A simplified v2–f model for near‐wall turbulence

A simplified version of the v²–f model is proposed that accounts for the distinct effects of low‐Reynolds number and near‐wall turbulence. It incorporates modified C_ε(1,2) coefficients to amplify the level of dissipation in non‐equilibrium flow regions, thus reducing the kinetic energy and length scale magnitudes to improve prediction of adverse pressure gradient flows, involving flow separation and reattachment. Unlike the conventional v²–f, it requires one additional equation (i.e. the elliptic equation for the elliptic relaxation parameter f_µ) to be solved in conjunction with the k–ε model. The scaling is evaluated from k in collaboration with an anisotropic coefficient C_v and f_µ. Consequently, the model needs no boundary condition on and avoids free stream sensitivity. The model is validated against a few flow cases, yielding predictions in good agreement with the direct numerical simulation (DNS) and experimental data. Copyright © 2007 John Wiley & Sons, Ltd.     

Assessment of the finite volume method applied to the v2 − f model

The objective of this paper is to assess the accuracy of low‐order finite volume (FV) methods applied to the v² ? f turbulence model of Durbin (Theoret. Comput. Fluid Dyn. 1991; 3 :1–13) in the near vicinity of solid walls. We are not (like many others) concerned with the stability of solvers ‐ the topic at hand is simply whether the mathematical properties of the v² ? f model can be captured by the given, widespread, numerical method. The v² ? f model is integrated all the way up to solid walls, where steep gradients in turbulence parameters are observed. The full resolution of wall gradients imposes quite high demands on the numerical schemes and it is not evident that common (second order) FV codes can fully cope with such demands. The v² ? f model is studied in a statistically one‐dimensional, fully developed channel flow where we compare FV schemes with a highly accurate spectral element reference implementation. For the FV method a higher‐order face interpolation scheme, using Lagrange interpolation polynomials up to arbitrary order, is described. It is concluded that a regular second‐order FV scheme cannot give an accurate representation of all model parameters, independent of mesh density. To match the spectral element solution an extended source treatment (we use three‐point Gauss–Lobatto quadrature), as well as a higher‐order discretization of diffusion is required. Furthermore, it is found that the location of the first internal node need to be well within y⁺=1. Copyright © 2009 John Wiley & Sons, Ltd.     

A meshless method for numerical simulation of depth‐averaged turbulence flows using a k‐ϵ model

A three‐dimensional numerical model is developed to analyze free surface flows and water impact problems. The flow of an incompressible viscous fluid is solved using the unsteady Navier–Stokes equations. Pseudo‐time derivatives are introduced into the equations to improve computational efficiency. The interface between the two phases is tracked using a volume‐of‐fluid interface tracking algorithm developed in a generalized curvilinear coordinate system. The accuracy of the volume‐of‐fluid method is first evaluated by the multiple numerical benchmark tests, including two‐dimensional and three‐dimensional deformation cases on curvilinear grids. The performance and capability of the numerical model for water impact problems are demonstrated by simulations of water entries of the free‐falling hemisphere and cone, based on comparisons of water impact loadings, velocities, and penetrations of the body with experimental data. For further validation, computations of the dam‐break flows are presented, based on an analysis of the wave front propagation, water level, and the dynamic pressure impact of the waves on the downstream walls, on a specific container, and on a tall structure. Extensive comparisons between the obtained solutions, the experimental data, and the results of other numerical simulations in the literature are presented and show a good agreement. Copyright © 2015 John Wiley & Sons, Ltd.     

Near‐wall profiles of mean flow and turbulence quantities predicted by eddy‐viscosity turbulence models

This paper presents for the simple flow over a flat plate the near‐wall profiles of mean flow and turbulence quantities determined with seven eddy‐viscosity turbulence models: the one‐equation turbulence models of Menter and Spalart & Allmaras; the k‐ω two‐equation model proposed by Wilcox and its TNT, BSL and SST variants and the $k-\sqrt{k}L$ two‐equation model. The results are obtained at several Reynolds numbers ranging from 10⁷ to 2.5 × 10⁹. Sets of nine geometrically similar Cartesian grids are adopted to demonstrate that the numerical uncertainty of the finest grid predictions is negligible. The profiles obtained numerically have relevance for the application of so‐called ‘wall function’ boundary conditions. Such wall functions refer to assumptions about the flow in the viscous sublayer and the ‘log law’ region. It turns out that these assumptions are not always satisfied by our results, which are obtained by computing the flow with full near‐wall resolution. In particular, the solution in the ‘log‐law’ region is dependent on the turbulence model and on the Reynolds number, which is a disconcerting result for those who apply wall functions. Copyright © 2009 John Wiley & Sons, Ltd.     

Time‐dependent simulation of cavitating flow with k − ℓ turbulence models

A compressible, multiphase, one‐fluid Reynolds‐averaged Navier–Stokes solver has been developed to study turbulent cavitating flows. The interplay between turbulence and cavitation regarding the unsteadiness and structure of the flow is complex and not well understood. This constitutes a critical point to accurately simulate the dynamic behavior of sheet cavities. In the present study, different formulations based on a k ? ? transport‐equation model are investigated and a scale‐adaptive formulation is proposed. Numerical results are given for a Venturi geometry and comparisons are made with experimental data. The scale‐adaptive model shows several improvements compared with standard turbulence models. Copyright © 2011 John Wiley & Sons, Ltd.     

Compound wall treatment with low‐Re turbulence model

A generalized treatment for the wall boundary conditions relating to turbulent flows is developed that blends the integration to a solid wall with wall functions. The blending function ensures a smooth transition between the viscous and turbulent regions. An improved low Reynolds number k?ε model is coupled with the proposed compound wall treatment to determine the turbulence field. The eddy viscosity formulation maintains the positivity of normal Reynolds stresses and Schwarz' inequality for turbulent shear stresses. The model coefficients/functions preserve the anisotropic characteristics of turbulence. Computations with fine and coarse meshes of a few flow cases yield appreciably good agreement with the direct numerical simulation and experimental data. The method is recommended for computing the complex flows where computational grids cannot satisfy a priori the prerequisites of viscous/turbulence regions. Copyright © 2011 John Wiley & Sons, Ltd.     

Low Reynolds number k–ε model for near‐wall flow

A wall‐distance free k–ε turbulence model is developed that accounts for the near‐wall and low Reynolds number effects emanating from the physical requirements. The model coefficients/functions depend non‐linearly on both the strain rate and vorticity invariants. Included diffusion terms and modified C_ε(1,2) coefficients amplify the level of dissipation in non‐equilibrium flow regions, thus reducing the kinetic energy and length scale magnitudes to improve prediction of adverse pressure gradient flows, involving flow separation and reattachment. The model is validated against a few flow cases, yielding predictions in good agreement with the direct numerical simulation (DNS) and experimental data. Copyright © 2005 John Wiley & Sons, Ltd.     

A robust near-wall elliptic-relaxation eddy-viscosity turbulence model for CFD

An eddy-viscosity model based on Durbin’s elliptic relaxation concept is proposed, which solves a transport equation for the velocity scales ratio instead of , thus making the model more robust and less sensitive to grid nonuniformities. Computations of flows and heat transfer in a plane channel, behind a step and in a round impinging jet show all satisfactory results.     

Assessment of variants of the k–ϵ turbulence model for engine flow applications

This paper is concerned with simulation of the mean flow and turbulence evolution in a model engine and comparison of the behaviour of certain important turbulence parameters, namely the intensity, length scale and dissipation time scale, as predicted by three variants of the k–? model developed for application to strongly compressible flows. The predictions pertain to the axisymmetric, disc-chamber, four-stroke, Imperial College model engine operating at 200 rpm and compression ratios of 3·5 and 6·7. The paper analyses the predicted variations of these parameters during the induction, compression and expansion strokes and identifies the versions that produce the most consistent and physically plausible variations. The significance, to the turbulence evolution, of the ratio of the turbulence dissipation time scale to the time scale of compression/expansion is also discussed. It is concluded that on these grounds the Morel–Mansour and El Tahry versions are, and the Watkins version is not, suitable for engine applications.     

Wall modeling via function enrichment within a high‐order DG method for RANS simulations of incompressible flow

We present a novel approach to wall modeling for the Reynolds‐averaged Navier‐Stokes equations within the discontinuous Galerkin method. Wall functions are not used to prescribe boundary conditions as usual, but they are built into the function space of the numerical method as a local enrichment, in addition to the standard polynomial component. The Galerkin method then automatically finds the optimal solution among all shape functions available. This idea is fully consistent and gives the wall model vast flexibility in separated boundary layers or high adverse pressure gradients. The wall model is implemented in a high‐order discontinuous Galerkin solver for incompressible flow complemented by the Spalart‐Allmaras closure model. As benchmark examples, we present turbulent channel flow starting from Re_τ=180 and up to Re_τ=100000 as well as flow past periodic hills at Reynolds numbers based on the hill height of Re_H=10595 and Re_H=19000.     

An immersed boundary method wall model for high‐Reynolds‐number channel flow over complex topography

High‐Reynolds‐number channel flows regularly encounter topographies composed of multiple length scales and that protrude into the boundary layer. Physically, the presence of immersed obstacles leads to increased velocity gradients, turbulence production, and manifestation of wakes. Considerable challenges are associated with numerically describing the presence of obstacles in channel flows. Common approaches include generation of a computational mesh that is uniquely designed for the flow and obstacle, the immersed boundary method, and terrain‐following coordinates. There are challenges and limitations associated with each of these techniques. Specification of boundary conditions representing the perimeter of solid obstacles is a primary challenge of the immersed boundary method. In this document, a simplistic canopy stress‐like wall model is used to impose boundary conditions. The model isolates aerodynamically relevant local frontal areas through evaluation of the gradient of the topographic height field. The gradient of the height field describes both the surface‐normal direction and the frontal area, making it ideal for detecting areas on which the flow impinges. The model is tested in numerical simulations of turbulent half‐channel flow over topographies with different obstacles affixed–right prisms, rectangular prisms, ellipsoidal mounds, and sinusoids. In all cases, the performance is strong relative to datasets presented in the literature. Results are finally presented for numerical simulation of flow over complex synthetic fractal‐like topography and a synthetic city. These results show interesting trends in how the turbulent multiscale flow field responds to multiscale topography. Copyright © 2012 John Wiley & Sons, Ltd.     

A two‐scale low‐Reynolds number turbulence model

In this study, a two‐scale low‐Reynolds number turbulence model is proposed. The Kolmogorov turbulence time scale, based on fluid kinematic viscosity and the dissipation rate of turbulent kinetic energy (ν, ε), is adopted to address the viscous effects and the rapid increasing of dissipation rate in the near‐wall region. As a wall is approached, the turbulence time scale transits smoothly from a turbulent kinetic energy based (k, ε) scale to a (ν, ε) scale. The damping functions of the low‐Reynolds number models can thus be simplified and the near‐wall turbulence characteristics, such as the ε distribution, are correctly reproduced. The proposed two‐scale low‐Reynolds number turbulence model is first examined in detail by predicting a two‐dimensional channel flow, and then it is applied to predict a backward‐facing step flow. Numerical results are compared with the direct numerical simulation (DNS) budgets, experimental data and the model results of Chien, and Lam and Bremhorst respectively. It is proved that the proposed two‐scale model indeed improves the predictions of the turbulent flows considered. Copyright © 2000 John Wiley & Sons, Ltd.     

An extended κ–ϵ finite element model

An extended κ–? model (to include low-Reynolds-number regions) employing weighting functions is presented. Wall functions for the near-wall zones are developed giving correct boundary values for the Shear stress and κ–?. A finite element model using a penalty formulation for incompressible turbulent flow is applied to Solve a flow between two plates. Results with mesh boundaries situated in the near-wall region and a: the wall are compared with measured values.     

SPH computation of plunging waves using a 2‐D sub‐particle scale (SPS) turbulence model

The paper presents a 2‐D large eddy simulation (LES) modelling approach to investigate the properties of the plunging waves. The numerical model is based on the smoothed particle hydrodynamics (SPH) method. SPH is a mesh‐free Lagrangian particle approach which is capable of tracking the free surfaces of large deformation in an easy and accurate way. The Smagorinsky model is used as the turbulence model due to its simplicity and effectiveness. The proposed 2‐D SPH–LES model is applied to a cnoidal wave breaking and plunging over a mild slope. The computations are in good agreement with the documented data. Especially the computed turbulence quantities under the breaking waves agree better with the experiments as compared with the numerical results obtained by using the k–ε model. The sensitivity analyses of the SPH–LES computations indicate that both the turbulence model and the spatial resolution play an important role in the model predictions and the contributions from the sub‐particle scale (SPS) turbulence decrease with the particle size refinement. Copyright © 2006 John Wiley & Sons, Ltd.     

Near‐wall turbulence modelling with enhanced dissipation

A low‐Reynolds number k–ε turbulence model is proposed that incorporates diffusion terms and modified C_ε(1,2) coefficients to amplify the level of dissipation in non‐equilibrium flow regions, thus reducing the kinetic energy and length scale magnitudes to improve prediction of adverse pressure gradient flows, involving flow separation and reattachment. Unlike the conventional k–ε model, it requires no wall function/distance parameter that bridges the near‐wall integration. The model is validated against a few flow cases, yielding predictions in good agreement with the direct numerical simulation (DNS) and experimental data. Copyright © 2003 John Wiley & Sons, Ltd.     

适用于浸入边界法的大涡模拟紊流壁面模型

基于近壁定常剪切应力假设,提出了一种新的适用于浸入边界法的大涡模拟紊流壁面模型。通过引入壁面滑移速度,修正了线性速度剖面计算得到的壁面剪切应力,使之满足Werner-Wengle模型。将其应用于平板紊流和高Re数圆管紊流的数值模拟,对比采用和不采用壁面模型的结果得知,采用此模型的速度剖面与实验值吻合良好,验证了此模型的有效性。研究了不同欧拉/拉格朗日网格相对位置对结果的影响,证明了此模型具有较好的鲁棒性,以及可根据局部流动状态和网格精度自动开闭的特点。     

A 5‐equation,transient, hyperbolic, 1‐dimensional model for slug capturing in pipes

A novel numerical scheme for slug capturing in pipes using a 1‐dimensional transient hyperbolic 5‐equation 2‐fluid model is presented. Previous work has shown that 1‐dimensional 2‐fluid models are able to capture slug flow automatically. In this work, a similar approach is further developed using a new numerical scheme, applied to a hyperbolic 5‐equation 2‐fluid model. Starting from a finite volume discretisation of a 5‐equation 2‐fluid hyperbolic model and adding appropriate closure relations, a second‐order code is implemented and applied to air‐water flows in horizontal pipes, simulating the 2‐phase to 1‐phase flow process. The code is evaluated in some common standard test cases. A slug capturing application is also discussed. We show, in an air/water horizontal pipe, slug initiation, growth, and development. Moreover, a grid refinement analysis is performed showing that the method is grid independent and we show the code capability to take into account eventual surface tension effects, through the instantaneous pressure relaxation process. Finally, a prediction of flow regime transitions is shown and compared with a well‐known theoretical flow pattern map in addition to a preliminary comparison of computed slug characteristics against well‐known empirical correlations.     

Implicit meanflow–multigrid algorithms for Reynolds stress model computation of 3‐D anisotropy‐driven and compressible flows

The present paper investigates the multigrid (MG) acceleration of compressible Reynolds‐averaged Navier–Stokes computations using Reynolds‐stress model 7‐equation turbulence closures, as well as lower‐level 2‐equation models. The basic single‐grid SG algorithm combines upwind‐biased discretization with a subiterative local‐dual‐time‐stepping time‐integration procedure. MG acceleration, using characteristic MG restriction and prolongation operators, is applied on meanflow variables only (MF–MG), turbulence variables being simply injected onto coarser grids. A previously developed non‐time‐consistent (for steady flows) full‐approximation‐multigrid (s–MG) is assessed for 3‐D anisotropy‐driven and/or separated flows, which are dominated by the convergence of turbulence variables. Even for these difficult test cases CPU‐speed‐ups r_CPUSUP∈[3, 5] are obtained. Alternative, potentially time‐consistent approaches (unsteady u–MG), where MG acceleration is applied at each subiteration, are also examined, using different subiterative strategies, MG cycles, and turbulence models. For 2‐D shock wave/turbulent boundary layer interaction, the fastest s–MG approach, with a V(2, 0) sawtooth cycle, systematically yields CPU‐speed‐ups of 5±½, quasi‐independent of the particular turbulence closure used. Copyright © 2008 John Wiley & Sons, Ltd.     

An eddy viscosity model with near‐wall modifications

An extended version of the isotropic k–ε model is proposed that accounts for the distinct effects of low‐Reynolds number (LRN) and wall proximity. It incorporates a near‐wall correction term to amplify the level of dissipation in nonequilibrium flow regions, thus reducing the kinetic energy and length scale magnitudes to improve prediction of adverse pressure gradient flows, involving flow separation and reattachment. The eddy viscosity formulation maintains the positivity of normal Reynolds stresses and the Schwarz' inequality for turbulent shear stresses. The model coefficients/functions preserve the anisotropic characteristics of turbulence. The model is validated against a few flow cases, yielding predictions in good agreement with the direct numerical simulation (DNS) and experimental data. Comparisons indicate that the present model is a significant improvement over the standard eddy viscosity formulation. Copyright © 2005 John Wiley & Sons, Ltd.