# Predicted and experimental steady and unsteady transonic flows about a biconvex airfoil

01 Feb 1981-

TL;DR: The results of computer code time dependent solutions of the two dimensional compressible Navier-Stokes equations and the results of independent experiments are compared to verify the Mach number range for instabilities in the transonic flow field about a 14 percent thick biconvex airfoil at an angle of attack of 0 deg and a Reynolds number of 7 million.

Abstract: Results of computer code time dependent solutions of the two dimensional compressible Navier-Stokes equations and the results of independent experiments are compared to verify the Mach number range for instabilities in the transonic flow field about a 14 percent thick biconvex airfoil at an angle of attack of 0 deg and a Reynolds number of 7 million. The experiments were conducted in a transonic, slotted wall wind tunnel. The computer code included an algebraic eddy viscosity turbulence model developed for steady flows, and all computations were made using free flight boundary conditions. All of the features documented experimentally for both steady and unsteady flows were predicted qualitatively; even with the above simplifications, the predictions were, on the whole, in good quantitative agreement with experiment. In particular, predicted time histories of shock wave position, surface pressures, lift, and pitching moment were found to be in very good agreement with experiment for an unsteady flow. Depending upon the free stream Mach number for steady flows, the surface pressure downstream of the shock wave or the shock wave location was not well predicted.

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01 Mar 1987TL;DR: The importance of mixed attached and separated flow modeling for aeroelastic analysis is discussed and recent calculations of periodic aerodynamic oscillations for an 18 percent thick circular arc airfoil are given.

Abstract: Computational methods for unsteady transonic flows are surveyed with emphasis on prediction. Computational difficulty is discussed with respect to type of unsteady flow; attached, mixed (attached/separated) and separated. Significant early computations of shock motions, aileron buzz and periodic oscillations are discussed. The maturation of computational methods towards the capability of treating complete vehicles with reasonable computational resources is noted and a survey of recent comparisons with experimental results is compiled. The importance of mixed attached and separated flow modeling for aeroelastic analysis is discussed, and recent calculations of periodic aerodynamic oscillations for an 18 percent thick circular arc airfoil are given.

84Â citations

01 Jan 1984

TL;DR: The original formulation of an aerodynamic response in terms of nonlinear functionals is shown to be compatible with a derivation based on the use of non linear functional expansions.

Abstract: Basic concepts involved in the mathematical modeling of the aerodynamic response of an aircraft to arbitrary maneuvers are reviewed. The original formulation of an aerodynamic response in terms of nonlinear functionals is shown to be compatible with a derivation based on the use of nonlinear functional expansions. Extensions of the analysis through its natural connection with ideas from bifurcation theory are indicated.

58Â citations

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TL;DR: In this article, the authors investigated the origin of shock oscillations on an 18% thick biconvex aerofoil using a thin-layer Navier-Stokes code.

26Â citations

01 Jan 1982

TL;DR: In this paper, the state of the art in computer simulations for transonic flowfields requiring solutions for the Navier-Stokes equations is assessed and the choice of coordinate systems and dependent variables for simulating the flow around airfoils is discussed, with particular attention to curvilinear coordinates.

Abstract: The state of the art in computer simulations for transonic flowfields requiring solutions for the Navier-Stokes equations is assessed. It is noted that current simulations of transonic flowfields require comparisons with experimental results because the simulations are not free from discretization errors. Cases of turbulence are treated with weighted variables in a time-averaged scheme to yield Reynolds averaged Navier-Stokes equations. The turbulence is modeled in a first-order approach with a Reynolds stress tensor or a second-order approach where the tensor is obtained from the Navier-Stokes equations. The choice of coordinate systems and dependent variables for simulating the flow around airfoils is discussed, with particular attention to curvilinear coordinates. The determinations of boundary conditions is examined, along with numerical methods related to physical phenomena

24Â citations

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19 Jun 1995

TL;DR: Gillan et al. as discussed by the authors used an explicit cell-vertex-centred Navier-Stokes code in conjunction with a hyperbolic C-grid generator to predict self-excited shock induced oscillations on an 18% thick circular-arc airfoil.

Abstract: A numerical study investigating the ability of the full massaveraged Navier-Stokes equations to accurately predict selfexcited shock induced oscillations on an eighteen per cent thick circular-arc airfoil is presented. An explicit cell-vertex-centred Navier-Stokes code is employed in conjunction with a hyperbolic C-grid generator. Turbulence closure is accomplished using the zero equation algebraic Baldwin-Lomax model. The code accurately predicts the shock induced oscillation onset boundary, reduced frequencies and hysteresis region. Comparing these results with those obtained from a thin-layer version of the code highlights the limitations of the latter technique. Only by employing the full mass averaged Navier-Stokes equations, operating on a suitably fine grid, can the dynamic shear layers be adequately resolved. Introduction The need to accurately model the evolution of unsteady aerodynamic phenomena, such as buffet, flutter, limit cycle oscillations and buzz has thrust unsteady computational fluid dynamics to the forefront of modern research. The classical experiments performed by Tijdeman' on a NACA64A006 airfoil with a trailing-edge flap detected three types of shock motion. Tijdeman's results have proven equally valid for rigid airfoil studies and form the basis for the classification of self-excited shock induced oscillations, or SIO. Tijdeman type A SIO is were the shock wave remains distinct throughout the oscillation with a cyclic change in both the shock wave strength and location. In Tijdeman type B SIO the shock wave vanishes for part of the cycle, normally whilst the shock is propagating upstream. Finally, Tijdeman type C SIO describes the shock wave motion whereby the shock remains distinct as it propagates upstream past the leading-edge and into the on coming flow. Throughout the past two decades extensive experimental investigations into periodic flow over varying thickness circulararc airfoils have been p e r f ~ r m e d ~ ~ . These experiments, which were conducted over a wide range of free stream Mach and Reynolds numbers, have detected all three types of SIO. References 2 and 3 identified Tijdeman type C SIO, with a reduced frequency of, k = 0.49, for an 18 % thick circular-arc airfoil at zero incidence over a narrow range of Mach numbers, namely: 0.73 < M, < 0.78. Small regions of Tijdeman type A SIO were also detected in the extremities of the periodic flow band. Figure 1 indicates that the extent of this periodic band decreases considerably as the Reynolds number approaches 'ResearchTechnicalEngineer, Nacelle Systems, Member AIAA. Copyright " 1995 by Mark Gillan. Published by the American Institute of Aeronautics and Astronautics, Inc. with permission. 3 x 10'. A similar phenomenon was detected by Mabey et a1.' whereby the SIO vanished within the Reynolds number range of: 3 x lo6 < Re, < 5 x 10'. Furthermore, the region of flow hysteresis which McDevitt et al.' discovered was later verified by Seegmiller et aL4 and LevyS (fig.1). Finally, Gibbg, who conducted a comprehensive study of flow over a 14% thick circular-arc airfoil, detected strong Tijdeman type B SIO, with small regions of Tijdeman type A SIO, occurring within a narrow Mach number band, namely: 0.84 < M, < 0.88. Following various steady-state c~mputations~.'~-", Levy5 eventually produced the first unsteady Navier-Stokes computation over an 18 7% thick circular-arc airfoil that was capable of reproducing the intrinsically asymmetric periodic flow. Levy used a modified version of the code employed by DeiwertloaL' which included solid-wall test section inviscid boundary conditions. Despite employing an extremely coarse 78 X 35 grid and a 1 % chord nose radius (to alleviate numerical difficulties), the impulsively started computation produced SIO with a reduced frequency of, k=0.4, comparable to that produced experimentally by McDevitt et a1.I of, k=0.49. Furthermore, in an attempt to show that the periodic flow solution was the result of viscous effects alone and not due to spurious numerical inaccuracies, Levy ran the periodic test case for various different configurations. Both an inviscid test case and a viscous computation, with a 114 chord trailing-edge splitter plate, which effectively prevents pressure wave communication across the wake, produced steady flow solutions, thereby corroborating Levy's hypothesis. Seegmiller et aL4 developed Levy's research further by presenting an in-depth explanation regarding the exact nature of the periodic mechanism. Furthermore, they4 astutely recognised that the period of the SIO depended on the time taken for the flow to adjust to, and counteract, the airfoil's effective change in camber due to the asymmetry of the shock-induced separation. This reasoning is supported by the earlier experimentation of ~ i n k e ' ~ . Subsequent Navier-Stokes computations over circular-arc airfoils have been performed by Steger", Edwards and ThomasL4, Gerteisenl' and illa an^^^'^. Steger" computed periodic flow with a reduced frequency of, k=0.41, at a free stream Mach number of, M, =0.783. Edwards and tho ma^'^ employed anupwind-biased Navier-Stokes scheme and computed Tijdeman type B SIO with a reduced frequency of, k=0.406, at a free-stream Mach number comparable to Steger's, of M, =0.78. GerteisenLs failed to reproduce either Tijdeman type B, or type C, SIO. Gertesin believed that this was due to a combination of insufficient shock resolution, inadequate turbulence modelling and the existence of transitionally dominated flow. Gillan16 computed periodic flow over an 18 % thick circular-arc airfoil at 0' angle of attack and a free stream Mach number of M, =0.771 (see fig.2). This Mach number lies within the periodic region which was detected both experimentally and computationally by Levy5 as depicted in fig. 1. No attempt was made during the computational analysis to detect the hysteresis region depicted in this figure. A 320x64 cells was employed, with 256 cells placed on the airfoil's surface. A far-field boundary of 20 chord lengths was enforced, with the initial near-wall normal grid spacing corresponding to a value of y+ < 2 for the assigned Reynolds number of Re, = 1 1 X 10.~. Moreover, a 1 % leading-edge radius was introduced in an attempt to avoid any unnecessary computational difficulties. The computational test case, which was impulsively started from free stream conditions, produced an unsteady periodic Tijdeman type B motion with a reduced frequency of, k = 0.396. Although this reduced frequency is approximately 21 % lower than that obtained experimentally by McDevitt et aL2 it compares favourably with the computed values of 0.4 and 0.406 obtained by Levy5 and Edwards and Thomasr4 respectively. In an attempt to introduce some non-equilibrium history effects into periodic flow computations various pre-eminent researchers have temporally 'abandoned' the use of NavierStokes solvers. Le Balleur and Girodroux-La~igne'~'~ used a small-disturbance potential method with a two-equation integral viscous solver to compute SIO with a reduced frequency of, k=0.34, at a free-stream Mach number of, M ~ 0 . 7 6 . Although this frequency is significantly lower than previous results, it indicates the ability of a non-Navier-Stokes solver to qualitatively reproduce unsteady periodic flows. Finally, Edwards2" has recently employed a new lag-entrainment integral boundary layer method coupled with a transonic small disturbance potential code to compute periodic flow over a range of configurations. Edwards accurately modelled the SIO period for an 18% thick circular-arc airfoil, detecting a seduced frequency of, k=0.47, at a free-stream Mach number of, M, =0.76. Furthermore, he reproduced, for the first time, the hysteresis region which was initially discovered experimentally by McDevitt et al.'. The primary objective of this paper is to investigate the ability of the full mass-averaged Navier-Stokes equations, operating on a suitably fine grid, to accurately predict SIO on an 18% thick circular-arc airfoil. In order to perform this analysis a hyperbolic grid generation code and an explicit finite volume cell vertex-centred Navier-Stokes code have been employed. The following section briefly describes both the numerical model and the results. Numerical Model A recently developed explicit Navier-Stokes finite volume code (MGENS2D) has been used in conjunction with an orthogonal boundary conforming hyperbolic C-grid. A detailed description of both the Navier-Stokes scheme and the grid generation code (MGHYPR) is given in ref. 16. If Q represents an arbitrary control volume, with a domain boundary, an, and a unit outward normal, n, then, neglecting body forces and external heat addition, the integral form of the 2-0 non-dimensional mass-averaged Navier-Stokes equations may be written as: where W is the vector of the conserved quantities: with p,u,v and E denoting the density, the cartesian velocities and the specific total internal energy respectively. The flux tensor P is then split into a convective part, Few, and a viscous part, F,,,: P = PCMV Fam

10Â citations

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01 Jan 1977

TL;DR: In this paper, a simple algebraic eddy viscosity model is used for the description of the turbulent transport process of transonic flows, taking into account a biconvex circular arc and a shockless lifting airfoil.

Abstract: An algorithm developed by MacCormack (1971) and applied to transonic flows by Deiwert (1974) is used in the reported investigation. The investigation is concerned with flows of aerodynamic interest. However, many of the concepts apply equally to flows in turbomachinery. Turbulent transonic flows are considered, taking into account a biconvex circular arc and a shockless lifting airfoil. A simple algebraic eddy viscosity model is used for the description of the turbulent transport process.

4Â citations

01 Feb 1975

TL;DR: In this article, the time-splitting explicit numerical method of MacCormack is applied to separated turbulent boundary layer flow problems, and numerical solutions of high-Reynolds-number separated flows over an airfoil and shock-separated flows over a flat plate are obtained.

Abstract: The time-splitting explicit numerical method of MacCormack is applied to separated turbulent boundary layer flow problems. Modifications of this basic method are developed to counter difficulties associated with complicated geometry and severe numerical resolution requirements of turbulence model equations. The accuracy of solutions is investigated by comparison with exact solutions for several simple cases. Procedures are developed for modifying the basic method to improve the accuracy. Numerical solutions of high-Reynolds-number separated flows over an airfoil and shock-separated flows over a flat plate are obtained. A simple mixing length model of turbulence is used for the transonic flow past an airfoil. A nonorthogonal mesh of arbitrary configuration facilitates the description of the flow field. For the simpler geometry associated with the flat plate, a rectangular mesh is used, and solutions are obtained based on a two-equation differential model of turbulence.

2Â citations