Filtering, averaging, and scale dependency in homogeneous variable density turbulence Free

Turbulent flows are often characterized using statistics obtained from averaging or from filtering fields of interest. Averaging is used to obtain a statistical description of the flow, whereby a notional ensemble of realizations is averaged at a given point in space-time $(x,y,z,t)$⁠. By averaging the Navier–Stokes equations, in the statistical description one obtains a set of governing equations for primitive variables. These Reynolds-averaged Navier–Stokes (RANS) equations have additional terms that lead to a closure problem, and unclosed terms are modeled with varying levels of complexity. When using the filtering approach, on the other hand, the Navier–Stokes equations are filtered, leading also to a closure problem, where models are used to represent the effects of small, unresolved scales on the resolved fields. When the filter size is such that the resolved fields represent the largest eddies of the flow, this approach to modeling turbulence is referred to as large eddy simulations (LES), in contrast to RANS modeling where all scales of the flow are modeled. For more details on RANS and LES modeling, refer to Ref. 1. Furthermore, a broad range of modeling approaches exists in which varying levels of resolution of scales, between LES and RANS, are addressed.² These approaches are sometimes referred to as scale resolving simulations (e.g., see the review in Ref. 3 and references therein), and, for complex, realistic flows, provide a more accurate representation than RANS models do, while being less computationally expensive than LES. Some scale resolving simulation strategies, referred to as hybrid models, use a weighted combination of LES subclosures and RANS models. For more information on hybrid models, refer to the recent reviews in Refs. 4 and 5.

Most often, each choice of LES,^1,6 hybrid,^4,5,7,8 or RANS¹ model is closely associated with a level of resolution of the length scales that are relevant to the flow. However, the idea that different characterizations (namely, either filtering and LES, or averaging and RANS), and different models are appropriate for discrete, distinct levels of resolution of the flow can seem somewhat arbitrary. Indeed, in practice it is often times difficult to draw a clear line that delimits where different models are valid and others are not. For a given flow, and perhaps for a given type of flows, it is reasonable to expect that it would be possible to have a characterization that is valid at any scale relevant to the flow, and a model of the flow dynamics that can be applied at an arbitrary level of resolution. An example of this can be found in Refs. 9 and 10, where a self-adapting model was presented and used to simulate decaying homogeneous isotropic turbulence; the model was able to calculate total kinetic energy with the same fidelity regardless of whether the resolution corresponded to DNS, RANS, or something in between.

In this work, we investigate the formal relationships between filtering and averaging, and present a generalized, scale-resolving (SR) statistical description for homogeneous turbulent flows that is valid at an arbitrary length scale, filter width, or resolution, i.e., any resolution from the smallest to the largest length scales of the flow. Traditionally, variables are defined using central moments in the RANS statistical approach and generalized central moments¹¹ when considering filtering quantities, which are constructed by expressing the central moments as residuals. We express both (RANS and filtered quantities) as central moments of generalized fluctuating quantities, thus expressing both approaches in the same general form. Generalized central moments in the filtering approach are expressed as inner products of generalized fluctuating quantities, $q′(ξ,x)=q(ξ)−q¯(x)$⁠, which represent fluctuations of a field variable q at points ξ with respect to its filtered value at a point x. The SR statistical description of the flow is consistent with the Navier–Stokes (NS) equations. We derive realizability conditions for SR statistics, and we show that, for filter kernels that are positive definite, these realizability conditions are equivalent to the realizability conditions of their RANS counterparts and that the latter constitute a special case of the former in the limit of large filter widths or length scales.

The SR statistical description of flows will be useful to inform our understanding of turbulence flow processes, by quantifying how processes depend on scale and on scale interactions. This framework can be applied to in-homogeneous turbulent flows with directions of homogeneity by filtering along these directions, using a positive filtering kernel chosen such that it varies only along directions of homogeneity. Furthermore, this framework will also be useful for justifying the choice of scale resolving simulation strategies such as hybrid models and to inform the development of scale resolving simulation strategies for complex, realistic flows.

We illustrate these concepts by deriving SR statistics equivalent to those often used for RANS characterization and modeling of variable density (VD) turbulence using Favre averaging,¹² namely, the density-specific volume covariance, b, the turbulence mass flux velocity, a_i, the Favre averaged Reynolds stress tensor, $Tij$⁠, and the large scale kinetic energy, k_r, along with governing equations for each of these variables, and investigate variable density effects using this formulation. We diagnose these variables and the terms in the governing equations for $b,a1,kr$ using data from recent DNS of homogeneous variable density turbulence (HVDT).¹³ We discuss how some terms in the governing equations are related to length scales. For example: for length scales that are larger than the vertical integral length scale, the variables in the SR statistics, as well as their governing equations, are fully represented by the RANS description; the rate of transfer of energy between resolved and unresolved kinetic energy, in a volume integrated sense, peaks at length scales of the order of the horizontal Taylor micro-scale.

This work is consistent with some aspects and concepts of hybrid modeling and can be used to further the development of models for scale resolving simulations. The results strongly suggest that the dynamics and processes relevant to the turbulence physics in HVDT transition smoothly, as a function of length scale, from the NS limit to the RANS limit. The dynamical processes represented by the terms in the balance equations of the SR variables that we diagnose for HVDT, $b,a1,kr$⁠, are all trivially zero in the NS limit, correspond to the RANS balance for this flow in the RANS limit, where some are active and some are not, and are all active at intermediate length scales. For example, in HVDT, only destruction is active in the RANS governing equation for b at scales approximately equal to or larger than the integral length scale, while at intermediate length scales and scales below the integral length scale for this flow, there is a balance between production, redistribution, destruction, and transport. From the perspective of modeling, this work supports the notion of a generalized, length-scale adaptive model in terms of the SR variables, that converges to DNS at high resolutions, and to classical RANS statistics at coarse resolutions. A model that relies on computing SR variables in terms of RANS statistics alone, for example by scaling the RANS statistics,⁷ would not be able to capture the full SR dynamics.

We begin by recalling the equations used for filtering and averaging in Sec. II, followed by a derivation of the realizability conditions for the filtered variables in Sec. II A. The flow that we use for diagnostics is described in Sec. III, where we present the governing equations for HVDT, and we derive the equations for the SR variables. Then, in Sec. IV, we present these diagnostics. Finally, in Sec. V, we provide a summary and discussion on the merits of this work for investigating physics underlying turbulence at different scales and for model development.

The Reynolds-averaged Navier–Stokes (RANS) method, also commonly referred to as the statistical approach, is used to compute flow statistics for diagnostic analysis of DNS and experiments of turbulent flows and for turbulence modeling. In this approach, ensemble averages are used to calculate central moments in terms of fluctuating quantities, where the latter are defined as departures from means, e.g., $q=⟨q⟩+q′$⁠, or using Favre, or density (ρ) weighted averages, $q=⟨ρq⟩/⟨ρ⟩+q″$⁠, for some variable q. Invoking ergodicity, such averages can be calculated along space-time directions of homogeneity in the flow. Here, we consider spatial homogeneity only and indicate the volume average of a spatially and temporally varying quantity $q(x,y,z,t)$ by angle brackets, such that

with $⟨q⟩$ retaining time dependency. When it is necessary to make the distinction between RANS quantities and filtered quantities, we use non-italicized roman subscript “$e$” to denote RANS statistical quantities defined using central moments of fluctuating quantities. Using the statistical approach for variable density turbulence (VDT) flows, several statistical quantities (referred to as the BHR statistics¹²) have been identified for playing an important role in the dynamics of the mean flow, and subsequently used for RANS modeling of VDT flows.^14–16 We will now introduce these quantities, and we will discuss them throughout this paper.

The Reynolds stress tensor

where ρ is density and u_i is velocity in direction x_i, plays an important role in modulating the exchange of momentum and kinetic energy between the mean and fluctuating portions of the flow.¹⁷ The turbulent mass flux velocity

has the effect of moderating the production of Reynolds stresses $Rij$ and turbulent kinetic energy.¹⁷ The density-specific volume covariance

where v is the specific volume, is a measure of mixing in VDT, and affects the production of $aei$⁠.¹⁷ 

In the filtering approach to modeling turbulence, variables are filtered in space using

where $G(ξ,x;σ)$ is a prescribed filtering kernel with parametric dependence on a filter width $w=σ$⁠. In general, the filtering kernel can be a function of space-time, but here we restrict it to spatial filters. Favre filtered quantities, commonly used in variable density turbulence, are defined as

Dynamical variables obtained through filtering are often times defined by analogy to the residual form of the RANS quantities,¹¹ e.g., as in (2). This way, in the context of scale resolving filtered flows, we define the density-specific volume covariance, the turbulent mass-flux velocity, and the turbulent stress tensor as

and

respectively.

Generalized central moments $φ(ui,uj)=uiuj¯−u¯i u¯j$ of the flow field u_i can be used to obtain transport equations for filtered quantities. This way, the equations that result from filtering the Navier–Stokes (NS) equations are mathematically identical to their RANS counterparts—a property referred to as the averaging invariance of the filtered NS equations¹¹—with the variables in the equations having different interpretations. These developments have since been extensively used in developing methods used in large eddy simulations (LES).

When employing RANS statistics, the flow is decomposed into a mean component and a fluctuating component, while, as remarked in Ref. 11, when using filtering and generalized central moments, representations of the flow field at different levels of filtering are compared. In this paper, we will write the above definitions for b, a_i, and $Tij$ in Eqs. (7)–(9) in terms of generalized moments of fluctuating quantities with respect to filtered fields.

The two approaches of employing RANS statistics and using filtering, and the associated generalized central moments to represent turbulent flows are closely related. This becomes clear when the moments from both approaches are expressed in terms of inner products.

In Ref. 18, RANS statistical moments are expressed in terms of inner products to derive realizability conditions for turbulent stresses in incompressible flows. Similarly, in the filtering approach, generalized central moments can be expressed in terms of inner products of generalized fluctuating quantities, which in turn represent moments of generalized fluctuating quantities, and which, under special conditions, at large filtering length scales become the RANS central moments. Thus, it is important to note the connections between generalized central moments and inner products, between filtering and averaging, and the notion of generalized fluctuations. This will be the subject of this section, and in Sec. IV, we will illustrate these concepts by diagnosing some quantities of interest and their budgets, from DNS of HVDT flow. This flow is described in Sec. III.

To obtain the realizability conditions for the turbulent stress tensor, we express $Tij$ as an inner product. Starting from (9) and using (5) and (6), with the notation $(·̃)(x)$⁠, e.g., as in $uiuj̃(x)$ and $ũi(x)$⁠, to indicate functions of space $x=(x1,x2,x3)$⁠, we write

where

For positive ρ and G, it can be shown that (11) is an inner product,¹⁸ or more specifically a density-weighted inner product, which is positive semi-definite. It can also be interpreted as a density weighted convolution. We have used the definition of a generalized fluctuating quantity, namely,

which represents fluctuations of a field variable $ui(ξ)$ at points ξ, with respect to its filtered value $ũi(x)$ at a point x. Note that the generalized fluctuating quantity $u´´i(ξ,x)$ depends on a separation distance from x, namely, $ζ=ξ−x$⁠. In terms of ζ,

which is an alternative expression for the generalized fluctuations.

Similar to the discussion in Ref. 18, $u´´i(ξ,x)≠ui″(x)$⁠, since $u´´i(ξ,x)=ui(ξ)−ũi(x)$ in general is a two-point quantity that depends on ξ and x, while $ui″(x)=ui(x)−⟨ρ(x)ui(x)⟩/⟨ρ(x)⟩$ is a single point quantity that depends only on x. However, as the filtering length scale becomes large and approaches or exceeds some dynamically relevant integral length scale $L$⁠, the x dependency can be dropped so that the generalized fluctuating quantity becomes the fluctuating quantity defined in the statistical approach,

and, in this limit, it becomes a single-point quantity. This is true in general, for homogeneous as well as for in-homogeneous flows, if the filtering is performed along directions of homogeneity, i.e., the positive filtering kernel $G(ξ,x)$ is chosen such that it varies only along directions of homogeneity. This is in analogy to the calculation of RANS statistics by averaging along directions of homogeneity in space and time.

Similar to $u´´i(ξ,x)$⁠, we define generalized fluctuations based on non-density weighted filtered quantities,

and

Applying the integral Schwarz inequality to the product of the filtered density and filtered specific volume, using the definition of the filter, we can obtain a realizability condition for $b=ρ¯ v¯−1$⁠, as follows:

A similar derivation can be used in the RANS limit to show that $be≥0$⁠.¹⁹ Using a similar approach as for the stress tensor, it can be shown that

Now, we note that $ai=ũi−u¯i$ can also be written as an inner product

Alternatively,

As before, when the filtering length scale is large, similar to a dominating length scale, $w∼L$⁠, the generalized density fluctuation becomes the RANS density fluctuation $ρ´→ρ−⟨ρ⟩=ρ′$⁠, and a_i becomes the RANS quantity used in the statistical approach,¹² $ai→aei=⟨ρ′ui′⟩/ρ=⟨ρ′ui″⟩/ρ$⁠.

With this, we can write a realizability condition for $aα$ in terms of b and $Tαα$ (where double Greek letter indices imply no summation), using the Schwartz integral inequality as follows:

which leads to

Again, as the filter length scale increases and becomes comparable to a relevant integral length scale, $w∼L$⁠, the filtering operation converges to a volume average; in this limit, the relationship above becomes $aeα2≤beRij$⁠, the realizability condition in the statistical approach.¹² 

The above realizability conditions are true only for positive kernels, such as the Gaussian filter kernel or the box filter kernel. Negative kernels, while mathematically adequate, and while sometimes desirable for validation of LES (e.g., Ref. 20), will yield different realizability conditions and do not preserve scalar bounds. For example, the sharp spectral filter is non-local, and its kernel oscillates around zero in physical space. As a result, a sharp spectral filter can lead to $ρ¯≤ρ1$ in VDT flows in which two pure fluids with different densities $ρ1≤ρ2$ mix, leading to values of b that violate realizability conditions. Furthermore, in flows with moderate to large Atwood numbers, this can lead to $ρ¯≤0$⁠, and to numerical issues and artifacts in the definition of Favre filtered quantities using (6) when $ρ¯$ is small. The realizability conditions that we present above are most attractive here due to (i) their physical interpretations, e.g., positive kinetic energy $Tii/2$⁠, positive densities $ρ¯$ and positive b, and (ii) because they converge to the realizability conditions for their counterparts in the RANS statistical description of the flow. For these reasons, we limit ourselves to filtering with positive kernels, and we use a Gaussian filter, with filter kernels in physical and spectral spaces given by

and

respectively. Note that the variance of the Gaussian filter commonly used in LES (⁠$σL2$⁠) is 24 × the variance of the Gaussian filter we use here, $σL2=24 σ2$⁠. For this filter, the filtered density remains bounded, i.e., $0≤ρ1≤ρ¯≤ρ2$⁠.

Inner products have several defining properties for real, two-point quantities f and g (e.g., Ref. 21). They are commutative in f, g,

distributive, or linear in the first argument

and positive definite

This way, the SR variables in (17), (19), (20), and (10), can be expressed as

in terms of the generalized fluctuations defined in (12), (15), and (16).

Realizability conditions are given by the properties of the inner product,²¹ namely,

In the limit as the filter width becomes small, $w→0$⁠, the filter approaches a delta function, and filtered quantities become close to the pointwise values of the underlying quantities so that the instantaneous, or Navier–Stokes flow fields are recovered, e.g.,

Since integration limits much larger than the filter width do not change the integral, the effective parts (i.e., the parts that contribute to inner products) of the generalized fluctuations, $u´´i(ξ,x), u´i(ξ,x),ρ´(ξ,x)$ and the generalized filtered quantities $Tij$⁠, a_i, and b all approach zero. Thus, we will refer to this limit as the Navier–Stokes limit (NS limit) or the Navier–Stokes description of the flow. We will refer to the statistical description corresponding to intermediate length scales or filter widths, where the generalized filtering statistical description $Tij$⁠, a_i, and b is nontrivial, as the scale-resolving (SR) description of the flow.

By filtering along directions of homogeneity, using a positive filtering kernel $G(ξ,x)$ chosen such that it varies only along directions of homogeneity, the SR framework can be used to obtain a statistical description of homogeneous as well as in-homogeneous turbulent flows. We can expect the behavior of the SR description of the flow to vary smoothly between the NS and RANS descriptions. At the largest scales, $Tij$⁠, a_i, and b become equal to the RANS statistical description, and they are non-zero, while in the NS limit they are zero. At the smallest dissipative and diffusive scales, the flow field is smooth and $Tij$⁠, a_i, and b in the SR description can therefore be expected to transition smoothly from zero to their RANS values as the length-scales increase. Similarly, the processes affecting the balances of $Tij$⁠, a_i, and b, e.g., production, dissipation, destruction, and transport of the quantities in the SR description of the flow, can be expected to vary smoothly between the two limits.

In what follows, we will systematically investigate the SR description, and the transitions between the NS and RANS descriptions of the flow, by diagnosing variables in the SR statistical description, $Tij$⁠, a_i, and b, using DNS of homogeneous variable density turbulence.

Buoyancy driven homogeneous variable density turbulence (HVDT) was first introduced in Refs. 22 and 23, and further developed and discussed in Refs. 13, 19, 24, and 25. HVDT is a canonical flow configuration in which two incompressible, compositionally different fluids with densities ρ₁ and ρ₂, with $ρ2>ρ1$⁠, are randomly distributed within an accelerated, triply periodic cube with side dimension $L=2π$⁠. Here, we consider flows in which the initial probability density function (PDF) of density is initially symmetrical, with $⟨ρ⟩=(ρ1+ρ2)/2$⁠, resembling a double-delta distribution with two peaks at densities $ρ=ρ1$ and $ρ=ρ2$⁠. The highly non-linear evolution of the flow can be divided into four distinct regimes^13,26 based on the sign of the time derivatives of the turbulent kinetic energy, where A and Fr are the Atwood and Froude numbers (defined below), as depicted in Fig. 1. The flow is initially quiescent, and at time t = 0, due to instability to buoyancy forces, the flow starts moving as available potential energy is converted to kinetic energy, which, at first, rapidly increases with time.^13,19 This first regime is dubbed the explosive growth regime. As time advances and the flow develops, it transitions to a turbulent state and, as a result, turbulent kinetic energy and the rates of mixing and dissipation increase. Following this stage, as a result of increasing turbulence and mixing, the density PDF is populated at intermediate densities (⁠$ρ1≤ρ≤ρ2$⁠). The lighter fluid has less inertia, so it is stirred more and mixes faster than the heavy fluid,^13,19,24,25 and the density PDF quickly becomes asymmetrical as the lighter fluid densities (⁠$ρ≤⟨ρ⟩$⁠) are populated more than the heavier fluid densities (⁠$ρ≤⟨ρ⟩$⁠)—these effects become more pronounced as A increases. As a result, buoyancy forces decrease and the rate of conversion from potential energy to kinetic energy peaks, as dissipation increases. The peak of kinetic energy production by buoyancy forces marks the beginning of the saturated growth regime, during which kinetic energy continues to grow at an increasingly slower rate, as turbulence develops, leading to more mixing and dissipation, and consequently less production. Eventually, the rate of dissipation overcomes the rate of production of kinetic energy, and the kinetic energy peaks. After this point, kinetic energy starts decaying rapidly during the fast decay regime and then more slowly during the gradual decay regime.

References 13 and 26 discuss how the identification of these regimes makes it possible to draw parallels between HVDT and more complex VDT flows, such as Rayleigh–Taylor instability (RTI),^27,28 RTI under variable-acceleration,^29–33 Richtmyer—Meshkov instability (RMI),^34,35 including reshock,^36–38 variable density mixing layers,³⁹ and variable density jets,⁴⁰ among others. Recent reviews of some of these flows and the relevant variable density (VD) processes involved may be found in Refs. 32, 36, and 37. For this reason, and since HVDT reaches much larger Reynolds numbers and is relatively simple to post-process and analyze thanks to spatial homogeneity, we will use it here to diagnose and investigate scale dependence of the dynamical processes in VD flows.

The flow is governed by the equations for conservation of mass and momentum

where ρ is density, u_i and g_i are velocity and the acceleration of gravity in the direction x_i, respectively, p is pressure, and the viscous stress tensor for the Newtonian fluids and strain rate tensor are given by

with δ_ij representing the Kronecker delta function. The divergence of the velocity field is not zero because of the effect that mixing of VD fluids has on the specific volume,³² 

There is a degree of freedom in the mean pressure gradient due to the triply periodic boundary conditions, which is constrained by requiring that the mean pressure gradient maximizes the production of the total kinetic energy from conversion of the available potential energy.^13,19 As a result, the mean pressure gradient is given in terms of fluctuating quantities $q′=q−⟨q⟩$ for a variable q, by

Note that due to homogeneity, the mean pressure gradient is constant in space, varying only in time. The definition of the mean pressure gradient also leads to $⟨ui⟩=0$ at all times during the flow evolution.

The total turbulent kinetic energy is defined as

and the total (available) potential energy^13,19 is given by

(no summation implied).

Relevant non-dimensional numbers are Atwood number A, Froude number Fr, computational Reynolds number Re₀, and Schmidt number Sc, defined as

where μ₀ is the reference dynamic viscosity, $ρ0=(ρ2+ρ1)/2$ is the reference density, and L₀, U₀, μ₀, and D₀ are the reference length, velocity, viscosity, and diffusion coefficient scales, respectively. All cases considered here have unity Fr and Sc numbers. The simulations analyzed here represent a subset of the cases presented in Ref. 13: the computational Reynolds numbers are $Re0=104$ and $Re0=1563$ for the low and high Atwood number cases, A = 0.05 and A = 0.75, respectively. Following Refs. 13 and 19, the results are further normalized using the velocity and times scales, $Ur=A/Fr2$ and $tr=Fr2/A$⁠, so that the kinetic energy scale is $kr=k*=Ur2=A/Fr2$⁠.

We obtain the governing, or transport, equations for the generalized statistics in the scale resolving description of the flow in the same way as for the RANS statistical description of the flow. Applying the filtering operations defined in (5) and (6) on the NS equations (36) and (37) results in

Transport equations for the filtered density-specific volume covariance b, turbulent mass-flux velocity a_i and turbulent stresses $Tij$ can be obtained following the same procedure used to derive the transport equations for the RANS BHR statistics¹² described in Sec. II, by applying the filtering operations (5) and (6) on the NS equations (36) and (37), and using the definitions (7)–(9). Alternatively, replacing central moments with their corresponding generalized central moments,¹¹ the transport equations for the unclosed RANS variables^12,14,15 can be converted to equations for the filtered quantities, using the following rules. The generalized central moments of relevance here are

where the arrows indicate the corresponding central moments in the RANS statistical formulation. Either way, the resulting transport equations are

and

The sub-scale kinetic energy is related to the turbulence stress tensor by

Here, we use the term sub-scale kinetic energy for k_s, but in the context of filtering it can be referred to as sub-filter kinetic energy, and in the context of LES it can also be referred to as unresolved kinetic energy. Complementing the sub-scale kinetic energy is the scale-resolved, or resolved, kinetic energy,

such that the sum of the two is the total kinetic energy

It is useful to investigate the sub-scale kinetic energy equation, for its physical significance, its ties to the stress tensor, and also for its role in modeling. The transport equation for the sub-scale kinetic energy is given by, following the form used in Ref. 14,

For completeness, the Favre averaged scale-resolved kinetic energy is

We decompose the pressure gradient using the RANS definition, as it is used in the DNS, as described above. As a result, here we use

The average molecular viscous stress tensor is given by

and the molecular dissipation is

where the resolved strain is given by

and μ is the dynamic viscosity. We define the kinetic energy transfer between resolved and sub-filter scale kinetic energies as

From Eq. (56), the sub-scale kinetic energy k_s transitions to 0 in the NS limit, and to the finite RANS quantity in the RANS limit, as $Tij$ does. In the NS limit, the Favre averaged velocity $ũi$ converges to the instantaneous velocity $ũi→ui$⁠, and $uiuj̃→uiuj$⁠, so the scale-resolved kinetic energy k_r transitions to the total kinetic energy in the NS limit, $kr→uiuj/2$⁠. In the RANS limit, k_r converges to the RANS mean kinetic energy.

As in Sec. II B, for smooth flows, we can expect the governing equations for the scale resolved statistics b, a_i, $Tij$⁠, and k_s in Eqs. (53)–(55) and (59) to have similar transitions to NS and RANS limits as the variables themselves. As a result, we expect the governing equations for the SR variables to transition to zero in the NS limit and to the governing equations for their RANS counterparts in the RANS limit. The governing equation for the scale-resolved kinetic energy k_r transitions to the governing equation for the total kinetic energy in the NS limit and to the governing equation for the RANS mean kinetic energy in the RANS limit.

We now investigate the scale-resolving generalized statistics using direct numerical simulations (DNS) of homogeneous variable density turbulence from Ref. 13. The HVDT DNS at the times indicated in Table I is filtered using a Gaussian kernel as defined in Eq. (23). The filter width w is given by

where $f0=1/π$ and $f1=512$ are parameters used to control the lower and upper bounds of w. We perform diagnostics at N_w = 15 filter widths normalized by the box size $L=2π$⁠, varying between a fraction of the grid size, where i = 1 and $w/L=(Δx/π)/L=3.1×10−4$⁠, and half the box size, where $i=Nw=15$ and w = 1/2, such that $(Δx/π)/L≤w/L≤1/2$⁠, as listed in Table II, as this range is observed to be large enough to contain the transition of the SR quantities between the NS and the RANS limits.

The DNS for this flow have many degrees of freedom, as the spatial resolution is $N3=10243$⁠, and there are several dynamical variables of interest. To simplify the analysis, a number of diagnostics are used to investigate scale dependence of the VDT statistics presented in Sec. II and of the budgets in the governing equations for some of these statistics. Since the flow is homogeneous, RANS statistics and, in general, volume averages have no spatial variability. For this reason, we will first investigate volume-averaged SR statistics. However, scale resolving statistics do vary in space, and this will be investigated too by looking at probability density function distributions as a function of filter width.

The evolution of the kinetic energy $Rii/2$ and kinetic energy production are shown in Fig. 1. Throughout this section, we will be looking at diagnostics from HVDT DNS with A = 0.05 and A = 0.75 at the four times $t/tr$ (with $tr=Fr2/A$⁠) listed in Table I, corresponding to the four regimes in HVDT, plotted in Fig. 1. The simulations considered here have Fr = 1. These four times correspond to (i) when the kinetic energy production peaks, at the end of the explosive growth regime, (ii) when the kinetic energy peaks, at the end of the saturated growth regime, (iii) the end of the fast decay regime when the net rate of decay of kinetic energy starts decreasing, and (iv) at a time during the gradual decay regime.

Volume integrated SR variables are plotted, normalized by their RANS statistics counterparts, in Fig. 2. For the small Atwood number A = 0.05 snapshot at the end of the explosive growth period, the first time shown, the statistics collapse reasonably well to a single curve. However, at later times this is no longer the case for either A = 0.05 or A = 0.75 cases, and the spread between the curves increases until the rapid decay regime, after which the spread seems to either stabilize or slightly decrease. The horizontal components of the turbulent stress tensor $T22$ and $T33$ have the same scaling. However, the vertical component $T11$ and the horizontal components $T22,T33$ of the stress tensor, the vertical component a₁, and b, each have different length scalings, and these scalings vary in time.

We compare the scaling of the SR variables to smallest Taylor micro-scale, which for this flow corresponds to the horizontal Taylor micro-scale,

and to the largest integral length scale, namely, based on the vertical velocity,

which are shown as vertical lines in Fig. 2. The volume integrated SR variables converge to their RANS values when the filter width is comparable to the vertical integral length scale, $w≈Lv$⁠, which is about an order of magnitude smaller than the box size for these DNS. The NS values are reached at length scales that are at least one order of magnitude smaller than λ_h. In the low Atwood number simulation, the NS limit is reached at larger length scales at early times, and at smaller length scales at later times, reflecting the population of smaller length scales as the turbulence develops in time. However, in the large Atwood number simulation, the NS limit is reached at more or less the same length scale for the first three times, and at slightly larger length scales for the last snapshot. Note that the volume integrated density $⟨ρ¯⟩$ is not shown in Fig. 2, as $⟨ρ¯⟩=⟨ρ⟩=(ρ1+ρ2)/2$ is constant throughout the flow.

At any given time, at intermediate filter widths, the SR statistical quantities have large variability in space. To quantify this, we compute the probability density function of b, a₁, $T11$⁠, and $T12$⁠, normalized by their RANS counterparts, shown in Fig. 3 for A = 0.75 at the time when $Rii$ peaks. The realizability conditions (33) and (32) for b and $Tij$⁠, respectively, are satisfied by the Gaussian filter we use here. The yellow line corresponding to i = 1 shows that at the NS limit, where the filter size is a fraction of a grid cell, $w/L=(Δx/π)/L=3.1×10−4$⁠, all quantities are zero everywhere in the domain, and the PDFs correspond to a delta function centered at 0. As the filter width increases, the range of values in the PDFs broadens until the filter width at which the widest PDF is observed is reached, namely, i = 9, $w/L=2.1×10−2$⁠, for b, where the filter width is just around the horizontal Taylor micro-scale λ_h (Fig. 2), i = 6, $w/L=4.3×10−3$ for a₁ and $T11$⁠, and i = 5, $w/L=2.6×10−3$ for $T12$⁠. At the broadest shape of the PDFs, the variability is such that values of about 3, 4, and 6 times the RANS values are observed in b, a₁, and $T11/R11$⁠, respectively, while for $T12/R12$⁠, this variability is much larger, and values of up to 100 times the RANS values are observed. The off diagonal stress $R12$ is two orders of magnitude smaller than $R11$ (not shown); however, the magnitude of the largest fluctuations in $T12$ and in $T11$ is about the same order of magnitude, resulting in the large normalized $T12$ values seen in Fig. 3. Since $R12$ is associated with shear stress production, this indicates that, while locally the shear stress can be the same order as the normal stress, overall there is no bias toward a certain direction, and negative and positive $T12$ values cancel out in the volume average. This is expected for a buoyancy driven flow, with no mean shear that the volume average of $T12$ is a small number that results from the sum of large numbers. Subsequently, as w/L continues to increase, the PDFs become narrower until eventually they again become a delta function at the RANS limit, $w/L=1/2$⁠. In the RANS limit, illustrated by the dark blue lines at i = 15 corresponding to $w/L=1/2$⁠, the SR quantities have no spatial variability as they reach their RANS values, where $b/be=1, a1/ae1=1, T11/R11=1$ and $T12/R12=1$⁠. Similar behavior is observed at other times as well as for A = 0.05.