Fully Compressible Low-Mach Number Simulations of Carbon-dioxide at Supercritical Pressures and Trans-critical Temperatures

This work investigates fully developed turbulent flows of carbon-dioxide close to its vapour-liquid critical point in a channel with a hot and a cold wall. Two direct numerical simulations are performed at low Mach numbers, with the trans-critical transition near the channel centre and the cold wall, respectively. An additional simulation with constant transport properties is used to selectively investigate the effect of the non-linear equation of state on turbulence. Compared to the case where the pseudo-critical transition occurs in the channel center, the case with the pseudo-critical transition close to the cold wall reveals that compressibility effects can exist in the near-wall region even at low Mach numbers. An analysis of the velocity streaks near the hot and the cold walls also indicates a greater degree of streak coherence near the cold wall. A comparison between the constant and variable viscosity cases at the same Reynolds number, Mach number and having the same isothermal wall boundary conditions reveals that variable viscosity increases turbulence near the cold wall and also causes higher velocity gradients near the hot wall. We also show that the extended van Driest transformation results in a better agreement of the velocity profile with the log-law of the wall compared to the standard van Driest transformation. The semi-locally scaled turbulent velocity fluctuations and the turbulent kinetic energy budgets on the hot and the cold sides of the channel collapse on top of each other, thereby establishing the validity of Morkovin’s hypothesis.     

Large Eddy Simulations Using an Unstructured Grid Compressible Navier - Stokes Algorithm

Large Eddy Simulation (LES) of the decay of isotropic turbulence and of channel flow has been performed using an explicit second-order unstructured grid algorithm for tetrahedral cells. The algorithm solves for cell-averaged values using the finite volume form of the unsteady compressible Jittered Navier-Stokes equations. The inviscid fluxes are obtained from Godunov's exact Riemann solver. Reconstruction of the flow variables to the left and right sides of each face is performed using least squares or Frink's method. The viscous fluxes and heat transfer are obtained by application of Gauss' theorem. LES of the decay of nearly incompressible isotropic turbulence has been performed using two models for the SGS stresses: the Monotone Integrated Large Eddy Simulation (MILES) approach, wherein the inherent numerical dissipation models the sub-grid scale (SGS) dissipation, and the Smagorinsky SGS model. The results using the MILES approach with least squares reconstruction show good agreement with incompressible experimental data. The contribution of the Smagorinsky SGS model is negligible. LES of turbulent channel flow was performed at a Reynolds number (based on channel height and bulk velocity) of 5600 and Mach number of 0.5 (at which compressibility effects are minimal) using Smagorinsky's SGS model with van Driest damping. The results show good agreement with experimental data and direct numerical simulations for incompressible channel flow. The SGS eddy viscosity is less than 10% of the molecular viscosity, and therefore the LES is effectively MILES with molecular viscosity.     

Large-eddy simulation of turbulent flow and heat transfer in plane and rib-roughened channels

Large-eddy simulation results are presented and discussed for turbulent flow and heat transfer in a plane channel with and without transverse square ribs on one of the walls. They were obtained with the finite-difference code Harwell-FLOW3D, Release 2, by using the PISOC pressure-velocity coupling algorithm, central differencing in space, and Crank-Nicolson time stepping. A simple Smagorinsky model, with van Driest damping near the walls, was implemented to model subgrid scale effects. Periodic boundary conditions were imposed in the streamwise and spanwise directions. The Reynolds number based on hydraulic diameter (twice the channel height) ranged from 10 000 to 40 000. Results are compared with experimental data, k-? predictions, and previous large-eddy simulations.     

Balance equation of the generalised sub-grid scale (SGS) turbulent kinetic energy in a new tensorial dynamic mixed SGS model

A new dynamic model is proposed in which the eddy viscosity is defined as a symmetric second rank tensor, proportional to the product of a turbulent length scale with an ellipsoid of turbulent velocity scales. The employed definition of the eddy viscosity allows to remove the local balance assumption of the SGS turbulent kinetic energy formulated in all the dynamic Smagorinsky-type SGS models. Furthermore, because of the tensorial structure of the eddy viscosity the alignment assumption between the principal axes of the SGS turbulent stress tensor and the resolved strain-rate tensor is equally removed, an assumption which is employed in the scalar eddy viscosity SGS models. The proposed model is tested for a turbulent channel flow. Comparison with the results obtained with other dynamic SGS models (Dynamic Smagorinsky Model, Dynamic Mixed Model and Dynamic K-equation Model) shows that the tensorial definition of the eddy viscosity and the removal of the local balance assumption of the SGS turbulent kinetic energy considerably improves the agreement between results obtained with Large Eddy simulation (LES) and Direct Numerical Simulations (DNS), respectevely. Received August 26, 1999     

Nonlinear turbulence models for predicting strong curvature effects

Prediction of the characteristics of turbulent flows with strong streamline curvature, such as flows in turbomachines, curved channel flows, flows around airfoils and buildings, is of great importance in engineering applications and poses a very practical challenge for turbulence modeling. In this paper, we analyze qualitatively the curvature effects on the structure of turbulence and conduct numerical simulations of a turbulent Uduct flow with a number of turbulence models in order to assess their overall performance. The models evaluated in this work are some typical linear eddy viscosity turbulence models, nonlinear eddy viscosity turbulence models （NLEVM） （quadratic and cubic）, a quadratic explicit algebraic stress model （EASM） and a Reynolds stress model （RSM） developed based on the second-moment closure. Our numerical results show that a cubic NLEVM that performs considerably well in other benchmark turbulent flows, such as the Craft, Launder and Suga model and the Huang and Ma model, is able to capture the major features of the highly curved turbulent U-duct flow, including the damping of turbulence near the convex wall, the enhancement of turbulence near the concave wall, and the subsequent turbulent flow separation. The predictions of the cubic models are quite close to that of the RSM, in relatively good agreement with the experimental data, which suggests that these models may be employed to simulate the turbulent curved flows in engineering applications.     

New Molecular Transport Model for FDF/LES of Turbulence with Passive Scalar

The paper addresses the issue of modelling and computation of wall-bounded turbulent flows with passive scalars. In the present approach, the large eddy simulation (LES) method is used to compute the velocity field in the near-wall zone. The LES is coupled with the Lagrangian filtered density function (FDF) model for the transport of a passive scalar. In the paper, we propose two models to account for the molecular transport near the wall and investigate their behaviour in the limit case of small filter widths. One of the models is tested numerically, and computational results for a heated channel flow are compared with the available DNS data.     

Modifications for an explicit algebraic stress model

An extension of the explicit algebraic stress model, developed by Gatski and Speziale [Gatski TB, Speziale CG. On the explicit algebraic stress models for complex turbulent flows. Journal of Fluid Mechanics 1993; 254: 59–78] is proposed. The extension implicates some essential characteristics of second‐order closure models. The strain‐dependent coefficients are modified, resulting in an alleviation of the numerical instabilities involved in the model. A new near‐wall damping function f_μ in the eddy viscosity relation is introduced. To enhance dissipation in near‐wall regions, the model constant C_ϵ1 is modified and an extra positive source term is included in the dissipation equation. In addition, a realizable time scale is incorporated to remove the wall singularity. Computed results show that the modified Gatski–Speziale (MGS) model predictions are in better agreement with the direct numerical simulation (DNS) and experimental data than those of the original Gatski–Speziale (OGS) model. Copyright © 2001 John Wiley & Sons, Ltd.     

Analysis of high-order velocity moments in a strained channel flow

In the current study, model expressions for fifth-order velocity moments obtained from the truncated Gram-Charlier series expansions model for a turbulent flow field probability density function are validated using data from direct numerical simulation (DNS) of a planar turbulent flow in a strained channel. Simplicity of the model expressions, the lack of unknown coefficients, and their applicability to non-Gaussian turbulent flows make this approach attractive to use for closing turbulent models based on the Reynolds-averaged Navier-Stokes equations. The study confirms validity of the model expressions. It also shows that the imposed flow deformation improves an agreement between the model and DNS profiles for the fifth-order moments in the flow buffer zone including when the flow reverses its direction. The study reveals sensitivity of particularly odd velocity moments to the grid resolution. A new length scale is proposed as a criterion for the grid generation near the wall and in the other flow areas dominated by high mean velocity gradients when higher-order statistics have to be collected from DNS.     

Calculation of complex near-wall turbulent flows with a low-Reynolds-number k-ϵ model

An improved low-Reynolds-number k-? model has been formulated and tested against a range of DNS (direct numerical simulation) and experimental data for channel and complex shear layer flows. The model utilizes a new form of damping function adopted to account for both wall proximity effects and viscosity influences and a more flexible damping argument based on the gradient of the turbulent kinetic energy on the wall. Additionally, the extra production of the inhomogeneous part of the viscous dissipation near a wall has been added to the dissipation equation with significantly improved results. The proposed model was successfully applied to the calculation of a range of wall shear layers in zero, adverse and favourable pressure gradients as well as backward-facing-step separated flows.     

Prediction of turbulent wall shear flows directly from wall

The fully elliptic Reynolds-averaged Navier–Stokes equations have been used together with Lam and Bremhorst's low-Reynolds-number model, Chen and Patel's two-layer model and a two-point wall function method incorporated into the standard k-? model to predict channel flows and a backward-facig step flow. These flows enable the evaluation of the performance of different near-wall treatments in flows involving streamwise and normal pressure gradients, flows with separation and flows with non-equilibrium turbulence characteristics. Direct numerical simulation (DNS) of a channel flow with Re =3200 further provides the detailed budgets of each modelling term of the k and ?-transport equations. Comparison of model results with DNS data to evaluate the performance of each modelling term is also made in the present study. It is concluded that the low-Reynolds-number model has wider applicability and performs better than the two-layer model and wall function approaches. Comparison with DNS data further shows that large discrepancies exist between the DNS budgets and the modelled production and destruction terms of the ? equation. However, for simple channel flow the discrepancies are similar in magnitude but opposite in sign, so they are cancelled by each other. This may explain why, even when employing such an inaccurately modelled ?-equation, one can still predict satisfactorily some simple turbulent flows.     

Hybrid LES-RANS: An approach to make LES applicable at high Reynolds number

The main bottle neck for using large eddy simulations (LES) at high Reynolds number is the requirement of very fine meshes near walls. Hybrid LES-Reynolds-averaged Navier-Stokes (RANS) was invented to get rid of this limitation. In this method, unsteady RANS (URANS) is used near walls and away from walls LES is used. The matching between URANS and LES takes place in the inner log-region. In the present paper, a method to improve standard LES-RANS is evaluated. The improvement consists of adding instantaneous turbulent fluctuations (forcing conditions) at the matching plane in order to provide the equations in the LES region with relevant turbulent structures. The fluctuations are taken from a DNS of a generic boundary layer. Simulations of fully developed channel flow and plane asymmetric diffuser flow are presented. Hybrid LES-RANS is used both with and without forcing conditions.     

Exact transport equation for eddy diffusivity in turbulent shear flow

Two-equation models that treat the transport equations for two variables are typical models for the Reynolds-averaged Navier–Stokes equation. Compared to the equation for the turbulent kinetic energy, the equation for the second variable such as the dissipation rate does not have a theoretical analogue. In this work, the exact transport equation for the eddy diffusivity was derived and examined for better understanding turbulence and improving two-equation models. A new length scale was first introduced, which involves the response function for the scalar fluctuation. It was shown that the eddy diffusivity can be expressed as the correlation between the velocity fluctuation and the new length scale. The transport equations for the eddy diffusivity and the length-scale variance were derived theoretically. Statistics such as terms in the transport equations were evaluated using the direct numerical simulation of turbulent channel flow. It was shown that the streamwise component of the eddy diffusivity is greater than the other two components in the whole region. In the transport equation for the eddy diffusivity, the production term due to the Reynolds stress is a main positive term, whereas the pressure–length-gradient correlation term plays a role of destruction. It is expected that the analysis of the transport equations is helpful in developing better turbulence models.     

Regularization modeling for LES of separated boundary layer flow

Regularization models for the turbulent stress tensor are applied to mixing and separated boundary layers. The Leray and the NS-α models in large-eddy simulation (LES) are compared to direct numerical simulation (DNS) and (dynamic) eddy-viscosity models. These regularization models are at least as accurate as the dynamic eddy-viscosity model, and can be derived from an underlying dynamic principle. This allows one to maintain central transport properties of the Navier-Stokes equations in the model and to extend systematically toward complex applications. The NS-α model accurately represents the small-scale variability, albeit at considerable resolution. The Leray model was found to be much more robust, allowing simulations at high Reynolds number. Leray simulations of a separated boundary layer are shown for the first time. The strongly localized transition to turbulence that arises under a blowing and suction region over a flat plate was captured accurately, quite comparable to the dynamic model. In contrast, results obtained with the Smagorinsky model, either with or without Van Driest damping, yield considerable errors, due to its excessive dissipation.     

Finite element,stream function-vorticity solution of secondary flow in the channels of a rod bundle

A finite element method has been applied to predict the overall features of the fully developed turbulent flow in the non-circular channels of a rod bundle. The finite element discretization is based on the conventional Galerkin method using an isoparametric quadrilateral element with mixed interpolation. The primary axial flow and turbulent kinetic energy distributions have been predicted for fully developed turbulent flow conditions right up to the wall. The secondary velocity is represented by the stream function-vorticity formulation and the no-slip boundary conditions are explicitly introduced in the nonlinear equations by a boundary vorticity formula. The Newton-Raphson method is applied to the stream function-vorticity equations and solved simultaneously by the frontal solution technique. A one-equation eddy viscosity model of turbulence and an algebraic stress transport model have been used to predict primary axial velocity, secondary velocities and turbulent kinetic energy. The predictions obtained for a central subchannel of an equilateral-triangular rod array with p/d= 1.3 are in reasonable agreement with experimental data.     

A RESIDUAL-BASED UNRESOLVED-SCALE FINITE ELEMENT MODELLING FOR IMPLICT LARGE EDDY SIMULATION 1)

对应于湍流的大尺度与小尺度流场信息, 本文在有限元的框架下, 假设Navier-Stokes方程的解的形函数由大尺度和不可解尺度形函数叠加组成, 引入对应的权函数, 将Navier-Stokes方程的有限元变分形式分解为大尺度和不可解尺度系统. 根据不可解尺度系统, 构建基于Navier-Stokes大尺度方程残差的不可解尺度模型, 将其代入Navier-Stokes方程的大尺度系统, 进而数值求解大尺度系统得到Navier-Stokes方程的大尺度解. 该方法无需像传统的大涡模拟方法那样对方程的解进行过滤, 通过对形函数进行尺度分解实现解的尺度分解. 本文使用该方法的自编程序代码开展了槽道湍流的数值模拟. 通过与有限差分大涡模拟、DNS计算结果的比较, 发现在使用较少网格情况下该方法预测的平均流向速度在近壁区与DNS数据吻合, 在黏性外层略偏高; 该方法对雷诺应力预测偏低导致从流向向垂向方向上湍动能输运略偏低. 流向速度等值面图显示该方法有效捕捉到了大尺度旋涡结构; 同时在近壁区可以观察到明显的低速条带结构.     

Large Eddy Simulation of Flow around a Rectangular Cylinder Using the Finite Element Method

In previous papers, we proposed finite element schemes based on the Petrov-Galerkin weak formulation using exponential weighting functions for solving accurately, and in a stable manner, the flow field of an incompressible viscous fluid. In this paper, we present the Petrov-Galerkin finite element scheme for turbulent flow fields based on large eddy simulation using the standard Smagorinsky model with the Van Driest damping function. The filtered incompressible Navier-Stokes equations are numerically integrated in time by using a fractional step strategy with second-order accurate Adams-Bashforth explicit differencing for both convection an diffusion terms. In order to evaluate more accurately a mass matrix, the well-known multi-pass algorithm was also adopted in this study. Numerical results obtained are compared through flow around a rectangular cylinder at Re = 22,000 with the experimental data and other existing numerical data.     

A Priori Tests of RANS Models for Turbulent Channel Flows of a Dense Gas

Dense gas effects, encountered in many engineering applications, lead to unconventional variations of the thermodynamic and transport properties in the supersonic flow regime, which in turn are responsible for considerable modifications of turbulent flow behavior with respect to perfect gases. The most striking differences for wall-bounded turbulence are the decoupling of dynamic and thermal effects for gases with high specific heats, the liquid-like behavior of the viscosity and thermal conductivity, which tend to decrease away from the wall, and the increase of density fluctuations in the near wall region. The present work represents a first attempt of quantifying the influence of such dense gas effects on modeling assumptions employed for the closure of the Reynolds-averaged Navier–Stokes equations, with focus on the eddy viscosity and turbulent Prandtl number models. For that purpose, we use recent direct numerical simulation results for supersonic turbulent channel flows of PP11 (a heavy fluorocarbon representative of dense gases) at various bulk Mach and Reynolds numbers to carry out a priori tests of the validity of some currently-used models for the turbulent stresses and heat flux. More specifically, we examine the behavior of the modeled eddy viscosity for some low-Reynolds variants of the \(k-\varepsilon \) model and compare the results with those found for a perfect gas at similar conditions. We also investigate the behavior of the turbulent Prandtl number in dense gas flow and compare the results with the predictions of two well-established turbulent Prandtl number models.     

Shear‐improved Smagorinsky modeling of turbulent channel flow using generalized Lattice Boltzmann equation

Generalized Lattice Boltzmann equation (GLBE) was used for computation of turbulent channel flow for which large eddy simulation (LES) was employed as a turbulence model. The subgrid‐scale turbulence effects were simulated through a shear‐improved Smagorinsky model (SISM), which is capable of predicting turbulent near wall region accurately without any wall function. Computations were done for a relatively coarse grid with shear Reynolds number of 180 in a parallelized code. Good numerical stability was observed for this computational framework. The results of mean velocity distribution across the channel showed good correspondence with direct numerical simulation (DNS) data. Negligible discrepancies were observed between the present computations and those reported from DNS for the computed turbulent statistics. Three‐dimensional instantaneous vorticity contours showed complex vortical structures that appeared in such flow geometries. It was concluded that such a framework is capable of predicting accurate results for turbulent channel flow without adding significant complications and the computational cost to the standard Smagorinsky model. As this modeling was entirely local in space it was therefore adapted for parallelization. Copyright © 2010 John Wiley & Sons, Ltd.     

基于Navier-Stokes方程残差的隐式大涡模拟有限元模型

对应于湍流的大尺度与小尺度流场信息, 本文在有限元的框架下, 假设Navier-Stokes方程的解的形函数由大尺度和不可解尺度形函数叠加组成, 引入对应的权函数, 将Navier-Stokes方程的有限元变分形式分解为大尺度和不可解尺度系统. 根据不可解尺度系统, 构建基于Navier-Stokes大尺度方程残差的不可解尺度模型, 将其代入Navier-Stokes方程的大尺度系统, 进而数值求解大尺度系统得到Navier-Stokes方程的大尺度解. 该方法无需像传统的大涡模拟方法那样对方程的解进行过滤, 通过对形函数进行尺度分解实现解的尺度分解. 本文使用该方法的自编程序代码开展了槽道湍流的数值模拟. 通过与有限差分大涡模拟、DNS计算结果的比较, 发现在使用较少网格情况下该方法预测的平均流向速度在近壁区与DNS数据吻合, 在黏性外层略偏高; 该方法对雷诺应力预测偏低导致从流向向垂向方向上湍动能输运略偏低. 流向速度等值面图显示该方法有效捕捉到了大尺度旋涡结构; 同时在近壁区可以观察到明显的低速条带结构.      

Large Eddy Simulation of Scalar Mixing in a Coaxial Confined Jet

The highly turbulent flow occurring inside gas-turbine combustors requires accurate simulation of scalar mixing if CFD methods are to be used with confidence in design. This has motivated the present paper, which describes the implementation of a passive scalar transport equation into an LES code, including assessment/testing of alternative discretisation schemes to avoid over/undershoots and excessive smoothing. Both second order accurate TVD and higher order accurate DRP schemes are assessed. The best performance is displayed by a DRP method, but this is only true on fine meshes; it produces similar (or larger) errors to a TVD scheme on coarser meshes, and the TVD approach has been retained for LES applications. The unsteady scalar mixing performance of the LES code is validated against published DNS data for a slightly heated channel flow. Excellent agreement between the current LES predictions and DNS data is obtained, for both velocity and scalar statistics. Finally, the developed methodology is applied to scalar transport in a confined co-axial jet mixing flow, for which experimental data are available. Agreement with statistically averaged fields for both velocity and scalar, is demonstrated to be very good, and a considerable improvement over the standard eddy viscosity RANS approach. Illustrations are presented of predicted time-resolved information e.g. time histories, and scalar pdf predictions. The LES results are shown, even using a simple Smagorinsky SGS model, to predict (correctly) lower values of the turbulent Prandtl number in the free shear regions of the flow, compared to higher values in the wall-affected regions. The ability to predict turbulent Prandtl number variations (rather than input these as in combustor RANS CFD models) is an important and promising feature of the LES approach for combustor flow simulation since it is known to be important in determining combustor exit temperature traverse.