Two-step Lax-Wendroff Method Research Articles

Novel algorithm for compressible Newtonian axisymmetric thermal flow

In this paper, we mainly focus on the development of the Taylor-Galerkin/Pressure-Correction method (TG/PC) method to solve the flow of a compressible Newtonian fluid under nonisothermal conditions. The process of development needs applying the two-step Lax–Wendroff method on the energy equation, after which two new steps for energy equations will be obtained within the TG/PC algorithm. In addition, for the density component, the Tait equation of state is adopted to relate density to pressure, which also yields a new step within the novel method to give the correction formula of compressible pressure. To perform the analysis of the new algorithm, Poiseuille flow through axisymmetric rectangular channel for the Newtonian thermal flow is utilized as a simple problem test. The effect of nondimensional factors such as Reynolds number (Re) and Prandtl number (Pr) is discussed in this study. The influence of thermal conductivity [Formula: see text] and viscosity [Formula: see text] on the solution components is presented as well. Also, comparison for both compressible and incompressible flows is provided in addition to a comparison between both cases’ convergence, as it turns out that the convergence of the incompressible case is better than the compressible case. All the results of the effects presented in this study agree well with the approved physical trend.

1D-2D mathematical modeling of wave refraction and wave attenuation by mangrove forests

This paper presents a theoretical and numerical analysis based on a model of mangrove forests incorporating stems and roots to determine the nearshore wave angle and wave height for one-dimensional (1-D) and two-dimensional (2-D) problems. The shallow water equations are solved using the two-step Lax-Wendroff method. The resulting wave height and wave direction vectors are displayed and analyzed to assess the impact of mangrove effects on various bathymetries, including constant, sloping, and parabolic profiles. This procedure is also applied to actual bathymetric data obtained from the Hatiya channel to analyze the influence of mangrove forests and shoreline geometry on the amplitude and wave directionality of nearshore waves. The obtained numerical results demonstrate that both coastal geometry and vegetation exhibit non-linear characteristics in wave behavior. Furthermore, these factors significantly contribute to the directionality and attenuation of nearshore waves.

Comparative Review on Computational Performance of Multistep Schemes in Solving One-Dimensional Linear Wave Equation

Among several numerical methods used to solve the hyperbolic model of the linear wave equation, single-step algorithms can be the more popular ones. However, these algorithms are time-consuming while incurring numerical inaccuracy. Thus, multistep methods can be a suitable option as it has a high order of accuracy. This study aims to investigate and compare the computational performance of these multistep schemes in solving hyperbolic model based on one-dimensional linear wave equation. The techniques studied in this paper comprise the two-step Lax-Wendroff method, MacCormack method, second-order upwind method, Rusanov-Burstein-Mirin method, Warming-Kutler-Lomax method, and fourth-order Runge-Kutta method. Finite difference method is applied in discretisation. Our simulation found that although higher-order multistep methods are more stable than single-step algorithm, they suffer numerical diffusion. The two-step Lax-Wendroff method outperforms other schemes, although it is relatively simple compared with the other three and four steps schemes. The second-order upwind method is attractive as well because it is executable even with a high Courant number.

Development of a transient gas dynamic model for the simulation of pulsation in reciprocating compressor piping systems

The subject of this paper is to develop a nonlinear transient dynamic model for simulating the pressure pulsation in reciprocating compressor piping systems. The model allows the interaction between the piping response and compressor processes. The two-step Lax–Wendroff method is employed to obtain solutions to the unsteady flow equations at internal points in a pipe, and the Trapezoidal version of the method of characteristics is adopted for handling boundary conditions. The compressor, which serves as a boundary condition for predicting the piping's pressure pulsation, is modeled comprehensively on the basis of the first law of thermodynamics, the valve dynamics assumed as a one-degree-of-freedom system and the flow through valve by introducing an effective area for flow through the valve throat. It is worth to note that, to solve the compressor boundary condition, all the related equations are necessary to be solved simultaneously to obtain the pulsating pressure in the pipe end, pressure variation in the cylinder, valve response, mass flow rate, indicated power, and other compressor performance parameters. Numerical results based on these proposed modeling techniques show a good agreement with previous measured data.

Numerical analysis of gas pulsation attenuation characteristics of a perforated tube in a reciprocating compressor piping system

This paper presents a non-linear mathematical model to simulate the pulsating flow in a compressor piping system with perforated tube geometries. The governing equations are numerically solved by a composite scheme incorporating the two-step Lax–Wendroff method with the trapezoidal integration version of the method of characteristics at the pipe boundaries. The validity of the mathematical model and the methodology was verified by comparing the analytical predictions with experimental results. On basis of the confirmed model, the effects of a perforated tube on pressure pulsations in the piping system were investigated. A further investigation on the effects of the geometrical parameters and compressor functioning conditions was also conducted. The results demonstrated that pressure fluctuations at the compressor cylinder nozzle could be significantly reduced with a ratio of 53.1% when a perforated tube was installed between compressor and surge volume, but it had little influence on the pressure pulsations in the piping downstream of the surge volume. It is worth mentioning that a longer perforated tube yielded a better damping performance while the hole diameter had little effect on the pressure pulsations. The magnitude to which the perforated tube could damp the pulsations changed under different functioning conditions and the damping capacity was less sensitive to the mean pressure in the piping than to the compressor running speed.

State-space estimation with a Bayesian filter in a coupled PDE system for transient gas flows

The accuracy of the first-principle models describing the evolution of gas dynamics in pipelines is sometimes limited by the lack of understanding of the gas transport phenomena. In this paper, a stochastic filtering approach is proposed based on a sequential Monte Carlo method to provide real-time estimates of the state in gas pipelines. After constructing a state-space model of the compressible single-phase flow based on the laws of conservation of mass and momentum, the optimal sequential importance resampling filter (SIR) is implemented. The state variables are updated with simulated measurements. The two-step Lax–Wendroff method is used for the discretization of the partial differential equations describing the gas model in both space and time to obtain finite-dimensional discrete-time state-space representations. The system states are then combined into an augmented state vector. The resulting nonlinear state-space model is used for the design of the particle filter that provides real-time estimations of the system states. Simulation results for a coupled PDE system describing an unsteady isothermal gas flow demonstrate the effectiveness of the proposed method. A sensitivity analysis is conducted to examine the performance of the filter for different model and observation error covariances and observation intervals.

Analysis of shock capturing methods for chemical species transport in unsteady compressible flow

This paper presents a chemical species transport model to account for variable composition and gas properties along the flow path in internal combustion engines. The numerical solution to adapt the gas dynamic model to chemical species transport in boundary conditions by means of the Method of Characteristics and in volumes by means of a filling and emptying model is described. The performance for chemical species transport in 1D elements of shock-capturing methods, such as the two-step Lax–Wendroff method and the Sweby’s TVD scheme considering several flux limiter definitions, is carried out by means of shock-tube tests. The influence of the fluid modelling as perfect or non-perfect gas on the numerical methods features and the flow characteristics on shock-tube results are analysed.

Blood flow through compliant vessels after endovascular repair: wall deformations induced by the discontinuous wall properties

In this paper we derive a set of equations which can be used to study wall deformations and transmural pressure at the anchoring sites of endovascular prostheses. The equations are the jump conditions associated with the underlying model equations. The model equations are derived from the Navier–Stokes equations to describe the blood flow through compliant axi-symmetric vessels after endovascular repair. They are in the form of a quasilinear hyperbolic system of partial differential equations with discontinuous coefficients. Since the weak form of the equations contains the product of the Dirac delta distribution with the Heaviside function, the jump conditions and the weak form cannot be obtained using the standard distribution theory. Driven by the undelying application in mind, we present a preliminary analysis leading to the jump conditions by interpreting the ambiguous products as a mean value with respect to the measure obtained in the limit of the “regularizing kernels” [17]. We show that the numerical solution obtained by using the Richtmyer two-step Lax–Wendroff method satisfies the weak form of the equations associated with a symmetric regularizing kernel in which case the weak form is independent of the particular choice of the kernel. We give an example (treatement of aortic abdominal aneurysm using multiple overlapping stents) where the conditions obtained in this paper can be used in the optimal design of an endovascular procedure.

Transient Response of a Cross-Flow Charge Air Intercooler and Its Influence on Engine Operation

To examine the influence of intercooler thermal inertia on transient engine operation, a 2D model of an air-to-air, cross-flow heat exchanger has been developed. Finned passages of heat exchanger core have been subdivided into separate channels of charge and cooling air, respectively. A two-step Lax-Wendroff differential method has been used to solve one-dimensional, nonhomentropic, unsteady, compressible fluid flow in each channel. Simultaneously, the Saul’yev explicit method has been applied to compute the 2D temperature distribution along the dividing plate. The Wieting correlation has been used to compute the local convection heat transfer coefficient and friction factor. The model has been verified against steady-state experimental data and then incorporated into an engine cycle simulation program based on “Filling-Emptying” method. Two engine transients have been simulated; the acceleration of the engine from idle to rated engine speed at constant load, and deceleration from rated power to rated torque by increasing the load torque. The first example shows the warming up of the intercooler, while in the second example intercooler temperatures are decreasing. The results have been compared with the predictions of an additional set of simulations, where the NTU-effectiveness method has been used to simulate the intercooler, and the thermal inertia of intercooler has been neglected. The results of both sets of simulations are discussed in the paper. [S0022-0434(00)02003-7]

Modelling one-dimensional unsteady flows in ducts: Symmetric finite difference schemes versus galerkin discontinuous finite element methods

The Euler equations for one-dimensional unsteady flows in ducts have been solved resorting to classical symmetric shock-capturing methods with second-order accuracy and to the recent discontinuous Galerkin finite-element method, with second- and third-order accuracy. In particular, the finite difference techniques adopted are the two-step Lax—Wendroff method and the MacCormack predictor—corrector method, with the addition of the flux corrected transport (FCT) or of the Davis nonupwind TVD scheme to suppress the spurious oscillations in the vicinity of discontinuous solutions. The finite-element method adopted is based on the weak formulation of the Euler equations, which are solved by introducing a discontinuous finite-element space discretization. A dissipative mechanism has been considered to supplement the FEM with a “discontinuity capturing” operator, adding a “viscous like” term to damp minor numerical overshoots arising in proximity of steep gradients of the solution. The numerical tests chosen to carry out a comparison between these schemes are the shock-tube problem and the shock—turbulence interaction problem. Both the test cases considered show the superiority of third-order FEM calculations, whereas the comparison between the computer run times points out the greater computational effort required.

The simulation of gas dynamics in engine manifolds using non-linear symmetric difference schemes

Symmetric difference schemes, such as the two-step Lax—Wendroff method or MacCormack's method, are used extensively in engine simulation codes in order to achieve second-order accuracy. These schemes, however, produce spurious oscillations when shock waves and contact surfaces are encountered and require the addition of non-linear terms (flux limiters) to cope with such phenomena. The paper reviews briefly the general theory behind the formulation of such nonlinear algorithms and examines the performance of two schemes in particular. The flux corrected transport (FCT) technique has been adopted by several research groups for this purpose and appears to operate satisfactorily for many flows in engine manifolds. It is shown in the present work that, under some circumstances, this method can distort the solution and produce large errors in mass conservation. The symmetric flux limiter due to Davis, based on the total variation diminishing (TVD) criterion, is shown here to avoid such problems and requires similar computational effort to the FCT approach.

Computational simulation of blood flow in human systemic circulation incorporating an external force field

A quasi-one-dimensional non-linear mathematical model for the computation of the blood flow in the human systemic circulation is constructed. The morphology and physical modelling of the whole system (arteries, capillaries and veins) are completed by different methods for the different vessel generations. A hybrid method is used to solve the problem numerically, based on the governing equation (continuity, momentum and state equations), the input boundary conditions and the predetermined initial conditions. The two-step Lax-Wendroff finite-difference method is used to compute variables for each individual vessel, and the characteristic method is employed for the computation of internal boundary conditions of the vessel connection and the input and output system boundary conditions. Using this approach, blood flow, transmural pressure and blood velocity are computed at all vessel sites and for each time step. The pressure and flow waveforms obtained show reasonable agreement with clinical data and results reported in the literature. When an external conservative force field is applied to the system, the results computed from the model are intuitively correct. The term representing the external pressure added to the system by the muscle, which represents active control on the cardiovascular system, is also embodied in this model.

An implicit method for transient gas flows in pipe networks

This paper describes a fully implicit finite-difference method for calculating the unsteady gas flow in pipeline networks. The algorithm for solving the finite-difference equations of a pipe is based on the Newton-Raphson method. The Von Neumann stability analysis on the finite-difference equations of a pipe shows that the equations are unconditionally stable. An iterative convergence method is applied to the calculation of node pressure at junctions in networks. The parameter study on the convergence shows that the stability depends on the convergence tolerance. Calculation results of a few sample cases are compared with those of the method of characteristics and the two-step Lax-Wendroff method. An excellent agreement between the methods is obtained when a small time step is used. Computation time can be greatly reduced by using the implicit method.

An Implicit Method for Transient Flow in Pipe Networks.

This paper describes a fully implicit finite-difference method for calculating unsteady gas flow in pipeline networks. The algorithm for solving the finite-difference equations of a pipe is based on the iterative Newton-Raphson method. The standard Neumann stability analysis of the finite-difference equations of a pipe shows that the equations are unconditionally stable. For the calculation of node pressure at junctions in networks, an iterative convergence method is applied. Studies on the convergence are performed parametrically. The result shows that the stability depends on the convergence tolerance. Calculation results of a few sample cases are compared with those of the method of characteristics and the two-step Lax-Wendroff method. An excellent agreement is obtained in a small time step. Computation time can be reduced by the method.

A study of high-speed pulsed water jets. 2nd report, Numerical analysis of unsteady flow in straight nozzles.

Pulsed water jets at the velocities of 300m/s to 500m/s produced by piston impact are examined by numerical analysis. Straight nozzles are used with the outlet diameters of 5 and 10mm. The two-step Lax-Wendroff method is employed for the unsteady axisymmetric two-phase flow in the nozzles using a Flux-Corrected Transport algorithm to limit artificial diffusion. After short transition, the flow in the nozzles arrives at a quasi-steady state. In the nozzle outlets, the contract flow and the separation with strong cavitation are formed. The vapor cavities also spread into the water column. Because of the short transition and contract flow, the jet speed of the straight nozzles is higher than that of the taper nozzles with the same nozzle outlet diameter. On the assumption that the flow in the nozzles is steady, the maximum jet speed can be approximated. It is illustrated that the nozzle shape affects the jet speed.

Reconnection and fluctuations in compressible MHD: A comparison of different numerical methods

Methods used in the past in order to solve numerically the resistive and compressible magnetohydrodynamic equations as an initial value problem are the two-step Lax-Wendroff method and the spectral method. Furthermore, it has recently been proposed to use direct finite space differences with a high-order time advance. This method should have considerably less numerical damping at small wavelengths than the Lax-Wendroff method. We will compare these three methods by solving the problem of reconnection in a periodic double current sheet system with superimposed magnetic field fluctuations. For the time advance we use the third-order Adams-Bashforth method. In addition to these methods we will study the behaviour of a fourth method based on the third-order Adams-Bashforth time advance and a determination of the spatial derivatives by cubic spline functions whose derivatives up to second order are continous at the grid points. In order to evaluate the various methods at small scalelengths, we analyze the time behaviour of the enstrophy (mean square vorticity) and of the maximum current density in the system. We found that the spectral method with wavenumbers |k| ≤32 and the spline method with a 64 × 64 grid give comparable results. The Lax-Wendroff method with a 128 × 128 grid (which contains the values at the half and at the full time step) gives results similar to the 64 × 64 finite difference method. However, both these methods have a large numerical dispersion and lead to unreasonably large values of the enstrophy and maximum current density. The finite difference method, although having less damping, has similar numerical dispersion properties as the Lax-Wendroff method. We then used the spline method in order to investigate the dependence of the reconnection process on the initial noise level. The maximum enstrophy produced during the magnetic island generation is directly proportional to the energy in the initial fluctuations.

Magnetic fields interacting with nonlinear compressible convection

Two-dimensional numerical simulations are used to study fully compressible convection in the presence of an imposed magnetic field. Highly nonlinear flows are considered that span multiple density scale heights. The convection tends to sweep the initially uniform vertical magnetic field into concentrated flux sheets with significant magnetic pressures. These flux sheets are partially evacuated, and effects of buoyancy and Lorentz forces there can serve to suppress motions. The flux sheets can be surrounded by a sheath of descending flow. If the imposed magnetic field is sufficiently strong, the convection can become oscillatory. The unstably stratified fluid layer has an initial density ratio (bottom to top of layer) of 11. Surveys of solutions at fixed Rayleigh number sample Chandrasekhar numbers from 1 to 1000 and magnetic Prandtl numbers from 1/16 to 1. These nonlinear simulations utilize a two-dimensional numerical scheme based on a modified two-step Lax-Wendroff method.

Comparison of unsteady flow calculations in a pipe by the method of characteristics and the two-step differential Lax-Wendroff method

This paper introduces a comparison of two of the methods which are most frequently used, at present, for the solution of an unsteady flow in the pipe system of internal combustion engines. These are (i) the method of characteristics and (ii) the two-step differential Lax-Wendroff method. In the case of homentropic flow a comparison is made between the lattice-point and mesh method of characteristics, and the two-step differential method is shown. In the case of non-homentropic flow the mesh method and two-step differential methods are also compared. The results show that for non-homentropic flow the two-step Lax-Wendroff method is faster and for engineering applications it is easier to implement on the computer.

Comparison of unsteady flow calculations in a pipe by the method of characteristics and the two-step differential Lax-Wendroff method

This paper introduces a comparison of two of the methods which are most frequently used, at present, for the solution of an unsteady flow in the pipe system of internal combustion engines. These are (i) the method of characteristics and (ii) the two-step differential Lax-Wendroff method. In the case of homentropic flow a comparison is made between the lattice-point and mesh method of characteristics, and the two-step differential method is shown. In the case of non-homentropic flow the mesh method and two-step differential methods are also compared. The results show that for non-homentropic flow the two-step Lax-Wendroff method is faster and for engineering applications it is easier to implement on the computer.

Interaction between a Shock-Heated Plasma and a Nozzle-Shaped Magnetic Field

The magnetohydrodynamic (MHD) equations for the interaction of a shock-heated plasma flow with a nozzle-shaped magnetic field are solved numerically with the two-step Lax-Wendroff method using the time-splitting technique. The results are compared with the experimental data obtained with an arc-driven shock tube. The measured output voltage of pickup coils and the measured luminosity profiles in the shock front agree well to calculations.