A Nonlocal Spectral Transfer Model and New Scaling Law for Scalar Turbulence

Time-evolution for the spectrum of a conserved scalar is governed by (see e.g., (Pope, 2001; Hill, 1978))

where $\mathcal{P}(\boldsymbol{k},t)$ denotes the spectral content for the large-scale production rate of the scalar variance, and $T(\boldsymbol{k},t)$ is the scalar spectral transfer function. The unknown nature of the $T(\boldsymbol{k},t)$ causes a closure problem in (6); thus, a proper modeling for the spectral transfer function is required. $T(\boldsymbol{k},t)$ could be defined as the rate of spectral flux function, $F(\boldsymbol{k},t)$, per unit wavenumber as (see e.g., (Corrsin, 1964; Hill, 1978))

By integrating (7) over the 3-D spectral domain, the spectral flux function is obtained.

As a well-known assumption, $\mathcal{P}(\boldsymbol{k},t)$ mainly contributes to the energy-containing scales directly, while it is obvious that $\boldsymbol{k}^{2}\,E_{\phi}(\boldsymbol{k},t)$ is mainly considerable in the small scales of the turbulent cascade where diffusion is the dominant transport mechanism (Pope, 2001). Therefore, integrating (6) over the wavenumber space yields the following:

In the statistically stationary state, the second and third integrals in (2.1) are approximately zero (Pope, 2001; Hill, 1978). In other words, within the inertial-convective subrange $\partial F(\boldsymbol{k})/\partial\boldsymbol{k}=0$ by the constant cascade assumption, and for the diffusive subrange $\partial F(\boldsymbol{k})/\partial\boldsymbol{k}=-2\,\mathcal{D}\,\boldsymbol{k}^{2}\,E_{\phi}(\boldsymbol{k})$. As a result, for the wavenumbers in the inertial-convective subrange $F(\boldsymbol{k})=\chi$, integrating with respect to $\boldsymbol{k}$ and employing (5). Subsequently, (2.1) is rewritten as:

This picture motivates the concept of modeling for the turbulent cascade process, here specifically for the case of scalar transport. This approach was initially introduced by Onsager in (Onsager, 1945, 1949), and later was generalized by (Corrsin, 1964) to the cascade mechanism of other systems such as passive scalars. In the cascade transfer, assuming the geometric doubling at each wavenumber step would approximate the step length as $\Delta\boldsymbol{k}\approx\boldsymbol{k}$. Therefore, the spectral flux function could be represented in the following form:

where $\tau(\boldsymbol{k})$ is the time-scale associated with the step at wavenumber $\boldsymbol{k}$ (Corrsin, 1964). Within the inertial-convective subrange, choosing $\tau(\boldsymbol{k})$ to be the local time-scale associated with the eddies of size $\ell=\boldsymbol{k}^{-1}$, reads as

In (11), $E(\boldsymbol{k})$ is the spectrum of the turbulent kinetic energy. Thus, according to the well-known Kolmogorov’s scaling for the inertial subrange, $E(\boldsymbol{k})\sim\varepsilon^{2/3}\,\boldsymbol{k}^{-5/3}$, where $\varepsilon$ is the dissipation rate of TKE.

Plugging (11) into (10), $F(\boldsymbol{k})=\chi\approx\boldsymbol{k}^{5/2}\,E(\boldsymbol{k})^{1/2}\,E_{\phi}(\boldsymbol{k})$, and according to the scaling of TKE spectrum, the scalar spectrum scales as

Direct numerical simulation of the scalar turbulence with a uniform imposed mean gradient, $\boldsymbol{G}=(0,1,0)$, advected on statistically stationary homogeneous isotropic turbulent (HIT) flow, provides a proper database to examine the scaling law in (12). In this study, a well-resolved DNS at the Taylor-scale Reynolds number $Re_{\lambda}=240$ for the case with $Sc=1$ is obtained. Resolving over sufficiently large time period in the statistically stationary state provides a rich sample space to obtain the ensemble-averaged spectra for the TKE and scalar. Figure 1 illustrates these time-averaged spectra over approximately 15 large-eddy turnover times, $\tau_{ET}$. Although the well-known Kolmogorov scaling for the TKE is achieved as shown in Figure 1a, Figure 1b reveals that the scalar spectrum does not obey the scaling law given in (12).

In Corrsin’s generalization to the Onsager’s cascade model, $\partial F/\partial\boldsymbol{k}$ is considered as the rate of gain or loss of the spectral content per unit wavenumber (Corrsin, 1964). Moreover, this generalized model could be applied to

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  non-conservative systems,

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  systems with different characteristic time-scales,

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  systems with different cascading mechanisms.

In fact, one can realize that the scaling given in (12) was obtained based upon this generalization. However, in the case of scalar spectrum, a main assumption that might be questioned is small-scale isotropy. Recently, studies showed that effects of large-scale anisotropic forcing do not vanish at the small-scales of turbulent transport of passive scalars. Moreover, these small-scale anisotropic fluctuations are identified to be highly intermittent due to intensified presence of nonlocal interactions in passive scalar turbulence. Anomalous scaling of high-order scalar structure functions is a clear and well-known experimental evidence supporting this significant deviation from local isotropy at small-scales of transport. In fact these effects are available in the inertial-convective subrange. Accordingly, we also observed that the scaling law for the scalar spectrum has a notable discrepancy with the ensemble-averaged spectrum obtained from DNS.

Based on the Corrsin’s generalization, we propose that the effects of the anisotropy could be effectively modeled in the spectral transfer function when the nonlocality in the cascade of scalar variance is properly considered. Inclusion of an added power-law behavior into the eddies of size $\ell$ mathematically enables modeling of the long-range interactions (nonlocality), and would return a modification in the local time-scale given in (11). As a result, we propose

where $\mathcal{C}_{\alpha}$ is a non-negative model coefficient, and $\mathcal{C}_{\alpha}=0$ yields the limit case $\ell=\boldsymbol{k}^{-1}$. Consequently, the modified time-scale is derived as

Plugging this nonlocal time-scale into (10) yields the following modified scaling law

Testing this modified scaling law on the time-averaged scalar spectrum shows a promising agreement through proper parameterization of $\alpha$ and $\mathcal{C_{\alpha}}$. Figure 2 illustrates that with $\alpha=0.65$ and $\mathcal{C}_{\alpha}\approx 3.9$ the universal scaling observed in the TKE spectrum is achieved for the scalar spectrum with respect to (15).

The multi-scale nature of the turbulent cascade process implies that the nonlocal transport effects induced by the large-scale anisotropy are fundamentally connected to the small-scales of motion through inter-scale interactions. Here, the inertial-convective subrange essentially plays the role of a meso-scale region for the turbulent cascade, where the presence of these nonlocal inter-scale interactions are highly pronounced. In fact, several fundamental studies focused on these nonlacal interactions and tried to unravel their nature by triad and tetrad dynamic models in the spectral domain (Chertkov et al., 1995, 1999; Waleffe, 1992; Cheung & Zaki, 2014). Therefore, a modification to the dissipation model (5) (initiated from the small-scale isotropy hypothesis) would compliment the nonlocal effects observed in larger scales of turbulent cascade. Subsequently, the total dissipation ($\chi_{T}$) is revised into

Defining $\mathcal{D}_{\alpha}:=\mathcal{D}\,\mathcal{C}_{\alpha}$, $\chi_{\alpha}$ characterizes the essence of having an underlying nonlocal diffusion operator in the advection-diffusion equation governing the turbulent transport of passive scalar. From a mathematical point of view, $\chi_{\alpha}$ directly stems from a fractional-order Laplacian operator, $(-\Delta)^{\alpha}(\cdot)$, acting on the scalar concentration; thus, the modified transport model reads as

A standard pseudo-spectral scheme based on the Fourier-collocation method is utilized to discretize and resolve the NS and AD equations. The triply periodic computational domain $\boldsymbol{\Omega}=[0,2\pi]^{3}$ is discretized in space by a uniform grid with $520^{3}$ grid points. A fourth-order Runge-Kutta (RK4) scheme is employed to perform the time-integration with a fixed $\Delta t=8\times 10^{-4}$, while the $\mathrm{CFL}<1$ condition is always checked; therefore, the numerical stability is ensured. In this study, we select a fully developed HIT field at $Re_{\lambda}=240$ as the initial state of our computational experiment, and investigate the development of the passive scalar concentration under the effect of a large-scale uniform imposed mean-gradient, $\boldsymbol{G}=(0,1,0)$. Concentration field, $\phi(\boldsymbol{x})$, is initiated from zero and is resolved for approximately 30 large-eddy turnover times under the standard model (3), and the introduced nonlocal model (17), while $Sc=1$. More detailed discussions about the numerical method, computational approach for the simulations, and the flow characteristics of the utilized HIT data have been presented in (Akhavan-Safaei & Zayernouri, 2020).

Transport of the scalar variance provides an important information about the evolution of the turbulence intensity. In fact, computational fluid dynamics approach make it possible to identify and keep track of the records of different influential mechanisms obtained from the mathematical modeling of the physics. It is well known that the multiplying both sides of the AD equation would yield the time-evolution equation for the turbulent intensity, $\phi^{2}$. Therefore, applying that to equation (17), after using the incompressibility condition and chain rule for the spatial derivatives, one can derive that

where $\boldsymbol{\mathcal{R}}(-\Delta)^{\alpha-1/2}(\cdot)$ denotes the fractional-order gradient obtained from the Riesz transform (Samiee et al., 2020; Akhavan-Safaei et al., 2021; Samiee et al., 2021). Due to the homogeneity of the scalar fluctuations, averaging over the spatial domain is equivalent to the ensemble-averaging operation $\langle\,\cdot\,\rangle$ (Pope, 2001). Thus, applying this averaging operation to (3.1) and considering that homogeneity of the fluctuating fields induces $\langle\nabla\cdot\,(\cdot)\rangle=\nabla\cdot\,(\langle\,\cdot\,\rangle)=0$, the evolution of scalar variance $\langle\phi^{2}\rangle$ is obtained as follows:

In (19), the rate of scalar variance is composed of a balance between the turbulent advection effects ($\mathcal{T}$), production by the imposed mean-gradient ($\mathcal{P}$), molecular diffusion ($\chi$), and the nonlocal diffusion ($\chi_{\alpha}$). It is clear that for the standard scalar transport model in which $\mathcal{D}_{\alpha}=0$, the nonlocal diffusion is consequently zero. According to the simulation considerations described in Section 2.3, we collect the records of the contributing terms in the right-hand side of (19), and Figure 3 illustrates these time records for the standard and nonlocal models. Moreover, during both of the simulations we compute the rate of the scalar variance, $\frac{d}{dt}\langle\phi^{2}\rangle$, using a forward-Euler finite difference scheme, and compare it with the record of the right-hand side of equation (19) constructed from the summation of the collected records. For both of the simulations an excellent match between these two computed quantities is observed during the entire simulation time as shown in Figure 3. In the current work, we are focused to examine the statistical behavior of the developed nonlocal model at the “turbulence equilibrium” state. In this context, equilibrium is interpreted when the rate of scalar variance is considered when the following condition statistically holds:

Figure 4 shows the displays the record of this quantity for standard and nonlocal models and we notice that after approximately two large-eddy turnover times from resolving the scalar concentration, $(\mathcal{P}+\mathcal{T})/(\chi+\chi_{\alpha})$ starts to fluctuate around 1. In order to make sure that the transient numerical effects are well past, we continue to simulate up to $t/\tau_{ET}=10$, and consider the rest of simulation statistics in the fully developed turbulent equilibrium state. Therefore, the time-averaging operations in our study is performed on a sample space over $10\leq t/\tau_{ET}\leq 30$. Accordingly, we can compute the time-averaged values of the contributing terms in evolution of scalar variance given in (19) as they are reported in Table 1. Comparing the time-averaged values of $\mathcal{P}$ from both of the models reveals that the nonlocal model approximately includes 2.5% more production rate of the scalar variance by the large-scale scalar mean-gradient. A reasonable interpretation for this observation is that once the nonlocal transfer of the scalar variance transfer in the cascade process is correctly modeled; therefore, this excessive 2.5% production rate is captured at the equilibrium state for scalar turbulence. In other words, devising a nonlocal turbulent dissipation model ($\chi_{\alpha}$) in the scalar variance cascade mechanism would enable a balance in the equilibrium state so that the nonlocal effects in turbulent transport originating from large-scale “anisotropy” source are better captured throughout the DNS.

Finally, we compute the time-averaged scalar variance spectrum obtained from the scalar field resolved by the nonlocal fractional-order model to examine the modified scaling law (15) a posteriori. Figure 5 depicts this spectrum and reveals that the scaling (15) seamlessly holds. It is worth emphasizing that the total turbulent dissipation is denoted by $\chi+\chi_{\alpha}$.

It is well-understood that statistics of the turbulence at the small-scales of the transport are represented through the central moments of the gradients of the fluctuating fields. Here, we are interested in discovering the small-scale statistics when the scalar field is sufficiently resolved with the proposed nonlocal scalar transport model. In fact, we seek to understand what would be the prediction of this model for the asymmetric and highly intermittent nature of passive scalar turbulence at the small scales. Therefore, we compute the skewness and flatness factors for the fluctuating concentration gradient, and due to the importance of the statistical behavior along the anisotropy direction $\parallel$, we focus on $\mathcal{S}_{\,\nabla_{\parallel}\phi}$ and $\mathcal{K}_{\,\nabla_{\parallel}\phi}$. Figure 6 illustrates the records of these two statistical quantities throughout the entire simulation for the standard and nonlocal models. Over the equilibrium state, explained in section 3.1, we obtain the mean values of the of these statistical quantities by time-averaging, and their values are reported in Table 2. These time-averaged values show that the nonlocal model yields the skewness factor 10% more than the standard model, and the flatness factor is approximately 7% higher in the nonlocal transport model. This evidently implies that an appropriate modeling of the nonlocal turbulent scalar transfer via the fractional-order model properly reflects the well-known statistical features of highly non-Gaussian behavior of the passive scalar turbulence reported in the literature (Warhaft, 2000; Sreenivasan, 2019).

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Structure functions of order $n$ for a turbulent field such as scalar concentration are defined as:

In (21), $\boldsymbol{r}=r\boldsymbol{e}$ where $r$ is the increment of spatial shift, and $\boldsymbol{e}$ denotes a unit vector along a direction of interest. In fact, the structure functions would provide the $n$th-order statistics of spatial increments in the fluctuating field, which are interesting metrics in studying the nonlocality. In this study, we are interested in analyzing the behavior of the second- and third order structure function of $\phi$ along the direction of large-scale anisotropy, i.e. $\boldsymbol{e}=(0,1,0)$, and regrading the size of the DNS grid, $r=2\,\eta$. Accordingly, Figure 7 shows the time-averaged (over the equilibrium turbulent region) structure functions of order 2 and 3 obtained from the simulations from standard and nonlocal scalar transport models. In Figure 7(a), one can observe that for $r>40\,\eta$ the nonlocal second-order structure function starts to exhibit higher values compared to the one computed from the simulations using standard model. For the third-order structure function values, the two models behave similarly up to $r/\eta=10$; however, after that the nonlocal model shows higher values within the spatial shift domain associated with the inertial-convective and integral-scale domain. It is apparent that the maximum value of the time-averaged $\langle(\delta_{r}\phi)^{3}\rangle$ in the nonlocal model is approximately 10 times higher than the standard model both occurring at $r/\eta\approx 200$.

As initially introduced in (Yaglom, 1949), mixed “velocity-scalar” third-order structure function is an importasnt two-point statistical quantity measuring the advective turbulent transport in passive scalars. In the presence of large-scale anisotropy (imposed mean scalar gradient), the longitudinal contribution to this mixed structure function plays the dominant role in the advective transport (Iyer & Yeung, 2014), and its functional form obtained as $-\langle\delta_{r}u_{L}(\delta_{r}\phi)^{2}\rangle$. Here, the subscript $L$ indicates the velocity component along the longitudinal direction with respect to spatial shift direction $\boldsymbol{r}$, where in our computational setup it would be $u_{2}$. Similar to the the second- and third-order scalar structure functions, we compute a time-averaged value for the $-\langle\delta_{r}u_{L}(\delta_{r}\phi)^{2}\rangle$ over the stationary time domain. Figure 8 shows that for the dissipative range ($r/\eta<6$) this structure function scales with $r^{3}$ in both standard and nonlocal models, while for almost the entire range of $6<r/\eta<200$ the mixed structure function obtained from the DNS with the nonlocal transport model scales with $r^{2}$. Unlike this universal-range scaling, one can observe that similar behavior is not necessarily seen in the $-\langle\delta_{r}u_{L}(\delta_{r}\phi)^{2}\rangle$ when the scalar field is resolved with the standard model. However, in the standard model, a scaling with $r$ could be identified within $20<r/\eta<60$. This comparison suggests that the considering the nonlocal effects in the turbulent cascade could result in emergence of more universal behavior in the two-point statistics of the advective transport, which inherently reveal high-order statistics of the nonlocality.