2.1 Amplitude modulation of the velocity fluctuations

Following the discussion in § 1.3, according to the HRAP formalism, the near-wall velocity may be modelled as

(2.1) $$\begin{eqnarray}u_{i}^{\prime }=a_{1}+a_{2}+\cdots +a_{N_{i}}.\end{eqnarray}$$

We group the addends to large scales

(2.2) $$\begin{eqnarray}u_{i,L}^{\prime }=a_{1}+\cdots +a_{N_{o}}\end{eqnarray}$$

and small scales

(2.3) $$\begin{eqnarray}u_{i,S}^{\prime }=a_{N_{o}+1}+\cdots +a_{N_{i}},\end{eqnarray}$$

where $u_{i}^{\prime }=u_{i,L}^{\prime }+u_{i,S}^{\prime }$ . According to the HRAP model, $u_{i,L}^{\prime }$ is also the velocity fluctuation at a wall-normal height $y_{o}\sim \unicode[STIX]{x1D6FF}/2^{N_{o}}$ and therefore, the large-scale fluctuation at a near-wall location may be modelled using velocity fluctuations at a wall-normal location further away from the wall. This is an important insight, which will be used in the later sections. The distinction between the small- and large-scale components is arbitrary as long as $1<N_{o}<N_{i}$ .

We model $u_{S}^{\prime }$ given a large-scale fluctuation $u_{L}^{\prime }$ . The subscript $i$ is dropped here for brevity. We will not start with (1.1), but instead, we will show (1.1) is the end point of the following analysis. First, because $u_{S}^{\prime }$ is a fluctuating quantity whose mean is removed, $\langle u_{S}^{\prime }|u_{L}^{\prime }\rangle =0$ . Second, $\langle {u_{S}^{\prime }}^{2}|u_{L}^{\prime }\rangle$ is

(2.4) $$\begin{eqnarray}\langle u_{S}^{\prime 2}|u_{L}^{\prime }\rangle =\langle (a_{N_{o}+1}+\cdots +a_{N_{i}})^{2}|u_{L}^{\prime }\rangle \sim \langle (N_{i}-N_{o}-1)a^{2}|u_{L}^{\prime }\rangle \sim \langle u_{\unicode[STIX]{x1D70F}}^{\prime 2}|u_{L}^{\prime }\rangle \cdot \log \left(\frac{y_{o}}{y_{i}}\right)D,\end{eqnarray}$$

where $y_{i}\sim \unicode[STIX]{x1D6FF}/2^{N_{i}}$ , and $D$ is a damping function (its detailed functional form will not be relevant here). It follows from (2.4) that the large scales affect the energetics of the small scales through the local velocity scale $u_{\unicode[STIX]{x1D70F}}$ , a mechanism that was discussed earlier in Zhang & Chernyshenko (Reference Zhang and Chernyshenko2016) and Baars et al. (Reference Baars, Hutchins and Marusic2017). The variance $\langle u_{\unicode[STIX]{x1D70F}}^{\prime 2}|u_{L}^{\prime }\rangle$ may be estimated by invoking the law of the wall

(2.5) $$\begin{eqnarray}\langle u_{\unicode[STIX]{x1D70F}}^{\prime 2}|u_{L}^{\prime }\rangle =\left[\frac{\unicode[STIX]{x1D705}(U(y_{o})+u_{L}^{\prime })}{\log (y_{o}/\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D708}})}\right]^{2},\end{eqnarray}$$

where $U(y_{o})$ is the mean velocity at the wall-normal height $y_{o}$ , and $\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D708}}$ is an inner, viscous length scale defined as $\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D708}}=\unicode[STIX]{x1D708}/\langle u_{\unicode[STIX]{x1D70F}}\rangle \cdot \exp (-\unicode[STIX]{x1D705}B)$ for smooth walls. $B$ is the additive constant in the law of the wall. Equation (2.5) is often used in a wall-modelled LES context, where wall shear stress is modelled as a function of the velocity away from the wall (Piomelli & Balaras Reference Piomelli and Balaras2002; Bose & Park Reference Bose and Park2017). Equations (2.4) and (2.5) provide estimates of the variance of $u_{S}^{\prime }$ . The same procedure may be followed for estimating the higher-order statistics, but for now we model $u_{S}^{\prime }$ as

(2.6) $$\begin{eqnarray}u_{S}^{\prime }|u_{L}^{\prime }=u_{\unicode[STIX]{x1D70F}}\cdot G(\unicode[STIX]{x1D707},\unicode[STIX]{x1D70E}^{2})=\frac{\unicode[STIX]{x1D705}(U(y_{0})+u_{L}^{\prime })}{\log (y_{o}/\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D708}})}G(\unicode[STIX]{x1D707},\unicode[STIX]{x1D70E}^{2}),\end{eqnarray}$$

where $G(\unicode[STIX]{x1D707},\unicode[STIX]{x1D70E}^{2})$ is a stochastic quantity whose mean $\unicode[STIX]{x1D707}$ and variance $\unicode[STIX]{x1D70E}^{2}$ are known. We rewrite (2.6) as

(2.7) $$\begin{eqnarray}u_{S}^{\prime +}|u_{L}^{\prime +}=\left(1+\frac{1}{U^{+}(y_{o})}u_{L}^{\prime +}\right)G(\unicode[STIX]{x1D707},\unicode[STIX]{x1D70E}^{2}),\end{eqnarray}$$

which conforms to (1.1), and provides us with a prediction of the modulation coefficient  $\unicode[STIX]{x1D6FD}$ ,

(2.8) $$\begin{eqnarray}\unicode[STIX]{x1D6FD}_{HRAP}=1/U^{+}(y_{o}).\end{eqnarray}$$

Both (2.7) and (2.8) rely on the response function being $f=\unicode[STIX]{x1D6FD}u_{L}^{\prime }$ . The detailed functional form of the response function will be determined in § 3.1. For now, we compare (2.8) and (1.2) to the measurements from a $Re_{\unicode[STIX]{x1D70F}}=2000$ channel flow direct numerical simulation (DNS) (Hoyas & Jiménez Reference Hoyas and Jiménez2006). The modelled large-scale velocity fluctuation at the inner location is $u_{i,L,m}^{\prime }=u_{o}^{\prime }$ . The outer velocity is directly used as the modelled inner large-scale velocity without filtration at a pre-specified length scale. The modulation coefficient $\unicode[STIX]{x1D6FD}$ is obtained using (1.5) and (1.6). The reader is directed to Mathis et al. (Reference Mathis, Hutchins and Marusic2011) for detailed procedures for determining $\unicode[STIX]{x1D6FD}$ from data. Additionally, the reader is directed to Zhang & Chernyshenko (Reference Zhang and Chernyshenko2016) for procedures for obtaining the theoretical prediction of $\unicode[STIX]{x1D6FD}$ . Figure 3 shows the measured and model predicted modulation coefficient $\unicode[STIX]{x1D6FD}$ as functions of both the inner and the outer location. Within the viscous sublayer, $\unicode[STIX]{x1D6FD}$ is independent of the inner location, and the measurements agree reasonably well with (2.8). For a given $y_{i}$ , both (2.8) and (1.2) predict that $\unicode[STIX]{x1D6FD}$ decreases as a function of $y_{o}$ , which bears out in the measurements. Without using the optimal filtered defined in Zhang & Chernyshenko (Reference Zhang and Chernyshenko2016), equation (1.2) is slightly less accurate than (2.8).

2.2 Amplitude modulation of wall shear stresses

To use (2.8) in WMLES, we will need to consider the amplitude modulation of the wall shear stresses. Amplitude modulation of wall shear stresses was previously considered in Mathis et al. (Reference Mathis, Marusic, Chernyshenko and Hutchins2013) as:

(2.9) $$\begin{eqnarray}\unicode[STIX]{x1D70F}_{w}^{\prime }=fu_{o,L}^{\prime }+[1+f]\unicode[STIX]{x1D70F}_{w,S,m}^{\ast }.\end{eqnarray}$$

The amplitude modulation response function is again denoted as $f$ and following Mathis et al. (Reference Mathis, Marusic, Chernyshenko and Hutchins2013), the modelling parameter $\unicode[STIX]{x1D6FC}$ equals $f$ . Without loss of generality, $f$ in (2.9) is a nonlinear function of $u_{o,L}^{\prime }$ .

3.1 Empirical response function

For the present measurements, we use boundary-layer flow data at friction Reynolds numbers $Re_{\unicode[STIX]{x1D70F}}=6500$ , 10 000 and 13 000. Details of the data sets can be found in Hutchins et al. (Reference Hutchins, Nickels, Marusic and Chong2009) and Talluru et al. (Reference Talluru, Baidya, Hutchins and Marusic2014). We focus on spatial amplitude modulation using the local velocity as the convective velocity (Yang & Howland Reference Yang and Howland2018). Following Mathis et al. (Reference Mathis, Hutchins and Marusic2009a ), we define the large and small scales using a pre-specified length scale $l^{+}=4000$ . We will also use a filtering length scale $l^{+}=7000$ to show the robustness of the measurements. The large and the small scales in (1.7) are at the same wall-normal height. We will not take into consideration the phase difference between the large and the small scales considering the high $Re_{\unicode[STIX]{x1D70F}}$ of the present data (Baars et al. Reference Baars, Hutchins and Marusic2017).

Figure 4(a–c) shows the response function $f=\langle \text{env}_{L}(u_{S}^{\prime })|u_{L}^{\prime }\rangle$ as functions of the large-scale velocity fluctuation at three wall-normal locations, $y^{+}\approx 5$ , $y^{+}\approx 75$ and $y^{+}\approx 200$ . Results of channel flow DNS at $Re_{\unicode[STIX]{x1D70F}}=2000$ and 5200 (Hoyas & Jiménez Reference Hoyas and Jiménez2006; Lee & Moser Reference Lee and Moser2015) are included for comparison. The DNS results agree reasonably well with the results obtained using the hot-wire data at similar Reynolds numbers. We make a few observations. First, the response function depends on the wall-normal distance but not $Re_{\unicode[STIX]{x1D70F}}$ when the Reynolds number is sufficiently high (e.g. $Re_{\unicode[STIX]{x1D70F}}\gtrsim 2000$ ). Second, the response function is linear, $f=\unicode[STIX]{x1D6FD}u_{L}^{\prime }$ , only within the viscous sublayer ( $y^{+}<5$ ). Third, beyond the viscous sublayer, the small-scale fluctuation responds differently to the large-scale fluctuation depending on whether $u_{L}^{\prime }$ is positive or negative, as also noted in Hwang et al. (Reference Hwang, Lee, Sung and Zaki2016). In the logarithmic region ( $y^{+}\approx 100{-}200$ ), $\langle \text{env}_{L}(u_{S}^{\prime })|u_{L}^{\prime }\rangle$ and $u_{L}^{\prime }$ are positively correlated for $u_{L}^{\prime }<0$ , and are negatively correlated for $u_{L}^{\prime }>0$ , leading to positive and negative amplitude modulation, respectively. Last, the response function $f(u_{L}^{\prime })$ is approximately a linear function of $u_{L}^{\prime }$ for $u_{L}^{\prime }<0$ and $u_{L}^{\prime }>0$ at all wall-normal locations. Figure 5(a–c) shows $\langle \text{env}_{L}(u_{S}^{\prime })|u_{L}^{\prime }\rangle$ at wall-normal locations $y/\unicode[STIX]{x1D6FF}\approx 0.1$ , 0.2 and 0.4, and the same observations can be made. In addition, the response function depends only weakly on the wall-normal location between $y/\unicode[STIX]{x1D6FF}\approx 0.1$ and 0.4. Results are similar when the analysis is repeated using $u_{S}^{\prime 2}$ (or $|u_{S}^{\prime }|$ ) as the metric of the small scale energetics (not shown for brevity).

Following the above discussion, we may re-write the response function as follows

(3.1a,b ) $$\begin{eqnarray}f(u_{L}^{\prime })=\unicode[STIX]{x1D6FD}_{{<}0}u_{L}^{\prime },\quad \text{for }u_{L}^{\prime }<0;\quad f(u_{L}^{\prime })=\unicode[STIX]{x1D6FD}_{{>}0}u_{L}^{\prime },\quad \text{for }u_{L}^{\prime }>0.\end{eqnarray}$$

Because the small-scale fluctuation responds differently to the large-scale fluctuation depending on the sign of $u_{L}^{\prime }$ , and because $R\sim \unicode[STIX]{x1D6FD}$ if $f\sim u_{L}^{\prime }$ , it is useful to consider a conditional amplitude modulation metric

(3.2) $$\begin{eqnarray}R_{{>}0}\equiv \frac{\langle \text{env}_{L}(u_{S}^{\prime })u_{L}^{\prime }|u_{L}^{\prime }>0\rangle }{\text{STD}(\text{env}_{L}(u_{S}^{\prime })|u_{L}^{\prime }>0)\text{STD}(u_{L}^{\prime }|u_{L}^{\prime }>0)}\end{eqnarray}$$

and similarly $R_{{<}0}$ . The conditional $R$ defined in (3.2) is shown in figure 4(d) as functions of $y^{+}$ for boundary-layer and channel flows at various Reynolds numbers. Figure 5(d) shows the same quantities as a function of $y/\unicode[STIX]{x1D6FF}$ . A few observations can be made. First, within the viscous sublayer, $R_{{<}0}\approx R_{{>}0}$ . Second, $R_{{>}0}$ is 0 at $y^{+}\approx 20$ , and $R_{{<}0}$ is 0 at $y^{+}\approx 500$ .

If the true response function follows (3.1), we may compute $\unicode[STIX]{x1D6FD}_{{<}}$ and $\unicode[STIX]{x1D6FD}_{{>}}$ similarly as $\unicode[STIX]{x1D6FD}$ in (1.1), except that, here, the constraint (1.6) will be applied for $u_{L}^{\prime }<0$ and $u_{L}^{\prime }>0$ , respectively:

(3.3) $$\begin{eqnarray}\langle \text{env}_{L}(u_{i,S,m}^{\ast })u_{i,L,m}^{\prime }|u_{i,L,m}^{\prime }\lessgtr 0\rangle =0.\end{eqnarray}$$

The response function $f(u_{i,L,m}^{\prime })$ , as defined in (3.1), is such that the universal signal, computed according to (1.5), satisfies (3.3).

To account for the nonlinearity of the response function $f$ as a function of $u_{L}^{\prime }$ , we can re-cast (2.8) to

(3.4) $$\begin{eqnarray}\unicode[STIX]{x1D6FD}_{\lessgtr 0}=1/\langle U^{+}(y_{o})+u_{o}^{\prime }|u_{o}^{\prime }\lessgtr 0\rangle .\end{eqnarray}$$

A consequence of (3.4) is that the predicted amplitude modulation coefficient is larger for $u_{L}^{\prime }<0$ than for $u_{L}^{\prime }>0$ , which is consistent with figure 4. Figure 6 shows the measured $\unicode[STIX]{x1D6FD}_{\lessgtr 0}$ as functions of the inner location $y_{i}$ for two outer locations ( $y_{o}^{+}=215$ and $400$ ). According to figure 6, $\unicode[STIX]{x1D6FD}_{{<}0}\approx \unicode[STIX]{x1D6FD}_{{>}0}$ in the viscous sublayer. Equation (3.4), however, is still successful only within the viscous sublayer.

For WMLES, the quantity of interest is the wall shear stress, which is a quantity in the viscous sublayer, where $f\sim u_{L}^{\prime }$ . Since the response function $f\sim u_{L}^{\prime }$ , the usefulness of (1.3) in the context of WMLES depends only on whether the large scales can be commensurate with the LES grids. To test the sensitivity of the results on the separating length scale, we repeat the above analysis for a different filtering length scale, $l^{+}=7000$ . Figure 7 shows the measured response function as functions of $u_{L}^{\prime }$ and very similar results are obtained compared to the $l^{+}=4000$ case.

Finally, we comment on the phase difference between the large scales and the small scales, and its connections to the amplitude modulation. Chung & McKeon (Reference Chung and McKeon2010) and Jacobi & McKeon (Reference Jacobi and McKeon2013) argued that amplitude modulation (measured using the correlation coefficient $R$ ) is but a measure of the phase difference between the large-scale fluctuations and the envelope of the small scales. A counter argument was proposed in Baars et al. (Reference Baars, Hutchins and Marusic2017) where the authors argued that the phase shift is a finite Reynolds number effect. It is worth noting that the phase difference between small and large scales is a potential explanation for the deviation from linearity of $f$ . While a full discussion of this point is beyond of the scope of this work, we note that the phase difference between the large and small scales is not relevant to WMLES, where one typically uses the instantaneous LES velocity directly above the wall to predict the local wall shear stress (Bose & Park Reference Bose and Park2017).

3.2 Positive and negative amplitude modulation

The small scales in the near-wall region are subjected to changes in the local Reynolds numbers, which are modulated by the large-scale fluctuations. Equations (2.8) and (3.4) are models for such effects. According to HRAP and the sketch in figure 2, a large-scale eddy affects the flow beneath it (see e.g. Yang et al. Reference Yang, Meneveau, Marusic and Biferale2016c ; Yang & Lozano-Durán Reference Yang and Lozano-Durán2017; Yang et al. Reference Yang, Baidya, Johnson, Marusic and Meneveau2017), and therefore it is unlikely that amplitude modulation due to large-scale modulation of the local Reynolds numbers would depend on the inner location. This expectation bears out in both (2.8) and (3.4), where the predicted response function is not a function of $y_{i}$ . However, in addition to the above mentioned mechanism that leads to positive amplitude modulation, there are also mechanisms that could lead to negative amplitude modulation. Negative amplitude modulation is defined as the decrease of small-scale energetics with increasing large-scale velocity. Negative modulation manifests as a negative slope in $f$ . Jacobi & McKeon (Reference Jacobi and McKeon2013) argued that negative amplitude modulation arises as a result of the phase difference between the envelope of the small scales and the large-scale fluctuations. Zhang & Chernyshenko (Reference Zhang and Chernyshenko2016), on the other hand, argued that the negative amplitude modulation is a combined effect of the modulation of the wall-normal coordinate and the decrease of STD $(u^{\prime })$ as a function of the wall-normal distance. Agostini & Leschziner (Reference Agostini and Leschziner2016b ) argued that negative modulation is the result of the splatting motions which are the result of sweeps towards the wall. Last, Baars et al. (Reference Baars, Talluru, Hutchins and Marusic2015, Reference Baars, Hutchins and Marusic2017) argued that the negative modulation is due to intrusions of non-/less-turbulent free stream flow. These mechanisms compete with the mechanism that is responsible for positive amplitude modulation, and lead to the overall behaviour of the response function in figures 4, 5 and 7.

Figure 2.

Schematic of the modelled boundary layer. The attached eddies are space filling. On a vertical plane cut, as shown here, the number of eddies doubles as the size halves. An eddy affects the shaded region below it. The velocity at a generic point in the flow field is a result of the additive superposition of all of the eddy-induced velocity fields there.

Figure 3.

(a) Symbols: measured amplitude modulation coefficients $\unicode[STIX]{x1D6FD}$ as functions of $y_{i}^{+}$ for outer locations, $y_{o}^{+}=200$ , 400 and 500 (which correspond to $y_{o}/\unicode[STIX]{x1D6FF}=0.1$ , 0.2 and 0.25) for $Re_{\unicode[STIX]{x1D70F}}=2000$ channel flow DNS (Hoyas & Jiménez Reference Hoyas and Jiménez2006). Different colours are used for different $y_{o}$ . Dashed lines: predictions from (1.2). Thin solid lines: (2.8). (b) Same as (a) but for $\unicode[STIX]{x1D6FD}$ as functions of $y_{o}^{+}$ for $y_{i}^{+}=1,$ 5 and 10.

Figure 4.

The response function $\langle \text{env}_{L}(u_{S}^{\prime })|u_{L}^{\prime }\rangle$ as a function of the large-scale fluctuation $u_{L}^{\prime }$ at wall-normal locations $y^{+}\approx 5$ (a), $y^{+}\approx 75$ (b) and $y^{+}\approx 200$ (c). Solid lines are linear fits of the data for $u_{L}^{\prime }>0$ and $u_{L}^{\prime }<0$ , respectively. The slopes are 0.32, $-0.01$ and $-0.04$ for $u_{L}^{\prime }>0$ in (a–c), and are 0.32, 0.06 and 0.04 for $u_{L}^{\prime }<0$ . The dashed black line indicates $u_{L}^{\prime }=0$ . The $y$ scale is magnified in (b,c) for better visualization. (d) Conditional correlation coefficient $R_{\lessgtr }$ as functions of $y^{+}$ . Solid lines represent $R_{{>}0}$ and dashed lines represent $R_{{<}0}$ . Solid vertical lines represent the buffer layer of $30<y^{+}<100$ .

Figure 5.

The response function $\langle \text{env}_{L}(u_{S}^{\prime })|u_{L}^{\prime }\rangle$ as functions of the large-scale fluctuation $u_{L}^{\prime }$ at wall-normal locations $y/\unicode[STIX]{x1D6FF}=0.1$ (a), 0.2 (b) 0.4 (c). The slope is $-0.05$ for $u_{L}^{\prime }>0$ in (a–c) and is $-0.04$ for $u_{L}^{\prime }<0$ . (d) $R_{\lessgtr 0}$ as functions of $y/\unicode[STIX]{x1D6FF}$ . Solid lines represent $R_{{>}0}$ and dashed lines represent $R_{{<}0}$ .

Figure 6.

The measured $\unicode[STIX]{x1D6FD}_{\lessgtr 0}$ as functions of $y_{i}^{+}$ for channel flow at $Re_{\unicode[STIX]{x1D70F}}=2000$ ( $+$ symbols) at (a) $y_{o}^{+}=215$ and (b) 400. The solid lines are the predictions using (3.4).

Figure 7.

(a–c) Same as in figure 4 except with $l^{+}=7000$ . (d) Same as in (a) except with $y/\unicode[STIX]{x1D6FF}=0.2$ . The linear fits are the same as in figures 4 and 5.

Since $\unicode[STIX]{x1D6FD}_{\gtrless 0}$ is a quantity that measures the overall effects of both the positive amplitude modulation and the negative amplitude modulation, $\unicode[STIX]{x1D6FD}_{\gtrless 0}$ itself is not independent of the inner location. If positive amplitude modulation is fully accounted for in (3.4), we could remove this form of interscale interaction from the response function using (1.5), (3.1) and (3.4), i.e. $u_{i,d,S}^{\prime }\equiv u_{i,S}^{\prime }-(1+\unicode[STIX]{x1D6FD}_{\gtrless 0}u_{o}^{\prime })u_{i,S}^{\ast }$ is only negatively modulated by the large scales. This expectation is confirmed in figure 8, where $\langle \text{env}_{L}(u_{i,d,S}^{\prime })|u_{L}^{\prime }\rangle$ are shown as functions of $u_{L}^{\prime }$ for a few inner locations and a fixed outer location $y_{o}^{+}=270$ (in a $Re_{\unicode[STIX]{x1D70F}}=5200$ channel (Lee & Moser Reference Lee and Moser2015)).

Figure 8.

The response functions $\langle \text{env}_{L}(u_{i,d,S}^{\prime })|u_{L}^{\prime }\rangle$ and $\langle \text{env}_{L}(u_{S}^{\prime })|u_{L}^{\prime }\rangle$ measured using $Re_{\unicode[STIX]{x1D70F}}=5200$ channel flow DNS. The vertical axis for (a–c) is $\langle \text{env}_{L}(\cdot )|u_{L}^{\prime }\rangle$ where $(\cdot )$ denotes either $u_{i,S}^{\prime }$ or $u_{i,d,S}^{\prime }$ . The outer location is at $y_{o}^{+}=270$ . The inner location is at $y_{i}=10$ for (a), $y_{i}^{+}=70$ for (b) and $y_{i}^{+}=140$ for (c).

3.3 Spatial and temporal amplitude modulation

The discussion to this point has been limited to spatial amplitude modulation. In this subsection, we will repeat the analysis in § 3.1 for the temporal amplitude modulation process. Hot-wire measurements are used directly for this purpose. Temporal amplitude modulation is the measure of amplitude modulation using Taylor’s hypothesis and a mean convective velocity for the conversion of a temporal signal to space (Yang & Howland Reference Yang and Howland2018). For brevity we only show results of the $Re_{\unicode[STIX]{x1D70F}}=13\,000$ boundary layer. Figure 9 shows the measured response function $\langle \text{env}_{L}(u_{S}^{\prime })|u_{L}^{\prime }\rangle$ as functions of the large-scale velocity fluctuation at wall-normal locations $y^{+}\approx 10$ , 75 and 200 (results at other wall-normal locations are similar and are not shown here for brevity). At $y^{+}\approx 10$ , spatial and temporal modulations are not distinguishably different according to this metric. Beyond $y^{+}\approx 10$ , the two processes lead to quantitatively different results. Temporal amplitude modulation is stronger than the same process in spatial coordinates, i.e. for $u_{L}^{\prime }<0$ , the small scales are more subdued, and for $u_{L}^{\prime }>0$ , the small scales are more energetic. Nevertheless, the response function can still be approximated using a piecewise linear function (3.1), and therefore, the discussion in § 3.1 is equally useful for temporal amplitude modulation.

Figure 9.

$\langle \text{env}_{L}(u_{S}^{\prime })|u_{L}^{\prime }\rangle$ as functions of $u_{L}^{\prime }$ at $y^{+}\approx 10$ (a), 75 (b) and 200 (c) for a boundary-layer flow at $Re_{\unicode[STIX]{x1D70F}}=13\,000$ . Orange: temporal data; blue: spatial data (see Yang & Howland (Reference Yang and Howland2018) for a detailed discussion on the conversion of temporal hot-wire data to spatial coordinates).

Figure 10 shows the $R_{\lessgtr 0}$ measured using the temporal data as functions of the wall-normal distance. Spatial amplitude modulation and temporal amplitude modulation are only quantitatively different. As $R_{\lessgtr 0}$ are measures of $\unicode[STIX]{x1D6FD}_{\lessgtr 0}$ , figures 9 and 10 and suggest that the temporally measured $R_{\lessgtr 0}$ are generally greater than their spatial counterparts in the buffer layer and in the logarithmic region.

Figure 10.

Conditional $R_{\lessgtr 0}$ as a function of $y^{+}$ for $Re_{\unicode[STIX]{x1D70F}}=13\,000$ boundary-layer flow. (a) $R_{{>}0}$ (b) $R_{{<}0}$ . Orange: temporal data; blue: spatial data.

5.1 Model formulation

Following the discussion in § 3, we can approximate the response function within the viscous sublayer as a linear function of the large-scale velocity fluctuation $f(y_{i},y_{o},u_{o,L}^{\prime })=\unicode[STIX]{x1D6FD}u_{o,L}^{\prime }$ . As a result, equation (2.9) reduces to

(5.1) $$\begin{eqnarray}\unicode[STIX]{x1D70F}_{w}^{\prime }=\unicode[STIX]{x1D6FD}u_{o,L}^{\prime }+(1+\unicode[STIX]{x1D6FD}u_{o,L}^{\prime })\unicode[STIX]{x1D70F}_{w,S,m}^{\ast }.\end{eqnarray}$$

WMLES requires the filtered wall shear stresses to integrate the LES equations in the bulk region (Piomelli & Balaras Reference Piomelli and Balaras2002; Choi & Moin Reference Choi and Moin2012). If the large scales are commensurate with the LES grids, filtering both sides of (5.1) leads to

(5.2) $$\begin{eqnarray}\widetilde{\unicode[STIX]{x1D70F}_{w}^{\prime }}=\unicode[STIX]{x1D70F}_{w,L}^{\prime }=\unicode[STIX]{x1D6FD}u_{o,L}^{\prime },\end{eqnarray}$$

and there is no need to specify the universal small-scale wall shear stress fluctuation. Here $\widetilde{\cdot }$ is the LES filtering operation. The modelling parameter $\unicode[STIX]{x1D6FD}$ is specified according to (2.8), $\unicode[STIX]{x1D6FD}=1/U^{+}(y_{o})$ . It then follows from (5.2) that

(5.3) $$\begin{eqnarray}\widetilde{\unicode[STIX]{x1D70F}_{w}}=\langle \unicode[STIX]{x1D70F}_{w}\rangle +\widetilde{\unicode[STIX]{x1D70F}_{w}^{\prime }}=\langle \unicode[STIX]{x1D70F}_{w}\rangle +\frac{1}{U(y_{1})}\widetilde{u^{\prime }}(y=y_{1}),\end{eqnarray}$$

where $y_{1}$ is the distance between the first off-wall grid point and the wall. If the first off-wall grid point is in the logarithmic range, equation (5.3) reduces to

(5.4) $$\begin{eqnarray}\widetilde{\unicode[STIX]{x1D70F}_{w}}=\langle \unicode[STIX]{x1D70F}_{w}\rangle +\frac{\unicode[STIX]{x1D705}\widetilde{u^{\prime }}}{\log (y_{1}/\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D708}})},\end{eqnarray}$$

where $\unicode[STIX]{x1D705}\approx 0.4$ is the von Kármán constant, $\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D708}}=\unicode[STIX]{x1D708}/u_{\unicode[STIX]{x1D70F}}\cdot \exp (-\unicode[STIX]{x1D705}B)$ for smooth walls, and $B\approx 5$ is the additive constant in the law of the wall. Different from (5.4), the equilibrium wall model reads

(5.5) $$\begin{eqnarray}\widetilde{\unicode[STIX]{x1D70F}_{w}}=\left[\frac{\unicode[STIX]{x1D705}u}{\log (y_{1}/\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D708}})}\right]^{2}\approx \langle \unicode[STIX]{x1D70F}_{w}\rangle +\frac{2\unicode[STIX]{x1D705}\widetilde{u^{\prime }}}{\log (y_{1}/\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D708}})},\end{eqnarray}$$

leading to a slightly different model for the wall shear stress fluctuations. Before we proceed and compare the two models in WMLES, we make a connection between (5.4) and the recently developed slip wall model (Bose & Moin Reference Bose and Moin2014), where a slip velocity is applied at the wall in replacement of the no-slip condition. The slip velocity is specified according to the following Robin-type boundary condition,

(5.6) $$\begin{eqnarray}\widetilde{u_{s}}-l_{p}\frac{\unicode[STIX]{x2202}\widetilde{u_{s}}}{\unicode[STIX]{x2202}n}=0,\end{eqnarray}$$

where $\widetilde{u_{s}}$ is the slip velocity, $n$ is the wall-normal direction and $l_{p}$ is the slip length. Bose & Moin (Reference Bose and Moin2014) used a dynamic procedure for determining the slip length, here, we turn to well-established Reynolds Averaged Navier–Stokes (RANS) closures for an estimate of $l_{p}$ . We start by considering the equilibrium momentum equation at the first off-wall grid point

(5.7) $$\begin{eqnarray}u_{\unicode[STIX]{x1D70F}}^{2}-\unicode[STIX]{x1D708}_{T}\frac{\unicode[STIX]{x2202}\langle u\rangle }{\unicode[STIX]{x2202}n}=0,\end{eqnarray}$$

where $u_{\unicode[STIX]{x1D70F}}$ is the friction velocity, $u_{\unicode[STIX]{x1D70F}}=\unicode[STIX]{x1D705}\langle u\rangle /\log (y_{1}/\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D708}})$ according to the law of the wall. The eddy viscosity is $\unicode[STIX]{x1D708}_{T}=\unicode[STIX]{x1D705}u_{\unicode[STIX]{x1D70F}}y_{1}$ . Dividing both sides of (5.7) by $u_{\unicode[STIX]{x1D70F}}$ and replacing the remaining $u_{\unicode[STIX]{x1D70F}}$ with $\unicode[STIX]{x1D705}\langle u\rangle /\log (y_{1}/\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D708}})$ , we obtain

(5.8) $$\begin{eqnarray}\frac{\unicode[STIX]{x1D705}\langle u\rangle }{\log (y_{1}/\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D708}})}-\unicode[STIX]{x1D705}y_{1}\frac{\unicode[STIX]{x2202}\langle u\rangle }{\unicode[STIX]{x2202}n}=0.\end{eqnarray}$$

By re-arranging the terms, equation (5.8) leads to

(5.9) $$\begin{eqnarray}\langle u\rangle -y_{1}\log (y_{1}/\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D708}})\frac{\unicode[STIX]{x2202}\langle u\rangle }{\unicode[STIX]{x2202}n}=0.\end{eqnarray}$$

While $\langle \cdot \rangle$ is not equivalent to $\widetilde{\cdot }$ , equation (5.9) provides us with an estimate of the slip length

(5.10) $$\begin{eqnarray}l_{p}=y_{1}\log (y_{1}/\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D708}}).\end{eqnarray}$$

A different, physics-based interpretation of the slip wall model formulation may be found in Lozano-Durán et al. (Reference Lozano-Durán, Bae, Bose and Moin2017). The arguments in Lozano-Durán et al. (Reference Lozano-Durán, Bae, Bose and Moin2017) are not detailed here as providing a physical interpretation to the slip model is not the focus of this paper. Here, we use (5.6) and (5.10) to estimate the wall shear stress predicted by the slip model,

(5.11) $$\begin{eqnarray}\widetilde{\unicode[STIX]{x1D70F}_{w}}=\unicode[STIX]{x1D708}_{T}\frac{\unicode[STIX]{x2202}\widetilde{u_{s}}}{\unicode[STIX]{x2202}n}=\frac{\unicode[STIX]{x1D708}_{T}}{l_{p}}\widetilde{u_{s}}=\frac{\unicode[STIX]{x1D705}u_{\unicode[STIX]{x1D70F}}}{\log (y_{1}/\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D708}})}\widetilde{u_{s}}.\end{eqnarray}$$

Normalizing the quantities in (5.11) with wall units and dropping the $+$ superscript for brevity, equation (5.11) leads to

(5.12) $$\begin{eqnarray}\widetilde{\unicode[STIX]{x1D70F}_{w}}=\frac{\unicode[STIX]{x1D705}y_{1}}{\log (y_{1}/\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D708}})}\widetilde{u_{s}}=\langle \unicode[STIX]{x1D70F}_{w}\rangle +\frac{\unicode[STIX]{x1D705}\widetilde{u^{\prime }}}{\log (y_{1}/\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D708}})},\end{eqnarray}$$

which happens to be the same form as (5.4). While detailed discussion falls out of the scope of this paper, this result may be the reason why the slip wall model outperforms the equilibrium model in channel flow (Bae et al. Reference Bae, Lozano-Duran, Bose and Moin2018).

5.2 LES set-up

We test the new wall model (5.4) in WMLES and compare its performance with that of the equilibrium wall model. The in-house pseudo-spectral code LESGO is used. Details of this code may be found in Bou-Zeid, Meneveau & Parlange (Reference Bou-Zeid, Meneveau and Parlange2005) and Anderson & Meneveau (Reference Anderson and Meneveau2011). Here, we briefly summarize the main features of the code. The code solves the filtered Navier–Stokes equation in a half-channel with periodic boundary conditions in both the streamwise and the spanwise directions. The half-channel is driven by an imposed pressure gradient. A zero-stress condition is imposed at the top boundary, and the bottom wall is modelled using a wall model. A pseudo-spectral scheme is used in both the streamwise and the spanwise direction, and a second-order finite difference scheme is used for spatial discretization in the wall-normal direction. The subgrid-scale stresses are modelled using the scale-dependent Lagrangian-averaged Smagorinsky model (Bou-Zeid et al. Reference Bou-Zeid, Meneveau and Parlange2005). For the calculations here, the computational domain is $2\unicode[STIX]{x03C0}\unicode[STIX]{x1D6FF}\times 1\unicode[STIX]{x1D6FF}\times 2\unicode[STIX]{x03C0}\unicode[STIX]{x1D6FF}$ in the streamwise, wall-normal and spanwise directions respectively. Two grids are used: $64^{3}$ and $128^{3}$ . All the statistics are averaged for 10 flow throughs after a statistically stationary state is reached. The flow through time is $t_{f}=L_{x}/u_{o}$ with $L_{x}$ and $u_{o}$ being the extent of the computational domain in the streamwise direction and the volume-averaged velocity. Here, we compare the performance of the new wall model in (5.3) to that of the equilibrium wall model given by (5.5). For the particular flow here, the mean wall shear stress is known from the mean momentum balance; therefore, the wall shear stress fluctuation is the quantity of interest. The wall model input $y_{o}=3.22\times 10^{-5}\unicode[STIX]{x1D6FF}$ , which corresponds to a $Re_{\unicode[STIX]{x1D70F}}=4200$ boundary layer.

5.3 WMLES results

Both the equilibrium wall model and the amplitude modulation based model capture the mean velocity profile correctly (see figure 13). The two models only differ in high-order statistics. WMLES often leads to deficiencies in its predictions of the variance of streamwise velocity (Park & Moin Reference Park and Moin2016; Park et al. Reference Park, Howland, Lozano-Duran and Moin2016). Figure 14 shows the variance of the streamwise velocity fluctuation for the WMLES using both the presently developed modulation-based wall model and the commonly used equilibrium wall model. While both models are fairly accurate when the resolution is high ( $128^{3}$ ), the modulation-based wall model is slightly more accurate for low-resolution calculations. In WMLES of channel flow, one can typically expect the grid convergence of the streamwise velocity variance from the centre of the channel to the wall (Yang et al. Reference Yang, Marusic and Meneveau2016a ,Reference Yang, Meneveau, Marusic and Biferale c ). In figure 14, this expectation only bears out for WMLES with the modulation-based model. Therefore, we may conclude that the modulation-based wall model leads to more accurately resolved bulk structures.

Figure 13.

(a) Mean velocity profiles as functions of the wall-normal distance for WMLES with the modulation-based model. The grid sizes are $64^{3}$ and $128^{3}$ . (b) Same as (a) but for WMLES that use the equilibrium wall model.

Figure 14.

(a) Variance of the streamwise velocity fluctuations as functions of the wall-normal distance. The DNS results are obtained from Lozano-Durán & Jiménez (Reference Lozano-Durán and Jiménez2014) and the two WMLES use the model in (5.3). The grid sizes are $64^{3}$ and $128^{3}$ . (b) Same as (a) but for WMLES that use the equilibrium wall model.