Eddy Diffusivity Model Research Articles

Modelling dust distributions in the atmospheric boundary layer on Mars

A time and height dependent eddy diffusion model is used to investigate possible scenarios for the size distribution of dust in the lower atmosphere of Mars. The dust is assumed to either have been advected from a distant source or to have originated locally. In the former case, the atmosphere is assumed to initially contain dust particles with sizes following a modified gamma distribution. Larger particles are deposited relatively rapidly while small particles are well mixed up to the maximum height of the afternoon boundary layer and are deposited more slowly. In other cases, a parameterization of the dust source at the surface is proposed. Model results show that smaller particles are rapidly mixed within the Martian boundary layer, while larger particles (r > 10 μm) are concentrated near the ground with a stronger diurnal cycle. In all simulations we assume that the initial concentration or surface source depend on a modified gamma function distribution. For small particles (cross-sectional area weighted mean radius, reff = 1.6 μm) distributions retain essentially the same form, though with variations in the mean and variance of the area-weighted radius, and the gamma function can be used to represent the particle size distribution reasonably well at most heights within the boundary layer. In the case of a surface source of larger particles (mean radius 50 μm) the modified gamma function does not fit the resulting particle size distribution. All results are normalised by a scaling factor that can be adjusted to correspond to an optical depth for assumed particle optical scattering properties.

A General Dynamic Linear Tensor-Diffusivity Subgrid-Scale Heat Flux Model for Large-Eddy Simulation of Turbulent Thermal Flows

In this article, a general dynamic linear tensor diffusivity model is proposed for representing the subgrid-scale (SGS) heat flux (HF). The tensor diffusivity for the model is an inhomogeneous linear function of the resolved strain and rotation rate tensors, and includes three conventional dynamic SGS HF modeling approaches as special cases. In contrast to the dynamic SGS eddy diffusivity modeling approach, the proposed model admits more degrees of freedom for representing the SGS thermal diffusivity, allows for nonalignment between the SGS HF and resolved temperature gradient, and consequently provides a more realistic geometric representation of the SGS heat flux. To validate the proposed modeling approach, numerical simulations have been performed based on a combined forced- and natural-convention flow in a vertical channel with a Reynolds number and a Grashof number Gr = 9.6 × 105. In comparison with the reported direct numerical simulation data and the results obtained using the conventional dynamic SGS eddy diffusivity model, it is shown that the proposed model is able to provide good predictions of various flow quantities at the resolved scale and, more important, offer new insights into near-wall flow physics at the subgrid scale.

Large eddy simulation of scalar turbulence using a new subgrid eddy diffusivity model

A new subgrid eddy diffusivity model is used for large eddy simulation of scalar turbulence. The new model is based on the modified Yaglom equation, i.e., the dynamical equation for the structure function of resolved-scale scalar fluctuations, by which the scalar transport between resolved grid scale and unresolved subgrid scale can be described exactly. The new model has been applied in the large eddy simulations of scalar transport in both decaying isotropic turbulence and fully developed turbulent channel flow. By the comparison with Smagorinsky model and the dynamical model, it is found that the new model can achieve similar accuracy as the dynamical model, and can reduce the computational cost about 20% in terms of CPU time relative to the dynamical model.

Nonlinear eddy diffusivity model for wall-bounded flow with arbitrary rotating axes

Rotating flows have been encountered in many engineering-relevant applications such as a flow in turbomachinery. Recently, rotating channel flows with arbitrary rotating axes have been investigated by direct numerical simulation (DNS). Although numerical studies of spanwise rotating channel flow have been reported, there are few reports on streamwise and wall-normal rotating channel flows. In the present study, first, we conducted DNS of various rotational channel flows in order to understand phenomena of rotating wall-bounded flows and to make databases for turbulence modelling. It is found from the DNS results that cases of streamwise rotating channel flow which have a mean spanwise velocity caused by rotation involve the counter gradient turbulent diffusion. Thus, since it is well-known that the conventional eddy diffusivity turbulence model (EDM) cannot predict this case, we have to consider the reconstruction of a nonlinear eddy viscosity turbulence model (NLEDM) based on a cubic model. Consequently, we evaluated existing nonlinear eddy viscosity turbulence models based on our DNS of streamwise and wall-normal rotating channel flows. Using the results of evaluation, we have improved the modeled expression for Reynolds stress, in which a new cubic-type nonlinear eddy viscosity model has been proposed. The proposed nonlinear two-equation turbulence model can accurately predict rotating channel flows with arbitrary rotating axes.

Nonlinear eddy diffusivity models reflecting buoyancy effect for wall-shear flows and heat transfer

The main objective of this study is to construct new nonlinear eddy diffusivity models reflecting buoyant effects in wall-bounded turbulent shear flows and heat transfer. It is now well known that the turbulent heat-fluxes, which are the key quantities for the prediction of turbulent flows with buoyancy, are not modeled accurately by employing the conventional turbulence models using eddy diffusivities. In order to appropriately predict wall-bounded turbulent flows with buoyancy, an innovative turbulence heat-transfer model with eddy diffusivities which are composed of the k– ε two-equation model for velocity field and the k θ – ε θ two-equation model for thermal field, must be constructed. Consequently, we should improve the modeled expressions for Reynolds stresses and turbulent heat-fluxes reflecting the buoyant effect in wall-bounded turbulent shear flows. The existing two-equation turbulence models were evaluated on the basis of the DNS data of channel flows with buoyancy. Using the results of evaluation, we constructed new modeled expressions for Reynolds stresses and turbulent heat-fluxes in explicit algebraic models, and reconstructed the nonlinear two-equation turbulence models for the buoyancy-affected wall-shear flows and heat transfer, including the newly proposed nonlinear eddy diffusivity for a momentum model (NLEDM) and the nonlinear eddy diffusivity for heat model (NLEDHM). The proposed nonlinear two-equation turbulence models reflecting buoyancy effect appropriately predict wall-bounded turbulent shear flows with buoyancy given by DNS.

Dynamic subgrid‐scale models for momentum and scalar fluxes in large‐eddy simulations of neutrally stratified atmospheric boundary layers over heterogeneous terrain

The accuracy of large‐eddy simulations (LESs) of the atmospheric boundary layer (ABL) over complex terrain relies on the ability of the subgrid‐scale (SGS) models to capture the effect of subgrid turbulent fluxes on the resolved fields of velocity and scalars (e.g., heat, water vapor, and pollutants). A common approach consists of parameterizing the SGS stresses and fluxes using eddy viscosity and eddy diffusivity models, respectively. These models require the specification of two parameters: the Smagorinsky coefficient in the eddy viscosity model and, in addition, the SGS Schmidt/Prandtl number in the eddy diffusivity model. This is complicated by the dependence of the coefficients on local conditions such as distance to the ground, mean shear, and atmospheric stability. In this study, scale‐dependent dynamic SGS models are used in conjunction with Lagrangian averaging to compute both the Smagorinsky coefficient and the SGS Schmidt (or Prandtl) number dynamically as the flow evolves in both space and time based on the local dynamics of the resolved scales. These tuning‐free models are implemented in LES of both homogeneous and heterogeneous neutral atmospheric boundary layers with surface fluxes of a passive scalar. In the homogeneous simulations the models are shown to accurately predict the resolved flow statistics (mean profiles and spectra of velocity and scalar concentration) and spatial distributions of the SGS model coefficients and parameters. In simulations over heterogeneous surfaces both coefficients adjust in a self‐consistent way to horizontal flow inhomogeneities associated with changes in surface conditions. For smooth‐to‐rough (rough‐to‐smooth) abrupt changes in surface roughness the Smagorinsky coefficient decreases (increases) in response to increased (decreased) mean shear and flow anisotropy associated with these transitions. The SGS Schmidt number also adjusts to inhomogeneities in the scalar field associated with changes in surface scalar flux. This illustrates the need for local calculation of model coefficients and brings into question the common practice of using a constant SGS Schmidt/Prandtl number in LES of the ABL.

Subgrid Scale Modeling of Turbulence for the Dynamic Procedure Using Finite Difference Method and Its Assessment on the Thermally Stratified Turbulent Channel Flow

Abstract Previously proposed methods for subgrid-scale (SGS) stress modeling were re-investigated and extended to SGS heat-flux modeling, and various anisotropic and isotropic eddy viscosity/diffusivity models were obtained. On the assumption that they are used in a finite-difference (FD) simulation, the models were constructed in such a way that they are insensitive to numerical parameters on which calculated flows are strongly dependent in the conventional Smagorinsky model. The models obtained, as well as those previously proposed, were evaluated a priori in a stably stratified open channel flow, which is considered to be a challenging application of large eddy simulation and suitable for testing both SGS stress and heat-flux models. The most important feature of the models proposed is that they are insensitive to the discretized test filtering parameter required in the dynamic procedure of Germano et al. (1991, Phys. Fluids, 3, pp. 1760–1765) in FD simulation. We also found in SGS heat-flux modeling that the effect of the grid (resolved)-scale (GS) velocity gradient plays an important role in the estimation of the streamwise heat flux, and an isotropic eddy diffusivity model with the effect of the GS velocity is proposed.

Mixed convection turbulent film condensation on a sphere

The present paper reports a research on condensation heat transfer of an isothermal sphere with an external flow of vapor. The high tangential velocity of the vapor flow is determined from potential flow theory. The transition criterion of the onset turbulence has been given in the local film Reynolds number ( Re Γ ). An eddy diffusivity model along with an expression by [H. Kato, N.N. Shiwaki, M. Hirota, On the turbulent heat transfer by free convection from a vertical plate, Int. J. Heat Mass Transfer, 11(1968) 1117–1125] is used to model turbulence. And the local liquid–vapor interfacial shear which occurs for high velocity vapor flow across a sphere surface is defined by the Colburn analogy. The paper then presents analytical analysis for the local dimensionless film thickness and heat transfer characteristics for the film condensation. And a comparison with those generated by previous theoretical of laminar condensation is discussed. The comparison shows the heat transfer coefficient of turbulent film condensation is higher than laminar film condensation under the high vapor velocity.

A Numerical and Experimental Investigation of the Slot Film-Cooling Jet With Various Angles

Numerical simulations coupled with laser Doppler velocimetry (LDV) experiments were carried out to investigate a slot jet issued into a cross flow, which is relevant in the film cooling of gas turbine combustors. The film-cooling fluid injection from slots or holes into a cross flow produces highly complicated flow fields. In this paper, the time-averaged Navier-Stokes equations were solved on a collocated body-fitted grid system with the shear stress transport k−ω, V2F k−ϵ, and stress-ω turbulence models. The fluid flow and turbulent Reynolds stress fields were compared to the LDV experiments for three jet angles, namely, 30, 60, and 90 deg, and the jet blowing ratio is ranging from 2 to 9. Good agreement was obtained. Therefore, the present solution procedure was also adopted to calculations of 15 and 40 deg jets. In addition, the temperature fields were computed with a simple eddy diffusivity model to obtain the film-cooling effectiveness, which, in turn, was used for evaluation of the various jet cross-flow arrangements. The results show that a recirculation bubble downstream of the jet exists for jet angles larger than 40 deg, but it vanishes when the angle is <30deg, which is in good accordance with the experiments. The blowing ratio has a large effect on the size of the recirculation bubble and, consequently, on the film cooling effectiveness. In addition, the influence of boundary conditions for the jet and cross flow are also addressed in the paper.

Predictions of flow and heat transfer in multiple impinging jets with an elliptic-blending second-moment closure

We present numerical computations of flow and heat transfer in multiple jets impinging normally on a flat heated surface, obtained with a new second-moment turbulence closure combined with an elliptic blending model of non-viscous wall blocking effect. This model provides the mean velocity and turbulent stress fields in very good agreement with PIV measurements. The exploration of several simpler closures for the passive thermal field, conducted in parallel, confirmed that the major prerequisite for the accurate prediction of the temperature field and heat transfer is to compute accurately the velocity and stress fields. If this is achieved, the conventional anisotropic eddy-diffusivity model can suffice even in complex flows. We demonstrate this in multiple-impinging jets where such a model combination provided the distribution of Nusselt number over the solid plate in good agreement with experiments. Extension of the elliptic blending concept to full second-moment treatment of the heat flux and its truncation to a quasi-linear algebraic model is also briefly discussed.

The heat flux and the temperature gradient in the lower atmosphere

Most parameterizations used in Large Eddy Simulations of the atmospheric boundary layer are based on the assumption that subgrid‐scale fluxes are aligned against spatial gradients of transported quantities (down‐gradient closures). Based on field experiments, we determine the distribution and most probable relative orientations of the subgrid‐scale (SGS) heat flux relative to parameterizations based on the temperature gradient. We show that, under neutral and unstable atmospheric stability, the SGS heat flux most likely lies within the geometric span of the so called mixed tensor eddy diffusivity model.

A Scale-Dependent Dynamic Model for Scalar Transport in Large-Eddy Simulations of the Atmospheric Boundary Layer

An important challenge in large-eddy simulations of the atmospheric boundary layer is the specification of the subgrid-scale (SGS) model coefficient(s) and, in particular, how to account for factors such as position in the flow, grid/filter scale and atmospheric stability. A dynamic SGS model (that assumes scale invariance of the coefficients) is implemented in simulations of a neutral boundary layer with a constant and uniform surface flux of a passive scalar. Results from our simulations show evidence that the lumped coefficient in the eddy-diffusion model computed with the dynamic procedure depends on scale. This scale dependence is stronger near the surface, and it is more important for the scalar than for the velocity field (Smagorinsky coefficient) due to the stronger anisotropic behaviour of scalars. Based on these results, a new scale-dependent dynamic model is developed for the eddy-diffusion lumped coefficient. The new model, which is similar to the one proposed earlier for the Smagorinsky coefficient, is fully dynamic, thus not requiring any parameter specification or tuning. Simulations with the scale-dependent dynamic model yield the expected trends of the coefficients as functions of position and filter/grid scale. Furthermore, in the surface layer the new model gives improved predictions of mean profiles and turbulence spectra as compared with the traditional scale-invariant dynamic model.

Nonlocal Eddy-Diffusivity Model for Scalar in Turbulence

Nonlocal Eddy-Diffusivity Model for Scalar in Turbulence

Heat Transfer Enhancement in Square Ducts With V-Shaped Ribs

This paper concerns a numerical investigation of the heat and fluid flow in V-shaped ribbed ducts. The Navier-Stokes equations and the energy equation are solved in conjunction with a low Reynolds number k–ε turbulence model. The Reynolds turbulent stresses are computed with an explicit algebraic stress model (EASM) while the turbulent heat fluxes are calculated with a simple eddy diffusivity model (SED). Detailed velocity and thermal field results have been used to explain the effects of the V-shaped ribs and the mechanisms of the heat transfer enhancement.

805 円管内沸騰流の熱伝達と圧力損失 : 強制対流蒸発域の理論解析

Heat transfer coefficient and pressure drop were simultaneously measured for vertical upward boiling flow of water in a round tube with an inner diameter of 8mm. Frictional pressure drop and heat transfer coefficient were compared with correlations of Chisholm-Laird and the authors in the form of Φ_1^2 and h_<TP>/h_<Lf>/(1-α) against Maltinelli parameter 1/X_<tt>. It was found that the analogy between heat transfer and fluid friction was valid for boiling flow. Numerical calculation of heat transfer coefficient was carried out for annular flow model. Liquid film thickness and wall shear stress was determined from Smith's correlation and our present correlation, respectively. Velocity and temperature profiles were calculated using eddy diffusivity model. Agreement between the experimental and calculated values of heat transfer coefficient was fairly good.

Coherent Doppler Profiler measurements of near‐bed suspended sediment fluxes and the influence of bed forms

This paper reports on remote acoustic observations of vertical turbulence intensity and vertical suspended sediment flux profiles on a planar beach in 3–4 m water depth. The measurements of suspended sediment concentration and velocity are colocated and simultaneous and extend through the wave bottom boundary layer to the bed with 0.7 cm vertical resolution. Normalized cospectra of the suspended sediment flux and the vertical velocity for different bed states (irregular ripples, cross ripples, linear transition ripples, and flat bed) indicate a small but significant peak at incident wave frequencies but are otherwise rather flat, with weak redness. Estimates of the vertical flux components indicate a general balance between upward fluxes due to waves and turbulence and downward settling. Two exceptions to this balance are found immediately above the bed and for nonmigrating irregular ripples. The contribution from the high‐frequency turbulent component is small. Wave phase averages for low‐energy bed states exhibit near‐bed peaks in the suspended sediment flux following wave phase reversal. Wave phase averages for the high‐energy cases do not exhibit a diffusive signature. Observed vertical profiles of turbulence intensity for different bed states reveal that the near‐bed turbulence levels are relatively independent of bed state. Friction velocity predictions from presently available models, including a bed stress model and a sediment eddy diffusion model, are compared to measured values of near‐bed turbulence intensity. Reasonable agreement is found between measured and predicted bottom friction velocities when wave friction factors from Tolman [1994] are used.

An explicit WET-type model for the turbulent scalar fluxes

A new explicit WET model for the turbulent heat fluxes is derived and developed from the transport equation of the scalar passive flux. In order to study the performance of the model itself, all other turbulent quantities are taken from a DNS channel flow data-base and thus the error source has been minimized. This anisotropic eddy diffusivity model can be solved with a two-equation model for the temperature field based on the temperature variance and its dissipation rate. The model can theoretically be used for a range of Prandtl numbers or a range of different fluids. The results are compared with the DNS channel flow and fairly good agreement is achieved

On a subgrid-scale heat flux model for large eddy simulation of turbulent thermal flow

A non-linear subgrid-scale (SGS) heat flux model is introduced in large eddy simulation for turbulent thermal flows. Unlike the linear isotropic eddy diffusivity model, the proposed model accounts for the SGS heat flux in terms of the large-scale strain-rate tensor and the temperature gradients. This is equivalent to using a tensor diffusivity. The model is to some extent similar to a scale-similarity model subjected to a Taylor expansion for the filtering operation. The formulation leading to the present proposal is discussed. The model is examined in LES for a buoyant flow in an infinite vertical channel with two differentially heated side walls. It is shown that the proposed model reproduces reasonable results as compared with the isotropic SGS diffusivity model and DNS data.

On diffusion theory in turbulence

The Fokker-Planck equation for the probability density of fluid particle position in inhomogeneous unsteady turbulent flow is derived. The equation is obtained starting from the general kinematic relationship between velocity and displacement of a fluid particle and applying exact asymptotic analysis. For (almost) incompressible flow the equation reduces to the convection diffusion equation and the equation pertaining to the scalar gradient hypothesis. In this way the connection is established with eddy diffusivity models, widely used in numerical codes of computational fluid dynamics. It is further shown that within the accuracy of the approximation scheme of the diffusion limit, diffusion constants can equally be based on coarse-grained Lagrangian statistics as defined by Kolmogorov or on Eulerian statistics in a frame that moves with the mean Eulerian velocity as proposed by Burgers. The results presented for diffusion theory are the leading terms of asymptotic expansions. Truncated terms are higher-order spatial derivatives of the probability density or of the scalar mean value with coefficients based on cumulants higher than second order of fluid velocities and their derivatives. The magnitude of these terms has been assessed by employing scaling rules of turbulent flows in pipes and channels, turbulent boundary layers, turbulent jets, wakes and mixing layers, grid turbulence, convective layers and canopy turbulence. It reveals that a true diffusion limit does not exist. Although truncated terms can be of limited magnitude, a limit process by which these terms become vanishingly small and by which the diffusion approximation would become exact does not occur for any of the cases of turbulent flow considered. Applying the concepts of diffusion theory resorts to employing approximate methods of analysis.

ONE-POINT CLOSURE MODELS FOR BUOYANCY-DRIVEN TURBULENT FLOWS

▪ Abstract Turbulent flows driven by thermal buoyancy are featured by phenomena that pose a special challenge to conventional one-point closure models. Inherent unsteadiness, energy nonequilibrium, counter-gradient diffusion, strong pressure fluctuations, and lack of universal scaling, all believed to be associated with distinct large-scale coherent eddy structures, are hardly tractable by Reynolds-type averaging. Second-moment closures, though inadequate for providing information on eddy structure, offer better prospects than eddy-viscosity models for capturing at least some of the phenomena. For some configurations (e.g., with heating from below), unsteady computational solutions of ensemble-averaged equations, using a one-point closure as the subscale model, may be unavoidable for accurate prediction of flow details and wall heat transfer. This article reviews the rationale and some specific modeling issues related to buoyant flows within the realm of one-point closures. The inadequacy of isotropic eddy-diffusivity models is discussed first, followed by the rationale of the second-moment modeling and its term-by-term scrutiny based on direct numerical simulations (DNS). Algebraic models based on a rational truncation of the differential second-moment closure are proposed as the minimum closure level for complex flows. These closures are also recommended as subscale models for transient statistical modeling (T-RANS) and very large eddy simulations (VLES). Examples of applications illustrate some recent achievements.