Kui Ou

Further Studies of Airfoils Supporting Non-Unique Solutions in Transonic Flow

Non-unique solutions of the Euler equations were originally discussed by Jameson in 1991 for several highly cambered airfoils which were the result of aggressive shape optimization. In 1999 Hafez and Guo found non-unique solutions for a symmetric parallel sided airfoil, and subsequently Kuzmin and Ivanova have discovered some fully convex symmetric airfoils that provide non-unique solutions. In this article four new symmetric airfoils, all of which exhibit non-unique solutions in a narrow band of transonic Mach numbers, were studied. The first, NU4 was the result of shape optimization. The second, JF1 is an extremely simple parallel sided airfoil. The third JB1, is also parallel sided but has continuous curvature over the entire profile. The fourth, JC6, is convex and C∞ continuous. CL − α plots of these airfoils exhibit three branches of zero angle of attack, the P, Z and N-branches with positive, zero and negative lift respectively. At some Mach numbers no stable Z-branch could be found. When the P-branch is continued to negative α in some cases there is a transition to the Z-branch, while in other cases there is a direct transition from the P to N-branch.

3D Flapping Wing Simulation with High Order Spectral Difference Method on Deformable Mesh

In this paper we carry out computational studies of three-dimensional flow over flapping wings. The problems we have investigated include, firstly, three-dimensonal simulation of flow over an extruded SD7003 airfoil in plunging motion at transitional Reynolds number and, secondly, flow over a pair of flapping rectangle wings with constant NACA0012 cross-sectional airfoil profile at low Reynolds number. The three-dimensional flapping wing simulations are performed using high-order spectral difference method at low Mach number. The high-order method allows a very coarse starting mesh to be used. By using high order solution, fine flow features in the vortex-dominated flow field are effectively captured. For the plunging SD7003 airfoil, we examine the laminar to turbulence flow transitional behavior at Re 40,000. For the NACA0012 rectangular wing, we analyze and compare the flow fields and aerodynamic efficiencies of several flapping wing motions at Re 2000. The flapping motions considered include wing plunging, twisting and pitching. Some of the flapping wing motions are accommodated through dynamic mesh deformation.

High-Order Spectral Difference Method for the Navier-Stokes Equation on Unstructured Moving Deformable Grid

In this paper the high-order accurate spectral difference method for the Navier-Stokes equations is applied to moving boundary problems. Boundary movements are achieved, firstly, through rigid displacement of the entire flow domain. In order to account for the dynamic rigid mesh motion, the Navier-Stokes equations are modified through an unsteady coordinate transformation. Airfoils in pitching and plunging motions are studied. In both cases, computation results are compared with existing experimental data, and favorable results have been obtained. Secondly, spectral difference method is extended to include capability for handling dynamic deforming grids. The physical boundary movement is achieved through a time dependent unsteady transformation that allows part of the flow domain to be rigidly displacing, part of it fixed, and the rest deforming smoothly in between. The time dependent transformation preserves spectral difference method’s high order accuracy by solving the governing equations in a steady reference domain where the same shape functions are used, and introducing the unsteady perturbation in the physical space only through the changes in the transformation metrics and Jacobian. The blended deforming mesh allows the far field boundary or some desirable portions of the flow domain to be unaltered. These together make the overall solver accurate, flexible, and simple to implement. The order of accuracy of the spectral difference method in highly distorted mesh has been demonstrated through simulation of euler vortex problem. Simulations for flow over a plunging cylinder with rigid displacing and dynamic deforming meshes have yielded nearly identical results.

High-Order Spectral Difference Simulation of Laminar Compressible Flow Over Two Counter-Rotating Cylinders

Flow past a single rotating cylinder has been studied, both numerically and experimentally, by many authors. In contrast is the flow past rotating cylinders in a side-by-side arrangement, which has only very limited numerical and experimental data available, despite the pioneering theoretical study by Jeffery 1 as early as 1922. In this paper, we discuss a numerical investigation of the steady laminar viscous flow past two infinite rotating cylinders in a side-by-side configuration. The solution of the compressible two-dimensional Navier-Stokes equations is determined numerically using the high-order spectral difference scheme over an unstructured qudralateral grid. Third order accurate results in both time and space were obtained and compared with existing data. In addition to obtaining a highorder accurate result for flow past two rotating cylinders, we extend the current numerical effort to investigate the effects of Reynolds number, compressibility, and high rotation speed, which have not been comprehensively studied in the past.

Airfoils Supporting Non-unique Transonic Solutions for Unsteady Viscous Flows

Non-unique numerical solutions of transonic flows have been found, first for potential flow equation, and later for Euler and RANS equations. The studies conducted so far have been mostly limited to steady state flow simulations. It is believed that unsteady simulations are needed to gain a better understanding of the evolution and stability of these flows. This paper re-examined a set of four recently designed symmetric airfoils that have been found to support non-unique solutions in steady transonic flows in a narrow band of transonic Mach numbers. Unsteady RANS solutions have been performed for these four airfoils in the transonic regime. Results indicate that all four airfoils exhibit unsteady non-unique transonic solutions. The unsteady non-unique solutions occur over a wider band of transonic Mach numbers than the steady non-unique solutions.

Unsteady Adjoint Method for the Optimal Control of Advection and Burger's Equations using High-Order Spectral Difference Method

In this study, we investigate the method for adjoint-based optimal control of linear and non-linear equations. In particular, we are interested in the formulation of the unsteady adjoint method based on the high-order spectral difference method. For the linear equation, we consider the simple 1D advection equation. For the non-linear equation, we consider the 1D viscous Burger’s equation. In both cases, the inverse design of the target solution is achieved through control of the initial condition. The focus of the study is the formulation of the unsteady adjoint method using the high-order spectral difference method. The paper demonstrates that the minimal numerical dissipation inherent in the high order method is very beneficial for adjoint type problem where backward integration in time is involved. The combination of adjoint approach and high order method will lead to an useful tool for optimization in the field of aeroacoustic simulation and design. I. Introduction While much work has been done in the field of steady adjoint approach for optimal control and inverse design problems using traditional CFD methods 1–4 such as finite volume methods, the current trend and demand for more time-accurate and spatial-accurate methods entail further development of the adjoint-based approach. Recent research has seen the combination of the unsteady adjoint method with the high order methods such as the Discontinous Galerkin (DG) method. 5–7 Method of this type can be a significant aid to the field of aeroacoustic design and optimization, since the capture of acoustic signiture often requires the time-accuracy and spatial-accuracy of the high order methods which have very small numerical dissipations. In this study, we investigate the formulation of unsteady adjoint approach based on high-order spectral difference (SD) method, and explore methods that can lead to better computational and optimization efficiency. We consider the problems of the optimal control of the advection and Burger’s equations, by treating the initial conditions as the control inputs, and matching the final solutions with design target profiles.

Flow Induced Cylinder Oscillation and Its Control With High Order Spectral Difference Method on Deformable Mesh

In the recent work by the authors, high order spectral difference (SD) method has been formulated in a framework with dynamic deformable meshes, and been demonstrated to preserve the design accuracy of the underlying high order temporal and spatial discretization methods. In this study, the current SD method has been extended to solve fluid structure interaction problems on deforming meshes. The SD method with collocated solution and flux points is used as the spatial discretization method, while the explicit fourth order five-stage Runge-Kutta is used to advance the flow in time. The solid structure, which is the cylinder, is modelled as a spring-mass system. The movement of the solid cylinder is handled with a dynamic deforming mesh using an algebraic high order blending polynomial. The present work has focused on flow induced cylinder oscillation, bluff body wake and cylinder interaction, and control of flow induced oscillation through mechanism of counter-rotating cylinders. Numerical experiment for flow over counter-rotating cylinder pair shows that significant drag reduction and wake suppression can be obtained. Numerical calculation for a free-stream flow over a free-floating cylinder shows that the wake symmetry of a cylinder can not be maintained. The resultant wake instability propels the cylinder in the cross-flow direction, but the cylinder was found to eventually locked in a flow induced self-oscillation at an equilibrium position. Active control of the cylinder vibration through the mechanism of counter-rotation of solid bodies is finally investigated.Copyright

Studies of Wings Supporting Non-unique Solutions in Transonic Flows

Non-unique numerical solutions of transonic flows over airfoils have been found, first for potential flow equation, and later for Euler and RANS equations. The studies have also been extended to unsteady flow simulations, and non-unique numerical solutions continue to be demonstrated. The question of whether three-dimensional effect can have a further influence on the uniqueness of the transonic flow solution remain an important one. Hitherto wings supporting non-unique solutions in transonic solutions have not been studied. Research in this direction will further our understanding of the behavior of non-unique transonic flows. This paper studied a set of four wings based on recently designed symmetrical airfoils that have been found to support non-unique transonic solutions in both steady and unsteady flows in a narrow band of transonic Mach numbers. The aspect ratios of the wings have been varied as a way to control the extend of the three-dimensional effect. For certain of these airfoils, the non-unique solutions cease to exist when extended to a full wing, while for others, non-unique solutions continue to exist depending on the choice of the aspect ratios of the wings. The flow conditions that support non-unique solutions also tend to change when the airfoils are extended to wings of different aspect ratios. The scope of the study is, at present, limited to Euler solutions.

Dynamic Mesh Deformation for Adaptive Grid Resolution Enhancement with Staggered Spectral Difference and Finite Volume Mesh

Better computational efficiency and reduced computational cost for high order CFD method are areas of active research. In this study, we investigate the potential of using dynamic deforming mesh to locally refine a region of interest in the flow domain. With given number of nodes or degree of freedoms, this has the advantage of achieving an optimal distribution of mesh points, concentrating the computational resource on areas of high gradients, and leaving areas of low gradient with less points. This can be very useful for high order finite element methods that usually have difficulties capturing clean shocks due to the large size of the mesh element, despite the large number of solution and flux poitns within the element. With deforming mesh, the mesh elements can be deformed and gathered near solution with steep gradient, hence leading to high resolution solution of discontinuity. In low gradient regions, despite the diminishing cells, the high order methods are very efficient at resolving the flow with small number of mesh cells.

Optimization of Flow Past a Moving Deformable Airfoil using Spectral Difference Method

In this paper we examine the low Reynolds number flow past a harmonically plunging, pitching, and deforming airfoil, both individually and in combination. The lift and thrust characteristics of the combined flapping airfoil motion of plunging, pitching and deforming are investigated using a high-fidelity flow solver. The high-order Navier-Stokes solver based on Spectral Difference (SD) method is used to analyze the moving airfoils. An algebraic mesh deformation algorithm is used to accommodate airfoil deformation and movement. The thrust characteristic of a harmonically oscillating and deforming airfoil is investigated. For a deformation function defined by two parameters, i.e. maximum camber Ac and the location of the maximum camber Xc, a particular deformation profile is identified that produces significant thrust.