Godunov Method Research Articles

Analysis of Wellbore Drilling Hydraulics Applying a Transient Godunov Scheme Considering Variations of Injected Flow Rates

A new application of the Godunov scheme to describe dynamic oil-well behavior is presented. The numerical model is able to capture discontinuities associated with surface flow-rate variations. The finite volume method and Riemann problems are utilized for building the unsteady discrete solution. Initial and boundary conditions are related to cases of static, steady and transient well condition. Well data used in simulation are taken from true operational conditions and well mechanical configuration. The results of Godunov’s modeling describe the behavior of transient pressure and transient flow rate inside drill pipe and annulus. These profiles are commonly caused by turning on, adjusting mud flow rate and turning off the rig pumps. The evaluated rig indicators are: back pressure, pumping pressure, bottomhole pressure and injected flow rate. Calculated transient profiles are physically consistent and in good agreement with published well data. Therefore, engineering contribution is the application of first-order Godunov method to evaluate the transient hydraulics whereas variations of mud flow rate; also, the analysis and interpretation of the dynamic pressure behavior travelling inside the well. The Godunov scheme has robust engineering applications for modeling the transient drilling hydraulics, e.g., managed pressure drilling, hydraulics of pipe connections, and foam cementing, as well.

Liquid temperature dependence of the shock formation in a cavitation bubble

The dependence of the radially convergent shock wave formation in a cavitation bubble on the surrounding liquid temperature TL in the range from 273.15 to 400 K is investigated at the liquid pressure equal to 50 bar. Realistic mathematical model is applied, in which the effects of the liquid compressibility, the heat conductivity of the vapor and liquid, the evaporation and condensation on the bubble surface are taken into account, wide-range equations of state are utilized. The governing equations of the vapor and liquid dynamics are solved numerically using a modification of the Godunov method of the second order of accuracy. It has been found that a radially convergent shock wave arises in the bubble in 273.15≤T_L≤375 К. In this interval, the distance between the shock wave formation position and the bubble surface decreases with decreasing the liquid temperature. The possibility of using a known simplified criterion of the formation of a shock wave inside a bubble to estimate its formation position under the studied conditions is considered. It is shown that with applying that criterion the shock wave formation position turns out to be correctly predicted at T_L≈325 К, while at T_L>325 К and T_L<325 К it is predicted closer to and more distant from the bubble surface, respectively.

Schemes of the second order accuracy for computing the dynamics of disturbances in an elastic body

The efficiency of the second-order accurate UNO- and TVD-modifications of the Godunov method is compared using a number of problems on the propagation of linear waves in an elastic body, their interaction with each other and with the surface of the body. In particular, one-dimensional problems having analytic solutions and a two-dimensional problem of the dynamics of a body in the vicinity of the impact domain on its free surface are considered. It is shown that if in the problems there are well-marked extrema or short waves, then the UNO-scheme is more effective, since in such cases decrease in the accuracy of the TVD-scheme becomes apparent due to strictly satisfying the TVD condition. Because of approximately satisfying the TVD condition, the UNO-scheme can lead to the appearance of oscillations of numerical nature at the level of approximation errors. However, this does not reduce the efficiency of the UNO scheme since the amplitude of those oscillations decreases with refinement of the grid.

Testing the First Order Accurate Godunov Method on Some Prototype and Applied Problems

The Godunov method of first order accuracy continues to attract the attention of researchers both because of its simple implementation and its sufficiently high numerical accuracy subject to relatively modest resource requirements. However, it is necessary to constantly verify that the method indeed produces acceptable accuracy. We test the method on a number of one- and two-dimensional prototype problems and compare the results with analytical and numerical solutions obtained by higher accuracy schemes. The results for the method tested are quite acceptable. We also compare the solutions of applied problems on flows in supersonic, freely expanding jets and on supersonic flow past blunt bodies. The results for these problems are also positive.

Performance Evaluation of Various Phase Change Materials for Thermal Energy Storage of A Solar Cooker via Numerical Simulation

In this paper, thermal performance of various phase change materials (PCMs) used as thermal energy storage in a solar cooker has been investigated numerically. Heat conduction equations in cylindrical domain are used to model heat transfer of the PCMs. Mathematical model of phase change problem in the PCM storage encompasses heat conduction equations in solid and liquid region separated by moving solid-liquid interface. The phase change problem is solved by reformulating heat conduction equations with emergence of moving boundary into an enthalpy equation. Numerical solution of the enthalpy equation is obtained by implementing Godunov method and verified by analytical solution of one-dimensional case. Stability condition of the numerical scheme is also discussed. Thermal performance of various PCMs is evaluated via the stored energy and temperature history. The simulation results show that phase change material with the best thermal performance during the first 2.5 hours of energy extraction is shown by erythritol. Moreover, magnesium chloride hexahydrate can maintain temperature of the PCM storage in the range of 110-116.7°C for more than 4 hours while magnesium nitrate hexahydrate is effective only for one hour with the PCM storage temperature around 121-128°C. Among the PCMs that have been tested, it is only erythritol that can cook 10 kg of the loaded water until it reaches 100°C for about 3.5 hours.Article History: Received June 22nd 2016; Received in revised form August 26th 2016; Accepted Sept 1st 2016; Available onlineHow to Cite This Article: Tarwidi, D., Murdiansyah, D.T, Ginanja, N. (2016) Performance Evaluation of Various Phase Change Materials for Thermal Energy Storage of A Solar Cooker via Numerical Simulation. Int. Journal of Renewable Energy Development, 5(3), 199-210.http://dx.doi.org/10.14710/ijred.5.3.199-210

The calculation of weakly non-spherical cavitation bubble impact on a solid

The effect of small spheroidal non-sphericity of a cavitation bubble touching a solid at the beginning of its collapse on its impact on the solid of a copper-nickel alloy is investigated. The impact on the solid is realized by means of a high-speed liquid jet arising at collapse on the bubble surface. The shape of the jet, its velocity and pressure are calculated by the boundary element method. The spatial and temporal characteristics of the pressure pulses on the solid surface are determined by the CIP-CUP method on dynamically adaptive grids without explicitly separating the gas-liquid interface. The solid surface layer dynamics is evaluated by the Godunov method. The results are analyzed in dimensionless variables obtained with using the water hammer pressure, the time moment and the jet-solid contact area radius at which the jet begins to spread on the solid surface. It is shown that in those dimensionless variables, the dependence of the spatial and temporal characteristics of the solid surface pressure pulses on the initial bubble shape non-sphericity is relatively small. The nonsphericity also slightly influences the main qualitative features of the dynamic processes inside the solid, whereas its effect on their quantitative characteristics can be significant.

Non-reflecting coupling method for one-dimensional finite difference/finite volume schemes based on spectral error analysis

Abstract For the compressible coupled simulation of piping systems including fluid machinery, an embedment of 3D finite volume schemes for active components such as pumps and compressors into 1D finite difference characteristic methods for passive components (e.g. pipes, valves) offers a sophisticated 3D investigation of the machinery with a moderate simulation effort for the entire system. While a 3D-1D coupling method for finite volume Godunov schemes is available from preliminary studies, in the present paper, a new non-reflecting coupling method for 1D finite volume Godunov and 1D finite difference characteristic methods is presented, and the mechanisms for spurious reflections at the coupling interface are revealed. The proposed methodology is based on spectral error analysis and is in general applicable to the coupling of any fundamentally different numerical schemes. It is demonstrated that by matching the numerical diffusion and speed of sound in both coupled domains, reflections can be minimized.

The Godunov Method for Calculating Multidimensional Flows of a One-Velocity Multicomponent Mixture

Computational formulas of the Godunov method are given for the equations of a generalized-equilibrium model of a heterogeneous medium on a curvilinear grid; with the use of this method, the problems of interaction of air shock waves with bubbles of various gases are investigated. Flow of a gas–liquid mixture in a nozzle mouthpiece is considered. The Prandtl–Meyer flow of a water–air mixture is calculated and compared with a self-similar solution.

Solving two-phase shallow granular flow equations with a well-balanced NOC scheme on multiple GPUs

A two-phase shallow granular flow model consists of mass and momentum equations for the solid and fluid phases, coupled together by conservative and non-conservative momentum exchange terms. Development of classic Godunov methods based on Riemann problem solutions for such a model is difficult because of complexity in building appropriate wave structures. Non-oscillatory central (NOC) differencing schemes are attractive as they do not need to solve Riemann problems. In this paper, a staggered NOC scheme is amended for numerical solution of the two-phase shallow granular flow equations due to Pitman and Le. Simple discretization schemes for the non-conservative and bed slope terms and a simple correction procedure for the updating of the depth variables are proposed to ensure the well-balanced property. The scheme is further corrected with a numerical relaxation term mimicking the interphase drag force so as to overcome the difficulty associated with complex eigenvalues in some flow conditions. The resultant NOC scheme is implemented on multiple graphics processing units (GPUs) in a server by using both OpenMP-CUDA and multistream-CUDA parallelization strategies. Numerical tests in several typical two-phase shallow granular flow problems show that the NOC scheme can model wet/dry fronts and vacuum appearance robustly, and can treat some flow conditions associated with complex eigenvalues. Comparison of parallel efficiencies shows that the multistream-CUDA strategy can be slightly faster or slower than the OpenMP-CUDA strategy depending on the grid sizes.

Using the PPML approach for constructing a low-dissipation, operator-splitting scheme for numerical simulations of hydrodynamic flows

An approach for constructing a low-dissipation numerical method is described. The method is based on a combination of the operator-splitting method, Godunov method, and piecewise-parabolic method on the local stencil. Numerical method was tested on a standard suite of hydrodynamic test problems. In addition, the performance of the method is demonstrated on a global test problem showing the development of a spiral structure in a gravitationally unstable gaseous galactic disk.

Solving numerically nonlinear systems of balance laws by multivariate sigmoidal functions approximation

A numerical method for solving systems of nonlinear one-dimensional balance laws, based on multivariate sigmoidal functions approximation, is developed. Constructive approximation theorems are first established in both, uniform and $$L^p$$ norms. A priori as well as a posteriori error estimates are derived for the numerical solutions, when various kinds of sigmoidal functions, such as unit step, logistic and Gompertz functions, are chosen. The residual of the numerical method is also estimated. Numerical examples are given to test the performance of the algorithm, a comparison with the Godunov method is made concerning accuracy and computational cost. Finally, the numerical stability of the method is analyzed.

Study of a New Asymptotic Preserving Scheme for the Euler System in the Low Mach Number Limit

This article deals with the discretization of the compressible Euler system for all Mach numbers regimes. For highly subsonic flows, since acoustic waves are very fast compared to the velocity of the fluid, the gas can be considered as incompressible. From the numerical point of view, when the Mach number tends to zero, the classical Godunov type schemes present two main drawbacks: they lose consistency and they suffer of severe numerical constraints for stability to be guaranteed since the time step must follow the acoustic waves speed. In this work, we propose and analyze a new unconditionally stable an consistent scheme for all Mach number flows, from compressible to incompressible regimes, stability being only related to the flow speed. A stability analysis and several one and two dimensional simulations confirm that the proposed method possesses the sought characteristics. 1. Introduction. Almost all fluids can be said to be compressible. However, there are many situations in which the changes in density are so small to be considered negligible. We refer to these situations saying that the fluid is in an incompressible regime. From the mathematical point of view, the difference between compressible and incompressible situations is that, in the second case, the equation for the conservation of mass is replaced by the constraint that the divergence of the velocity should be zero. This is due to the fact that when the Mach number tends to zero, the pressure waves can be considered to travel at infinite speed. From the theoretical point of view, researchers try to fill the gap between those two different descriptions by determining in which sense compressible equations tend to incompressible ones [2, 20, 21, 22, 33]. In this article we are interested in the numerical solution of the Euler system when used to describe fluid flows where the Mach number strongly varies. This causes the gas to pass from compressible to almost incompressible situations and consequently it causes most of the numerical methods build for solving compressible Euler equations to fail. In fact, when the Mach number tends to zero, it is well known that classical Godunov type schemes do not work anymore. Indeed, they lose consistency in the incompressible limit. This means that when close to the limit, the accuracy of theses schemes is not sufficient to describe the flow. Many efforts have been done in the recent past in order to correct this main drawback of Godunov schemes, for instance by using preconditioning methods [34] or by splitting and correcting the pressure on the collocated meshes [5], [9, 10], [12], [13, 14, 30], [15], [23, 24], [26, 27, 28], or instead by using staggered grids like in the famous MAC scheme, see for instance [3], [16], [17], [18], [19], [31]. Unfortunately, even if these approaches permit to bypass the consistency problem of Godunov methods, they all need to resolve the scale of the acoustic waves in the fluid in order to remain stable. This means that they suffer from a restrictive CFL (Courant-Frierichs-Levy) condition which is inversely proportional to the Mach number value. In this work, we derive a method which

On the Riemann Problem Simulation for the Drift-Flux Equations of Two-Phase Flows

This work presents computational simulations and analytical techniques for solving the drift-flux two-phase flow model. The model equations are formulated to describe the exact solution of the Riemann problem. The solution is constructed by solving the conservation of mass for each phase and the mixture conservation momentum equation of the two phases under isothermal conditions. Particular attention is given to address the expressions for jump relationships and the Riemann invariants. The performance of the developed Riemann solver is assessed with respect to different test cases selected from the literature. Comparisons with Godunov methods of centred-type are provided to demonstrate the use of the proposed exact and computational framework. Excellent agreement is observed between analytical results and numerical predictions.

Traffic Lights or Roundabout? Analysis using the Modified Kinematic LWR Model

AbstractTraffic flow is treated as a continuum governed by the kinematic LWR model and the Greenshield flux function. The model is modified to describe traffic flow on a road with traffic lights or a roundabout. Parameters introduced determine the traffic flow behaviour and queue formation, and numerical simulations based on the Godunov method are carried out. The numerical procedure is shown to converge, and produces results consistent with previous analytic results.

One-Dimensional Numerical Framework for Shock Compaction of Cellular Foams

AbstractA one-dimensional (1D) finite-volume implementation, based on the second-order Godunov method for predicting dynamic response of foams that exhibit irreversible compaction, is presented. Cellular foams, with an upward concave stress-strain relationship associated with densification of the material resulting from collapse of the cell structure, have the possibility of a strong discontinuity with shock-type characteristics. An approximate solution to the local Riemann problem is developed considering all possible wave structure(s) in the material based on the quasi-static response of the material. The prediction of dynamic compaction response of the foam subjected to solid impact is shown to compare favorably with experimental results. For an applied blast pressure loading, attenuation of transmitted stress wave in the foam is shown to be a result of the energy dissipation provided by compaction of the foam.

Modifications of the Godunov method for computing disturbance propagation in an elastic body

Numerical investigation of efficiency of UNO- and TVD-modifications of the Godunov method of the second order accuracy for computation of linear waves in an elastic body in comparison with the classical Godunov method is carried out. To this end, one-dimensional cylindrical Riemann problems are considered. It is shown that the both modifications are considerably more accurate in describing radially converging as well as diverging longitudinal and shear waves and contact discontinuities both in one- and two-dimensional problem statements. At that the UNO-modification is more preferable than the TVD-modification because exact implementation of the TVD property in the TVD-modification is reached at the expense of “cutting” solution extrema.

Inflow Outflow Effect and Shock Wave Analysis in a Traffic Flow Simulation

This paper investigates the effect of inflow, outflow and shock waves in a single lane highway traffic flow problem. A constant source term has been introduced to demonstrate the inflow and outflow. The classical Lighthill Whitham and Richards (LWR) model combined with the Greenshields model is used to obtain analytical and numerical solutions. The model is treated as an IBVP and numerical solutions are presented using Lax Friedrichs scheme. Godunov method is also used to present shock wave analysis. The numerical procedures adopted in this investigation yield results which are very much consistent with real life scenario in terms of traffic density and velocity.

Hydrodynamic modeling of QGP expansion using an exact solution of Riemann problem

Hydrodynamic modelling of quark-gluon plasma requires sophisticated numerical schemes that have low numerical viscosity and are able to cope with high gradients of energy density that may appear in initial conditions. We propose to use the Godunov method with an exact Riemann solver for ideal hydrodynamic modelling to meet these conditions. We present the results of numerical tests of the method, such as the sound wave propagation and the shock tube problem, which show both high precision and low numerical viscosity.

Stability Analysis of Discontinuous Galerkin Discrete Shock Profiles for Steady Scalar Conservation Laws

We present a stability analysis of stationary discrete shock profiles for a discontinuous Galerkin method approximating scalar nonlinear hyperbolic conservation laws with a convex flux. We consider the Godunov method to evaluate the numerical flux and use either an explicit third-order Runge--Kutta scheme or an implicit backward Euler scheme for the time integration. Applying a linear stability analysis, we show that the steady solutions may become unstable when the numerical flux is not differentiable. In particular, the situation of a shock at an interface of the mesh corresponds to an unstable solution for a space discretization accuracy higher than second-order, whatever the time integration method and the physical flux. Spectra of the linearized operator indicate that the fourth-order numerical scheme with the explicit time integration is also unstable for a continuous range of shock positions around an interface.

Modeling of the Interaction of an Underwater Shock Wave and an Obstacle in the Presence of a Bubble Screen

The author has described a modified Godunov method intended to integrate hyperbolic equations of a generalized equilibrium model in divergent form. With the indicated method, the author has investigated the problem of interaction of an underwater shock wave and an obstacle in the presence of a bubble screen.