A hybrid Cartesian grid and gridless method for compressible flows

This paper presents the development of finite-volume multiscale methods for quadrilateral and triangular unstructured grids. Families of Darcy-flux approximations have been developed for consistent approximation of the general tensor pressure equation arising from Darcy's law together with mass conservation. The schemes are control-volume distributed (CVD) with flow variables and rock properties sharing the same control-volume location and are comprised of a multipoint flux family formulation (CVD-MPFA). The schemes are used to develop a CVD-MPFA based multiscale finite-volume (MSFV) formulation applicable to both structured and unstructured grids in two dimensions. The basis functions are a key component of the MSFV method, and are a set of local solutions, usually defined subject to Dirichlet boundary conditions. A generalization of the Cartesian grid Dirichlet basis functions described in [P. Jenny, S. H. Lee, and H. A. Tchelepi, J. Comput. Phys., 187 (2003), pp. 47--67] is presented here for unstructured grids. Whilst the transition from a Cartesian grid to an unstructured grid is largely successful, use of Dirichlet basis functions can still lead to pressure fields that exhibit spurious oscillations in areas of strong heterogeneity. New basis functions are proposed in an attempt to improve the pressure field solutions where Neumann boundary conditions are imposed almost everywhere, except corners which remain specified by Dirichlet values.

In this work, a novel technique called normal ray refinement (NRR) is developed, implemented and investigated. Normal ray refinement is designed to allow for the viscous fluid flow simulations using an unstructured Cartesian grid framework in a computationally efficient manner. A key benefit of using a Cartesian grid method is that the grid can be automatically generated, thereby saving a vast amount of time and effort for complex geometries. The main drawback of using the Cartesian grid method is the large number of cells required to resolve viscous boundary layers, and it is this problem that the NRR approach addresses. The NRR approach relies on the use of refined normal rays of cells emanating from the body surface and spanning the boundary layer. Separating these rays along the body surface are relatively large cells too coarse to accurately capture viscous gradients. The heart of the NRR approach lies in the inter-ray communication strategies used between the normal rays that allow the accurate simulation of boundary layers even though the cells separating the rays are large. This yields a large reduction in the number of cells in the grid, which reduces the computational cost of simulation. This paper provides a background on different viscous Cartesian grid-based methods, followed by an explanation of the NRR approach, then some initial 2D results obtained using NRR for Reynolds numbers up to 1 million. It is shown that NRR can yield substantial reduction in computational cost relative to the standard Cartesian approach.

The Euler equations describe the ow phenomena of compressible inviscid gas dynamics We simulate such ows using a higher order Cartesian grid method together with a special treatment for the cells cut by the boundary of a body We describe a new method for the treatment of the boundary where these cut boundary cells are maintained as whole cells rather than as cut cells thus avoiding stability problems The method is second order accurate but not strictly conservative but we can show that this error in the conservation does not lead to spurious phenomena on some representative test calculations The advantages of the new boundary treatment are that it is second order accurate that it is independent of the applied method and that it can easily be extended to three dimensional calculations

A high-order accurate flow solution method for quadrilateral elements is developed and tested. The method employs the flux reconstruction (FR) approach proposed by Huynh. Ideal orders of accuracy are achieved for several problems on structured-based grids while some deterioration is observed on purely unstructured grids. Cares for curved wall boundary are also shown. The localized artificial diffusivity (LAD) method is incorporated to capture discontinuities that is crucial for compressible flow simulations. The FR+LAD works very well for 1D test cases but multi-dimensional shock problems requires further modifications to confine the artificial diffusivity to the shock location. Hybrid scheme of the FR and finite difference method taking advantage of 1D-feature alleviates the instability. I. Introduction Expanding demands for the computational fluid dynamics (CFD) requires more reliable numerical schemes. Specifically, turbulent flow simulations, acoustic simulation, combustion simulations and their combined problems. The key words in such simulations may be high accuracy and high resolution. In structured grid CFD, a compact finite difference scheme satisfies both requirements at higher level than existing other methods. If we put more emphasis on easier handling of complex geometries and flow-data dependent grid generation (grid adaptation), then unstructured grid method should also be explored and it should depart from 2nd-order scheme with limiter. Recent efforts in this area are development of the discontinuous Galerkin (DG) method, 1,2 the spectral finite volume (SV) method 3 and the staggered-grid multi-domain method spectral finite difference (SG) method 4,5 and its evolutional method called a spectral difference method. 6 The common and maybe the only way for those unstructured methods is to add degrees of freedom in each cell to achieve high order since the wide stencil of the original grid is not possible due to unstructured nature. The ENO-type method that uses original degrees of freedom in the grids and extends adaptive stencils to the flow solutions is also explored but the difficulty of extending stencils due to undirectional nature of unstructured grids and the unsuitability for parallel computing prohibits wide applications. A flux reconstruction (FR) scheme 7,8 studied in the paper is the new one among the former group. It uses differential form of conservation laws like the SD method but it is more general and includes equivalent counter part of the schemes in the group. In this paper, the performance of the FR is tested by a newly developed Navier-Stokes solver. Realistic flow applications of the FR in the Navier-Stokes equations are seldom seen since it is relatively new scheme. The Localized artificial diffusivity (LAD) 9–13 is incorporated for the shock capturing in the FR. The last item is motivated by the challenge and success of combining the SD and LAD in the recent study. 11,12 Wang 14 developed the lifting collocation penalty (LCP) method to extend the FR to triangular or tetrahedral grids and applied the LCP-FR combined method for hybrid unstructured grids. In the present study, only quadrilateral cells are treated to directly apply the LAD method that was developed on structured grids. The shock capturing by the FR+LAD high order scheme is tried.

An unstructured grid method is presented to compute three-dimensional compressible turbulent flows for complex geometries. The Navier-Stokes equations, along with the one-equation turbulence model of Spalart-Allmaras are solved by the use of a parallel, matrix-free implicit method on unstructured tetrahedral grids. The developed method has been used to predict drags in the transonic regime for both DLR-F4 and DLR-F6 configurations to assess the accuracy and efficiency of the method

A quadtree-based adaptive Cartesian grid generator and flow solver were developed. The grid adaptation based on pressure or density gradient was performed and a gridless method based on the least-square fashion was used to treat the wall surface boundary condition, which is generally difficult to be handled for the common Cartesian grid. First, to validate the technique of grid adaptation, the benchmarks over a forward-facing step and double Mach reflection were computed. Second, the flows over the NACA 0012 airfoil and a two-element airfoil were calculated to validate the developed gridless method. The computational results indicate the developed method is reasonable for complex flows.

A coupled regional and local model is required when groundwater flow and solute transport are to be simulated in local areas of interest with a finer grid while regional aquifer boundary and major stresses should be retained with a coarser grid. The coupled model should also maintain interactions between the regional and local flow systems. In the Beijing Plain (China), assessment of managed aquifer recharge (MAR), groundwater pollution caused by rivers, capture zone of well fields, and land subsidence at the cone of depression requires a coupled regional and local model. This study evaluates three methods for coupling regional and local flow models for simulating MAR in the Chaobai River catchment in the Beijing Plain. These methods are the conventional grid refinement (CGR) method, the local grid refinement (LGR) method and the unstructured grid (USG) method. The assessment included the comparison of the complexity of the coupled model construction, the goodness of fit of the computed and observed groundwater heads, the consistency of regional and local groundwater budgets, and the capture zone of a well filed influenced by the MAR site. The results indicated that the CGR method based on MODFLOW-2005 is the easiest to implement the coupled model, capable of reproducing regional and local groundwater heads and budget, and already coupled with density and viscosity dependent model codes for transport simulation. However, the CGR method inherits shortcomings of finite difference grids to create multiple local models with inefficient computing efforts. The USG method based on MODFLOW-USG has the advantage of creating multi-scale models and is flexible to simulate rivers, wells, irregular boundaries, heterogeneities and the MAR site. However, it is more difficult to construct the coupled models with the unstructured grids, therefore, a good graphic user interface is necessary for efficient model construction. The LGR method based on MODFLOW-LGR can be used to create multiple local models in uniform aquifer systems. So far, little effort has been devoted to upgrade the LGR method for complex aquifer structures and develop the coupled transport models.

In this study, the added resistance of ships in short waves is systematically studied by using two different numerical methods - Rankine panel method and Cartesian grid method – and existing asymptotic and empirical formulae. Analysis of added resistance in short waves has been preconceived as a shortcoming of numerical computation. This study aims to observe such preconception by comparing the computational results, particularly based on two representative three-dimensional methods, and with the existing formulae and experimental data. In the Rankine panel method, a near-field method based on direct pressure integration is adopted. In the Cartesian grid method, the wave-body interaction problem is considered as a multiphase problem, and volume fraction functions are defined in order to identify each phase in a Cartesian grid. The computational results of added resistance in short waves using the two methods are systematically compared with experimental data for several ship models, including S175 containership, KVLCC2 and Series 60 hulls (C_B = 0.7, 0.8). The present study includes the comparison with the established asymptotic and empirical formulae in short waves.

In this paper, we consider the construction of a generalized algorithm for calculating diffuse radiation configuration factors (CFs). The generalized algorithm is based on dividing the existing computational algorithms into two groups (paired and one-to-many), as well as the results of studying the possible combinations of grid and computational methods (exact, special, numerical integration, and combined) for average, local, and elementary CFs. Grid methods are presented as the set of the object on which the grid is applied (surface, contour, solid angle), the grid type (the 1st and 2nd grid types are distinguished), and the method of its construction (deterministic, statistical, hybrid or hierarchical method). The general scheme of applying grid methods in the algorithms for calculating the CF is obtained. The generalized algorithm for calculating the CF is presented in the form of a directed graph that is actually a generalized block diagram containing practically all existing and possible algorithms for calculating the CF. The results may be used as the base for the software package for calculating the CFs for a wide class of problems.

The added resistance of a ship in waves can be related to ship-generated unsteady waves. In the present study, the unsteady wave-pattern analysis is applied to calculate the added resistance in waves for two modified Wigley models using a Cartesian-grid method. In the present numerical method, a first-order fractional-step method is applied to the velocity-pressure coupling in the fluid domain, and one of volume-of-fluid (VOF) methods is adopted to capture the fluid interface. A ship is embedded in a Cartesian grid, and the volume fraction of the ship inside the grid is calculated by identifying whether each grid is occupied by liquid, gas, and solid body. The sensitivity to the location of measuring position of unsteady waves as well as the number of solution grids is examined. The added resistance computed by direct pressure integration and wave pattern analysis is compared with experimental data. In addition, nonlinear characteristics of the added resistance in waves are investigated by detailed analyses of unsteady flow field and resulting wave pattern.

A parallel Navier-Stokes solver based on dynamic overset unstructured grids method is presented to simulate the unsteady turbulent flow field around helicopter in forward flight. The grid method has the advantages of unstructured grid and Chimera grid and is suitable to deal with multiple bodies in relatively moving. Unsteady Navier-Stokes equations are solved on overset unstructured grids by an explicit dual time-stepping, finite volume method. Preconditioning method applied to inner iteration of the dual-time stepping is used to speed up the convergence of numerical simulation. The Spalart-Allmaras one-equation turbulence model is used to evaluate the turbulent viscosity. Parallel computation is based on the dynamic domain decomposition method in overset unstructured grids system at each physical time step. A generic helicopter Robin with a four-blade rotor in forward flight is considered to validate the method presented in this paper. Numerical simulation results show that the parallel dynamic overset unstructured grids method is very efficient for the simulation of helicopter flow field and the results are reliable.

The capability of an unstructured hybrid grid method to compute the compressible Navier-Stokes equations is discussed for vortical flows over slender delta wings at low subsonic, high-alpha conditions as a basic study of offdesign aerodynamics of HSCT (High-Speed Civil Transport) configurations. Computations of these flows require high resolutions of the boundary layers on the wing and of the vortical flow region above the wing. The unstructured hybrid grid, which is composed of prismatic grid in boundary layers and tetrahedral and pyramidal grids for the remaining region, is used to treat such viscous flows. The compressible Navier-Stokes equations are solved on the hybrid grid by a cell-vertex, upwind finite volume method. The LU-SGS implicit time integration scheme is used for the convergence acceleration to the steady state. The numerical accuracy of the present method is discussed by comparing with the experimental data of the same configurations. It is demonstrated that the present hybrid grid method is efficient and accurate enough to predict the vortical flow fields over the delta wings. The capability of the method is also shown for a HSCT configuration at a low subsonic, high alpha condition.

An automatic adaptive hybrid Cartesian grid generation and solution system is presented together with applications for turbulent flows around complicated three-dimensional geometries. The primary computational grid is an octree Cartesian grid with a cut-cell approach. A quasi-prismatic grid may be added for resolving the boundary layer region of viscous flow around the solid body. For external flow simulations the flow solver TAU from the Deutsches Zentrum fiir Luftund Raumfahrt is integrated in the simulation system. Coarse grids are generated automatically, which are required for multilevel smoothing. The initial hybrid Cartesian grid is improved with help of several adaptation features, which are optimized in order to avoid any user interaction. Therefore, the final grid depends only on the particular geometry and flow structure (selforganized grid).

An automatic adaptive hybrid Cartesian grid generator is presented together with solutions of flows around complicated two- and three-dimensional geometries. The primary computational grid is a Cartesian grid, which is generated based on an octree-data structure. A secondary grid may be added for resolving the boundary layer region of viscous flow around the solid body. It consists of quasi triangular-prismatic cells generated by marching the body surface triangulation outward. A modified unstructured flow solver is applied for the compressible Euler and Navier Stokes equations. For turbulent flows, the eddy viscosity is calculated using the Spalart-Allmaras turbulence model. The hybrid Cartesian grid is improved due to manifold adaptation features, which are optimized in order to avoid any user interaction. Therefore, the final grid depends only on the particular geometry and flow structure (self-organized grid).

Many problems in fluid dynamics require the representation of complicated internal or external boundaries of the flow. Here we present a method for calculating time-dependent incompressible inviscid flow which combines a projection method with a "Cartesian grid" approach for representing geometry. In this approach, the body is represented as an interface embedded in a regular Cartesian mesh. The advection step is based on a Cartesian grid algorithm for compressible flow, in which the discretization of the body near the flow uses a volume-of-fluid representation. A redistribution procedure is used to eliminate time-step restrictions due to small cells where the boundary intersects the mesh. The projection step uses an approximate projection based on a Cartesian grid method for potential flow. The method incorporates knowledge of the body through volume and area fractions along with certain other integrals over the mixed cells. Convergence results are given for the projection itself and for the time-dependent algorithm in two dimensions. The method is also demonstrated on flow past a half-cylinder with vortex shedding.