Theoretical and Computational Fluid Dynamics

The aerodynamic and aeroacoustic performance of a low-aspect-ratio ( $$\hbox {AR}=0.2$$ ) pitching foil during dynamic stall are investigated numerically with focus on the effects of trailing edge serrations. A hybrid method coupling an immersed boundary method for incompressible flows with the Ffowcs Williams–Hawkings acoustic analogy is employed. Large eddy simulation and turbulent boundary layer equation wall model are also employed to capture the turbulent effects. A modified NACA0012 foil with a rectangular trailing edge flap attached to the trailing edge (baseline case) undergoing pitching motion is considered. Trailing edge serrations are applied to the trailing edge flap and their effects on the aerodynamic and aeroacoustic performance of the oscillating airfoil are considered by varying the wave amplitude ( $$2h^*= 0.05, 0.1$$ , and 0.2) at a Reynolds number of 100,000 and a Mach number of 0.05. It is found that the reduction of the sound pressure level at the dimensionless frequency band $$St_{b}\in [1.25,4]$$ can be over 4 dB with the presence of the trailing edge serrations ( $$2h^*=0.1$$ ), while the aerodynamic performance and its fluctuations are not significantly altered except a reduction around 10% in the negative moment coefficient and it fluctuations. This is due to the reduction of the average spanwise coherence function and the average surface pressure with respect to that of the baseline case, suggesting the reduction of the spanwise coherence and the noise source may result in the noise reduction. Analysis of the topology of the near wake coherent structure for $$2h^*=0.1$$ reveals that the alignment of the streamwise-oriented vortex with the serration edge may reduce the surface pressure fluctuation.

The Clark model for the turbulent stress tensor in large-eddy simulation is investigated from a theoretical and computational point of view. In order to be applicable to compressible turbulent flows, the Clark model has been reformulated. Actual large-eddy simulation of a weakly compressible, turbulent, temporal mixing layer shows that the eddy-viscosity part of the original Clark model gives rise to an excessive dissipation of energy in the transitional regime. On the other hand, the model gives rise to instabilities if the eddy-viscosity part is omitted and only the “gradient” part is retained. A linear stability analysis of the Burgers equation supplemented with the Clark model is performed in order to clarify the nature of the instability. It is shown that the growth-rate of the instability is infinite in the inviscid limit and that sufficient (eddy-)viscosity can stabilize the model. A model which avoids both the excessive dissipation of the original Clark model as well as the instability of the “gradient” part, is obtained when the dynamic procedure is applied to the Clark model. Large-eddy simulation using this new dynamic Clark model is found to yield satisfactory results when compared with a filtered direct numerical simulation. Compared with the standard dynamic eddy-viscosity model, the dynamic Clark model yields more accurate predictions, whereas compared with the dynamic mixed model the new model provides equal accuracy at a lower computational effort.

We present a computational study evaluating the effectiveness of the nonlinear Galerkin method for dissipative evolution equations. We begin by reviewing the theoretical estimates of the rate of convergence for both the standard spectral Galerkin and the nonlinear Galerkin methods. We show that the rate of convergence in both cases depends mainly on how well the basis functions of the spectral method approximate the elements in the space of solutions. This in turn depends on the degree of smoothness of the basis functions, the smoothness of the solutions, and on the level of compatibility at the boundary between the basis functions of the spectral method and the solutions. When the solutions are very smooth inside the domain and very compatible with the basis functions at the boundary, there may be little advantage in using the nonlinear Galerkin method. On the other hand, for less smooth solutions or when there is less compatibility at the boundary with the basis functions, there is a significant improvement in the rate of convergence when using the nonlinear Galerkin method. We demonstrate the validity of our assertions with numerical simulations of the forced dissipative Burgers equation and of the forced Kuramoto-Sivashinsky equation. These simulations also demonstrate that the analytical upper bounds derived for the rates of convergence of both the standard Galerkin and the nonlinear Galerkin are nearly sharp.

Proper orthogonal decomposition (POD) has been used to develop a reduced-order model of the hydrodynamic forces acting on a circular cylinder. Direct numerical simulations of the incompressible Navier–Stokes equations have been performed using a parallel computational fluid dynamics (CFD) code to simulate the flow past a circular cylinder. Snapshots of the velocity and pressure fields are used to calculate the divergence-free velocity and pressure modes, respectively. We use the dominant of these velocity POD modes (a small number of eigenfunctions or modes) in a Galerkin procedure to project the Navier–Stokes equations onto a low-dimensional space, thereby reducing the distributed-parameter problem into a finite-dimensional nonlinear dynamical system in time. The solution of the reduced dynamical system is a limit cycle corresponding to vortex shedding. We investigate the stability of the limit cycle by using long-time integration and propose to use a shooting technique to home on the system limit cycle. We obtain the pressure-Poisson equation by taking the divergence of the Navier–Stokes equation and then projecting it onto the pressure POD modes. The pressure is then decomposed into lift and drag components and compared with the CFD results.

This paper presents a numerical analysis of the hole-tone phenomenon (Rayleigh’s bird-call), based on a three-dimensional discrete vortex method. Evaluation of the sound generated by the self-sustained flow oscillations is based on the Powell–Howe theory of vortex sound and a boundary integral/element method. While the fundamental problem can be modeled well under the assumption of axial symmetry, the purpose of employing a full three-dimensional model is to investigate the influence of non-axisymmetric perturbations of the jet on the sound generation (with a view to flow control). Experimentally, such perturbations can be applied at the jet nozzle via piezoelectric or electro-mechanical actuators, placed circumferentially inside the nozzle at its exit. In the mathematical/numerical model, this is simulated by wave motions of a deformable nozzle. Both standing and traveling (rotating) waves are considered. It is shown that a considerable reduction of the sound generation is possible.

Large-eddy simulation (LES) results for laminar-to-turbulent transition in a spatially developing boundary layer are presented. The disturbances are ingested into a laminar flow through an unsteady suction-and-blowing strip. The filtered, three-dimensional time-dependent Navier–Stokes equations are integrated numerically using spectral, high-order finite-differences, and a three-stage low-storage Runge–Kutta/Crank–Nicolson time-advancement method. The buffer-domain technique is used for the outflow boundary condition. The localized dynamic model used to parametrize the subgrid-scale (SGS) stresses begins to have a significant impact at the beginning of the nonlinear transition (or intermittency) region. The flow structures commonly found in experiments are also observed in the present simulation; the computed linear instability modes and secondary instability $\Lambda$-vortex structures are in agreement with the experiments, and the streak-like structures and turbulent statistics compare with both the experiments and the theory. The physics captured in the present LES are consistent with the experiments and the full Navier–Stokes simulation (DNS), at a significant fraction of the DNS cost. A comparison of the results obtained with several SGS models shows that the localized model gives accurate results both in a statistical sense and in terms of predicting the dynamics of the energy-carrying eddies, while requiring fewer ad hoc adjustments than the other models.

The dynamic subgrid-scale model is used in finite-difference computations of turbulent flow in a plane channel, for a range of Reynolds numbers (based on friction velocity and channel half-width) between 200 and 5000. Adoption of approximate wall boundary conditions allows the use of very coarse grids in all directions. The comparison of first- and second-order moments with the reference data is satisfactory, despite the mesh coarseness. Turbulent kinetic energy budgets also compare well with DNS data. Near the wall, the dynamic formulation gives improved results over the Smagorinsky model, as observed in previous simulation. In the core of the flow where, at high Reynolds number, the turbulent eddies obey inertial-range dynamics, the Smagorinsky and dynamic models give similar results. The behavior of the model, its implementation when approximate wall boundary conditions are used, and the effect of numerical resolution are discussed.