Dynamic Grid Adaptation Research Articles

Numerical Simulation of Shock Wave Reflecting Patterns for Different Flow Conditions

The numerical experiment has been conducted to investigate the unsteady shock wave reflecting phenomena. The cell-vertex finite-volume, Roe's upwind flux difference splitting method with unstructured grid is implemented to solve unsteady Euler equations. The <TEX>$4^{th}$</TEX>-order Runge-Kutta method is applied for time integration. A linear reconstruction of the flux vector using the least-square method is applied to obtain the <TEX>$2^{nd}$</TEX>-order accuracy for the spatial derivatives. For a better resolution of the shock wave and slipline, the dynamic grid adaptation technique is adopted. The new concept of grid adaptation technique, which is much simpler than that of conventional techniques, is introduced for the current study. Three error indicators (divergence and curl of velocity, and gradient of density) are used for the grid adaptation procedure. Considering the quality of the solution and the numerical efficiency, the grid adaptation procedure was updated up to <TEX>$2^{nd}$</TEX> level at every 20 time steps. For the convenience of comparison with other experimental and analytical results, the case of interaction between the straight incoming shock wave and a sharp wedge is simulated for various flow conditions. The numerical results show good agreement with other experimental and analytical results, in the shock wave reflecting structure, slipline, and the trajectory of the triple points. Some critical cases show disagreement with the analytical results, but these cases also have been proven to show hysteresis phenomena.

Dynamic Grid Adaptation Using the MPDATA Scheme

A dynamic grid adaptation (DGA) scheme is developed using various combinations of the multidimensional positive definite advection transport algorithm (MPDATA) to show the applicability of DGA with the MPDATA scheme to solve advection problems. A one-dimensional model is used to show the effects of varying the number of grid points and the parameters that control the grid redistribution scheme, to determine a stability criteria for the scheme and to investigate the effect of several MPDATA options. A two-dimensional model is used to show the applicability of the scheme in multiple dimensions and to illustrate the effects of DGA in combination with MPDATA options. Diffusion errors are reduced by more than 90% using DGA when compared to static, uniformly spaced grid computations. Phase errors are reduced using certain MPDATA options by more than 25%.

The Two-Mesh Grid Refinement Method for the B2 Code

We introduce version 6.0 of the 2D multispecies plasma fluid code B2, to which a flexible dynamic grid adaptation module has been added. The method followed in this paper involves the creation of two grid structures. A fine 2D mesh produced from the magnetic flux surface structure serves as a scaffolding on which a rougher, adaptable mesh is constructed depending on the physical situation. The conditions for grid refinement/coarsening involve normalized gradients of the plasma parameters. The algorithm and its numerical implementation are described in detail. As a sample application, the detachment behaviour of an ASDEX-Upgrade L-Mode discharge has been studied. A comparison of the results from the old and new B2 code versions is presented. Similar results in the overall plasma behaviour are seen but differences appear in the detachment zone, due to the grid adapting to the strong local gradients. Increased (decreased) gas puffing leads to an upward (downward) shift of the detachment front as well as to a shift of the zone of maximum grid density. The new code allows one to increase the grid density in the zone of interest without a strong increase CPU runtime, thus enabling a more realistic modelling of the detachment front.

Grid Adaption and Its Effect on Entrainment in anE–lModel of the Atmospheric Boundary Layer

Abstract A one-dimensional numerical model KOLUM is introduced that demonstrates the use of continuous dynamic grid adaption in modeling the atmospheric boundary layer. The entrainment rates of KOLUM are compared against recent calibrations for smoke clouds and water clouds derived from large-eddy simulations. The simulations performed with KOLUM support the claim that turbulent kinetic energy–diagnostic length scale (E–l) models overpredict entrainment by smoke and water clouds, independent of the use of grid adaption. The benefits of grid adaption are slight. Curiously, the simulations did not confirm spurious entrainment of E–l models in episodes of subsidence.

Simulation of a reacting pollutant puff using an adaptive grid algorithm

A new dynamic solution adaptive grid algorithm, DSAGA‐PPM, has been developed for use in air quality modeling. In this paper, this algorithm is described, and is evaluated with a test problem. Cone‐shaped distributions of various chemical species undergoing chemical reactions are rotated to simulate the transport and chemistry processes that occur in the atmosphere. The results obtained by using DSAGA‐PPM are more accurate than those obtained from a static grid with the same number of nodes. The computational cost associated with a static grid solution with the same level of accuracy is prohibitive. Because of its efficient use of computational resources, DSAGA‐PPM has the potential to improve the accuracy, or efficiency, or a combination of both in air quality models.

Simulation of dispersion of a power plant plume using an adaptive grid algorithm

A new dynamic adaptive grid algorithm has been developed for use in air quality modeling. This algorithm uses a higher order numerical scheme—the piecewise parabolic method (PPM)—for computing advective solution fields; a weight function capable of promoting grid node clustering by moving grid nodes; and a conservative interpolation equation using PPM for redistributing the solution field after movement of grid nodes. Applications of the algorithm to a model problem, in which emissions from a point source disperse through the atmosphere in time, reflect that the algorithm is able to capture not only the regional ozone plume distribution, but also the small-scale plume structure near the source. In contrast, the small-scale plume structure was not captured in the corresponding static grid solution. Performance achieved in model problem simulations indicates that the algorithm has the potential to provide accurate air quality modeling solutions at costs that may be significantly less than those incurred in obtaining equivalent static grid solutions.

3D simulations of turbulent deflagrations using dynamic grid adaptation

AbstractWe present a CFD code for simulating turbulent deflagrations in gas mixtures for studies of explosion accidents. A dynamic grid adaptation method is used to increase the accuracy of the calculation in regions of interest, such as at flame and shock fronts. The numerical simulations compared with experimental data and is shown to yield strongly improved results at at modest cost in computational time.

An Adaptive Grid Algorithm for Air-Quality Modeling

A new dynamic adaptive grid algorithm has been developed for use in air-quality modeling. This algorithm uses a higher order numerical scheme—the piecewise parabolic method (PPM)—for computing advective solution fields, a weight function capable of promoting grid node clustering by moving grid nodes, and a conservative interpolation equation using PPM for redistributing the solution field after movement of grid nodes. Applications of the algorithm to model problems show that the algorithm provides solutions more accurate than those obtained with static grids. Performance achieved in model problem simulations indicates that the algorithm has the potential to provide accurate air-quality modeling solutions at costs that may be significantly less than those incurred in obtaining equivalent static grid solutions.

極超音速の熱的・化学的非平衡流れ場における円柱と斜め衝撃波の干渉 Ｅｄｎｅｙ問題 ＩＶ

A shock-shock hot interaction problem is studied numerically. A high-order Godunov-type scheme coupled with solution-adaptive unstructured grids has been employed to compute a hypersonic thermochemically nonequilibrium viscous flow; the impinging shock generated by a wedge interacts with the bow shock around a 2-D circular cylinder. The dynamic grid adaptation to solution and the high-resolution methods contribute to a considerable improvement in both accuracy and efficiency. High-temperature real gas effects have been evaluated by a thermal and chemical non-equilibrium model. Type IV interaction of Edney near the transition between Type III and Type IV is obtained. The shock interaction structure is clearly captured with high resolution.

Node-based dynamic adaptive grid with quadrilateral and hexahedral elements

A node-based data structure is developed for use in either two-dimensional (2-D) or three-dimensional (3-D) finite element method (FEM) problems. This grid uses quadrilateral (2-D) or hexahedral (3-D) elements and is dynamically adaptive according to the 1-irregular requirement for refinement–derefinement. Implemented in C++, the grid consists of node objects linked with pointers to neighboring node objects. The algorithm used to traverse the grid starts at the first node of the grid and constructs each FEM element by following the links of the nodes. This algorithm uses a stack to maintain the correct order of elements during the grid traversals. Because the elements and their connectivity are not permanently stored, the computer storage requirements for the grid are reduced significantly in 3-D compared with an element-based approach. Although the node-based approach may increase the CPU-time and complexity to access an element, it reduces the complexity and time required to refine and derefine an element. This implementation is applicable to many FEM domains.

Numerical simulation of explosion phenomena in industrial environments

The code REACFLOW developed at the JRC Ispra combines advanced numerical techniques for the simulation of transient, multi-dimensional, multi-component gas flows undergoing chemical reactions to a unique tool. It uses a true 2-D discretisation with an unstructured triangular grid to ensure a maximum of flexibility for the representation of complex geometries. The numerical discretisation uses a finite volume scheme based on an approximate Riemann solver. Explicit, implicit and semi-implicit methods cover the whole range of time scales. Compressible and incompressible flow is treated with an arbitrary number of components. Chemical reactions are calculated fully implicitly. Diffusion processes are also modelled using a finite volume equivalence to the finite element Galerkin method. A κ-ε turbulence model is currently being implemented. A system for dynamic grid adaptation automatically detects locations of refinement and coarsement based on local gradients of flow variables. The code capability will be demonstrated by various applications, including a hydrogen/air explosion in a containment and a ‘tulip’ flame calculation.

A novel wavelet based time domain simulation approach

A novel time domain simulation approach for analyzing three-dimensional (3-D) passive microwave structures is presented. The electromagnetic fields of the structure under investigation are expanded into a hierarchical system of wavelets and scaling functions for all three dimensions leading to a stable, multiscale algorithm. By neglecting small wavelet coefficients and thereby reducing the computational effort, this method is equivalent to a finite difference scheme with a time dynamic space adaptive grid. A brief overview of the main ideas and advantages of the new method as well as the results of two numerical examples are given.

Combustion and pressure wave interaction in enclosed mixtures initiated by temperature nonuniformities

The dynamics of one-dimensional combustion waves originating from hot-spot ignition was studied numerically by solving the corresponding conservation equations for mass, species, energy, and momentum. The mathematical model included a multispecies transport model and a detailed reaction mechanism. The numerical solution of the time-dependent conservation equations was based on the method of lines and on a dynamic adaptive grid technique. Due to the stiffness of the resulting system of ordinary differential equations, the BDF-code DASSL was used to integrate the equations. The results of the simulations show that the interaction of combustion and pressure wave can lead to deflagration to detonation transition. Depending on the temperature levels and the shaple of the temperature nonuniformity, ordinary flame propagation, deflagration to detonation transition or to deflagration waves, depending on the initial conditions.

Modeling scramjet combustor flowfields with a grid adaptation scheme

The accurate description of flow features associated with the normal injection of fuel into supersonic primary flows is essential in the design of efficient engines for hypervelocity aerospace vehicles. The flow features in such injections are complex with multiple interactions between shocks and between shocks boundary layers. Numerical studies of perpendicular sonic N2 injection and mixing in a Mach 3.8 scramjet combustor environment are discussed. A dynamic grid adaptation procedure based on the equilibration of spring-mass system is employed to enhanced the description of the complicated flow features. Numerical results are compared with experimental measurements and indicate that the adaptation procedure enhances the capability of the modeling procedure to describe the flow features associated with scramjet combustor components.

Solving partial differential equations using recursive grids

In this paper we discuss numerical methods and software tools for solving partial differential equations with variable local resolution. The emphasis of our work is on finding approaches which will greatly simplify developing applications using dynamic adaptive grids. First, we investigate a class of adaptive grids that looks particularly promising for use in general-purpose solvers and propose a discretization method suitable for this class of grids. Second, we describe an interactive programming environment which can be used, among other things, as a geometrically-oriented debugger for developing applications using adaptive grids. Finally, we briefly illustrate an application of the above-mentioned methods and software to flame–flow interaction problems.

Structured and unstructured grid adaptation schemes for numerical modeling of field problems

The use of grid adaptation procedures for compressible flows and heat transfer is presented in this paper. Grid adaptation schemes are presented for structured and unstructured grids in the context of finite-difference and finite-element methods. For compressible flow problems with regular domains, the use of finite-difference methods has proven to be quite effective. For such analyses, a dynamic grid adaptation scheme is employed with a MacCormack predictor-corrector scheme to describe compressible flow features. The adaptation procedure uses a grid relocation stencil that is valid at both the interior and boundary points of the finite-difference grid. Linear combinations of spatial derivatives of specific flow variables, calculated with finite-element interpolation functions, are used as adaptation measures. The impact of the grid relocation procedure is demonstrated by simulating the effects of shock impingement on a three-dimensional unsymmetric axial corner. When the domain to be modeled is quite complex, unstructured finite-elementor finite-volume formulations are preferred. An unstructured mesh adaptation scheme with refinement and coarsening based on refinement indicators is used to describe inviscid and viscous flow features. This adaptation scheme uses quadrilateral and triangular elements to add grid density to regions of high gardients and remove elements where these gardients are low. Modifications of the mesh adaptation procedure enables its utilization in the analyses of heat transfer problems commonly associated with hypervelocity flows. The strategies used for mesh adaptation, computing refinement indicators, and time marching are described. The effectiveness of this procedure is reflected in good solution accuracy, reduction in the number of elements used, and computational efficiency.

A Fast Dynamic Grid Adaption Scheme for Meteorological Flows

Abstract The continuous dynamic grid adaption (CDGA) technique is applied to a compressible, three-dimensional model of a rising thermal. The computational cost, per grid point per time step, of using CDGA instead of a fixed, uniform Cartesian grid is about 53% of the total cost of the model with CDGA. The use of general curvilinear coordinates contributes 11.7% to this total, calculating and moving the grid 6.1%, and continually updating the transformation relations 20.7%. Costs due to calculations that involve the gridpoint velocities (as well as some substantial unexplained costs) contribute the remaining 14.5%. A simple way to limit the cost of calculating the grid is presented. The grid is adapted by solving an elliptic equation for gridpoint coordinates on a coarse grid and then interpolating the full finite-difference grid. In our application, the additional costs per grid point of CDGA are shown to be easily offset by the savings resulting from the reduction in the required number of grid points. ...

Application of Continuous Dynamic Grid Adaption Techniques to Meteorological Modeling. Part II: Efficiency

Abstract The continuous dynamic adaptive grid (CDGA) technique has been shown to yield significant improvements in solution accuracy over equivalent fixed-grid methods. In this paper we consider the related question of efficiency; for a given degree of solution accuracy is the CDGA method faster than the fixed-grid approach, and if so, by how much? A new grid-generation algorithm based on discrete equidistribution principles is proposed, which, while it has the disadvantage of producing nonorthogonal grids, is extremely simple to implement and fast to execute. The algorithm is applied to a barotropic model designed to examine the motion of interacting multiple vortices. Numerical tests confirm the efficacy of the proposed algorithm and show that the CDGA method can produce solutions as accurate as those obtained from a fixed-grid model in only one-third the time.

Application of Continuous Dynamic Grid Adaption Techniques to Meteorological Modeling. Part I: Basic Formulation and Accuracy

Abstract The continuous dynamic grid adaption (CDGA) technique developed in astrophysics and aeronautics is applied, to our knowledge, for the first time to meteorological modeling. The aim of CDGA is to improve the accuracy of numerical solutions of partial differential equations (typically those governing fluid flow) by the use of nonuniform grids that have higher local resolution in regions where the numerical error is presumed to be large. Conceptually, CDGA has some relationship to the well-known technique of grid stretching, but its power lies in its ability to determine an appropriate spatial distribution of grid points automatically and to update this distribution in response to changes in the evolving numerical solution. Application of the technique is facilitated by transforming the governing equations from physical space in which the grid is nonuniform, nonorthogonal and for which the individual grid points are in continuous motion to computational space, which by definition has both a regular ...

Simulation of ignition and combustion wave propagation in enclosed C 2H 6/O 2/N 2 mixtures

The dynamics of one-dimensional C2H6/Air combustion waves originating from hot-spot ignition was studied numerically by solving the corresponding conservation equations for mass, species, energy and momentum. The model included a detailed reaction mechanism and a multispecies transport model. The numerical solution was based on the method of lines and on a dynamic adaptive grid technique. Due to the stiffness of the resulting differential equations, a BDF-code was used to integrate the system. In this work, hot-spot ignition was studied for a reaction system under high pressure and high temperature conditions. The results show that hot-spot ignition can cause different combustion phenomena depending on the shape of the temperature nonuniformity. A supersonic combustion wave was observed when the velocity of the temperature-gradient based autoignition wave was much higher than the velocity of pressure waves. Detonation wave initiation was the result when an interaction between ignition process and pressure wave was possible. Ordinary flame propagation occurred, when the mixture outside the hot-spot was far away from autoignition.