Transport and modeling of subgrid-scale turbulent kinetic energy in channel flows

Figures 1(a)–(d) show the mean velocity profiles at $\mathrm{Re}_{\tau}=180$ and $400$ for each grid resolution. Here and hereafter, values with a superscript “$+$” denote those normalized by $u_{\tau}$ and $\nu$. In addition to the models provided in Table 1, we have performed the simulations without any SGS model, which is denoted as “nomodel.” The SMM predicts reasonable mean velocity profiles for all grid resolutions and Reynolds numbers demonstrated in the present simulation cases. This result is consistent with the previous studies of SMM.Abe (2013); Ohtsuka and Abe (2013); Abe (2014); Kobayashi (2018); Inagaki and Kobayashi (2020) Moreover, in Fig. 1(c), the SMM of the LD case at $\mathrm{Re}_{\tau}=400$ provides almost the same result as the LR case. Hence, the domain size does not affect the statistics even in the LR. The EVM and DSM overestimate the mean velocity for the LR case at both Reynolds numbers. In contrast, for the MR, the EVM and DSM almost overlap each other and predict reasonable mean velocity profiles at both Reynolds numbers. For the MR at $\mathrm{Re}_{\tau}=400$, the nomodel provides a good prediction. However, as discussed later, the nomodel overestimates the turbulent kinetic energy. Hence, the grid resolution of the MR is insufficient as a DNS. Furthermore, the result that the DSM and EVM provide a worse prediction than the nomodel does not suggest the eddy viscosity is unnecessary. Because a large amount of turbulent energy is transferred from the GS to the SGS as shown later in Sec. III.3.2, the eddy viscosity is essential to expressing the energy transfer. Table 2 shows the bulk mean velocity $U_{\mathrm{b}}$ normalized by the DNS value $U_{\mathrm{b}}^{\mathrm{DNS}}$ for each simulation where $U_{\mathrm{b}}$ is defined by

Note that $U_{\mathrm{b}}^{\mathrm{DNS}}$ at $\mathrm{Re}_{\tau}=180$ and $400$ yields $15.8$ and $17.8$, respectively. in the present simulation. The errors of the prediction of SMM are within 4%, whereas those of other cases exceed approximately 10% for either the LR or MR.

The Reynolds shear stress is decomposed into the GS and SGS components as

where $f^{\prime}(=f-\langle f\rangle)$ denotes the fluctuation of $f$ around its mean value. The decomposition provided by Eq. (18) is unique, independent of the selected filter when the filter operation is imposed only in the directions in which the turbulence is statistically homogeneous and the average $\langle\cdot\rangle$ is taken over these directions. In our LESs, we implicitly impose the filter operation on the inhomogeneous or $y$ direction because the grid resolution in the $y$ direction is coarser than that of the reference DNSs. Thus, the decomposition provided by Eq. (18) is not necessarily unique. However, in this study, we assume that the decomposition of Eq. (18) is valid even in our LESs.

Figures 2(a)–(d) show the profiles of the GS and SGS Reynolds shear stresses at $\mathrm{Re}_{\tau}=400$ for each grid resolution. For the LR case, the EVM and DSM provide the large GS Reynolds stress compared with the fDNS. However, they overestimate the mean velocity profile because the SGS component is small. In contrast, the SMM predicts reasonable profiles for both the GS and SGS Reynolds shear stresses. As a qualitative trend, the SMM and EVM that employ the SGS energy transport equation model reproduce the large intensity of the SGS shear stress in the near-wall region observed in the fDNS with SCF. However, the DSM does not predict the large intensity in the near-wall region. As a result, the total Reynolds shear stress of the DSM is smaller than the turbulent shear stress of the nomodel despite the presence of effective viscosity. Hence, the DSM predicts a larger mean velocity than the nomodel. Even in the MR case, the DSM provides a small SGS shear stress in the near-wall region compared with the fDNS, SMM, and EVM. This result suggests that the dynamic procedure does not necessarily provide excellent performance in predicting the near-wall behavior of the eddy viscosity. Although the EVM provides a large SGS shear stress compared with the DSM in the near-wall region for the LR, the total Reynold shear stress is not enough large to predict the mean velocity profile. The SMM provides a smaller GS Reynolds shear stress than other models and fDNSs for the MR. The large SGS shear stress compensates for this underestimation. For the nomodel, the LR case provides a slightly smaller Reynolds shear stress than the MR, which leads to the increase of the mean velocity. This result indicates that the generation mechanism of the Reynolds shear stress becomes weak for the nomodel in the LR. AbeAbe (2019) and Inagaki and KobayashiInagaki and Kobayashi (2020) showed that the eddy-viscosity-based models are insufficient for reproducing the generation mechanism of the GS Reynolds shear stress in coarse grid cases.

Figures 3(a) and (b) show the profiles of the mean SGS energy $\langle k^{\mathrm{sgs}}\rangle$ at $\mathrm{Re}_{\tau}=180$ and $400$, respectively, for each grid resolution or filter length scale. For all cases, the profiles of the filtered DNS (fDNS) depend on the selected filter. The differences are relatively small at $\mathrm{Re}_{\tau}=400$ compared with those at $\mathrm{Re}_{\tau}=180$. For an a posteriori test or an LES run, one is not aware of the filter selected in the simulation. In other words, the filter operation $\overline{\cdot}$ is an implicit one. Hence, we cannot determine which filter is appropriate when we compare an LES and fDNS. As mentioned in Sec. II.2, the SGS energy $k^{\mathrm{sgs}}$ must be locally positive semi-definite in SGS energy transport equation models. Thus, the GF that guarantees the positive semi-definiteness seems to be appropriate. However, note that for the present analysis, the mean SGS energy $\langle k^{\mathrm{sgs}}\rangle$ is positive semi-definite in fDNS even when the SCF is adopted because it reads $\langle(\bm{u}-\overline{\bm{u}})^{2}\rangle$ in the SCF. As the eddy viscosity predicts the mean energy transfer rate and not the instantaneous one, it may be sufficient to predict the mean SGS energy in the concept of statistical evaluation of an SGS model.Meneveau (1994); Pope (2004); Moser, Haering, and Yalla (2021) The results of the SMM are close to those of the fDNS with SCF for the LR at both Reynolds numbers. However, for the MR, the results of SMM differ from both the SCF and GF results of the fDNS. Hence, we can confirm that the prediction of the SGS energy transport equation model with SMM is close to neither SCF nor GF. Here, we note again that the domain size does not affect the statistics even in the LR because the SMM of the LD at $\mathrm{Re}_{\tau}=400$ provides almost the same result as the LR. The results of the EVM are slightly lower than those of the SMM for the LR although the mean velocity is overestimated. Furthermore, for the MR, the EVM and SMM predict different profiles of the mean SGS energy even in the similar mean velocity. This result suggests that the SMM provides different turbulent fluctuation structures compared with the EVM. For the LR, the EVM predicts longer streak structures of streamwise velocity fluctuation than the SMM.Inagaki and Kobayashi (2020)

To avoid ambiguity due to the selected filter, it is useful to observe the total turbulent kinetic energy $K(=\langle\bm{u}{}^{\prime}{}^{2}\rangle/2)$ (see PopePope (2004)):

This decomposition is unique in the fDNS the same as the Reynolds stress (18) because the filter operation is imposed only in the $x$ and $z$ directions. Figures 4(a)–(d) show the profiles of the total turbulent kinetic energy $K$ at $\mathrm{Re}_{\tau}=180$ and $400$, respectively, for each grid resolution. At both Reynolds numbers, the EVM, SMM, and nomodel excessively overestimate the total turbulent energy. Furthermore, the DSM and nomodel depict only the GS component of turbulent energy because $k^{\mathrm{sgs}}$ is not determined or is absent in these cases. Even though the EVM and DSM employ the eddy viscosity, they overestimate the turbulent energy as large as the nomodel that is less dissipative. However, the mechanism of the overestimation observed in the EVM and DSM differs from that for the nomodel. We discuss this point in Appendix B. Overestimation of the turbulent energy is often observed in several EVMs even when the prediction of the mean velocity is reasonable. In contrast, the SMM predicts reasonable predictions of the total turbulent energy compared with the DNS for all cases. Again, the LD provides almost the same result as the LR for the SMM at $\mathrm{Re}_{\tau}=400$. Hence, we can interpret that the SMM adequately predicts the basic statistical properties of turbulent channel flows.

The ratio of the SGS energy to the total turbulent energy $\langle k^{\mathrm{sgs}}\rangle/K$ is a simple measure of turbulence resolution for exploring the tolerance of an adaptive LES.Pope (2004) The EVM provides reasonable mean velocity profiles for the MR at both $\mathrm{Re}_{\tau}=180$ and $400$. The peak value of the SGS energy for the EVM for the MR at $\mathrm{Re}_{\tau}=400$ is approximately $0.8$, whereas the peak value of the total turbulent energy for DNS is $4$. Therefore, the tolerance of the EVM can be evaluated as $\langle k^{\mathrm{sgs}}\rangle/K\leq 0.2$. In contrast, the SMM predicts the total turbulent energy well even when $\langle k^{\mathrm{sgs}}\rangle/K\simeq 0.5$ at the peak of turbulent energy for the LR at $\mathrm{Re}_{\tau}=400$. Hence, the SMM is more adaptive to grid resolutions than the EVM.

The energy transfer rate from the GS to the SGS field is a significant statistical index for evaluating the SGS models.Moser, Haering, and Yalla (2021) In the SGS energy transport equation, the production term $P^{\mathrm{sgs}}$ provided by Eq. (7a) represents the energy transfer. For fDNS and SGS energy transport equation models, we can quantitatively observe the contributions of the other terms in the SGS energy transport. Figure 5 shows the statistical average of the budget for the SGS energy transport equation provided by Eqs. (6) and (7a)–(7e) for several cases of the LR. For the SMM and EVM, the dissipation term is provided by Eq. (10a), and the sum of the turbulent and pressure diffusions is modeled using Eq. (10b) in terms of the gradient diffusion approximation. The latter is represented by the SGS diffusion in Fig. 5(d). Comparing Figs. 5(b), (c), and (d), the profile of the SGS diffusion is qualitatively similar to the sum of the turbulent and pressure diffusions. Hence, the gradient diffusion model can be the first approximation of the sum of the turbulent and pressure diffusions. At both Reynolds numbers, the diffusion terms are prominent only in the near-wall region, where $y^{+}<50$. In the region away from the wall where $y^{+}>50$, the production and dissipation terms are dominant, and they are almost balanced. In addition, for the SMM, the dissipation without the wall-correction term is dominant in $y^{+}>40$. These trends are observed in all other cases including the MR, although the figures are omitted. Hence, we can assume that the statistical average of the production term balances with that of the dissipation term in the SGS energy transport, except for the near-wall region. This result is slightly different from the budget for the total turbulent kinetic energy (see e.g. Lee and MoserLee and Moser (2015) and their DNS database). The turbulent diffusion term becomes dominant in the channel center region in the budget for the total turbulent kinetic energy. In contrast, the diffusion terms are negligible in the channel center region in the SGS energy transport. This result suggests the dominance of the energy cascade from the GS turbulent energy to the SGS energy. We will discuss this point later. Notably, this does not necessarily suggest that the production term balances with the dissipation locally in time and space. However, we expect that the modeling based on the instantaneous balance or local equilibrium between the production and dissipation terms can predict the statistical energy transfer rate from the GS to the SGS field.

Figure 6 shows the production term in the mean SGS energy transport equation at $\mathrm{Re}_{\tau}=180$ and $400$ for each grid resolution or filter scale. In contrast with the SGS energy shown in Fig. 3, the difference resulting from the choice of filter between the SCF and GF is relatively small for the mean production term. In particular, at $\mathrm{Re}_{\tau}=400$, the profiles of the fDNS with SCF and GF almost overlap in $y^{+}>50$ for each filter scale. Moreover, the large mean SGS energy does not necessarily correspond to the large energy transfer rate. For example, for the LR at both Reynolds numbers, the mean SGS energy of the SMM is slightly larger than that of the fDNS with the SCF in the region away from the wall, whereas the mean production term or energy transfer rate of the SMM is slightly smaller than that of the fDNS with the SCF. However, the SMM provides reasonable predictions of the mean production term in $y^{+}>50$ compared with the fDNS results. This result suggests that the SMM reasonably predicts the mean energy transfer from the GS to the SGS in the region away from the wall. The profiles of the mean production for the EVM and DSM are not significantly different from those of the SMM in $y^{+}>50$ even in the LR cases in which the mean velocity is overestimated. This result suggests that the mean production modeled by the eddy viscosity, as provided by Eq. (9), predicts a reasonable energy transfer rate compared with the fDNS. However, for LESs to predict the statistical properties of inhomogeneous turbulent flows, the mean SGS stress should be reproduced in addition to the energy transfer rate.Meneveau (1994); Moser, Haering, and Yalla (2021) Otherwise, the mean velocity is overestimated as in the LR cases of the EVM and DSM (Fig. 1). The SMM succeeds in predicting the mean velocity at least in two scenarios: One is the additional contribution of the extra anisotropic term $\tau^{\mathrm{eat}}_{ij}$ in Eq. (14), which enhances the mean SGS stress. The other is the recovery of the SGS stress–velocity gradient correlation in the GS Reynolds shear stress transport, which disappears in EVMs.Abe (2019); Inagaki and Kobayashi (2020)

To further investigate the SGS energy production, we consider the energy transfer pathway. The mean production term in the SGS energy transport equation is decomposed into two terms:

Here, $\xi^{\mathrm{SGS}}$ denotes the exchange rate of the turbulent energy between the GS and SGS because it appears in the transport equation for the GS turbulent energy in the opposite sign. In contrast, the second term on the right-hand side of Eq. (20) denotes the exchange rate of energy between the kinetic energy of the mean velocity and SGS energy. That is,

Because $\xi^{\mathrm{SGS}}$ denotes the turbulent energy exchange across the cutoff scale, it involves an energy cascade. When $-\xi^{\mathrm{SGS}}$ is positive and dominant compared with $\langle\tau^{\mathrm{sgs}}_{ij}\rangle S_{ij}$, we can infer that most of the SGS energy is transferred by the energy cascade. Figure 7 shows the ratio of $-\xi^{\mathrm{SGS}}$ to the mean production of the SGS energy $\langle P^{\mathrm{sgs}}\rangle$. For both the fDNSs and LESs, the ratio is small in the near-wall region where $y^{+}<50$ because both the mean velocity gradient $\partial U_{x}/\partial y$ and mean SGS shear stress $\langle\tau^{\mathrm{sgs}}_{xy}\rangle$ are large. For the SCF, a negative sign appears at $y^{+}=10$ for the LR at $\mathrm{Re}_{\tau}=180$ and $\mathrm{Re}_{\tau}=400$, which indicates a backward energy transfer or inverse cascade. The inverse cascade in the near-wall region has been discussed by several studies, e.g., Cimarelli et al.Cimarelli, Angelis, and Casciola (2013) and HambaHamba (2018). As it moves away from the wall, the ratio $-\xi^{\mathrm{SGS}}/\langle P^{\mathrm{sgs}}\rangle$ increases and reaches approximately 60%. The ratio is relatively small for the LR of the fDNS with GF at $\mathrm{Re}_{\tau}=180$. Generally, LR cases provide a smaller ratio than the MR, regardless of fDNS or LES. A simple interpretation is that the LR cases involve larger scales than the MR. Therefore, a large amount of SGS energy is transferred from the mean velocity even when the energy transfer is relatively local in scale. Consequently, the ratio $-\xi^{\mathrm{SGS}}/\langle P^{\mathrm{sgs}}\rangle$ for the LR becomes smaller than that for the MR. This scenario can also account for the GF cases providing a smaller ratio than the SCF. Owing to the smooth profile of the filter kernel of the GF around the cutoff scale, the SGS energy via the GF involves larger scales than that via the SCF. Therefore, the ratio of the SGS energy resulting from the mean velocity increases when the GF is employed instead of the SCF. Consequently, $-\xi^{\mathrm{SGS}}/\langle P^{\mathrm{sgs}}\rangle$ in the GF case decreases compared with that in the SCF.

Interestingly, even in the coarse grid or large filter scale of $\Delta x^{+}=105$ and $\Delta z^{+}=79$ at $\mathrm{Re}_{\tau}=400$ (LR), most of the SGS energy is transferred from the GS turbulent energy. This large ratio indicates that the energy cascade is responsible for most of the production mechanism of the SGS energy in the region away from the wall. This statement is consistent with Motoori and Goto,Motoori and Goto (2021) which elucidated that the energy cascade due to vortex stretching generates hierarchical vortex structures in the log layer of a high-Reynolds-number turbulent channel flow. The energy cascade is essential in most high-Reynolds-number turbulent flows. Hence, the dominance of the energy cascade in the production term suggests that the local balance between the production and dissipation in the SGS energy transport is also prominent in turbulent flows other than the channel flow. Furthermore, in most LESs, the production term is expressed simply by the eddy viscosity using Eq. (9). Therefore, we expect that the SGS energy production in terms of the eddy viscosity can be reasonable for expressing the energy transfer rate even in other turbulent flows.

In fDNS, the backscatter or energy transfer from the GS to the SGS frequently occurs.Piomelli et al. (1991); Aoyama et al. (2005) The backscatter is physically natural in a kinematic sense because it is observed around an elliptic Burgers vortex.Kobayashi (2018) The occurrence of backscatter is inconsistent with the local production–dissipation equilibrium assumption that is a basis of the Smagorinsky model. However, in the conventional SGS energy transport equation models, the backscatter is not possible because the production term is provided by the eddy viscosity as Eq. (9). Hence, the dynamics is different between the fDNS and LES with the SGS energy transport equation models. To investigate further details on the relation between production and dissipation in LES, we calculate the correlation between them. The correlation coefficient between two variables $f$ and $g$ is defined by

Figure 8 shows the correlation coefficient between the production and dissipation terms $\mathrm{cor}(P^{\mathrm{sgs}},\varepsilon^{\mathrm{sgs}})$. The correlation is low for the fDNS. That is, the GF cases provide approximately 40% and the SCF cases provide only 20% in $y^{+}>20$. The reason for the low correlation for the GF cases is probably because the production term can be negative, whereas the dissipation term is positive semi-definite. For the SCF, the dissipation term can be also negative. Thus, the SCF predicts lower correlations than the GF. In contrast, the LESs of the SMM and EVM provide a high correlation exceeding approximately 80% in an entire region regardless of the grid resolution and Reynolds number. Note that this high correlation is not a numerical artifact due to the small domain size or grid points because the LD at $\mathrm{Re}_{\tau}=400$ also provides a high correlation. Intuitively, we expected that the SGS energy transport equation models fairly account for the nonequilibrium effects owing to the convection and diffusion terms. However, as the present result suggests, we revealed that the amounts of nonequilibrium effects are essentially small in the SGS energy transport equation models. In other words, the production instantaneously or locally balances with the dissipation in the LESs. Figures 9(a) and (b) show the contour maps of the production and dissipation terms, respectively, in the SGS energy transport equation at $y^{+}=100$ for the SMM in the LD at $\mathrm{Re}_{\tau}=400$. The pattern of the production is almost the same as that of the dissipation for the SMM. Figures 9(c) and (d) show the production and dissipation terms, respectively, for the fDNS with GF at the same height and Reynolds number. For the fDNS with the GF, the contour map for the production differs from that of the dissipation in contrast with the SMM. Moreover, for the fDNS, the production exhibits the opposite color patches that depict the negative production or inverse cascade. In contrast, for the fDNS, the sign of the dissipation term is always positive because of the use of the GF. In addition, the dissipation in the fDNS exhibits finer structures than the production, which yields a low correlation.

The excessively high correlation results of the LESs compared with the fDNSs suggest that the SGS energy transport equation models do not reproduce the exact transport mechanism of the SGS energy. In other words, the conventional SGS energy transport equation models are not effective in representing the nonequilibrium effect due to the convection and diffusion terms. Moreover, the production term based on the eddy viscosity employed in SGS energy transport equation models cannot reproduce the backscatter. Nevertheless, the eddy-viscosity-based production quantitatively predicts the mean energy transfer rate from the GS to SGS, as discussed in Sec. III.3.2. In the context that the statistical property is significant in SGS modeling,Meneveau (1994); Pope (2004); Moser, Haering, and Yalla (2021) it may be sufficient to accurately predict the mean energy transfer rate. Thus, the production term based on the eddy viscosity provided by Eq. (9) can be a reasonable approximation for SGS modeling.

The high correlation between the production and dissipation terms observed in the LES of the SGS energy transport equation suggests that the SGS energy can be reduced to a local algebraic expression. For the SMM, in particular, the production term yields $2\nu^{\mathrm{sgs}}\overline{s}^{2}$ (9), and the dissipation is modeled by $C_{\varepsilon}(k^{\mathrm{sgs}})^{3/2}/\overline{\Delta}$ (10a) in the region away from the wall. The local balance between them with the eddy-viscosity expression (5) finally yields $k^{\mathrm{sgs}}=(2C_{\mathrm{sgs}}/C_{\varepsilon})\overline{\Delta}^{2}\overline{s}^{2}$ (12). Figure 10 shows the coefficient between the SGS energy $k^{\mathrm{sgs}}$ and the GS strain rate $\overline{s}^{2}$. For the fDNS, the correlation is not high in the entire region regardless of the selected filter or filter scale. One of the causes of this low correlation is the local imbalance between production and dissipation in the SGS energy transport equation, as observed in Figs. 8 and 9. In contrast, SMM and EVM provide a good correlation between $k^{\mathrm{sgs}}$ and $\overline{s}^{2}$. $\mathrm{cor}(k^{\mathrm{sgs}},\overline{s}^{2})$ for the SGS energy transport equation models is lower than the correlation between the production and dissipation $\mathrm{cor}(P^{\mathrm{sgs}},\varepsilon^{\mathrm{sgs}})$ (Fig. 8) because the production term is weighted by $\sqrt{k^{\mathrm{sgs}}}$ compared with $\overline{s}^{2}$. Therefore, when the dissipation $\propto(k^{\mathrm{sgs}})^{3/2}$ is large (small), the production $\propto\sqrt{k^{\mathrm{sgs}}}$ also becomes large (small). In $\mathrm{cor}(k^{\mathrm{sgs}},\overline{s}^{2})$, the preferable relation in $\mathrm{cor}(P^{\mathrm{sgs}},\varepsilon^{\mathrm{sgs}})$ disappears. Nevertheless, $\mathrm{cor}(k^{\mathrm{sgs}},\overline{s}^{2})$ is still higher than 60% in the region away from the wall. This result suggests that an algebraic expression $k^{\mathrm{sgs}}\propto\overline{s}^{2}$ is a physically reasonable approximation for the SMM to some extent.

The local production–dissipation equilibrium relation $k^{\mathrm{sgs}}=C\overline{\Delta}^{2}\overline{s}^{2}$ with a constant $C$ (12) does not exhibit proper near-wall behavior $k^{\mathrm{sgs}}=O(y^{2})$. An empirical near-wall damping function $f_{k}$ can correct the near-wall behavior, as discussed in Sec. II.3. As a first step of the modeling of SGS energy, we consider the damping function that reproduces the proper near-wall behavior. A conventional damping function is an exponential damping function based on the wall-friction velocity and kinematic viscosity, that is, $f_{k}=1-\exp[-(y^{+}/A)^{2}]$ with a constant $A$. Note that the wall-friction velocity is not always appropriate as a representative velocity scale for damping functions because it often yields zero, e.g., at a separation point. The vanishing of the wall-friction velocity results in unphysical damping $f_{k}=0$, regardless of the distance from the wall. In addition, when we use the wall-friction velocity in the damping function, we must calculate the nearest reference wall point to which each grid point corresponds in advance. To avoid these defects, we consider an alternative normalized distance from the solid wall. According to Refs. Inagaki, 2011; Inagaki and Abe, 2017, a damping function should be independent of the grid resolutions in turbulent channel flows, such as that based on the wall-friction velocity. Hence, the filter scale $\overline{\Delta}$ is not suitable for normalizing the distance from the solid wall. Here, we consider the Kolmogorov length scale based on the GS velocity field; $\eta^{\mathrm{gs}}=[\nu^{3}/(\nu\overline{s}^{2})]^{1/4}=\nu^{1/2}/(\overline{s}^{2})^{1/4}$. In the vicinity of the solid wall in turbulent channel flows, the mean velocity gradient is dominant in $\overline{s}^{2}$. Hence, the grid dependence of $\overline{s}^{2}$ is expected to be small in the near-wall region. Furthermore, in the LES of turbulent flows, $\overline{s}^{2}$ does not vanish even if the wall-friction velocity of the nearest reference point vanishes owing to the turbulent fluctuation. Therefore, $\eta^{\mathrm{gs}}$ can be a representative for normalizing the distance from the solid wall. Consequently, we define the normalized distance from the wall based on $\eta^{\mathrm{gs}}$ as

This is rewritten as $y^{s}=u_{\varepsilon}^{\mathrm{gs}}y/\nu$ where $u_{\varepsilon}^{\mathrm{gs}}\{=[\nu(\nu\overline{s}^{2})]^{1/4}\}$ corresponds to the Kolmogorov velocity scale based on the GS velocity field. The distance from the solid wall based on the Kolmogorov velocity scale was first suggested by Abe et al.Abe, Kondoh, and Nagano (1994) in RANS modeling and this method is employed in the LESInagaki (2011) including the SMMAbe (2013); Inagaki and Abe (2017) (15). Equation (15) employs the SGS Kolmogorov velocity scale based on the SGS dissipation rate $(\nu\varepsilon^{\mathrm{sgs}})^{1/4}$. However, $\varepsilon^{\mathrm{sgs}}$ cannot be used in the damping function of the SGS energy because $\varepsilon^{\mathrm{sgs}}$ involves the SGS energy [see Eq. (10a)]. The use of the GS Kolmogorov velocity or length scale is a primitive alternative for the damping function of the SGS energy.

Figure 11 shows the profiles of $y^{s}$ with respect to the conventional distance from the wall based on the viscous unit $y^{+}$ for several LESs of the SMM. Because $\eta^{\mathrm{gs}}$ fluctuates, we plot the mean value of the new distance $\langle y^{s}\rangle$. Note that the statistical average of a quantity solely determined by $\overline{s}^{2}$ is calculated using the probability density function (PDF) of $\overline{s}^{2}$ obtained in the SMM. That is, when we write $\sigma=\overline{s}^{2}$, the statistical average of $q(\overline{s}^{2})$ reads

where $q(\sigma)[=q(\overline{s}^{2})]$ and $f^{\mathrm{PDF}}(\sigma)[=f^{\mathrm{PDF}}(\overline{s}^{2})]$ denote an arbitrary function solely determined by $\overline{s}^{2}$ and the PDF of $\overline{s}^{2}$, respectively. The damping function is typically required in the region approximately $y^{+}<50$. In the region $y^{+}<50$, the profiles of $\langle y^{s}\rangle$ almost collapse to a unique curve. Hence, we can employ $y^{s}$ instead of $y^{+}$ to construct a damping function that is robust to grid resolutions.

Using the statistics of the SMM, we search for the form of the damping function $f_{k}$ for an algebraic model of the SGS energy that reproduces proper near-wall behavior. We refer to this analysis that uses the statistics of the SMM as the semi-a priori test to distinguish it from the a priori analysis based on the DNS. Figure 12 shows the ratio of the mean SGS energy to that of the local equilibrium solution with respect to $\langle y^{s}\rangle$ in several LESs of the SMM. In the near-wall region $\langle y^{s}\rangle<10$, which corresponds to $y^{+}<40$, the profiles of the ratio almost collapse to a unique curve. As the Reynolds number increases, the flat region close to unity increases. Hence, we fix the value $2C_{\mathrm{sgs}}/C_{\varepsilon}$, such as that provided by the parameters used in the original SMM that employs the SGS energy transport equation model. That is, $C_{\mathrm{sgs}}=0.075$ and $C_{\varepsilon}=0.835$.

To construct the best-fit curve in the near-wall region, we consider the following damping function:

In this damping function, the numerator of $f_{k}$, which is similar to the conventional damping function, reproduces the near-wall asymptote $k^{\mathrm{sgs}}\sim O(y^{2})$, whereas the denominator of $f_{k}$ is employed to realize the strong damping of the SGS energy. Three parameters $a_{s}$, $b_{s}$, and $c_{s}$ are determined to minimize the squared difference between the mean SGS energy obtained using the transport equation in the SMM and the semi-a priori test of the model calculated from the same simulation., i.e., we determine $a_{s}$, $b_{s}$, and $c_{s}$ such that they minimize the following value $\mathcal{S}$:

with Eq. (24). By employing the technique described in Eq. (24) that uses the PDF of $\overline{s}^{2}$, we can perform this minimization procedure significantly faster than using the velocity fields to calculate the statistical average. The best fit values yield $a_{s}=0.6$, $b_{s}=0.77$, and $c_{s}=7.3$. The black dashed line with crosses in Fig. 12 represents the semi-a priori profile of the mean damping function $\langle f_{k}\rangle$ with the best-fit parameters. The semi-a priori profile of the mean damping function $\langle f_{k}\rangle$ collapses well to the curve of the ratio in the near-wall region $\langle y^{s}\rangle<10$. Because the correlation between production and dissipation is not perfect, the instantaneous properties of LES alter when replacing the transport equation with the algebraic model for SGS energy. Consequently, the a posteriori test, which is a numerical simulation of the ZE-SMM employing the algebraic SGS energy model with this damping function, predicts a slightly decreased mean velocity profile compared with the original SMM. Therefore, in the a posteriori test, we refine the parameter as $c_{s}=7.6$ to predict a more accurate mean velocity profile. The solid gray line with pluses in Fig. 12 represents the semi-a priori profile of the refined damping function.

To reproduce the near-wall asymptote, the dynamic procedureGermano (1986); Marstorp et al. (2009) or invariants of velocity gradient tensorNicoud and Ducros (1999); Kobayashi (2005); Nicoud et al. (2011); Trias et al. (2015); Silvis, Remmerswaal, and Verstappen (2017) may be more convenient. However, we could not find the universal function independent of the grid resolution and Reynolds number that reproduces the profile of SGS energy in terms of the dynamic procedure or invariants of velocity gradient tensor. Namely, these models do not reproduce the large intensity of the SGS energy in the near-wall region, similar to the SGS Reynolds shear stress of the DSM as in Fig. 2. We will discuss the modeling of the SGS energy in terms of the invariants of velocity gradient tensor in a future study.

Here, we discuss the a posteriori results of the ZE-SMM that employs the SGS energy provided by Eqs. (13) and (25) with $a_{s}=0.6$, $b_{s}=0.77$, and $c_{s}=7.6$ instead of solving the transport equation (6). Figures 13(a)–(d) show the mean velocity profiles of the ZE-SMM compared with the DNS and original SMM at each Reynolds number and grid resolution. We also plot the result of the ZE-SMM with $c_{s}=7.3$, which is the best-fit value in the semi-a priori test, at $\mathrm{Re}_{\tau}=400$ for the LR; it slightly underestimates the mean velocity compared with the original SMM. For all Reynolds numbers and grid resolutions provided in this study, the ZE-SMM provides almost the same results as the original SMM, which is based on the SGS energy transport equation model. Table 3 shows the bulk mean velocity of the ZE-SMM compared with those of the DNS and SMM. Although the ZE-SMM tends to slightly underestimate the mean velocity at $\mathrm{Re}_{\tau}=180$, the difference between the ZE-SMM and SMM is seemingly small. In addition, at $\mathrm{Re}_{\tau}=1000$, the resolution dependence of the bulk mean velocity for the ZE-SMM decreases compared with that for the SMM.

Figures 14(a)–(d) show the profiles of the total turbulent energy for the ZE-SMM compared with the DNS and original SMM at each Reynolds number and grid resolution. We also depict the profiles of the mean SGS energy in the insets. For the LR and VLR cases, the ZE-SMM predicts slightly large values of the total turbulent energy at the peak position compared with the SMM. In contrast, the mean SGS energy for the ZE-SMM decreases in the entire region compared with that for the SMM. Hence, in the ZE-SMM, the GS turbulent fluctuation becomes slightly healthy compared with the SMM. The ZE-SMM with $c_{s}=7.3$ provides a slightly large mean SGS energy at $\mathrm{Re}_{\tau}=400$ for the LR compared with the ZE-SMM. This results in a slightly large eddy viscosity such that it underestimates the mean velocity profile shown in Fig. 13(b).

The difference in the total turbulent energy between the ZE-SMM and SMM seems to be small, similar to the mean velocity profiles shown in Fig. 13. Hence, we can interpret that the ZE-SMM is almost equivalent to the SMM within the Reynolds numbers and grid resolutions provided in this study. However, both the ZE-SMM and SMM underestimate the total turbulent energy in the region away from the wall at $\mathrm{Re}_{\tau}=1000$ and $2000$ compared with the DNS. This underestimation results from the modeling of the SGS stress, but not the zero-equation reduction of the SMM. Thus, future research can provide further improvements for the SMM.

Figures 15(a) and (b) show the profiles of the GS and SGS Reynolds shear stress, respectively, for the ZE-SMM compared with the SMM and fDNS. The GS Reynolds shear stress becomes large compared with the original SMM, similar to the turbulent kinetic energy. On the other hand, the SGS shear stress decreases because the SGS energy decreases in the ZE-SMM compared with that in the SMM as in Fig. 13. Note that the mean velocity is reproduced because the total shear stress is unchanged. For the LR case, the ZE-SMM reproduces the large intensity of the SGS shear stress in the near-wall region observed in the fDNS with SCF and SMM, although the intensity decreases. Hence, the ZE-SMM provides a qualitatively similar profile for the SGS shear stress.

AbeAbe (2019) and Inagaki and KobayashiInagaki and Kobayashi (2020) demonstrated that the SMM predicts a large intensity of the energy spectrum in the high-wavenumber or low-wavelength region close to the cutoff scale, in contrast to conventional EVMs. Furthermore, Inagaki and KobayashiInagaki and Kobayashi (2020) clarified that the extra anisotropic term in the SGS stress significantly contributes to the generation of such low-wavelength turbulent fluctuations, which aids in reproducing the more accurate streaky structures of turbulent wall-shear flow in contrast with conventional EVMs. To confirm whether the ZE-SMM retains this preferable property, we investigate the GS Reynolds stress spectrum $E^{\mathrm{GS}}_{ij}(k_{x},y)$:

where $k_{x}^{\mathrm{max}}=\pi N_{x}/L_{x}$, $\Delta k_{x}=2\pi/L_{x}$, $\tilde{\overline{u}}_{i}^{\prime}$ denotes the Fourier coefficient of the velocity fluctuation $\overline{u}_{i}^{\prime}$ defined by

and the superscript $*$ denotes the complex conjugate. Here, we consider Fourier decomposition only in the streamwise direction.

Figures 16(a)–(d) show the profiles of the premultiplied GS Reynolds stress spectrum for the streamwise component for the fDNS, SMM, EVM, and ZE-SMM at $y^{+}=20$ and $y^{+}=100$, respectively, in $\mathrm{Re}_{\tau}=400$. Note that the damping function operates at $y^{+}=20$, whereas it is saturated at $y^{+}=100$. To focus on the small scales, we use the wavelength $\lambda_{x}(=2\pi/k_{x})$ as the horizontal axis instead of the wavenumber $k_{x}$. For the LR at $y^{+}=20$, both the ZE-SMM and SMM predict the large intensity similar to the fDNS results even in the low-wavelength region close to the cutoff scale. For the MR at $y^{+}=20$ and both the LR and MR at $y^{+}=100$, the spectra of the ZE-SMM and SMM decrease in the low-wavelength region compared with the fDNS results. However, in contrast with the SMM and ZE-SMM, the spectra of the EVM are slightly accumulated in the large scale or high-wavelength region. The LD cases for the ZE-SMM and SMM provide almost the same results as that of the LR, which indicates that the domain size does not affect the spectra of the LR. In all cases, the ZE-SMM provides quantitatively the same spectra as the SMM.

A small difference between the ZE-SMM and SMM is observed in the shear stress spectrum. Figures 17(a)–(d) show the profiles of the premultiplied GS Reynolds stress spectrum for the shear component for the fDNS, SMM, EVM, and ZE-SMM at $y^{+}=20$ and $y^{+}=100$, respectively, in $\mathrm{Re}_{\tau}=400$. Although both the ZE-SMM and SMM provide large intensities of spectra close to the cutoff wavelength scale, the lines for the SMM disappear in the wavelength region twice larger than the cutoff scale at $y^{+}=20$. That is, the lines for the LR and MR cases of the SMM disappear at $\lambda_{x}^{+}=400$ and $200$, respectively, which correspond to $\lambda_{x}^{+}=2\lambda_{x}^{\mathrm{max}}{}^{+}=4\Delta x^{+}$. This disappearance is caused because the GS Reynolds shear stress spectrum yields a positive value, $E^{\mathrm{GS}}_{xy}>0$. In contrast, the spectra for the ZE-SMM do not disappear because the GS Reynolds shear stress spectrum always yields a negative. This difference between the ZE-SMM and SMM is simply due to the GS turbulent velocity fluctuation in the ZE-SMM being healthier than that in the SMM. Owing to this property, the probability of a positive GS Reynolds shear stress spectrum decreases. In contrast with the EVM, in which the spectra disappear in the relatively high-wavelength region, both the ZE-SMM and SMM provide a wide range of negative GS Reynolds shear stress spectra as the fDNS. This result is the same as that for the streamwise spectra. Hence, the ZE-SMM retains the preferable property of the SMM, which enhances the GS turbulent fluctuations close to the cutoff wavelength scale.

Because the ZE-SMM predicts almost the same spectra as the SMM, the conventional SGS energy transport equation model that employs the transport equation for the SGS energy provided by Eqs. (6), (9), (10a), and (10b) seems to be unnecessary for predicting the statistical properties of turbulent flows. The same statistics can be predicted by employing the algebraic expression based on the local equilibrium assumption with a damping function instead of solving the transport equation. This is because in SGS energy transport equation models, the production term expressed by the eddy viscosity and strain rate often locally balances with the dissipation rate in the region away from the wall. Owing to this local balance, the basic statistics are reproduced even when the SGS energy is replaced with the algebraic model expressed by the strain rate and filter length scale. The present study suggests that the reduction of SGS energy transport equation model into the zero-equation model by employing the algebraic expression of SGS energy based on the local equilibrium between production and dissipation does not decrease the performance of the LES if the local production–dissipation equilibrium is dominant in the transport equation.