Canadian Aeronautics and Space Institute

Spatial and temporal eigensolution analysis of the discontinuous Galerkin (DG) method shows the favorable dispersion and diffusion properties of such methods when high order polynomial approximations are used, making them very suitable for application to under-resolved turbulent flows. The recent introduction of split forms of the inviscid fluxes for the discontinuous Galerkin method improves the robustness of the scheme, such that stable simulations are obtained even for very high polynomial orders and severely under-resolved cases. The present work investigates the robustness and accuracy of very high order split form DG schemes applied to turbulent channel flows for severely and moderately under-resolved configurations. In addition to using two distinct approximate Riemann solvers for the interface fluxes, a central formulation is also employed and analyzed. Improved results are obtained when a subgrid scale model is added to the equations in order to increase the dissipation at the smallest resolved scales. The remarkable stability observed for the scheme even for the dissipation-free, central formulation enables the development of tailored subgrid scale turbulence models that improve the results of practical, under-resolved engineering simulations.

The work presents comparative results between wall friction and pressure coefficient predictions from Large Eddy Simulation (LES) and Reynolds-averaged Navier-Stokes (RANS) computations in channel flows with an adverse pressure gradient created by a 2-D bump. LES results are obtained using the semi-spectral code, MFLOPS3D, developed at Laboratoire de Mecanique de Lille, while RANS solutions are computed using a commercial code based on 2nd-order finite volume discretization with typical two-equation turbulence models and one seven-equation Reynolds stress transport model. Reynolds numbers range from Reτ = 617 to 2000, based on the friction velocity at the channel inlet. It is found that LES produces very consistent results when compared to DNS or other highly resolved LES. Two-equation and seven-equation RANS models are found to correctly predict friction coefficients in zero and favorable pressure gradient regions, but they fail in regions of adverse pressure gradients. The Reynolds stress transport model shows a better performance when compared to two-equation eddy-viscosity models, particularly in the higher Reynolds number cases. However, none of the RANS models is able to capture the full physics of turbulence related with adverse pressure gradients. The limitations of RANS models are observed both in separated and attached flow conditions at the rear portion of the bump. Such limitations seem to be related to the fact that these models do not correctly describe the evolution of the turbulent kinetic energy close to the walls in adverse pressure gradient regions.

Aeroacoustic predictions of a model airframe noise problem are conducted in order to assess the effects of wake and boundary layer interaction on both the flow and acoustic fields. Simulations of unsteady laminar flows, including both noise generation and its subsequent propagation to the far field, are performed for a two-dimensional configuration composed of a cylinder placed above a NACA 0012 airfoil at 5 deg. angle of incidence. The Reynolds number based on the airfoil chord is set at Rec = 5000 and the Reynolds number based on the diameter of the cylinder is Red = 200. An assessment of cylinder position and freestream Mach number effects on sound radiation is presented for M∞ = 0.1, 0.3 and 0.5. The investigation of the noise sources for airfoil and cylinder vortex shedding frequencies is also presented using a hybrid methodology which employs direct calculation for near field source computations and the Ffowcs Williams-Hawkings (FW-H) equation as the acoustic analogy formulation. In order to verify the numerical solutions, acoustic prediction results obtained by direct noise calculation (DNC) are compared to those computed by the FW-H equation and good agreement is observed. The separate contributions of the terms composing the quadrupole sources are analyzed. Even for the low Reynolds number flows investigated, Reynolds stresses are the dominant quadrupole noise sources and viscous effects as well as entropy fluctuations can be neglected. It is also found that while quadrupole sources can be neglected in noise predictions by the FW-H equation for the M∞ = 0.1 flow calculations, they should be included for the M∞ = 0.3 and 0.5 acoustic predictions at low and high frequencies. For moderate Mach number flows, at the cylinder vortex shedding frequency, dipole sources are related to both acoustic scattering and diffraction phenomena that occur along the airfoil surface and trailing edge, respectively. However, quadrupole sources are more specifically related to diffraction effects along the airfoil trailing edge. When the cylinder vortex shedding is the driving noise source mechanism, intense dipolar interference occurs for all configurations analyzed.