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The immersed interface method for the Navier–Stokes equations with singular forces
 J. Comput. Phys
"... Peskin’s Immersed Boundary Method has been widely used for simulating many fluid mechanics and biology problems. One of the essential components of the method is the usage of certain discrete delta functions to deal with singular forces along one or several interfaces in the fluid domain. However, t ..."
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Cited by 83 (5 self)
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Peskin’s Immersed Boundary Method has been widely used for simulating many fluid mechanics and biology problems. One of the essential components of the method is the usage of certain discrete delta functions to deal with singular forces along one or several interfaces in the fluid domain. However, the Immersed Boundary Method is known to be firstorder accurate and usually smears out the solutions. In this paper, we propose an immersed interface method for the incompressible Navier–Stokes equations with singular forces along one or several interfaces in the solution domain. The new method is based on a secondorder projection method with modifications only at grid points near or on the interface. From the derivation of the new method, we expect fully secondorder accuracy for the velocity and nearly secondorder accuracy for the pressure in the maximum norm including those grid points near or on the interface. This has been confirmed in our numerical experiments. Furthermore, the computed solutions are sharp across the interface. Nontrivial numerical results are provided and compared with the Immersed Boundary Method. Meanwhile, a new version of the Immersed Boundary Method using the level set representation of the interface is also proposed in this paper. c ○ 2001 Academic Press Key Words: Navier–Stokes equations; interface; discontinuous and nonsmooth solution; immersed interface method; immersed boundary method; projection method; level set method. 1.
Simulation of a flapping flexible filament in a flowing soap film by the immersed boundary method
 J Comput Phys
"... This paper reports the computer simulation of a flapping flexible filament in a flowing soap film using the immersed boundary method. Our mathematical formulation includes filament mass and elasticity, gravity, air resistance, and the two wires that bound the flowing soap film. The incompressible v ..."
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Cited by 73 (9 self)
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This paper reports the computer simulation of a flapping flexible filament in a flowing soap film using the immersed boundary method. Our mathematical formulation includes filament mass and elasticity, gravity, air resistance, and the two wires that bound the flowing soap film. The incompressible viscous Navier–Stokes equations, which are used to describe the motion of the soap film and filament in our formulation, are discretized on a fixed uniform Eulerian lattice while the filament equations are discretized on a moving Lagrangian array of points which do not necessarily coincide with the fixed Eulerian mesh points of the fluid computation. The interaction between the filament and the soap film is handled by a smoothed approximation to the Dirac delta function. This delta function approximation is used not only to interpolate the fluid velocity and to apply force to the fluid (as is commonly done in immersed boundary computations), but also to handle the mass of the filament, which is represented in our calculation as delta function layer of fluid mass density supported along the immersed filament. Because of this nonuniform density, we need to use a multigrid method for solving the discretized fluid equations.
The immersed boundary method: a projection approach.
 J. Comput. Phys.,
, 2007
"... Abstract A new formulation of the immersed boundary method with a structure algebraically identical to the traditional fractional step method is presented for incompressible flow over bodies with prescribed surface motion. Like previous methods, a boundary force is applied at the immersed surface t ..."
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Cited by 59 (12 self)
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Abstract A new formulation of the immersed boundary method with a structure algebraically identical to the traditional fractional step method is presented for incompressible flow over bodies with prescribed surface motion. Like previous methods, a boundary force is applied at the immersed surface to satisfy the noslip constraint. This extra constraint can be added to the incompressible NavierStokes equations by introducing regularization and interpolation operators. The current method gives prominence to the role of the boundary force acting as a Lagrange multiplier to satisfy the noslip condition. This role is analogous to the effect of pressure on the momentum equation to satisfy the divergencefree constraint. The current immersed boundary method removes slip and nondivergencefree components of the velocity field through a projection. The boundary force is determined implicitly without any constitutive relations allowing the present formulation to use larger CFL numbers compared to some past methods. Symmetry and positivedefiniteness of the system are preserved such that the conjugate gradient method can be used to solve for the flow field. Examples show that the current formulation achieves secondorder temporal accuracy and better than firstorder spatial accuracy in L 2 norms for oneand twodimensional test problems. Results from twodimensional simulations of flows over stationary and moving cylinders are in good agreement with those from previous experimental and numerical studies.
An adaptive, formally second order accurate version of the immersed boundary method
, 2006
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An immersed interface method for viscous incompressible flows involving rigid and flexible boundaries
 J. Comp. Phys
, 2006
"... We present an immersed interface method for the incompressible NavierStokes equations capable of handling rigid immersed boundaries. The immersed boundary is represented by a set of Lagrangian control points. In order to guarantee that the noslip condition on the boundary is satisfied, singular fo ..."
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Cited by 37 (3 self)
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We present an immersed interface method for the incompressible NavierStokes equations capable of handling rigid immersed boundaries. The immersed boundary is represented by a set of Lagrangian control points. In order to guarantee that the noslip condition on the boundary is satisfied, singular forces are applied on the fluid. The forces are related to the jumps in pressure and the jumps in the derivatives of both pressure and velocity, and are interpolated using cubic splines. The strength of the singular forces is determined by solving a small system of equations iteratively at each time step. The NavierStokes equations are discretized on a staggered Cartesian grid by a second order accurate projection method for pressure and velocity. Keywords: Immersed interface method, NavierStokes equations, Cartesian grid method, finite difference, fast Poisson solvers, irregular domains.
When vortices stick: an aerodynamic transition in tiny insects
 J. Exp. Biol
"... We have used computational fluid dynamics to study changes in lift generation and vortex dynamics for Reynolds numbers (Re) between 8 and 128. The immersed boundary method was used to model a twodimensional wing through one stroke cycle. We calculated lift and drag coefficients as a function of tim ..."
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Cited by 27 (4 self)
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We have used computational fluid dynamics to study changes in lift generation and vortex dynamics for Reynolds numbers (Re) between 8 and 128. The immersed boundary method was used to model a twodimensional wing through one stroke cycle. We calculated lift and drag coefficients as a function of time and related changes in lift to the shedding or attachment of the leading and trailing edge vortices. We find that the fluid dynamics around the wing fall into two distinct patterns. For Re�64, leading and trailing edge vortices are alternately shed behind the wing, forming the von Karman vortex street. For Re�32, the leading and trailing edge vortices remain attached to the Summary wing during each ‘half stroke’. In threedimensional studies, large lift forces are produced by ‘vortical asymmetry ’ when the leading edge vortex remains attached to the wing for the duration of each half stroke and the trailing edge vortex is shed. Our twodimensional study suggests that this asymmetry is lost for Re below some critical value (between 32 and 64), resulting in lower lift forces. We suggest that this transition in fluid dynamics is significant for lift generation in tiny insects. Key words: insect flight, Reynolds number, aerodynamics, computational fluid dynamics.
Penalty immersed boundary method for an elastic boundary with mass
 Physics of Fluids
"... The immersed boundary (IB) method has been widely applied to problems involving a moving elastic boundary that is immersed in fluid and interacting with it. But most of the previous applications of the IB method have involved a massless elastic boundary and used efficient Fourier transform methods f ..."
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Cited by 22 (3 self)
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The immersed boundary (IB) method has been widely applied to problems involving a moving elastic boundary that is immersed in fluid and interacting with it. But most of the previous applications of the IB method have involved a massless elastic boundary and used efficient Fourier transform methods for the numerical solutions. Extending the method to cover the case of a massive boundary has required spreading the boundary mass out onto the fluid grid and then solving the NavierStokes equations with a variable mass density. The variable mass density of this previous approach makes Fourier transform methods inapplicable, and requires a multigrid solver. Here we propose a new and simple way to give mass to the elastic boundary and show that the new method can be applied to many problems for which the boundary mass is important. The new method does not spread mass to the fluid grid, retains the use of Fourier transform methodology, and is easy to implement in the context of an existing IB method code for the massless case.