Lower-Dimensional Manifold (Algebraic) Representation of Reynolds Stress Closure Equations

For complex turbulent flows, Reynolds stress closure modeling (RSCM) is the lowest level at which models can be developed with some fidelity to the governing Navier–Stokes equations. Citing computational burden, researchers have long sought to reduce the seven-equation RSCM to the so-called algebraic Reynolds stress model which involves solving only two evolution equations for turbulent kinetic energy and dissipation. In the past, reduction has been accomplished successfully in the weak-equilibrium limit of turbulence. In non-equilibrium turbulence, attempts at reduction have lacked mathematical rigor and have been based on ad hoc hypotheses resulting in less than adequate models.?In this work we undertake a formal (numerical) examination of the dynamical system of equations that constitute the Reynolds stress closure model to investigate the following questions. (i) When does the RSCM equation system formally permit reduced representation? (ii) What is the dimensionality (number of independent variables) of the permitted reduced system? (iii) How can one derive the reduced system (algebraic Reynolds stress model) from the full RSCM equations? Our analysis reveals that a lower-dimensional representation of the RSCM equations is possible not only in the equilibrium limit, but also in the slow-manifold stage of non-equilibrium turbulence. The degree of reduction depends on the type of mean-flow deformation and state of turbulence. We further develop two novel methods for deriving algebraic Reynolds stress models from RSCM equations in non-equilibrium turbulence. The present work is expected to play an important role in bringing much of the sophistication of the RSCM into the realm of two-equation algebraic Reynolds stress models. Another objective of this work is to place the other algebraic stress modeling efforts in the lower-dimensional modeling context. Received 19 November 1999 and accepted 3 August 2000     

Upstream monotonic interpolation for scalar transport with application to complex turbulent flows

A new monotonic scheme for the approximation of steady scalar transport is formulated and implemented within a collocated finite-volume/pressure-correction algorithm for general turbulent flows in complex geometries. The scheme is essentially a monotonic implementation of the quadratic QUICK interpolation and uses a continuous and compact limiter to secure monotonicity. The principal purpose is to allow an accurate and fully bounded, hence stable, approximation of turbulence convection in the context of two-equation eddy viscosity and Reynolds stress transport modelling of two- and three-dimensional flows, both subsonic and transonic. Among other benefits, this capability permits an assessment to be made of the adequacy of approximating turbulence convection with first-order upwind schemes in conjunction with higher-order formulations for mean-flow properties—a widespread practice. The performance characteristics of the bounded scheme are illustrated by reference to computations for scalar transport, for a transonic flow in a Laval nozzle, for one separated laminar flow and for two separated turbulent flows computed with a non-linear RNG model and full Reynolds stress closure.     

Anisotropy Invariant Reynolds Stress Model of Turbulence (AIRSM) and its Application to Attached and Separated Wall-Bounded Flows

Numerical predictions with a differential Reynolds stress closure, which in its original formulation explicitly takes into account possible states of turbulence on the anisotropy-invariant map, are presented. Thus the influence of anisotropy of turbulence on the modeled terms in the governing equations for the Reynolds stresses is accounted for directly. The anisotropy invariant Reynolds stress model (AIRSM) is implemented and validated in different finite-volume codes. The standard wall-function approach is employed as initial step in order to predict simple and complex wall-bounded flows undergoing large separation. Despite the use of simple wall functions, the model performed satisfactory in predicting these flows. The predictions of the AIRSM were also compared with existing Reynolds stress models and it was found that the present model results in improved convergence compared with other models. Numerical issues involved in the implementation and application of the model are also addressed.     

End-to-end differentiable learning of turbulence models from indirect observations

The emerging push of the differentiable programming paradigm in scientific computing is conducive to training deep learning turbulence models using indirect observations. This paper demonstrates the viability of this approach and presents an end-to-end differentiable framework for training deep neural networks to learn eddy viscosity models from indirect observations derived from the velocity and pressure fields. The framework consists of a Reynolds-averaged Navier–Stokes(RANS) solver and a neuralnetwork-represented turbulence model, each accompanied by its derivative computations. For computing the sensitivities of the indirect observations to the Reynolds stress field, we use the continuous adjoint equations for the RANS equations, while the gradient of the neural network is obtained via its built-in automatic differentiation capability. We demonstrate the ability of this approach to learn the true underlying turbulence closure when one exists by training models using synthetic velocity data from linear and nonlinear closures. We also train a linear eddy viscosity model using synthetic velocity measurements from direct numerical simulations of the Navier–Stokes equations for which no true underlying linear closure exists. The trained deep-neural-network turbulence model showed predictive capability on similar flows.     

Modelling turbulence around and inside porous media based on the second moment closure

To predict turbulence in porous media, a new approach is discussed. By double (both volume and Reynolds) averaging Navier–Stokes equations, there appear three unknown covariant terms in the momentum equation. They are namely the dispersive covariance, the macro-scale and the micro-scale Reynolds stresses, in the present study. For the macro-scale Reynolds stress, the TCL (two-component-limit) second moment closure is applied whereas the eddy viscosity models are applied to the other covariant terms: the Smagorinsky model and the one-equation eddy viscosity model, respectively for the dispersive covariance and the micro-scale Reynolds stress. The presently proposed model is evaluated in square rib array flows and porous wall channel flows with reasonable accuracy though further development is required.     

An extension of the second moment closure model for turbulent flows over macro rough walls

An advanced second moment closure for rough wall turbulence is proposed. In contrast to previously proposed models relying on an empirical correlation based on equivalent sand grain roughness, the proposed model mathematically derives roughness effects by applying spatial and Reynolds averaging to the governing equations. The additional terms in the momentum equations are the drag force and inhomogeneous roughness density terms. The drag force term is modeled with respect to the plane porosity and plane hydraulic diameter. The two-component limit pressure-strain model is applied to the additional pressure-strain term, which is related to the external force terms. An evaluation of turbulence over surfaces with randomly distributed semi-spheres confirms that the developed model reasonably reproduces the effects of roughness on mean velocity, Reynolds stress, and energy dissipation. Turbulence over rough surfaces of marine paint is also simulated to assess the predictive performance for higher Reynolds number turbulent flows over real rough surfaces. The developed model successfully reproduces the dependence of the Reynolds number on roughness effects. Moreover, qualitative agreement of the skin friction increase with the experimental data is confirmed.     

Progress in the extension of a second-moment closure for turbulent environmental flows

An advanced second-moment closure for the double-averaged turbulence equations of porous medium and vegetation flows is proposed. It treats three kinds of second moments which appear in the double-averaged momentum equation. They are the dispersive covariance, the volume averaged (total) Reynolds stress and the micro-scale Reynolds stress. The two-component-limit pressure–strain correlation model is applied to model the total Reynolds stress equation whilst a novel scale-similarity non-linear $k''–''\epsilon$ two-equation eddy viscosity model is employed for the micro-scale turbulence. For the dispersive covariance, an algebraic relation is applied. Model validation in several fully developed homogeneous porous medium flows, porous channel flows and aquatic vegetation canopy flows is performed with satisfactory agreement with the data.     

A NON-ISOTROPIC MULTIPLE-SCALE TURBULENCE MODEL

This paper describes a newly developed non-isotropic multiple-scale turbulence model(MS/ASM)for complex flow calculations.This model focuses on the direct modeling of Reynolds stresses and utilizes split-spectrum concepts to model multiple-scale effects in turbulence.Validation studies on free shear flows,rotating flows and recirculating flows show that the current model performs significantly better than the single-scale k-εmodel.The present model is relatively inexpensive in terms of CPU time which makes in suitable for broad engineering flow applications.     

Anisotropic Organised Eddy Simulation for the prediction of non-equilibrium turbulent flows around bodies

The unsteady turbulent flow around bodies at high Reynolds number is predicted by an anisotropic eddy-viscosity model in the context of the Organised Eddy Simulation (OES). A tensorial eddy-viscosity concept is developed to reinforce turbulent stress anisotropy, that is a crucial characteristic of non-equilibrium turbulence in the near-region. The theoretical aspects of the modelling are investigated by means of a phase-averaged PIV in the flow around a circular cylinder at Reynolds number 1.4×10⁵. A pronounced stress–strain misalignment is quantified in the near-wake region of the detached flow, that is well captured by a tensorial eddy-viscosity concept. This is achieved by modelling the turbulence stress anisotropy tensor by its projection onto the principal matrices of the strain-rate tensor. Additional transport equations for the projection coefficients are derived from a second-order moment closure scheme. The modification of the turbulence length scale yielded by OES is used in the Detached Eddy Simulation hybrid approach. The detached turbulent flows around a NACA0012 airfoil (2-D) and a circular cylinder (3-D) are studied at Reynolds numbers 10⁵ and 1.4×10⁵, respectively. The results compared to experimental ones emphasise the predictive capabilities of the OES approach concerning the flow physics capture for turbulent unsteady flows around bodies at high Reynolds numbers.     

Numerical analysis of turbulent structure in compound meandering open channel by algebraic Reynolds stress model

The turbulent flow in a compound meandering channel with a rectangular cross section is one of the most complicated turbulent flows, because the flow behaviour is influenced by several kinds of forces, including centrifugal forces, pressure‐driven forces and shear stresses generated by momentum transfer between the main channel and the flood plain. Numerical analysis has been performed for the fully developed turbulent flow in a compound meandering open‐channel flow using an algebraic Reynolds stress model. The boundary‐fitted coordinate system is introduced as a method for coordinate transformation in order to set the boundary conditions along the complicated shape of the meandering open channel. The turbulence model consists of transport equations for turbulent energy and dissipation, in conjunction with an algebraic stress model based on the Reynolds stress transport equations. With reference to the pressure–strain term, we have made use of a modified pressure–strain term. The boundary condition of the fluctuating vertical velocity is set to zero not only for the free surface, but also for computational grid points next to the free surface, because experimental results have shown that the fluctuating vertical velocity approaches zero near the free surface. In order to examine the validity of the present numerical method and the turbulent model, the calculated results are compared with experimental data measured by laser Doppler anemometer. In addition, the compound meandering open channel is clarified somewhat based on the calculated results. As a result of the analysis, the present algebraic Reynolds stress model is shown to be able to reasonably predict the turbulent flow in a compound meandering open channel. Copyright © 2005 John Wiley & Sons, Ltd.     

Turbulent cascade modeling of single and bubbly two-phase turbulent flows

The analysis of turbulent two-phase flows requires closure models in order to perform reliable computational multiphase fluid dynamics (CMFD) analyses. A spectral turbulence cascade-transport model, which tracks the evolution of the turbulent kinetic energy from large to small liquid eddies, has been developed for the analysis of the homogeneous decay of isotropic single and bubbly two-phase turbulence. This model has been validated for the decay of homogeneous, isotropic single and two-phase bubbly flow turbulence for data having a 5 mm mean bubble diameter. The Reynolds number of the data based on bubble diameter and relative velocity is approximately 1400.     

Identification of local extinction topology in axisymmetric bluff‐body diffusion flames with a reactedness‐mixture fraction presumed probability density function model

The effects of finite‐rate chemistry, such as partial extinctions and re‐ignitions, are investigated in turbulent non‐pre‐mixed reacting flows stabilized in the wake of an axisymmetric bluff‐body burner. A two‐dimensional large‐eddy simulation procedure is employed that uses a partial equilibrium/two‐scalar reactedness mixture fraction probability density function (PDF) combustion sub‐model, which is applied at the sub‐grid scale (SGS) level. An anisotropic sub‐grid eddy–viscosity and two equations for the SGS turbulence kinetic and scalar energies complete the SGS closure model. The scalar covariances required in the joint PDF formulation are obtained from an extended scale‐similarity assumption between the resolved and the sub‐grid fluctuations. Extinction due to strong turbulence/chemistry interactions is recognized with the help of a ‘critical’, locally variable, turbulent Damkohler number criterion, while transient localized extinctions and re‐ignitions are treated with a Lagrangian transport equation for a reactedness progress variable. Comparisons with available experimental data suggested that the formulated approach was capable of identifying the effects of large‐scale vortex structure activity, which were inherent in the reacting wake and dominant in the counterpart isothermal flows that otherwise would have been obscured if a standard time‐averaged procedure had been used. Additionally, the post‐extinction and re‐ignition behaviour and its time‐varying interaction with the large‐scale structure dynamics were more appropriately addressed within the context of the present time‐dependent method. Copyright © 2001 John Wiley & Sons, Ltd.     

From single-scale turbulence models to multiple-scale and subgrid-scale models by Fourier transform

A theoretical method based on mathematical physics formalism that allows transposition of turbulence modeling methods from URANS (unsteady Reynolds averaged Navier–Stokes) models, to multiple-scale models and large eddy simulations (LES) is presented. The method is based on the spectral Fourier transform of the dynamic equation of the two-point fluctuating velocity correlations with an extension to the case of non-homogenous turbulence. The resulting equation describes the evolution of the spectral velocity correlation tensor in wave vector space. Then, we show that the full wave number integration of the spectral equation allows one to recover usual one-point statistical closure whereas the partial integration based on spectrum splitting gives rise to partial integrated transport models (PITM). This latter approach, depending on the type of spectral partitioning used, can yield either a statistical multiple-scale model or subfilter transport models used in LES or hybrid methods, providing some appropriate approximations are made. Closure hypotheses underlying these models are then discussed by reference to physical considerations with emphasis on identification of tensorial fluxes that represent turbulent energy transfer or dissipation. Some experiments such as the homogeneous axisymmetric contraction, the decay of isotropic turbulence, the pulsed turbulent channel flow and a wall injection induced flow are then considered as typical possible applications for illustrating the potentials of these models.      

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.     

On the nature of multitude scalings in decaying isotropic turbulence

We report multitude scaling laws for isotropic fully developed decaying turbulence through group theoretic method employing on the spectral equations both for modelling and without any modelling of nonlinear energy transfer. For modelling, besides the existence of classical power law scalings, an exponential decay of turbulent energy in time is obtained subject to exponentially decaying integral length scale at infinite Reynolds number limit. For the transfer without modelling, at finite Reynolds number, in addition to general power law decay of turbulence intensity with integral length scale growing as a square root of time, an exponential decay of energy in time is explored when integral length scale remains constant. Both the power and exponential decaying laws of energy agree to the theoretical results of George (1992), George and Wang (2009) and experimental results of fractal grid generated turbulence by Hurst and Vassilicos (2007). At infinite Reynolds number limit, a general power law scaling is obtained from which all classical scaling laws are recovered. Further, in this limit, turbulence exhibits a general exponential decaying law of energy with exponential decaying integral length scale depending on two scaling group parameters. The role of symmetry group parameters on turbulence dynamics is discussed in this study.     

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.     

Spatial resolution effects on the measurement of scalar variance and scalar gradient in turbulent nonpremixed jet flames

The effect of spatial averaging is important for scalar gradient measurements in turbulent nonpremixed flames, especially when the local dissipation length scale is small. Line imaging of Raman, Rayleigh and CO-LIF is used to investigate the effects of experimental resolution on the scalar variance and radial gradient in the near field of turbulent nonpremixed CH₄/H₂/N₂ jet flames at Reynolds numbers of 15,200 and 22,800 (DLR-A and B) and in piloted CH₄/air jet flames at Reynolds numbers of 13,400, 22,400 and 33,600 (Sandia flames C/D/E). The finite spatial resolution effects are studied by applying the Box filter with varying filter widths. The resulting resolution curves for both scalar variance and mean squared-gradient follow nearly the same trends as theoretical curves calculated from the model turbulence kinetic energy spectrum of Pope. The observed collapse of resolution curves of mean squared-gradient for nearly all studied cases implies the shape of the dissipation spectrum is approximately universal. Fluid transport properties are shown to have no effect on the dissipation resolution curve, which implies that the dissipation length scale inferred from the square gradient is equivalent to the length scale for the scalar dissipation rate, which includes the diffusion coefficient. With the Box filter, the required spatial resolution to resolve 98% of the mean dissipation rate is about one−two times of the dissipation cutoff length scale (analogous to the Batchelor scale in turbulent isothermal flows). The effects of resolution on the variances of mixture fraction, temperature, and the inverted Rayleigh signal are also compared. The ratio of the filtered variance to the true variance is shown to depend nearly linearly on the probe resolution. The inverted Rayleigh scattering signal can be used to study the resolution effect on temperature variance even when the Rayleigh scattering cross section is not constant. The experimental results also indicate that these laboratory scale turbulent jet flames have small effective Reynolds numbers, such that there is some direct interaction of the large (energy containing) and small (dissipative) scalar length scales, especially for the near field case at x/d = 7.5 of the piloted Sandia flames C/D/E.     

Calculation of principal characteristics of turbulent streams in a state of structural equilibrium

A review of articles on the study of turbulent streams having transverse displacement, in which a turbulent energy balance equation is used, is contained in [1]. Levin [2] proposed a certain development of Rotta's method [3] making it possible to determine the characteristics of the average flow and the radial distribution of pulsation magnitudes. However, in this article the scale of the turbulence (the quantityl) was given as an empirical function of the coordinates. At the same time it is clear that the distribution of the turbulence scale depends on the conditions of the problem. A special differential equation proposed in [4,5] describing the variation in time and space of the quantityl has the drawback that in deriving this equation it is necessary to invoke additional hypotheses which are difficult to test experimentally. In the present article, along with the velocity of the average flow, the pressure, and the pulsation magnitudes, the scale of the turbulence is considered as an important characteristic of the stream, determined by the reference system which consists of the Reynolds equations, continuity equations, and equations for the component of the Reynolds stress tensor. Rotta's approximate semiempirical relations and an experimental relation for the single-point correlation coefficient between the turbulent pulsations in velocity are used for closure of the system obtained. An approximate calculation is given for the principal average and pulsation characteristics of the flow for the region of the stream where the turbulence is in a state of structural equilibrium [6]. A comparison of the calculated and experimental data is presented.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 1, pp. 95–99, January–February, 1973.     

Direct Numerical Simulation of Self-Similar Turbulent Boundary Layers in Adverse Pressure Gradients

Direct numerical simulations of the Navier–Stokes equations have been carried out with the objective of studying turbulent boundary layers in adverse pressure gradients. The boundary layer flows concerned are of the equilibrium type which makes the analysis simpler and the results can be compared with earlier experiments and simulations. This type of turbulent boundary layers also permits an analysis of the equation of motion to predict separation. The linear analysis based on the assumption of asymptotically high Reynolds number gives results that are not applicable to finite Reynolds number flows. A different non-linear approach is presented to obtain a useful relation between the freestream variation and other mean flow parameters. Comparison of turbulent statistics from the zero pressure gradient case and two adverse pressure gradient cases shows the development of an outer peak in the turbulent energy in agreement with experiment. The turbulent flows have also been investigated using a differential Reynolds stress model. Profiles for velocity and turbulence quantities obtained from the direct numerical simulations were used as initial data. The initial transients in the model predictions vanished rapidly. The model predictions are compared with the direct simulations and low Reynolds number effects are investigated.