Two-scale Direct-interaction Approximation Research Articles

Bridging turbulent scales: A unified LES approach from low to high Reynolds numbers using TSDIA and GOY Shell models

Large Eddy Simulation (LES) has become a widely adopted method for modeling turbulence across various fields, such as aerospace, wind energy, and urban wind flow analysis. While traditional LES models are highly effective at high Reynolds numbers, they face challenges at low Reynolds numbers due to the absence of a clearly defined inertial subrange. This paper introduces an improved LES approach by integrating Yoshizawa's Two-Scale Direct Interaction Approximation (TSDIA) theory and the GOY shell model. This combined method extends the applicability of LES by capturing turbulence features across both inertial and dissipation ranges, overcoming limitations found in conventional models. By deriving model constants directly from the shell model, this approach avoids reliance on empirical values and enhances accuracy, particularly at low Reynolds numbers. Validation against DNS data demonstrates a significant improvement in prediction accuracy, offering a more comprehensive and adaptable solution for turbulent flow simulations across a wide range of practical applications. This method provides a more robust foundation for industrial fluid simulations and environmental modeling.

Cross-helicity effect onα-type dynamo in non-equilibrium turbulence

Turbulence is typically not in equilibrium, i.e. mean quantities such as the mean energy and helicity are typically time-dependent. The effect of non-stationarity on the turbulent hydromagnetic dynamo process is studied here with the use of the two-scale direct-interaction approximation, which allows one to explicitly relate the mean turbulent Reynolds and Maxwell stresses and the mean electromotive force to the spectral characteristics of turbulence, such as the mean energy, as well as kinetic and cross-helicity. It is demonstrated that the non-equilibrium effects can enhance the dynamo process when the magnetohydrodynamic turbulence is both helical and cross-helical. This effect is based on the turbulent infinitesimal-impulse cross-response functions, which do not affect turbulent flows in equilibrium. The evolution and sources of the cross-helicity in magnetohydrodynamic turbulence are also discussed.

Dust distribution study at the blast furnace top based on k–S –up model

The dust distribution law acting at the top of a blast furnace (BF) is of great significance for understanding gas flow distribution and mitigating the negative influence of dust particles on the accuracy and service life of detection equipment. The harsh environment inside a BF makes it difficult to describe the dust distribution. This paper addresses this problem by proposing a dust distribution k-Sε-u <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</sub> model based on interphase (gaspowder) coupling. The proposed model is coupled with a k-Sε model (which describes gas flow movement) and a model up (which depicts dust movement). First, the kinetic energy equation and turbulent dissipation rate equation in the k-Sε model are established based on the modeling theory and single-Green-function two-scale direct interaction approximation (SGF-TSDIA) theory. Second, a dust particle movement u <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</sub> model is built based up on a force analysis of the dust and Newton's laws of motion. Finally, a coupling factor that describes the interphase interaction is proposed, and the k-Sε-u <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</sub> model, with clear physical meaning, rigorous mathematical logic, and adequate generality, is developed. Simulation results and on-site verification show that the k-Sε-u <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</sub> model not only has high precision, but also reveals the aggregate distribution features of the dust, which are helpful in optimizing the installation position of the detection equipment and improving its accuracy and service life.

Analysis of destruction term in transport equation for turbulent energy dissipation rate

The K–\(\varepsilon \) model used in turbulence simulations involves the transport equations for the turbulent kinetic energy and its dissipation rate. In contrast to the turbulent energy equation derived from the Navier–Stokes equation, the transport equation for the dissipation rate has been modeled empirically and has less of a theoretical grounding. An analysis of the dependence of terms in the exact transport equation on the Reynolds number \(\textit{Re}\) has suggested that the two dominant \(O(\textit{Re}^{1/2})\) terms cancel out at the leading order and their O(1) difference yields the terms in the model equation. The two-scale direct interaction approximation (TSDIA) can be used to derive the model equation for inhomogeneous turbulence. In this study, as a first step toward deriving all the terms in the model equation, the two dominant terms in the exact transport equation were investigated theoretically and numerically for the case of homogeneous isotropic turbulence, in order to derive the destruction term. The two terms were first analyzed by using the TSDIA under the assumption of a simple energy spectrum form. However, the finite-width effect of the inertial range of the energy spectrum did not give the expected \(O(\textit{Re}^{-1/2})\) corrections to the leading-order terms. Then, the model equations of the Markovianized Lagrangian renormalized approximation (MLRA) were numerically solved to obtain the accurate energy spectrum profile of a decaying homogeneous isotropic turbulence. Thereby, it was shown that the deviation of the energy spectrum from the \(-5/3\) law was responsible for the \(O(\textit{Re}^{-1/2})\) corrections. By assuming the energy spectrum suggested by the MLRA results, the two dominant terms were analyzed again by using the TSDIA to successfully derive the destruction term in the model equation.

How turbulence fronts induce plasma spin-up.

A calculation which describes the spin-up of toroidal plasmas by the radial propagation of turbulence fronts with broken parallel symmetry is presented. The associated flux of parallel momentum is calculated by using a two-scale direct-interaction approximation in the weak turbulence limit. We show that fluctuation momentum spreads faster than mean flow momentum. Specifically, the turbulent flux of wave momentum is stronger than the momentum pinch. The scattering of fluctuation momentum can induce edge-core coupling of toroidal flows, as observed in experiments.

A statistical theory for inhomogeneous turbulent flow formulated in the mean-Lagrangian coordinates

An analytical way for investigating the inhomogeneous turbulent flow has been proposed by introducing Lagrangian-like view. The present method is developed based on TSDIA (Two-Scale Direct Interaction Approximation) which is one of the few theoretical works for the inhomogeneous turbulent flow formulated in the Eulerian frame work and has fundamental inconsistency with the general covariance under the coordinate transformation. The present work reached certain success in the following two aspects. First, the algebraic representation for the Reynolds stress derived by the present method has been proved to be generally covariant. Second the algebraic Reynolds stress is in much better agreement with the result of a DNS of the channel flow.

Mechanisms of countergradient diffusion in turbulent combustion

The mechanism of countergradient diffusion of chemical species and heat in turbulent combustion is sought with the aid of the results by the two-scale direct-interaction approximation. The deviation of the Reynolds stress and the turbulent fluxes of chemical species, heat, and mass from their gradient-diffusion representations is related to the Lagrange derivatives of mean velocity and scalars. Their relative magnitude to the gradient-diffusion parts paves the way for explaining the countergradient diffusion. Towards engineering applications, these theoretical findings are converted to a Reynolds-averaged model. It consists of a closed system of equations for the mean density, the mean velocity, the mean internal energy, and the mean scalar, with the turbulence equations for the kinetic energy and its dissipation rate supplemented. This system is tested in a turbulent premixed flame and is shown to reproduce some of the characteristics pointed out by the direct numerical simulation.

Theoretical investigation of some thermal effects in turbulence modeling

Fluid compressibility effects arising from thermal rather than dynamical aspects are theoretically investigated in the framework of turbulent flows. The Mach number is considered low and not to induce significant compressibility effects which here occur due to a very high thermal gradient within the flowfield. With the use of the Two-Scale Direct Interaction Approximation approach, essential turbulent correlations are derived in a one-point one-time framework. In the low velocity gradient limit, they are shown to directly depend on the temperature gradient, assumed large. The impact of thermal effects onto the transport equations of the turbulent kinetic energy and dissipation rate is also investigated, together with the transport equation for both the density and the internal energy variance.

Derivation of a Nonlinear Reynolds Stress Model Using Renormalization Group Analysis and Two-Scale Expansion Technique

Adopting Yoshizawa's two-scale expansion technique, the fluctuating field is expanded around the isotropic field. The renormalization group method is applied for calculating the covariance of the fluctuating field at the lower order expansion. A nonlinear Reynolds stress model is derived and the turbulent constants inside are evaluated analytically. Compared with the two-scale direct interaction approximation analysis for turbulent shear flows proposed by Yoshizawa, the calculation is much more simple. The analytical model presented here is close to the Speziale model, which is widely applied in the numerical simulations for the complex turbulent flows.

Investigation for a Compressible One-Equation-Type SGS Model Using the DNS Database of the Compressible Turbulent Mixing Layer

A new subgrid-scale (SGS) one-equation model of the large eddy simulation (LES) for compressible turbulent flows, which includes the coherent structure model function suggested by Kobayashi, is proposed by the theoretical analysis of the two-scale direct-interaction approximation. Using the direct numerical simulation (DNS) data of the compressible turbulent mixing layer, a priori test for the model expression is performed. In the LES, we show that the predictions of the present SGS model reproduce the DNS results.

Spatial and spectral evolution of turbulence

Spreading of turbulence as a result of nonlinear mode couplings and the associated spectral energy transfer is studied. A derivation of a simple two-field model is presented using the weak turbulence limit of the two-scale direct interaction approximation. This approach enables the approximate overall effect of nonlinear interactions to be written in the form of Fick’s law and leads to a coupled reaction-diffusion system for turbulence intensity. For this purpose, various classes of triad interactions are examined, and the effects that do not lead to spreading are neglected. It is seen that, within this framework, large scale, radially extended eddies are the most effective structures in promoting spreading of turbulence. Thus, spectral evolution that tends toward such eddies facilitates spatial spreading. Self-consistent evolution of the background profile is also considered, and it is concluded that the profile is essentially slaved to the turbulence in this phase of rapid evolution, as opposed to the case of avalanches, where it is the turbulence intensity that would be slaved to the evolving profile. The characteristic quantity describing the evolving background profile is found to be the mean “potential vorticity” (PV). It is shown that the two-field model with self-consistent mean PV evolution can be reduced to a single Fisher-like turbulence intensity transport equation. In addition to the usual nonlinear diffusion term, this equation also contains a “pinch” of turbulence intensity. It is also noted that internal energy spreads faster than kinetic energy because of the respective spectral tendencies of these two quantities.

Nonlinear Triad Interactions and the Mechanism of Spreading in Drift-Wave Turbulence

We present the results of a derivation of the fluctuation energy transport matrix for the two-field Hasegawa-Wakatani model of drift wave turbulence. The energy transport matrix is derived from a two-scale direct interaction approximation assuming weak turbulence. We examine different classes of triad interactions and show that radially extended eddies, as occurs in penetrative convection, are the most effective in turbulence spreading. We show that in the near-adiabatic limit internal energy spreads faster than the kinetic energy. Previous theories of spreading results are discussed in the context of weak turbulence theory.

Modeling of the turbulent magnetohydrodynamic residual-energy equation using a statistical theory

The difference between the kinetic and magnetic energies in a conducting fluid is investigated in the framework of magnetohydrodynamics. The deviation from equipartition is measured by the turbulent residual energy KR. With the aid of the two-scale direct-interaction approximation, a statistical analytical theory for inhomogeneous turbulence, expressions for the correlation tensors appearing in the evolution equation for the residual energy are derived. Using these results, we propose a model equation for KR evolution. Examination of the structure of this equation shows that the evolution of the scaled residual energy is related to the cross helicity (velocity-magnetic-field correlation) of turbulence coupled with the mean-field shears. An application to the solar wind shows that the scaled ∣KR∣ can be increased near the outside of the Alfvén point in the inner heliosphere whereas the almost stationary behavior of ∣KR∣ is suggested in the outer heliosphere. These results are consistent with observations of solar-wind turbulence.

A new methodology for Reynolds-averaged modeling based on the amalgamation of heuristic-modeling and turbulence-theory methods

A new methodology for the Reynolds-averaged Navier-Stokes modeling is presented on the basis of the amalgamation of heuristic-modeling and turbulence-theory methods. A characteristic turbulence time scale is synthesized in a heuristic manner through the combination of several characteristic time scales. An algebraic model of turbulent-viscosity type for the Reynolds stress is derived from the Reynolds-stress transport equation with the time scale embedded. It is applied to the state of weak spatial and temporal nonequilibrium, and is compared with its theoretical counterpart derived by the two-scale direct-interaction approximation. The synthesized time scale is justified through the agreement of the two expressions derived by these entirely different methods. The derived model is tested in rotating isotropic, channel, and homogeneous-shear flows. It is extended to a nonlinear algebraic model and a supersonic model. The latter is shown to succeed in reproducing the reduction in the growth rate of a free-shear layer flow, without causing wrong effects on wall-bounded flows such as channel and boundary-layer flows.

Numerical Prediction of Turbulent Flow in a Two-Dimensional Asymmetric Diffuser with a Third-Order Nonlinear K-.EPSILON. Model

A third-order nonlinear K-e model proposed by the authors is applied to turbulent flow through a two-dimensional asymmetric diffuser including separation and reattachment. The numerical results show that the present model predicts profiles of the mean velocities and Reynolds stresses better than a representative linear eddy-viscosity model but does not reproduce the flow separation. The model is improved by incorporating a strain-convection effect derived from a two-scale direct-interaction approximation analysis into the Reynolds stress to capture separation and reattachment.

Theoretical Investigation of an Eddy-Viscosity-Type Expression of the Reynolds Stress with Low-Reynolds-Number Effect

In low-Reynolds-number turbulent flows, the influence of the molecular viscosity is important. The turbulence models which are applied to those flows should include the low-Reynolds-number effect. In this study, turbulent flow with the molecular viscosity effect is analyzed theoretically with the aid of a two-scale direct-interaction approximation (TSDIA) and the energy spectrum and a new low-Reynolds-number-type eddy-viscosity representation are derived. An priori test for the model expression on the basis of the result of direct numerical simulation (DNS) for turbulent Couette flows is performed.

Statistical analysis of mean-flow effects on the pressure–velocity correlation

Mean-flow effects on the pressure–velocity correlation in the turbulent-energy equation are investigated with the aid of a two-scale direct-interaction approximation. Those effects are composed of strain, vorticity, and momentum-diffusion effects, which correspond to the modeling of the rapid term in the pressure–strain correlation of the Reynolds-stress equation. They are missing in the current turbulence modeling in striking contrast to the latter, and become important in turbulent flows accompanying large mean-velocity gradients and curvatures.

Generation of large-scale structures by strong drift turbulence

The previously existing quasilinear theory of the generation of a large-scale radial electric field by small-scale drift turbulence in a plasma is generalized for the case of strong turbulence which is usually observed in experiments. The geostrophic equation (i.e., the reduced Charney-Hasegawa-Mima equation) is used to construct a systematic theory in the two-scale direct interaction approximation. It is shown that, as in the quasilinear case, drift turbulence results in a turbulent viscosity effect and leads to the renormalization of the Poisson bracket in the Charney-Hasegawa-Mima equation. It is found that, for strong drift turbulence, the viscosity coefficient is represented as a sum of two parts, which are comparable in magnitude. As in quasilinear theory, the first part is determined by the second-order correlation functions of the turbulent field and is always negative. The second part is proportional to the third-order correlation functions, and the sign of its contribution to the turbulent viscosity coefficient depends strongly on the turbulence spectrum. The turbulent viscosity coefficient is calculated numerically for the Kolmogorov spectra, which characterize the inertial interval of the drift turbulence.

Turbulent transport modeling in low Mach number flows

Model representations for the Reynolds stress and the turbulent heat flux of flows at low Mach numbers have been theoretically derived by applying the two-scale direct-interaction approximation to their transport equations based on the assumption that the density variation is relatively small. The derived representations can be applied to general turbulent flows at low Mach numbers, whether the Boussinesq approximation holds in the flow or not. The model representation for the turbulent heat flux newly involves the differential effect of mean pressure gradients on hot, light fluids and on cold, heavy ones, in addition to the familiar effect of mean temperature diffusion. Certain a priori tests have shown that the model can reproduce the vertical turbulent heat flux in the natural convection along a heated vertical plate and the countergradient diffusion for the turbulent heat flux observed in a nonpremixed swirling flame, both of which can not be explained by the standard turbulence model of the gradient diffusion type.

Effects of pressure fluctuations on turbulence growth in compressible homogeneous shear flow

Direct numerical simulation results are used to study compressibility effects on turbulent energy growth in homogeneous shear flow. Normalized amplitude of the pressure fluctuations is shown to decrease as the turbulent Mach number increases. This decrease causes the reduction in the pressure–strain term, changes the anisotropy of the Reynolds stress, and reduces the growth rate of turbulent kinetic energy as is the case of mixing layers. The transport equation for the pressure variance is examined in detail. It is shown that when the turbulent Mach number is high the pressure-variance dissipation term is not negligible and contributes to the reduction in the pressure fluctuations. The pressure-variance dissipation is closely related to the high-wavenumber part of the pressure-variance spectrum. Profile of the spectrum is observed to depend on the turbulent Mach number. A statistical theory called the two-scale direct-interaction approximation is applied to obtain model expressions for pressure-related terms. The pressure–dilatation correlation is a dominant term in the pressure-variance equation; its model expression as derived contains the material derivative of turbulent kinetic energy. This quantity represents nonequilibrium properties of turbulent field and can explain the very small value of the pressure dilatation in boundary layers and channel flows. The model expression for the pressure–strain correlation is also derived; the normalized pressure variance can be a parameter that describes compressibility effect on the pressure strain.