A VERSATILE SHARP INTERFACE IMMERSED BOUNDARY METHOD FOR INCOMPRESSIBLE FLOWS WITH COMPLEX BOUNDARIES

Abstract

A sharp interface immersed boundary method for simulating incompressible viscous flow past three-dimensional immersed bodies is described. The method employs a multi-dimensional ghost-cell methodology to satisfy the boundary conditions on the immersed boundary and the method is designed to handle highly complex three-dimensional, stationary, moving and/or deforming bodies. The complex immersed surfaces are represented by grids consisting of unstructured triangular elements; while the flow is computed on non-uniform Cartesian grids. The paper describes the salient features of the methodology with special emphasis on the immersed boundary treatment for stationary and moving boundaries. Simulations of a number of canonical two- and three-dimensional flows are used to verify the accuracy and fidelity of the solver over a range of Reynolds numbers. Flow past suddenly accelerated bodies are used to validate the solver for moving boundary problems. Finally two cases inspired from biology with highly complex three-dimensional bodies are simulated in order to demonstrate the versatility of the method.

Fig. 1

Schematic describing the naming convention and location of velocity components employed in the spatial discretization of the governing equations.

Fig. 2

Example of the type of surface mesh with triangular elements used to represent all immersed bodies in the current solver. This particular body is based on a CT scan of a harbor porpoise (Phocoena phocoena).

Fig. 3

Representative example showing the harbor porpoise of Fig. 2 immersed in a non-uniform Cartesian grid.

Fig. 4

Schematic showing the procedure for computing whether a node is inside or outside the body. The cube represents a fluid cell and its node for which this determination is to be made. Vector $\overrightarrow{p}$ is the position vector between this node and the surface triangle closest to the node and $\widehat{n}$ is the outward pointing surface normal of this triangular element.

Fig. 5

2D schematic describing ghost-cell methodology used in the current solver. Schematic depicts an immersed boundary cutting through a Cartesian grid and identifies three particular ghost-cells (GC) that form the basis for discussion in this section. BI and IP denote body-intercept and image-point respectively.

Fig. 6

2D schematic showing two degenerate situations that can be encountered in the identification of the body-intercept point for a ghost-cell. (a) Case where there are two possible body-intercept points and (b) case where there is no body-intercept point detected on the body.

Fig. 7

Schematic showing the formation of fresh-cells due to boundary motion and the interpolation stencil (in grey) for one representative fresh-cell.

Fig. 8

(a) Contours of u₁ (line contours) and u₂ (greyscale contours) for numerical solution on the 630 × 630 grid. (b) Distribution of error in u₁ component of velocity on the 126 × 126 grid.

Fig. 9

L₁, L₂ and L_∞ norms of the error for the streamwise velocity u₁ and transverse velocity u₂ components versus the computational grid size.

Fig. 10

Non-uniform grid employed in the vicinity of the circular cylinder for the Re_d = 1000 simulations.

Fig. 11

Computed spanwise vorticity contour plots for (a) Re_d = 300 and (b) 1000 at one time-instant.

Fig. 12

Computed temporal variation of drag and lift coefficients for the (a) Re_d = 300 and (b) 1000 cases.

Fig. 13

Comparison of computed (a) vortex shedding Strouhal number (St) and (b) computed base suction coefficient (−C_pb with established computational and experimental results.

Fig. 14

Contour plot of spanwise vorticity at one time-instant for the NACA 0008 airfoil at Re_c = 6000 and α = 4°.

Fig. 15

Temporal variation of force coefficients for NACA 0008 airfoil at α = 4° for Re_c = 2000 and 6000 (a) Drag coefficient (b) Lift coefficient.

Fig. 16

(a) Computed streamline pattern on one plane of symmetry for Re_d = 100 sphere case. (b) Isosurface of enstrophy at one time-instance for Re_d = 350 sphere case.

Fig. 17

(a) Temporal variation of drag and side force coefficients on a sphere in a uniform flow for Re_d = 350. (b) Comparison of computed mean drag coefficient with experimental and numerical data.

Fig. 18

Computed spanwise vorticity contours for a suddenly started normal flat-plate at four stages in the start-up process. Upper and lower halves of each figure correspond to Re_h = 126 and 1000 respectively. (a) tU_o/h = 0.5 (b) 1 (c) 2 and (d) 3.

Fig. 19

Time evolution of computed bubble length behind flat plate at Re_h = 126 and 1000 compared to established experimental and computational results.

Fig. 20

Computed spanwise vorticity contours for a suddenly started cylinder at four stages in the start-up process. Upper and lower halves of each figure correspond to Re_h = 1000 and 550 respectively. (a) tU_o/d = 0.5 (b) 1 (c) 1.5 and (d) 2.

Fig. 21

Time evolution of computed drag coefficient for suddenly started cylinder at Re_d = 550 and 1000 compared to established experimental and computational results. Also included in the figure is the temporal variation of drag-coefficient for a suddenly started sphere at Re_d = 550.

Fig. 22

Grid employed in the pectoral fin simulations and fin configuration at three stages in its motion.

Fig. 23

Isosurfaces of ∧_i and corresponding streamlines at three stages in the pectoral fin stroke of the bluegill sunfish. (a) t × f = 1/3 (b) t × f = 2/3 (c) t × f = 1. Body of the sunfish is shown for reference only and not included in the simulations.

Fig. 24

(a) Surface mesh used to define the geometry of the dragonfly body and wings. (b) Two-dimensional view of the dragonfly model immersed in the fluid grid.

Fig. 25

Isosurfaces of ∧_i at three stages in the flapping cycle of a modeled dragonfly. (a) t × f = 0.25 (b) t × f = 0.75 (c) t × f = 1.