Reynolds Stress Transport Equations Research Articles

A consistency condition for non-linear algebraic Reynolds stress models in turbulence

An important consistency condition for non-linear algebraic Reynolds stress models involving their dependence on rotational strains is discussed from a basic theoretical standpoint. A variety of recently proposed models are demonstrated to violate this condition which leads to physically inconsistent results and unrealizable solutions in rotating flows — an undesirable state of affairs that was avoided in the majority of earlier models. It is shown that when non-linear algebraic models are systematically derived from the Reynolds stress transport equation, such inconsistencies are automatically avoided. The implications of these results for turbulence modeling are demonstrated quantitatively by the illustrative example of isotropic turbulence in a rotating frame.

Reynolds-stress modelling of turbulent rotating flows

The present analysis shows that the EVM can not reflect the turbulence physics in non-inertial frame. The effects of Coriolis force on turbulence is embodied naturally in the Reynolds-stress transport equation. It is observed that the existing second-moment closure model with appropriate near-wall treatment can adequately predict flows in rotating channel and in axially rotating pipe for moderate rotation rate.

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.

Modeling of the Reynolds Stress Transport Equation

An empirical strategy for improving modeling of the energy dissipation rate e ij and the velocity-pressure gradient Π ij terms in the transport equations for the Reynolds stresses is proposed on the basis of available direct numerical simulations of the turbulent boundary layer and fully developed turbulent channel flow. The dissipation-rate model takes into account the anisotropy of the dissipation rate not only near the wall but elsewhere in the flow as well. When modeling the velocity-pressure gradient term, the usual split between pressure-transport and redistributive terms is avoided. One consequence of this strategy is the elimination of wall reflection or near-wall correction terms. Wall-topography parameters, such as the distance to the wall and wall-normal vectors, are also absent. Calculations of a turbulent boundary layer over a smooth wall confirm the potential improvements that can be achieved with this approach.

Toward consistent formulation of Reynolds stress and scalar flux closures

Langevin stochastic differential equations provide a consistent basis for Reynolds stress, scalar transport and p.d.f. models. However, the stochastic equations must be capable of representing existing closures, like the General Linear Model, or the Rotta and Monin return to isotropy formulations. A consistent approach to derive both Reynolds stress and scalar flux transport equations, starting from a stochastic differential equation for velocity fluctuations, is presented here. A set of algebraic relations for the dispersion tensor is derived for homogeneous shear flow and for the log-layer.

Some developments in computational modeling of turbulent flows

In this paper, some recent developments of two turbulence closure schemes at ICOMP, NASA Lewis will be discussed. One is the Reynolds-stress algebraic equation model and the other is the Reynolds-stress transport equation model. Various model constraints required by the rapid distortion theory, the invariant theory and the realizability principle, etc. will be described in the model development. The models discussed are for high-turbulent Reynolds number flows, so that the near-wall turbulence and the low-Reynolds-number turbulence are not discussed here.

Reynolds Stress Transport Equations in a Momentumless Wake: Experiments and Models

The determination of the transfer terms in the Reynolds stress transport equations is an important issue in the improvement of closure models in turbulence. A part of these efforts is to provide accurate experimental information on each term of these equations and, particularly, on the pressure-strain term. The self-similar region of the momentumless wake of an axisymmetric, propeller-driven body is studied to enable the required balances of the Reynolds stress transport equations. The axial and azimuthal balances show that the flow is mainly dominated by convection. For the shear equations, there is a quasiequilibrium between production and pressure strain, except in the center of the wake. Comparisons are made between the experimentally determined distributions of pressure-strain terms and the corresponding distributions predicted by second-order closure models.

Reynolds stress transport equations in a momentumless wake - Experiments and models

Reynolds stress transport equations in a momentumless wake - Experiments and models

Modeling of the Reynolds stress transport equation

Modeling of the Reynolds stress transport equation

Simulation of turbulent square-duct flow - Dissipation and small-scale motion

Results and the database from a direct numerical simulation of turbulent flow in a square duct are used to investigate the dissipation processes in a flow with two inhomogeneous directions. First, the effect of the corner on small-scale topological patterns is identified, and second, information for the assessment and development of turbulence closure models is provided by calculating all of the terms in the transport equations for the turbulent dissipation rate and enstrophy. It is shown that, with the exception of two production terms, these budgets have the same dynamics down to the wall. The effects of the corner are manifested by vanishing turbulence production in the viscous sublayer very close to the corner, and reduced dissipation of turbulent vorticity along the diagonal, farther away from the corner. IRECT numerical simulation (DNS) of simple shear flows has proven to be an effective and important tool for studying turbulence structures and near-wall effects.1 The present paper extends this idea to complex turbulent flows by using the database from a recent DNS of turbulent square-duct flow2 to investigate the influence of the corner on the structure of small-scale turbulence. These effects are manifested by vanishing turbulence production as well as reduced turbulence energy and dissipation along the diagonal (corner bisector).2 The imbalance between the turbulence along the corner bisector and the midchannel (wall bisector) creates the stress-driven secondary flow, defined as secondary flow of the second kind.24 Of particular interest in this paper is the turbulence dissipationrate (and enstrophy) budget because the dissipation rate contributes significantly to the Reynolds stress balance.3 An order-of-magnitude analysis of the dissipation process is provided by Tennekes and Lumley5 by estimating the terms in the turbulence enstrophy budget. Mansour et al.6 presented all of the terms in the dissipation-rate budget from the DNS of turbulent channel flow and demonstrated the significance of these terms in a wall-bounded flow. The work of Rodi and Mansour7 also indicates the importance of all terms in the dissipation-rate budget near the wall, including the transport and diffusion. Because of their connection to the dissipation processes,8 the small-scale topological structures also are considered. By displaying scatter plots of the invariants of the local deformation tensor obtained in the homogeneous direction and in time, we can deduce the preferred small-scale topologies and dissipation production processes in the square duct. The present results document all of the terms in the turbulence dissipation rate and enstrophy equations that are essential for both the modeling of Reynolds stress transport equations and the development of subgrid-scale models.

Predictions of axisymmetric and two-dimensional impinging turbulent jets

The k − ε turbulence model and a version of a second-moment closure, modified to include the effect of pressure reflections from a solid surface, have been used as the basis of predictions of the flow that results from the orthogonal impingement of circular and two-dimensional (2-D) jets on a flat surface. Comparison of model predictions has been made with velocity measurements obtained in the stagnation and wall jet regions of the impinging flows. Results, in general, confirm the superiority of the Reynolds stress transport equation model for predicting mean and fluctuating velocities within the latter regions of such flows. In particular, modifications to the second-moment closure to account for the influence of the surface in distorting the fluctuating pressure field away from the wall successfully predict the damping of normal-to-wall velocity fluctuations throughout the impinging flows. In contrast, results derived from the eddy-viscosity-based approach do not, in general, accurately reproduce experimental observations.

Reynolds stress closure applied to axisymmetric, impinging turbulent jets

A second-order, single-point closure model for calculating the transport of momentum in turbulent flows is extended to cover flows that are close to solid surfaces. In such flows the proximity of a solid boundary directly influences the fluctuating pressure field within the main body of the flow and leads to a dampening of velocity fluctuations normal to the wall. These effects are accommodated through the incorporation of an additional contribution in the modelled form of the redistributive fluctuating pressure term used in the Reynolds stress transport equation. Predictions of the extended closure model are compared with available data in configurations where an air jet impinges orthogonally onto a plane surface. The inclusion of the wall reflection model is shown to result in superior predictions of mean velocities, and normal and shear stresses. In particular, normal-to-wall velocity fluctuations and shear stresses are successfully damped resulting in agreement with observations.

Fully explicit and self-consistent algebraic Reynolds stress model

A fully-explicit, self-consistent algebraic expression for the Reynolds stress, which is the exact solution to the Reynolds stress transport equation in the `weak equilibrium'' limit for two-dimensional mean flows for all linear and some quasi-linear pressure-strain models, is derived. Current explicit algebraic Reynolds stress models derived by employing the `weak equilibrium'' assumption treat the production-to-dissipation (P varepsilon) ratio implicitly, resulting in an effective viscosity that can be singular away from the equilibrium limit. In the present paper, the set of simultaneous algebraic Reynolds stress equations are solved in the full non-linear form and the eddy viscosity is found to be non-singular. Preliminary tests indicate that the model performs adequately, even for three dimensional mean flow cases. Due to the explicit and non-singular nature of the effective viscosity, this model should mitigate many of the difficulties encountered in computing complex turbulent flows with the algebraic Reynolds stress models.

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.

Towards the development of second-order closure models for nonequilibrium turbulent flows

A new approach for the development of second-order closure models suitable for nonequilibrium turbulent shear flows is presented. The central idea is based on the implementation of a relaxation time approximation to the nonequilibrium extension of an explicit algebraic stress model. Here, the algebraic stress model is systematically derived from the full Speziale et al. 1991 (SSG) second-order closure in the equilibrium limit of homogeneous turbulence. It is then extended to nonequilibrium turbulent flows by means of a Padé approximation, whereby approximate consistency with the rapid distortion theory (RDT) solution for homogeneous shear flow is established. The resulting model is tested in homogeneous shear flow turbulence under a wide range of shear rates. Substantially improved results are obtained in comparison to conventional second-order closure models that are based on the direct modeling of terms in the Reynolds stress transport equation.

Computational method of second-moment turbulence closures in complex geometries

This paper presents a finite volume numerical method to calculate turbulent flows using second-moment turbulence closures. The numerical method is formulated for a collocated grid arrangement and on body-fitted coordinate grids. A special interpolation scheme is derived to resolve the potential decoupling between momentum and Reynolds stress transport equations. This interpolation scheme naturally leads to stability-promoting terms in the momentum equations in the form of anisotropic eddy viscosities. Issues such as the implementation of boundary conditions with a body-fitted coordinate grid are also discussed. Applications are given to demonstrate the robustness of the numerical method and the performance of a second-moment turbulence closure. Computational results are compared with available experimental data and to those of the standard two-equation turbulence model. Good results have been obtained with the second-moment turbulence closure.

Shear-free turbulent boundary layers. Part 1. Physical insights into near-wall turbulence

Direct numerical simulation is used to examine the interaction of turbulence with a wall in the absence of mean shear. The influence of a solid wall on turbulence is analysed by first considering two ‘simpler’ types of boundaries. The first boundary is an idealized permeable wall. This boundary isolates and elucidates the viscous effects created by the wall. The second boundary is an idealized free surface. This boundary complements the first by allowing one to isolate and investigate the kinematic effects that occur near boundaries. The knowledge gained from these two simpler flows is then used to understand how turbulence is influenced by solid walls where both viscous and kinematic effects occur in combination.Examination of the instantaneous flow fields confirms the presence of previously hypothesized structures (splats), and reveals an additional class of structures (antisplats). Statistical analysis of the Reynolds stresses and Reynolds stress transport equations indicates the relative importance of dissipation, intercomponent energy transfer, and energy transport. It is found that it is not the structures themselves, but the imbalance between structures which leads to intercomponent energy transfer. Remarkably, this imbalance (and hence near-wall intercomponent energy transfer) is controlled by viscous processes such as dissipation and diffusion. The analysis presented herein is a departure from past notions of how boundaries influence turbulence. The efficacy of these qualitative physical concepts is demonstrated in Part 2 where improved near-wall turbulence models are derived based on these ideas.

Shear-free turbulent boundary layers. Part 2. New concepts for Reynolds stress transport equation modelling of inhomogeneous flows

Models for the dissipation tensor and (slow) pressure–strain terms of the Reynolds stress transport equations are presented which are applicable near boundaries. These models take into account the large inhomogeneity and anisotropy that can be present near walls and surfaces, and are inspired by the physical insights developed in Part 1 of this paper. The dissipation tensor model represents a fundamentally new approach to dealing with turbulence inhomogeneities. The pressure–strain model shows how the classic return-to-isotropy model of Lumley (1978) can be adapted to the near-wall region. The closure hypotheses underlying these two models are tested in an a priori fashion using direct numerical simulation (DNS) data.

A Calculation Method for Developing Turbulent Flow in Rectangular Ducts of Arbitrary Aspect Ratio

This paper describes a full Reynolds stress transport equation model for predicting developing turbulent flow in rectangular ducts. The pressure-strain component of the model is based on a modified form of the Launder, Reece and Rodi pressure-strain model and the use of a linear wall damping function. Predictions based on this model are compared with predictions referred to high Reynolds number and low Reynolds number k–ε transport equation models and with experimental data taken in square and rectangular ducts. The results indicate that the proposed model yields improved predictions of primary flow development and Reynolds stress behavior in a square duct. The proposed model also yields Reynolds stress anisotropy and secondary flow levels that are compatible and agree well with experiment, without recourse to a quadratic damping function to model near-wall pressure-strain behavior.

Comparison of variants of the bi‐conjugate gradient method for compressible Navier‐Stokes solver with second‐moment closure

AbstractVariants of the bi‐conjugate gradient (Bi‐CG) method are used to resolve the problem of slow convergence in CFD when it is applied to complex flow field simulation using higher‐order turbulence models. In this study the Navier‐Stokes and Reynolds stress transport equations are discretized with an implicit, total variation diminishing (TVD), finite volume formulation. The preconditioning technique of incomplete lower‐upper (ILU) factorization is incorporated into the conjugate gradient square (CGS), bi‐conjugate gradient stable (Bi‐CGSTAB) and transpose‐free quasi‐minimal residual (TFQMR) algorithms to accelerate convergence of the overall itertive methods. Computations have been carried out for separated flow fields over transonic bumps, supersonic bases and supersonic compression corners. By comparisons of the convergence rate with each other and with the conventional approximate factorization (AF) method it is shown that the Bi‐CGSTAB method gives the most efficient convergence rate among these methods and can speed up the CPU time by a factor of 2·4–6·5 as compared with the AF method. Moreover, the AF method may yield somewhat different results from variants of the Bi‐CG method owing to the factorization error which introduces a higher level of convergence criterion.