Structure-based transient models for scalar dissipation rate in homogeneous turbulence

https://doi.org/10.1016/j.ijheatfluidflow.2020.108557Get rights and content

Highlights

  • We develop a structure-based set of transport models for scalar dissipation rate.

  • The proposed models are derived from a structure-sensitized transport model for the large-scale field of enstrophy and by applying the weak equilibrium assumption.

  • The Interactive Particle Representation model (IPRM) is used to estimate turbulent scalar-flux vector and a set of turbulence tensors.

  • Good agreement is achieved with available DNS results.

Abstract

We present, for the first time, transient models for scalar-dissipation rate that carry information about the morphology of the turbulence structures. The proposed models are derived from a structure-sensitized transient model for the large-scale field of enstrophy. Special attention is given to the model equation of Yoshizawa (1988), from which alternative closures for scalar-dissipation rate are derived. Assuming negligible transient variations of the turbulent anisotropies, lead to an expression similar to the one used by traditional transient models for scalar dissipation rate. To reduce model uncertainties, the turbulent scalar-flux vector and the turbulence structure tensors are obtained from the same closure models for all cases considered. The estimation performance of the models is evaluated on eight cases of deformation in both fixed and rotating frames, showing encouraging results. The proposed models achieve fair agreement with homogeneous Direct Numerical Simulations (DNS) predictions in the presence of either transverse or streamwise mean scalar gradient.

Introduction

Scalar dissipation rate ϵϕ is associated with the transfer rate of scalar energy from the largest eddies to the smallest ones, where scalar diffusion becomes important. Thus, accurate modeling of this quantity requires reliable predictions for the turbulent time scales. A common modeling approach for the scalar dissipation rate is to assume that the integral time scales of velocity and scalar fluctuations are proportional, which results in simple algebraic models. The simplest algebraic model for the scalar dissipation rate assumes that the scales of the velocity and the scalar field are equal, known as the “equal-scale” model. Despite its computational and implementational attractiveness, the “equal-scale” assumption is the primary source of performance limitations. For example, experimental works on turbulent jets and turbulent diffusion flames (Chevray and Tutu, 1978, Panchapakesan, Lumley, 1993) suggest a varying velocity to scalar time-scale ratio RT, which was also observed in buoyant axisymmetric plumes (Shabbir and George, 1994) and axisymmetric jets (Pietri et al., 2000). As a result, several transport models have been developed to overcome this limitation (Elghobashi and Launder, 1983, Nagano and Kim, 1988, Newman et al., 1981, Shih et al., 1987), achieving successful predictions for a variety of heat transfer problems (Hanjalic, 1994). However, there are still difficulties that pose major uncertainties both numerically and physically. The major source of these uncertainties stems from the fact that the transport equation for the turbulent scalar dissipation rate involves many small-scale terms that are typically modelled in terms of large-scale quantities. As a result, most strategies for modelling the scalar dissipation rate transport equation are based on semi-empirical scaling arguments in which individual terms in the exact equations are modelled together as simple expressions representing transport, production and dissipation. The intent is to mimic the properties of the scalar variance transport equation, whose terms have a more transparent physical meaning. A way to tackle these issues is to consider the modelling of the turbulent enstrophy ω2¯=ωiωi¯ (where the overbar denotes an averaging process, ωi denotes the vorticity vector and the prime symbol denotes its fluctuation) which is directly related to the turbulent energy dissipation rate ϵκ in the limit of homogeneous turbulence. This option is elegant, because the turbulent enstrophy equation possesses many similarities with the turbulent scalar dissipation rate equation, i.e. lacks expressions involving pressure, which has been proven to be largely intractable. Consequently, several of the production terms in the enstrophy equation can be analyzed through formal vorticity transport analysis, resulting into a more careful account of production effects than it has been so far feasible in the scalar dissipation rate equation. Reynolds, Langer and Kassinos (hereafter RLK02) Reynolds et al., 2002 proposed a transient model for enstrophy equation that was sensitized to the anisotropy of the large-scale turbulent structure, called Large-Scale Enstrophy (LSE) model. To achieve that, they decomposed the turbulent field into a large-scale and a small-scale part. Then, they modelled the production and dissipation terms in terms of invariants of a set of tensors describing the morphology of the turbulent structures, the so-called turbulence structure tensors (Kassinos et al., 2001). The resulting structure-based model (SBM) exhibited encouraging results for different types of flows, in both stationary and rotating frames. This is attributed to the fact that SBMs do not require ad-hoc expressions to account for rotational corrections in various limiting states since this information is implicitly incorporated in the model through the invariants of the structure tensors. Panagiotou and Kassinos (here-after PK16) Panagiotou and Kassinos, 2016, Panagiotou and Kassinos, 2017 recently proposed a structure-based model for turbulent scalar fields based on observations suggesting that a similar cascade mechanism exists for scalar field. Their model was successfully tested for various flow configurations, ranging from simple passive scalars to strongly stably stratified flows. Furthermore, it was proven to be suitable for use as a diagnostic tool for the morphology of highly anisotropic turbulent structures.

In this study, we propose two transient models for the scalar dissipation rate in the framework of structure-based modelling. In order to accomplish this, we assume that the transient variations of the components of the turbulence anisotropy tensors are negligible compared to the scales’ variations, an approach called the “weak-equillibrium assumption” (WEA) Rodi (1972). Starting from the LSE model, this assumption reduces the complexity of the resulting expressions, which exhibit a functional form similar to that found in traditional models. We have also considered the “two-scale direct interaction approximation” (TSDIA) proposed by Yoshizawa (1988), who expressed the transport equation for ϵϕ in terms of the transport equations for scalar-variance ϕ2¯ and dissipation rate ϵκ. Starting from Yoshizawa’s transport equation (here-after YS) and adopting WEA, we propose an alternative SBM model. The performance of the resulting SBM models, along with that of the original Yoshizawa’s model, is tested in several shear flows, involving stationary and rotational frames.

The governing equations for the velocity and scalar fields are introduced in Section 2, where we also provide the definitions of the turbulence structure tensors. In Section 3, we briefly introduce the main idea behind the formulation of the large-scale model, while Section 4 provides details on the modelling method used to evaluate the structure tensors and the scalar-flux vector. In Section 5, we briefly discuss Yoshizawa’s model for scalar dissipation rate, while we introduce the algebraic “equal-scale” model. In Section 6, we derive several structure-based transient models for scalar dissipation rate based on the “weak-equilibrium” assumption. In Sections 7 and 8, the performance of the proposed models is evaluated on eight cases, yielding promising results. Summary and conclusions are given in Section 9.

Section snippets

Governing equations

In incompressible flow, the transport of a passive scalar is described by the dimensionless continuity, momentum and advection-diffusion equations,uj,j=0,tui+ujui,j=1ρp,i+1Reui,jj,tϕ+ujϕ,j=1Peϕ,jj, where ρ is the density of the fluid, ui, p, ϕ are the instantaneous velocity, pressure and passive scalar fields respectively, Re is the Reynolds number and Pe is the Peclet number. Hereafter, we are using index notation whereby repeated indexes imply summation; an index following a comma denotes

Turbulence scales in structure-based modeling

In the framework of two-equation structure-based modeling, RLK02 have proposed the use of the LSE equation to provide one of the turbulence scales. In the LSE equation, closure is achieved in terms of the one-point structure tensors, thus sensitizing the turbulence scales to the effects of the turbulence structure. The main idea behind the formulation of the large-scale enstrophy model is that the turbulent kinetic energy is transferred from large to small scales through a cascade mechanism

An interacting particle representation model (IPRM) for scalar transport

It is readily deduced from Eq. (13) that the determination of the structure tensors and scalar-flux vector uiϕ¯ is required in order to bring the ELSE model into a closed form. In the current study these quantities are obtained from the model developed by Kassinos and Reynolds (1994), called Particle Representation Model (PRM), which is proven to be a numerically efficient and affordable method for carrying out exact Rapid Distortion Theory (RDT) computations. The PRM is a reduced

Classical “equal scales” model

As mentioned in the introduction, the simplest model for ϵλ is obtained if the time (or length) scales between the velocity and the scalar field are assumed to be equal. Using dimensional analysis, we obtain that the time and the length scales of the velocity field are τvκ/ϵκ and vκ3/2/ϵκ respectively. Similarly for the scalar field, the corresponding scales are τϕ ~ λλ and ϕ(λϵλ)κ1/2 respectively. Setting τv=τϕ (or v=ϕ) yields a linear algebraic relationship between the velocity and

Scalar dissipation rate transport in structure-based modelling

In this section we provide details on the development of structure-based evolution models for scalar dissipation rate based on the “weak-equilibrium” assumption. The “weak-equillibrium” assumption, proposed by Rodi (1972), states that the advection and diffusion of the normalized turbulent anisotropies are negligible, thus their transient variations can be neglected. Taking the time derivative of the algebraic expression for ϵκ in (13) and applying Rodi’s assumption yields an evolution model

Numerical conditions

In order to ascertain the performance of the aforementioned models, we consider the case of homogeneous shear in fixed and rotating frames at high Reynolds and Peclet numbers. Both transverse and streamwise mean scalar gradients are considered. These simple canonical cases are of particular interest because of their relevance to geophysical flows and industrial applications, such as turbomachinery flows. The configuration of the mean flow is given byGij=Sδi1δj2,Ωif=Ωfδi3,Λi=Sϕδiα,where greek

Results

As mentioned in the introduction, our goal is to propose transient models for the dissipation rates. Consequently, this section discusses the estimation ability of SBM-E+WEA-ϵλ and SBM-E+SBM-YS, denoted as Class C models in Table 1, while the remaining models are also considered to facilitate the discussion regarding the impact that the equilibrium assumption has on performance. We consider several validation cases, involving both transverse and streamwise mean scalar gradients, in both fixed

Summary and conclusions

Structure-based transient models have been proposed which, for the first time, provide predictions for the scalar dissipation rate in the homogeneous limit. The model parameters are expressed in terms of invariants of the structure tensors, thus being sensitized to the large-scale anisotropy of the turbulent field. The weak-equillibrium assumption has been adopted in order to bring the proposed models, denoted as Class C, to a functional form comparable to the standard forms that exist in

CRediT authorship contribution statement

C.F. Panagiotou: Conceptualization, Methodology, Formal analysis, Investigation, Software, Validation, Writing - review & editing. F.S. Stylianou: Conceptualization, Writing - review & editing, Writing - original draft. S.C. Kassinos: Conceptualization, Supervision, Writing - review & editing, Writing - original draft.

Declaration of Competing Interest

No conflict of interest exists.

References (32)

  • R. Chevray et al.

    Intermittency and preferential transport of heat in a round jet

    J. Fluid Mech.

    (1978)
  • S.E. Elghobashi et al.

    Turbulent time scales and the dissipation ratio of temperature variance in the thermal mixing layer

    Phys. Fluids

    (1983)
  • K. Hanjalic

    Achievements and limitations in modelling and computation of buoyant turbulent flows and heat transfer

    Heat Transfer 1994, Proceedings of the 10th International Heat Transfer Conference, Brighton, GB, Aug 14–18

    (1994)
  • S.C. Kassinos et al.

    The transport of a passive scalar in magnetohydrodynamic turbulence subjected to mean shear and frame rotation

    Phys. Fluids

    (2007)
  • S.C. Kassinos et al.

    A structure-Based Model for the Rapid Distortion of Homogeneous Turbulence.

    (1994)
  • S.C. Kassinos et al.

    An extended structured-based model based on a stochastic eddy-axis evolution equation

    Annual Research Briefs, Center for Turbulence Research, California, USA

    (1995)
  • This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

    View full text