Nested Cartesian grid method in incompressible viscous fluid flow

In this work, the local grid refinement procedure is focused by using a nested Cartesian grid formulation. The method is developed for simulating unsteady viscous incompressible flows with complex immersed boundaries. A finite-volume formulation based on globally second-order accurate central-difference schemes is adopted here in conjunction with a two-step fractional-step procedure. The key aspects that needed to be considered in developing such a nested grid solver are proper imposition of interface conditions on the nested-block boundaries, and accurate discretization of the governing equations in cells that are with block-interface as a control-surface. The interpolation procedure adopted in the study allows systematic development of a discretization scheme that preserves global second-order spatial accuracy of the underlying solver, and as a result high efficiency/accuracy nested grid discretization method is developed. Herein the proposed nested grid method has been widely tested through effective simulation of four different classes of unsteady incompressible viscous flows, thereby demonstrating its performance in the solution of various complex flow–structure interactions. The numerical examples include a lid-driven cavity flow and Pearson vortex problems, flow past a circular cylinder symmetrically installed in a channel, flow past an elliptic cylinder at an angle of attack, and flow past two tandem circular cylinders of unequal diameters. For the numerical simulations of flows past bluff bodies an immersed boundary (IB) method has been implemented in which the solid object is represented by a distributed body force in the Navier–Stokes equations. The main advantages of the implemented immersed boundary method are that the simulations could be performed on a regular Cartesian grid and applied to multiple nested-block (Cartesian) structured grids without any difficulty. Through the numerical experiments the strength of the solver in effectively/accurately simulating various complex flows past different forms of immersed boundaries is extensively demonstrated, in which the nested Cartesian grid method was suitably combined together with the fractional-step algorithm to speed up the solution procedure.     

High-order compact schemes for incompressible flows: A simple and efficient method with quasi-spectral accuracy

In this paper, a finite difference code for Direct and Large Eddy Simulation (DNS/LES) of incompressible flows is presented. This code is an intermediate tool between fully spectral Navier–Stokes solvers (limited to academic geometry through Fourier or Chebyshev representation) and more versatile codes based on standard numerical schemes (typically only second-order accurate). The interest of high-order schemes is discussed in terms of implementation easiness, computational efficiency and accuracy improvement considered through simplified benchmark problems and practical calculations. The equivalence rules between operations in physical and spectral spaces are efficiently used to solve the Poisson equation introduced by the projection method. It is shown that for the pressure treatment, an accurate Fourier representation can be used for more flexible boundary conditions than periodicity or free-slip. Using the concept of the modified wave number, the incompressibility can be enforced up to the machine accuracy. The benefit offered by this alternative method is found to be very satisfactory, even when a formal second-order error is introduced locally by boundary conditions that are neither periodic nor symmetric. The usefulness of high-order schemes combined with an immersed boundary method (IBM) is also demonstrated despite the second-order accuracy introduced by this wall modelling strategy. In particular, the interest of a partially staggered mesh is exhibited in this specific context. Three-dimensional calculations of transitional and turbulent channel flows emphasize the ability of present high-order schemes to reduce the computational cost for a given accuracy. The main conclusion of this paper is that finite difference schemes with quasi-spectral accuracy can be very efficient for DNS/LES of incompressible flows, while allowing flexibility for the boundary conditions and easiness in the code development. Therefore, this compromise fits particularly well for very high-resolution simulations of turbulent flows with relatively complex geometries without requiring heavy numerical developments.     

A smoothing technique for discrete delta functions with application to immersed boundary method in moving boundary simulations

The effects of complex boundary conditions on flows are represented by a volume force in the immersed boundary methods. The problem with this representation is that the volume force exhibits non-physical oscillations in moving boundary simulations. A smoothing technique for discrete delta functions has been developed in this paper to suppress the non-physical oscillations in the volume forces. We have found that the non-physical oscillations are mainly due to the fact that the derivatives of the regular discrete delta functions do not satisfy certain moment conditions. It has been shown that the smoothed discrete delta functions constructed in this paper have one-order higher derivative than the regular ones. Moreover, not only the smoothed discrete delta functions satisfy the first two discrete moment conditions, but also their derivatives satisfy one-order higher moment condition than the regular ones. The smoothed discrete delta functions are tested by three test cases: a one-dimensional heat equation with a moving singular force, a two-dimensional flow past an oscillating cylinder, and the vortex-induced vibration of a cylinder. The numerical examples in these cases demonstrate that the smoothed discrete delta functions can effectively suppress the non-physical oscillations in the volume forces and improve the accuracy of the immersed boundary method with direct forcing in moving boundary simulations.     

A kernel-free boundary integral method for elliptic boundary value problems

This paper presents a class of kernel-free boundary integral (KFBI) methods for general elliptic boundary value problems (BVPs). The boundary integral equations reformulated from the BVPs are solved iteratively with the GMRES method. During the iteration, the boundary and volume integrals involving Green’s functions are approximated by structured grid-based numerical solutions, which avoids the need to know the analytical expressions of Green’s functions. The KFBI method assumes that the larger regular domain, which embeds the original complex domain, can be easily partitioned into a hierarchy of structured grids so that fast elliptic solvers such as the fast Fourier transform (FFT) based Poisson/Helmholtz solvers or those based on geometric multigrid iterations are applicable. The structured grid-based solutions are obtained with standard finite difference method (FDM) or finite element method (FEM), where the right hand side of the resulting linear system is appropriately modified at irregular grid nodes to recover the formal accuracy of the underlying numerical scheme. Numerical results demonstrating the efficiency and accuracy of the KFBI methods are presented. It is observed that the number of GMRES iterations used by the method for solving isotropic and moderately anisotropic BVPs is independent of the sizes of the grids that are employed to approximate the boundary and volume integrals. With the standard second-order FEMs and FDMs, the KFBI method shows a second-order convergence rate in accuracy for all of the tested Dirichlet/Neumann BVPs when the anisotropy of the diffusion tensor is not too strong.     

Towards oscillation-free implementation of the immersed boundary method with spectral-like methods

It is known that, when the immersed boundary method (IBM) is implemented within spectral-like methods, the Gibbs oscillation seriously deteriorates the calculation of derivatives near the body surface. In this paper, a radial basis function (RBF) based smoothing technique is proposed with the intention of eliminating or efficiently reducing the Gibbs oscillation without affecting the flow field outside the body. Based on this technique, a combined IBM/spectral scheme is developed to solve the incompressible Navier–Stokes equations. Numerical simulations of flow through a periodic lattice of cylinders of various cross sections are performed. The results demonstrate that the proposed methodology is able to give accurate and nearly oscillation-free numerical solutions of incompressible viscous flows.     

A Brinkman penalization method for compressible flows in complex geometries

To simulate flows around solid obstacles of complex geometries, various immersed boundary methods had been developed. Their main advantage is the efficient implementation for stationary or moving solid boundaries of arbitrary complexity on fixed non-body conformal Cartesian grids. The Brinkman penalization method was proposed for incompressible viscous flows by penalizing the momentum equations. Its main idea is to model solid obstacles as porous media with porosity, , and viscous permeability approaching zero. It has the pronounced advantages of mathematical proof of error bound, strong convergence, and ease of numerical implementation with the volume penalization technique. In this paper, it is extended to compressible flows. The straightforward extension of penalizing momentum and energy equations using Brinkman penalization with respective normalized viscous, η, and thermal, η_T, permeabilities produces unsatisfactory results, mostly due to nonphysical wave transmissions into obstacles, resulting in considerable energy and mass losses in reflected waves. The objective of this paper is to extend the Brinkman penalization technique to compressible flows based on a physically sound mathematical model for compressible flows through porous media. In addition to penalizing momentum and energy equations, the continuity equation for porous media is considered inside obstacles. In this model, the penalized porous region acts as a high impedance medium, resulting in negligible wave transmissions. The asymptotic analysis reveals that the proposed Brinkman penalization technique results in the amplitude and phase errors of order O((η)^1/2) and O((η/η_T)^1/43/4), when the boundary layer within the porous media is respectively resolved or unresolved. The proposed method is tested using 1- and 2-D benchmark problems. The results of direct numerical simulation are in excellent agreement with the analytical solutions. The numerical simulations verify the accuracy and convergence rates.     

An adaptive version of ghost-cell immersed boundary method for incompressible flows with complex stationary and moving boundaries

An adaptive version of immersed boundary method for simulating flows with complex stationary and moving boundaries is presented.The method employs a ghost-cell methodology which allows for a sharp representation of the immersed boundary.To simplify the implementation of the methodology,a volume-of-fluid method is introduced to identify the immersed boundary.In addition,the domain is spatially discretized using a tree-based discretization which is relatively simple to implement a fully flexible adaptive refi...     

A high-order cut-cell method for numerical simulation of hypersonic boundary-layer instability with surface roughness

Laminar-turbulent transition of hypersonic boundary layers can be affected significantly by the existence of surface roughness. Currently many important mechanisms of roughness-induced transition are not well understood. In recent years, direct numerical simulation (DNS) has been extensively applied for investigating instability and transition mechanisms of hypersonic boundary layers. Most of the past DNS studies, however, have been based on body-fitted grids for smooth surfaces without roughness. Due to complex geometry of embedded roughness, the use of body-fitted grids can be very difficult for flow with arbitrary surface roughness. In this paper, we present a new high-order cut-cell method to overcome the natural complexities in grid generation around arbitrary surface of roughness. The new method combines a non-uniform-grid finite-difference method for discrete grid points near the solid boundary and a shock-fitting method for the treatment of the bow shock. The non-uniform-grid finite-difference formulas are expressed in a general explicit form so that they can be applied to different multi-dimensional problems without any modification. The computational accuracy of new algorithms of up to O(h⁴) are tested on several one- and two-dimensional elliptic equations in irregular domains. In addition, the new method is applied to the simulation of the receptivity process of a Mach 5.92 flow over a flat plate under the combined effect of an isolated surface roughness element and surface blow and suction. A good agreement is found between the unsteady flow results and those obtained by a Linear Stability Theory (LST).