Reduced-order modeling of transonic flows around an airfoil submitted to small deformations

A reduced-order model (ROM) is developed for the prediction of unsteady transonic flows past an airfoil submitted to small deformations, at moderate Reynolds number. Considering a suitable state formulation as well as a consistent inner product, the Galerkin projection of the compressible flow Navier–Stokes equations, the high-fidelity (HF) model, onto a low-dimensional basis determined by Proper Orthogonal Decomposition (POD), leads to a polynomial quadratic ODE system relevant to the prediction of main flow features. A fictitious domain deformation technique is yielded by the Hadamard formulation of HF model and validated at HF level. This approach captures airfoil profile deformation by a modification of the boundary conditions whereas the spatial domain remains unchanged. A mixed POD gathering information from snapshot series associated with several airfoil profiles can be defined. The temporal coefficients in POD expansion are shape-dependent while spatial POD modes are not. In the ROM, airfoil deformation is introduced by a steady forcing term. ROM reliability towards airfoil deformation is demonstrated for the prediction of HF-resolved as well as unknown intermediate configurations

Coupling strategies for solving the RANS equations

For the present work two implicit methods of coupling the compressible Reynolds Averaged Navier-Stokes equations in conjunction with the one equation Spalart-Allmaras turbulence model have been developed. The first approach, known as fully coupled technique, strongly couples the two different systems of equations, and accordingly solves for a single system. The second technique has been defined as weakly coupled approach. On the one hand, it also solves for a single set of equations. On the other hand, the full Jacobian is not build by excluding the evaluation of the cross derivatives. The latter approach must be understood in the sense of an intermediate step between the loosely and fully coupled techniques, allowing to evaluate the coupling solution strategy. The subject of this thesis is to examine whether it is advantageous to solve the systems of equations in a mathematically consistent coupled manner or loosely coupled. For the space discretization, an unstructured finite volume scheme based on node-centered dual mesh is used. The solution procedure is based on a nonlinear agglomeration multigrid technique combined with a multistage line implicit Runge-Kutta smoother. The inner system of equations is solved through a Block Symmetric Gauss-Seidel scheme. The assessment of the newly developed methodologies is obtained by a comparative study with a loosely coupled solution strategy along with experimental data. The attention is focused on the accuracy of the results, the number of overall cycles and convergence rates of the solution method. Several numerical computations have been carried out in four two-dimensional and three-dimensional well known benchmark test cases: the CASE 9, MDA30P30N, DPW5CRM and the NASA Trap Wing. The obtained results evidence that no improvement is obtained regarding accuracy but demonstrate superiorities and inferiorities in the convergence rate for the weakly coupled and fully coupled strategies.Outgoin

On the eddy-resolving capability of high-order discontinuous Galerkin approaches to implicit LES / under-resolved DNS of Euler turbulence

AbstractWe present estimates of spectral resolution power for under-resolved turbulent Euler flows obtained with high-order discontinuous Galerkin (DG) methods. The ‘1% rule’ based on linear dispersion–diffusion analysis introduced by Moura et al. (2015) [10] is here adapted for 3D energy spectra and validated through the inviscid Taylor–Green vortex problem. The 1% rule estimates the wavenumber beyond which numerical diffusion induces an artificial dissipation range on measured energy spectra. As the original rule relies on standard upwinding, different Riemann solvers are tested. Very good agreement is found for solvers which treat the different physical waves in a consistent manner. Relatively good agreement is still found for simpler solvers. The latter however displayed spurious features attributed to the inconsistent treatment of different physical waves. It is argued that, in the limit of vanishing viscosity, such features might have a significant impact on robustness and solution quality. The estimates proposed are regarded as useful guidelines for no-model DG-based simulations of free turbulence at very high Reynolds numbers

Study of hybrid air vehicles stability using computational fluid dynamics

This paper uses Computational Fluid Dynamics to predict aerodynamic damping of airships or hybrid air vehicles. This class of aircraft is characterised by large lifting bodies combining buoyancy and circulatory lift. Damping is investigated via forced oscillations of the vehicle in pitch and yaw. The employed method is verified using data for lighter than air vehicles. The use of fins and stabilisers was found to be beneficial. The rear part of the body was dominated by separated flow that containedmore frequencies than the forcing frequency imposed on the body. The final design is seen to be dynamically stable across a range of conditions for small pitch angles

Computational Aerodynamics on unstructed meshes

New 2D and 3D unstructured-grid based flow solvers have been developed for simulating steady compressible flows for aerodynamic applications. The codes employ the full compressible Euler/Navier-Stokes equations. The Spalart-Al Imaras one equation turbulence model is used to model turbulence effects of flows. The spatial discretisation has been obtained using a cell-centred finite volume scheme on unstructured-grids, consisting of triangles in 2D and of tetrahedral and prismatic elements in 3D. The temporal discretisation has been obtained with an explicit multistage Runge-Kutta scheme. An "inflation" mesh generation technique is introduced to effectively reduce the difficulty in generating highly stretched 2D/3D viscous grids in regions near solid surfaces. The explicit flow method is accelerated by the use of a multigrid method with consideration of the high grid aspect ratio in viscous flow simulations. A solution mesh adaptation technique is incorporated to improve the overall accuracy of the 2D inviscid and viscous flow solutions. The 3D flow solvers are parallelised in a MIMD fashion aimed at a PC cluster system to reduce the computing time for aerodynamic applications. The numerical methods are first applied to several 2D inviscid flow cases, including subsonic flow in a bump channel, transonic flow around a NACA0012 airfoil and transonic flow around the RAE 2822 airfoil to validate the numerical algorithms. The rest of the 2D case studies concentrate on viscous flow simulations including laminar/turbulent flow over a flat plate, transonic turbulent flow over the RAE 2822 airfoil, and low speed turbulent flows in a turbine cascade with massive separations. The results are compared to experimental data to assess the accuracy of the method. The over resolved problem with mesh adaptation on viscous flow simulations is addressed with a two phase mesh reconstruction procedure. The solution convergence rate with the aspect ratio adaptive multigrid method and the direct connectivity based multigrid is assessed in several viscous turbulent flow simulations. Several 3D test cases are presented to validate the numerical algorithms for solving Euler/Navier-Stokes equations. Inviscid flow around the M6 wing airfoil is simulated on the tetrahedron based 3D flow solver with an upwind scheme and spatial second order finite volume method. The efficiency of the multigrid for inviscid flow simulations is examined. The efficiency of the parallelised 3D flow solver and the PC cluster system is assessed with simulations of the same case with different partitioning schemes. The present parallelised 3D flow solvers on the PC cluster system show satisfactory parallel computing performance. Turbulent flows over a flat plate are simulated with the tetrahedron based and prismatic based flow solver to validate the viscous term treatment. Next, simulation of turbulent flow over the M6 wing is carried out with the parallelised 3D flow solvers to demonstrate the overall accuracy of the algorithms and the efficiency of the multigrid method. The results show very good agreement with experimental data. A highly stretched and well-formed computational grid near the solid wall and wake regions is generated with the "inflation" method. The aspect ratio adaptive multigrid displayed a good acceleration rate. Finally, low speed flow around the NREL Phase 11 Wind turbine is simulated and the results are compared to the experimental data

An a posteriori-implicit turbulent model with automatic dissipation adjustment for Large Eddy Simulation of compressible flows

In this work we present an a posteriori high-order finite volume scheme for the computation of compressible turbulent flows. An automatic dissipation adjustment (ADA) method is combined with the a posteriori paradigm, in order to obtain an implicit subgrid scale model and preserve the stability of the numerical method. Thus, the numerical scheme is designed to increase the dissipation in the control volumes where the flow is under-resolved, and to decrease the dissipation in those cells where there is excessive dissipation. This is achieved by adding a multiplicative factor to the dissipative part of the numerical flux. In order to keep the stability of the numerical scheme, the a posteriori approach is used. It allows to increase the dissipation quickly in cells near shocks if required, ensuring the stability of the scheme. Some numerical tests are performed to highlight the accuracy and robustness of the proposed numerical scheme

Aeroelastic Analysis of a Wind Turbine Blade Using the Harmonic Balance Method

Most current wind turbine aeroelastic codes rely on the blade element momentum method with empirical corrections to compute aerodynamic forces on the wind turbine blades. While efficient, this method relies on experimental data and does not allow designers much flexibility for alternative blade designs. Unsteady solutions to the Navier-Stokes equations offer a significant improvement in aerodynamic modeling, but these are currently too computationally expensive to be useful in a design situation. However, steady-state solutions to the Navier-Stokes equations are possible with reasonable computation times. The harmonic balance method provides a way to represent unsteady, periodic flows through coupled a set of steady-state solutions. This method offers the possibility of unsteady flow solutions at a computational cost on the order of a few steady-state solutions. By coupling a harmonic balance driven aerodynamic model with a mode shape-based structural dynamics model, an efficient aeroelastic model for a wind turbine blade driven by the Navier-Stokes equations is developed in this dissertation. For wind turbine flows, turbulence modeling is essential, especially in the transition of the boundary layer from laminar to turbulent. As part of this dissertation, the Spalart-Allmaras turbulence model and the gamma-Re\_theta-t transition model are included in the aerodynamic model. This marks the first time that this transition model, turbulence model, and the harmonic balance method have been coupled to study unsteady wind turbine aerodynamics. Results show that the transition model matches experimental data more closely than a fully turbulent model for the onset of both static and dynamic stall. Flutter is of particular interest as turbines continue to increase in size, and longer and softer blades continue to enter the field. In this dissertation, flutter is investigated for the 1.5 MW WindPACT rotor blade. The aeroelastic model created, which incorporates the harmonic balance method and a fully turbulent aerodynamic model, is the first of its kind for wind turbine flutter analysis. Predictions match those of other aeroelastic models for the 1.5 MW WindPACT blade, and the first flapwise and edgewise modes are shown to dominate flutter for the rotor speeds considered