Differential Stress Model Research Articles

A revisitation of White−Metzner viscoelastic fluids

The famous White–Metzner (WM) constitutive equation expresses a relatively simple nonlinear viscoelastic fluid of polymer melts. However, such a differential stress model, substantial with strong hyperbolic and singular problems, has hitherto always obtained unsatisfactory simulations of corner vortex in a typical contraction flow, especially for high Weissenberg numbers. A modified WM model useful for viscoelastic fluid computations is, therefore, proposed herein. As a validation, this model primarily fits the first normal stress difference for characterizing a fluid's elasticity, as well as shear viscosity and extensional viscosity. It is significant to discuss the vortex formation and growth, with the predicted vortex sizes in good agreement with the experimental data.

Performance of second-moment differential stress and flux models for natural convection in an enclosure

In this paper the relative performance of two different differential stress and flux models for computing natural convection in an enclosure is examined. One model is the low-Reynolds number differential stress and flux model and the other model is a recently developed elliptic blending based differential stress and flux model. The primary emphasis of the study is placed on investigating the accuracy of the both models for the turbulent natural convection in an enclosure. Both models are used to predict turbulent natural convections in a square cavity with hot and cold side walls and conducting top and bottom walls (Ra = 1.58 × 109). The relative performance between both models is examined through comparing their solutions with the experimental data. It is shown that both models predict the vertical mean velocity component and mean temperature well. It is also shown that the low-Reynolds number differential stress and flux model slightly under-predicts the turbulent kinetic energy and vertical turbulent heat flux. It is also noted that the implementation of the elliptic blending second moment model is easier since this model does not contain any wall-related parameters and damping functions.

Developing theory of probability density function for stochastic modeling of turbulent gas-particle flows

Turbulent gas-particle flows are studied by a kinetic description using a probability density function (PDF). Unlike other investigators deriving the particle Reynolds stress equations using the PDF equations, the particle PDF transport equations are directly solved either using a finite-difference method for two-dimensional (2D) problems or using a Monte-Carlo (MC) method for three-dimensional (3D) problems. The proposed differential stress model together with the PDF (DSM-PDF) is used to simulate turbulent swirling gas-particle flows. The simulation results are compared with the experimental results and the second-order moment (SOM) two-phase modeling results. All of these simulation results are in agreement with the experimental results, implying that the PDF approach validates the SOM two-phase turbulence modeling. The PDF model with the SOM-MC method is used to simulate evaporating gas-droplet flows, and the simulation results are in good agreement with the experimental results.

CFD Prediction of Heat and Fluid Flow through U-Bends using High Reynolds-number EVM and DSM Models

The cooling system of gas turbine blades understanding and improvement is increasing desires for computational techniques which can accurately model the flow field and heat transfer characteristics of blade cooling passage designs under realistic operating conditions. The present study discusses the comparisons between modeling and measuring gas turbine blade-cooling applications of the flow development and heat transfer in a stationary square cross-sectioned U-bend of strong curvature of Rc/D = 0.65. A turbulence model of the differential stress model (DSM) is used in combination with three different wall treatments such as a standard form of the wall function (SWF), an analytical wall function (AWF) and a numerical wall function (NWF). The combination of DSM with the numerical wall function (DSM/NWF) has improved markedly the Nusselt number predictions along the outer wall after the bend exit, but did not show any other distinctive predictive advantages over the other DSM models.

Computation of a turbulent natural convection in a rectangular cavity with the elliptic-blending second-moment closure

A numerical study of a turbulent natural convection in an enclosure with the elliptic-blending second-moment closure (EBM) is presented. The primary emphasis of the study is placed on an investigation of the accuracy and numerical stability of the elliptic-blending second-moment closure for the turbulent natural convection flow. The turbulent heat fluxes in this model are treated by the general gradient diffusion hypothesis (GGDH). The model is applied to the prediction of a natural convection in a rectangular cavity and the computed results are compared with the experimental data commonly used for a validation of the turbulence models. The results are also compared with those by the two-layer model, the SST model, the V2-f model and the second-moment differential stress and flux model. It is shown that the elliptic blending model predicts as good as or better than the existing models for the mean velocity and turbulent quantities although this model employs a simpler GGDH for treating the turbulent heat fluxes.

Computational Study of Thermal Striping in an Upper Plenum of Kalimer

A computational study of thermal striping in an upper plenum of the Korea Advanced Liquid-Metal Reactor (KALIMER) was performed. The primary objective of the present study was to find the distribution of the amplitude of temperature fluctuation in the hot pool of KALIMER. The computations were performed using the CFX-4 code with the differential stress and flux turbulence model and the Van Leer convection scheme. Two cases with different outlet velocity of the control rod fuel assemblies are considered. The distributions of the velocity vector, temperature, and temperature fluctuation were obtained from the calculation. In order to quantitatively understand the amplitude of temperature fluctuation at the bottom wall of the upper internal structure (UIS), the amplitude of the fluctuation of temperature in the radial and angular directions was investigated. The amplitude of temperature fluctuation at the UIS bottom plate was largely dependent on the magnitude of the outlet velocity of the control rod fuel assemblies. From the calculated results, it was found that the largest temperature fluctuation occurred at the radial edge of the UIS bottom in the KALIMER design. Since thermal striping is dependent on the amplitude of temperature fluctuations and frequency, the region of the UIS bottom edge needs to be analyzed with a detailed unsteady calculation.

Computation of a turbulent natural convection in a rectangular cavity with the low-reynolds-number differential stress and flux model

A numerical study of a natural convection in a rectangular cavity with the low-Reynoldsnumber differential stress and flux model is presented. The primary emphasis of the study is placed on the investigation of the accuracy and numerical stability of the low-Reynolds-num ber differential stress and flux model for a natural convection problem. The turbulence model considered in the study is that developed by Peeters and Henkes (1992) and further refined by Dol and Hanjalic (2001), and this model is applied to the prediction of a natural convection in a rectangular cavity together with the two-layer model, the shear stress transport model and the time-scale bound v2--f modeI, all with an algebraic heat flux model. The computed results are compared with the experimental data commonly used for the validation of the turbulence models. It is shown that the low-Reynolds-num ber differential stress and flux model predicts well the mean velocity and temperature, the vertical velocity fluctuation, the Reynolds shear stress, the horizontal turbulent heat flux, the local Nusselt number and the wall shear stress, but slightly under-predicts the vertical turbulent heat flux. The performance of the v-~--f model is comparable to that of the low-Reynolds-num ber differential stress and flux model except for the over-prediction of the horizontal turbulent heat flux. The two layer model predicts poorly the mean vertical velocity component and under-predicts the wall shear stress and the local Nusselt number. The shear stress transport model predicts well the mean velocity, but the general performance of the shear stress transport model is nearly the same as that of the two-layer model, under-predicting the local Nusselt number and the turbulent quantities.

The computation of flow and heat transfer through square‐ended U‐bends, using low‐Reynolds‐number models

This study is concerned with the computation of turbulent flow and heat transfer in U‐bends of strong curvature. Following the earlier studies within the authors' group on flows through round‐ended U‐bends, here attention is turned to flows through square‐ended U‐bends. Flows at two Reynolds numbers have been computed, one at 100,000 and the other at 36,000. In the heat transfer analysis, the Prandtl number was either 0.72 (air) or, in a further departure from our earlier studies, 5.9 (water). The turbulence modelling approaches examined, include a two‐layer and a low‐Re k‐ε model, a two‐layer and a low‐Re version of the basic differential stress model (DSM) and a more recently developed, realisable version of the differential stress model that is free of wall‐parameters. For the low‐Re effective viscosity model (EVM) and DSMs, an alternative, recently proposed length‐scale correction term, independent of wall distance has also been tested. Even the simplest model employed – two‐layer EVM – reproduces the mean flow development with reasonable accuracy, suggesting that the mean flow development is mainly influenced by mean pressure rather than the turbulence field. The heat transfer parameters, on the other hand, show that only the low‐Re DSMs produce reliable Nusselt number predictions for both Prandtl numbers examined.

Development and validation of a non-linear k-ε model for flow over a full-scale building

At present the most popular turbulence models used for engineering solutions to flowrnproblems are the k- e and Reynolds stress models. The shortcoming of these models based on the isotropicrneddy viscosity concept and Reynolds averaging in flow fields of the type found in the field of WindrnEngineering are well documented. In view of these shortcomings this paper presents the implementationrnof a non-linear model and its evaluation for flow around a building. Tests were undertaken using thernclassical bluff body shape, a surface mounted cube, with orientations both normal and skewed at 45 o tornthe incident wind. Full-scale investigations have been undertaken at the Silsoe Research Institute with arn6 m surface mounted cube and a fetch of roughness height equal to 0.01 m. All tests were originallyrnundertaken for a number of turbulence models including the standard, RNG and MMK k- e models andrnthe differential stress model. The sensitivity of the CFD results to a number of solver parameters wasrntested. The accuracy of the turbulence model used was deduced by comparison to the full-scale predictedrnroof and wake recirculation zone lengths. Mean values of the predicted pressure coefficients were used tornfurther validate the turbulence models. Preliminary comparisons have also been made with availablernpublished experimental and large eddy simulation data. Initial investigations suggested that a suitablernturbulence model should be able to model the anisotropy of turbulent flow such as the Reynolds stressrnmodel whilst maintaining the ease of use and computational stability of the two equations models.rnTherefore development work concentrated on non-linear quadratic and cubic expansions of the Boussinesqrneddy viscosity assumption. Comparisons of these with models based on an isotropic assumption are presentedrnalong with comparisons with measured data.

Calculation of fully developed turbulent flow in rectangular ducts with two opposite roughened walls

The predictive capabilities of three transport-type turbulence models are analyzed by comparing predictions with experimental data obtained for fully developed turbulent flow in a square duct (Case 1) and in a 2:1 aspect ratio rectangular duct (Case 2), each with two opposite rib-roughened walls. The models include the Demuren-Rodi (DR) k-ϵ model, the Sugiyama et al. (S) k-ϵ model, and the differential stress (DS) model proposed recently by the authors. For Case 1, the S model is unable to predict experimentally observed axial mean velocity distortion induced by secondary flow in the cross plane. For both test cases, the DR and DS models exhibit similar levels of performance, except in the vicinity of a corner formed by the juncture of adjacent smooth and rib-roughened walls, where the DS model outperforms the DR model. The results are analyzed to explain why the DR model leads to the formation of a spurious secondary flow cell near this corner that is not present in the experimental flow.

Calculation of fully developed turbulent flow in rectangular ducts with two opposite roughened walls

The predictive capabilities of three transport-type turbulence models are analyzed by comparing predictions with experimental data obtained for fully developed turbulent flow in a square duct (Case 1) and in a 2:1 aspect ratio rectangular duct (Case 2), each with two opposite rib-roughened walls. The models include the Demuren-Rodi (DR) k- ϵ model, the Sugiyama et al. (S) k- ϵ model, and the differential stress (DS) model proposed recently by the authors. For Case 1, the S model is unable to predict experimentally observed axial mean velocity distortion induced by secondary flow in the cross plane. For both test cases, the DR and DS models exhibit similar levels of performance, except in the vicinity of a corner formed by the juncture of adjacent smooth and rib-roughened walls, where the DS model outperforms the DR model. The results are analyzed to explain why the DR model leads to the formation of a spurious secondary flow cell near this corner that is not present in the experimental flow.

Calculation of Fully-Developed Turbulent Flow in Rectangular Ducts With Nonuniform Wall Roughness

The predictive capabilities of four transport-type turbulence models are analyzed by comparing predictions with experimental data for fully-developed flow in (1) a rectangular duct with a step change in roughness on one wall (Case 1), and (2) a square duct with one rib-roughened wall (Case 2). The models include the Demuren-Rodi (DR) k-ε model, the Sugiyama et al. (S) k-ε model, the Launder-Li (LL) Reynolds stress transport equation model, and the differential stress (DS) model proposed recently by the authors. For the first flow situation (Case 1), the results show that the DS model yields improved agreement between predicted and measured primary and secondary mean velocity distributions in comparison to the DR and LL models. For the second flow situation (Case 2), the DS model is superior to the DR and S models for predicting experimentally observed mean velocity, turbulence kinetic energy, and Reynolds stress anisotropy behavior, especially in the vicinity of a corner formed by the juncture of adjacent smooth and rough walls. The results are analyzed in order to explain why the DR model leads to the formation of a spurious secondary flow cell near this corner that is not present in the experimental flow.

Differential stress modelling of turbulent flows in model reciprocating engines

A study is made here of the application of a differential stress model (DSM) of turbulence to flows in two model reciprocating engines. For the first time this study includes compressive effects. An assessment between DSM and k-ɛ results is made comparing with laser Doppler anemometry experimental data of the mean flow and turbulence intensity levels during intake and compression strokes. A well-established two-dimensional finite-volume computer code is employed. Two discretization schemes are used, namely the HYBRID scheme and the QUICK scheme. The latter is found to be essential if differentiation is to be made between the turbulence models. During the intake stroke the DSM results are, in general, similar to the k-ɛ results in comparison to the experimental data, except for the turbulence levels, which the DSM seriously underpredicts. This is in contrast to a parallel set of calculations of steady in-flow, which showed significant gains from using the DSM, particularly at the turbulence field level. The increased number of grid lines employed in those calculations contribute to this apparent difference between steady and unsteady flows, but cycle- to-cycle variations are more likely to be the primary cause, resulting in too high levels of turbulence intensity being measured. However, during the compression stroke the DSM returns vastly superior results to the k-ɛ model at both the mean flow and turbulence intensity levels. This is because the DSM generates an anisotropic shear stress field during the early stages of compression that suppresses the main vortical structure, in line with the experimental data.

風洞実験及びLES,DSMとの比較によるASMの精度・問題点の検討 : ASMによる立方体周辺の非等方乱流場の数値解析

A three - dimensional turbulent flowfield around a cube placed in a surface boundary layer is predicted using Algebraic Stress Model (ASM) and Differential Stress Model (DSM). The results from these models are compared with those of wind tunnel tests conducted by the authors, and Large Eddy Simulation (LES). The overestimation of turbulent energy values around the frontal corner occurred in the k-e model are corrected by ASM. But ASM cannot reproduce anisotropic properties of each component of normal stresses in the area above the frontal corner while these discrepancies do not appear in the results of DSM. The differences between the results from two models are derived from the expression of convection and diffusion terms of transport equation of <u_i'u_j'>. Algebraic approximation in the case of ASM is examined precisely by the numerical data given by LES.

Experimental investigation of an adverse pressure gradient wake and comparison with calculations

The wake of a flat plate was subjected to a strong adverse pressure gradient, causing local flow reversal and high turbulence intensities. The mean flow and turbulence quantities were measured in the wake and in the boundary layer of the plate by using a three-component laser-Doppler anemometer. The experimental data were used to determine some of the terms in the kinetic energy equation. In addition, numerical solutions of the Reynolds-averaged Navier-Stokes equations were compared with the experimental results. The comparison showed that both a k − ϵ model and a differential stress model correctly predicted the spreading rate of the wake. However, the mean velocity and the kinetic energy on the wake centerline were poorly predicted. Neither model was able to predict the measured mean-flow reversal near the centerline.

Turbulent flow in closed and free-surface unbaffled tanks stirred by radial impellers

The three-dimensional turbulent flow field in unbaffled tanks stirred by radial impellers was numerically simulated by a finite-volume method on body-fitted, co-located grids. Simulations were run, with no recourse to empirical input, in the rotating reference frame of the impeller. Free-surface problems were also simulated, in which the profile of the central vortex was computed as part of the solution by means of an iterative technique. Predicted velocity and turbulence fields in the whole vessel and power consumptions were assessed against available literature data; free-surface profiles were also compared with original experimental data obtained in a model tank. Both the eddy-viscosity k− ε turbulence model and a second-order differential stress (DS) model were used and compared: satisfactory results were obtained only by using the latter model. The need for including source terms arising from fluctuating Coriolis forces in the Reynolds stress transport equations is highlighted.

Turbulence modeling of rotating confined flows

We propose here an original differential stress model which takes into account some of the implicit effects of rotation on the turbulence. It is applied to the numerical prediction of the turbulent flow inside a shrouded rotor-stator system. The results of this model are compared to those obtained with four other turbulence models and to the experimental data. The advanced second-order models produce quite correct predictions for the mean flow, but it is shown that it is necessary to take account of the rotation effects in order to obtain satisfactory behaviour of the Reynolds stress.

Computational analysis of a single jet impingement ground effect lift loss

A study was carried out on single round jet ground effect lift loss. The jet exit Mach number and velocity were 0.71 and 240 m/s, respectively. The effects of ground height, baffle plate edge, and jet exit turbulent intensity were assessed, and Navier-Stokes equations were solved using computational fluid dynamics. Turbulence closure was achieved using a k-e model, and the result compared with calculations obtained with a differential stress model. Three baffle plate edges were tested (rounded, squared, and chamfered), and the ground heights varied from rj = 0.15 to 0.8. It was found that flow mechanisms varied significantly with ground heights. A coherent vortex existed between the baffle plate and the ground at the low ground heights (17 0.25), the vortex disappeared and separation at the plate edge played an important part in determining the lift loss. The baffle plate edge was found to account for as much as 14% of the ground effect lift loss. The stress model was found to improve the accuracy of the prediction.

High-speed GSMAC-FEM for wind engineering

A stable and cost-efficient finite element formulation, the high-speed GSMAC-FEM (generalized simplified marker and cell finite element method) for unsteady incompressible viscous flow analysis is developed. In this paper, the high-speed GSMAC-FEM with the differential stress turbulence model, which includes curve effect in Rotta's term as a pressure-strain correlation, is identified by computing the channel flow and the flow over a backward-facing step. Then, the superiority of the differential stress model to the kt-e turbulence model for the prediction of the pressure distributions on a building in a turbulent stream is demonstrated.

Numerical analysis of wind around building using high-speed GSMAC-FEM — Validation of differential stress model —

A stable and cost-efficient finite element formulation, the high-speed GSMAC-FEM (generalized simplified marker and cell finite-element method), for unsteady incompressible viscous flow analysis is developed. In this paper, the high-speed GSMAC-FEM with the differential stress turbulence model, which includes curve effect in Rotta's term as a pressure-strain correlation, is validated by computing the channel flow and the flow over a backward-facing step. After that, the superiority of the differential stress model to the k - ϵ turbulence model for the prediction of the pressure distributions on a building in a turbulent stream is demonstrated.