R. K. Lin

A differentially interpolated direct forcing immersed boundary method for predicting incompressible Navier-Stokes equations in time-varying complex geometries

A dispersion-relation-preserving dual-compact scheme developed in Cartesian grids is applied together with the immersed boundary method to solve the flow equations in irregular and time-varying domains. The artificial momentum forcing term applied at certain points in cells containing fluid and solid allows an imposition of velocity condition to account for the motion of solid body. We develop in this study a differential-based interpolation scheme which can be easily extended to three-dimensional simulation. The results simulated from the proposed immersed boundary method agree well with other numerical and experimental results for the chosen benchmark problems. The accuracy and fidelity of the IB flow solver developed to predict flows with irregular boundaries are therefore demonstrated.

AN INCOMPRESSIBLE NAVIER-STOKES MODEL IMPLEMENTED ON NONSTAGGERED GRIDS

The present study aims to develop an effective finite-difference model for solving incompressible Navier-Stokes equations. For the sake of programming simplicity, discretization of equations is made on nonstaggered grids without oscillatory solutions arising from the decoupling of the velocity and pressure fields. For the sake of computational efficiency, both segregated and alternating direction implicit (ADI) solution algorithms are employed to reduce the matrix size and, in turn, the CPU time. For the sake of numerical accuracy, a convection-diffusion-reaction finite-difference scheme is employed to provide nodally exact solutions in each ADI solution step. The convective instability problem is thus eliminated, since each convective term is modeled analytically even in multidimensional cases. The validity of the proposed numerical model is rigorously justified by solving one- and two-dimensional problems, which are amenable to analytical solutions. The simulated solutions for the scalar prototype equation agree well with the exact solutions and provide a very high spatial rate of convergence. The same is true for the simulated results of the Navier-Stokes equations.

Development of a dispersion relation-preserving upwinding scheme for incompressible Navier-Stokes equations on nonstaggered grids

ABSTRACT In this article a scheme which preserves the dispersion relation for convective terms is proposed for solving the two-dimensional incompressible Navier–Stokes equations on nonstaggered grids. For the sake of computational efficiency, the splitting methods of Adams-Bashforth and Adams-Moulton are employed in the predictor and corrector steps, respectively, to render second-order temporal accuracy. For the sake of convective stability and dispersive accuracy, the linearized convective terms present in the predictor and corrector steps at different time steps are approximated by a dispersion relation-preserving (DRP) scheme. The DRP upwinding scheme developed within the 13-point stencil framework is rigorously studied by virtue of dispersion and Fourier stability analyses. To validate the proposed method, we investigate several problems that are amenable to exact solutions. Results with good rates of convergence are obtained for both scalar and Navier–Stokes problems.

Simulation of Portal Hemodynamic Changes in a Donor After Right Hepatectomy

Remnant livers will be regenerated in live donors after a large volume resection for transplantation. How the structures and hemodynamics of portal vein will evolve with liver regeneration remains unknown. This prompts the present hemodynamic simulation for a 25 year-old man who received a right donor lobectomy. According to the magnetic resonance imaging/computed tomography images taken prior to the operation and one month after the operation, three sequential models of portal veins (pre-op, immediately after the operation, and one-month post-op) were constructed by AMIRA and HYPERMESH, while the immediately after the operation model was generated by removing the right branch in the pre-op model. Hemodynamic equations were solved subject to the sonographically measured inlet velocity. The simulated branch velocities were compared with the measured ones. The predicted overall pressure in the portal vein after resection was found to increase to a magnitude that has not reached to an extent possibly leading to portal hypertension. As expected, blood pressure has a large change only in the vicinity of the resection region. The branches grew considerably different from the original one as the liver is regenerated. Results provide useful evidence to justify the current computer simulation.

An effective explicit pressure gradient scheme implemented in the two-level non-staggered grids for incompressible Navier-Stokes equations

In this paper, an improved two-level method is presented for effectively solving the incompressible Navier-Stokes equations. This proposed method solves a smaller system of nonlinear Navier-Stokes equations on the coarse mesh and needs to solve the Oseen-type linearized equations of motion only once on the fine mesh level. Within the proposed two-level framework, a prolongation operator, which is required to linearize the convective terms at the fine mesh level using the convergent Navier-Stokes solutions computed at the coarse mesh level, is rigorously derived to increase the prediction accuracy. This indispensable prolongation operator can properly communicate the flow velocities between the two mesh levels because it is locally analytic. Solution convergence can therefore be accelerated. For the sake of numerical accuracy, momentum equations are discretized by employing the general solution for the two-dimensional convection-diffusion-reaction model equation. The convective instability problem can be simultaneously eliminated thanks to the proper treatment of convective terms. The converged solution is, thus, very high in accuracy as well as in yielding a quadratic spatial rate of convergence. For the sake of programming simplicity and computational efficiency, pressure gradient terms are rigorously discretized within the explicit framework in the non-staggered grid system. The proposed analytical prolongation operator for the mapping of solutions from the coarse to fine meshes and the explicit pressure gradient discretization scheme, which accommodates the dispersion-relation-preserving property, have been both rigorously justified from the predicted Navier-Stokes solutions.

Prediction of a temperature‐dependent electroosmotically driven microchannel flow with the Joule heating effect

Purpose – A convection‐diffusion‐reaction scheme is proposed in this study to simulate the high gradient electroosmotic flow behavior in microchannels. The equations governing the total electric field include the Laplace equation for the effective electrical potential and the Poisson‐Boltzmann equation for the electrical potential in the electric double layer.Design/methodology/approach – Mixed electroosmotic/pressure‐driven flow in a straight microchannel is studied with the emphasis on the Joule heat in the equations of motion. The nonlinear behaviors resulting from the hydrodynamic, thermal and electrical three‐field coupling and the temperature‐dependent fluid viscosity, thermal conductivity, electrical permittivity, and conductivity of the investigated buffer solution are analyzed.Findings – The solutions computed from the employed flux discretization scheme for the hydrodynamic, thermal and electric field equations have been verified to have good agreement with the analytical solution. Parametric st...

Computer Simulation of Hemodynamic Changes After Right Lobectomy in a Liver with Intrahepatic Portal Vein Aneurysm

BACKGROUND/PURPOSE Intrahepatic portal vein aneurysm is rare and its natural history is unknown. A 22-year-old healthy man, who wished to donate part of his liver to his diseased father, was incidentally diagnosed to have an intrahepatic portal vein aneurysm. The surgical decision of performing live donor hepatectomy for such a patient is normally difficult. We combined modern imaging reconstruction technologies with scientific computing as a new modality to foresee the risks of surgical complications. METHODS Cross-sectional computed tomography images were used to reconstruct the three-dimensional image of portal vein distribution using the 3D-Doctor v3.5 software. The reconstructed images were further employed to generate surface and interior meshes with CFX software. Simulated hemodynamic changes in velocity, pressure, and wall stress were determined for the right lobectomy case pre- and postoperatively. RESULTS The simulation results indicated that aneurismal pressure would be elevated significantly to 12.03 mmHg after operation. The left segmental portal venous blood flow would increase from 2.95- to 4.25-fold. The area near the branch point of one left segmental portal vein, which supplies blood to liver segment 4, and the portal vein aneurysm would endure high shear stress gradient. The resulting elevated aneurismal pressure may cause the thin wall to enlarge and rupture, while the high shear stress gradient would lead to vascular endothelial cell injury. Living donor surgery was not recommended hemodynamically based on the simulated results. CONCLUSION Scientific computing and modern imaging technologies can be applied together to aid surgeons to make the best decision in difficult clinical situations.

A FOUR-STEP TIME SPLITTING SCHEME FOR CONVECTION-DIFFUSION EQUATION

In this article we develop a four-step time splitting scheme for solving the two-dimensional convection-diffusion equation by using a fourth-order-accurate Pade approximation. The resulting temporal scheme includes two explicit equations and two coupled implicit equations. For constructing an equal-order scheme for the explicit equations, we approximate the second derivative terms using the fourth-order-accurate centered scheme. In the approximation of the first-derivative term, it is essential that the fourth-order-accurate scheme takes solutions at the upwind side into favorable consideration. For constructing an efficient scheme for the implicit equations, we apply the alternating direction implicit scheme of Peaceman and Rachford. For the sake of accuracy, in each sweep we apply a three-point, nodally exact, one-dimensional convection-diffusion-reaction (CDR) scheme. As is standard practice, we validate the proposed method by solving several problems that are amenable to exact solutions. Results with good rate of convergence are obtained for the investigated one- and two-dimensional problems.

On An Equal Fourth-Order-Accurate Temporal/Spatial Scheme for the Convection-Diffusion Equation

In this article, the convection-diffusion equation is discretized using the Pade method for the temporal derivative term and the wavenumber-extended method for the spatial derivative term. These temporal and spatial approximations result in two explicit equations and two implicitly coupled equations. To construct an equal-order scheme for the solution obtained at n Δt, both temporal/spatial derivatives are approximated to render fourth-order accuracy without using solutions obtained previously at (n − 2)Δt, (n − 3)Δt, etc. When approximating the first-order derivative term, it is essential to take the upwind nodal points into consideration. For revealing the dispersion and dissipation natures of the proposed scheme, both von Neumann (Fourier) and dispersion analyses were conducted. We validate the proposed method by solving several problems that are amenable to exact solutions. Results with theoretical rates of convergence are obtained for each of the one- and two-dimensional problems investigated.

DEVELOPMENT OF A LOCALLY ANALYTIC PROLONGATION OPERATOR IN THE TWO-GRID, THREE-LEVEL METHOD FOR THE NAVIER-STOKES EQUATIONS

In this article, two three-level methods employing the same prolongation operator are proposed for efficiently solving the incompressible Navier-Stokes equations in a two-grid system. Each method involves solving one smaller system of nonlinear equations in the coarse mesh. The chosen Newton- or Oseen-type linearized momentum equations along with a correction step are solved only once on the fine mesh. Within the three-level framework, the locally analytic prolongation operator needed to bridge the convergent Navier-Stokes solutions obtained at the coarse mesh and the interpolated velocities at the fine mesh is developed to improve the prediction quality. To increase prediction accuracy, the linearized momentum equations are discretized within the alternating direction implicit context using our previously developed nodally exact convection-diffusion-reaction finite-difference scheme. Two proposed three-level methods are rigorously assessed in terms of simulated accuracy, nonlinear convergence rate, and elapsed CPU time.