scholarly journals Immersed Boundary Method for Simulating Interfacial Problems

Mathematics ◽  
2020 ◽  
Vol 8 (11) ◽  
pp. 1982
Author(s):  
Wanho Lee ◽  
Seunggyu Lee
Keyword(s):  
Delta Function ◽  
Navier Stokes ◽  
Immersed Boundary ◽  
Navier Stokes Equation ◽  
Boundary Problems ◽  
Flow Around A Cylinder ◽  
Dirac Delta ◽  
Lagrangian Variables ◽  
Delta Functions ◽  
Ib Method

We review the immersed boundary (IB) method in order to investigate the fluid-structure interaction problems governed by the Navier–Stokes equation. The configuration is described by the Lagrangian variables, and the velocity and pressure of the fluid are defined in Cartesian coordinates. The interaction between two different coordinates is involved in a discrete Dirac-delta function. We describe the IB method and its numerical implementation. Standard numerical simulations are performed in order to show the effect of the parameters and discrete Dirac-delta functions. Simulations of flow around a cylinder and movement of Caenorhabditis elegans are introduced as rigid and flexible boundary problems, respectively. Furthermore, we provide the MATLAB codes for our simulation.

scholarly journals The immersed boundary method

Acta Numerica ◽  
2002 ◽  
Vol 11 ◽  
pp. 479-517 ◽  
Author(s):  
Charles S. Peskin
Keyword(s):  
Biological Fluid ◽  
Mathematical Structure ◽  
Delta Function ◽  
Dirac Delta Function ◽  
Future Research ◽  
Immersed Boundary ◽  
Dirac Delta ◽  
Least Action ◽  
Lagrangian Variables ◽  
Ib Method

This paper is concerned with the mathematical structure of the immersed boundary (IB) method, which is intended for the computer simulation of fluid–structure interaction, especially in biological fluid dynamics. The IB formulation of such problems, derived here from the principle of least action, involves both Eulerian and Lagrangian variables, linked by the Dirac delta function. Spatial discretization of the IB equations is based on a fixed Cartesian mesh for the Eulerian variables, and a moving curvilinear mesh for the Lagrangian variables. The two types of variables are linked by interaction equations that involve a smoothed approximation to the Dirac delta function. Eulerian/Lagrangian identities govern the transfer of data from one mesh to the other. Temporal discretization is by a second-order Runge–Kutta method. Current and future research directions are pointed out, and applications of the IB method are briefly discussed.

High-Order Immersed-Boundary Simulation and Error Analysis for Flow Around a Porous Structure

2017 ◽  
Author(s):  
Meihua Zhang ◽  
Z. Charlie Zheng
Keyword(s):  
Porous Structure ◽  
Three Dimensional ◽  
Time Discretization ◽  
Navier Stokes ◽  
Immersed Boundary ◽  
Dimensional Sphere ◽  
Navier Stokes Equation ◽  
Flow Resistivity ◽  
Fifth Order ◽  
Ib Method

In this study, a two-dimensional cylinder or a three-dimensional sphere is used as an example for a porous structure in the flow field. Immersed-boundary (IB) methods have been used to simulate flow around a cylinder/sphere, which is a typical problem to verify the effectiveness and accuracy of the methods. The flow was previously simulated by modeling the solid obstacles as a porous medium with flow resistivity. The current study makes an extension of the previous IB method. The fifth order WENO-Z scheme and third order Runge-Kutta method are used for space and time discretization, respectively. The Navier–Stokes equation is used for the fluids region and a ZK type of source term is applied in place of the forcing term for the porous domain. These two equations are solved simultaneously. Thus, there is no need to specify the interface conditions directly. In addition, the errors are estimated in terms of the flow resistivity that is zero for fluid, finite for porous, and infinite for non-porous solid. The aim of this paper is to establish estimates of the error induced by such an IB method. Numerical tests are performed to confirm the improvement of this method.

Computation of Flow Through a Three-Dimensional Periodic Array of Porous Structures by a Parallel Immersed-Boundary Method

2014 ◽  
Vol 136 (4) ◽  
Author(s):  
Zhenglun Alan Wei ◽  
Zhongquan Charlie Zheng ◽  
Xiaofan Yang
Keyword(s):  
Porous Media ◽  
Parallel Implementation ◽  
Flow Behavior ◽  
Three Dimensional ◽  
Flow Simulation ◽  
Periodic Array ◽  
Immersed Boundary ◽  
Navier Stokes Equation ◽  
Flow Through ◽  
Ib Method

A parallel implementation of an immersed-boundary (IB) method is presented for low Reynolds number flow simulations in a representative elementary volume (REV) of porous media that are composed of a periodic array of regularly arranged structures. The material of the structure in the REV can be solid (impermeable) or microporous (permeable). Flows both outside and inside the microporous media are computed simultaneously by using an IB method to solve a combination of the Navier–Stokes equation (outside the microporous medium) and the Zwikker–Kosten equation (inside the microporous medium). The numerical simulation is firstly validated using flow through the REVs of impermeable structures, including square rods, circular rods, cubes, and spheres. The resultant pressure gradient over the REVs is compared with analytical solutions of the Ergun equation or Darcy–Forchheimer law. The good agreements demonstrate the validity of the numerical method to simulate the macroscopic flow behavior in porous media. In addition, with the assistance of a scientific parallel computational library, PETSc, good parallel performances are achieved. Finally, the IB method is extended to simulate species transport by coupling with the REV flow simulation. The species sorption behaviors in an REV with impermeable/solid and permeable/microporous materials are then studied.

Cylinder rolling on a wall at low Reynolds numbers

2011 ◽  
Vol 685 ◽  
pp. 461-494 ◽  
Author(s):  
Alain Merlen ◽  
Christophe Frankiewicz
Keyword(s):  
Three Dimensional ◽  
Reynolds Numbers ◽  
Navier Stokes ◽  
Navier Stokes Equation ◽  
Flow Around A Cylinder ◽  
Low Reynolds Numbers ◽  
Initial Location ◽  
Upstream Pressure ◽  
Small Reynolds Numbers ◽  
Onset Of Cavitation

AbstractThe flow around a cylinder rolling or sliding on a wall was investigated analytically and numerically for small Reynolds numbers, where the flow is known to be two-dimensional and steady. Both prograde and retrograde rotation were analytically solved, in the Stokes regime, giving the values of forces and torque and a complete description of the flow. However, solving Navier–Stokes equation, a rotation of the cylinder near the wall necessarily induces a cavitation bubble in the nip if the fluid is a liquid, or compressible effects, if it is a gas. Therefore, an infinite lift force is generated, disconnecting the cylinder from the wall. The flow inside this interstice was then solved under the lubrication assumptions and fully described for a completely flooded interstice. Numerical results extend the analysis to higher Reynolds number. Finally, the effect of the upstream pressure on the onset of cavitation is studied, giving the initial location of the phenomenon and the relation between the upstream pressure and the flow rate in the interstice. It is shown that the flow in the interstice must become three-dimensional when cavitation takes place.

An Aid to Learn Computational Fluid Dynamics: Immersed-Boundary-Based Simulation of 2D Flow

Volume 1 ◽  
2004 ◽  
Author(s):  
Sungsu Lee ◽  
Kyung-Soo Yang ◽  
Jong-Yeon Hwang
Keyword(s):  
Complex Geometry ◽  
Mass Conservation ◽  
Computational Method ◽  
Staggered Grid ◽  
Navier Stokes ◽  
Immersed Boundary ◽  
Step Method ◽  
Fractional Step Method ◽  
Navier Stokes Equation ◽  
Finite Volume Approach

Development of geometry-independent computational method and educational codes for simulation of 2D flows around objects of complex geometry is presented. Referred as immersed boundary method, it introduces virtual forcing to governing equations to represent the effect of physical boundaries. The present method is based on a finite-volume approach on a staggered grid with a fractional-step method to solve Navier-Stokes equation and continuity equation. Both momentum and mass forcings are introduced on and inside the object to satisfy no-slip condition and mass conservation. Since Cartesian grid lines in general do not coincide with the immersed boundaries, several interpolation schemes are employed. Several examples are simulated using the method presented in this study and the results agree well with other results. Both user-friendly preprocessor with GUI and FORTRAN-based solver are open to the public for educational purposes.

The Flow of A Falling Ellipse: Numerical Method and Classification

2015 ◽  
Vol 31 (6) ◽  
pp. 771-782 ◽  
Author(s):  
R.-J. Wu ◽  
S.-Y. Lin
Keyword(s):  
Reynolds Number ◽  
Stokes Equations ◽  
Correction Method ◽  
Navier Stokes ◽  
Immersed Boundary ◽  
Two Dimensional ◽  
Inertia Moment ◽  
Navier Stokes Equations ◽  
Direct Forcing ◽  
Ib Method

AbstractA modified direct-forcing immersed-boundary (IB) pressure correction method is developed to simulate the flows of a falling ellipse. The pressure correct method is used to solve the solutions of the two dimensional Navier-Stokes equations and a direct-forcing IB method is used to handle the interaction between the flow and falling ellipse. For a fixed aspect ratio of an ellipse, the types of the behavior of the falling ellipse can be classified as three pure motions: Steady falling, fluttering, tumbling, and two transition motions: Chaos, transition between steady falling and fluttering. Based on two dimensionless parameters, Reynolds number and the dimensionless moment of inertia, a Reynolds number-inertia moment phase diagram is established. The behaviors and characters of five falling regimes are described in detailed.

scholarly journals Ordinary differential equations with delta function terms

2012 ◽  
Vol 91 (105) ◽  
pp. 125-135 ◽  
Author(s):  
Marko Nedeljkov ◽  
Michael Oberguggenberger
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Delta Function ◽  
Limiting Distribution ◽  
Nonlinear Ordinary Differential Equations ◽  
Smooth Solutions ◽  
Dirac Delta ◽  
Delta Functions ◽  
Derivatives Of

This article is devoted to nonlinear ordinary differential equations with additive or multiplicative terms consisting of Dirac delta functions or derivatives thereof. Regularizing the delta function terms produces a family of smooth solutions. Conditions on the nonlinear terms, relating to the order of the derivatives of the delta function part, are established so that the regularized solutions converge to a limiting distribution.

scholarly journals Direct Numerical Simulation of Gas–Particle Flows with Particle–Wall Collisions Using the Immersed Boundary Method

2018 ◽  
Vol 8 (12) ◽  
pp. 2387 ◽  
Author(s):  
Yusuke Mizuno ◽  
Shun Takahashi ◽  
Kota Fukuda ◽  
Shigeru Obayashi
Keyword(s):  
Immersed Boundary Method ◽  
Three Dimensional ◽  
Flow Simulation ◽  
Navier Stokes ◽  
Immersed Boundary ◽  
Navier Stokes Equation ◽  
Level Set Function ◽  
Boundary Method ◽  
The One ◽  
Energy And Momentum

We investigated particulate flows by coupling simulations of the three-dimensional incompressible Navier–Stokes equation with the immersed boundary method (IBM). The results obtained from the two-way coupled simulation were compared with those of the one-way simulation, which is generally applied for clarifying the particle kinematics in industry. In the present flow simulation, the IBM was solved using a ghost–cell approach and the particles and walls were defined by a level set function. Using proposed algorithms, particle–particle and particle–wall collisions were implemented simply; the subsequent coupling simulations were conducted stably. Additionally, the wake structures of the moving, colliding and rebounding particles were comprehensively compared with previous numerical and experimental results. In simulations of 50, 100, 200 and 500 particles, particle–wall collisions were more frequent in the one–way scheme than in the two-way scheme. This difference was linked to differences in losses in energy and momentum.

Dirac Delta Functions Via Nonstandard Analysis

1975 ◽  
Vol 18 (5) ◽  
pp. 759-762 ◽  
Author(s):  
A. H. Lightstone ◽  
Kam Wong
Keyword(s):  
Continuous Function ◽  
Real Number ◽  
Nonstandard Analysis ◽  
Short Interval ◽  
Delta Function ◽  
Number System ◽  
Dirac Delta Function ◽  
Dirac Delta ◽  
Delta Functions ◽  
Real Number System

We recall that a Dirac delta function δ(x) in the real number system is the idealization of a function that vanishes outside a "short" interval and satisfies It is conceived as a function δ for which δ(0)=+ ∞, δ(t)=0 if t≠0, and This function should possess the "sifting property" for any continuous function f. Even though certain sequences of functions are used, via a limit operation, to approximate a Dirac delta function (for details, see [3] and [4]), no function in has these properties.

A Very Simple Immersed Boundary Method Applied to Three-Dimensional Incompressible Navier-Stokes Solvers Using Staggered Grid

2007 ◽  
Author(s):  
Satoru Yamamoto ◽  
Koichiro Mizutani
Keyword(s):  
Immersed Boundary Method ◽  
Computational Algorithm ◽  
Three Dimensional ◽  
Staggered Grid ◽  
Navier Stokes ◽  
Immersed Boundary ◽  
Surface Mesh ◽  
Flow Problems ◽  
Smac Method ◽  
Ib Method

A very simple computational algorithm for solving flows over complex geometries is presented. This algorithm is based on a kind of the Immersed Boundary (IB) Methods. The complex 3D geometries are generated by the 3DCG software as a surface mesh of triangle polygons. A 3D Cartesian grid is used for the flow field. Only the points stuck with grid lines on triangle polygons are detected and flow velocities are modified using the points with the IB method. In this paper, this approach is applied to an incompressible Navier-Stokes solver based on the Simplified Marker and Cell (SMAC) method. Some practical flow problems assuming low Reynolds number are calculated and the results are compared with the experimental and the numerical results.

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