Immersed boundary method for flow around an arbitrarily moving body (original) (raw)
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International Journal for Numerical Methods in Fluids, 2006
In this paper, a new immersed-boundary method for simulating ows over complex immersed, moving boundaries is presented. The ow is computed on a ÿxed Cartesian mesh and the solid boundaries are allowed to move freely through the mesh. The present method is based on a ÿnite-di erence approach on a staggered mesh together with a fractional-step method. It must be noted that the immersed boundary is generally not coincident with the position of the solution variables on the grid, therefore, an appropriate strategy is needed to construct a relationship between the curved boundary and the grid points nearby. Furthermore, a momentum forcing is added on the body boundaries and also inside the body to satisfy the no-slip boundary condition. The immersed boundary is represented by a series of interfacial markers, and the markers are also used as Lagrangian forcing points. A linear interpolation is then used to scale the Lagrangian forcing from the interfacial markers to the corresponding grid points nearby. This treatment of the immersed-boundary is used to simulate several problems, which have been validated with previous experimental results in the open literature, verifying the accuracy of the present method.
An immersed boundary technique for simulating complex flows with rigid boundary
Computers & Fluids, 2007
A new immersed boundary (IB) technique for the simulation of flow interacting with solid boundary is presented. The present formulation employs a mixture of Eulerian and Lagrangian variables, where the solid boundary is represented by discrete Lagrangian markers embedding in and exerting forces to the Eulerian fluid domain. The interactions between the Lagrangian markers and the fluid variables are linked by a simple discretized delta function. The numerical integration is based on a second-order fractional step method under the staggered grid spatial framework. Based on the direct momentum forcing on the Eulerian grids, a new force formulation on the Lagrangian marker is proposed, which ensures the satisfaction of the no-slip boundary condition on the immersed boundary in the intermediate time step. This forcing procedure involves solving a banded linear system of equations whose unknowns consist of the boundary forces on the Lagrangian markers; thus, the order of the unknowns is one-dimensional lower than the fluid variables. Numerical experiments show that the stability limit is not altered by the proposed force formulation, though the second-order accuracy of the adopted numerical scheme is degraded to 1.5 order. Four different test problems are simulated using the present technique (rotating ring flow, lid-driven cavity and flows over a stationary cylinder and an in-line oscillating cylinder), and the results are compared with previous experimental and numerical results. The numerical evidences show the accuracy and the capability of the proposed method for solving complex geometry flow problems both with stationary and moving boundaries.
An Immersed Boundary Method Based on the Kinematic Relation of the Velocity-Vorticity Formulation
Journal of Mechanics, 2014
An immersed boundary method is proposed for the simulation of the interaction of an incompressible flow with rigid bodies. The method is based on a new interpretation of velocity-vorticity formulation and no longer includes the force term which is an essential issue of common immersed boundary methods. The system is considered in an Eulerian frame and retrieving the vorticity in this formulation enforces continuity at the fluid-solid interface and rigid motion of the solid. The method focuses on the mutual kinematic relations between the velocity and vorticity fields and with retrieving the vorticity field and recalculating the velocities yields the solenoidal velocity field. The method is applied to the two dimensional problems and the results show that the solenoidality is satisfied acceptably. The comparisons with 2D test cases are provided to illustrate the capabilities of the proposed method.
Lecture Notes in Computational Science and Engineering, 2019
The method for numerical simulation of flows over moving solid bodies of complex shapes on unstructured meshes is presented. The mathematical model is based on the compressible Navier-Stokes equations. The immersed boundary penalty method, namely the Brinkman penalization method, is used to mimic the influence of the solid on the flow. This method provides a possibility to operate in simply connected domains covering the streamlined bodies and, therefore, does not require a traditional “body-fitted” mesh. The relaxation source terms (i.e. penalty functions) are added to the governing equations, to provide the required boundary condition on the fluid-solid interface. The original level set technique of tracking the moving solid boundary over the computational domain is developed. The results of numerical simulation of flow over pitching and plunging airfoil demonstrate the efficiency of the method.
An Eulerian Immersed Boundary Method for flow simulations over stationary and moving rigid bodies
Journal of the Brazilian Society of Mechanical Sciences and Engineering, 2010
The fluid flow over bodies with complex geometry has been the subject of research of many scientists and widely explored experimentally and numerically. The present study proposes an Eulerian Immersed Boundary Method for flows simulations over stationary or moving rigid bodies. The proposed method allows the use of Cartesians Meshes. Here, twodimensional simulations of fluid flow over stationary and oscillating circular cylinders were used for verification and validation. Four different cases were explored: the flow over a stationary cylinder, the flow over a cylinder oscillating in the flow direction, the flow over a cylinder oscillating in the normal flow direction, and a cylinder with angular oscillation. The time integration was carried out by a classical 4 th order Runge-Kutta scheme, with a time step of the same order of distance between two consecutive points in x direction. High-order compact finite difference schemes were used to calculate spatial derivatives. The drag and lift coefficients, the lock-in phenomenon and vorticity contour plots were used for the verification and validation of the proposed method. The extension of the current method allowing the study of a body with different geometry and three-dimensional simulations is straightforward. The results obtained show a good agreement with both numerical and experimental results, encouraging the use of the proposed method.
A robust immersed boundary method for semi-implicit discretizations of the Navier–Stokes equations on curvilinear grids is presented. No-slip conditions are enforced via momentum forcing, and mass conservation at the immersed boundary is satisfied via a mass source term developed for moving bodies. The errors associated with an explicit evaluation of the momentum forcing are analysed, and their influence on the stability of the underlying Navier–Stokes solver is examined. An iterative approach to compute the forcing term implicitly is proposed, which reduces the errors at the boundary and retains the stability guarantees of the original semi-implicit discretization of the Navier–Stokes equations. The implementation in generalized curvilinear coordinates and the treatment of moving boundaries are presented, followed by a number of test cases. The tests include stationary and moving boundaries and curvilinear grid problems (decaying vortex problem, stationary cylinder, flow in 90 bend in circular duct and oscillating cylinder in fluid at rest).
A numerical method is developed for solving the 3D, unsteady, incompressible Navier-Stokes equations in Cartesian domains containing immersed boundaries of arbitrary geometrical complexity moving with prescribed kinematics. The governing equations are discretized on a hybrid staggered/non-staggered grid layout using second-order accurate finitedifference formulas. The discrete equations are integrated in time via a second-order accurate dual-time-stepping, artificial compressibility iteration scheme. 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. At every instant in time, 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. Grid convergence tests are carried out for the flow induced by an oscillating sphere in a cubic cavity, which show that the method is second-order accurate. The method is validated by applying it to calculate flow in a Cartesian domain containing a rigid sphere rotating at constant angular velocity as well as flow induced by a flapping wing. The ability of the method to simulate flows in domains with arbitrarily complex moving bodies is demonstrated by applying to simulate flow past an undulating fish-like body and flow past an anatomically realistic planktonic copepod performing an escape-like maneuver.
The immersed boundary method: A projection approach
Journal of Computational Physics, 2007
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 no-slip constraint. This extra constraint can be added to the incompressible Navier-Stokes 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 no-slip condition. This role is analogous to the effect of pressure on the momentum equation to satisfy the divergence-free constraint. The current immersed boundary method removes slip and non-divergence-free 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 positive-definiteness 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 second-order temporal accuracy and better than first-order spatial accuracy in L 2 -norms for oneand two-dimensional test problems. Results from two-dimensional simulations of flows over stationary and moving cylinders are in good agreement with those from previous experimental and numerical studies.
This paper details immersed boundary method (IBM) to facilitate numerical investigation of the laminar forced fluid flow over a forward-backward facing step through a 2D channel by treating step as an immersed boundary (IB). The present numerical method is based on a finite volume approach on a staggered grid together with a fractional step method. When an IB (step) is present, both momentum forcing and mass source terms are applied on the body surface to satisfy the no-slip boundary condition on the immersed boundary and to satisfy the continuity for the mesh containing the immersed boundary respectively. In the IBM, the necessity of an accurate interpolation scheme satisfying the no-slip condition on the immersed boundary is important because the grid lines generally do not coincide with the immersed boundary. Results are presented with respect to Reynolds numbers for different sizes of the step.
A collocated finite volume embedding method for simulation of flow past stationary and moving body
Computers & Fluids, 2009
A simple and conservative numerical scheme is introduced in this paper to simulate unsteady flow around stationary and moving body. Based on the embedding method (immersed boundary (IB) + volume of fluid (VOF)) implemented in the finite-volume framework, flow past the arbitrarily complex geometry can be readily computed on any existing mesh system. Flow variables stored at cell centers, including those residing within the immersed body, are computed where the induced effect on the flow due to the immersed body is realised via a simple acceleration term (forcing function) derived based on the VOF value. In the current work, an identical VOF value is used for all momentum equations, in contrast to that of the pre-existing method, whereby numerical interpolation is required. The method is verified with a number of flow cases, including flow in a 2D square cavity, flow past a stationary and oscillating cylinder and flow induced by a flapping ellipse in an enclosure.