Modelling stellar convective transport with plumes: I. Non-equilibrium turbulence effect in double-averaging formulation

There are several ways to extract the local spatiotemporal structures from the random fluctuations. The conditional averaging procedure is one of such ways. For example in the turbulent boundary-layer study, we consider the streamwise and wall-normal velocity fluctuation components, $u^{\prime}$ and $v^{\prime}$, and divide the whole value domain of fluctuations into the four quadrants depending on the signs of $(u^{\prime},v^{\prime})=(+,+),(-,+),(-,-),(+,-)$, and examine the statistical properties in each quadrant. For example, the motions belonging to the quadrant $(+,+)$ (or $(-,-)$) represents an event at which the turbulent flow is moving along (or opposite to) the streamwise direction while departing (approaching) from the wall. In this sense, the statistics based on the four quadrants should correspond to a conditional averaging linked to the type of events, such as the bursting, sweeping, etc. in the turbulent boundary layer [see Walles (2013) and references cited therein]. For the convective turbulence with plumes, a conditional averaging within the four quadrants based on the combination of the vertical component of the velocity fluctuation, $w^{\prime}$, and the temperature fluctuation $\theta^{\prime}$, $(w^{\prime},\theta^{\prime})$ is possible. In this case, for example, the motions with $(w^{\prime},\theta^{\prime})=(+,+)$ represents an ascending plume with being heated, and $(-,-)$ does a descending plume with being cooled.

Another way to represent the coherent and incoherent fluctuations is to adopt a double-averaging methodology (Finnigan & Shaw, 2008; Pokrajac, McEwan & Nikora, 2008; Dey, 2014; Dey, Paul & Padhi, 2020). We regard convection plumes as a turbulent coherent fluctuation, and investigate the properties of the fluctuations. There are several types of the double averaging method. The representative double-averaging procedure is the double filtering ubiquitously adopted in the data processing and the large-eddy simulations (LESs) of turbulent flows. By a sequential application of two or more filters with varied filtering levels (in frequency or length scale) to the raw data, slowly varying fluctuating motions are extracted from the fast varying fluctuating ones (Sagaut, Deck & Terracol, 2006).

In this work, we consider a combination of the time and space averaging. Depending on the sequential order of time and space averaging, there are two ways of the time and space double averaging method: (i) the time–space averaging, where the spatial averaging is taken to the already temporary-averaged variables; and (ii) the space–time averaging, where the temporal averaging is taken to the already spatial averaged variables. The first one, the time–space averaging is more appropriate for the purpose of extracting the plume structures. As is usual for the case of double averaging or filtering procedures, the averaging should be taken first with a finer resolution manner, then a “coarse grained” averaging should be taken. Otherwise, the coarse grained averaging makes the fine structures be smeared out. In this work, the averaging window for the time average is set much shorter than the counterpart for the space average.¹¹1The fraction of the lifetime of a plume, $\tilde{\tau}$, to the averaging time window $T$ is defined by $\Delta_{\rm{time}}=\tilde{\tau}/T$, while the fraction of the occupation domain area of a plume, $\tilde{s}=b^{2}$ ($b$: horizontal dimension of the plume), to the space averaging window $\Delta S=\Delta_{x}\Delta_{y}$ is $\Delta_{\rm{space}}=\tilde{s}/\Delta S$. If these fractions are much smaller than unity, the plume effects are statistically smeared out by the averaging. On the other hand, if the fraction is comparable to unity, the plume effects are detectable after the averaging. Because of the spatial localisation of a plume, $\Delta_{\rm{space}}$ is very small ($\Delta_{\rm{space}}\ll 1$), whereas $\Delta_{\rm{time}}$ can be comparable to unity if we set the averaging time window $T$ similar to the lifetime $\tilde{\tau}$ ($T\simeq\tilde{\tau}$). This is the reason why we should take the time averaging first. So, we should first take the temporal average over a time period during which a plume persistently exists, then perform the space average of the already time-averaged quantities. We further assume that the space averaging provide a surrogate of the ensemble averaging.

As for the time average, the usual time average defined by

is not suitable at all for detecting the plume structure since their structures are smeared out during the long averaging time $T\to\infty$. In order to catch a plume structure, which is local in time and space, we adopt a time filter defined by

Here, as for the appropriate averaging time $T$, we adopt a time which is much longer than the eddy turn-over time of turbulence, $\tau$, and much shorter than the time scale of the mean-field evolution, $\Xi$, as

Of course, all the statistics depend on the value of the averaging time $T$. It is difficult to determine the averaging time $T$ in a general manner. Here, we adopt $T$ which is similar to the plume lifetime (time of the persistent presence of a plume). The eddy turn-over time of the turbulence $\tau$ and the time scale of the mean-field evolution $\Xi$ are determined by the density scale height, intensity of turbulence, the depth of the convection zone, etc. As will be discussed at the end of § 6.2, the averaging time window $T$ should be put similar to the characteristic lifetime of the descending plume. The lifetime of a plume can be estimated from the characteristic turbulent velocity $v$ and the dimension of density stratification. This is much longer than the eddy turnover time $\tau$ estimated from $v$ and mixing length. So, it is expected to be possible to define an averaging time that satisfies the condition (24).

On the other hand, as for the space averaging, we consider

Here $\Delta S$ is the area of the averaging surface, which is spanned by the homogeneous directions. In case the statistical quantity is inhomogeneous in the vertical or $z$ direction, and homogeneous in the horizontal or $x$-$y$ directions, we put $\Delta S=\Delta_{x}\Delta_{y}$ and the space averaging is defined with the horizontal averaging as

In this work, the statistical quantities are assumed to be homogeneous in the horizontal surface, and the horizontal averaging is regarded as a surrogate of the ensemble averaging.

We adopt the time–space double averaging methodology. In this procedure, we first take the time average of a physical quantity $f$ and denote it as $\overline{f}$. Then we take the space average of the time averaged quantity and denote it as $\langle{\overline{f}}\rangle$. With this double-averaging procedure, a field quantity $f$ is decomposed into

where $\tilde{f}$ is defined by

It is often called the dispersion in analogy with the wave decomposition procedure. If we rewrite this in the form of

we see that a time averaged quantity $\overline{f}$ is divided into the space averaged part $\langle{\overline{f}}\rangle$ and the deviation from the space average, $\tilde{f}$. Namely, $\tilde{f}$ represents the part of a time-averaged quantity $\overline{f}$ deviating from the space average $\langle{\overline{f}}\rangle$, and $f^{\prime\prime}$ represents the deviation from the time average $\overline{f}$. On the other hand, from the viewpoint of the space averaging, both $\tilde{f}$ and $f^{\prime\prime}$ are treated as the fluctuations. The fluctuation in the space averaging procedure, $f^{\prime}$, is divided into $\tilde{f}$ and $f^{\prime\prime}$ as

In other words, $\tilde{f}$ represents the persistent or coherent part of the spatial fluctuations $f^{\prime}$, while $f^{\prime\prime}$ is the random or incoherent counterpart.

Of course, in order to distinguish the coherent fluctuation from the incoherent one, it is required to use much more elaborated mathematical tools. In the wavelet analysis of turbulence, a velocity field is decomposed on the basis of scaling functions and wavelets of different families are often applied. The part of fluctuation is called coherent if it is expressed in terms of such an expansion, and incoherent otherwise (Farge, et al., 2003; Goldstein & Vasilyev, 2004). The notion of coherency does not directly correspond to the usual decomposition between the large and small scales. Coherent motions exist even at small scales, and random motions are observed even at large scales. In this work, however, we do not delve into such detailed formulations of coherency, but denote $\tilde{f}$ as the coherent fluctuation and $f^{\prime\prime}$ as the incoherent fluctuation in the present time–space double averaging procedure.

In the present work, we assume the following relations among the time and space averaging:

Among these relations, (31a) is an assumption that the space and time averaging satisfies the Reynolds rule. Since time averaging (23) contains information of the past time during the averaging period $T$, so strictly speaking, we have $\overline{\overline{f}}\neq\overline{f}$ for a finite $T$. However, we approximate that $\overline{\overline{f}}=\overline{f}$. Relation (31c) implies that both the space averaging of the already time-averaged, $\langle{\overline{f}}\rangle$, and the time averaging of the already space-averaged, $\overline{\langle{f}\rangle}$, are equivalent to the space averaged one $\langle{f}\rangle$. This is a consequence of the fact that the time averaging and space averaging are defined independently. Relation (31g) is derived from (31a) and (31c) as

These relations (31) are fully utilised in deriving the evolution equations of the turbulent correlations in § 4.

We adopt the time–space double averaging procedure. In this framework, a dispersion $\tilde{f}$ belongs to the averaged field $\overline{f}(=\langle{\overline{f}}\rangle+\tilde{f})$ in the time-averaging sense, while it belongs to a fluctuation field $f^{\prime}(=\tilde{f}+f^{\prime\prime})$ and represents the coherent components of fluctuations in the space-averaging sense. In order to see the structure of the turbulence model we adopted in this work, we present a diagram similar to the coherency diagram first introduced by Goldstein & Vasilyev (2004). In the context of the present double-averaging method, this diagram illustrates the relationship among the field quantities decomposed as (27). It shows that the fluctuations are decomposed into the coherent and incoherent fields. It simultaneously exhibits the separation between the space-averaged mean fields and the residual fluctuation fields (Fig. 1).

In the present work, a time-averaged quantity $\overline{f}$ is divided into its space-averaged part $\langle{\overline{f}}\rangle$ and its deviation or dispersion part $\tilde{f}(=\overline{f}-\langle{\overline{f}}\rangle)$. The space averaging of the incoherent fluctuation vanishes by definition. This can be seen from (31) as $\langle{f^{\prime\prime}}\rangle=\langle{f-\overline{f}}\rangle=\langle{f}\rangle-\langle{\overline{f}}\rangle=0$. The criterion for separating the coherent component from incoherent one depends on the averaging time $T$ in (23). If we set the averaging time $T$ sufficiently large (e.g., much larger than the lifetime of the characteristic coherent fluctuation such as a plume), the coherent motions are smeared out and the portion of the coherent component is reduced. This situation is not suitable for properly describing the effects of plume motions, which is one of the essential ingredients of convective flow.

In summary, with the time–space averaging procedure, a field quantity $f$ is decomposed as

In the convective turbulent flows, not only the turbulent transport due to the usual eddy or random/incoherent fluctuation but also the transport due to the structural/coherent fluctuation plays a key role. In the presence of plume structures, effective transports of the mass, momentum, and energy are greatly enhanced, compared to the case without the plume structures or to the case with the structures rapidly disappearing. Therefore, how to incorporate the effects of coherent fluctuations such as plumes is of primary importance in modelling the convective turbulent flows.

In the $K-\varepsilon$ model, one of the most widely used Reynolds-averaged turbulence models, the eddy viscosity is expressed as

where $C_{\nu}$ is the model constant, $K$ is the turbulent kinetic energy defined by (19), and $\varepsilon$ is its dissipation rate defined by (20). They are the two one-point turbulent statistical quantities chosen to describe the dynamic and statistical properties of turbulent fields. The eddy-viscosity model (41) is very simple and useful, but is known to have several drawbacks. One is the overestimate of $K$ and $\varepsilon$ when applied to the homogeneous shear turbulence. In order to alleviate such a drawback, various ways for improving the model (41) have been proposed. The non-equilibrium effect is one of such methods and is expected to be valid in deriving the turbulent transport that deviates from the simple eddy-viscosity representation (Yoshizawa & Nisizima, 1993).

In order to see the background of the non-equilibrium model of turbulent transport, here we briefly show how to derive the non-equilibrium effect from the analytical statistical theory of inhomogeneous turbulence. With the aid of the multiple-scale renormalisation perturbation expansion (Yoshizawa, 1984; Yokoi, 2020), the turbulent energy with the non-equilibrium effect implemented is expressed as

where $\ell_{\rm{C}}$ is the scale of the largest eddy, $C_{Kn}$ ($n=1-3$) are the model constants, and the Lagrangian derivative along the mean velocity ${\bf{U}}(=\langle{\bf{u}}\rangle)$ is defined as

We solve (42) by iteration. The lowest-order expression is

leading to $\ell_{\rm{C}}$ as

($C_{\ell 1}$: model constant). This expression, obtained without recoursing to the non-equilibrium part, corresponds to the simplest eddy-viscosity expression. Substituting (45) into (42), and using the iteration, we have

($C_{\ell 2}$ and $C_{\ell 3}$: model constants). The second and third terms in (46) represent the non-equilibrium effect. Equation (46) can be approximated as

where $\ell_{\rm{E}}$ is the equilibrium length scale defined by

($C^{\prime}_{\rm{N}}$: model constant). It follows from (47) that the time scale of turbulence is expressed as

where the non-equilibrium correction factor $\Gamma$ is defined by

which can be either positive or negative. Here, $\tau_{\rm{E}}$ is the equilibrium time scale defined by

As we see from (47) and (49), the non-equilibrium effect can be represented by the variation of the turbulent energy $K$ and its dissipation rate $\varepsilon$ along the fluid motion. From (49), the turbulent viscosity is expressed as

where $\nu_{\rm{TE}}$ is the equilibrium eddy viscosity without the non-equilibrium effect related to $\Gamma$ ($C_{\rm{N}}$: model constant).

Expression (54) implies that in case that the turbulent energy $K$ and its dissipation rate $\varepsilon$ vary along the mean flow, namely in the case of non-equilibrium turbulence, the effective turbulent viscosity $\nu_{\rm{T}}$ may deviate from its equilibrium value. Equation (54) shows that the effective viscosity is suppressed if $\Gamma>0$ or equivalently $D(K^{2}/\varepsilon)/Dt=D(K\tau)/Dt>0$ with the time scale $\tau=K/\varepsilon$, and that $\nu_{\rm{T}}$ is enhanced if $\Gamma<0$ or equivalently $D(K^{2}/\varepsilon)/Dt=D(K\tau)/Dt<0$.

Actually, it was established by recent experiments with single-phase fluid jets and with two-phase fluid buoyant bubble plumes and negatively buoyant particle plumes that the turbulent kinetic energy $K$ and its dissipation rate $\varepsilon$ vary along the plume and jet stream direction (Bordoloi, et al., 2020; Charonko & Prestridge, 2017; Lai & Socolofsky, 2019a, b). For instance, in the round jet experiment, it was reported that the ratio of the axial fluctuation intensity $\langle{u^{\prime}{}^{2}}\rangle$ to the root of the dissipation rate, $\sqrt{\varepsilon}$, showed a decreasing tendency with the axial or streamwise direction (Lai & Socolofsky, 2019a). In this case, since the non-equilibrium contribution $\Gamma$ in (50) is negative, $D(K^{2}/\varepsilon)/Dt<0$, it is anticipated from (54) that the turbulent viscosity will be effectively enhanced. This matches the requirement for improving the current turbulence model by adjusting the model constants. For another example, in the buoyant bubble plume experiment, it was reported that the turbulent energy is increased along the ascending bubble motion. This tendency is much more dominant in the adjustment region, where the non-equilibrium property is much more prominent, than in the asymptotic region (Lai & Socolofsky, 2019b). In another experiment, the turbulent energy is reported to be enhanced in the direction of the negatively buoyant particle plume, while the energy dissipation does not show significant changes (Bordoloi, et al., 2020). In these two cases of the buoyant bubble plume and negatively buoyant particle plume, $D(K^{2}/\varepsilon)/Dt$ is positive, so the turbulent viscosity is expected to be reduced by the non-equilibrium effect. As we see from these jet and plume experiments, the inhomogeneities of the turbulent kinetic energy and its dissipation rate along the flow direction may lead to a significant alteration of the turbulent transport coefficients due to the non-equilibrium effect (Yokoi, 2022).

For the sake of simplicity of argument, we further assume that the non-equilibrium property along a flow for the energy dissipation rate $\varepsilon$ is much smaller than the counterpart on the turbulent energy $K(=\langle{{\bf{u}}}^{\prime}{}^{2}\rangle/2)$ in the sense

Under this condition, the non-equilibrium eddy viscosity (54) is approximately expressed as

where $\tau$ is the eddy turn-over time, and $C_{\rm{N}}$ the model constant. Inequality (55) does not necessarily meet the flow conditions in all the real turbulence cases. However, we adopt (56) as a starting point, since the estimate of dissipation rate in practical flows is often much harder than that of the turbulent energy.

With the analogy of the non-equilibrium effect on the Reynolds stress in the ensemble- or space-averaging procedure, (56), the turbulent mass flux $\overline{\rho^{\prime\prime}{\bf{u}}^{\prime\prime}}$ in the time-averaging procedure may be modelled as

where $\kappa^{\prime\prime}_{\rm{E}}$ is the equilibrium effective diffusivity due to the random or incoherent fluctuations, $C^{\prime\prime}$ the model constant, $\tau^{\prime\prime}$ the time scale of the incoherent fluctuation, $\overline{D}/\overline{D}t(=\partial/\partial t+\overline{\bf{u}}\cdot\nabla)$ the Lagrange or material derivative along the time-averaged velocity $\overline{\bf{u}}$, and $k^{\prime\prime}$ the time-averaged turbulent energy of the incoherent fluctuation defined by

By a similar analogy, the dispersion turbulent mass flux $\overline{\tilde{\rho}\tilde{\textbf{u}}}$ may be modelled as

In relation to (27) and (29), the total time-averaged turbulent energy is defined by

If we assume that the coherent and incoherent fluctuations have no correlation in time: $\overline{\tilde{f}g^{\prime\prime}}=0$ [see (31g)], the total turbulent energy and the coherent and incoherent fluctuation energies are related to each other as²²2The assumption of no correlation between the coherent and incoherent fluctuations in time is equivalent to the present formulation (27) with the decomposition of a field quantity $f^{\prime}$ into $\tilde{f}$ and $f^{\prime\prime}$ (30). In this sense, assumption of no-correlation between $\tilde{f}$ and $g^{\prime\prime}$ (or $f^{\prime\prime}$) is just the restatement of the adoption of the decomposition (30). Note that this does not deny the interaction between the coherent and incoherent fluctuations at all. As (40) shows, the production of the incoherent fluctuation energy results from the sink of the coherent fluctuation energy.

In the time–space double averaging procedure, the turbulent mass flux $\langle{\rho^{\prime}{\bf{u}}^{\prime}}\rangle(=\left\langle{\overline{\rho^{\prime}{\textbf{u}}^{\prime}}}\right\rangle)$ is divided into the contribution from the coherent fluctuations and the incoherent ones. Assuming that each contribution to the turbulent mass flux is expressed as (57) and (59), we can write the turbulent mass flux as

The first or unity-related part in the parentheses of each of these two terms gives the usual eddy diffusivity contribution to the turbulent mass flux. They represent the equilibrium effect in the turbulent mass flux and are written as

where $\kappa_{\rm{E}}=\tilde{\kappa}_{\rm{E}}+\kappa^{\prime\prime}_{\rm{E}}$ is the equilibrium turbulent eddy diffusivity. The rest parts of two terms of Eq. (62) or the $\overline{D}/\overline{D}t$-related parts in the parentheses represent the non-equilibrium effect. They are written as

where the time derivative part $\partial/\partial t$ of the Lagrangian derivative was dropped since the time derivatives of the time-averaged quantities are much smaller than the advective derivative part $\tilde{\bf{u}}\cdot\nabla$.³³3The effects of the time derivatives are different between the fast and slow time scales. To see this we introduce two-scale time variables for time $t$, the fast and slow variables are given by $\tau(=t)$ and $T(=\delta t)$, respectively, where $\delta(\ll 1)$ is a scale parameter. A time-averaged quantity $\overline{f}$ depends only on the slow variable $T$ as $\overline{f}=\overline{f}(T)$. Under this two-time analysis, the time derivative is written as $\partial/\partial t=\partial/\partial\tau+\delta\partial/\partial T$. Then the time derivative of the time-averaged quantity is expressed as $\partial\overline{f}/\partial t=(\partial/\partial\tau+\delta\partial\overline{f}/\partial T)\overline{f}=\delta\partial\overline{f}/\partial T$. Because of the smallness of $\delta$, the time derivative of the time-averaged quantity is eventually very small. On the other hand, the space derivative of the time-averaged quantity is not small since the scales associated with the space derivative is much larger than the counterparts of the time averaging. According to the relation (31), the double-averaging procedure obeys