Adaptive Cartesian Immersed Boundary Method for Simulation of Flow over Flexible Lifting Surfaces

Transcription

1 20th AIAA Computational Fluid Dynamics Conference June 2011, Honolulu, Hawaii AIAA Adaptive Cartesian Immersed Boundary Method for Simulation of Flow over Flexible Lifting Surfaces Robert E. Harris * and Vladimir I. Kolobov CFD Research Corporation, Huntsville, AL Abstract An efficient solver for the incompressible Navier-Stokes equations in which solid surfaces are represented using an Immersed Boundary Method (IBM) is presented. Adaptive Cartesian mesh refinement and coarsening are employed both to resolve immersed boundaries and to dynamically resolve important flow features as they are developing. Dynamically evolving immersed boundaries are tracked using Lagrangian marker points that are governed by direct momentum forcing using either simple force laws or explicitly prescribed motion in a framework that permits simulation of both rigid and flexible lifting surfaces. The adaptive Cartesian IBM capability presented here is well suited for low Reynolds number flows over flexible lifting surfaces as encountered in modern flapping-wing flyers, flow over complex geometries undergoing large-scale deformations, and vortex-dominated flows. The implementation of both the incompressible Navier-Stokes solver and the IBM solution algorithm are described in detail, and validation studies are presented which demonstrate the effectiveness of the method for resolving flow over flexible lifting surfaces. Results for several representative test cases are presented, and favorable comparisons are made with published data and with results obtained using a Volume of Fluid (VOF) advection scheme in the same solver framework. I. Introduction To meet modern engineering requirements, uninhabited low Reynolds number flyers are becoming the focus of significant research and development efforts. These aircraft operate in low Reynolds number (Re= ) 1 regimes in which there exist strong interactions between flow separation and transition to turbulence 2. This presents unique challenges which have lead researchers to shift focus from traditional aircraft designs to less conventional approaches. Due to the small scale and lack of passenger imposed limitations, low Re flyers represent a rich opportunity to explore novel concepts which may benefit from exploiting non-traditional flow physics. Particularly, utilization of large deformation fluid-structure interactions 3, 4, 5 as well as active and passive wing morphing 6, 7 can increase performance and aid in delaying separation. There are many examples found in nature where wing morphing is used to exploit unsteady aerodynamics in order to delay separation and increase performance. Experiments show that bats use the same tricks as bumble-bees for flow control and lift enhancements. 8 The details of the wing topography and deformation are aerodynamically important for engineering applications of flapping flight. 9 Computational Fluid Dynamics (CFD) can provide valuable insight for intelligent designs of low Reynolds number flyers that embrace unsteady flow physics inspired by nature. Determining the proper choice of computational grid is an important first step toward achieving this task. Three types of grid methodologies have been used for simulations of insect and bird aerodynamics: deforming mesh methods, 10 composite Chimera mesh methods, 11 and Cartesian mesh methods with Immersed Boundaries/Interfaces. 12 The first 3D CFD model of an insect with deforming wings, which was reported in Young et al., 9 employed * Senior Engineer, Aerospace & Defense Division, reh@cfdrc.com, AIAA Member. Technical Fellow, Aerospace & Defense Division, vik@cfdrc.com, AIAA Senior Member. 1 Copyright 2011 by Robert E. Harris. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

2 deforming mesh methods. Insufficient grid resolution prevented capturing the detail of the wake instabilities behind the trailing edge. We have selected a hybrid approach combing Adaptive Mesh Refinement (AMR) with octree Cartesian mesh technology and an Immersed Boundary Method (IBM) for the present study. Decoding the aerodynamic secrets of insect flight brings significant new computational challenges that are difficult to address with existing computer codes. To accurately predict the flowfield, and to gain a better understanding of transition mechanisms and their impact on flow separation, complex flow features should be locally resolved to the Kolmogorov scale. 13 This requirement poses serious problems for conventional methods employing body-fitted grids, including both overset methods and deforming mesh methods, with heavily anisotropic boundary layer refinement. Considering the large deformations involved during flapping-wing flight, 14, 15 the grid can very quickly become distorted to an unacceptable degree, 16 which can lead to an invalid mesh. The high frequency of re-meshing required to prevent such an overly distorted grid will likely render this method cost-prohibitive and impractical. Furthermore, in the low Reynolds number regime of interest here, boundary layers are much thicker and do not necessarily require an anisotropically-refined body-fitted grid, as is needed to accurately predict the flow at higher Reynolds numbers. In this situation, a more intelligent approach utilizing an adaptive isotropic mesh is better-suited for efficient and accurate prediction of boundary layers and localized regions of turbulent flow. Additionally, with the geometry represented using IBM, rather than a body-fitted grid, costly re-meshing and hole-cutting operations are avoided entirely. The paper is organized as follows. In Section 2, we present the basic formulation of the incompressible Navier-Stokes solver utilized for this work. After that, the formulation for the Immersed Boundary (IBM) is described in detail in Section 3, along with the methodology for computing the force densities at the interface using direct momentum forcing. Numerical results including comparisons of IBM and cut-cell VOF approaches with published data for several well known test cases are given in Section 4. Finally, conclusions and recommendations for future work are summarized in Section 5. II. Formulation of incompressible Navier-Stokes solver The computational framework leveraged for the current effort is the Gerris Flow Solver (GFS). 17 GFS is a modular, extensible, and heavily object-oriented solver for the incompressible Navier-Stokes equations, written in the C programming language. The GFS source code is freely available for use under the GNU GPL. 18 GFS is a state-of-the-art adaptive Cartesian octree-based solver framework for efficient numerical simulation of low Mach number flows with many powerful features. Among these features are: Explicit Navier-Stokes solver for time-dependent variable-density flows Adaptive mesh refinement: the resolution is adapted dynamically to the flow features Entirely automatic mesh generation in complex geometries Second-order accurate in both space and time Unlimited number of advected/diffused passive tracers Flexible specification of additional source terms Volume of Fluid (VOF) advection scheme for interfacial flows GFS has been employed by many researchers for different problems in fluid mechanics, such as moving body problems, 19 two-phase flows with electro-hydrodynamic forces, 20 multiscale simulation of primary atomization, 21 growth of instabilities in two-phase mixing layers, 22 simulation of droplets, bubbles, and waves, 23 and many others. Details and features of the native incompressible Navier-Stokes solver in GFS, including some representative results, are presented in the next few sections. Additional details on this formulation can be found in Popinet. 17 In this paper, we describe an extension of the GFS framework that makes use of the Immersed Boundary Method (IBM) where surface force distributions are utilized for imposing solid wall boundary conditions. The incompressible Navier-Stokes equations with external forces are given by u 2 ρ + ( u ) u = p + µ u + F, (1a) t 2

3 u = 0, (1b) where u is the velocity vector, ρ is the density, p is the pressure, µ is the dynamic viscosity, and F is an applied force. These equations are valid for a constant density, incompressible, viscous fluid. The applied forcing term is zero in the bulk of the flow, but is taken to represent the force due to the presence of a boundary immersed in the fluid in our implementation of IBM. This IBM approach will be discussed in greater detail in Section 3 below. A. Spatial Discretization GFS utilizes a quadtree (2D), or octree (3D), spatial discretization in which the computational grid is comprised of either square, or cubic finite volumes arranged hierarchically, respectively. A schematic illustrating the spatial discretization and corresponding tree representation is shown in Figure 1. Figure 1. Example of quadtree discretization and corresponding tree representation. 17 The length of a cell edge is denoted by h. Each cell may be the parent of up to four children (eight in 3D). The root cell is the base of the tree and a leaf cell is a cell without any child. The level of a cell is defined by starting from zero for the root cell and by adding one every time a group of four descendant children is added. Each cell C has a direct neighbor at the same level in each direction d (four in 2D, six in 3D), noted N d. Each of these neighbors is accessed through a face of the cell, noted C d. In order to handle embedded solid boundaries, we also define mixed cells, which are cut by a solid boundary. To simplify the calculations required at the cell boundaries, we add the constraints illustrated in Figure 2: (a) The levels of direct neighboring cells cannot differ by more than one. (b) The levels of diagonally neighboring cells cannot differ by more than one. (c) All the cells directly neighboring a mixed cell must be at the same level. Figure 2. Additional constraints on the quadtree discretization. 17 The refinement necessary to conform to the given constraint is indicated by the dotted lines. 3

4 While not fundamentally necessary, these constraints greatly simplify the gradient and flux calculations for the VOF advection scheme. Constraints (a) and (b) have little impact on the flexibility of the discretization (they only impose gradual refinement by increments of two). Constraint (c) is more restrictive as it forces all the cells cut by the interface to be at the same level (i.e. the whole solid boundary must be described at the same resolution). It is also important to note that the quad/octree structure imposes a locally spatially isotropic refinement, which may not be suitable for high Reynolds number flows, but is an attractive approach for the resolution of low Reynolds number flows that typically involve much thicker boundary layers. B. Temporal Discretization Eq. (1) is advanced in time using a classical fractional-step projection method. At any given time step n, we assume that the velocity at time instance n, u n and the fractional step pressure p n 1/2 are known at the cell centers. In a first step, a provisional velocity u ** is computed by applying a 2 nd -order accurate Godunov procedure 24 to Eq. (1a) and taking a single time step. This provisional velocity u ** does not necessarily satisfy Eq. (1b), so an approximate projection operator is applied to obtain the new divergencefree velocity u n+1. This operation also yields the fractional step pressure p n+1/2. The temporal advancement of the diffusion terms utilizes a family of schemes, at the limits of which the scheme becomes either Crank- Nicholson or forward-euler. C. Poisson Equation The projection method relies on the Hodge decomposition of the velocity field as u ** = u + φ, (2) where u = 0 on the solid surface Ω. Taking the divergence of Eq. (2) yields the Poisson equation 2 φ = u **. (3) The divergence-free velocity field is then defined as u = u ** φ, where φ is obtained as the solution of the Poisson problem given by Eq. (3). This defines the projection of the velocity u ** onto the space of divergence-free velocity fields. In the context of the approximate projection method we are using here, the discrete formulation of the projection operator will depend on where the velocity field is discretized relative to the pressure field. We will use both an exact projection for face-centered advection velocities and an approximate projection for the final projection of the cell-centered velocities. In practice, the spatially discretized Poisson problem results in a linear system of equations with the pressure at cell centers as unknowns, which can be solved using iterative methods (Jacobi, Gauss-Seidel, etc.) using a relaxation operator. The methodology for computing the relaxation operator is omitted here for brevity, but is given in detail in Popinet. 17 This procedure is second-order accurate in space at coarse/fine boundaries and uses consistent flux estimation. We employ a multigrid acceleration technique to greatly improve the efficiency of the Poisson solution described above. An illustration of a typical multilevel hierarchy for a quadtree grid with local refinement is shown in Figure 3. Using this multilevel hierarchy, we apply a classical multigrid V-cycle until the L norm of the residual has converged to 3 orders-of-magnitude and we apply 4 iterations of the relaxation operator at each level. D. Adaptive Mesh Refinement The adaptive mesh refinement strategy employed here is described in detail in Popinet, 17 but we review the procedure here for completeness. In the first step, all the leaf cells which satisfy a given criterion are refined. This includes neighboring cells when necessary, in order to satisfy the aforementioned neighboring cell depth requirements of the Octree grid. In the second step, the parent cells of all the leaf cells are considered, and all of these cells which do not satisfy the refinement criterion are coarsened. The solution 4

5 quantities for newly coarsened cells are computed as volume-weighted averages from their former children, and for newly created cells the solution quantities are computed from a linear interpolation using the parent cell value and its gradients. Figure 3. Example of simple multilevel hierarchy for use in solving Poisson equation. 17 The refinement criteria can consist of any function of pressure, velocity, or their derivatives. Additionally, geometric refinement criteria can be used to provide greater resolution at specified spatial or temporal locations. Here, we generally use a geometric refinement criterion based on distance to the solid surface, and we combine that with a criteria based on vorticity. At each adaptation step, the adaptive cell cost is computed from the combined local adaptation criteria multiplied by the cell size. Then if the adaptive cell cost exceeds a user-defined maximum allowable cell cost, the cell is refined. Additionally, the grid will not be refined past a user-defined maximum refinement level or maximum number of cells. The computational cost of this algorithm is small compared to the cost of the Navier-Stokes solution. It can be applied at every time step with a negligible overall penalty of less than 5% of the total cost, while in practice every 10 or 100 steps is usually sufficient depending on the application. E. Results and Validation of Incompressible Solver To serve as a test suite and to assist in benchmarking and validation of the subsequent implementation of IBM within GFS, we now employ the native GFS solver with cut-cell boundary theatment for simulations of several different test problems. All cases are run with adaptive mesh refinement, with the grid locally adapting to regions of high vorticity. Static/Moving cylinder at Re=400 First we consider flow over a stationary circular cylinder at Re=400. The computational grid and vorticity contours at a point where the flow is nearly fully developed are shown in Figure 4. The von Karman vortex street is clearly visible from the vorticity contours. To compare with the static cylinder case, we now consider the problem of a cylinder moving through a quiescent flow at Re=400. The vorticity contours at several time instances are plotted in Figure 5, and a comparison of computed lift coefficient for 5

6 both the static and moving cylinder case is shown in Figure 6 it is apparent that the lift coefficient agrees very well between the static and moving cylinder cases. Figure 4. Grid (left) and vorticity contours (right) for static cylinder case at Re=400. Figure 5. Vorticity contours at various time steps for moving cylinder case at Re=400. Figure 6. Comparison of lift coefficient time history for both stationary and moving cylinder cases. Pitching/plunging NACA 0012 airfoil As an additional test of the moving body capabilities of the native GFS solver, we now consider flow over a NACA0012 airfoil that is undergoing plunging motion at an amplitude of 20% chord and frequency of 20 radians per second. The computational grids for this case are shown in Figure 7. There exists an analytical solution for this case 25,26 which derives from potential flow theory. Both the computed and 6

7 analytical values of lift coefficient for this case are plotted in Figure 8. The computed lift agrees very well with the analytical solution. Figure 7. Grids at various times for NACA 0012 airfoil undergoing plunging motion. Figure 8. Computed lift coefficient compared with analytical solution. 25,26 Next we consider flow over a NACA0012 airfoil that is undergoing pitching motion at +/- 4 degrees angle-of-attack and at a frequency of 20 radians per second. The computational grids for this case are shown in Figure 9. Both the computed and analytical values of lift coefficient for this case are plotted in Figure 10. Again, the computed lift agrees very well with the analytical solution even with the fairly coarse grids employed here. The results presented above give us some confidence in the moving body capabilities 7

8 of the GFS solver, and serve as a preliminary test suite for benchmarking and validation of the Immersed Boundary Method (IBM) and Immersed Interface Method (IIM) under development. Figure 9. Grids at various times for NACA 0012 airfoil undergoing pitching motion. Figure 10. Computed lift coefficient compared with analytical solution. 25,26 Thin flexible flapping wing In order to simulate flow over surfaces undergoing prescribed structural deformation, we have developed a module to generate surface geometries for wings with both chordwise and spanwise flexibility. The families of wings that we consider here have been shown to accurately simulate flapping flight with 8

9 flexible wings or fins for a variety of animals. 27,28,29 Wing flexibility is specified by the speed of a wave traveling backwards on the wing, with lower wave speeds resulting in a more flexible wing. Planform wing shape can also be specified by adjusting the chord length distribution along the span. The y coordinate on the wing surface is given by y(x, z,t) = ξz{ h 0 +ε ( x +1) exp{ iω t ( x +1) / c } γ ( x +1) }. As a preliminary test of the capabilities of GFS for simulating flow over flexible surfaces, we now consider flow over the flexible flapping wing described above. The initial wing geometry for this problem is computed and output from our module as a pre-processing step, and the subsequent wing motion is prescribed in the GFS input file. The flexible surfaces, as well as vorticity contours and computational grids for this case at various times are displayed in Figure 11. We carry out this simulation in 2D for several periods of motion to illustrate how the flow becomes increasingly vortex-dominated over time. Figure 11. Vorticity (left) and computational grids (right) at various time instances. We can clearly observe the leading-edge vortex during the upstroke, as well as many vortices being shed from the trailing edge throughout the simulation. The compute lift and drag coefficients are displayed in Figure 12. As a final test case to further demonstrate the capabilities of the VOF advection scheme in GFS, 17 we consider revisit the above flexible flapping wing case in 3D. This case was executed on a parallel machine with 8 CPUs. Figure 13 shows vorticity isosurfaces, as well as the wing surface and computational grid for three different time instances during the simulation. The large starting vortex is observed being shed from 9

10 the trailing edge, and the large scale tip vortices are also very clearly defined. The grid is re-adapted to vorticity and re-load-balanced every 10 time steps, for which the finest grid cells are 9e-4 units in length. The grid contains over 2M cells after about 400 time steps. Figure 12. Time history of lift (left) and drag (right)) coefficients for flexible flapping wing case. Figure 13. Vorticity isosurfaces, including surface and a slice of the computational grid colored by vorticity for 3D flexible flapping wing case at three different time instances. Despite the complexity of the above 3D deforming body simulation, the GFS input file is very simple and requires minimal user input. The complete input file is given in Figure 14 below. 1 0 GfsSimulationMoving GfsBox GfsGEdge {} { GfsTime { end = 1. } GfsRefineSolid 12 GfsSolidMoving { istep = 1 } wing.gts { tz = -.2 } { level = 12 } GfsAdaptVorticity { istep = 1 } { cmax = 6e-2 maxlevel = 12 } GfsInit {} { U = 1 } GfsSurfaceBc V Dirichlet { gdouble h0 = 0.18, f = 0.2, c = 0.306, ts = 30., sc = 4.0; gdouble h = -z*h0*sin(2.*m_pi*f*(ts*t-(sc*x+1.)/c))*2.*m_pi*f*ts; return z > 0.? h : -h; } GfsOutputTime { istep = 50 } stdout GfsOutputSimulation { istep = 50 } test-%06ld.vtk { format = VTK } } GfsBox { left = GfsBoundaryInflowConstant 1 right = GfsBoundaryOutflow } Figure 14. GFS input file for flexible flapping wing case. The initial surface geometry file wing.gts is the only other file required, and is specified using the GfsSolidMoving object. The entire grid generation process is directed by GfsRefineSolid and 10

11 GfsAdaptVorticity, and the solid body motion and deformation is prescribed using GfsSurfaceBc. The criterion for stopping execution of the simulation is given in GfsTime, the solution quantities are initialized in GfsInit, and the farfield boundary conditions are specified using GfsBox. The remaining GfsOutput events are only needed for output of different quantities during post-processing. The above demonstrations and validations give us confidence in the capabilities of the underlying GFS framework. In particular, GFS offers high flexibility, efficiency, and adequate accuracy for simulations of incompressible viscous flows in the presence of moving and deforming bodies. Another important strength of GFS is the ease of use and minimal required user input. The input file shown in Figure 14 above is evidence of that. Even for a complicated 3D flexible flapping wing problem, the adaptive mesh refinement is governed entirely by a single line in the input file, and the movement and deformation of the flexible flapping surface requires only 5 lines of input. This paradigm of minimalism with respect to the required input, while providing a high degree of flexibility through object-oriented programming practices has been strictly adhered to throughout this work, as will be described in subsequent sections. III. Immersed Boundary Method The native incompressible Navier-Stokes solver of the GFS framework described in the previous section makes use of cell-cutting and a Volume-Of-Fluid (VOF) advection scheme to handle moving and deforming solid surfaces. While this is certainly a viable approach for many applications, difficulties can arise if the solid surface is very thin or if the structural deformations are large, and solution accuracy can be degraded near the solid surface. For these reasons, the Immersed Boundary Method (IBM) is a desirable alternative to the cut-cell VOF method, and has been an active area of research. There are two versions of IBM, one using ghost cells inside solids, and another using surface forces for imposing boundary conditions. In the latter approach, zero-thickness bodies can be easily handled and there are no restrictions with respect to large-scale structural deformations. Furthermore, since there are no physical solid surfaces present in these methods, octree adaptive mesh refinement is a very attractive approach for robustly and efficiently resolving geometric features of immersed flexible moving surfaces, and it offers self-aware physics with dynamic adaptive mesh refinement of local flow features. In this manner, flow features of even the finest detail can be captured and preserved automatically as the flow develops. Our implementation of the Immersed Boundary Method (IBM) 30,31,32 represents solid surfaces embedded in the fluid as applied forces in the momentum equations. These forces are computed to ensure the enforcement of the no-slip boundary condition at the solid surface. The IBM approach can suffer from losses of accuracy at the interface due to smearing out of the solution caused by the spreading of the solid force between the Eulerian grid and the Lagrangian markers. Many ways to remedy this obstacle have been 12,33.34,35,36, 37,38,31 presented in the literature. These approaches typically involve modifications to the numerical schemes employed in the Navier-Stokes solver in the form of correction terms to the primitive variables and their derivatives. This is in contrast to the IBM approach where the force is spread to the Eulerian grid and included directly in the momentum equations as a source term. These approaches are generally referred to as Immersed Interface Methods (IIM). We now present the technical details of the IBM in a formulation that is readily extended to a more accurate IIM formulation. The basic formalism to be presented is the essential element common to most Immersed Boundary and Immersed Interface Methods. A. Surface Geometry Representation In the Immersed Boundary Method (IBM), the Eulerian computational grid does not contain any physical definition of the surface geometry. The approaches for representing solid surfaces in these methods typically involve either a level set representation, 39 in which the solid surface is represented by the zero level of a signed distance function, or using a set of distributed Lagrangian marker points. 36,12,40,31 Researchers have obtained very promising results using both approaches. Here we implement the latter method based on Lagrangian markers. This approach is very well suited for implementation within the GFS framework. In the Lagrangian marker approach, a set of discrete points is defined in order to track and manipulate the solid surface positions throughout the simulation. In order to update both the flow solution due to solid 11

12 forces, and the position of the Lagrangian marker points, it is necessary to be able to integrate various quantities over the surface. To this end, we introduce a triangulated surface to provide connectivity to the Lagrangian markers. The basic idea is to construct a triangulated surface with similar resolution to that of the maximum resolution of the intended Eulerian CFD grid. In fact, the ratio of the marker spacing to characteristic grid length can greatly affect the computation of the solid forces, as well as the overall stability of the method. This will be discussed in greater detail in the next Section 3.2. Figure 15 shows an example Lagrangian marker distribution and corresponding surface triangulation for a flexible wing geometry. Figure 15. Lagrangian marker distribution (left) and surface triangulation (right) for a thin flexible wing surface geometry. A common operation in IBM formulations is the computation of a quantity locally on the Eulerian grid that is due to a distribution over the Lagrangian markers. Typically, the quantity of interest is either a solid force or a velocity. Consider a time-varying interface Γ(t) immersed in a domain Ω. The applied body force F( x, t) on the fluid in the Eulerian domain due to the presence of the solid represented by the interface Γ(t) can be expressed as δ ˆ ˆ, (4) F( x, t) = f ( sˆ, t) x X ( sˆ, t) S where x is a coordinate in the Eulerian grid, X ( sˆ, t) of parameter ŝ, f ( sˆ, t) X ˆ ( sˆ, t), and x Xˆ ( sˆ, t ) ( ) 12 ( ) ds ˆ represents a coordinate on the surface of Γ(t) in terms is the force density per unit area along Γ exerted by the interface at the point δ is the Dirac-delta function which is evaluated as the product of three 1D Dirac-delta functions, one in each spatial direction. So, if the force density on the interface is known, the solid force at any point in the Eulerian grid can be readily computed via Eq. (4). Typically the Dirac-delta function in Eq. (4) is replaced by a suitable discrete version; two examples of which are given in Eqs. (5)- (6) below δ w ( r) 1 ( ) ( 1+ cos( πr / 2w) ), r < 2w δ r = w 4w 0, r 2w r / w r / w 4 r / w, 8w r w 1 = r / w r / w 4 r / w, w < r 2w 8w 0, otherwise where w is a constant typically chosen to be comparable to the minimum cell size in the Eulerian grid. Eq. (5) and Eq. (6) are known to give very similar results, so Eq. (5) is often used for simplicity. It should also be mentioned that recently Yang et al. 41 showed that moving body simulations may be performed more accurately using smoothed versions of Eqs. (5)-(6), which help eliminate non-physical oscillations in the (5) (6)

13 forces. In order to ensure that the grid resolution of the Eulerian grid is consistent along the entire interface, and within a distance 2w from the interface, we have developed a module to ensure that there are no hanging nodes in the octree grid near the interface. The grid adaptation module written to support this effort can guarantee regions of uniform grid that are N cells thick and smoothly transition the interface to the farfield. As a demonstration of this module, Figure 16 shows a grid without the use of this new technique (left), and a grid where the new technique has been employed (right). It is evident that the old approach has significant non-uniformity near the wall, while new approach provides a sufficiently uniform grid near the interface. Figure 16. Computational grid exhibiting non-uniformity near the interface (left), and one exhibiting smoothly varying regions of uniform grid (right). Another common operation in IBM formulations is the determination of the velocity at the interface U ( Xˆ ( sˆ, t), t) due to the Eulerian velocity field u ( x, t). This is carried out in a manner very similar to Eq. (4) as follows U ( Xˆ ( sˆ, t), t) = u( x t) ( x Xˆ Ω, δ ( sˆ, t) ) dx, (7) where the integral in Eq. (7) is now over the Eulerian grid, rather than the interface. It should be noted that the interface generally only affects the Eulerian grid cells that it directly intersects, so only cells in the immediate vicinity of the interface need be included in the evaluation of Eq. (7). As an alternative to Eq. (7), the velocity at the surface can be obtained by simple trilinear interpolation from the Eulerian grid. This tends to give very similar results to Eq. (7), and is less expensive to compute. With the above utilities in place, information can now be readily transferred from the Lagrangian markers to the Eulerian grid, and vice-versa. B. Determination of Boundary Force Densities The evaluation of Eq. (4) above depends on a force density that is not generally known in advance, so it typically must be computed in some feedback manner since it depends on the flow quantities. Essentially, the force density must be constructed so that the velocity comes to a rest at the surface Γ(t) to satisfy the no-slip boundary condition. There are numerous methods in the literature for computing the force density, and we consider two of them here. The first approach uses a feedback forcing mechanism, and the second approach utilizes direct momentum forcing with an implicit force calculation. These approaches are discussed below in more detail. Feedback Forcing The first approach considered here for computing the forces uses a feedback mechanism in which the interface is represented by a system of stiff springs. This approach is outlined in detail in Xu and Wang. 36 The basic idea is that the computational interface is allowed to move very slightly away from the known physical interface, but is attached to the physical interface by stiff springs. Thus, the Lagrangian markers are advected by the local fluid velocity. The position of the Lagrangian markers is advanced at each time step using the local flow velocity given by Eq. (7) and the current time step t, i.e. 13

14 ( t) X ˆ ( s t t) Xˆ ˆ, + = ( sˆ, t) + tu Xˆ ( sˆ, t),. (8) For this approach, the very small movement of the interface is necessary so that a finite force field will be generated, but generally a large spring constant is selected so that the movement is so small that it is not visually noticeable. For this scenario, the force density is computed via ( ) ˆ E f sˆ, t = κ X ( sˆ, t) Xˆ ( sˆ, t) where κ is the spring constant, and X E ( sˆ, t) ( ), (9) ˆ is the known exact position of the interface. This approach is illustrated in the schematic in Figure 17 below. This demonstrates the system of springs anchoring the computational surface to the known physical surface, and the analog of a surface fixed in place by a spring and damper. Figure 17. Schematic illustrating feedback spring system for representation of an interface. 36 We have implemented the above feedback forcing mechanism, and tested the formulation for flow over a rigid circular cylinder at Re=150. This case is primarily intended to demonstrate that the no-slip condition is getting satisfied when using the feedback spring system outlined in Eq. (9) with a spring constant of 1e5. The results for this case are shown in Figure 18 below. It is evident that the applied forcing given by Eq. (9) is indeed producing a cylindrical obstacle which the flow is unable to penetrate. While these results indicate that this technique may have some promise, there are considerable numerical challenges to be dealt with for practical applications. For the simple cylinder case considered here, small oscillations in the flow were observed and proper convergence was never obtained. Figure 18. Flow over stationary cylinder at Re=150 using IBM approach; Velocity contours (left), Velocity vectors and grid (right). This approach is known to necessitate a very small time step in order to ensure stability, and researchers typically utilize a more complicated implicit update of the Lagrangian markers to improve the stability 14

15 requirements. 36,33 Furthermore, the spring constant required for the computational boundary to closely represent the physical boundary may be exceedingly high and call for data limiting to maintain stability. This sets up a vibrational time scale which can also negatively impact the stability requirements. In addition, Xu and Wang 36 showed that the advection of the Lagrangian markers with the local fluid velocity can cause the boundary to contain irregular truncation errors, which may necessitate smoothing or Fourier filtering in order to maintain a smooth shape throughout the simulation. Direct Momentum Forcing The second approach considered here for computing the forces utilizes a direct momentum forcing approach. Both explicit and implicit formulations are discussed below. Explicit method As a very simple demonstration of this approach, we consider an explicitly prescribed forced rotation of a circular cylinder, in which the force takes the form f ( sˆ, t) = Cτˆ where τˆ is a unit vector in a direction tangent to the interface and C is a constant. The results of this case are shown in Figure 19, at Re=10 (left) and Re=1,000 (right). It is evident that the Re=10 case entrains more fluid than the Re=1,000 case due to increased viscosity. Obviously for practical problems, such an explicit forcing is not possible since the surface forces are heavily dependent on local flow quantities. Figure 19. Flow over rotating cylinder at Re=10 (left) and 1,000 (right) using forced rotation. Implicit method Direct momentum forcing via implicit methods has been successfully demonstrated by many different researchers, 42,32,43 and has been shown to greatly reduce the restrictive time step requirements common to many indirect feedback type approaches. The basic idea is that the velocity of the Lagrangian markers is f sˆ, t are obtained by solving a banded system of linear explicitly prescribed, and then the forces ( ) equations at each time step to enforce the no-slip condition on the current velocity field. n u In 2D, the system of equations can be reduced to a tri- or penta-diagonal system. 32 This approach is very attractive for the current study, as it allows for the simulation of moving and deforming bodies by explicitly prescribing spatially and temporally varying surface velocities. The complete details of this approach are given in Su et al. 32 The current work utilizes a slightly modified version of this approach, which is outlined below. Initially, the fluid velocity is interpolated to the Lagrangian markers using Eq. (7) to obtain the current interface velocity, i.e. 15

16 n ( ˆ n ( ˆ) ) ( ) ( ˆ U X s = Ω u x δ x X ( sˆ )) dx. (10) Next, we must obtain the velocity of the markers at time n+1, U n +1 Xˆ ( sˆ ). This can be explicitly ( ) prescribed as in the current work, or calculated as part of the solution by coupling with a Fluid Structure Interaction (FSI) module for aeroelastic effects. Using this velocity, we can write n+ 1 ( ) ( ( )) ( ˆ n ( ˆ) ) ( ˆ n ˆ U X s U X ( sˆ )) Ω F x δ x X sˆ dx =, (11) t where F n ( x) is the applied interface force at Eulerian grid coordinate x, which is still an unknown at this point. Eq. (11) states that we require the acceleration of Lagrangian marker X ˆ ( sˆ ) due to the applied force to be equal to the temporal rate-of-change of the marker velocity. Combining Eqs. (4) and (11), the Lagrangian boundary force can be obtained as n+ 1 ( ) ( ( )) ( ˆ n ( ˆ) ) ( ˆ ˆ U X s U X ( sˆ )) drˆ δ x X sˆ dx =, n f ( rˆ ) x Yˆ ( rˆ ) Ω S F n x δ (12) ( ) Now, rearranging the left-hand side of Eq. (12), we arrive at n+ 1 { ( ( )) ( ( )) } ( ) ( ˆ n ( ˆ) ) ( ˆ ˆ ˆ n U X s U X ( sˆ )) x Y rˆ x X sˆ dx f rˆ drˆ = S Ω δ δ. (13) t Eq. (13) is a system of M equations for the M Lagrangian marker forces f ( ˆ ) t s k, k=1,2,,m. Since the support of the discrete delta function is the same as the grid spacing h, the coefficient matrix given in Eq. (13) is banded. In fact, for a closed boundary it is a periodic banded matrix. 32 For the solution of the system of equations given by Eq. (13), we employ the GNU scientific library, 44 which ships with most Linux distributions, along with the heavily optimized ATLAS 45 (Automatically Tuned Linear Algebra Software) library. For simplicity, the Lagrangian markers are generally uniformly distributed along the immersed boundary. As demonstrated by Su et al., 32 the optimum marker spacing ( ŝ) scales with the grid spacing h. This implies that any non-uniformity in the marker spacing will likely necessitate a corresponding nonuniformity in the Eulerian grid resolution in the same region. The above formulation is slightly different than that of Su et al., 32 in that we use the interpolated surface velocity at the current time step, rather than an intermediate velocity from an advection operation. Additionally, the forces f ( ˆ ) s k obtained by solving the system of equations given by Eq. (13) are spread to the grid via Eq. (4) and included in the momentum equations as source terms. We have implemented both procedures and found that the current formulation produces very similar results to that of Su et al., 32 but is considerably easier to implement since the underlying projection method is not modified. C. Addition of Source Term in Momentum Equations The force terms computed by solving the linear system given in Eq. (13), which represent the forces that the interface exerts on the fluid, are added to the momentum equations as source terms. The module that houses the above formulation is completely self-contained in the sense that it is written and compiled into a library that is completely separate from the GFS source code. To access this IBM functionality, the library just needs to be placed in a location where Gerris looks for such libraries. Also, the module has been developed such that only a single line of the input file needs to be modified to switch between the native VOF advection scheme and the new IBM approach. To illustrate this, an input file for flow over a cylinder is shown in Figure 20, with the important parts shown in red. The file for the VOF advection scheme and IBM scheme are given in Figure 20a and Figure 20b, respectively. This demonstrates the ease at which one can switch between solvers. 16

17 2 1 GfsSimulation GfsBox GfsGEdge {} { GfsTime { end = 10 } GfsRefineSolid 6 GfsSolid cylinder.gts { scale = 0.25 } GfsInit {} { U=1 } GfsSourceViscosity { beta = 1} GfsOutputSimulation { step=1 } Output-%06ld.vtk { format = VTK } } GfsBox { left = GfsBoundaryInflowConstant 1 } GfsBox { right = GfsBounaryOutflow } 1 2 right (a) 2 1 GfsSimulation GfsBox GfsGEdge {} { GfsTime { end = 10 } GfsRefineSolid 6 GModule GfsIBM GfsImmersedBoundary cylinder.gts { scale = 0.25 } GfsInit {} { U=1 } GfsSourceViscosity { beta = 1} GfsOutputSimulation { step=1 } Output-%06ld.vtk { format = VTK } } GfsBox { left = GfsBoundaryInflowConstant 1 } GfsBox { right = GfsBounaryOutflow } 1 2 right (b) Figure 20. Gerris input files for simulation of flow over a cylinder using native GFS VOF advection scheme (a), and new IBM scheme (b). D. Solution Procedure The full solution procedure carried out by the new IBM module for GFS is as follows: 1. Read input file, generate initial Eulerian grid refined to surface(s) n 2. Initialize velocity u n 1 / 2 and pressure p at cell centers 3. Advect Lagrangian markers via Eq. (8) using prescribed marker velocity at current step U ( X ( s) ) n ˆ ˆ, and current t (if surface is moving) 4. Construct matrix given in LHS of Eq. (13) (only if surface is moving or on initialization) 5. Compute RHS of Eq. (13) using current marker velocity U ( X ( s) ) n ˆ ˆ calculated from Eq. (10) and U n +1 Xˆ sˆ ( ) prescribed marker velocity at step n+1, ( ) 6. Solve linear system of equations given by Eq. (13) for Lagrangian marker forces, f ( sˆ k ) 7. Spread Lagrangian marker forces f ( sˆ k ) given by Eq. (5) or (6), to obtain F n ( x) to Eulerian grid via Eq. (4), using discrete delta function 8. Solve Navier-Stokes equations given by Eq. (1) in entire flow domain using procedure of Popinet 17 outlined in Section 2, and using body forces F n ( x) as source terms in Eq. (1a) 9. Adapt Eulerian grid to desired flow quantities to ensure dynamic resolution of small scale flow features 10. Repeat steps 3-9, or stop and post-process if maximum time step or convergence reached 17

18 IV. Results and Validation We now employ the implicit direct forcing IBM methodology in Section 3 for the simulation of several different types of problems. In particular, we consider flow over stationary rigid surfaces, moving rigid surfaces, and moving flexible surfaces. We employ the adaptive mesh refinement methodology outlined in Section 2 for all cases considered here, using vorticity as the adaptation criterion with a specified maximum allowable cell cost of C MAX =5e-2. Additionally, we employ a geometric refinement based on the distance to the immersed boundary. Thus, the initial Eulerian grid is generated such that the cells in the immediate vicinity of the immersed boundary are at the maximum refinement level L MAX, and the refinement level then decays with distance from the immersed boundary, leaving N BAND cells at each coarser level until the domain boundary is reached. L MAX provides a limit on the number of recursive subdivisions that can occur starting from the unit root octant (taken to be at level zero), so it is also indicative of the length scale of the smallest cells (1/2^L MAX ). With this methodology in place, the dynamic Eulerian grid is completely defined by the current vorticity field, and the three user input parameters (C MAX, L MAX, N BAND ). An example of an initial grid for an airfoil generated in this fashion is shown in Figure 16 (right). The Lagrangian marker point distribution ( ŝ) on the surface is chosen such that it scales with the grid spacing set up by the maximum refinement level, i.e. h MIN =(1/2^L MAX ). In practice, we have found that using a marker spacing that corresponds to the minimum cell diagonal length, namely ( s) 2hMIN (2D) and ( s) 3hMIN (3D), results in a well-conditioned, stable and convergent scheme. ˆ = ˆ = A. Stationary Body Problems We first employ the new IBM capability for simulation of flows over stationary bodies at low Reynolds number in order to demonstrate and benchmark the technology in terms of accuracy, efficiency, and robustness. For all cases considered here, the number of cells in each grid level away from the body is set to N BAND =4. This ensures that all Eulerian cells in the vicinity of the interface that are involved in the matrix calculation via Eq. (13) are at the same level throughout the simulation. With this in mind, and using the fact that the immersed boundary is not moving, we need only compute the matrix once on initialization, and the upper triangular part of the matrix can then be stored. Then for the duration of the simulation, the forces on the immersed boundary are obtained very efficiently via back substitution. All simulations are carried out using a time step calculated from a CFL number of 0.3, and backward Euler temporal advancement for the diffusion terms. Re=40 flow over stationary cylinder We now consider Re=40 flow over a stationary circular cylinder of radius R= 1/16. The cylinder is centered at the origin, the extents of the computational domain are (x 1, y 1 )x(x 2, y 2 )=[-1/2,-1/2]x[3/2,1/2], and the simulation is carried out until time t=15. The velocity and pressure are initialized everywhere to (1,0), and 0, respectively. At the left boundary, the velocity is set to V=(1,0) and the pressure gradient is set to dp/dx=0, while at the right boundary the pressure is set to P=0 and the velocity gradient is set to du/dx=0. A symmetry condition is used at the top and bottom domain boundaries. To ensure grid independent solutions, we consider three different refinement levels, where L MAX = (9, 10, 11), for which the Lagrangian marker point distribution is 75, 150, and 300, respectively, and also we include comparisons with the Gerris VOF approach. Streamlines of the flow are displayed in Figure 21, along with a schematic defining some widely presented parameters that are characteristic of this flowfield. Of particular relevance are the length of the recirculation zone L, the horizontal distance from the trailing edge to the recirculation bubbles a, and the vertical distance between the recirculation bubbles, b. A comparison of the current IBM and VOF results for these length scales, along with the computed drag coefficient, is provided in Table 1. Very good agreement with published data is observed. Velocity magnitude and pressure contours, as well as computational grids, are shown in Figure 22 for both the IBM approach (a) and the VOF approach (b), for L MAX =11. Twenty even contours of velocity between 0 and 1.3, and 20 even contours of pressure between -0.4 and 0.7 are presented for both the IBM 18

19 and VOF approaches. The IBM and VOF grids contain 37,378 and 31,452 cells, respectively, at the final time step. This disparity exists because cells that have been cut out of the domain for the VOF simulation are necessary for the IBM simulation and are retained. It is evident that the results, and computational grids at t=15 are nearly indistinguishable. Figure 21. Streamlines for Re=40 flow over a stationary cylinder using IBM approach. A schematic displaying important length scales for this problem is also shown at the bottom. Table 1. Comparison of computed length scales and drag coefficient for present study of Re=40 flow over stationary cylinder with published results. L a b CD Present (IBM) Present (VOF) Linnick and Fasel Xu Contanceau and Bouard Silva et al Ye et al Kim et al The convergence history for drag coefficient for this case is presented in Figure 23. The final converged values for the drag coefficient are 1.54 and 1.53 for the IBM and VOF methods, respectively, which agree 19

20 very well with published data, as presented in Table 1. For a qualitative look at the flowfield, the velocity vectors are presented on a contour plot of velocity magnitude in Figure 24 (left), and the forces on the immersed boundary are presented on a contour plot of vorticity in Figure 24 (right). The solid forces have the largest magnitude in the vicinity of the stagnation point, and the forces have the smallest magnitude in the recirculation zone near the trailing edge. (a) (b) Figure 22. Velocity contours (top), pressure contours (middle), and computational grids (bottom) for Re=40 flow over a stationary cylinder using IBM (a) and Gerris VOF (b) methods. Figure 23. Convergence history of drag coefficient for simulation of flow over a stationary cylinder at Re=40 using IBM (left) and cut-cell VOF (right) methods. Final converged values are indicated. 20

21 Figure 24. Velocity contours with vectors (left), and vorticity contours with solid force vectors (right) for simulation of flow over a stationary cylinder at Re=40 using IBM. Re=100 flow over stationary cylinder We now consider Re=160 flow over a stationary cylinder. This is an unsteady problem that is known to produce the well known Von-Karman vortex street. The geometry, domain extents, initial and boundary conditions are the same as for the Re=40 case. Figure 25 shows the vorticity contours (top) and computational grid (bottom) for this case. The expected periodic shedding is indeed predicted with the new IBM capability. To better illustrate the unsteady separation, the velocity vectors at a particular time instance are shown in Figure 26. Figure 25. Vorticity contours (top), and computational grid (bottom) for simulation of flow over a stationary cylinder at Re=160 using IBM. The time history of the lift and drag coefficients are plotted in Figure 27 for both the IBM and VOF approaches, and the minimum and maximum values are presented in Table 2. The results for both lift and drag coefficient agree very well with those obtained from several different researchers for this case. 21

22 Figure 26. Velocity contours and vectors for simulation of flow over a stationary cylinder at Re=160 using IBM. Figure 27. Time history of drag (left) and lift (right) coefficients for simulation of Re=160 flow over stationary cylinder. IBM and VOF method results are shown in red and green, respectively. Table 2. Comparison of lift and drag coefficient for present study of Re=160 flow over stationary cylinder with published results. CL CD Present (IBM) (+/-) (+/-) 0.09 Present (VOF) (+/-) (+/-) 0.10 Su et al Lai and Peskin Kim et al Silva et al Xu et al Re=500 flow over stationary NACA0012 airfoil As a final test of the new IBM approach for stationary body problems, we consider a Re=500 flow over a NACA 0012 airfoil. The airfoil chord length is 0.05 and the domain is a unit box with the airfoil quarterchord at the origin. We employ the same boundary conditions as in the previous cylinder cases. To ensure 22

23 grid independent solutions, we utilize 4 different maximum refinement levels, L MAX = (11, 12, 13, 14), for which the Lagrangian marker point resolution is 50, 100, 200, and 400, respectively. The convergence history of drag coefficient and the pressure coefficient are plotted in Figure 28. The computed pressure coefficient with the present IBM formulation agrees reasonably well with the IBM results of Ping et al., 52 as well as recent experimental results. 52 Figure 28. Convergence history of drag coefficient (left) and pressure coefficient (right) for simulation of Re=500 flow over stationary NACA 0012 airfoil using IBM solver. Displayed in Figure 29 are 20 even contours of velocity between 0 and 1.1, as well as 20 even contours of pressure between and 0.55 for the level 14 case. Both a zoomed-in and a zoomed-out view of the final computational grid, which contains 58,821 cells, are shown in Figure 30. Figure 29. Velocity (left), and pressure (right) contours for simulation of Re=500 flow over stationary NACA 0012 airfoil using IBM. Figure 30. Computational grid for simulation of Re=500 flow over stationary NACA 0012 airfoil using IBM; (left) zoomed-out view; (right) magnified view displaying grid near surface. 23

24 The velocity vector plot provided in Figure 31 gives a clear picture of the boundary layer profile and small region of separated flow near the trailing edge of the airfoil. Figure 31. Velocity contours and vectors for simulation of Re=500 flow over stationary NACA 0012 airfoil using IBM. B. Moving Body Problems To demonstrate the moving body capabilities of the new AMR-IBM capability, we now employ the Re=40 cylinder and Re=500 NACA0012 airfoil cases described above. In this scenario, the bodies move through a flow that is at rest. For these and subsequent cases, the matrix in Eq. (13) must be reconstructed at each and every time step since the surfaces are moving through the grid. This only adds a small expense, since less than 1% of the total simulation cost comes from this operation. The cases considered here employ a domain that is 8 units long and 1 unit high, with an outflow boundary condition similar to all previous cases, and with a symmetry boundary condition on the top, bottom, and inlet. The bodies will be taken as moving from right-to-left in a zero freestream. Thus, the velocity contours will only compare with the stationary cases if plotted in a non-inertial body-fixed frame, rather than the inertial frame. Re=40 flow over moving cylinder Vorticity and velocity contours for the Re=40 moving cylinder at three different time instances are presented in Figure 32. The velocity field here looks quite different from that shown in Figure 24, due to the fact that the body is moving and the vector field presented here is in the inertial frame. The vorticity contours agree very well with the static case. Figure 32. Vorticity contours (left) and velocity contours/vectors (right) at three different time instances for simulation of Re=40 flow over a moving cylinder. 24

25 Computational grids for this case are shown at three different time instances in Figure 33. The adaptive mesh refinement is successfully carrying out the geometric refinement based on the distance to the moving surface, as well as the vorticity based refinement to capture the vorticity downstream of the cylinder. The computed drag coefficient for this case is 1.54, which is identical to that obtained for the stationary cylinder at Re=40. This is very encouraging, and gives confidence in the present AMR-IBM implementation for moving body problems. Figure 33. Computational grids at three different time instances for simulation of Re=40 flow over a moving cylinder. Re=500 flow over moving NACA0012 airfoil We now consider the Re=500 flow over a NACA0012 airfoil translating through a stationary fluid. Pressure and velocity contours are displayed at 4 different time instances in Figure even contours of pressure between and 0.51, and 20 even contours of velocity between 0 and 1 are displayed. The immersed airfoil surface as well as the flow quantities are adequately preserved as the airfoil translates through the fluid. Close-up views of the pressure, velocity, and grid at the final step are shown in Figure 35, and 4 time instances of the computational grid are displayed in Figure 36. The computed drag coefficient for this case is 1.81, which agrees very well with the stationary case in which the computed drag is The large wake observed in the computational grids is indicative of the fidelity of the method, as there is still finite vorticity being preserved more than 10 chord lengths down steam of the airfoil. If such resolution of the vortical structures is not needed, pressure-based adaptation criteria could be readily deployed. While preliminary, results from the described moving body simulations serve as a proof-of-concept and give us confidence that the new AMR-IBM methodology is functioning properly, and most of the necessary infrastructure is in place for simulations of these types of problems. All of these simulations were carried out using a time step computed from a CFL number of 0.3, which is only possible because of the improved stability properties of the implicit direct force calculation via Eq. (13) as opposed to an explicit approach. C. Flexible Deforming Body Problems Re=4,630 flow over zero-thickness flexible flapping wing As a final test of the new AMR-IBM capability, we revisit the flexible flapping wing case introduced in Section 1. The main difference here is that with the IBM approach, we can actually simulate this as a body of zero-thickness, whereas the VOF approach requires bodies of finite thickness. 25

26 Figure 34. Pressure (left) and velocity (right) at four different time instances for Re=500 flow over a NACA 0012 airfoil translating through a stationary fluid. Figure 35. Close-up of velocity (left), pressure (middle) and grid (right) at final step for Re=500 flow over a NACA 0012 airfoil translating through a stationary fluid. Figure 36. Computational grids at four different time instances for Re=500 flow over a NACA 0012 airfoil translating through a stationary fluid. 26

27 In fact, many different approaches are unable to handle zero-thickness bodies. In particular, ghost cell IBM approaches, 53 VOF methods, 17 and many body-fitted grid methods, to name a few. This is one of the great strengths of the IBM/IIM technology. The boundary conditions employed here are the same as those used for the above airfoil and cylinder cases. The geometry is governed by the following equation [ h + ε ( x + 1) ] exp{ iω[ t ( x + 1) / c] } [ γ ( 1) ] y( x, z, t) = z{ x + }. 0 ξ (14) in which both chordwise and spanwise flexibility can be prescribed. This simple equation has been shown to accurately simulate the flapping flight of many different birds and insects. 28,29 The Reynolds number used for this case is Re=4,630, and the simulation is carried out for 4 complete flapping periods, which is sufficient to observe the periodicity of the flow in terms of the lift and drag coefficients. Figure 37 and Figure 38 show the vorticity contours and velocity vectors, respectively, at six different time instances throughout the simulation. The well known leading-edge vortices are clearly observed, and they subsequently convect downstream and have strong interactions with the trailing edge vortices. This is a highly vortex-dominated flow in which the behavior and interaction of the vortices plays a key role in understanding the underlying flow physics. In fact, recent works by Shyy et al. 54 have shown that tip vortices, which are typically seen as phenomena that decrease lift and induce drag, can actually increase lift for low aspect-ratio flapping wings by creating a low-pressure region near the wing tip and by anchoring the leading-edge vortex to delay or even prevent it from shedding. Learning how to manipulate such vortical flows is the key to controlling flow separation for flexible flapping wing vehicles. The computational grids colored by vorticity are shown, along with the deforming wing surface, at 4 different time instances in Figure 39. It is evident that vorticity-based adaptive mesh refinement strategy employed here is very effective at resolving the vortical structures in an efficient manner. The vortices are still preserved several chord lengths downstream of the wing with minimal dissipation observed. To illustrate that the flow has reached a fully developed periodic state, we present the time history of the lift and drag coefficients, as well as the lift versus drag in Figure 40. It is immediately apparent that while there is a small net lift, there is a rather significant net thrust being generated (negative drag) by this configuration. Figure 37. Vorticity contours at six different time instances for simulation of Re=4,630 flow over a zero-thickness flexible flapping wing. 27

28 Figure 38. Velocity contours and vectors at six different time instances for simulation of Re=4,630 flow over a zero-thickness flexible flapping wing. Figure 39. Computational grids colored by vorticity at four different time instances for simulation of Re=4,630 flow over a zero-thickness flexible flapping wing. 28