3.1. Database

The model is validated against eight direct numerical simulations (DNS) or LES flow fields, with the overall ranges of parameters presented in table 2, where $ \textit{Re}_\theta = {u_e \theta }/{\nu }$ denotes the momentum thickness Reynolds number. The flow fields are named as follows. The cases b1n, b2n and b14 correspond to TBLs subjected to relatively smooth, constant APGs, with Clauser parameters $\beta =1.0, 2.0, 1.4$ , respectively. The cases m13n, m16n and m18n denote flows in which the free-stream velocity follows a power-law distribution of the form $u_e(x)=C(x-x_0)^m$ , and $m=-0.13, -0.16, -0.18$ , respectively. The flows LSM2017 and LSM2018 simply refer to datasets used to analyse large-scale motions (LSMs) that were published in 2017 and 2018, respectively.

Among them, flow m18n has the strongest PG, and also the strongest local disequilibrating parameter $\gamma$ , while flow b14 has the highest Reynolds number. The tested flow fields cover a wide range of cases, with Clauser PG $\beta$ reaching maximum $4.53$ , $ \textit{Re}_\theta$ peaking at $9.65\times 10^3$ , and the non-dimensional second-order PG $\gamma$ ranging from $-4.57\times 10^{-3}$ to $2.82\times 10^{-4}$ . Flow data for LSM2017 are incomplete, therefore only the numerical results within its equilibrium region are used for validation.

The streamwise development of flow parameters is illustrated in figure 1. Here and throughout the rest of the paper, streamwise location $x$ is non-dimensionalised by the mean nominal boundary layer thickness $\delta _{99}$ within the computational domain, $\hat x=x/\langle \delta _{99}\rangle$ . As shown in figure 1(a), Clauser PG $\beta$ in all flow fields undergoes a process of first increasing and then decreasing. Correspondingly, in figure 1(b), $\gamma$ is first positive and then negative. Negative $\gamma$ can reach very large negative values, with its absolute magnitude far exceeding the maximum positive values. This is due to the smaller wall friction, and therefore smaller $u_\tau$ in the downstream region, which also indicates that downstream regions are more susceptible to local disequilibrating effects. Figures 1(c) and 1(d) illustrate that the streamwise evolution of the Reynolds number is relatively uniform and is not strongly related to variations of PG.

The markers in figure 1(a) indicate the locations of the profiles used for validation in § 3.3, which span a broad parameter space. However, limited validation exists for regions with $\gamma \gt 0$ , due to the fact that the flow fields in these regions may have relatively low Reynolds numbers, and be more susceptible to inflow conditions.

Figure 1.

An overall description of seven flow fields along streamwise direction. The data are obtained directly through DNS or LES (Bobke et al. Reference Bobke, Vinuesa, Orlu and Schlatter2017; Yoon, Hwang & Sung Reference Yoon, Hwang and Sung2018; Pozuelo et al. Reference Pozuelo, Li, Schlatter and Vinuesa2022). The horizontal axis denotes the streamwise coordinate $\hat x=x/\langle \delta _{99}\rangle$ , where $x$ is non-dimensionalised by the nominal boundary layer thickness $\delta _{99}$ averaged over the computational domain. Plots show (a) Clauser PG $\beta$ , (b) streamwise derivative of PG $\gamma$ as defined in (2.15), with a zoomed-in view of the positive $\gamma$ region, (c) friction Reynolds number $ \textit{Re}_\tau$ , (d) momentum thickness Reynolds number $ \textit{Re}_\theta$ . The markers in (a) indicate the locations of the profiles used for validation in § 3.3: the green markers are tested in figure 3; the light blue markers are tested in figure 4; the red markers are tested in figure 5; the two blue triangle markers are tested in figure 7.

3.2. Verification of similarity

This subsection demonstrates that the new scaling method in § 2.1 effectively collapses the non-dimensional mixing length profiles (2.7), which play a fundamental role in the overall method.

We calculate $\lambda ^*$ directly from flow fields according to (2.5), and the results are shown in figure 2. Due to the relatively larger uncertainty of $\lambda ^*$ in the outer region, a one-dimensional Gaussian smoothing with window length 5 points (approximately $y^+=30{-}50$ in the outer region) is applied along the wall-normal direction. Here, the critical ratio $n$ of criterion (2.21) is set to $2$ . This is because at the outer edge of the boundary layer, $\partial u /\partial y$ is really small, leading to an ill-defined $\lambda ^*$ . Yet for the same reason, the magnitude of $\lambda ^*$ in these regions is not important, as $\partial u/\partial y$ is negligibly small, and the integration in this region has little impact on the mean profile of $u^*$ or $\tau ^*$ .

Twenty-five $\lambda ^*$ profiles in five flow fields are presented in figures 2(a) and 2(c). As shown in figure 2(a), in the inner region, even in the viscous sublayer where $y^+\lt 5$ , the new scaling method collapses all the mixing length profiles significantly better than the traditional friction-based scaling, which is shown in figure 2(b). In the outer region, the new scaling actually has the same form as the traditional friction-based scaling. As shown in figure 2(c), the horizontal axis is $\eta = y^*/Re_\tau ^*=y^+/Re_\tau =y/\delta$ , while the vertical axis is $\lambda ^*/Re_\tau ^*=\lambda ^+/Re_\tau ^+ = \lambda /\delta$ . All the profiles have similar growth trends, and their final stable values are closely aligned, showing obvious similarity in $\lambda ^*$ in different regions across different flow fields. In flow fields with stronger PG, such as flow m18n, $\lambda ^*$ in the outer region is less stable, but all the profiles end up at approximately $\lambda ^*/Re_\tau ^*= 0.08$ , which agrees well with our current model (2.7) at $y^* =Re_\tau ^*$ , as long as $ \textit{Re}_\tau ^* \gg a$ and $\ b\ll 1$ so that $\lambda ^*(Re_\tau ^*) \approx kb\, Re_\tau ^*$ , indicating that $kb\approx 0.08$ . This is also consistent with the optimal parameters in table 1.

Figure 2.

Profiles of non-dimensional mixing length $\lambda ^*$ at different streamwise locations, in different flow fields. The data are obtained directly through DNS or LES (Bobke et al. Reference Bobke, Vinuesa, Orlu and Schlatter2017; Lee Reference Lee2017; Yoon et al. Reference Yoon, Hwang and Sung2018). The results have been processed with Gaussian smoothing, and the critical ratio $n$ in criterion (2.21) is set to $2$ . Within the same flow field, lines of lighter colour indicate locations further downstream. (a) Profiles under new inner scaling $\lambda ^*\sim y^*$ . (b) Profiles under traditional inner scaling $\lambda ^+\sim y^+$ . (c) Profiles under outer scaling $\lambda ^*/Re_\tau ^*\sim \eta$ (also $\lambda ^+/Re_\tau \sim \eta$ ).

3.3. Verification of the estimation–correction model

In this subsection, it will be shown that the current estimation–correction model largely captures the shape of the profile and provides good predictive capability. To begin with, $\tau ^*$ from (2.17) can be decomposed into four parts, as shown in (3.1), namely $\tau _{0}^*, \ \tau _{\beta }^*, \tau _{e}^*,\ \tau _{\gamma }^*$ :

(3.1) \begin{align} \tau ^*=\tau _{0}^*+\tau _{\beta }^*+\tau _{e}^*+\tau _{\gamma }^*, \\[-28pt] \nonumber \end{align}

(3.2) \begin{align} \begin{cases} \tau _{0}^* = 1-G(y^*),\\[5pt]\displaystyle \tau _{\beta }^*= \frac {\beta }{1+\beta }\frac {1}{\delta _1^*}\left (\theta ^*(1-G(y^*))-\int _{y^*}^{Re_\tau ^*}\left (1-\frac {\widetilde u^{*2}}{u_e^{*2}}\right )\mathrm d y^*\right ),\\[12pt]\displaystyle \tau _{e}^*=\gamma \frac {\partial u_e^*}{u_e^*\,\partial \beta }\left (-\int _0^{y^*}\widetilde u^{*2}\,\mathrm dy^*+G(y^*)\int _0^{Re_\tau ^*}\widetilde u^{*2}\,\mathrm dy^*\right ),\\[12pt]\displaystyle \tau ^*_{\gamma }=\gamma \left ( \int _0^{y^*}\frac {\partial \widetilde u^{*2}}{\partial \beta }\,\mathrm d y^* -\widetilde u^*\int _0^{y^*}\frac {\partial \widetilde u^*}{\partial \beta }\,\mathrm d y^* -G(y^*)\right. \\[12pt]\qquad \left.\left ( \int _0^{Re_\tau ^*} \frac {\partial \widetilde u^{*2}}{\partial \beta }\, \mathrm d y^* -u_e^*\int _0^{Re_\tau ^*}\frac {\partial \widetilde u^*}{\partial \beta }\,\mathrm d y^* \right ) \right ). \end{cases} \\[0pt] \nonumber \end{align}

The first part, $\tau _{0}^*$ , is the shear stress introduced by the natural increase in the friction Reynolds number, independent of PG. It is also referred to as the Reynolds number effect, exactly the same form as (1.9), which is the balanced ZPG TBL case. From (2.17), it is easy to verify that $G(0)=0$ and $G(Re_\tau ^*)=1$ , yielding $\tau _{0}^*(0)=1$ and $\tau _{0}^*(Re_\tau ^*)=0$ , which is consistent with ZPG boundary conditions.

The second part, $\tau _{\beta }^*$ , demonstrates the effect of equilibrium PG on the entire profile, which is independent of non-equilibrium term $\gamma$ . The $(1+\beta )$ in the denominator is the result of scaling by $u_\beta ^{2} = u_\tau ^2 (1+\beta )$ , and actually, if scaled by the traditional viscous coordinate, $\tau _{\beta }^+$ is directly proportional to $\beta$ . At the boundary, $\tau _{\beta }^*(0)=-\beta /(1+\beta )$ and $\tau _{\beta }^*(Re_\tau ^*)=0$ . Together with $\tau _{0}^*$ , we have $\tau _{0}^*(0)+\tau _{\beta }^*(0)=1/(1+\beta )$ , which is consistent with wall conditions $\tau ^*(0)=\tau /(\rho u_\beta ^{2})=1/(1+\beta )$ .

The third part, $\tau _{e}^*$ , and the fourth part, $\tau _{\gamma }^*$ , are both proportional to $\gamma$ , which do not appear in the RANS equations, and these two parts are considered as the representation of first-order local disequilibrating effects. Also, $\tau _{e}^*$ is proportional to $\partial u_e^{*}/\partial \beta$ , therefore representing the coupling effect between PG and the outer flow development, while $\tau _{\gamma }^*$ is related to the integral of $\partial u^*/\partial \beta$ , which indicates the real local non-equilibrium effect induced by the varying PG. We have $\tau _{e}^*(0)=\tau _{e}^*(Re_\tau ^*)=\tau _{\gamma }^*(0)=\tau _{\gamma }^*(Re_\tau ^*)=0$ , indicating that the local disequilibrating effect only impacts regions far from the wall and far from the outer edge.

The validation results are shown in figures 3, 4 and 5. Among them, figure 3 shows the results of flow b1n, a rather mild APG flow, figure 4 shows the results of flow m16n, a moderate APG flow, and figure 5 shows the results of flow m18n, a rather strong APG flow. In all three figures, the four plots on the left represent the validation results of mean velocity profiles, where the black line shows reference data obtained directly from DNS or LES, the green line shows original model by Subrahmanyam et al. (Reference Subrahmanyam, Cantwell and Alonso2022), the blue dashed line denotes the estimation-step result $\widetilde u^*$ derived from (2.9), and the red line denotes the correction-step result $u^*$ derived from (2.19). Note that the parameters $(k,a,m,b,n)$ of the Subrahmanyam et al. (Reference Subrahmanyam, Cantwell and Alonso2022) model that we call here are the averages of the parameters given in their original paper (§ 6, table 5). The four plots on the right represent the validation results of total shear stress, where the black line still shows reference data from DNS or LES, the red line denotes the correction-step result $\tau ^*$ from (2.17), and the blue line denotes its decomposition $\tau _{0}^*+\tau _{\beta }^*$ from (3.1), which is the part independent of $\gamma$ . The discrepancy between the red line and the blue line is actually the first-order local disequilibrating effect that is proportional to $\gamma$ given by our estimation–correction model. All the figures are converted to the commonly used friction-based scaling.

Figure 3.

Velocity and total shear stress profiles in flow field b1n (Bobke et al. Reference Bobke, Vinuesa, Orlu and Schlatter2017), showing different streamwise locations. (a,c,e,g) Profiles of mean velocity. The black solid line represents the reference LES data, the green solid line represents the original model by Subrahmanyam et al. (Reference Subrahmanyam, Cantwell and Alonso2022), the blue dashed line represents the estimation $\widetilde u^*$ obtained from (2.9), and the red solid line represents the correction $u^*$ obtained from (2.19). (b,d,f,h) Profiles of total shear stress. The black line represents the reference LES data, the red line represents the present model $\tau ^*$ from (2.17), and the blue line represents its decomposition $\tau _{0}^*+\tau _{\beta }^*$ from (3.1). All the quantities are converted to the commonly used friction-based scaling. The specific flow parameters are (a,b) $\hat x=21, \beta = 1.1, \gamma = -1.4\times 10^{-5}, Re_\tau = 448$ , (c,d) $\hat x=29, \beta = 1.0, \gamma = -1.2\times 10^{-5}, Re_\tau = 568$ , (e,f) $\hat x=41, \beta = 0.9, \gamma = -0.9\times 10^{-5}, Re_\tau = 716$ , (g,h) $\hat x=52, \beta = 0.6, \gamma = -14.7\times 10^{-5}, Re_\tau = 851$ .

Figure 4.

Velocity and total shear stress profiles in flow field m16n (Bobke et al. Reference Bobke, Vinuesa, Orlu and Schlatter2017), showing different streamwise locations. The legend follows the same convention as in figure 3. Note that in (b), the blue line and the red line are so close that they almost collapse. The specific flow parameters are (a,b) $\hat x=17, \beta = 2.8, \gamma = -1.1\times 10^{-5}, Re_\tau = 422$ , (c,d) $\hat x=23, \beta = 2.5, \gamma = -4.0\times 10^{-5}, Re_\tau = 539$ , (e,f) $\hat x=32, \beta = 2.1, \gamma = -4.8\times 10^{-5}, Re_\tau = 704$ , (g,h) $\hat x=41, \beta = 0.9, \gamma = -50.0\times 10^{-5}, Re_\tau = 861$ .

Figure 5.

Velocity and total shear stress profiles in flow field m18n (Bobke et al. Reference Bobke, Vinuesa, Orlu and Schlatter2017), showing different streamwise locations. The legend follows the same convention as in figure 3. Note that in (d), the blue line and the red line are so close that they almost collapse. The specific flow parameters are (a,b) $\hat x=15, \beta = 4.1, \gamma = 14.6\times 10^{-5}, Re_\tau = 402$ , (c,d) $\hat x=21, \beta = 4.5, \gamma = -2.4\times 10^{-5}, Re_\tau = 503$ , (e,f) $\hat x=30, \beta = 3.5, \gamma = -15.9\times 10^{-5}, Re_\tau = 669$ , (g,h) $\hat x=38, \beta = 0.8, \gamma = -95.4\times 10^{-5}, Re_\tau = 858$ .

In figures 3(a,c,e,g), it can be seen that the prediction of the correction-step result $u^*$ is highly accurate at all locations, generally superior to the original model, and far superior to the estimation-step result $\widetilde u^*$ . Three models all behave well in the inner region. In the buffer region, the original model is larger, while $\widetilde u^*$ is smaller, and in the wake region, the original model and $\widetilde u^*$ are both smaller than the LES result. This is because the initial estimation $\widetilde \tau ^*$ in (2.8), which is obtained based on channel flow results, is a straight line, whereas in an APG TBL, real shear stress is a concave function that exceeds the initial $\widetilde \tau ^*$ at all points except the boundaries, as shown in figures 3(b,d,f,h). As a result, the Subrahmanyam et al. (Reference Subrahmanyam, Cantwell and Alonso2022) original model and $\widetilde u^*$ which are based on $\widetilde \tau ^*$ both underestimate. Meanwhile, in order to balance this outer-region error, based on the min-square-error criterion, the original model forcibly obtains the optimal parameters, resulting in the non-physical overestimation in the buffer region. In short, the advantage of our model $u^*$ is significant. However, the accuracy of $u^*$ on flow b1n is within expectation, as the tunable parameters $(k,a,m,b,n)$ of our current model were determined based on this dataset.

In figures 3(b,d,f,h), it can be seen that the correction-step prediction $\tau ^*$ is also very close to the LES results. In the inner region, at locations with smaller $|\gamma |$ where the boundary layer is in near-equilibrium state, such as figures 3(d,f), $\tau ^*$ agrees better with LES data compared to figures 3(b,h), where the local non-equilibrium effect is rather significant. In the outer region, none of the profiles matches the data well. One of the key reasons for this is that although we have set the critical ratio $n = 5$ in criterion (2.21) to ensure that the shear stress tends to zero outside the boundary layer, it remains at approximately $0.1$ as shown by the black lines. However, in theoretical analysis, $\tau ^*(Re_\tau ^*)$ should accurately be zero. Furthermore, shown by the discrepancy between the red and blue lines in figures 3(b,d,f,h), the term $\tau _{e}^* + \tau _{\gamma }^*$ that indicates the local disequilibrating effect always adjusts $\tau ^*$ to the correct direction. In particular, in figure 3(h), the effect of the streamwise derivative of PG on $\tau ^*$ profile is clearly visible.

Except for flow field b1n in figure 3, all the results shown in other figures are part of the validation set rather than the calibration set.

In figure 4, the results are generally consistent with those in figure 3. As shown in figures 4(a,c,e,g), the correction-step result $u^*$ maintains high accuracy across all locations, and its improvement over the estimation $\widetilde u^*$ and the original model is even more evident. In figures 4(b,d,f,h), although the final result $\tau ^*$ exhibits some numerical deviation, it still captures the overall trend of the LES data well.

Figure 5 still presents almost the same results as figures 3 and 4. In figures 5(a,c,e,g), the prediction of $u^*$ is still accurate at all locations, far better than $\widetilde u^*$ and the original model. In figures 5(b,d,f,h), $\tau ^*$ still captures the general shape and trend of the LES data, despite exhibiting non-negligible deviations in magnitude. In general, for flow m18n, the prediction deviates more from LES data compared to flow b1n or m16n. This can be attributed partially to the stronger APG effect, which induces more significant local non-equilibrium effects. However, as a validation set, the observed error of $u^*$ in figures 5(e,g) remains within an acceptable range.

In figures 5(b,d,f,h), the local disequilibrating effect terms $\tau _{e}^*$ and $\tau _{\gamma }^*$ still adjust $\tau ^*$ to the correct direction, and especially in figure 5(h) where $|\gamma |$ is considerably large, the local disequilibrating effect plays a significantly vital role in the total shear stress profile. However, the current stress profile predictions still leave room for improvement. One possible reason for the overprediction in figures 5(b,d) may be the relative low Reynolds number( $ \textit{Re}_\tau =402,503$ ), which leads to incomplete turbulence development, and consequently an overestimation of the Reynolds stress $\langle u^\prime \,v^\prime\rangle$ .

To avoid excessive discussion, the validation results for other flow fields are not presented in such detail. Instead, figure 6 shows the comparison of the cumulative relative error of velocity prediction given by the original model and the current model. The relative error is defined as

(3.3) \begin{align} \mathrm{Error} = \frac {\int _0^{Re_\tau }|u^+-u_0^+|\,\mathrm{d}y^+}{\int _0^{Re_\tau } u_0^+\,\mathrm{d}y^+}, \end{align}

where $u^+$ is the prediction value, and $u_0^+$ is the true value given by DNS/LES. As shown by figure 6, our model performs generally better than the original model. In flow fields where $\beta$ is relatively large or varies rapidly, such as m18n, m16n and b2n, the predictive accuracy of our model is significantly higher than that of the original model. When $\beta$ is small, our model still performs slightly better than the original model, such as in flows b1n and m13n. However, at higher Reynolds numbers, the prediction error of our model increases noticeably and becomes comparable to that of the original model, such as in flow b14. A possible reason is that our calibration dataset is based on flow field b1n, which corresponds to a relatively low Reynolds number. This suggests a possible Reynolds number dependence of the model parameters.

Figure 6.

Cumulative relative error of velocity prediction. The crosses show the prediction error given by the original model of Subrahmanyam et al. (Reference Subrahmanyam, Cantwell and Alonso2022), while the dots show the prediction error given by the current model. The error is given by (3.3).

To intuitively demonstrate the role of the local disequilibrating effect, we extract one profile each from the flow fields m13n and m18n. These two profiles share the same $ \textit{Re}_\tau$ and $\beta$ , but differ significantly in $\gamma$ , with ratio approximately 9. The streamwise locations of the selected profiles are indicated by the blue triangle markers in figure 1(a). As shown by figures 7(a,b), it is evident that both the velocity and shear stress distributions differ significantly between the two profiles. Encouragingly, our model, through the correction involving $\gamma$ , successfully captures the differences between the two profiles.

As shown by figures 7(a,b), a large negative value of $\gamma$ leads to reduced velocities in the near-wall region, increased velocities in the outer region, and an overall increase in total shear stress. This is because a larger absolute value of $\gamma$ indicates a more rapid variation of the PG, while the velocity and stress profiles (especially the stress profile) cannot adjust to such rapid changes, exhibiting a lag, as described by Bobke et al. (Reference Bobke, Vinuesa, Orlu and Schlatter2017). Regarding the current cases, if the cases with small $|\gamma |$ are considered approximately in equilibrium, then the cases with larger negative $\gamma$ largely reflect the flow under a stronger PG. Therefore, both the stress and velocity profiles are closer to those observed at higher $\beta$ . As is well known, a strong APG leads to reduced velocities in the near-wall region, increased velocities in the outer region, and an overall increase in total shear stress; thus a large negative $\gamma$ also produces these effects. However, as stated above, since only local disequilibrating effects are obtained, the results remain subject to non-negligible deviations.

Figure 7.

Two profiles extracted from flow fields m13n and m18n (Bobke et al. Reference Bobke, Vinuesa, Orlu and Schlatter2017), with the same $\beta$ and $ \textit{Re}_\tau$ , but different $\gamma$ : (a) mean velocity profiles, and (b) total shear stress profiles. The blue lines indicate flow m13n, $\hat {x} = 48$ , $\beta = 0.77$ , $\gamma = -11.1 \times 10^{-5}$ , $ \textit{Re}_\tau = 862$ . The red lines indicate flow m18n, $\hat {x} = 37.8$ , $\beta = 0.77$ , $\gamma = -97.6 \times 10^{-5}$ , $ \textit{Re}_\tau = 861$ .

Finally, a detailed analysis was performed on the contributions of each decomposed term of $\tau ^*$ in (3.1), based on the data calculated at six different streamwise locations of flow m18n. The corresponding results are shown in figure 8.

Figure 8(a) shows the profile of $\tau _{0}^*$ , which is the part that is completely independent of PG. Its profile is similar to the ZPG TBL case, or even the straight-line profile of the channel flow case. This explains why the channel flow shear stress profile can be used directly in ZPG TBLs, and can be used as the initial estimation in our model. All six profiles start from $\tau _{0}^*(0)=1$ at the wall, and monotonically decrease to $\tau _{0}^*(Re_\tau ^*)=0$ at the outer edge. The profiles are highly similar.

Figure 8(b) shows the profile of $\tau _{\beta }^*$ , which is the part that is directly proportional to $\beta$ and independent of non-equilibrium effect $\gamma$ . All six profiles start from $\tau _{\beta }^*(0)=-\beta /(1+\beta )$ , reach a peak at approximately $0{-}1$ , and finally drop to zero at the outer edge. The peak value gradually decreases along the streamwise direction. Despite the difference in $\beta$ , the six profiles still exhibit strong similarity.

Figures 8(c,d) show the profiles of $\tau _{e}^*$ and $\tau _{\gamma }^*$ , respectively, which are the parts that are directly proportional to $\gamma$ . These two terms are almost equal in magnitude but opposite in sign, achieving a delicate balance between the interaction of PG with outer flow development, and the local pressure non-equilibrium effect. All the profiles vary from zero back to zero, undergoing a transition from positive to negative, or the opposite, depending on the sign of $\gamma$ .

Figures 8(e,f) show the profile of $\tau _{0}^* +\tau _{\beta }^*$ , which is the part that is independent of $\gamma$ , and the profile of $\tau _{e}^*+\tau _{\gamma }^*$ , which is the part that is proportional to $\gamma$ and shows a first-order local disequilibrating effect. A comparison between the two plots reveals that the non-history effect component of the total stress is relatively consistent across all profiles, while the local-disequilibrating-effect term varies sensitively with the magnitude of $\gamma$ . The local disequilibrating effect peaks at approximately $1.3$ , contributing approximately half of the total stress.

From the analysis of figure 8, we can see that the height and location of the shear stress peak are determined by the $\tau _\beta ^*$ term, because $\tau _0^*$ does not exhibit a peak, while the other two terms are very small in magnitude most of the time. Therefore, the mismatch of total shear stress profiles shown in figures 3(b,d,f,h), 4(b,d,f,h) and 5(b,d,f,h), and also the mismatch of the peak location shown in figure 7(b), are probably caused by $\tau _\beta ^*$ . Unfortunately, due to the lack of true values, further analysis is challenging. We will continue to devote our efforts to investigating this issue.

Figure 8.

Profiles of the components of the correction-step result $\tau ^*$ , calculated at different streamwise locations of flow field m18n (Bobke et al. Reference Bobke, Vinuesa, Orlu and Schlatter2017): (a,b,c,d) contributions of $\tau _{0}^*,\tau _{\beta }^*,\tau _{e}^*,\tau _{\gamma }^*$ , respectively; (e) $\tau _{0}^* +\tau _{\beta }^*$ ; (f) $\tau _{e}^*+\tau _{\gamma }^*$ . All the quantities are converted to the commonly used friction-based scaling.