilinear coordinate systems are optimized for viscous flows about arbitrary airfoils at angles of attack such that boundary-layer-dependent grid systems for high Reynolds numbers are generated efficiently. The grid systems are utilized in implicit finite-difference solutions. Solution of a one-dimensional model equation is compared with the theoretical solution. The unsteady Navier-Stokes equations are solved for the incompressible flow around a cylinder and around NACA airfoils approaching stall. The predicted flows around a NACA 6412 airfoil near stall at Reynolds numbers of 4x 10 4 and 2x 10 5 are compared with the experimental observations obtained in a smoke tunnel. Nomenclature= amplitude coefficients for contracting mesh systems = drag coefficient = lift coefficient = pressure coefficient = contours in a doubly connected region = velocity divergence = coordinate-transformation parameter, Eq.(11) = decay coefficients for contracting mesh systems = number of £ lines in transformed plane = Jacobian determinant = number of 17 lines in transformed plane = number of points in boundary layer = characteristic length = nondimensional pressure (C p /2) = summations of exponential functions for contraction of mesh systems = two-dimensional physical region = Reynolds number = Strouhal number based on thickness = nondimensional time (Ut/L ) = nondimensional velocity component in x direction = freestream velocity magnitude = nondimensional velocity component in y direction = nondimensional physical field coordinates = coordinate-transformation parameter, Eq.(2) = coordinate-transformation parameter, Eq.(2) (2) 17 = transformed coordiante £ = tranformed coordinate a = coordinate-transformation parameter, Eq. (U) r = coordinate-transformation parameter, Eq. (U) 0 = angle of attack <f> . = acceleration parameter IJL = function definition, Eq. (6) Superscripts N (s) * Subscripts U cs = time step index = iterate number = the transformed plane = field position in mesh systems = central-space difference = first partial differentiation = second partial differentiation = cross partial differentiation = value at infinity

Analysis of Two-Dimensional Incompressible Flow Past Airfoils Using Unsteady Navier-Stokes Equations

“…For the same Joukowski airfoil configuration, they showed improved computational efficiency. Hodge et al [7] solved the unsteady Navier-Stokes equations using 0-grids and predicted flows around NACA airfoils at stall Re of Uxio 1 * and 2*10 5 and compared their results qualitatively with experimental data.…”

An experimental investigation of an airfoil pitching at moderate to high rates to large angles of attack

“…A number of approaches have been taken with regard to the prediction of unsteady stalled airfoils. In general, these approaches range from empirical models [8][9][10][11] to models based on the Navier-Stokes equations [12][13].…”

A method for computing three dimensional flows using non‐orthogonal boundary‐fitted co‐ordinates

“…4. The P'-equation is then solved, and the solution used to correct U" and V* to U and V through equations (14) and (15). The corrected velocities conserve mass.…”