One-dimensional Model Equations Research Articles

Adaptive energy group division in the few-group cross-section generation for full spectrum reactor modeling with deterministic method

Advanced nuclear reactors, especially the newly developed small and micro-reactors have complex neutron spectrum, which makes the deterministic reactor core calculations sensitive to the energy group structure of few-group cross-sections. To avoid significantly increasing the cost of energy discretization in the core calculation, two energy group structures with 31 groups and 33 groups were adopted for typical thermal and fast reactor cores, respectively. Then, an adaptive scheme of group division for reactor cores with a medium neutron spectrum was proposed. The works were based on the full spectrum nuclear reactor analysis code SARAX/TULIP. An equivalent one-dimensional model of the core was proposed to capture the key neutron spectrum features of the reactor core. Such features were used to adaptively determine a few-group structure for the following reactor core calculations. Then, the neutron spectrum in different zones with more details was calculated. With this spectrum, the cross-sections were condensed into the determined energy groups. Three tests based on different neutron spectrum were calculated to verify the schemes. The results show that using the adaptive energy group division scheme, the following core calculation can meet the accuracy requirement of different reactors with different neutron spectra.

A sonic black hole of a rectangular cross-section

In this study, a rectangular cross-section sonic black hole (SBH) is explored and a one-dimensional model equation for the acoustic field inside it is presented. The model equation's applicability is verified through a comparison with the corresponding two-dimensional Helmholtz equation. A general analytical solution, which takes into account the SBH's quadratically varying internal cross-section, is derived using linearly independent confluent Heun functions. To investigate the acoustic wave behavior in the SBH, the wave splitting method is utilized to obtain forward and backward acoustic pressure waves, and the reflection coefficient is calculated. To confirm the validity of the analytical solution, a comparison is made with numerical results obtained from the Riccati equation. Within the context of this study, a detailed discussion is provided regarding the applicability of the analytical solution.

Derivation and stability analysis of two-fluid model equations for bubbly flow with bubble oscillations and thermal damping

The two-fluid model with bubble oscillations, proposed by Egashira et al. (2004), can explain the properties of cavitating bubbly flow and pressure wave propagation in the bubbly liquid. However, the viscous effect as well as energy conservation leading to temperature changes inside the bubble with bubble oscillations have not yet been considered. Hence, this study aimed to incorporate the viscous (bulk viscosity and drag) and thermal effects to the previously proposed two-fluid model with bubble oscillations. Bulk viscosity was considered by averaging the shear stress term in the single-phase momentum conservation for a Newtonian fluid, and the drag was introduced by transforming the interfacial shear stress. We derived the averaged energy conservation for a general two-phase flow with a thermal conduction inside bubbles and heat transfer between the two phases, and limited this equation to that for a bubbly flow by closing the interfacial temperature gradient term via constitutive equations for a single bubble. Furthermore, we investigated the stability of our proposed one-dimensional model equations using the dispersion analysis. This analysis provided the following insight: (i) The difference in the temperature gradient models had a slight effect on the stability of the proposed model equations; (ii) the thermal conduction inside the bubbles was dominant in the thermal damping in bubbly flows rather than the heat transfer between the two phases; (iii) incorporating both the bulk viscosity and drag stabilized the proposed model equations. Our results provide insights into the development of mathematical models to investigate the thermal effects in bubbly flow with bubble oscillations, such as cavitating bubbly flow and wave propagation in bubbly liquids.

Implementation of low-storage Runge-Kutta time integration schemes in scalable asynchronous partial differential equation solvers

The asynchronous computing method based on finite-difference schemes has shown promise in significantly improving the scalability of time-dependent partial differential equation (PDE) solvers by either relaxing data synchronization or avoiding communication between processing elements (PEs) on massively parallel machines. This method uses high-order accurate asynchrony-tolerant (AT) schemes coupled with appropriate time integration schemes to provide the desired accuracy required by high-fidelity solvers such as the direct numerical simulations for fluid flow. For time integration, Runge-Kutta (RK) schemes, particularly the low-storage implementation, are widely used due to their ability to provide good stability properties and be computationally efficient. However, the implementation of AT schemes with multi-stage RK schemes necessitates an over-decomposition of the spatial domain in a parallel setting, leading to increased message sizes for communication and redundant computations. In this paper, we propose a novel method to couple asynchrony-tolerant and low-storage explicit RK schemes in solving time-dependent PDEs that would result in a significant reduction in communications and relaxed synchronizations. We develop new asynchrony-tolerant schemes for ghost or buffer point updates that are necessary to maintain desired order of accuracy. The accuracy of this method is investigated, both theoretically and numerically, using simple one-dimensional linear model equations. Thereafter, we demonstrate the scalability of the proposed numerical method through three-dimensional simulations of decaying Burgers' turbulence, performed using two different asynchronous algorithms: communication-avoiding and synchronization-avoiding algorithms. The scalability studies up to 27,000 cores were found to yield speed-ups up to 6× compared to a baseline synchronous algorithm. Overall, the proposed approach shows the potential to improve the scalability of exascale PDE solvers significantly.

An Efficient Dynamic Modeling Technique for a Central Tie Rod Rotor

Compared with the three-dimensional rotor model for a central tie rod rotor, an equivalent one-dimensional model can greatly improve the computational efficiency in rotor dynamics analysis with a certain accuracy. However, little research work can be found on improving the modeling accuracy of one-dimensional models using experimental data. In this paper, a one-dimensional discrete mass model considering pretightening force is proposed for central tie rod rotors to achieve the purpose of both efficient and accurate modeling. Experimental testing and three-dimensional model analysis are used as reference and verification approaches. A sensitivity-based method is adopted to update the proposed one-dimensional model via minimizing the error in the critical speed comparing with the corresponding three-dimensional finite element model which has been verified by a modal test. Prediction of damped unbalanced response is conducted to show the practicality of the updated one-dimensional model. Results show that the method presented in this research work can be used to simulate a complex preloaded rotor system with high efficiency and accuracy.

Sharp Lower Bound for the First Eigenvalue of the Weighted p-Laplacian I

We establish sharp lower bounds for the first nonzero eigenvalue of the weighted p-Laplacian with $$1< p< \infty $$ on a compact Bakry–Emery manifold $$(M^n,g,f)$$ satisfying $${\text {Ric}}+\nabla ^2 f \ge \kappa \, g$$ , provided that either $$1<p \le 2$$ or $$\kappa \le 0$$ . For $$1<p \le 2$$ , we provide a simple proof via the modulus of continuity estimates. The proof for the $$\kappa \le 0$$ case is based on a sharp gradient comparison theorem for the eigenfunction together with a careful analysis of the underlying one-dimensional model equation.

Coupled method for the solution of a one-dimensional heat equation with axial symmetry

In this article, we implement the Projected Differential Transform Method (PDTM) coupled with Laplace Transform Method (LTM), hereby referred to as LPDTM, to solve a one-dimensional heat model (equation) with axial symmetry. Using the proposed framework (LPDTM), the exact solution (results) are obtained with convenience. It is noted that the suggested methodology performs very well and gives rapidly converging series solutions with less computational activities.

Solute transport model equation for mobile phase in semi-infinite porous media

An analytical solution of one-dimensional solute transport model equation for mobile phase is derived for semi-infinite heterogeneous porous media. The transport equation considers advection, longitudinal dispersion, linear isotherm, and porosity in its formulation. The dispersion is assumed to be Fickian, characterized by a constant dispersion coefficient. Temporally dependent velocity patterns are employed for the prediction of contaminant concentration. The analytical solution is compared with two reported solutions and can be used for benchmarking numerical codesand also as a preliminary predictive tool for groundwater management. Also, the obtained analytical solution of the proposed model may be helpful in the groundwater hydrology areas.

Upscaled equations for two-phase flow in highly heterogeneous porous media: Varying permeability and porosity

In this article we consider a two-phase flow model in a highly heterogeneous porous column. The porous column consists of homogeneous blocks, where the porosity and permeability vary from one block to the other. The flow direction is perpendicular to the layering of the porous column, and hence can be approximated by one-dimensional model equations. The periodic change in porosity and absolute permeability enforce the fluid to be trapped at the interface between the blocks, leading to a highly varying saturation. In order to capture the effective behavior, upscaled equations for the average saturation are derived via homogenization. This technique relies on a notion of periodicity and allows averaging over any number of blocks that may have any internal distributions of the rock parameters. Moreover, the present article also derives effective equations for randomly distributed layers of different porosity and absolute permeability. Numerical experiments are performed which show good agreement between the averaged solutions of the original micro-scale equations and the solutions of the upscaled equations, also in the case of randomly distributed layers. In particular these numerical experiments show how the internal distribution of the permeability and porosity affect the effective behavior of the flow.

Morphodynamic model of the lower Yellow River: flux or entrainment form for sediment mass conservation?

Abstract. Sediment mass conservation is a key factor that constrains river morphodynamic processes. In most models of river morphodynamics, sediment mass conservation is described by the Exner equation, which may take various forms depending on the problem in question. One of the most widely used forms of the Exner equation is the flux-based formulation, in which the conservation of bed material is related to the stream-wise gradient of the sediment transport rate. An alternative form of the Exner equation, however, is the entrainment-based formulation, in which the conservation of bed material is related to the difference between the entrainment rate of bed sediment into suspension and the deposition rate of suspended sediment onto the bed. Here we represent the flux form in terms of the local capacity sediment transport rate and the entrainment form in terms of the local capacity entrainment rate. In the flux form, sediment transport is a function of local hydraulic conditions. However, the entrainment form does not require this constraint: only the rate of entrainment into suspension is in local equilibrium with hydraulic conditions, and the sediment transport rate itself may lag in space and time behind the changing flow conditions. In modeling the fine-grained lower Yellow River, it is usual to treat sediment conservation in terms of an entrainment (nonequilibrium) form rather than a flux (equilibrium) form, in consideration of the condition that fine-grained sediment may be entrained at one place but deposited only at some distant location downstream. However, the differences in prediction between the two formulations have not been comprehensively studied to date. Here we study this problem by comparing the results predicted by both the flux form and the entrainment form of the Exner equation under conditions simplified from the lower Yellow River (i.e., a significant reduction of sediment supply after the closure of the Xiaolangdi Dam). We use a one-dimensional morphodynamic model and sediment transport equations specifically adapted for the lower Yellow River. We find that in a treatment of a 200 km reach using a single characteristic bed sediment size, there is little difference between the two forms since the corresponding adaptation length is relatively small. However, a consideration of sediment mixtures shows that the two forms give very different patterns of grain sorting: clear kinematic waves occur in the flux form but are diffused out in the entrainment form. Both numerical simulation and mathematical analysis show that the morphodynamic processes predicted by the entrainment form are sensitive to sediment fall velocity. We suggest that the entrainment form of the Exner equation might be required when the sorting process of fine-grained sediment is studied, especially when considering relatively short timescales.

Simulation of a closed low-pressure honeycomb adsorber for thermal energy storage

The efficient implementation of renewable energy sources necessitates thermal energy storages. For domestic as well as industrial applications thermal energy storages based on closed adsorption are studied. Against this background, a closed low-pressure honeycomb adsorber is numerically examined in this work. The examined adsorber contains stacked layers of honeycomb blocks with rectangular channels which are separated by heat exchanger plates. Zeolite 13X and water is assumed as the adsorption pair. The focus of this work is solely on the adsorption process. The numerical model applies a one-dimensional model for the single channels of the honeycomb blocks. The one-dimensional model has been presented in a previous work of the authors. To account for transversal heat conduction in the honeycomb cross-section, the one-dimensional model equations are extended by heat source/sink terms. In addition, the mass transport equation is modified for rectangular channel flow. The results demonstrate that the heat and mass transfer and the adsorption processes are strongly coupled and can be only understood by their interaction. Regarding modelling aspects, it is found that the spatial variations of temperature and pressure as well as the local deviation from adsorption equilibrium are significant. Hence, no equilibrium assumptions should be made. Further, the minor rarefaction effect of slip should be considered. With respect to the application, the analysis yields, that the thermal power can be optimized by variation of the honeycomb geometry parameters, e.g. channel size. The local optimum is a result of the inverse dependencies of the external and internal mass transfer resistance on the channel size. Interestingly, the optimum for peak and mean power do not coincide in general. Finally, it is found that the thermal power can be controlled effectively by the inlet pressure.

One- and Two-dimensional Solitary Wave States in the Nonlinear Kramers Equation with Movement Direction as a Variable

We study self-propelled particles by direct numerical simulation of the nonlinear Kramers equation for self-propelled particles. In our previous paper, we studied self-propelled particles with velocity variables in one dimension. In this paper, we consider another model in which each particle exhibits directional motion. The movement direction is expressed with a variable $\phi$. We show that one-dimensional solitary wave states appear in direct numerical simulations of the nonlinear Kramers equation in one- and two-dimensional systems, which is a generalization of our previous result. Furthermore, we find two-dimensionally localized states in the case that each self-propelled particle exhibits rotational motion. The center of mass of the two-dimensionally localized state exhibits circular motion, which implies collective rotating motion. Finally, we consider a simple one-dimensional model equation to qualitatively understand the formation of the solitary wave state.

A new smoothing approach for accelerating the convergence of power-law preconditioning method in steady and unsteady flows simulation

A new explicit smoothing approach is presented to improve the convergence acceleration of laminar viscous steady-state and unsteady flows simulation over airfoils (with/without considering surface air injection). This approach is developed based on Jameson's residual smoothing technique, and it is called here as a Solution Smoothing Method (SSM). This scheme is accompanied by the progressive locally power-law preconditioning method. The steady/unsteady preconditioned governing equations are solved using Jameson's cell-centered finite volume method in which the stabilization is achieved by means of the second- and fourth-order artificial dissipation terms. An implicit dual-time algorithm is used for simulation of unsteady cases and an explicit four-stage Runge–Kutta time integration with local time step is applied to achieve the steady-state condition in pseudo-time in an inner loop. Stability of SSM is analyzed on a one-dimensional linear model equation, and the effects of different Courant numbers, artificial dissipation coefficients, and smoothing parameters are investigated. Numerical results in comparison with other classic preconditioning methods show a reduction of the iteration number required to compute the steady-state and unsteady solution. Results show about 55 ∼ 77 and 40 ∼ 72% reduction in iteration number for steady and unsteady flows simulation, respectively.

An exponential time-integrator scheme for steady and unsteady inviscid flows

An exponential time-integrator scheme of second-order accuracy based on the predictor–corrector methodology, denoted PCEXP, is developed to solve multi-dimensional nonlinear partial differential equations pertaining to fluid dynamics. The effective and efficient implementation of PCEXP is realized by means of the Krylov method. The linear stability and truncation error are analyzed through a one-dimensional model equation. The proposed PCEXP scheme is applied to the Euler equations discretized with a discontinuous Galerkin method in both two and three dimensions. The effectiveness and efficiency of the PCEXP scheme are demonstrated for both steady and unsteady inviscid flows. The accuracy and efficiency of the PCEXP scheme are verified and validated through comparisons with the explicit third-order total variation diminishing Runge–Kutta scheme (TVDRK3), the implicit backward Euler (BE) and the implicit second-order backward difference formula (BDF2). For unsteady flows, the PCEXP scheme generates a temporal error much smaller than the BDF2 scheme does, while maintaining the expected acceleration at the same time. Moreover, the PCEXP scheme is also shown to achieve the computational efficiency comparable to the implicit schemes for steady flows.

Unsteady free surface flow in porous media: One-dimensional model equations including vertical effects and seepage face

This note examines the two-dimensional unsteady isothermal free surface flow of an incompressible fluid in a non-deformable, homogeneous, isotropic, and saturated porous medium (with zero recharge and neglecting capillary effects). Coupling a Boussinesq-type model for nonlinear water waves with Darcy's law, the two-dimensional flow problem is solved using one-dimensional model equations including vertical effects and seepage face. In order to take into account the seepage face development, the system equations (given by the continuity and momentum equations) are completed by an integral relation (deduced from the Cauchy theorem). After testing the model against data sets available in the literature, some numerical simulations, concerning the unsteady flow through a rectangular dam (with an impermeable horizontal bottom), are presented and discussed.

On the well-posedness of various one-dimensional model equations for fluid motion

We consider 1D equations with nonlocal velocity of the form wt+uwx+δuxw=−νΛγwwhere the nonlocal velocity u is given by (1) u=(1−∂xx)−βw, β>0 or (2) u=Hw (H is the Hilbert transform). In this paper, we address several local well-posedness results with blow-up criteria for smooth initial data. We then establish the global well-posedness by using the blow-up criteria.

Evaluation of finite difference approximations of absorption and dispersion implemented in sonic boom propagation model equations

Recently developed one-dimensional far-field sonic boom propagation model equations are variants of an augmented Burgers equation. The Gulfstream Aerospace Corporation far-field sonic boom propagation tool, GACBoom, employs a numerical approach that consists of operator splitting in the time-domain. The mechanisms in the equation include geometric spreading, atmospheric stratification, nonlinearity, absorption, and dispersion. For propagation of sonic booms in the atmosphere, the predominant dissipative mechanisms are thermoviscous attenuation and molecular relaxation due to diatomic oxygen and nitrogen, the latter of which also contribute to dispersion. One method for utilizing predictions with these mechanisms is to employ a finite difference approximation to the dissipative and dispersive terms in the sonic boom propagation model equation. This paper uses a discrete Fourier analysis to investigate both spatial and temporal discretization ramifications for the dissipative and dispersive components inclu...

JFNK method with a physics-based preconditioner for the fully implicit solution of one-dimensional drift-flux model in boiling two-phase flow

The Jacobian-free Newton-Krylov (JFNK) method with an efficient physics-based preconditioner is applied for the numerical solution of the one-dimensional drift-flux model with the closure constitutive equations. Additional closure correlations, including the flow pattern dependent heat transfer correlations and the flow pattern independent kinematic constitutive correlation, are used to close the governing equations of one-dimensional drift-flux model. The governing equations have been discretized using the first-order upwind method for spatial discretization and the fully implicit method for temporal discretization. An efficient physics-based preconditioner derived from the semi-implicit solution of governing equations is used in the JFNK method to improve the efficiency and numerical stability. The numerical verification and code validation have been performed for subcooled boiling two-phase flow in a vertical tube. By comparing with the other methods (JFNK method without preconditioner, the Broyden method and the Broyden-Fletcher-Goldfarb-Shanno (BFGS) method), the preconditioning JFNK method shows the robustness and a good computational efficiency. Moreover, by comparing the numerical simulation results with the experimental results, it is found that the JFNK method with a physics-based preconditioner shows the good accuracy for the numerical simulation for one-dimensional boiling two-phase flow.

One-dimensional model equations for hyperbolic fluid flow

In this paper we study the singularity formation for two nonlocal 1D active scalar equations, focusing on the hyperbolic flow scenario. Those 1D equations can be regarded as simplified models of some 2D fluid equations.

On simulations of spinning processes with a stationary one-dimensional upper convected Maxwell model

Abstract This work deals with the behavior of viscoelastic jets under gravitational forces described by an asymptotic upper convected Maxwell (UCM) model, system of partial differential equations. Considering fiber spinning, we show that the one-dimensional model equations in general allow for the simulation of drawing processes with and without die swell effect. But, as the model is of hyperbolic type and the run of the characteristics crucially depend on the physical parameters, the existence regimes of the stationary solutions associated to certain boundary conditions turn out to be limited. We investigate the regimes for gravitational uniaxial and 2d spinning scenarios numerically.