Usability of explicit filtering in large eddy simulation with a low‐order numerical scheme and different subgrid‐scale models

Based on a priori tests, in large eddy simulation (LES) of turbulent fluid flow, the numerical error related to low‐order finite‐difference‐type methods can be large in comparison with the effect of subgrid‐scale (SGS) model. Explicit filtering has been suggested to reduce the error, and it has shown promising results in a priori studies and in some simulations with fourth‐order method. In this paper, the effect of explicit filtering on the total simulation error is studied together with a second‐order scheme, where the numerical error should be even larger. The fully developed turbulent channel flow between two parallel walls is used as a test case. Rather simple SGS models are applied, because these models are most likely used in practical applications of LES. Explicit filtering is here applied to the non‐linear convection term of the Navier–Stokes equations, four three‐dimensional filter functions are applied, and the effect of filtering is separated from the effect of SGS modelling. It is shown that the effect of filtering is rather large and smooth filters introduce an additional error component that increases the total simulation error. Finally, filtering via subfilter‐scale modelling is applied, and it is shown that this approach performs better. However, the large‐frequency components of the resolved flow field are not as effectively damped as when the non‐linear convection term is filtered. Copyright © 2007 John Wiley & Sons, Ltd.     

Evaluation of Smagorinsky‐based subgrid‐scale models in a finite‐volume computation

Smagorinsky‐based models are assessed in a turbulent channel flow simulation at Re_b=2800 and Re_b=12500. The Navier–Stokes equations are solved with three different grid resolutions by using a co‐located finite‐volume method. Computations are repeated with Smagorinsky‐based subgrid‐scale models. A traditional Smagorinsky model is implemented with a van Driest damping function. A dynamic model assumes a similarity of the subgrid and the subtest Reynolds stresses and an explicit filtering operation is required. A top‐hat test filter is implemented with a trapezoidal and a Simpson rule. At the low Reynolds number computation none of the tested models improves the results at any grid level compared to the calculations with no model. The effect of the subgrid‐scale model is reduced as the grid is refined. The numerical implementation of the test filter influences on the result. At the higher Reynolds number the subgrid‐scale models stabilize the computation. An analysis of an accurately resolved flow field reveals that the discretization error overwhelms the subgrid term at Re_b=2800 in the most part of the computational domain. Copyright © 2002 John Wiley & Sons, Ltd.     

Assessment of stretched vortex subgrid‐scale models for LES of incompressible inhomogeneous turbulent flow

The physical space version of the stretched vortex subgrid scale model is tested in LES of the turbulent lid‐driven cubic cavity flow. LES is carried out by using a higher order finite‐difference method. The effects of different vortex orientation models and subgrid turbulence spectrums are assessed through comparisons of the LES predictions against DNS. Three Reynolds numbers 12000, 18000, and 22000 are studied. Good agreement with the DNS data for the mean and fluctuating quantities is observed. Copyright © 2013 John Wiley & Sons, Ltd.     

A general optimal formulation for the dynamic Smagorinsky subgrid‐scale stress model

In this paper, a general optimal formulation for the dynamic Smagorinsky subgrid‐scale (SGS) stress model is reported. The Smagorinsky constitutive relation has been revisited from the perspective of functional variation and optimization. The local error density of the dynamic Smagorinsky SGS model has been minimized directly to determine the model coefficient C_S. A sufficient and necessary condition for optimizing the SGS model is obtained and an orthogonal condition (OC), which governs the instantaneous spatial distribution of the optimal dynamic model coefficient, is formulated. The OC is a useful general optimization condition, which unifies several classical dynamic SGS modelling formulations reported in the literature. In addition, the OC also results in a new dynamic model in the form of a Picard's integral equation. The approximation tensorial space for the projected Leonard stress is identified and the physical meaning for several basic grid and test‐grid level tensors is systematically discussed. Numerical simulations of turbulent Couette flow are used to validate the new model formulation as represented by the Picard's integral equation for Reynolds numbers ranging from 1500 to 7050 (based on one half of the velocity difference of the two plates and the channel height). The relative magnitudes of the Smagorinsky constitutive parameters have been investigated, including the model coefficient, SGS viscosity and filtered strain rate tensor. In general, this paper focuses on investigation of fundamental mathematical and physical properties of the popular Smagorinsky constitutive relation and its related dynamic modelling optimization procedure. Copyright © 2005 John Wiley & Sons Ltd.     

Multi‐scale time integration for transient conjugate heat transfer

The quasi‐steady assumption is commonly adopted in existing transient fluid–solid‐coupled convection–conduction (conjugate) heat transfer simulations, which may cause non‐negligible errors in certain cases of practical interest. In the present work, we adopt a new multi‐scale framework for the fluid domain formulated in a triple‐timing form. The slow‐varying temporal gradient corresponding to the time scales in the solid domain has been effectively included in the fluid equations as a source term, whilst short‐scale unsteadiness of the fluid domain is captured by a local time integration at a given ‘frozen’ large scale time instant. For concept proof, validation and demonstration purposes, the proposed methodology has been implemented in a loosely coupled procedure in conjunction with a hybrid interfacing treatment for coupling efficiency and accuracy. The present results indicate that a much enhanced applicability can be achieved with relatively small modifications of existing transient conjugate heat transfer methods at little extra cost. Copyright © 2016 John Wiley & Sons, Ltd.     

LBE simulation of coherent vortex motion and heat transfer in jets of cross flow

Lattice Boltzmann equation method is used to simulate the coherent vortex motions and interactions and the heat transfer characteristics of jets in cross flow (JICFs) via TD2G9 model. After validation, the characteristics of cross flow under different Reynolds numbers are illustrated, including the mean profiles, the Reynolds stress tensor, the vortex and temperature fields, the temperature gradients near the walls, and the coherent correlation of vortex motions. The results show that the velocity profiles in JICF can be characterized by three basic regions, which are mainly caused by the mergence of JICF with the main flow. The temperature gradient near the walls can also be categorized by four basic regions, which are caused mainly by the impulse of JICFs too. Coherent vortex motions are found in JICF for Re = 3000, which are proved by strong periodic correlation of flow variables over a fixed area.     

A ‘poor man's Navier–Stokes equation’: derivation and numerical experiments—the 2‐D case

We present a systematic derivation of a discrete dynamical system directly from the two‐dimensional incompressible Navier–Stokes equations via a Galerkin procedure and provide a detailed numerical investigation (covering more than 107 cases) of the characteristic behaviours exhibited by the discrete mapping for specified combinations of the four bifurcation parameters. We show that this simple 2‐D algebraic map, which consists of a bilinearly coupled pair of logistic maps, can produce essentially any (temporal) behaviour observed either experimentally or computationally in incompressible Navier–Stokes flows as the bifurcation parameters are varied in pairs over their ranges of stable behaviours. We conclude from this that such discrete dynamical systems deserve consideration as sources of temporal fluctuations in synthetic‐velocity forms of subgrid‐scale models for large‐eddy simulation. Copyright © 2004 John Wiley & Sons, Ltd.     

On the influence of numerical schemes and subgrid–stress models on large eddy simulation of turbulent flow past a square cylinder

Influence of finite difference schemes and subgrid‐stress models on the large eddy simulation calculation of turbulent flow around a bluff body of square cylinder at a laboratory Reynolds number, has been examined. It is found that the type and the order of accuracy of finite‐difference schemes and the subgrid‐stress model for satisfactory results are dependent on each other, and the grid resolution and the Reynolds number. Using computational grids manageable by workstation‐level computers, with which the near‐wall region of the separating boundary layer cannot be resolved, central‐difference schemes of realistic orders of accuracy, either fully conservative or non‐conservative, suffer stability problems. The upwind‐biased schemes of third order and the Smagorinsky eddy‐viscosity subgrid model can give reasonable results resolving much of the energy‐containing turbulent eddies in the boundary layers and in the wake and representing the subgrid stresses in most parts of the flow. Noticeable improvements can be obtained by either using higher order difference schemes, increasing the grid resolution and/or by implementing a dynamic subgrid stress model, but each at a cost of increased computational time. For further improvements, the very small‐scale eddies near the upstream corners and in the laminar sublayers need to be resolved but would require a substantially larger number of grid points that are out of the range of easily accessible computers. Copyright © 2002 John Wiley & Sons, Ltd.     

On flow of a fourth‐grade fluid with heat transfer

The influence of heat transfer on the steady flow of a fourth‐grade fluid between two stationary parallel porous plates is studied. The flow is engendered under the application of a constant pressure gradient. The concept of homotopy analysis method is utilized for the series solution of the governing problem. Numerical solution has been also carried out. In addition, both analytic and numerical solutions are compared. The variations of embedded parameters into the solution are predicted through the graphical representations. Copyright © 2010 John Wiley & Sons, Ltd.     

Effect of heat transfer on the flow of a second‐grade fluid in divergent/convergent channel

This paper studies the effects of a second‐grade fluid on the flow and heat transfer characteristics in a divergent/convergent channel. The momentum and energy equations are first given in a nondimensional form and then solved analytically using the method of homotopy analysis method. Convergence of derived series solutions is shown. Graphical results for the velocity and the temperatures are presented and discussed for various emerging parameters. Copyright © 2009 John Wiley & Sons, Ltd.     

Numerical prediction of turbulent flow structure generated by a synthetic cross‐jet into a turbulent boundary layer

A detailed numerical study using large‐eddy simulation (LES) and unsteady Reynolds‐averaged Navier–Stokes (URANS) was undertaken to investigate physical processes that are engendered in the injection of a circular synthetic (zero‐net mass flux) jet in a zero pressure gradient turbulent boundary layer. A complementary study was carried out and was verified by comparisons with the available experimental data that were obtained at corresponding conditions with the aim of achieving an improved understanding of fluid dynamics of the studied processes. The computations were conducted by OpenFOAM C++, and the physical realism of the incoming turbulent boundary layer was secured by employing random field generation algorithm. The cavity was computed with a sinusoidal transpiration boundary condition on its floor. The results from URANS computation and LES were compared and described qualitatively and quantitatively. There is a particular interest for acquiring the turbulent structures from the present numerical data. The numerical methods can capture vortical structures including a hairpin (primary) vortex and secondary structures. However, the present computations confirmed that URANS and LES are capable of predicting current flow field with a more detailed structure presented by LES data as expected. Copyright © 2011 John Wiley & Sons, Ltd.     

Melting heat transfer in the stagnation‐point flow of an upper‐convected Maxwell (UCM) fluid past a stretching sheet

The steady laminar boundary layer flow and heat transfer past a stretching sheet arre considered. Upper‐convected Maxwell (UCM) fluid is treated as a rheological model. The resulting nonlinear differential system is solved by homotopy analysis method (HAM). The influence of melting parameter (M), Prandtl number (Pr), Deborah number (β) and stretching ratio (A = a/c) on the velocity and temperature profiles is thoroughly examined. It is noticed that fields are effected appreciably with the variation of parameters. Furthermore, it is seen that the local Nusselt number is a decreasing function of melting parameter. Copyright © 2011 John Wiley & Sons, Ltd.     

Wall properties and heat transfer analysis of the peristaltic motion in a power‐law fluid

A mathematical analysis has been carried out to investigate the effect of elasticity of the flexible walls on the peristaltic flow of a power‐law fluid. The heat transfer analysis is further examined. Expressions of stream function, velocity, and temperature distributions are presented in closed form under long wavelength and low Reynolds number approximations. The effects of the various parameters entering into the mathematical analysis are sketched and discussed. Copyright © 2012 John Wiley & Sons, Ltd.     

On the performance of sudden‐expansion pipe without/with cross‐flow injection: experimental and numerical studies

The paper explores the possibilities that different turbulence closures offer, for in‐depth analysis of a complex flow. The case under investigation is steady, turbulent flow in a pipe with sudden expansion without/with normal‐to‐wall injection through jets. This is a typical geometry where generation of turbulence energy takes place, due to sudden change in boundary conditions. This study is aimed at investigating the capability of a developed computational program by the present authors with three different turbulence models to calculate the mean flow variables. Three two‐equation models are implemented, namely the standard linear k ? ε model, the low Reynolds number k ? ε model and the cubic nonlinear eddy viscosity (NLEV) k ? ε model. The performance of the chosen turbulence models is investigated with regard to the available data in the literature including velocity profiles, turbulent kinetic energy and reattachment position in a pipe expansion. In order to further assess the reliability of the turbulence models, an experimental program was conducted by the present authors also at the fluid mechanics laboratory of Menoufiya University. Preliminary measurements, including the surface pressure along the two walls of the expansion pipe and the pressure drop without and with the presence of different arrangements of wall jets produced by symmetrical or asymmetrical fluid cross‐flow injection, are introduced. The results of the present studies demonstrate the superiority of the cubic NLEV k ? ε model in predicting the flow characteristics over the entire domain. The simple low Reynolds number k ? ε model also gives good prediction, especially when the reattachment point is concerned. The evaluation of the reattachment point and the pressure‐loss coefficient is numerically addressed in the paper using the cubic NLEV k ? ε model. The results show that the injection location can control the performance of the pipe‐expansion system. It is concluded that the introduction of flow injection can increase the energy loss in the pipe expansion. The near‐field turbulence structure is also considered in the present study and it is noticed that the turbulence level is strongly affected by the cross‐flow injection and the jet location. Copyright © 2011 John Wiley & Sons, Ltd.     

Implementation of CLEAR algorithm on non‐orthogonal curvilinear co‐ordinates for solution of incompressible flow and heat transfer

In this paper, the CLEAR (coupled and linked equations algorithm revised) algorithm is extended to non‐orthogonal curvilinear collocated grids. The CLEAR algorithm does not introduce pressure correction in order to obtain an incompressible flow field which satisfies the mass conservation law. Rather, it improves the intermediate velocity by solving an improved pressure equation to make the algorithm fully implicit since there is no term omitted in the derivation process. In the extension of CLEAR algorithm from a staggered grid system in Cartesian coordinates to collocated grids in non‐orthogonal curvilinear coordinates, three important issues are appropriately treated so that the extended CLEAR can lead to a unique solution without oscillation of pressure field and with high robustness. These three issues are (1) solution independency on the under‐relaxation factor; (2) strong coupling between velocity and pressure; and (3) treatment of the cross pressure gradient terms. The flow and heat transfer problems in a rectangular enclosure with an internal eccentric circle and the flow in a lid‐driven inclined cavity are computed by using the extended CLEAR. The results show that the extended CLEAR can guarantee the solution independency on the under‐relaxation factor, the smoothness of pressure profile even at very small under‐relaxation factor and good robustness which leads to a converged solution for the small inclined angle of 5^° only with 5‐point computational molecule while the extended SIMPLE‐series algorithm usually can get a converged solution for the inclined angle larger than 30^° under the same condition. Copyright © 2006 John Wiley & Sons, Ltd.     

Conjugate model for heat and mass transfer of porous wall in the high temperature gas flow

Heat and mass transfer of a porous permeable wall in a high temperature gas dynamical flow is considered. Numerical simulation is conducted on the ground of the conjugate mathematical model which includes filtration and heat transfer equations in a porous body and boundary layer equations on its surface. Such an approach enables one to take into account complex interaction between heat and mass transfer in the gasdynamical flow and in the structure subjected to this flow. The main attention is given to the impact of the intraporous heat transfer intensity on the transpiration cooling efficiency. The project supported by the National Natural Science Foundation of China (19889209) and Russian Foundation for Basic Research (97-02-16943)     

Effects of heat and mass transfer on peristaltic flow of Williamson fluid in a non‐uniform channel with slip conditions

The combined influence of heat and mass transfer has been explored in a study of peristaltic transport of magnetohydrodynamic Williamson fluid in a non‐uniform channel with flexible walls. The slip conditions are paid due attention and long wavelength and small Reynolds number assumptions are adopted in the problem formulation. The obtained results are valid for small Weissenberg number. A detailed study of involved key parameters in the obtained solutions is made by the sketched graphs. Copyright © 2010 John Wiley & Sons, Ltd.