Numerical solutions of Reeks-Hall equation for particulate concentration in recirculating turbulent fluid flow

2014 ◽  
Author(s):  
◽  
Giang N. Nguyen
Keyword(s):  
Power Plants ◽  
Numerical Solutions ◽  
Numerical Results ◽  
Normal Operation ◽  
Time Step ◽  
Time Step Size ◽  
First Order ◽  
Resuspension Rate ◽  
Particulate Concentration ◽  
Rock N Roll

[ACCESS RESTRICTED TO THE UNIVERSITY OF MISSOURI AT AUTHOR'S REQUEST.] Source term is an important issue in safety assessment of nuclear power plants. Therefore, modeling of particulate concentration in reactor coolant systems during normal operation and hypothesized or real accidents is of continuing interest. We report here on exploration of a numerical solution of the Reeks and Hall equation with the use of fractional resuspension rate in its original integral form. The numerical results for particulate concentration are compared with those obtained from the exact expression given by Williams and experimental data provided by Wells et al. The numerical results agree very well with exact results and also agree well with the data of Wells et al. Applications of numerical method to problems with time dependent resuspension rate (for which exact solutions are not available), are explored and some typical results are reported. Research is carried out for three related resuspension models: Reeks, Reed, and Hall (1988), Vainshtein et al (1997), and Rock 'n Roll (2001). Results from Rock 'n Roll model show some advantages over the other two models. Since the advanced numerical technique we used may not be entirely suitable for use in large integrated computed codes, we have also explored use of a first order finite difference scheme for solving the Reeks-Hall equation. This first order scheme is sensitive to time-step size, but can work in some cases.

Numerical Investigation of Sub-Iteration Criterions for Unsteady Flow Computations with Dual-Time Method

2013 ◽  
Vol 432 ◽  
pp. 189-195
Author(s):  
Guang Ning Li ◽  
Min Xu
Keyword(s):  
Unsteady Flow ◽  
Unsteady Flows ◽  
Numerical Results ◽  
Iteration Step ◽  
Convergence Criterion ◽  
Step Size ◽  
Time Step ◽  
Time Step Size ◽  
Physical Time ◽  
New Criterion

The convergence of sub-iteration with the dual-time method is very important for the prediction of unsteady flow field. The influence of sub-iteration step number, criterion of sub-iteration convergence and the choice of physical time step size on the calculation results are discussed by solving of the two-dimensional unsteady Euler equations. A new convergence criterion (named residual criterion) of sub-iteration for unsteady flows is proposed, and the unsteady flow test case AGARD-CT5 is calculated to verify the new criterion. The results show that, with the same criterion of sub-iteration, the results from different physical time step sizes are in agreement with each other. The difference between the experiment data and the numerical results are small, and if the sub-iteration criterion used is reasonable and small enough, the dependence of numerical results of unsteady flows on the physical time step will be decreased as possible. The new criterion of sub-iteration for dual-time step unsteady calculations can be used for engineering problem.

scholarly journals A Set of New Stable, Explicit, Second Order Schemes for the Non-Stationary Heat Conduction Equation

Mathematics ◽  
2021 ◽  
Vol 9 (18) ◽  
pp. 2284
Author(s):  
Endre Kovács ◽  
Ádám Nagy ◽  
Mahmoud Saleh
Keyword(s):  
Numerical Solutions ◽  
Finite Difference Methods ◽  
Second Order ◽  
Stiff Systems ◽  
Implicit Methods ◽  
Step Size ◽  
Time Step ◽  
Two Stage ◽  
Time Step Size ◽  
New Methods

This paper introduces a set of new fully explicit numerical algorithms to solve the spatially discretized heat or diffusion equation. After discretizing the space and the time variables according to conventional finite difference methods, these new methods do not approximate the time derivatives by finite differences, but use a combined two-stage constant-neighbour approximation to decouple the ordinary differential equations and solve them analytically. In the final expression for the new values of the variable, the time step size appears not in polynomial or rational, but in exponential form with negative coefficients, which can guarantee stability. The two-stage scheme contains a free parameter p and we analytically prove that the convergence is second order in the time step size for all values of p and the algorithm is unconditionally stable if p is at least 0.5, not only for the linear heat equation, but for the nonlinear Fisher’s equation as well. We compare the performance of the new methods with analytical and numerical solutions. The results suggest that the new algorithms can be significantly faster than the widely used explicit or implicit methods, particularly in the case of extremely large stiff systems.

scholarly journals Direct integrator of block type methods with additional derivative for general third order initial value problems

2020 ◽  
Vol 12 (10) ◽  
pp. 168781402096618
Author(s):  
Mohammed Yousif Turki ◽  
Fudziah Ismail ◽  
Norazak Senu ◽  
Zarina Bibi Ibrahim
Keyword(s):  
Initial Value Problems ◽  
Numerical Solutions ◽  
Numerical Results ◽  
Order System ◽  
Block Type ◽  
Initial Value ◽  
Third Order ◽  
First Order ◽  
Block Methods ◽  
Hermite Interpolating Polynomial

This paper presents the construction of the two-point and three-point block methods with additional derivatives for directly solving [Formula: see text]. The proposed block methods are formulated using Hermite Interpolating Polynomial and approximate the solution of the problem at two or three-point concurrently. The block methods obtain the numerical solutions directly without reducing the equation into the first order system of initial value problems (IVPs). The order and zero-stability of the proposed methods are also investigated. Numerical results are presented and comparisons with other existing block methods are made. The performance shows that the proposed methods are very efficient in solving the general third order IVPs.

scholarly journals New Numerical Aspects of Caputo-Fabrizio Fractional Derivative Operator

Mathematics ◽  
2019 ◽  
Vol 7 (4) ◽  
pp. 374 ◽  
Author(s):  
Sania Qureshi ◽  
Norodin A. Rangaig ◽  
Dumitru Baleanu
Keyword(s):  
Fractional Derivative ◽  
Numerical Approximation ◽  
Absolute Difference ◽  
Step Size ◽  
Time Step ◽  
Derivative Operator ◽  
Time Step Size ◽  
First Order ◽  
Forward Difference ◽  
Difference Formula

In this paper, a new definition for the fractional order operator called the Caputo-Fabrizio (CF) fractional derivative operator without singular kernel has been numerically approximated using the two-point finite forward difference formula for the classical first-order derivative of the function f ( t ) appearing inside the integral sign of the definition of the CF operator. Thus, a numerical differentiation formula has been proposed in the present study. The obtained numerical approximation was found to be of first-order convergence, having decreasing absolute errors with respect to a decrease in the time step size h used in the approximations. Such absolute errors are computed as the absolute difference between the results obtained through the proposed numerical approximation and the exact solution. With the aim of improved accuracy, the two-point finite forward difference formula has also been utilized for the continuous temporal mesh. Some mathematical models of varying nature, including a diffusion-wave equation, are numerically solved, whereas the first-order accuracy is not only verified by the error analysis but also experimentally tested by decreasing the time-step size by one order of magnitude, whereupon the proposed numerical approximation also shows a one-order decrease in the magnitude of its absolute errors computed at the final mesh point of the integration interval under consideration.

Numerical Investigation on Vortex Induced Motions of a Tension Leg Platform With Circular Columns

2021 ◽  
Author(s):  
Pengfei Zhi ◽  
Xinshu Zhang
Keyword(s):  
Numerical Investigation ◽  
Sensitivity Analyses ◽  
Normal Operation ◽  
Floating Platform ◽  
Tension Leg Platform ◽  
Step Size ◽  
Time Step ◽  
Time Step Size ◽  
Spacing Ratio ◽  
Lock In

Abstract Vortex Induced Motion (VIM) of multi-column floating platforms, such as Tension Leg Platform (TLP) and semi-submersible (SEMI), in current is well-acknowledged. Substantial VIM response of the multi-column floating platform may cause fatigue failure of mooring and riser systems, which can affect the normal operation of the platform. The present paper focuses on the numerical investigation on VIM of a TLP with circular columns using Computational Fluid Dynamics (CFD). Sensitivity analyses (e.g., mesh size, the number of prism layers and time-step size) for the VIM responses of the TLP are conducted. The effects of the current heading and mooring stiffness on the VIM are investigated. The three degrees of freedom VIM responses (in-line, transverse and yaw responses) and corresponding amplitude spectra are computed and analyzed. Motion trajectories are plotted to understand the VIM behaviors. Regarding the effect of the current heading, the largest transverse response is examined at 15° current heading and the corresponding maximum nominal amplitude is around 0.43. The difference of the maximum nominal amplitudes between the cases at 0° and 15° current headings is less than 5%. For 15°, 30° and 45° current headings, the nominal transverse amplitudes decrease as the current heading increases in the lock-in range. For the four studied current headings, the maximum width of the lock-in range is found at 0° current heading and narrows as the current incidence increases. The largest yaw response is observed at 0° current heading and the maximum nominal amplitude is around 9.1°. Regarding the effect of the mooring stiffness, the lock-in ranges and the maximum nominal amplitudes of the transverse motions have little difference for the four mooring stiffnesses. The maximum nominal transverse and yaw responses are around 0.25 and 5.1°, respectively, which occur when the mooring stiffness reaches the maximum. The flow pattern analyses indicate that the flow interference between the upstream and downstream columns may have significant effects on the VIM responses and a stronger interference at the present spacing ratio may lead to a larger VIM response. The contours of the vertical vorticity in the horizontal plane show that the mean positions of the flow separation points are always on highest or lowest (in the transverse direction perpendicular to the current heading) points of the columns, which is the reason that the VIM trajectories for the TLP with circular columns are always along the direction perpendicular to the current heading.

Efficient Time-Stepping/Spectral Methods for the Navier-Stokes-Nernst-Planck-Poisson Equations

2017 ◽  
Vol 21 (5) ◽  
pp. 1408-1428 ◽  
Author(s):  
Xiaoling Liu ◽  
Chuanju Xu
Keyword(s):  
Stability Condition ◽  
Time Discretization ◽  
Equation System ◽  
Second Order ◽  
Navier Stokes ◽  
Step Size ◽  
Time Step ◽  
Time Step Size ◽  
First Order ◽  
Time Stepping

AbstractThis paper is concerned with numerical methods for the Navier-Stokes-Nernst-Planck-Poisson equation system. The main goal is to construct and analyze some stable time stepping schemes for the time discretization and use a spectral method for the spatial discretization. The main contribution of the paper includes: 1) an useful stability inequality for the weak solution is derived; 2) a first order time stepping scheme is constructed, and the non-negativity of the concentration components of the discrete solution is proved. This is an important property since the exact solution shares the same property. Moreover, the stability of the scheme is established, together with a stability condition on the time step size; 3) a modified first order scheme is proposed in order to decouple the calculation of the velocity and pressure in the fluid field. This new scheme equally preserves the non-negativity of the discrete concentration solution, and is stable under a similar stability condition; 4) a stabilization technique is introduced to make the above mentioned schemes stable without restriction condition on the time step size; 5) finally we construct a second order finite difference scheme in time and spectral discretization in space. The numerical tests carried out in the paper show that all the proposed schemes possess some desirable properties, such as conditionally/unconditionally stability, first/second order convergence, non-negativity of the discrete concentrations, and so on.

Field Methods for Studying the Radiation Situation in the Area of the Belarusian NPP in the Pre-Operational Period

ANRI ◽  
2020 ◽  
pp. 31-44
Author(s):  
Aleksey Ekidin ◽  
Aleksey Vasil'ev ◽  
Maksim Vasyanovich ◽  
Evgeniy Nazarov ◽  
Mariya Pyshkina ◽  
...  
Keyword(s):  
Dose Rate ◽  
Nuclear Power ◽  
Power Plants ◽  
Nuclear Power Plants ◽  
Field Studies ◽  
Radiation Monitoring ◽  
Normal Operation ◽  
Construction Site ◽  
External Radiation ◽  
Field Methods

The article presents the results of field studies in the area of the Belarusian NPP in the pre-operational period. The «background» contents of gamma-emitting radionuclides in individual components of the environment are determined. The main array of dose rate measurements in the area of the NPP construction site is in the range 0.048 ÷ 0.089 μSv/h. External radiation in the surveyed area is formed at 96% due to 40K, 226Ra and 232Th. The information obtained can be used to correctly interpret the data of future radiation monitoring during normal operation of nuclear power plants.

scholarly journals A First-Order Explicit-Implicit Splitting Method for a Convection-Diffusion Problem

2020 ◽  
Vol 20 (4) ◽  
pp. 769-782
Author(s):  
Amiya K. Pani ◽  
Vidar Thomée ◽  
A. S. Vasudeva Murthy
Keyword(s):  
Splitting Method ◽  
Diffusion Problem ◽  
Euler Method ◽  
Second Order ◽  
Time Step ◽  
Convection Diffusion ◽  
First Order ◽  
Backward Euler ◽  
Strang Splitting ◽  
Spatially Periodic

AbstractWe analyze a second-order in space, first-order in time accurate finite difference method for a spatially periodic convection-diffusion problem. This method is a time stepping method based on the first-order Lie splitting of the spatially semidiscrete solution. In each time step, on an interval of length k, of this solution, the method uses the backward Euler method for the diffusion part, and then applies a stabilized explicit forward Euler approximation on {m\geq 1} intervals of length {\frac{k}{m}} for the convection part. With h the mesh width in space, this results in an error bound of the form {C_{0}h^{2}+C_{m}k} for appropriately smooth solutions, where {C_{m}\leq C^{\prime}+\frac{C^{\prime\prime}}{m}}. This work complements the earlier study [V. Thomée and A. S. Vasudeva Murthy, An explicit-implicit splitting method for a convection-diffusion problem, Comput. Methods Appl. Math. 19 2019, 2, 283–293] based on the second-order Strang splitting.

scholarly journals Verification of Convergence Rates of Numerical Solutions for Parabolic Equations

2019 ◽  
Vol 2019 ◽  
pp. 1-10
Author(s):  
Darae Jeong ◽  
Yibao Li ◽  
Chaeyoung Lee ◽  
Junxiang Yang ◽  
Yongho Choi ◽  
...  
Keyword(s):  
Parabolic Equations ◽  
Convergence Rates ◽  
Numerical Solutions ◽  
Second Order ◽  
Order Convergence ◽  
Numerical Convergence ◽  
Verification Method ◽  
First Order ◽  
Convergence Test ◽  
Second Order Convergence

In this paper, we propose a verification method for the convergence rates of the numerical solutions for parabolic equations. Specifically, we consider the numerical convergence rates of the heat equation, the Allen–Cahn equation, and the Cahn–Hilliard equation. Convergence test results show that if we refine the spatial and temporal steps at the same time, then we have the second-order convergence rate for the second-order scheme. However, in the case of the first-order in time and the second-order in space scheme, we may have the first-order or the second-order convergence rates depending on starting spatial and temporal step sizes. Therefore, for a rigorous numerical convergence test, we need to perform the spatial and the temporal convergence tests separately.

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