Computation of Vortex Shedding and Radiated Sound for a Circular Cylinder

The Lighthill acoustic analogy approach combined with Reynolds-averaged Navier Stokes is used to predict the sound generated by unsteady viscous ow past a circular cylinder assuming a correlation length of ten cylinder diameters. The twodimensional unsteady ow eld is computed using two NavierStokes codes at a low Mach number over a range of Reynolds numbers from 100 to 5 million. Both laminar ow as well as turbulent ow with a variety of eddy viscosity turbulence models are employed. Mean drag and Strouhal number are examined, and trends similar to experiments are observed. Computing the noise within the Reynolds number regime where transition to turbulence occurs near the separation point is problematic: laminar ow exhibits chaotic behavior and turbulent ow exhibits strong dependence on the turbulence model employed. Comparisons of fareld noise with experiment at a Reynolds number of 90,000, therefore, vary signi cantly, depending on the turbulence model. At a high Reynolds number outside this regime, three di erent turbulence models yield self-consistent results. INTRODUCTION The sound generated by the unsteady viscous ow over a circular cylinder is representative of several blu body ows found in engineering applications (e.g., automobile antenna noise, aircraft landing gear noise, etc.). Although extensive experimental studies have been performed on circular cylinder ows (see, e.g., Morkovin), many questions about the physical processes that occur still exist, particularly at higher Reynolds numbers. Numerically, many computational uid dynamics (CFD) studies have also been performed, but primarily only at low Reynolds numbers with purely laminar ow. The ability to accurately compute cylinder noise across a broad range of Reynolds numbers with numerical methods would enhance the understanding of blu body noise generation mechanisms. A complete understanding of circular cylinder ow is particularly elusive because transition from laminar to turbulent ow occurs in a distinct succession over an enormous range of Reynolds numbers, and each transition state is sensitive to extremely small disturbances. These disturbances, such as free stream turbulence and surface roughness, can signi cantly alter the range of Reynolds numbers over which each transition state occurs. Especially at Reynolds numbers at and above roughly 100,000, experiments can show widely di erent behavior due to di erences in the experimental ow conditions. The ow around a circular cylinder is often characterized into three distinct ow regimes: subcritical, supercritical, and transcritical). Subcritical ow indicates purely laminar boundary layer separation. In this regime, regular vortex shedding at a Strouhal number of about 0.2 is observed over a range of Reynolds numbers from roughly 200 to 100,000. The supercritical regime, from Reynolds numbers of roughly 100,000 to 4 million, is characterized by either a dramatic rise in the Strouhal number or else a loss of organized vortex shedding altogether. Also, the wake is noticeably narrower and the forces are much smaller in magnitude. It is somewhere in this regime that transition to turbulence begins to occur on the body at or near the point of separation. In the transcritical regime, above a Reynolds number of roughly 4 million, periodic vortex shedding re-establishes at a higher Strouhal number of 0.26 0.30. 7 The cylinder now experiences fully turbulent boundary layer separation and higher force coe cients than in the supercritical regime. Any attempt to numerically model circular cylinder ow is complicated by the fact that the ow above a Reynolds number of around 180 is three-dimensional, raising doubts about the applicability of two-dimensional simulations. Additionally, transition occurs o -body in the wake or shear layer at Reynolds numbers between roughly 200 and the supercritical regime. Without performing very expensive direct numerical simulations (DNS), this behavior is not captured by numerical methods that solve the Navier-Stokes equations on typical grids used for aerodynamic analysis. This de ciency may or may not be important at lower Reynolds numbers, depending on how far behind the cylinder transition occurs and what feature of the ow is of interest. But it certainly has an adverse e ect at higher Reynolds numbers for which transition occurs at or near the separation point on the cylinder. For Reynolds numbers at and above the supercritical regime, Reynolds-averaging with the use of a turbulence model is one way to introduce the important e ect of turbulence into a numerical simulation. However, without an accurate built-in transition model, it is di cult, if not impossible, to model the important e ects of transition, particularly when it occurs on or near the body. It is not surprising, then, that most numerical studies of ow around a circular cylinder have focused primarily on low Reynolds number ows less than about 1000. At Reynolds numbers of roughly 200 or less, many researchers have successfully computed the Strouhal number and mean drag over a circular cylinder (see, for example references, 10{14). At higher Reynolds numbers, however, two-dimensional numerical methods cannot predict the lift and drag forces accurately, due to the increasingly prominent three-dimensionality of the the real ow eld. Nonetheless, it is still important to try to understand and characterize the capabilities and limitations of existing two-dimensional numerical methods at higher Reynolds numbers (where most blu -body noise sources of interest occur), since these methods may yield a deeper insight into the physics at a relatively low cost. In this work, an investigation is made into the ability to numerically predict the aerodynamic properties of a circular cylinder across a range of Reynolds numbers from 100 to 5 million. The unsteady viscous ow eld is computed by two di erent Reynolds-averaged Navier Stokes (RANS) ow solvers with a variety of turbulence models, as well as at lower Reynolds numbers using the laminar Navier-Stokes equations. The computations are two-dimensional and time-accurate. The oweld computations are emphasized because they are the most challenging aspect of the noise prediction problem. The noise is calculated with an acoustic prediction code that uses the calculated unsteady surface pressures as input. The acoustic code uses an advanced acoustic analogy integral formulation developed by Farassat. In this paper, results are presented at a single observer location for each of the oweld computations. The noise predictions are compared with the experiment of Revell. A companion paper by the authors deals more fully with the acoustic aspects of the problem. No attempt is made in this paper to predict broadband noise. To do this, it maybe necessary to resolve the small-scale structures and uctuations using DNS, rather than model the global e ects of turbulence using an e ective eddy viscosity as is done currently using RANS.