Sharp-Interface Cartesian Method for Simulating Flow past 3D Flexible Bodies

Transcription

1 Sharp-Interface Cartesian Method for Simulating Flow past 3D Flexible Bodies ANVAR GILMANOV & FOTIS SOTIROPOULOS School of Civil and Environmental Engineering Georgia Institute of Technology Atlanta, GA USA Abstract: - A second-order accurate, finite-difference numerical method is developed for solving the 3D, unsteady, incompressible Navier -Stokes equations in Cartesian domains containing geometrically complex, flexible, immersed boundaries. Unstructured, triangular meshes are employed to discretize complex immersed boundaries. The nodes of the surface mesh constitute a set of Lagrangian control points used to track the motion of the flexible body. The influence of the body on the flow is accounted for by applying boundary conditions at Cartesian grid nodes located in the exterior but in the immediate vicinity of the body by reconstructing the solution along the local normal to the body surface. The capabilities of the method are demonstrated by applying it to simulate flow past a swimming and maneuvering fish-like body.. Key-Words: - Cartesian methods, immersed boundaries, fluid-structure interaction, aquatic locomotion. 1 Introduction Many flows of technological and/or biological significance take place in multi-connected domains with complex, flexible, immersed boundaries. Typical examples range from flows in natural rivers with flexible vegetation, to flows in the human cardiovascular system, and flows past swimming and flying animals and insects. In recent years, "non-boundary conforming" numerical methods are attracting attention in simulations of such problems due to their increased versatility as compared to boundary-fitted techniques based on the so called Arbitrary Lagrangian Eulerian (ALE) [1]. ALE methods are better suited for carrying out high Reynolds number simulations but, due to the need for the mesh to conform to the body at all times, they are inherently limited to problems with moderate body deformations. This limitation can be mitigated by solving the governing equations on a fixed Cartesian grid and accounting for the effect of a stationary or moving body, which no longer coincides with a grid surface, via proper treatment of the solution variables at grid cells in the vicinity of the boundary. Depending on the approach adopted to satisfy the boundary conditions on the body, Cartesian-grid methods are typically classified in three categories: a) immersed boundary methods [2]; b) Cartesian grid methods [3]; and c) hybrid Cartesian grid/immersed boundary methods [4]. In this paper, we develop a sharp interface, hybrid approach for simulating flows past biologically-inspired, flexible bod ies. Our method employs a novel interpolation scheme to reconstruct the solution near the immersed boundary [5], which allows for the relatively straightforward modeling of flows around bodies of arbitrary geometrical complexity. The capabilities of the method are demonstrated by applying it to simulate flow past a realistic fish-like body: 1) swimming in-line via body undulations ; and 2) executing a C-shaped turning maneuver. The simulated flow patterns are shown to be in good qualitative agreement with laboratory observations and well known theoretical results. The paper is organized as follows. First we describe the basic numerical method for solving the governing equations and the method for tracking the motion and accounting for the effect of complex immersed boundaries on a Cartesian mesh. Subsequently we present our simulations for flow past an undulating fish-like body. We show that the method can capture the well known effect of slip velocity ratio (ratio of flow velocity to the phase speed of the undulatory body wave) on the structure of the wake. Then we present a small sample of results from the application of the method to simulate the flow past a maneuvering fish. Finally, we summarize our findings and present conclusions and recommendations for future work.

2 2 The Numerical Method 2.1 The base flow solver We solve the 3D, unsteady, incompressible Navier - Stokes equations in Cartesian coordinates. The governing equations are discretized using second order accurate finite-difference approximations. A novel hybrid staggered/non-staggered grid layout is developed, which simplifies the implementation of boundary conditions near complex, immersed boundaries (see below) while ensuring the accurate satisfaction of the discrete continuity equation (see [6] for a detailed description of this approach). The time der ivative in the momentum equations is discretized using the second-order accurate, threepoint backward finite difference formula. The discrete governing equations are integrated in time using the dual (or pseudo) time stepping artificial compressibility approach (see [6] for details). Boundary conditions at the outflow and side boundaries of the computational domain are specified using a characteristics-based approach. The specification of boundary conditions on immersed boundaries is described in the subsequent section. 2.2 Treatment of immersed boundaries The approach for treating complex, flexible immersed boundaries consists of the following twostep procedure: 1) Lagrangian tracking of the immersed boundary, which requires the precise description of the boundary shape and its relation to the fixed Cartesian grid; and 2) local reconstruction of the solution near the immersed boundary such that the physical boundary conditions are satisfied exactly on the body at every instant in time. The reconstruction algorithm we employ herein has already been presented in [5]. For the sake of completeness, however, we summarize below the entire formulation with more emphasis on algorithmic issues pertaining to the simulation of flows with geometrically complex, flexible, immersed boundaries. In the Cartesian/Immersed boundary formulation proposed in [5] the immersed boundary is treated as a sharp interface. Boundary cond itions are applied at nodes in the immediate vicinity of the immersed boundary by reconstructing the solution along the well-defined normal to the body direction using information from interior nodes and the known boundary conditions on the body. To facilitate the reconstruction of the solution in the vicinity of arbitrarily complex immersed boundaries, the immersed boundary is discretized using an unstructured, triangular mesh with M elements of size similar to the Cartesian grid spacing in the vicinity of the body. Fig. 1. Schematic depicting the reconstruction of the solution at an immersed boundary node b by interpolating along the normal to the surface of the body. Fig. 2. Schematic illustrating the decomposition of a complex body into a set of convex regions. The first step in the implementation of the algorithm is to determine the location and shape of the immersed boundary. Since in this work we assume that the motion of the body is prescribed, the location of the body at every physical time step can be readily determined by integrating the equation of motion for each node of the surface mesh. Having established the shape and location of the interface, the next step is to identify the near-boundary nodes of the Cartesian (such as node b in Fig. 1) where the velocity vector needs to be reconstructed in order to obtain boundary conditions we shall denote such nodes as immersed boundary (IB) nodes. For a convex body i.e. a body that contains all lines connecting any two points on its surface the grid node (i,j,k) will be internal to the body (solid node) if the following condition is satisfied (no summation over repeated indices): r n r r ( ) < 0 m =, M m i,j,k m 1 (1) where r r is the position vector. Nodes that do not satisfy the above conditions for all m will be either fluid or IB nodes. To determine the IB nodes, we examine the 6 grid nodes (i±1,j±1,k±1) surrounding each solid node (i,j,k). Each such node that is not a

3 solid node is classified as an IB node. For bodies of more complex concave shape, Eqn. (1) will not in general be satisfied at all internal nodes. Such situation is illustrated in Fig. 2, which depicts a schematic of the flexible, fishlike body we discuss in a subsequent section of this paper. To handle such complex body shapes, we partition the body into a set of geometrically simpler regions (A, B, C, D, E, F, and G) as shown in Fig. 2. Each subregion is a convex, closed subset of the body defined such that adjacent subregions do not overlap but share a common intersection plane, denoted with a dash line in Fig. (2). Given the convex decomposition of the body, Eqn. (1) can now be applied in each convex subregion and the above stated criterion for determining the internal nodes is generalized as follows: a Cartesian grid node will be internal to the body if it satisfies Eqn. (1) for all surface nodes of at least one of the convex subregions that compr ise the body. Since in this work we consider bodies with prescribed motion, the convex decomposition for each instantaneous shape of the body is performed only once at the start of the calculation by inspection of the body shape and stored for subsequent use. Generalizing the method, however, to fully-coupled, fluid/structure interaction problems, will require the implementation of an efficient algorithm for determining the convex decomposition of a general, three-dimensional body. Fig. 3. Surface mesh for the reconstructed mackerel body. After the IB nodes have been determined, boundary conditions are prescribed at these nodes by reconstructing the solution along the line that passes through the specific IB node and is normal to the body. With reference to Fig. 1 and assuming the solution at all interior nodes is known at the previous time step, boundary conditions at the IB node b are constructed by linearly interpolating the flow variables along points a and c, where the line a-c is the normal to the body passing through point b. Point a is located on the body and, thus, the flow variables there can be prescribed from known boundary conditions. Point c on the other hand is an interior node and the solution there can be computed from the previous iteration. With boundary conditions prescribed at all IB nodes the solution in all fluid nodes can be advanced to the next iteration. A more detailed description of the algorithm can be found in [5, 6]. 3 Results and Discussion In this section we report results from the application of the method to simulate the flow past a fishlike, flexible, three-dimensional object. The shape of the three-dimensional body is based on the actual anatomy of a mackerel fish. An actual mackerel was frozen and sliced and its cross-sectional dimensions were carefully measured and used to construct the body geometry shown in Fig. 3. With the exception of the caudal fin (fish tail), all other fins of the fish are not modeled. The body shape is discretized with approximately 1100 triangular elements and is shown in Fig. 3. The governing equations are made nondimensional by scaling lengths with the length of the fish, L, velocities with free-stream velocity of the ambient flow, U, and time with 1/f, where f is characteristic frequency of the body kinematics (see below). Simulations are carried out for two types of body kinematics. The first corresponds to straight, in-line swimming with a prescribed undulatory wave propagating along the fish body. The second case simulates the flow induced by a fish performing a rapid turning maneuver. The results for each case are presented in the subsequent section. 3.1 Undulatory swimming Fishlike swimming motion is prescribed by specifying the lateral displacement of the fish backbone as a function of time. We consider biologic ally inspired kinematics mimicking body and caudal fin locomotion, which is the most frequently encountered swimming mode in fishes [7]. To mimic such motion, we prescribe the lateral displacement of the fish body in terms of a traveling wave of varying amplitude [7] as follows: y( x,t) = a( x) sin( kx ωt ) (2) where a(x) is the wave amplitude that is assumed to vary non-linearly along the fish body, the k=2π/λ is the wave number, corresponding to wave length λ, and ω (=2πf) is the circular frequency of oscillation. The wave amplitude is a quadratic function of x: 2 a( x ) = ao + a x + a (3) 1 2x

4 where the constants a o, a 1, and a 3 are selected to ensure that a(x) becomes maximum at the tail. Slip = 1.1 Slip = 0.9 Slip = 0.6 Fig. 4. Inviscid flow past an undulating mackerel: Effect of slip velocity on wake structure. White arrows indicate the direction of the wake flow. Fig. 5. Principle wake shapes, consisting of vortices (circles) and jets (wavy lines) observed in [8] (from [7]). The Reynolds number is based on the fish length and swimming speed, Re=UL/ν. The Strouhal number is based on the mean lateral excursion of the caudal fin at the trailing edge A = 2amax, and the tailbeat frequency f, St= Af / U. Another important governing parameter is the so-called slip ratio S, defined as the ratio of the ambient flow velocity U to the phase velocity of the undulatory wave (V = ω/k): S = U/V. Laboratory experiments with flapping flat plates [8] and with live actively swimming fish [9] have established the important of the slip velocity in the structure of the wake flow (see Fig. 5). More specifically, it has been shown that body undulations with S>1 produce a wake flow with a classical Karman street and a net drag force on the body. For S near unity, the wake consists of a series of vortices one behind the other with essentially zero axial force on the body. Finally, for S<1 areversed Karman street is produced with a strong thrust jet flow [9]. As a first demonstration of the capabilities of the method we apply it to simulate the flow past the undulating fishlike body for three different slip velocities: S = 1.1, 0.9, and 0.6. Since all available experimental data have been obtained for high Reynolds numbers (typically 10 5 to 10 6 ) we carry out inviscid flow simulations by allowing the flow to slip on the fish body and setting the normal velocity component to the body equal to zero. The simulations have been carried out on a grid, clustered in the vicinity of the body. A snapshot of the calculated instantaneous streamlines and pressure contours around the fish for all three slip ratios and at the same instant in time is shown in Fig. 4. In agreement with the available experiments (see Fig. 5), we also find that for S>1 the wake has a strong drag component and a classical Karman street while a reverse Karman street, thrust wake is produced for S<1. The calculated wake structure for the S=0.9 is remarkably similar to that reported in [9] for S=1, consisting of an array of in-line vortices with no net axial flow in between them. These results are encouraging as they demonstrate that our method can capture, at least in a qualitative sense, important physics of undulatory locomotion. For the inviscid simulations reported above, we were able to obtain grid-insensitive results on relatively coarse meshes (approximately 200,000 nodes were used to obtain the results shown in Fig. 4). For viscous calculations on the other, the wake structure is found to be very sensitive to the mesh resolution near the body. In Figs. 5 and 6 we present results from a set of viscous flow computations (imposing the no-slip condition on the body) for Re=3,000 and St = 0.5. Calculations were carried out on two grids, a coarse grid with a total of nodes and a fine mesh with nodes. The coarse and fine mesh results are compared with each other in terms of instantaneous streamlines and pressure contours at the horizontal plane of symmetry and at the same instant within the period

5 Coarse mesh 3.2 Fish maneuvering In this section we report some results to demonstrate the ability of the method to simulate the flow in a domain with a flexible body that undergoes a rather complex deformation. We simulate the flow induced by the same fishlike body used in the previous section as it executes a C-shaped, turning maneuver. The kinematics of the motion is broadly based on the experim ental observations reported in [7]. The computational domain is discretized with a mesh and viscous flow simulations are carried out for Re = 100. Fine mesh Fig. 5. Viscous flow past an undulating mackerel (Re=3000, St=0.5): Effect of grid refinement on wake structure. White arrows indicate the direction of the wake flow. Coarse mesh Fine mesh Fig. 6. Viscous flow past an undulating mackerel (Re=3000, St=0.5): Effect of grid refinement on vorticity field. in Fig. 5. As seen, on the coarse grid the wake flow is a classical Karman street (drag wake). On the fine mesh, on the other hand, a very pronounced thrust wake with reverse Karman street is produced. To identify the reasons for this dramatic discrepancy, we compare in Fig. 6 a snapshot of the vertical vorticity field on the two grids. Even though the two calculated flowfields are broadly similar, it is clearly evident that on the coarse mesh the thickness of the viscous layer is over-predicted while the strength of the wake vortices is considerably underpredicted. Fig. 7. Viscous flow past a turning mackerel (Re=100): Iso-surface of vorticity magnitude and instantaneous stream-lines. Red arrow marks the direction of fish turning. Blue arrows indicate the direction of the jet flow induced by the vortex rings. An instantaneous snapshot of the calculated flowfield during the turning maneuver is shown in Fig. 7 in terms of streamlines and an iso-surface of the vorticity magnitude. The red arrows indicate the

6 direction of turning while the blue arrows mark the direction of the flow. As the fish turns a vortex ring of elliptical shape is shed in the vicinity of the caudal fin and is advected away from the turning body. The structure of the vortex ring is made clearly visib le by the vorticity iso-surface. The jetlike flow induced by this ring is directed away from the fish and produces the turning thrust force that helps the fish to maneuver. Our calculations further reveal a second vortex ring linked with the first one with the combined flow by the two rings further contributing to the production of turning thrust. These results are in broad qualitative agreement with recent measurements with live fish, which have indeed revealed the shedding of a vortex ring during turning, which induces a jet flow directed away from the fish and, thus, assists in the turning maneuver [10]. 4 Conclusion We have presented a numerical method for simulating unsteady, 3D flows in domains containing complex, flexible immersed boundaries. The method is a hybrid Cartesian/Immersed Boundary approach, which treats immersed boundaries as sharp interfaces using a novel reconstruction algorithm. Application of the model to simulate flow past a swimming and maneuvering fish yield ed encouraging results, which are in broad qualitative agreement with experimental observations with live fish. A more comprehensive validation of the numerical method is currently under way using the experimental data of [11] for flow induced by a robotic fly wing is currently under way. These results will be reported in a future journal publication. Acknowledgments This work was supported by NSF Career grant , a grant from Oak Ridge National Laboratory and DOE, and NIH grant RO1-HL [3] Udaykumar H, Mittal R, Rampunggoon P, Khanna A., A sharp interface Cartesian grid method for simulating flows with complex moving boundaries. Journal of Computational Physics, Vol. 174, 2001, pp [4] Fadlun EA, Verzicco R, Orlandi P, Mohd-Yusof J., Combined immersed-boundary finitedifference methods for 3D complex flow simulations. Journal of Computational Physics. Vol. 161, 2000, pp [5] Gilmanov A, Sotiropoulos F, Balaras E. A General Reconstruction Algorithm for Simulating Flows with Complex 3D Immersed Boundaries on Cartesian Grids. Journal of Computational Physics. Vol. 191(2), 2003, pp [6] Gilmanov, A., and Sotiropoulos, F. A Cartesian Grid Algorithm for Simulating Fluid -Structure Interaction Problems with Application to Fishlike Swimming, submitted, [7] Wolfgang, M. J., Anderson, J. M., Grosenbaugh, M. A., Yue, D. K. P., and Triantafyllou, M. S., Near-Body Flow Dynamics in Swimming Fish, Journal of Experimental Biology, Vol. 202, 1999, pp [8] Hertel, H., Structure, form, movement, Reinhold Publishing Corp., New York, [9] Muller, U. K., Stamhuis, E. J., and Videler, J. J., Riding the Waves: the Role of the Body Wave in Undulatory Fish Swimming, Integrative Comparative Biology, Vol. 42, 2002, pp [10] Sakakibara, J., Nakagawa, M., and Yoshida, M., Stereo-PIV Study of Flow Around a Maneuvering Fish, Experiments in Fluids, Vol. 36, 2002, pp [11] Dickinson, M. H., Lehmann, F. O., and Sane, S. P., Wing Rotation and the Aerodynamic Basis of Insect Flight, Science, Vol. 284, 1999, pp References: [1] Ramamurti R., Sandberg WC, Löhner R, Walker JA, Westneat MW. Fluid dynamics of flapping aquatic flight in the bird wrasse: threedimensional, unsteady computations with fin deformation. Journal of Experimental Biology Vol. 205, 2002, pp [2] Peskin C. The immersed boundary method. Acta Numerica 2002, pp