Preface to the special issue 'High order methods for CFD problems'

Since a few years, there has been a renew in interest in (very) high order schemes for compressible fluid dynamics for steady and unsteady problems. Within Europe and the US, several conferences deal with these issues, some of them have been launched only recently. Several special issues of recent or future AIAA conferences are specially devoted to that topic. The goal of these researches is to design cheaper and more efficient numerical methods able to handle very large and very complex problems

Development of a Robust and Efficient Parallel Solver for Unsteady Turbomachinery Flows

The traditional design and analysis practice for advanced propulsion systems relies heavily on expensive full-scale prototype development and testing. Over the past decade, use of high-fidelity analysis and design tools such as CFD early in the product development cycle has been identified as one way to alleviate testing costs and to develop these devices better, faster and cheaper. In the design of advanced propulsion systems, CFD plays a major role in defining the required performance over the entire flight regime, as well as in testing the sensitivity of the design to the different modes of operation. Increased emphasis is being placed on developing and applying CFD models to simulate the flow field environments and performance of advanced propulsion systems. This necessitates the development of next generation computational tools which can be used effectively and reliably in a design environment. The turbomachinery simulation capability presented here is being developed in a computational tool called Loci-STREAM [1]. It integrates proven numerical methods for generalized grids and state-of-the-art physical models in a novel rule-based programming framework called Loci [2] which allows: (a) seamless integration of multidisciplinary physics in a unified manner, and (b) automatic handling of massively parallel computing. The objective is to be able to routinely simulate problems involving complex geometries requiring large unstructured grids and complex multidisciplinary physics. An immediate application of interest is simulation of unsteady flows in rocket turbopumps, particularly in cryogenic liquid rocket engines. The key components of the overall methodology presented in this paper are the following: (a) high fidelity unsteady simulation capability based on Detached Eddy Simulation (DES) in conjunction with second-order temporal discretization, (b) compliance with Geometric Conservation Law (GCL) in order to maintain conservative property on moving meshes for second-order time-stepping scheme, (c) a novel cloud-of-points interpolation method (based on a fast parallel kd-tree search algorithm) for interfaces between turbomachinery components in relative motion which is demonstrated to be highly scalable, and (d) demonstrated accuracy and parallel scalability on large grids (approx 250 million cells) in full turbomachinery geometries

High-Order Stabilized Finite Elements on Dynamic Meshes

The development of dynamic mesh capability for turbulent flow simulations using the Streamlined Upwind Petrov-Galerkin (SUPG) discretization is described. The current work extends previous research to include high-order spatial accuracy, including the satisfaction of the discrete geometric conservation law (GCL) on curved elements. Two closely-related schemes are described and the ability of these schemes to satisfy the GCL, while also maintaining temporal accuracy and conservation is assessed. Studies indicate that although one scheme discretizes the time derivative in conservative form, both schemes exhibit temporal conservation errors that decrease according to the expected design order of accuracy. The source of the temporal conservation errors is examined, and it is demonstrated that many finite-volume and finite-element schemes can also be expected to have difficulty strictly satisfying conservation in time. The effects on conservation are examined and, while present in the simulations, are seen to be negligible for the problems considered

Numerical Simulations of Hypersonic Aerothermoelastic Phenomena

This dissertation examines the following subjects important to hypersonic aerothermoelastic flows: aerodynamic heating on a double-wedge airfoil, active cooling on a double-wedge airfoil, and aerothermoelastic panel utter with deformable supports. To facilitate this examination, an aerothermoelastic solver was created by coupling solvers for the structural elasticity and thermal diffusion equations to a Reynolds averaged Navier-Stokes (RANS) solver. Validation studies were performed on the aerodynamic, aerothermal, and aerothermoelastic configurations of the solver, which were then used to investigate the aforementioned subjects. The aerodynamic solver was validated for hypersonic heating simulations by comparison to the Fay-Riddell equation for the peak heat flux on a circular cylinder and by comparison to compressible boundary layer ow for heat flux into a at plate. A novel investigation was then performed of the heat flux on a double-wedge airfoil, looking at variations of heat flux due to angle of attack, wall temperature, Mach number, and altitude. The aerothermal solver was validated using experimental and computational data from the hypersonic heating of a spherical protuberance on a flat plate. The solver was then used in a novel study to analyze the effect of active cooling on the steady-state skin temperature of a double-wedge airfoil in hypersonic flow. Active cooling using a piecewise continuous cooling distribution resulted in sufficient temperature reduction, but also results in significant chordwise temperature gradients. The aerothermoelastic solver was validated using computational data from the analysis of panel utter. The solver was then used to examine the effects of deformable structural supports on the utter dynamics of the panel for the first time. Deformable supports along existing simply-supported boundaries were shown to be ineffective at improving resistance to utter or buckling. Deformable supports within a panel were shown to increase the utter resistance threefold and buckling resistance almost fivefold

A stabilized finite element dynamic overset method for the Navier-Stokes equations

In terms of mesh resolution requirements, higher-order finite element discretization methods offer a more economic means of obtaining accurate simulations and/or to resolve physics at scales not possible with lower-order schemes. For simulations that may have large relative motion between multiple bodies, overset grid methods have demonstrated distinct advantages over mesh movement strategies. Combining these approaches offers the ability to accurately resolve the flow phenomena and interaction that may occur during unsteady moving boundary simulations. Additionally, overset grid techniques when utilized within a finite element setting mitigate many of the difficulties encountered in finite volume implementations. This research presents the development of an overset grid methodology for use within a streamline/upwind Petrov-Galerkin formulation for unsteady, viscous, moving boundary simulations. A novel hole cutting procedure based on solutions to Poisson equation is introduced and compared to existing techniques. A MPI-based parallel three-dimensional overset grid assembly framework is developed. Order of accuracy is examined via the method of manufactured solutions. The potential benefits of using Adaptive Mesh Refinement (AMR) in overset grid simulations are explored by combining the overset method with an AMR approach. The importance of considering linearization due to the overset boundaries within the preconditioning is studied. Numerical experiments are performed comparing an ILU(k) preconditioner with two proposed modifications referred to as “triangular inter-grid ILU(k)” and “Jacobi inter-grid ILU(k)”. The efficiency gains observed from the proposed modifications are also applicable to general parallel simulations on distributed memory machines, regardless of whether an overset grid approach is used. Overset grid results are presented for several inviscid and viscous, steady-state and time-dependent moving boundary simulations with linear, quadratic, and cubic elements

High-order arbitrary Lagrangian-Eulerian discontinuous Galerkin methods for the incompressible Navier-Stokes equations

This paper presents robust discontinuous Galerkin methods for the incompressible Navier-Stokes equations on moving meshes. High-order accurate arbitrary Lagrangian-Eulerian formulations are proposed in a unified framework for both monolithic as well as projection or splitting-type Navier-Stokes solvers. The framework is flexible, allows implicit and explicit formulations of the convective term, and adaptive time-stepping. The Navier-Stokes equations with ALE transport term are solved on the deformed geometry storing one instance of the mesh that is updated from one time step to the next. Discretization in space is applied to the time discrete equations so that all weak forms and mass matrices are evaluated at the end of the current time step. This design ensures that the proposed formulations fulfill the geometric conservation law automatically, as is shown theoretically and demonstrated numerically by the example of the free-stream preservation test. We discuss the peculiarities related to the imposition of boundary conditions in intermediate steps of projection-type methods and the ingredients needed to preserve high-order accuracy. We show numerically that the formulations proposed in this work maintain the formal order of accuracy of the Navier-Stokes solvers. Moreover, we demonstrate robustness and accuracy for under-resolved turbulent flows

Numerical and experimental investigations on unsteady aerodynamics of flapping wings

The development of a dynamic unstructured grid high-order accurate spectral difference (SD) method for the three dimensional compressible Navier-Stokes (N-S) equations and its applications in flapping-wing aerodynamics are carried out in this work. Grid deformation is achieved via an algebraic blending strategy to save computational cost. The Geometric Conservation Law (GCL) is imposed to ensure that grid deformation will not contaminate the flow physics. A low Mach number preconditioning procedure is conducted in the developed solver to handle the bio-inspired flow. The capability of the low Mach number preconditioned SD solver is demonstrated by a series of two dimensional (2D) and three dimensional (3D) simulations of the unsteady vortex dominated flow. Several topics in the flapping wing aerodynamics are numerically and experimentally investigated in this work. These topics cover some of the cutting-edge issues in flapping wing aerodynamics, including the wake structure analysis, airfoil thickness and kinematics effects on the aerodynamic performances, vortex structure analysis around 3D flapping wings and the kinematics optimization. Wake structures behind a sinusoidally pitching NACA0012 airfoil are studied with both experimental and numerical approaches. The experiments are carried out with Particle Image Velocimetry (PIV) and two types of wake transition processes, namely the transition from a drag-indicative wake to a thrust-indicative wake and that from the symmetric wake to the asymmetric wake are distinguished. The numerical results from the developed SD solver agree well with the experimental results. It is numerically found that the deflective direction of the asymmetric wake is determined by the initial conditions, e.g. initial phase angle. As most insects use thin wings (i. e., wing thickness is only a few percent of the chord length) in flapping flight, the effects of airfoil thickness on thrust generation are numerically investigated by simulating the flow fields around a series of plunging NACA symmetric airfoils with thickness ratio ranging from 4.0% to 20.0% of the airfoil chord length. The contribution of viscous force to flapping propulsion is accessed and it is found that viscous force becomes thrust producing, instead of drag producing, and plays a non-negligible role in thrust generation for thin airfoils. This is closely related to the variations of the dynamics of the unsteady vortex structures around the plunging airfoils. As nature flyers use complex wing kinematics in flapping flight, kinematics effects on the aerodynamic performance with different airfoil thicknesses are numerically studied by using a series of NACA symmetric airfoils. It is found that the combined plunging and pitching motion can outperform the pure plunging or pitching motion by sophisticatedly adjusting the airfoil gestures during the oscillation stroke. The thin airfoil better manipulates leading edge vortices (LEVs) than the thick airfoil (NACA0030) does in studied cases, and there exists an optimal thickness for large thrust generation with reasonable propulsive efficiency. With the present kinematics and dynamic parameters, relatively low reduced frequency is conducive for thrust production and propulsive efficiency for all tested airfoil thicknesses. In order to obtain the optimal kinematics parameters of flapping flight, a kinematics optimization is then performed. A gradient-based optimization algorithm is coupled with a second-order SD Navier-Stokes solver to search for the optimal kinematics of a certain airfoil undergoing a combined plunging and pitching motion. Then a high-order SD scheme is used to verify the optimization results and reveal the detailed vortex structures associated with the optimal kinematics of the flapping flight. It is found that for the case with maximum propulsive efficiency, there exists no leading edge separation during most of the oscillation cycle. In order to provide constructive suggestions to the design of micro-air-vehicles (MAVs), 3D simulations of the flapping wings are carried out in this work. Both the rectangular and bio-inspired wings with different kinematics are investigated. The formation process of two-jet-like wake patterns behind the finite-span flapping wing is found to be closely related to the interaction between trailing edge vortices and tip vortices. Then the effects of the wing planforms on the aerodynamics performance of the finite-span flapping wings are elucidated in terms of the evolution and dynamic interaction of unsteady vortex structures