scholarly journals What Causes the Divergences in Local Second-Order Closure Models?

2005 ◽  
Vol 62 (5) ◽  
pp. 1645-1651 ◽  
Author(s):  
V. M. Canuto ◽  
Y. Cheng ◽  
A. M. Howard
Keyword(s):  
Ad Hoc ◽  
Coefficient Of Thermal Expansion ◽  
Potential Temperature ◽  
Turbulence Models ◽  
Turnover Time ◽  
Second Order ◽  
Dimensionless Temperature ◽  
Gravitational Acceleration ◽  
Buoyancy Driven Flows ◽  
Large Eddies

Abstract It has been known for three decades that in the case of buoyancy-driven flows the widely used second-order closure (SOC) level-2.5 turbulence models exhibit divergences that render them unphysical in certain domains. This occurs when the dimensionless temperature gradient Gh (defined below) approaches a critical value Gh(cr) of the order of 10; thus far, the divergences have been treated with ad hoc limitations of the typewhere τ is the eddy turnover time scale, g is the gravitational acceleration, α is the coefficient of thermal expansion, T is the mean potential temperature, and z is the height. It must be noted that large eddy simulation (LES) data show no such limitation. The divergent results have the following implications. In most of the ∂T/∂z < 0 portion of a convective planetary boundary layer (PBL), a variety of data show that τ increases with z, −∂T/∂z decreases with z, and Gh decreases with z. As one approaches the surface layer from above, at some zcr (∼0.2H, H is the PBL height), Gh approaches Gh(cr) and the model results diverge. Below zcr, existing models assume the displayed equation above. Physically, this amounts to artificially making the eddy lifetime shorter than what it really is. Since short-lived eddies are small eddies, one is essentially changing large eddies into small eddies. Since large eddies are the main contributors to bulk properties such as heat, momentum flux, etc., the artificial transformation of large eddies into small eddies is equivalent to underestimating the efficiency of turbulence as a mixing process. In this paper the physical origin of the divergences is investigated. First, it is shown that it is due to the local nature of the level-2.5 models. Second, it is shown that once an appropriate nonlocal model is employed, all the divergences cancel out, yielding a finite result. An immediate implication of this result is the need for a reliable model for the third-order moments (TOMs) that represent nonlocality. The TOMs must not only compare well with LES data, but in addition, they must yield nondivergent second-order moments.

Nonlocal Convective PBL Model Based on New Third- and Fourth-Order Moments

2005 ◽  
Vol 62 (7) ◽  
pp. 2189-2204 ◽  
Author(s):  
Y. Cheng ◽  
V. M. Canuto ◽  
A. M. Howard
Keyword(s):  
Fourth Order ◽  
Potential Temperature ◽  
Turbulence Models ◽  
Second Order ◽  
Order Moment ◽  
Third Order ◽  
Mean Velocity ◽  
Eddy Simulation ◽  
Analytic Expressions ◽  
Aircraft Data

Abstract The standard approach to studying the planetary boundary layer (PBL) via turbulence models begins with the first-moment equations for temperature, moisture, and mean velocity. These equations entail second-order moments that are solutions of dynamic equations, which in turn entail third-order moments, and so on. How and where to terminate (close) the moments equations has not been a generally agreed upon procedure and a variety of models differ precisely in the way they terminate the sequence. This can be viewed as a bottom-up approach. In this paper, a top-down procedure is suggested, worked out, and justified, in which a new closure model is proposed for the fourth-order moments (FOMs). The key reason for this consideration is the availability of new aircraft data that provide for the first time the z profile of several FOMs. The new FOM expressions have nonzero cumulants that the model relates to the z integrals of the third-order moments (TOMs), giving rise to a nonlocal model for the FOMs. The new FOM model is based on an analysis of the TOM equations with the aid of large-eddy simulation (LES) data, and is verified by comparison with the aircraft data. Use of the new FOMs in the equations for the TOMs yields a new TOM model, in which the TOMs are damped more realistically than in previous models. Surprisingly, the new FOMs with nonzero cumulants simplify, rather than complicate, the TOM model as compared with the quasi-normal (QN) approximation, since the resulting analytic expressions for the TOMs are considerably simpler than those of previous models and are free of algebraic singularities. The new TOMs are employed in a second-order moment (SOM) model, a numerical simulation of a convective PBL is run, and the resulting mean potential temperature T, the SOMs, and the TOMs are compared with several LES data. As a final consistency check, T, SOMs, and TOMs are substituted from the PBL run back into the FOMs, which are again compared with the aircraft data.

scholarly journals Method for Measuring the Second-Order Moment of Atmospheric Turbulence

Atmosphere ◽  
2021 ◽  
Vol 12 (5) ◽  
pp. 564
Author(s):  
Hong Shen ◽  
Longkun Yu ◽  
Xu Jing ◽  
Fengfu Tan
Keyword(s):  
Atmospheric Turbulence ◽  
Weighting Function ◽  
Structure Constant ◽  
Turbulence Models ◽  
Second Order ◽  
Index Structure ◽  
Order Moment ◽  
Site Testing ◽  
Optical Effects ◽  
Novel Method

The turbulence moment of order m (μm) is defined as the refractive index structure constant Cn2 integrated over the whole path z with path-weighting function zm. Optical effects of atmospheric turbulence are directly related to turbulence moments. To evaluate the optical effects of atmospheric turbulence, it is necessary to measure the turbulence moment. It is well known that zero-order moments of turbulence (μ0) and five-thirds-order moments of turbulence (μ5/3), which correspond to the seeing and the isoplanatic angles, respectively, have been monitored as routine parameters in astronomical site testing. However, the direct measurement of second-order moments of turbulence (μ2) of the whole layer atmosphere has not been reported. Using a star as the light source, it has been found that μ2 can be measured through the covariance of the irradiance in two receiver apertures with suitable aperture size and aperture separation. Numerical results show that the theoretical error of this novel method is negligible in all the typical turbulence models. This method enabled us to monitor μ2 as a routine parameter in astronomical site testing, which is helpful to understand the characteristics of atmospheric turbulence better combined with μ0 and μ5/3.

Morphing Continuum Theory: Incorporating the Physics of Microstructures to Capture the Transition to Turbulence Within a Boundary Layer

2016 ◽  
Vol 139 (1) ◽  
Author(s):  
Louis B. Wonnell ◽  
James Chen
Keyword(s):  
Boundary Layer ◽  
Flat Plate ◽  
Ad Hoc ◽  
Constitutive Relations ◽  
Turbulence Models ◽  
Continuum Theory ◽  
Transition To Turbulence ◽  
Flow Turbulence ◽  
Good Agreement ◽  
Inherent Advantage

A boundary layer with Re = 106 is simulated numerically on a flat plate using morphing continuum theory. This theory introduces new terms related to microproperties of the fluid. These terms are added to a finite-volume fluid solver with appropriate boundary conditions. The success of capturing the initial disturbances leading to turbulence is shown to be a byproduct of the physical and mathematical rigor underlying the balance laws and constitutive relations introduced by morphing continuum theory (MCT). Dimensionless equations are introduced to produce the parameters driving the formation of disturbances leading to turbulence. Numerical results for the flat plate are compared with the experimental results determined by the European Research Community on Flow, Turbulence, and Combustion (ERCOFTAC) database. Experimental data show good agreement inside the boundary layer and in the bulk flow. Success in predicting conditions necessary for turbulent and transitional (T2) flows without ad hoc closure models demonstrates the theory's inherent advantage over traditional turbulence models.

Using Direct Numerical Simulations for Investigating Physics of Turbulence in Porous Media

2017 ◽  
Author(s):  
Y. Jin ◽  
A. V. Kuznetsov
Keyword(s):  
Porous Media ◽  
Turbulence Models ◽  
Maximum Size ◽  
Reynolds Numbers ◽  
Direct Numerical Simulations ◽  
Velocity Measurements ◽  
Turbulent Eddies ◽  
Size Limitation ◽  
Large Eddies ◽  
Convection In Porous Media

One of the most controversial topics in the field of convection in porous media is the issue of macroscopic turbulence. It remains unclear whether it can occur in porous media. It is difficult to carry out velocity measurements within porous media, as they are typically optically opaque. At the same time, it is now possible to conduct a definitive direct numerical simulation (DNS) study of this phenomenon. We examine the processes that take place in porous media at large Reynolds numbers, attempting to accurately describe them and analyze whether they can be labeled as true turbulence. In contrast to existing work on turbulence in porous media, which relies on certain turbulence models, DNS allows one to understand the phenomenon in all its complexity by directly resolving all the scales of motion. Our results suggest that the size of the pores determines the maximum size of the turbulent eddies. If the size of turbulent eddies cannot exceed the size of the pores, then turbulent phenomena in porous media differ from turbulence in clear fluids. Indeed, this size limitation must have an impact on the energy cascade, for in clear fluids the turbulent kinetic energy is predominantly contained within large eddies.

Periodic Behavior of a Nonlinear Third Order Vibrating System

2002 ◽  
Author(s):  
Gholamreza Nakhaie Jazar ◽  
Mohammad H. Alimi ◽  
Mohammad Mahinfalah ◽  
Ali Khazaei
Keyword(s):  
Differential Equation ◽  
Dynamical Systems ◽  
Ad Hoc ◽  
Order Differential Equation ◽  
Second Order ◽  
Order System ◽  
Third Order ◽  
Modeling Process ◽  
Third Order Differential Equation ◽  
Second Order Systems

In modeling of dynamical systems, differential equations, either ordinary or partial, are a common outcome of the modeling process. The basic problem becomes the existence of solution of these deferential equations. In the early days of the solution of deferential equations at the beginning of the eighteenth century the methods for determining the existence of nontrivial solution were so limited and developed very much on an ad hoc basis. Most of the efforts on dynamical system are related to the second order systems, derived by applying Newton equation of motion to dynamical systems. But, behavior of some dynamical systems is governed by equations falling down in the general nonlinear third order differential equation x″′+f(t,x,x′,x″)=0, sometimes as a result of combination of a first and a second order system. It is shown in this paper that these equations could have nontrivial solutions, if x, x′, x″, and f(t,x,x′,x″) are bounded. Furthermore, it is shown that the third order differential equation has a τ-periodic solution if f(t,x,x′,x″) is an even function with respect to x′. For this purpose, the concept of Green’s function and the Schauder’s fixed-point theorem has been used.

A FIRST- AND SECOND-ORDER TURBULENCE MODELS IN HYDROGEN NON-PREMIXED FLAM

2015 ◽  
Vol 33 (3) ◽  
pp. 27-34 ◽  
Author(s):  
N. Oumrani ◽  
M. Aouissi ◽  
A. Bounif ◽  
B. Yssaad ◽  
F. Tabet ◽  
...  
Keyword(s):  
Turbulence Models ◽  
Second Order

Hydrogen Distribution in a Nuclear Reactor Containment

2008 ◽  
Author(s):  
Deoras M. Prabhudharwadkar ◽  
Kannan N. Iyer ◽  
Nalini Mohan ◽  
S. S. Bajaj ◽  
S. G. Markandeya
Keyword(s):  
Concentration Distribution ◽  
Room Temperature ◽  
Nuclear Reactor ◽  
Structural Integrity ◽  
Practical Importance ◽  
Turbulence Models ◽  
Accident Analysis ◽  
Time Varying ◽  
Hydrogen Distribution ◽  
Buoyancy Driven Flows

The management of hydrogen in nuclear reactor containment after LOCA is of practical importance to preserve the structural integrity of the containment. This paper presents the results of systematic work carried out using the commercial software FLUENT to assess the concentration distribution of hydrogen in a typical Indian Nuclear Reactor Containment. Accurate turbulence modelling is important to predict the concentration distribution correctly. The turbulence models which were most commonly cited in the literature for modelling buoyancy driven flows were assessed for their suitability and it was found that the buoyancy modified Standard k-ε model is adequate for the purpose by comparing with some experimental data available in the literature. Subsequently, unstructured meshes were generated to represent the containment of a typical Indian nuclear reactor. Analyses were carried out to quantify the hydrogen distribution for three cases. These were (1) Uniform injection of hydrogen for a given period of time at room temperature, (2) Time varying injection as has been computed from an accident analysis code, (3) Time varying injection (as used in case (2)) at a high temperature. A parametric exercise was also carried out in case (1) where the effect of various inlet orientations and locations on hydrogen distribution was studied. Results of all these cases have been presented in this paper.

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