Generalized Richardson extrapolation procedures for estimating grid-independent numerical solutions

2016 ◽  
Vol 26 (3/4) ◽  
pp. 1121-1144 ◽  
Author(s):  
Bantwal R. (Rabi) Baliga ◽  
Iurii Yuri Lokhmanets
Keyword(s):  
Heat Transfer ◽  
Fluid Flow ◽  
Numerical Solutions ◽  
Richardson Extrapolation ◽  
Grid Refinement ◽  
Content Type ◽  
Extrapolation Procedure ◽  
Flow And Heat Transfer ◽  
Numerical Predictions ◽  
Hexahedral Elements

Purpose – The purpose of this paper is to present outcomes of efforts made over the last 20 years to extend the applicability of the Richardson extrapolation procedure to numerical predictions of multidimensional, steady and unsteady, fluid flow and heat transfer phenomena in regular and irregular calculation domains. Design/methodology/approach – Pattern-preserving grid-refinement strategies are proposed for mathematically rigorous generalizations of the Richardson extrapolation procedure for numerical predictions of steady fluid flow and heat transfer, using finite volume methods and structured multidimensional Cartesian grids; and control-volume finite element methods and unstructured two-dimensional planar grids, consisting of three-node triangular elements. Mathematically sound extrapolation procedures are also proposed for numerical solutions of unsteady and boundary-layer-type problems. The applicability of such procedures to numerical solutions of problems with curved boundaries and internal interfaces, and also those based on unstructured grids of general quadrilateral, tetrahedral, or hexahedral elements, is discussed. Findings – Applications to three demonstration problems, with discretizations in the asymptotic regime, showed the following: the apparent orders of accuracy were the same as those of the numerical methods used; and the extrapolated results, measures of error, and a grid convergence index, could be obtained in a smooth and non-oscillatory manner. Originality/value – Strict or approximate pattern-preserving grid-refinement strategies are used to propose generalized Richardson extrapolation procedures for estimating grid-independent numerical solutions. Such extrapolation procedures play an indispensable role in the verification and validation techniques that are employed to assess the accuracy of numerical predictions which are used for designing, optimizing, virtual prototyping, and certification of thermofluid systems.

Numerical solutions of non-alignment stagnation-point flow and heat transfer over a stretching/shrinking surface in a nanofluid

2016 ◽  
Vol 26 (6) ◽  
pp. 1747-1767 ◽  
Author(s):  
Ioan Pop ◽  
Kohi Naganthran ◽  
Roslinda Nazar
Keyword(s):  
Heat Transfer ◽  
Differential Equations ◽  
Stagnation Point ◽  
Numerical Solutions ◽  
Dual Solutions ◽  
Stagnation Point Flow ◽  
Content Type ◽  
Boundary Layer Equations ◽  
Flow And Heat Transfer ◽  
Shrinking Surface

Purpose – The purpose of this paper is to analyse numerically the steady stagnation-point flow of a viscous and incompressible fluid over continuously non-aligned stretching or shrinking surface in its own plane in a water-based nanofluid which contains three different types of nanoparticles, namely, Cu, Al2O3 and TiO2. Design/methodology/approach – Similarity transformation is used to convert the system of boundary layer equations which are in the form of partial differential equations into a system of ordinary differential equations. The system of similarity governing equations is then reduced to a system of first-order differential equations and solved numerically using the bvp4c function in Matlab software. Findings – Unique solution exists when the surface is stretched and dual solutions exist as the surface shrunk. For the dual solutions, stability analysis has revealed that the first solution (upper branch) is stable and physically realizable, while the second solution (lower branch) is unstable. The effect of non-alignment is huge for the shrinking surface which is in contrast with the stretching surface. Practical implications – The results obtained can be used to explain the characteristics and applications of nanofluids, which are widely used as coolants, lubricants, heat exchangers and micro-channel heat sinks. This problem also applies to some situations such as materials which are manufactured by extrusion, production of glass-fibre and shrinking balloon. In this kind of circumstance, the rate of cooling and the stretching/shrinking process play an important role in moulding the final product according to preferable features. Originality/value – The present results are original and new for the study of fluid flow and heat transfer over a stretching/shrinking surface for the problem considered by Wang (2008) in a viscous fluid and extends to nanofluid by using the Tiwari and Das (2007) model.

Effects of Guide Vanes on the Tip Heat Transfer Enhancement of a Turbine Blade

2011 ◽  
Author(s):  
Gongnan Xie ◽  
Bengt Sunde´n
Keyword(s):  
Heat Transfer ◽  
Fluid Flow ◽  
Heat Transfer Enhancement ◽  
Turbine Blade ◽  
Pressure Loss ◽  
Transfer Enhancement ◽  
Guide Vanes ◽  
Flow And Heat Transfer ◽  
Numerical Predictions ◽  
Blade Tip

Gas turbine blade tips encounter large heat load as they are exposed to the high temperature gas. A common way to cool the blade and its tip is to design serpentine passages with 180-deg turns under the blade tip-cap inside the turbine blade. Improved internal convective cooling is therefore required to increase the blade tip life time. This paper presents numerical predictions of turbulent fluid flow and heat transfer through two-pass channels with and without guide vanes placed in the turn regions using RANS turbulence modeling. The effects of adding guide vanes on the tip-wall heat transfer enhancement and the channel pressure loss were analyzed. The guide vanes have a height identical to that of the channel. The inlet Reynolds numbers are ranging from 100,000 to 600,000. The detailed three-dimensional fluid flow and heat transfer over the tip-walls are presented. The overall performances of several two-pass channels are also evaluated and compared. It is found that the tip heat transfer coefficients of the channels with guide vanes are 10∼60% higher than that of a channel without guide vanes, while the pressure loss might be reduced when the guide vanes are properly designed and located, otherwise the pressure loss is expected to be increased severely. It is suggested that the usage of proper guide vanes is a suitable way to augment the blade tip heat transfer and improve the flow structure, but is not the most effective way compared to the augmentation by surface modifications imposed on the tip-wall directly.

Verification of Finite Volume Computations on Steady-State Fluid Flow and Heat Transfer

2001 ◽  
Vol 124 (1) ◽  
pp. 11-21 ◽  
Author(s):  
J. Cadafalch ◽  
C. D. Pe´rez-Segarra ◽  
R. Co`nsul ◽  
A. Oliva
Keyword(s):  
Heat Transfer ◽  
Fluid Flow ◽  
Steady State ◽  
Finite Volume ◽  
Numerical Solutions ◽  
Three Dimensional ◽  
Independent Solution ◽  
Post Processing ◽  
Square Duct ◽  
Flow And Heat Transfer

This work presents a post-processing tool for the verification of steady-state fluid flow and heat transfer finite volume computations. It is based both on the generalized Richardson extrapolation and the Grid Convergence Index GCI. The observed order of accuracy and a error band where the grid independent solution is expected to be contained are estimated. The results corresponding to the following two and three-dimensional steady-state simulations are post-processed: a flow inside a cavity with moving top wall, an axisymmetric turbulent flow through a compressor valve, a premixed methane/air laminar flat flame on a perforated burner, and the heat transfer from an isothermal cylinder enclosed by a square duct. Discussion is carried out about the certainty of the estimators obtained with the post-processing procedure. They have been shown to be useful parameters in order to assess credibility and quality to the reported numerical solutions.

Fluid flow and heat transfer of a stratified system during natural convection – influence of chamfered corners

2019 ◽  
Vol 29 (2) ◽  
pp. 470-486 ◽  
Author(s):  
Alireza Rahimi ◽  
Aravindhan Surendar ◽  
Aygul Z. Ibatova ◽  
Abbas Kasaeipoor ◽  
Emad Hasani Malekshah
Keyword(s):  
Heat Transfer ◽  
Fluid Flow ◽  
Natural Convection ◽  
Entropy Generation ◽  
Volume Fraction ◽  
Solid Volume Fraction ◽  
Three Dimensional ◽  
Immiscible Fluids ◽  
Content Type ◽  
Flow And Heat Transfer

Purpose This paper aims to investigate the three-dimensional natural convection and entropy generation in the rectangular cuboid cavities included by chamfered triangular partition made by polypropylene. Design/methodology/approach The enclosure is filled by multi-walled carbon nanotubes (MWCNTs)-H2O nanofluid and air as two immiscible fluids. The finite volume approach is used for computation. The fluid flow and heat transfer are considered with combination of local entropy generation due to fluid friction and heat transfer. Moreover, a numerical method is developed based on three-dimensional solution of Navier–Stokes equations. Findings Effects of side ratio of triangular partitions (SR = 0.5, 1 and 2), Rayleigh number (103 < Ra < 105) and solid volume fraction (f = 0.002, 0.004 and 0.01 Vol.%) of nanofluid are investigated on both natural convection characteristic and volumetric entropy generation. The results show that the partitions can be a suitable method to control fluid flow and energy consumption, and three-dimensional solutions renders more accurate results. Originality/value The originality of this work is to study the three-dimensional natural convection and entropy generation of a stratified system.

Computation of flow and heat transfer through channels with periodic dimple/protrusion walls using low-Reynolds number turbulence models

2019 ◽  
Vol 29 (3) ◽  
pp. 1178-1207 ◽  
Author(s):  
Mohammad Fazli ◽  
Mehrdad Raisee
Keyword(s):  
Heat Transfer ◽  
Cooling System ◽  
Correction Term ◽  
Turbine Blades ◽  
Turbulence Models ◽  
Internal Cooling ◽  
Recirculation Bubble ◽  
Content Type ◽  
Flow And Heat Transfer ◽  
Numerical Predictions

PurposeThis paper aims to predict turbulent flow and heat transfer through different channels with periodic dimple/protrusion walls. More specifically, the performance of various low-Rek-ε turbulence models in prediction of local heat transfer coefficient is evaluated.Design/methodology/approachThree low-Re numberk-εturbulence models (the zonalk-ε, the lineark-εand the nonlineark-ε) are used. Computations are performed for three geometries, namely, a channel with a single dimpled wall, a channel with double dimpled walls and a channel with a single dimple/protrusion wall. The predictions are obtained using an in house finite volume code.FindingsThe numerical predictions indicate that the nonlineark-εmodel predicts a larger recirculation bubble inside the dimple with stronger impingement and upwash flow than the zonal and lineark-εmodels. The heat transfer results show that the zonalk-εmodel returns weak thermal predictions in all test cases in comparison to other turbulence models. Use of the lineark-εmodel leads to improvement in heat transfer predictions inside the dimples and their back rim. However, the most accurate thermal predictions are obtained via the nonlineark-εmodel. As expected, the replacement of the algebraic length-scale correction term with the differential version improves the heat transfer predictions of both linear and nonlineark-εmodels.Originality/valueThe most reliable turbulence model of the current study (i.e. nonlineark-εmodel) may be used for design and optimization of various thermal systems using dimples for heat transfer enhancement (e.g. heat exchangers and internal cooling system of gas turbine blades).

Temporally- and Spatially-Periodic Laminar Flow and Heat Transfer in Staggered-Plate Arrays

2007 ◽  
Author(s):  
Alexandre Lamoureux ◽  
B. Rabi Baliga
Keyword(s):  
Heat Transfer ◽  
Reynolds Number ◽  
Fluid Flow ◽  
Numerical Solutions ◽  
Initial Conditions ◽  
Plate Thickness ◽  
Geometrical Parameters ◽  
Reduced Pressure ◽  
Flow And Heat Transfer ◽  
Spatially Periodic

A computational investigation of temporally- and spatially-periodic laminar two-dimensional fluid flow and heat transfer in staggered-plate arrays is presented in this paper. The objective and the novel aspect of this study is the investigation of the influence (on the numerical solutions) of including single and multiple representative geometric modules in the calculation domain, with spatially-periodic boundary conditions imposed on the instantaneous velocity and temperature fields in both the streamwise and the lateral directions. The following geometrical parameters, normalized with respect to a representative module height, were studied: a dimensionless plate length equal to 1, and a dimensionless plate thickness of 0.250. This relatively high value of dimensionless plate thickness, compared to those commonly encountered in rectangular offset-fin cores of compact heat exchangers, was deliberately chosen to induce and enhance the unsteady features of the fluid flow and heat transfer phenomena. Different specified values of the time-mean modular streamwise gradient of the reduced pressure were investigated, yielding values of Reynolds number (Kays and London definition) in the range of 100 to 625. The Prandtl number was fixed at 0.7. In the multiple-module simulations, for Reynolds number values exceeding 400, it was found that multiple solutions are possible: the particular solution which is obtained in any one simulation depends on the specified initial conditions. The results presented include time-mean modular friction factors, modular Colburn factors, and Strouhal numbers.

Numerical and Experimental Prediction of Transitional Characteristics of Flow and Heat Transfer in an Array of Heated Blocks

1999 ◽  
Vol 121 (3) ◽  
pp. 202-208 ◽  
Author(s):  
Y. Asako ◽  
Y. Yamaguchi ◽  
M. Faghri
Keyword(s):  
Heat Transfer ◽  
Reynolds Number ◽  
Fluid Flow ◽  
Parallel Plate ◽  
Three Dimensional ◽  
Parallel Plates ◽  
Number Range ◽  
Flow And Heat Transfer ◽  
Numerical Predictions ◽  
Experimental Prediction

Three-dimensional numerical analysis, for transitional characteristics of fluid flow and heat transfer in periodic fully developed region of an array of the heated square blocks deployed along one wall of the parallel plates duct, is carried out by using Lam-Bremhorst low-Reynolds-number two equation turbulence model. Computations were performed for Prandtl number of 0.7, in the Reynolds number range of 200 to 2000 and for two sets of geometric parameters characterizing the array. The predicted transitional Reynolds number is lower than the value for the parallel plate duct and it decreases with increasing the height above the module. Experiments were also performed for pressure drop measurements and for flow visualization and the results were compared with the numerical predictions.

Numerical investigation of fluid flow structure and heat transfer in a passage with continuous and truncated V-shaped ribs

2015 ◽  
Vol 25 (1) ◽  
pp. 171-189 ◽  
Author(s):  
Shian Li ◽  
Gongnan Xie ◽  
Bengt Sunden
Keyword(s):  
Heat Transfer ◽  
Reynolds Number ◽  
Fluid Flow ◽  
Pressure Drop ◽  
Heat Transfer Enhancement ◽  
Thermal Performance ◽  
Internal Cooling ◽  
Transfer Enhancement ◽  
Content Type ◽  
Flow And Heat Transfer

Purpose – The employment of continuous ribs in a passage involves a noticeable pressure drop penalty, while other studies have shown that truncated ribs may provide a potential to reduce the pressure drop while keeping a significant heat transfer enhancement. The purpose of this paper is to perform computer-aided simulations of turbulent flow and heat transfer of a rectangular cooling passage with continuous or truncated 45-deg V-shaped ribs on opposite walls. Design/methodology/approach – Computational fluid dynamics technique is used to study the fluid flow and heat transfer characteristics in a three-dimensional rectangular passage with continuous and truncated V-shaped ribs. Findings – The inlet Reynolds number, based on the hydraulic diameter, is ranged from 12,000 to 60,000 and a low-Re k-e model is selected for the turbulent computations. The local flow structure and heat transfer in the internal cooling passages are presented and the thermal performances of the ribbed passages are compared. It is found that the passage with truncated V-shaped ribs on opposite walls provides nearly equivalent heat transfer enhancement with a lower (about 17 percent at high Reynolds number of 60,000) pressure loss compared to a passage with continuous V-shaped ribs or continuous transversal ribs. Research limitations/implications – The fluid is incompressible with constant thermophysical properties and the flow is steady. The passage is stationary. Practical implications – New and additional data will be helpful in the design of ribbed passages to achieve a good thermal performance. Originality/value – The results imply that truncated V-shaped ribs are very effective in improving the thermal performance and thus are suggested to be applied in gas turbine blade internal cooling, especially at high velocity or Reynolds number.

Performance of combined spectral collocation method and artificial compressibility method for 3D incompressible fluid flow and heat transfer

2020 ◽  
Vol 30 (12) ◽  
pp. 5037-5062
Author(s):  
Jing-Kui Zhang ◽  
Miao Cui ◽  
Ben-Wen Li ◽  
Ya-Song Sun
Keyword(s):  
Heat Transfer ◽  
Fluid Flow ◽  
Incompressible Flow ◽  
Three Dimensional ◽  
Function Form ◽  
Combined Method ◽  
Time Step ◽  
Content Type ◽  
Artificial Compressibility ◽  
Flow And Heat Transfer

Purpose The purpose of this paper is to develop a combined method for three-dimensional incompressible flow and heat transfer by the spectral collocation method (SCM) and the artificial compressibility method (ACM), and further to study the performance of the combined method SCM-ACM for three-dimensional incompressible flow and heat transfer. Design/methodology/approach The partial differentials in space are discretized by the SCM with Chebyshev polynomial and Chebyshev–Gauss–Lobbatto collocation points. The unsteady artificial compressibility equations are solved to obtain the steady results by the ACM. Three-dimensional exact solutions with trigonometric function form and exponential function form are constructed to test the accuracy of the combined method. Findings The SCM-ACM is developed successfully for three-dimensional incompressible flow and heat transfer with high accuracy that the minimum value of variance can reach. The accuracy increases exponentially along with time marching steps. The accuracy is also improved exponentially with the increasing of nodes before stable accuracy is achieved, while it keeps stably with the increasing of the time step. The central processing unit time increases exponentially with the increasing of nodes and decreasing of the time step. Research limitations/implications It is difficult for the implementation of the implicit scheme by the developed SCM-ACM. The SCM-ACM can be used for solving unsteady impressible fluid flow and heat transfer. Practical implications The SCM-ACM is applied for two classic cases of lid-driven cavity flow and natural convection in cubic cavities. The present results show good agreement with the published results with much fewer nodes. Originality/value The combined method SCM-ACM is developed, firstly, for solving three-dimensional incompressible fluid flow and heat transfer by the SCM and ACM. The performance of SCM-ACM is investigated. This combined method provides a new choice for solving three-dimensional fluid flow and heat transfer with high accuracy.

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