1/36 Gridless Method for Solving Moving Boundary Problems Wang Hong Department of Mathematical Information Technology University of Jyväskyklä

2/36 Content Introduction Principle of Gridless Method Steady Simulation of Euler Equations and Applications Unsteady Simulation of Euler Equations: Validation on Referenced Airfoils Conclusion and Future Research

3/36 At present, CFD (Computational Fluid Dynamics) community has many methods in solving moving boundary problems, for instance, dynamic mesh method and fictitious domain method and so on. The dynamic mesh method contains different techniques, for example, mesh reconstruction methods and mesh deformation methods. Introduction

4/36 Mesh reconstruction methods have been proposed based on the regeneration of mesh according to the moving boundaries. For example, when we use unstructured Cartesian mesh to solve moving boundary problems, we should fasten the mesh; the moving boundaries cut the mesh elements. These kinds of methods need to consider the cutting elements in every time step, and refine and coarsen the grids to satisfy the distribution requirements. Introduction

5/36 Introduction Mesh deformation methods are basically relied on the control models. Spring approximation model was firstly introduced by Batina to solve the vibrating airfoil flows. The basic idea is that treat every edge of mesh elements as a spring, the coefficients of the spring are related to the length of every edge of mesh elements; after the movement of the boundaries, the new positions of the mesh points are defined by solving the force equilibrium of the spring system.

6/36 A fast dynamic cloud method based on Delaunay graph mapping strategy is proposed in this presentation. A dynamic cloud method makes use of algebraic mapping principles and therefore points can be accurately redistributed in the flow field without any iteration. In this way, the structure of the gridless clouds is not necessary changed so that the clouds regeneration can be avoided successfully. Introduction

7/36 Gridless node definition Spatial discretisation Principle of Gridless Method

8/36 Global and close-up views of typical structure for gridless clouds Gridless Node Definition

9/36 If we keep the first order of, then we have the approximation Define the total error in the cloud of points as in order to minimize the total error, let Ax ＝ b Spatial Discretisation – Least Square Method

10/36 Steady Simulation of Euler Equations and Applications Governing Equations Boundary Conditions Spatial Discretisation Time Discretisation Steady Flow Simulation Results NACA0012 RAE2822

11/36 Where ρ is the density ， u and v are the velocity components ， p is the pressure, e is the total energy per unit volume, for an idea gas, it can be written as Governing Equations

12/36 Boundary Conditions – Solid Wall (1) Direct method (2) Mirror method

13/36 Boundary Conditions – Far Field (1) Subsonic inflow (2) Subsonic outflow (3) Supersonic inflow (4) Supersonic outflow

14/36 Spatial Discretisation

15/36 Center node i and satellite node k. (“+” denotes the right wave, “ － ” denotes the left wave) Roe Scheme Spatial Discretisation

16/36 represents the stage coefficients, and Time Discretisation

17/36 Global and close-up views of the computational domain for the NACA0012 airfoil. Steady Flow Simulation Results – NACA nodes on the airfoil and 5557 nodes in the flow field.

18/36 Flow field pressure coefficients and mach number distributions for NACA 0012 airfoil. Steady Flow Simulation Results – NACA0012

19/36 Steady Flow Simulation Results – NACA0012 Flow field pressure coefficients and mach number distributions for NACA 0012 airfoil.

20/36 Steady Flow Simulation Results – NACA0012 Flow field pressure coefficients and mach number distributions for NACA 0012 airfoil.

21/36 Steady Flow Simulation Results – NACA0012 Flow field pressure coefficients and mach number distributions for NACA 0012 airfoil.

22/36 Steady Flow Simulation Results – RAE2822 Global and close-up views of the computational domain for the RAE2822 airfoil. 335 nodes on the airfoil and 5842 nodes in the flow field.

23/36 Steady Flow Simulation Results – RAE2822 Flow field pressure coefficients and mach number distributions for RAE2822 airfoil.

24/36 Steady Flow Simulation Results – RAE2822 Flow field pressure coefficients and mach number distributions for RAE2822 airfoil.

25/36 Unsteady Simulation of Euler Equations and Validation A Fast Dynamic Cloud Method Unsteady Flow Simulation Results NACA0012 NACA64A010

26/36 (a) Global view (b) Close-up view Back ground mesh for NACA0012 airfoil based on Delaunay triangulation A Fast Dynamic Cloud Method

27/36 (a) Spring analogy strategy (b) Delaunay graph mapping strategy Moved gridless clouds of 30°pitching airfoil A Fast Dynamic Cloud Method

28/36 Close-up views of computational domain for the NACA0012 airfoil for 2.51° pitch. Unsteady Flow Simulation Results – NACA nodes on the airfoil and 5557 nodes in the flow field.

29/36 Comparisons of computed lift and moment coefficients with the experimental and Kirshman’s data for prescribed oscillation of NACA0012 airfoil. Unsteady Flow Simulation Results – NACA0012

30/36 Close-up views of computational domain for the NACA64A010 airfoil for 1.01° pitch. Unsteady Flow Simulation Results – NACA64A nodes on the airfoil and 4006 nodes in the flow field.

31/36 Comparisons of the first Fourier mode component of surface pressure coefficients with the experimental data for oscillating NACA64A010 airfoil. Unsteady Flow Simulation Results – NACA64A010

32/36 Gridless Method with Dynamic Clouds of Points for Solving Unsteady CFD Problems in Aerodynamics – accepted by International Journal for Numerical Methods in Fluids Mach number distribution for pitching airfoil (NACA0012 and NACA64A010)

33/36 Conclusion and Future Introduction Principle of Gridless Method Steady Simulation of Euler Equations Unsteady Simulation of Euler Equations Future Research Inverse problem with one NACA 0012 airfoil using gridless method Inverse problem with dual NACA 0012 airfoils using gridless method Mesh/gridless hybridized algorithms to solve other boundary moving problems Our method is both flexible and efficient, therefore it is quite suitable to solve optimization problems, such as: Multi element airfoil lift optimization with Navier-Stokes flows in Aerodynamics Antennas optimal position in Telecommunications

34/36 Inverse problem of NACA 0012 Genetic Algorithms Using a Gridless Euler Solver for 2-D Unsteady Inverse Problems in Aerodynamic Design – will be submitted to EUROGEN 2009

35/36 Subsonic and Transonic Simulations of dual airfoils Mach number distribution for dual aifoils

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