P. De Palma

An immersed boundary method for compressible flows using local grid refinement

This paper combines a state-of-the-art method for solving the three-dimensional preconditioned Navier-Stokes equations for compressible flows with an immersed boundary approach, to provide a Cartesian-grid method for computing complex flows over a wide range of the Mach number. Moreover, a flexible local grid refinement technique is employed to achieve high resolution near the immersed body and in other high-flow-gradient regions at a fraction of the cost required by a uniformly fine grid. The method is validated versus well documented steady and unsteady test problems, for a wide range of both Reynolds and Mach numbers. Finally, and most importantly, for the case of the laminar compressible steady flow past an NACA-0012 airfoil, a thorough mesh-refinement study shows that the method is second-order accurate.

Edge states in a boundary layer

The understanding of laminar-turbulent transition in shear flows has recently progressed along new paradigms based on the central role of nonlinear exact coherent states. We follow such paradigms to identify, for the first time in a spatially developing flow, localized flow structures living on the edge of chaos, which are the precursors of turbulence. These coherent structures are constituted by hairpin vortices and streamwise streaks. The results reported here extend the dynamical systems description of transition to spatially developing flows.

A numerical method for turbomachinery aeroelasticity

This work provides an accurate and efficient numerical method for turbomachinery flutter. The unsteady Euler or Reynolds-averaged Navier-Stokes equations are solved in integral form, the blade passages being discretised using a background fixed C-grid and a body-fitted C-grid moving with the blade. In the overlapping region data are exchanged between the two grids at every time step, using bilinear interpolation. The method employs Roes second-order-accurate flux difference splitting scheme for the inviscid fluxes, a standard second-order discretisation of the viscous terms, and a three-level backward difference formula for the time derivatives. The dual-time-stepping technique is used to evaluate the nonlinear residual at each time step. The state-of-the-art second-order accuracy of unsteady transonic flow solvers is thus carried over to flutter computations. The code is proven to be accurate and efficient by computing the 4th Aeroelastic Standard Configuration, namely, the subsonic flow through a turbine cascade with flutter instability in the first bending mode, where viscous effect are found practically negligible. Then, for the very severe 11th Aeroelastic Standard Configuration, namely, transonic flow through a turbine cascade at off-design conditions, benchmark solutions are provided for various values of the inter-blade phase angle.

Third-order-accurate fluctuation splitting schemes for unsteady hyperbolic problems

This paper provides a two-dimensional fluctuation splitting scheme for unsteady hyperbolic problems which achieves third-order accuracy in both space and time. For a scalar conservation law, the sufficient conditions for a stable fluctuation splitting scheme to achieve a prescribed order of accuracy in both space and time are derived. Then, using a quadratic space approximation of the solution over each triangular element, based on the reconstruction of the gradient at the three vertices, and a four-level backward discretization of the time derivative, an implicit third-order-accurate scheme is designed. Such a scheme is extended to the Euler system and is validated versus well-known scalar-advection problems and inviscid discontinuous flows.

The effects of non-normality and nonlinearity of the Navier–Stokes operator on the dynamics of a large laminar separation bubble

The effects of non-normality and nonlinearity of the two-dimensional Navier–Stokes differential operator on the dynamics of a large laminar separation bubble over a flat plate have been studied in both subcritical and slightly supercritical conditions. The global eigenvalue analysis and direct numerical simulations have been employed in order to investigate the linear and nonlinear stability of the flow. The steady-state solutions of the Navier–Stokes equations at supercritical and slightly subcritical Reynolds numbers have been computed by means of a continuation procedure. Topological flow changes on the base flow have been found to occur close to transition, supporting the hypothesis of some authors that unsteadiness of separated flows could be due to structural changes within the bubble. The global eigenvalue analysis and numerical simulations initialized with small amplitude perturbations have shown that the non-normality of convective modes allows the bubble to act as a strong amplifier of small dis...

A Numerical Method for Turbomachinery Aeroelasticity

This work provides an accurate and efficient numerical method for turbomachinery flutter. The unsteady Euler or Reynolds-averaged Navier–Stokes (RANS) equations are solved in integral form, the blade passages being discretised using a background fixed C-grid and a body-fitted C-grid moving with the blade. In the overlapping region data are exchanged between the two grids at every time step, using bilinear interpolation. The method employs Roe’s second-order-accurate flux difference splitting scheme for the inviscid fluxes, a standard second-order discretisation of the viscous terms, and a three-level backward difference formula for the time derivatives. The state-of-the-art second-order accuracy of numerical methods for unsteady compressible flows with shocks is thus carried over, for the first time to the authors knowledge, to flutter computations. The dual time stepping technique is used to evaluate the nonlinear residual at each time step, thus extending to turbomachinery aeroelasticity the state-of-the-art efficiency of unsteady RANS solvers. The code is proven to be accurate and efficient by computing the 4th Aeroelastic Standard Configuration, namely, the subsonic flow through a turbine cascade with flutter instability in the first bending mode, where viscous effect are found practically negligible. Then, the very severe 11th Aeroelastic Standard Configuration is computed, namely, the transonic flow through a turbine cascade at off-design conditions, where the turbulence model is found to be the critical feature of the method.Copyright

Residual distribution schemes for advection and advection-diffusion problems on quadrilateral cells

This paper provides a study of some difficulties arising when extending residual distribution schemes for scalar advection and advection-diffusion problems from triangular grids to quadrilateral ones. The Fourier and truncation error analyses on a structured mesh are employed and a generalized modified wavenumber is defined, which provides a general framework for the multidimensional analysis and comparison of different schemes. It is shown that, for the advection equation, linearity preserving schemes for quadrilaterals provide lower dissipation with respect to their triangle-based counterparts and very low or no damping for high frequency Fourier modes on structured grids; therefore, they require an additional artificial dissipation term for damping marginally stable modes in order to be employed with success for pure advection problems. In the case of advection-diffusion problems, a hybrid approach using an upwind residual distribution scheme for the convective fluctuation and any other scheme for the diffusion term is only first-order accurate. On the other hand, distributing the entire residual by an upwind scheme provides second-order accuracy; however, such an approach is unstable for diffusion dominated problems, since residual distribution schemes are characterized by undamped modes associated with the discretization of the diffusive fluctuation. The present analysis allows one to determine the conditions for a stable hybrid approach to be second-order accurate and to design an optimal scheme having minimum dispersion error on a nine-point stencil. Well-documented testcases for advection and advection-diffusion problems are used to compare the accuracy properties of several schemes.

An Implicit Fluctuation Splitting Scheme for Turbomachinery Flows

This paper describes an accurate, robust and efficient methodology for solving two-dimensional steady transonic turbomachinery flows. The Euler fluxes are discretized in space using a hybrid multidimensional upwind method, which, according to the local flow conditions, uses the most suitable fluctuation splitting (FS) scheme at each cell of the computational domain. The viscous terms are discretized using a standard Galerkin finite element scheme. The eddy viscosity is evaluated by means of the Spalart-Allmaras turbulence transport equation, which is discretized in space by means of a mixed FS-Galerkin approach. The equations are discretized in time using an implicit Euler scheme, the Jacobian being evaluated by two-point backward differences. The resulting large sparse linear systems are solved sequentially using a preconditioned GMRES strategy. The proposed methodology is employed to compute subsonic and transonic turbulent flows inside a high-turning turbine-rotor cascade.

A purely nonlinear route to transition approaching the edge of chaos in a boundary layer

The understanding of transition in shear flows has recently progressed along new paradigms based on the central role of coherent flow structures and their nonlinear interactions. We follow such paradigms to identify, by means of a nonlinear optimization of the energy growth at short time, the initial perturbation which most easily induces transition in a boundary layer. Moreover, a bisection procedure has been used to identify localized flow structureslivingontheedgeofchaos,foundtobepopulatedbyhairpinvortices and streaks. Such an edge structure appears to act as a relative attractor for the trajectory of the laminar base state perturbed by the initial finite-amplitude disturbances, mediating the route to turbulence of theflow, via the triggering of a regeneration cycle of 3 and hairpin structures at different space and time scales. Thesefindings introduce a new, purely nonlinear scenario of transition in a boundary-layerflow. (Somefigures may appear in colour only in the online journal)

A multi-dimensional solution adaptive multigrid solver for the Euler equations

The encouraging results, obtained using the simple wave decomposition approach in conjunction with the fluctuation splitting space discretization, stimulate further work to improve such a methodology. Some difficulties, as convergence stag nation, still need to be investigated. The explicit multigrid acceleration has proven effective in all cases considered so far.