Under-relaxation Factors Research Articles

Investigation of Under-Relaxation Factors in 2D Dam Break Simulation of OpenFOAM®

Computational fluid dynamics (CFD) is one of the most significant applied sciences in the industry. For this reason, many expensive CFD software is available in the market. OpenFOAM® is a free CFD software that has been used extensively in many ranges of fields. But, even with the free charge, it is still hard to get a good result with OpenFOAM® without extensive computational resources. This study aims to find a way to get good results with minimum computation time by investigating the under-relaxation effects. This study chose the dam break case due to its simple and readily available validation data. The simulation domain was a 200 × 200 mm2 dam filled with 50 × 100 mm2 water in the dam’s bottom left corner. The simulations were performed with under-relaxation factors, ranging from 0.3 to 1. By changing the under-relaxation factor to 0.95, the mean absolute percentage error became six times smaller, and the simulation was 366 times faster. It shows that the under-relaxation factor is essential in the quality of results and simulation time. However, the most optimal under-relaxation factor may differ from one case to another.

Comparison of collocated methods of MCIM and PWIM for different solution algorithms in CVFEM

The pressure-velocity coupling method of MCIM (Mass Corrected Interpolation Method) and different solution algorithms of this method were previously developed and tested for steady and unsteady incompressible flows. But the effect of under-relaxation factors in cell-face velocities and the main equations never have been investigated. On the other hand, this method and its different algorithms never have been compared with similar and more common methods in the formulation of primitive variables and non-staggered grid arrangements. In this paper, the accuracy, performance, and computational efficiency of the different solution algorithms of MCIM are compared with the well-known PWIM (Pressure Weighted Interpolation Method) of Rhie & Chow. In this study, a fully coupled, segregated approach without any pressure correction and SIMPLE-like segregated algorithms are proposed for two methods of MCIM and PWIM on the control volume-based finite element (CVFEM) framework. Also, for each method, the proper under-relaxation factors for the main equations and cell-face velocities are introduced. The accuracy and performance of the proposed algorithms are demonstrated by solving different steady and unsteady benchmark problems, and the results of both methods for different algorithms are compared. The results indicate that the both methods have spatial 2nd-order accuracy approximately and are not very different from each other. The results also show that the numerical dissipation of the MCIM is much less than PWIM. Comparing the computational cost of the different algorithms of both methods shows that the coupled algorithm of the MCIM for steady and unsteady problems with different time steps and various grid numbers is at least 10% better than other algorithms.

Compromising with corrector step of SIMPLE Algorithm

Abstaining from a corrector step of SIMPLE-like algorithms, a coupled pressure-based algorithm is designed using a cell-centered finite-volume Δ-approximation on a collocated grid to predict all-speed fluid flows. A consistently formulated cell-face dissipation scheme is invoked to unravel the occurrence of pressure–velocity decoupling. Modified Rusanov, Roe and Lax–Wendroff flux-difference schemes accompanied by a newly devised differentiable slope limiter function are used to compute inviscid fluxes over the entire spectrum of subsonic to hypersonic flow. Numerical experiments in reference to incompressible and compressible flows demonstrate that the proposed algorithm maintains an excellent consistency with analytic solutions and available literature data. The method is anticipated to be stable for an extensive range of Mach numbers with an avoidance of under-relaxation (UR) factors.

Numerical Computation of Low Reynolds Number Viscous Flow Past Bluff Bodies

This article presents a two-dimensional steady viscous flow simulation past circular and square cylinders at low Reynolds numbers (based on the diameter) by the finite volume method with a non-orthogonal body-fitted grid. Diffusive fluxes are discretized using central differencing scheme, and for convective fluxes upwind and central differencing schemes are blended using a ‘deferred correction’ approach. A simplified pressure correction equation is derived, and proper under-relaxation factors are used so that computational cost is reduced without adversely affecting the convergence rate. The governing equations are expressed in Cartesian velocity components and solution is carried out using the SIMPLE algorithm for collocated arrangement of variables. The mesh yielding grid-independent solution is then utilized to study, for the very first time, the effect of the Reynolds number on the separation bubble length, separation angle, and drag coefficients for both circular and square cylinders. Finally, functional relationships between the computed quantities and Reynolds number (Re) are proposed up to Re = <i>40</i>. It is found that circular cylinder separation commences between Re= <i>6.5-6.6</i>, and the bubble length, separation angle, total drag vary as Re, Re<i>−0.5</i>, Re<i>−0.5</i> respectively. Extrapolated results obtained from the empirical relations for the circular cylinder show an excellent agreement with established data from the literature. For a square cylinder, the bubble length and total drag are found to vary as Re and Re<i>−0.666</i>, and are greater than these for a circular cylinder at a given Reynolds number. The numerical results substantiate that a square shaped cylinder is more bluff than a circular one.

Numerical Analysis of Turbulent Flow Around Two-Dimensional Bodies Using Non-Orthogonal Body-Fitted Mesh

This paper deals with the numerical simulation of a turbulent flow around two-dimensional bodies by the finite volume method with non-orthogonal body-fitted grid. The governing equations are expressed in Cartesian velocity components and solution is carried out using the SIMPLE algorithm for collocated arrangement of scalar and vector variables. Turbulence is modeled by the <i>k</i>- ε turbulence model and wall functions are used to bridge the solution variables at the near wall cells and the corresponding quantities on the wall. A simplified pressure correction equation is derived and proper under-relaxation factors are used so that computational cost is reduced without adversely affecting the convergence rate. The numerical procedure is validated by comparing the computed pressure distribution on the surface of NACA 0012 and NACA 4412 hydrofoils for different angles of attack with experimental data. The grid dependency of the solution is studied by varying the number of cells of the C-type structured mesh. The computed lift coefficients of NACA 4412 hydrofoil at different angles of attack are also compared with experimental results to further substantiate the validity of the proposed methodology.

Low-head direct chill slab casting of aluminium alloy AA4045: 3-D coupled turbulent flow and heat transfer study

In the present study, a low-head direct chill (DC) industrial-scale caster for rolling ingots has been modelled using the advanced computational fluid dynamics approach. Specifically, the 3-D coupled turbulent melt flow and solidification heat transfer in a slab caster has been modelled for the aluminium AA4045 alloy. The turbulence generated in the melt pool is taken into account in the model through the implementation of a popular low-Reynolds (Re) number version of the k-ε eddy viscosity model. The solidification heat transfer for this complex opaque system has been considered through the employment of the well-known ‘enthalpy-porosity’ scheme. A staggered control-volume (CV)-based finite-difference scheme is used to discretise the governing transport equations and the associated boundary conditions. The discretised algebraic equations are then solved sequentially using the tridiagonal matrix algorithm with implicit under-relaxation factors. The SIMPLE algorithm was employed to resolve the velocity–pressure couplings in the momentum equations. Three important process parameters of this casting problem, namely, the casting speed, the pouring temperature (inlet melt superheat), and the convective heat transfer coefficient (HTC) at the metal–mould interface for the low-head 30-mm mould, are varied in their corresponding practical range of 60–180 mm min1, 16–64°C and 1000–4000 W m2°C, respectively. The predicted results are presented in the form of velocity and temperature fields, solid shell thickness at the mould exit, as well as the sump depth and mushy-region thickness at the centre of the ingot. The obtained results are then critically discussed.

Modelación CFD de casos básicos de convección en ambientes cerrados: Necesidades de principiantes en CFD para adquirir habilidades y confianza en la modelación CFD

Prediction of airflow pattern and velocities, temperature, moisture and pollutants concentration is required to design healthy and comfortable indoor environments. Computational Fluid Dynamics (CFD) is the most advanced technique to model and predict the airflow in enclosed environments. However, the main errors in CFD models and their results are linked to the human factor. Beginners on CFD modeling do not account with skills, experience and engineering judgment to generate robust and reliable CFD models. This process is not intuitive and new CFD users need guidance. This paper aims to provide more complete information on CFD modeling of basic natural, forced and mixed convection cases that would allow CFD beginners to acquire skills and confidence. CFD modeling includes mesh generation, setting convergence criteria and under-relaxation factors, and evaluating different turbulence models for each convection case. Results show that users´ expertise is needed in each step of CFD modeling, even for these basic convection cases. Rev. ing. constr. [online]. 2014, vol.29, n.1, pp. 22-45. ISSN 0718-5073 http://dx.doi.org/10.4067/S0718-50732014000100002

Assessment on Effects of Under-Relaxation Factors on 2D Incompressible Laminar Flow over a Backward-Facing Step (BFS)

In this paper, the effects of changing under-relaxation factors for different variables on the numerical solution of 2D incompressible laminar flow over a backward-facing step (BFS) are studied using PHOENICS commercial Computational Fluid Dynamics (CFD) software. This is conducted by changing under-relaxation factors for velocities and pressure during the 2D simulation. Ten different batches of under-relaxation factor for pressure ranging from 0.1 to 1.0 were used while the values of under-relaxation factor for velocities were manipulated between 0.1 and 1.0. For each batch of the computation, the error percentage of pressure and velocities were obtained. Based on this work, it is found that the recommended values of under-relaxation factor for pressure to achieve lower error percentage are between 0.6 and 0.8. Based on findings of the study, the appropriate values of under-relaxation factor for pressure and velocities can be selected to achieve the levels of error percentage permitted for computational studies.

J011025 VOFモデルと非反復時間進行アルゴリズムを用いたタンクスロッシング非定常問題の数値的研究

It is desirable that fuel is continuously supplied to the gasoline tank intake, even when we have got on vehicle with nearly empty fuel. As a means which reduces surface agitation by sloshing, giving baffles to a tank is often considered. In this study, we simulate sloshing phenomena based on the Navier-Stokes equation by using of volume of fluid (VOF) model and non iterative time advancement (NITA) scheme. We examine the accuracy and efficiency of calculation by using of NITA scheme, which varies depending on two kinds of algorithms and under-relaxation factors. In the simulation of flow visualization, the tank with baffles is analyzed first. Then, the baffles are switched from wall boundaries to interior boundaries and the tank without baffles is analyzed under the same conditions as the tank with baffles. Finally, liquid interface and velocity magnitude are indicated by a contour.

Numerical simulation of turbulent flow in a baffled stirred tank with an explicit algebraic stress model

An explicit algebraic stress model (EASM) was used to simulate anisotropic turbulent flows in baffled stirred tanks equipped with a standard Rushton turbine. The quantitative predictions of velocity components, turbulence kinetic energy, Reynolds stresses and turbulence energy dissipation rate in the context of anisotropic turbulence were conducted to assess the comprehensive performance of the EASM. A lot of efforts have been made to ensure numerical stability during the calculations such as using a good initial flow field, manipulating source terms and adjusting under-relaxation factors. The predicted results were also compared with experimental data and other simulation results obtained using the standard k– ε model, algebraic stress model (ASM), Reynolds stress model (RSM) and large eddy simulation (LES). All the simulations were run with in-house codes. The simulation results show that agreement between the EASM predictions and experimental values is satisfactory. The EASM is consistently superior to the standard k– ε model when predicting both peak values and trend of variation in velocities and turbulence quantities. In comparison to the RSM, the EASM has almost the same predictive accuracy. The EASM is inferior to the LES on the prediction of turbulence kinetic energy. Nevertheless, the computational cost of the EASM is significantly lower than that of the LES, which is an obvious advantage in practical applications.

Reply on comments on: “Influence of parent vessel dominancy on fluid dynamics of anterior communicating artery aneurysms” by T. Hassan, A. A. Hassan, and Y. Ahmed (2010) Acta Neurochirurgica

1.1 Numerical solution details: It is so familiar in many research works using computational fluid dynamics (CFD) to make a grid dependency study and introduce only into the paper the results of the final mesh, such as in [2, 6, 7, 9, 10, 14, 15, 27]. We have followed the same concept in our paper to Acta Neurochirurgica for each case. 1.2 We understand well the effects of the under-relaxation factors (URF) on the numerical solution, and usually, in the CFD world, there is no need for mentioning the values of the URF, such as in [1, 2, 6–10, 13–15, 18, 26, 27]. Nevertheless, the values of the URF that have been used in our paper (0.3 for pressure and 0.7 for momentum) were good enough to make sure that our final numerical results are physical and good. 1.3 We have used the area weighted average formula (Eq. 1) [4] for monitoring the pressure at different planes in the anterior communicating (ACOM) artery, and the integral formula (Eq. 2) [4] for monitoring the integrated values of the pressure and the WSS on the surfaces of the ACOM artery and the aneurysm, beside monitoring the residuals at each iteration of the solution steps. We end the simulation process when the solution no longer changes with additional iterations.

Validation of fully implicit method for simulation of flows with interfaces using primitive variables

This paper describes applications of the fully implicit procedure for computing flow of two immiscible fluids. Six problems in two and three dimensions are solved to highlight effects of grid size, pressure smoothing, TVD convection scheme and geometric and fluid dynamic evaluations of surface tension force. Free surface and cavity flows are considered in which effect of sloshing, interface merger and splitting as well as splashing are included. Wherever possible, present solutions are compared with results of previous experiments and/or numerical computations. Computational details such as grid size, time step, under-relaxation factors, mass/volume conservation are reported.

An Efficient Segregated Algorithm for Incompressible Fluid Flow and Heat Transfer Problems—IDEAL (Inner Doubly Iterative Efficient Algorithm for Linked Equations) Part II: Application Examples

In this article, comprehensive comparisons are made between the SIMPLER and IDEAL algorithms for four application examples. It is found that the IDEAL algorithm is efficient and stable not only for the simple, low-Re/Ra or coarse-mesh flow cases, but also for the complex, high-Re/Ra or fine-mesh flow cases. For the low-Re/Ra, coarse-mesh flow cases, the ratio of CPU time of IDEAL to that of SIMPLER ranges from 0.029 to 0.7. For the high-Re/Ra, fine-mesh flow cases, the IDEAL algorithm can obtain convergent results but the SIMPLER algorithm cannot, even though the underrelaxation factors are adjusted.

Implementation of CLEARER Algorithm on Three-Dimensional Nonorthogonal Curvilinear Coordinates and its Application

The newly proposed CLEARER algorithm is extended to three-dimensional nonorthogonal curvilinear coordinates. This algorithm can not only guarantee full coupling between the pressure and velocity terms, and solution independence of underrelaxation factors, it can also satisfy the physical and geometric conservation laws, hence it owns high efficiency and more robustness than its companion SIMPLE-like algorithms. To ease implementation of the CLEARER algorithm, the irregular physical domain is transformed to a regular computational domain, and correspondingly, the governing equations are also transformed. The mathematical formulations in both the predictor step and the corrector step are provided in detail, as well as the solution procedure. The CLEARER algorithm is adopted to simulate numerically the fluid flow and heat transfer in a triangular wavy fin-and-tube heat exchanger, and the influence of eccentricity and transverse tube pitch on the friction factor and Nusselt number is investigated in detail. It is found that the friction factor and Nusselt number increase with increasing eccentricity and decreasing transverse tube pitch.

IMPROVEMENT OF SIMPLER ALGORITHM FOR INCOMPRESSIBLE FLOW ON STAGGERED GRID SYSTEM

In this paper, the updating of the intermediate velocity in the correction stage in SIMPLE-like algorithms on staggered grid is discussed in detail, based on the discussion CLEARER algorithm that is extended from the non-staggered grid to the staggered grid. The performance of SIMPLER and CLEARER algorithms are compared using two numerical examples with reliable solutions. The results show that CLEARER can predict the numerical results as accurately as SIMPLER, and it can also enhance the convergence rate greatly under low under-relaxation factors. In some cases CLEARER algorithm can only need 27% of iteration number required by SIMPLER to reach the same convergence criterion.

Dynamic simulation of natural circulation steam generators with the use of finite-volume-algorithms – A comparison of four algorithms

The paper reports the results of a comparison of the four finite-volume-algorithms SIMPLE, SIMPLEC, SIMPLER and PISO. The analysis has been done to explore and compare the suitability of the algorithms for the dynamic simulation of natural circulation steam generators based on a so called tube-header-model. The suitability of the algorithms was tested with the results of two simulation examples – a heat recovery steam generator with a horizontal tube bank ( Fig. 3a) and a two-drum steam generator with a vertical tube bank ( Fig. 3b). The calculations are done at low system pressure and under hot start-up conditions for the boilers. The results of the computations demonstrate that the SIMPLER algorithm should be preferred for dynamic simulations which are based on the tube-header-model. The algorithm shows a high stability, an extended field of variability of the under-relaxation factors as well as small required under-relaxation. The strong dependence of the calculation time on the number of control volumes is the only unfavourable characteristic of the SIMPLER algorithm.

Conjugate natural convection in the roof cavity of heavy construction building during summer

This paper presents the results of a study of conjugate natural convection inside a building attic in the shape of a rectangular enclosure bounded by realistic walls made from composite construction materials under summer day boundary conditions. The effects of cavity aspect ratio, Rayleigh number ( Ra), and orientation of the external surfaces on the flow and heat transfer characteristics were the main focus of the investigation. The problem was formulated in terms of the vorticity-stream function procedure, and the governing equations for steady, laminar, two-dimensional conjugate natural convection heat transfer were solved by employing the Alternating Direction Implicit (ADI) control volume method along with under-relaxation factors for temperature, vorticity, and stream functions. For Ra ranging from 10 3 to 10 10, steady state results of the streamline and temperature contours in addition to local and mean Nusselt number at all surfaces of the cavity were obtained. The results show that the values of Ra and the aspect ratio have significant effect on the temperature and stream function contours within the enclosure. Another important finding of the study is that heat flux into the room increases with the increase of both the aspect ratio and Rayleigh number.

Optimization of the nonlinear convergence rate of the SIMPLER Picard algorithm for steady incompressible internal flows through enforcement of continuity projections

We discuss the performance optimization of the Semi-Implicit Method for Pressure-Linked Equations—Revised (SIMPLER) Picard algorithm for steady incompressible internal flows. We discuss the nonlinear convergence of the Picard iteration as a function of the pressure and scalar potential continuity projections stemming from the SIMPLER algorithm, for three example problems. In particular, we discuss the choice of under-relaxation method, and choice of under-relaxation factors; the choice of the projection algorithm; and the required tolerance for the linear solve of the generalized Poisson equations for the pressure and scalar potential equations that arise from the projection operations. We conclude that the convergence of the nonlinear Picard iteration can be effectively controlled by the optimal enforcement of the continuity projections. Copyright © 2005 John Wiley & Sons, Ltd.

Investigations of flows in a new infrared suppressor

The seven-hole probe was used to measure the mixing effectiveness of a lobe–ejector combined with a double-wall diffuser. The results show that the lobe ejected the air from surrounding between the double walls as well as the secondary flow. The effects of area ratio of the primary flow and the secondary flow as well as the speed of the primary flow on the ejected flow-rate ratio, the ejected flow-rate ratio from the gap of the double walls and the distribution of the mixing velocity have been obtained. The co-located grid, the great viscosity and the cooperation of the linear and simultaneous under-relaxation factors are applied to solve the coupling of fluid and solid.The numerical results show that the air is also ejected into the double wall section to form the cooling flow, the annular-like flow generated along the stream from the lobe enhances the mixing of the primary and secondary flows and the velocity is well distributed. The more ejected air cools the exhaust gas and the wall of diffuser, and it suppresses their infrared radiations by the exhaust gas from the lobe pipe and the wall of diffuser. The numerical results and experimental data show good agreement.

Grid generation and numerical analysis of multi-stream flow in the complex channel with a forced mixer lobe

The co-located grid, SIMPLEC and Chen-Kim modified k - e turbulence model are applied to investigate numerically the multi-stream flow and temperature fields in the complex channel with a forced mixer lobe at room temperature and at elevated temperature. The body-fitted coordinate grids are generated respectively in subdomains according to the shapes of the channel by solving Poisson’s equations to compose the whole grid of the domain. The large viscosity, linear and simultaneous under-relaxation factors are used to solve the coupling of fluid and solid. The solid grid is complemented at the upper inlet of the secondary flow to keep the same node number at the inlet and at double-wall sub-domains. The numerical results and experimental data show good agreement at room temperature. It is illustrated that the cooling air ejected into the slot between the double plates decreases the temperature of the wall.