Well-balanced compressible cut-cell simulation of atmospheric flow

2009 ◽  
Vol 367 (1907) ◽  
pp. 4559-4575 ◽  
Author(s):  
R. Klein ◽  
K. R. Bates ◽  
N. Nikiforakis
Keyword(s):  
Attractive Alternative ◽  
Test Problems ◽  
Time Step ◽  
Atmospheric Flow ◽  
Flow Equations ◽  
Cut Cell ◽  
Flux Approximation ◽  
Terrain Following ◽  
Lee Wave ◽  
Cell Simulation

Cut-cell meshes present an attractive alternative to terrain-following coordinates for the representation of topography within atmospheric flow simulations, particularly in regions of steep topographic gradients. In this paper, we present an explicit two-dimensional method for the numerical solution on such meshes of atmospheric flow equations including gravitational sources. This method is fully conservative and allows for time steps determined by the regular grid spacing, avoiding potential stability issues due to arbitrarily small boundary cells. We believe that the scheme is unique in that it is developed within a dimensionally split framework, in which each coordinate direction in the flow is solved independently at each time step. Other notable features of the scheme are: (i) its conceptual and practical simplicity, (ii) its flexibility with regard to the one-dimensional flux approximation scheme employed, and (iii) the well-balancing of the gravitational sources allowing for stable simulation of near-hydrostatic flows. The presented method is applied to a selection of test problems including buoyant bubble rise interacting with geometry and lee-wave generation due to topography.

scholarly journals Comparison of Terrain-Following and Cut-Cell Grids Using a Nonhydrostatic Model

2016 ◽  
Vol 144 (6) ◽  
pp. 2085-2099 ◽  
Author(s):  
James Shaw ◽  
Hilary Weller
Keyword(s):  
Velocity Field ◽  
Gravity Waves ◽  
Initial Conditions ◽  
Implicit Time ◽  
Test Results ◽  
Time Step ◽  
Induced Gravity ◽  
Time Stepping ◽  
Cut Cell ◽  
Terrain Following

Abstract Terrain-following coordinates are widely used in operational models but the cut-cell method has been proposed as an alternative that can more accurately represent atmospheric dynamics over steep orography. Because the type of grid is usually chosen during model implementation, it becomes necessary to use different models to compare the accuracy of different grids. In contrast, here a C-grid finite-volume model enables a like-for-like comparison of terrain-following and cut-cell grids. A series of standard two-dimensional tests using idealized terrain are performed: tracer advection in a prescribed horizontal velocity field, a test starting from resting initial conditions, and orographically induced gravity waves described by nonhydrostatic dynamics. In addition, three new tests are formulated: a more challenging resting atmosphere case, and two new advection tests having a velocity field that is everywhere tangential to the terrain-following coordinate surfaces. These new tests present a challenge on cut-cell grids. The results of the advection tests demonstrate that accuracy depends primarily upon alignment of the flow with the grid rather than grid orthogonality. A resting atmosphere is well maintained on all grids. In the gravity waves test, results on all grids are in good agreement with existing results from the literature, although terrain-following velocity fields lead to errors on cut-cell grids. Because of semi-implicit time stepping and an upwind-biased, explicit advection scheme, there are no time step restrictions associated with small cut cells. In contradiction to other studies, no significant advantages of cut cells or smoothed coordinates are found.

scholarly journals A Method for Imposing Surface Stress and Heat Flux Conditions in Finite-Difference Models with Steep Terrain

2007 ◽  
Vol 135 (3) ◽  
pp. 906-917 ◽  
Author(s):  
C. C. Epifanio
Keyword(s):  
Finite Difference ◽  
Linear Systems ◽  
Surface Stress ◽  
Test Problems ◽  
Time Step ◽  
Terrain Following ◽  
Thermally Driven ◽  
Stress Boundary Condition ◽  
Steep Terrain ◽  
Stress Boundary

Abstract A numerical implementation of the surface stress boundary condition is presented for finite-difference models in which the terrain slope and curvature cannot necessarily be considered small. The method involves reducing the discretized stress condition in terrain-following coordinates to a pair of coupled linear systems for the two horizontal velocity components at the boundary. The linear systems are then solved iteratively at each model time step to provide the unique boundary values of velocity consistent with the specified values of the stress. Similar methods are used to prescribe the normal flux of heat across the boundary. A related method for imposing stress conditions in two-dimensional vorticity–streamfunction models is also discussed. The effectiveness of the boundary conditions is demonstrated through a series of test problems involving topographic wake flows and thermally driven flows on steep slopes. It is shown that the use of the conventional flat-boundary approximation can lead to substantial errors when the resolved topography is sufficiently steep.

scholarly journals Performance of Adaptive Unstructured Mesh Modelling in Idealized Advection Cases over Steep Terrains

Atmosphere ◽  
2018 ◽  
Vol 9 (11) ◽  
pp. 444 ◽  
Author(s):  
Jinxi Li ◽  
Jie Zheng ◽  
Jiang Zhu ◽  
Fangxin Fang ◽  
Christopher. Pain ◽  
...  
Keyword(s):  
Unstructured Mesh ◽  
Computational Cost ◽  
Adaptive Mesh ◽  
Galerkin Finite Element Method ◽  
Adaptive Meshes ◽  
Galerkin Finite Element ◽  
Computational Costs ◽  
Cut Cell ◽  
Discontinuous Galerkin Finite Element ◽  
Terrain Following

Advection errors are common in basic terrain-following (TF) coordinates. Numerous methods, including the hybrid TF coordinate and smoothing vertical layers, have been proposed to reduce the advection errors. Advection errors are affected by the directions of velocity fields and the complexity of the terrain. In this study, an unstructured adaptive mesh together with the discontinuous Galerkin finite element method is employed to reduce advection errors over steep terrains. To test the capability of adaptive meshes, five two-dimensional (2D) idealized tests are conducted. Then, the results of adaptive meshes are compared with those of cut-cell and TF meshes. The results show that using adaptive meshes reduces the advection errors by one to two orders of magnitude compared to the cut-cell and TF meshes regardless of variations in velocity directions or terrain complexity. Furthermore, adaptive meshes can reduce the advection errors when the tracer moves tangentially along the terrain surface and allows the terrain to be represented without incurring in severe dispersion. Finally, the computational cost is analyzed. To achieve a given tagging criterion level, the adaptive mesh requires fewer nodes, smaller minimum mesh sizes, less runtime and lower proportion between the node numbers used for resolving the tracer and each wavelength than cut-cell and TF meshes, thus reducing the computational costs.

scholarly journals Forecasts covering one month using a cut-cell model

2013 ◽  
Vol 6 (4) ◽  
pp. 875-882 ◽  
Author(s):  
J. Steppeler ◽  
S.-H. Park ◽  
A. Dobler
Keyword(s):  
Cell Model ◽  
Control Model ◽  
Precipitation Forecast ◽  
Target Area ◽  
Strong Impact ◽  
Monthly Precipitation ◽  
Small Areas ◽  
Cut Cell ◽  
Terrain Following ◽  
The Impact

Abstract. This paper investigates the impact and potential use of the cut-cell vertical discretisation for forecasts covering five days and climate simulations. A first indication of the usefulness of this new method is obtained by a set of five-day forecasts, covering January 1989 with six forecasts. The model area was chosen to include much of Asia, the Himalayas and Australia. The cut-cell model LMZ (Lokal Modell with z-coordinates) provides a much more accurate representation of mountains on model forecasts than the terrain-following coordinate used for comparison. Therefore we are in particular interested in potential forecast improvements in the target area downwind of the Himalayas, over southeastern China, Korea and Japan. The LMZ has previously been tested extensively for one-day forecasts on a European area. Following indications of a reduced temperature error for the short forecasts, this paper investigates the model error for five days in an area influenced by strong orography. The forecasts indicated a strong impact of the cut-cell discretisation on forecast quality. The cut-cell model is available only for an older (2003) version of the model LM (Lokal Modell). It was compared using a control model differing by the use of the terrain-following coordinate only. The cut-cell model improved the precipitation forecasts of this old control model everywhere by a large margin. An improved, more transferable version of the terrain-following model LM has been developed since then under the name CLM (Climate version of the Lokal Modell). The CLM has been used and tested in all climates, while the LM was used for small areas in higher latitudes. The precipitation forecasts of the cut-cell model were compared also to the CLM. As the cut-cell model LMZ did not incorporate the developments for CLM since 2003, the precipitation forecast of the CLM was not improved in all aspects. However, for the target area downstream of the Himalayas, the cut-cell model considerably improved the prediction of the monthly precipitation forecast even in comparison with the modern CLM version. The cut-cell discretisation seems to improve in particular the localisation of precipitation, while the improvements leading from LM to CLM had a positive effect mainly on amplitude.

scholarly journals An orthogonal terrain-following coordinate and its preliminary tests using 2-D idealized advection experiments

2014 ◽  
Vol 7 (4) ◽  
pp. 1767-1778 ◽  
Author(s):  
Y. Li ◽  
B. Wang ◽  
D. Wang ◽  
J. Li ◽  
L. Dong
Keyword(s):  
Rotation Angle ◽  
Computational Grids ◽  
Time Step ◽  
Numerical Errors ◽  
Terrain Following ◽  
Steep Terrain ◽  
Definition Of ◽  
Σ Coordinate ◽  
Basis Vectors ◽  
Better Than

Abstract. We have designed an orthogonal curvilinear terrain-following coordinate (the orthogonal σ coordinate, or the OS coordinate) to reduce the advection errors in the classic σ coordinate. First, we rotate the basis vectors of the z coordinate in a specific way in order to obtain the orthogonal, terrain-following basis vectors of the OS coordinate, and then add a rotation parameter b to each rotation angle to create the smoother vertical levels of the OS coordinate with increasing height. Second, we solve the corresponding definition of each OS coordinate through its basis vectors; and then solve the 3-D coordinate surfaces of the OS coordinate numerically, therefore the computational grids created by the OS coordinate are not exactly orthogonal and its orthogonality is dependent on the accuracy of a numerical method. Third, through choosing a proper b, we can significantly smooth the vertical levels of the OS coordinate over a steep terrain, and, more importantly, we can create the orthogonal, terrain-following computational grids in the vertical through the orthogonal basis vectors of the OS coordinate, which can reduce the advection errors better than the corresponding hybrid σ coordinate. However, the convergence of the grid lines in the OS coordinate over orography restricts the time step and increases the numerical errors. We demonstrate the advantages and the drawbacks of the OS coordinate relative to the hybrid σ coordinate using two sets of 2-D linear advection experiments.

scholarly journals THE NUMERICAL TREATMENT OF NONLINEAR FRACTAL–FRACTIONAL 2D EMDEN–FOWLER EQUATION UTILIZING 2D CHELYSHKOV POLYNOMIALS

Fractals ◽  
2020 ◽  
Vol 28 (08) ◽  
pp. 2040042
Author(s):  
M. HOSSEININIA ◽  
M. H. HEYDARI ◽  
Z. AVAZZADEH
Keyword(s):  
Approximate Solution ◽  
Finite Difference ◽  
Test Problems ◽  
Operational Matrices ◽  
Algebraic Equations ◽  
Time Step ◽  
Discrete Method ◽  
Difference Formula ◽  
Fowler Equation ◽  
Chelyshkov Polynomials

This paper develops an effective semi-discrete method based on the 2D Chelyshkov polynomials (CPs) to provide an approximate solution of the fractal–fractional nonlinear Emden–Fowler equation. In this model, the fractal–fractional derivative in the concept of Atangana–Riemann–Liouville is considered. The proposed algorithm first discretizes the fractal–fractional differentiation by using the finite difference formula in the time direction. Then, it simplifies the original equation to the recurrent equations by expanding the unknown solution in terms of the 2D CPs and using the [Formula: see text]-weighted finite difference scheme. The differentiation operational matrices and the collocation method play an important role to obtaining a linear system of algebraic equations. Last, solving the obtained system provides an approximate solution in each time step. The validity of the formulated method is investigated through a sufficient number of test problems.

An Efficient Numerical Model of Pulsating Combustion and Its Experimental Validation

2009 ◽  
Author(s):  
Luca Casarsa ◽  
Pietro Giannattasio ◽  
Diego Micheli
Keyword(s):  
Numerical Model ◽  
Combustion Chamber ◽  
Second Order ◽  
Operating Conditions ◽  
Gas Dynamics Equations ◽  
Time Step ◽  
Flow Equations ◽  
Tail Pipe ◽  
Flux Difference ◽  
Pulse Jet

A simple and efficient numerical model is presented for the simulation of pulse combustors. It is based on the numerical solution of the quasi-1D unsteady flow equations and on phenomenological sub-models of turbulence and combustion. The gas dynamics equations are solved by using the Flux Difference Splitting (FDS) technique, a finite-volume upwind numerical scheme, and ENO reconstructions to obtain second-order accurate non-oscillatory solutions. The numerical fluxes computed at the cell interfaces are used to transport also the reacting species, their formation energy and the turbulent kinetic energy. The combustion progress in each cell is evaluated explicitly at the end of each time step according to a second-order overall reaction kinetics. In this way, the computations of gas dynamic evolution and heat release are decoupled, which makes the model particularly simple and efficient. A comprehensive set of measurements has been performed on a small Helmholtz type pulse-jet in order to validate the model. Air and fuel consumptions, wall temperatures, pressure cycles in both combustion chamber and tail-pipe, and instantaneous thrust have been recorded in different operating conditions of the device. The comparison between numerical and experimental results turns out to be satisfactory in all the working conditions of the pulse-jet. In particular, accurate predictions are obtained of the device operating frequency and of shape, amplitude and phase of the pressure waves in both combustion chamber and tail-pipe.

Reservoir Simulation of Variable Bubble-Point Problems

1976 ◽  
Vol 16 (01) ◽  
pp. 10-16 ◽  
Author(s):  
L.K. Thomas ◽  
W.B. Lumpkin ◽  
G.M. Reheis
Keyword(s):  
Reservoir Simulation ◽  
Water Injection ◽  
Material Balance ◽  
Point Problem ◽  
Time Step ◽  
Bubble Point ◽  
Free Gas ◽  
Flow Equations ◽  
Gas Saturation ◽  
Surrounding Areas

Abstract This paper presents the development of a general beta reservoir simulator that will model conventional (fixed bubble-point) problems as well as problems involving a variable bubble point, such as gas injection projects above the original bubble point and water injection projects resulting in a collapsing gas saturation, Provisions are included for allowing the pressure to cross the bubble point with the same relative ease as in a conventional simulator. Example problems are presented to demonstrate the utility of the model for gas and water injection problems. problems Introduction Many reservoir simulation problems involve treating a variable bubble point throughout the reservoir. For example, when gas is injected into an undersaturated reservoir, gas will go into solution, increasing the bubble point of the oil. As this oil moves away from the injector, the bubble point of surrounding areas also may increase point of surrounding areas also may increase because of mixing. During waterfloods of saturated reservoirs, the gas saturations in regions near the injectors frequently reduce to zero at pressures below the original bubble point. Thus, the bubble point will vary areally throughout the field. point will vary areally throughout the field.Recent publications have discussed certain aspects of the variable bubble-point problem. Most of these papers contain only a brief discussion of this problem. Ridings discusses the need for allowing saturation pressure to vary continuously throughout the reservoir as long as there is free gas associated with the oil. In the model presented by Cook et al., free gas saturation is monitored for saturated systems and the bubble point is set equal to the prevailing reservoir pressure when the gas saturation in a cell disappears. Bubble points for undersaturated cells are allowed to change because of the entrance of free gas and mixing. Spilletta et al. assume that a cell that is saturated or undersaturated at the beginning of a time step will remain so throughout the time step. They then solve their saturation equations for water and gas saturations. The bubble point of any undersaturated cell is adjusted to account for nonzero gas saturation, and the water saturation is modified to conserve oil. Steffensen and Sheffield devote their paper to the reservoir simulation of a collapsing gas saturation during waterflooding. In their model, blocks that have a free gas saturation at the beginning of a time step and have zero or negative gas saturations at the end of a time step are detected, and the bubble points for these cells are set equal to the estimated pressure where Sg reduced to zero. Gas saturation for these blocks is set equal to zero and S is set to 1 - S . The oil saturation in adjacent saturated grid blocks is then adjusted so that oil material balance is maintained. Mixing caused by flow between undersaturated blocks is neglected. This paper presents a comprehensive analysis of modeling variable bubble-point problems.* It treats the specific problems of simulating gas injection above the bubble point as well as waterflooding depleted reservoirs. It differs from previously reported work by accounting for the effect of bubble-point change on computed pressure change during a time step. Also, provisions are included that allow the pressure to cross the bubble point with the same relative ease as in a conventions! simulator In regard to waterflooding, mis paper differs from the work of Steffensen and Sheffield in mat it allows for bubble-point changes caused by mixing. DEVELOPMENT OF FLOW EQUATIONS To simulate the variable bubble-point problem, the expansion of the flow equations above the bubble point must include the effects of pressure and bubble point on fluid properties. Also, special consideration must be given to cells passing through the bubble point if both pressure and material-balance errors are to be eliminated. SPEJ P. 10

Multi-Element Stochastic Galerkin Method Based on Edge Detection for Uncertainty Quantification of Discontinuous Responses

2020 ◽  
Vol 5 (4) ◽  
Author(s):  
Shigetaka Kawai ◽  
Akira Oyama
Keyword(s):  
Edge Detection ◽  
Gibbs Phenomenon ◽  
Test Problems ◽  
Time Step ◽  
Computational Costs ◽  
Generalized Polynomial ◽  
Minimal Decomposition ◽  
Resolution Level ◽  
Stochastic Galerkin ◽  
Discontinuous Responses

Abstract We propose a new multi-element generalized polynomial chaos (MEgPC) method to minimize the computational costs required for the existing MEgPC to circumvent the Gibbs phenomenon in the presence of discontinuities in a random space. The proposed method uses edge detection to capture the discontinuous behavior of a solution with minimal decomposition. In contrast, the existing MEgPC iterates splitting the random space into two equal parts until achieving a sufficient resolution level. We take advantage of the fact that the stochastic Galerkin (SG) methods facilitate adaptive refinement of the decomposition at every time-step during a computation for the proposed method. The numerical experiments for two-test problems demonstrate the performance of the proposed method. The results show that the proposed method is consistently more accurate than conventional methods for sufficiently high polynomial orders with minimal additional computational costs to capture discontinuities.

Numerical simulation of higher-order diffusion-wave equations of variable coefficients using the meshless spectral method

2020 ◽  
pp. 2150010
Author(s):  
Manzoor Hussain ◽  
Sirajul Haq
Keyword(s):  
Wave Equations ◽  
Interpolation Method ◽  
Variable Coefficients ◽  
Higher Order ◽  
Implicit Time ◽  
Test Problems ◽  
Time Step ◽  
Diffusion Wave ◽  
Time Step Size ◽  
Time Stepping

In this paper, meshless spectral interpolation technique using implicit time stepping scheme is proposed for the numerical simulations of time-fractional higher-order diffusion wave equations (TFHODWEs) of variable coefficients. Meshless shape functions, obtained from radial basis functions (RBFs) and point interpolation method (PIM), are used for spatial approximation. Central differences coupled with quadrature rule of [Formula: see text] are employed for fractional temporal approximation. For advancement of solution, an implicit time stepping scheme is then invoked. Simulations performed for different benchmark test problems feature good agreement with exact solutions. Stability analysis of the proposed method is theoretically discussed and computationally validated to support the analysis. Accuracy and efficiency of the proposed method are assessed via [Formula: see text], [Formula: see text] and [Formula: see text] error norms as well as number of nodes [Formula: see text] and time step-size [Formula: see text].

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