# 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.

##### Citations

More filters

••

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.

84 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.

58 citations

••

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.

26 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

24 citations

••

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

10 citations

##### References

More filters

••

TL;DR: In this paper, an experimental and theoretical study of transonic flow over a thick airfoil, prompted by a need for adequately documented experiments that could provide rigorous verification of viscous flow simulation computer codes, is reported.

Abstract: An experimental and theoretical study of transonic flow over a thick airfoil, prompted by a need for adequately documented experiments that could provide rigorous verification of viscous flow simulation computer codes, is reported. Special attention is given to the shock-induced separation phenomenon in the turbulent regime. Measurements presented include surface pressures, streamline and flow separation patterns, and shadowgraphs. For a limited range of free-stream Mach numbers the airfoil flow field is found to be unsteady. Dynamic pressure measurements and high-speed shadowgraph movies were taken to investigate this phenomenon. Comparisons of experimentally determined and numerically simulated steady flows using a new viscous-turbulent code are also included. The comparisons show the importance of including an accurate turbulence model. When the shock-boundary layer interaction is weak the turbulence model employed appears adequate, but when the interaction is strong, and extensive regions of separation are present, the model is inadequate and needs further development.

189 citations

### "Predicted and experimental steady a..." refers background in this paper

...i-or a constant angle of Attack and Reynolds number, and at progressively higher froe-stream Mach number-,, tiles flow field about the airfoil can be (1) steady, j with trailing-edge boundary-layer separation, (2) unsteady, with aerodynamically self-excited periodic oscillations in shock-wave location and intensity and in the location of boundiry-laver separation, or (3) can be steady with shock-induced separation....

[...]

••

TL;DR: In this paper, an experimental and computational investigation of the steady and unsteady transonic flowfields about a thick airfoil is described, and an operational computer code for solving the two-dimensional, compressible NavierStokes equations for flow over airfoils was modified to include solid-wall, slip-flow boundary conditions to properly assess the code and help guide the development of improved turbulence models.

Abstract: An experimental and computational investigation of the steady and unsteady transonic flowfields about a thick airfoil is described. An operational computer code for solving the two-dimensional, compressible NavierStokes equations for flow over airfoils was modified to include solid-wall, slip-flow boundary conditions to properly assess the code and help guide the development of improved turbulence models. Steady and unsteady fiowfieids about an 18% thick circular arc airfoil at Mach numbers of 0.720, 0.754, and 0.783 and a chord Reynolds number of 11 x 10 are predicted and compared with experiment. Results from comparisons with experimental pressure and skin-friction distributions show improved agreement when including test-section wall boundaries in the computations. Steady-flow results were in good quantitative agreement with experimental data for flow conditions which result in relatively small regions of separated flow. For flows with larger regions of separated flow, improvements in turbulence modeling are required before good agreement with experiment will be obtained. For the first time, computed results for unsteady turbulent flows with separation caused by a shock wave were obtained which qualitatively reproduce the time-dependent aspects of experiments. Features such as the intensity and reduced frequency of airfoil surface-pressure fluctuations, oscillatory regions of trailing-edge and shock-induced separation, and the Mach number range for unsteady flows were all qualitatively reproduced.

152 citations

01 Jul 1976

TL;DR: In this article, a new numerical method used to drastically reduce the computation time required to solve the Navier-Stokes equations at flight Reynolds numbers is described, which makes it possible and practical to calculate many important three-dimensional, high Reynolds number flow fields on computers.

Abstract: A new numerical method used to drastically reduce the computation time required to solve the Navier-Stokes equations at flight Reynolds numbers is described. The new method makes it possible and practical to calculate many important three-dimensional, high Reynolds number flow fields on computers.

93 citations

••

TL;DR: In this article, four different algebraic eddy viscoisity models are tested for viability to achieve turbulence closure for the class of flows considered, ranging from an unmodified boundary-layer mixing-length model to a relaxation model incorporating special considerations for the separation bubble region.

Abstract: The two-dimensional Reynolds averaged compressible Navier-Stokes equations are solved using MacCormack's second-order accurate explicit finite difference method to simulate the separated transonic tur- bulent flowfield over an airfoil. Four different algebraic eddy viscoisity models are tested for viability to achieve turbulence closure for the class of flows considered. These models range from an unmodified boundary-layer mixing-length model to a relaxation model incorporating special considerations for the separation bubble region. Results of this study indicate the necessity for special attention to the separated flow region and suggest limits of applicability of algebraic turbulence models to these separated flowfield. each of these studies the time-dependent Reynolds averaged Navier-Stokes equations for two-dimensional compressive flow are used and tur- bulence closure is achieved by means of model equations for the Reynolds stresses. Wilcox1'2 used a first-order accurate numerical scheme and the two equation differential tur- bulence model of Saffman 12 to simulate the supersonic shock boundary-layer interaction experiment of Reda and Mur- phy 13 and the compression corner flow of Law.14 Good quan- titative agreement with the Reda and Murphy data was ob- tained, but only the qualitative features of the compression corner flow were well simulated. Using a more sophisticated second-order accurate numerical scheme, Baldwin3'4 con- sidered both the two equation differential model of Saffman and a simpler algebraic mixing-length model to simulate the hypersonic shock boundary-layer interaction experiment of Holden.15 He found the more elaborate model of Saffman to yield somewhat better results than the algebraic model, but at the cost of considerably more computing time. Good quan- titative agreement with experiment was not obtained with either model. Following Baldwin's approach all subsequent investigations have been performed using the more rigorous second-order accurate numerical scheme of Mac- Cormack.17'18 Deiwert5'6'11 considered an algebraic mixing- length model to simulate the transonic airfoil experiment of McDevitt et al. 16 while Horstman et al. 8 used a similar ap- proach to simulate their hypersonic shock boundary-layer ex- periment on an axisymmetric cylinder. In each of these studies, while qualitative features of the flows were described well, good quantitative agreement with experiment in the in- teraction regions was not obtained. Using a relaxing turbulence model Shang and Hankey7 simulated the compression corner flow of Law, and Baldwin and Rose10 simulated the flat plate flow of Reda and Murphy. In each of these studies the relaxing model was found to per- form significantly better than the simpler algebraic model and, according to Shang and Hankey, provided significantly better comparisons with measurements than were obtained by Wilcox using the two equation differential model of Saffman. In each of these studies it was essential that the full Navier- Stokes equations be considered to describe the viscous- inviscid interaction and the elliptic nature of separating-

90 citations

••

TL;DR: In this paper, an explicit finite-difference method with time splitting is used to solve the time-dependent equations for compressible turbulent flow, and a nonorthogonal computational mesh of arbitrary configuration facilitates the description of the flow field.

Abstract: A code has been developed for simulating high Reynolds number transonic flow fields of arbitrary configuration. An explicit finite-difference method with time splitting is used to solve the time-dependent equations for compressible turbulent flow. A nonorthogonal computational mesh of arbitrary configuration facilitates the description of the flow field. The code is applied to simulate the flow over an 18 percent thick circular-arc biconvex airfoil at zero angle of attack and free-stream Mach number of 0.775. A simple mixing-length model is used to describe the turbulence and chord Reynolds numbers of 1, 2, 4, and 10 million are considered. The solution describes in sufficient detail both the shock-induced and trailing-edge separation regions, and provides the profile and friction drag.

81 citations