4.1. Tripping effects

In the current section, an investigation is performed on whether the presence of a small spanwise curvature influences the downstream distance taken for the boundary layer to become a fully developed turbulent boundary layer. With that goal, the following criteria are considered.

1. (i) Statistical homogeneity of the velocity profiles in the span.

2. (ii) Presence of a logarithmic layer in the velocity profile.

3. (iii) Vanishing of residual harmonics of the trip in the spanwise spectra.

The first two conditions were previously developed by Plasseraud et al. (Reference Plasseraud, Kumar, Ma and Mahesh2022) to assess the effectiveness of a trip on the windward side of a prolate spheroid. These criteria verify that the resulting boundary layer is turbulent and that no signature of the trip is visible, ensuring that the state of the flow is not a function of the specific geometric characteristics of the trip. Figure 6 shows the instantaneous velocity field in the wake of the trips for case 2000 and illustrates how homogeneity is evaluated. The immediate vicinity of the trips is heterogeneous in the span, with low-speed streaks behind each trip and pockets of higher velocity between the trips. The mean velocity field then homogenises as the wakes of the trips diffuse. The homogeneity criterion is evaluated by taking velocity profiles along $z$ -planar slices behind and between each trip, shown in figure 6 with solid and dashed lines, respectively, for one trip. Profiles at the same streamwise stations along slices behind the trip are averaged together, and profiles at the same streamwise station along slices between the trip are averaged together. Once homogeneity is achieved, spanwise averaging is performed by considering data from every grid element in the span.

Figure 6. Contour plot of instantaneous transitionary streamwise velocity field for case 2000 at $n = 0.5h$ highlighting profile extraction locations.

Figure 7 shows the velocity profiles between ( $u_{bt}$ ) and behind the trip ( $u_{bh}$ ) for case 2000 at $s = 20h, 30h, 40h, 50h$ . The profiles behind the trips initially have a 15 % momentum deficit with respect to the profiles between the trips, which have a momentum surplus as the former are in the recirculation zones of the obstacles while the latter are in the accelerated region. The two profiles converge with increasing distance from the trip as diffusion homogenises the flow. Figure 7 also shows where a logarithmic layer develops in the profile, indicating that a canonical turbulent boundary layer is found between $s = 40h$ and $50h$ with ZPG. The same analysis is conducted for the case with spanwise curvature of radius $500h$ . A similar momentum deficit is visible in figure 8, and profiles behind and between the trips converge with increasing streamwise distance. However, the pressure gradient transports fluid elements in recirculation regions behind the trip into high-speed streaks between the trips, promoting mixing. Figure 8 shows convergence of the profiles between and behind the trips at a more upstream location of $s \approx 40h$ ; however, logarithmic layer development is not visible until $s \approx 50h$ , indicating that the transition location is unchanged.

Figure 7. Mean velocity profiles behind the trip (solid) and between the trips (dashed) for case 2000 at (a) $s = 20h$ , (b) $s = 30h$ , (c) $s = 40h$ , (d) $s = 50h$ with comparison to the law of the wall (blue).

Figure 8. Mean velocity profiles behind the trip (solid) and between the trips (dashed) for case 2150 at (a) $s = 20h$ , (b) $s = 30h$ , (c) $s = 40h$ , (d) $s = 50h$ with comparison to the law of the wall (blue).

A quantitative metric of the homogeneity of the flow is calculated as the root mean square (RMS) of the difference between velocity profiles between and behind the trip, compared graphically between cases 2000 and 2150 in figure 9:

(4.1) \begin{equation} \textit{RMS}_{bh/bt} = \sqrt {\frac {1}{N}\sum ^{N} \left(\int _0^{n_{\textit{max}}}(u_{bh}-u_{bt}) dn\right)^2}\!. \end{equation}

Figure 9. Comparison of $\textit{RMS}_{bh/bt}$ for case 2000 (black) and case 2150 (red).

The deviation for both cases is at a maximum in the near-trip region and then decreases downstream. The streamwise distance at which $\textit{RMS}_{bh/bt}$ plateaus is taken as the transition distance, $s_t$ , which corresponds to the streamwise location where a logarithmic layer forms in figures 7 and 8. Here $\textit{RMS}_{bh/bt}$ for case 2150 shows signs of faster convergence before tapering out, matching the prior observations. The noise in the RMS for case 2150 can be attributed to fluid elements travelling in the span. The smooth convergence of case 2000 exhibited in figure 9 is not present in the presence of curvature, as the higher velocity fluid elements between the trip and the slower velocity fluid elements behind the trip do not gradually diffuse.

Lastly, to further examine the effect of spanwise curvature on transition location, the spanwise spectra of velocity fluctuations is considered. A discrete Fourier transform is applied in the spanwise direction considering $N_z$ uniformly spaced points at $n=h$ using the same sampling frequency and period as in figure 3. The energy spectrum is defined as

(4.2) \begin{equation} E_{\textit{uu}}(k_z) = \frac {1}{N_z}\overline {|\hat {u'}(k_z, t)|^2},\end{equation}

where $\hat {u'}(k_z, t)$ is the Fourier transform of streamwise fluctuations. The energy spectrum for cases 2000 and 2150 at $s = 5h, 25h, 45h, 65h, 85h$ is shown in figure 10. In the immediate wake of the trips ( $s = 5h$ ), the spectra show elevated energy levels at low wavenumbers ( $k_z \approx 2{-}4$ ), a secondary peak at $k_z = 2\pi h$ and decay with increasing $k_z$ . With increasing streamwise distance, the spectral energy progressively decreases until $s_t$ where the spectral signature is similar for $s \gt s_t$ . This reflects the natural relaxation of the boundary layer towards an equilibrium turbulent boundary layer, as residual harmonics of tripping are only visible at $s= 5h$ . Both case 2000 and 2150 exhibit a similar behaviour at large wavenumbers.

Figure 10. Spanwise energy spectra of velocity fluctuations $E_{\textit{uu}}$ for case 2000 (solid) and case 2150 (dashed) progressing from $s = 5h$ to $s = 85h$ , incremented by $\Delta s = 20h$ .

4.2. Crossflow profile

The presence of transverse curvature introduces a crossflow (secondary flow), as the initially 2-D boundary layer gains a spanwise component. Figure 11 shows a schematic of a 3-D boundary layer. The secondary flow is manifested as a non-uniform $w$ component of velocity and is a consequence of the spanwise pressure gradient $\partial \overline {p}/\partial {\zeta }$ in the boundary layer coupled with a wall-normal velocity gradient.

Figure 11. Three-dimensional boundary layer schematic.

Figure 12. Streamwise (- - -) and spanwise (—) velocity profiles for case 2000 (black), case 2110 (green) and case 2150 (red) at $s = 140h$ .

Figure 12 shows the streamwise $\overline {u}_{99}^+$ and spanwise $\overline {w}_{99}^+$ velocity profiles at $s = 140h$ for case 2000 (ZPG at $\textit{Re}_h = 2000$ ), case 2110 (with $R = 1000h$ spanwise curvature) and case 2150 (with $R = 500h$ spanwise curvature). The streamwise velocity profiles are similar and have a logarithmic layer in all the cases although the maximum value is highest in case 2000. The difference is more pronounced in the spanwise velocity profile. The spanwise velocity is negligible in case 2000 while it is not in cases with spanwise curvature. Here $\overline {w}_{99}^+$ increases from zero at the wall to a maximum at $n^+ \approx 40$ . The location of this maximum corresponds to the bottom of the logarithmic layer. The spanwise velocity then decreases until the edge of the boundary layer. The formation of a crossflow in a spanwise pressure gradient field can be understood by considering the streamline-transverse momentum equation in the steady regime as listed in (3.3). Assuming homogeneity in the spanwise direction, a small variation in the streamwise direction and $v \approx 0$ :

(4.3) \begin{equation} 0 = \frac {\overline {u}_s^2}{r} + f + \nu\! \left(\frac {\partial {\overline {u}_{{\zeta }}}}{\partial {n^2}} - \frac {\overline {u}_{{\zeta }}}{R^2}\right)\! .\end{equation}

Viscous effects are negligible in the free stream, recovering the prescribed body force. Within the boundary layer, if $\overline {u}_{{\zeta }} = 0$ then $\overline {u}_s^2/r = -f$ ; however, since $f$ is a constant, this equation can only be valid if the left-hand side is constant. More specifically, it can only be valid if $\overline {u}_s$ is constant because if $r$ varies, $\overline {u}_{{\zeta }}$ cannot be constant, hence, a contradiction. Therefore, a gradient of $\overline {u}_s$ necessarily implies a non-zero and varying $\overline {u}_{{\zeta }}(n)$ within the boundary layer ( $\overline {w}_{99} = 0$ by definition). This non-zero $\overline {u}_{{\zeta }}(n)$ is visible as a crossflow, also called secondary flow. The crossflow can be analytically interpreted by considering (3.3) in local streamline coordinates and neglecting viscous and turbulent stresses:

(4.4) \begin{align} \frac {\overline {u}_s^2}{r(n)} = \frac {{U_\infty }^2}{R} \rightarrow r(n) = \frac {\overline {u}_s^2R}{{U_\infty }^2}. \end{align}

Intuition about the behaviour of $r(n)$ can be gained by considering the log layer where $\overline {u}_s^+ = ({1}/{\kappa })\ln (n^+) + B \Rightarrow r(n^+) = (Ru_\tau ^2/{U_\infty }^2)( ({1}/{\kappa })\ln (n^+)+B)^2$ , where $\kappa =0.41$ and $B=5.2$ are constants from the logarithmic law of the wall. The angle of the streamwise axis (where $s$ is the streamwise coordinate from the start of curvature) is defined as

(4.5) \begin{align} \alpha (s,n) = \frac {s}{r(n)}. \end{align}

Thus, at any point, the relative slip velocity (figure 14) between the free-stream velocity at $n \geqslant \delta _{99}$ and the velocity within the boundary layer at $n \lt \delta _{99}$ is

(4.6) \begin{align} \overline {w}_{99}(s,n) &= \overline {u}_{99}(s,n) \sin\! \left(\alpha (s,n)-\alpha \left(s, \delta _{99}\right)\right)\!, \end{align}

(4.7) \begin{align} \overline {w}_{99}(s,n) &= \overline {u}_{99}(s,n) \sin\! \left(\frac {s}{r(n)}-\frac {s}{R}\right)\! ,\end{align}

(4.8) \begin{align} \overline {w}_{99}(s,n) &= \overline {u}_{99}(s,n) \sin\! \left(\frac {s{U_\infty }^2}{\overline {u}_{99}(s,n)^2 R}-\frac {s}{R}\right)\!, \end{align}

(4.9) \begin{align} \overline {w}_{99}(s,n) &= \overline {u}_{99}(s,n) \sin\! \left(\frac {s}{R}\left(\frac {{U_\infty }^2}{\overline {u}_{99}(s,n)^2}-1\right)\right)\! . \end{align}

Similarly, the secondary flow can be derived in the viscous sublayer:

(4.10) \begin{equation} \overline {w}_{99}(s, n) = \frac {n^2}{2 R} \textit{Re}_h + n\frac {\partial \overline {w}_{99}}{\partial n}|_{n=0}. \end{equation}

Figure 13. Comparing measured (red) and analytical crossflow profiles (black) within the log layer (—, (4.9)) and viscous sublayer (- - -, (4.10)) for case 2150 at $s=145h$ .

Figure 13 shows a comparison of the derived components of the secondary flow with the measured crossflow for $R = 500h, \textit{Re}_h = 2000, s = 145h$ . Good agreement is obtained in the log layer (4.9) and the viscous sublayer (4.10). The positive wall-normal velocity gradient implies that the local radius of curvature $r$ must also increase to balance (4.3). Figure 15 shows the angle between the local velocity vector and the free-stream velocity vector along the domain for cases 2000, 2110 and 2150 (ZPG and prescribed spanwise free-stream curvature of $R = 500h$ and $R = 1000h$ at $\textit{Re}_h = 2000$ ) defined by Cooke (Reference Cooke1958) as $\epsilon = \cos ^{-1}(\langle \overline {u}_{99}(n), \overline {v}_{99}(n), \overline {w}_{99}(n) \rangle \boldsymbol{\cdot }\langle \overline {u}_{99}(\delta _{99}), \overline {v}_{99}(\delta _{99}), \overline {w}_{99}(\delta _{99}) \rangle )$ . As expected, the orientation of the velocity vector is constant throughout the entire profile in the absence of any external forcing. From the cases studied, $\epsilon$ scales linearly with the curvature. The orientation of the velocity vectors start to converge towards the free-stream orientation more rapidly within the log layer.

Figure 14. Comparison between the velocity vector within the boundary layer (blue) and at the boundary layer edge (red), emphasising that the orientation of the velocity vector within the boundary layer is not uniform in the presence of spanwise curvature.

Figure 15. Angle between local velocity vector and free-stream velocity vector for case 2000 (black), case 2110 (green) and case 2150 (red) at $s = 125h$ , showing increased curvature within the boundary layer.

4.3. Velocity profiles

Figure 18 shows the hodograph highlighting the spanwise component of velocity versus the streamwise component in the $\delta _{99}$ -aligned coordinate system at $s = 140h$ for cases 2000, 2110 and 2150. A large secondary flow is visible, which is maximum at $\overline {w}_{99}/{U_\infty } \approx 0.05$ at $\overline {u}_{99}/{U_\infty } \approx 0.65$ for case 2110 and $\overline {w}_{99}/{U_\infty } \approx 0.085$ at $\overline {u}_{99}/{U_\infty } \approx 0.7$ for case 2150. The spanwise velocity then decreases closer to the wall and outside the boundary layer.

Figure 16. Mean velocity profiles for case 2000 (black) and case 2150 (red) at (a) $s = 25h$ , (b) $s = 50h$ , (c) $s = 75h$ , (d) $s = 100h$ .

Figure 16 shows the velocity magnitude profiles downstream of the trip for cases 2000 and 2150 (base case and spanwise curvature, $\textit{Re}_h = 2000$ ). In both cases, the boundary layer is initially perturbed and transitional. As curvature effects compound, the two cases start to diverge from each other, with the spanwise curvature case exhibiting a fuller profile. Figure 17 shows the deviation between the base case and the spanwise curvature cases. This deviation is calculated as the RMS of the difference between the two curves, as in (4.1). For both radii of curvature, a local maximum is observed in the vicinity of the trip because of the increased mixing of the recirculation and accelerated region due to the spanwise pressure gradient. The deviation becomes minimal at $s \approx 20h$ and increases monotonically until the end of the domain. This suggests that the characteristic length necessary for the curvature to affect the profile is much longer than the length for the boundary layer to become turbulent.

Figure 17. Deviation between velocity profiles of case 1110 and case 1000 (green) and case 1150 and case 1000 (red).

Figure 18. Hodograph plot $\overline {w}_{99}$ vs $\overline {u}_{99}$ at $s = 140h$ for case 2000 (black), case 2110 (green) and case 2150 (red).

4.4. Integral boundary layer parameters

Figure 19 shows the evolution of the $99{{\rm th}}$ percentile boundary layer thickness $\delta _{99}$ , the displacement thickness $\delta ^{*}$ , the momentum thickness $\theta$ and the shape factor $H = \delta ^{*}/\theta$ , downstream of the trip for cases with and without spanwise curvature at $\textit{Re}_h = 2000$ . These quantities are computed using the methodology proposed by Griffin, Fu & Moin (Reference Griffin, Fu and Moin2021). Both cases 2000 and 2150 have similar values of $\delta _{99}$ , whereas $\delta ^{*}$ , $\theta$ and $H$ are lower after  $s_t$ in the presence of spanwise curvature compared with the baseline case. Here $\delta ^{*}$ , $\theta$ and $H$ exhibit no differences between the two cases until $s_t$ , matching the discussion in § 4.1. The sharp decrease in $H$ in the transitionary region until $s_t$ is consistent with the development of a turbulent boundary layer. The lower shape factor observed in the case with curvature is indicative of a fuller velocity profile (visible in figure 16). The lower shape profile of the case with spanwise curvature indicates that the three-dimensionality of the boundary layer allows the boundary layer to have higher momentum deficit in a smaller region.

Figure 19. Boundary layer evolution for case 2000 (black) and case 2150 (red) showing (a) $\delta _{99}$ thickness, (b) displacement thickness, (c) momentum thickness, (d) shape factor.

4.5. Turbulent stresses

The diagonal Reynolds stress components for cases 2000, 2110 and 2150 are shown in figure 20, and the off-diagonal Reynolds stress terms for the same cases are shown in figure 21. Here $\overline {u_n'^2}$ peaks at the bottom of the log layer with slightly decreasing magnitudes as the radius of curvature decreases; $\overline {u_n'^2}$ , $\overline {u_\zeta '^2}$ and $\overline {u_s'u_n'}$ all have a single maximum at the location of maximum crossflow (figure 12). This maximum is highest for case 2000, followed by case 2110 and lowest for case 2150. Here $\overline {u_n'^2}$ and $\overline {u_n'u_\zeta '}$ show deficits of $5\%$ in case 2110 and $10\,\%$ in case 2150 compared with case 2000, while $\overline {u_\zeta '^2}$ exhibits a reduction of $2\,\%$ and $4\,\%$ ; $\overline {u_s'u_{{\zeta }}'}$ and $\overline {u_n'u_{{\zeta }}'}$ show a reverse trend, with case 2150 having a maximum around twice the value of case 2110. In addition, these two components are not negligible compared with $\overline {u_s'u_n'}$ in the presence of spanwise curvature. The value of $\overline {u_n'u_{{\zeta }}'}$ is slightly lower than $\overline {u_s'u_{{\zeta }}'}$ that is ${\sim} 25\,\%$ of $\overline {u_s'u_n'}$ for case 2150. The maxima of $\overline {u_s'u_{{\zeta }}'}$ and $\overline {u_n'u_{{\zeta }}'}$ for case 2150 are around twice the value of case 2110, suggesting that the value of these components scales with the radius of curvature.

Figure 20. Reynolds stress profiles for case 2000 (black), case 2110 (green) and case 2150 (red), showing (a) $\overline {u_s'^2}^+$ , (b) $\overline {u_n'^2}^+$ , (c) $\overline {u_\zeta '^2}^+$ at $s = 145h$ .

Figure 21. Reynolds stress profiles for case 2000 (black), case 2110 (green) and case 2150 (red), showing (a) $\overline {u_s'u_n'}^+$ , (b) $\overline {u_s'u_{{\zeta }}'}^+$ , (c) $\overline {u_n'u_{{\zeta }}'}^+$ at $s = 145h$ .

Figure 22. Profiles of Reynolds stress tensor component $\overline {u_s'u_{{\zeta }}'}^+$ for case 2000 (black), case 2110 (green) and case 2150 (red) at (a) $s = 25h$ , (b) $s=50h$ , (c) $s=100h$ , (d) $s=145h$ .

Figure 22 shows the evolution of the profiles of $\overline {u_s'u_{{\zeta }}'}^+$ downstream of the trip for cases 2000, 2110 and 2150 (base case and spanwise curvatures $R = 1000h$ and $R = 500h$ at $\textit{Re}_h = 2000$ ). The magnitude near $\overline {u_s'u_{{\zeta }}'}^+$ is zero close to the wall where viscous stresses dominate. While $\overline {u_s'u_{{\zeta }}'}$ is negligible in the case without a pressure gradient, the term increases in the spanwise case near the end of the viscous sublayer starting from $n^+ \approx 5$ . Beyond $s_t$ , the magnitude of $\overline {u_s'u_{{\zeta }}'}$ increases as curvature effects compound.

Figure 23. Evolution of the integral of turbulent kinetic energy for case 2000 (black), case 2110 (green) and case 2150 (red) ( $\textit{Re}_h = 2000$ , baseline and spanwise curvature radii $R = 1000h, 500h$ ).

The total energy in the boundary layer can be studied via normalised turbulent kinetic energy (TKE) $k^+ = ({1}/{2}) \overline {u_i'u_i'}/u_\tau ^2$ integrated in the wall-normal direction. This quantity is shown in figure 23 for the baseline case and with spanwise curvature at $\textit{Re}_h = 2000$ . For all cases, $\int k^+$ is highest in the wake of the trip, decreases to a minimum at $s_t$ (determined in § 4.1) and increases until the end of the domain. The high initial values are due to the perturbation created by the trip. The decay is associated with a dissipation of the wake of the trips. The increase in $k^+$ for $s \gt {s_t}$ can be attributed to the development of the turbulent boundary layer. Turbulent kinetic energy within the boundary layer decreases in the presence of spanwise curvature, which follows the reduction of the diagonal stresses (figure 20). This analysis can be continued by examining the TKE budget equation, which in Cartesian coordinates, is

(4.11) \begin{equation} \frac {\partial k}{\partial t} + \bar {u}_i\partial _ik = \mathcal{P} - \epsilon - \partial _iT_i + \overline {f_iu_i'},\end{equation}

where $f_i$ is the externally applied body force and $T_i$ is the turbulent transport term encompassing viscous diffusion ( $-\nu \partial _i k$ ), turbulent convection ( $({1}/{2})\overline {u_j'u_j'u_i'}$ ) and density-normalised pressure transport ( $\overline {p'u_i'}$ ). The current application considers a steady flow with spanwise homogeneity and body forces in $x$ and $z$ that are constant in $y$ ; after applying continuity, (4.11) can be rewritten in the form

(4.12) \begin{equation} \frac {\partial (\bar {u}k)}{\partial x} + \frac {\partial (\bar {v}k)}{\partial y} = \mathcal{P} - \epsilon - \frac {\partial T_x}{\partial x} - \frac {\partial T_y}{\partial y} + \overline {f_xu'} + \overline {f_zw'}. \end{equation}

The total TKE along a profile can be evaluated by integrating in the wall-normal direction from the wall $y=0$ to some location in the free stream $y \gt \delta _{99} = y^*$ :

(4.13) \begin{align} \frac {\partial }{\partial x}\int _0^{y*}(\bar {u}k + T_x){\rm d}y + \bar {v}k|_0^{y*} &= \int _0^{y*}\! \mathcal{P}{\rm d}y - \int _0^{y*}\! \epsilon {\rm d}y - T_y|_0^{y*} &\quad + \int _0^{y*}\overline {f_xu'}{\rm d}y + \int _0^{y*}\overline {f_zw'}{\rm d}y. \end{align}

The no-slip and no-penetration boundary conditions at the wall and the laminar conditions in the free stream coupled with the growth of the boundary layer in $x$ occurring on a much larger length scale than the thickness of the boundary layer $\delta$ yield an integrated balance between production, dissipation and molecular diffusion due to the wall (shown to be small by figure 20):

(4.14) \begin{equation} \int _0^{y*}\! \mathcal{P}{\rm d}y = \int _0^{y*}\! \epsilon {\rm d}y + \nu \frac {\partial k}{\partial y}\Big|_{y=0}. \end{equation}

Changes in the magnitude of TKE can only occur due to sources and sinks in the original equation, which in this case are production $\mathcal{P}$ and dissipation $\epsilon$ . The other terms redistribute the turbulent energy in space or between components.

Figure 24. Breakdown of contributing terms to TKE production $\mathcal{P}_{\textit{ij}}^+$ for case 2000 (black), case 2110 (green) and case 2150 (red) at $s = 145h$ . The dominant term $\mathcal{P}_{\textit{sn}}^+$ is shown in full opacity and minor terms $\mathcal{O}(\lt 10^{-3})$ are shown faded for clarity.

Production of TKE is defined as $\mathcal{P}_{\textit{ij}} = \overline {u_i'u_j'}\partial _j\overline {u}_i$ and the contribution of each of the individual terms of this double summation are shown in wall units ( $\mathcal{P}_{\textit{ij}}^+ = \mathcal{P}_{\textit{ij}}\nu /u_\tau ^4$ ) in figure 24 and tabularly in table 3. In a ZPG TBL, production can be reduced to $\mathcal{P}_{\textit{sn}}$ with minimal contribution from the other terms. In the presence of spanwise curvature, the streamwise-normal fluctuation terms $\overline {u_s'u_{{\zeta }}'}$ and $\overline {u_n'u_{{\zeta }}'}$ are not negligible, as shown in figure 21, although their corresponding gradients $\partial \overline {u}_s/\partial \zeta$ , $\partial \overline {u}_\zeta /\partial s$ , $\partial \overline {u}_n/\partial \zeta$ and $\partial \overline {u}_\zeta /\partial n$ are small in comparison to $\partial \overline {u}_s/\partial n$ . The presence of crossflow does produce an appreciable wall-normal gradient of spanwise velocity (see $\mathcal{P}_{{\zeta }n}^+$ column in table 3); however, it is still negligible when compared with the wall-normal gradient of streamwise velocity. Thus, $\mathcal{P}_{\textit{sn}}^+$ is still the only dominant term and TKE production decreases, as $\overline {u_s'u_n'}$ is smaller in the presence of spanwise curvature, shown in figures 21, 24 and table 3. Consequently, TKE decreases in the presence of spanwise curvature.

Table 3. Scale of each contributing term to TKE production $P_{\textit{ij}}^+$ for case 2000, 2110 and 2150 at $s = 145h$ . Terms of the order of $10^{-3}$ or smaller are not shown fully are they are negligible compared with the magnitude of $\mathcal{P}_{\textit{sn}}^+$ .

Figure 25. Wall-normal distribution of the structure parameter $a_1$ for case 2150 (red) at $s=145h$ . The typical value for 2-D boundary layers is $a_1=0.15$ (Johnston & Flack Reference Johnston and Flack1996) and is shown with the black dashed line.

The decrease in TKE production can be understood by considering the structure parameter $a_1$ . Defined in (4.15), $a_1$ relates the turbulent shear stress to the TKE and is a metric for the efficiency of turbulent production processes:

(4.15) \begin{equation} a_1 = \frac {\sqrt {\overline {u_s'u_n'}^2 + \overline {u_n'u_{{\zeta }}'}^2}}{2k}. \end{equation}

In most 3-D turbulent boundary layers, $a_1$ is lower than the typical value of 0.15 found in 2-D turbulent boundary layers (Bradshaw & Sendstad Reference Bradshaw and Sendstad1990; Johnston & Flack Reference Johnston and Flack1996; Holstad et al. Reference Holstad, Andersson and Pettersen2010), explaining the drop in TKE production reported by Moin et al. (Reference Moin, Shih, Driver and Mansour1990) and in figure 24. This is because the cross-correlated fluctuations disrupt the random turbulent processes that generate turbulent stress; or in other words, crossflow inhibits turbulent stress. Figure 25 shows the evolution of $a_1$ for the baseline case and the case with a spanwise radius of $R = 500h$ . In the limit, $a_1$ is zero at the wall and then increases sharply in the viscous sublayer, reaching a maximum in the log layer and decaying in the free stream. When the boundary layer is three dimensional, $a_1$ is consistently below the 2-D threshold.

Figure 26. Wall-normal balance of TKE budget highlighting (a) production and dissipation and (b) balance between source and sink terms for case 2000 (black) and case 2150 (red) at $s=145h$ .

The source and sink terms in the TKE budget (4.11) are shown in figure 26. The production curves match those of figure 24. Dissipation is at a maximum at the wall, where it is balanced by molecular diffusion (Pope Reference Pope2000). Here $\epsilon$ rises sharply in the viscous sublayer until $n^+ = 5$ , at which point the case with spanwise curvature exhibits smaller magnitudes of $\epsilon$ than the ZPG case. The balance between production and dissipation is recovered at $n^+ \approx 40$ at the bottom of the log layer where the turbulent transport terms are 0 for both cases showing that an approximation of $\mathcal{P}=\epsilon$ is appropriate beyond the buffer layer. Lower production and TKE lead to less turbulent fluctuation intensity implying that the turbulence is less energetic in the presence of spanwise curvature.

Figure 27. Evolution of friction velocity for case 2000 (black), case 2110 (green) and case 2150 (red).

Figure 27 shows the friction velocity $u_{\tau } = \sqrt {\tau _w/\rho }$ (where $\rho$ is the density of the fluid and $\tau _w$ is the wall shear stress computed as $\rho \nu({{\rm d}\overline {u}}/{{\rm d}n}) |_{n=0}$ ) versus streamwise distance for the cases with and without spanwise curvature at $\textit{Re}_h = 2000$ . For all cases, $u_{\tau }$ increases rapidly in the wake of the trip and decays until the end of the domain. The drop in wall shear stress after transition can be attributed to an increase of the boundary layer thickness (increasing $\delta _{99}$ ) as the boundary layer develops. Thicker turbulent boundary layers have smaller velocity gradients in the wall-normal direction, leading to slower velocity in the near-wall region. While $u_{\tau }$ in the baseline case decreases monotonically, it is constant in the presence of spanwise curvature starting from $s_t$ . In the free stream, the prescribed body force is orthogonal to the velocity and, thus, produces no work; however, inside the boundary layer, the velocity vector has a different orientation. The body force is thus not orthogonal to the velocity inside the boundary layer and produces some work. This could explain the higher values of wall shear stress in the curvature cases compared with the baseline case. The asymptotic value of $u_{\tau }$ increases as the applied curvature decreases.

4.6. Alignment between shear stresses and turbulent stresses

Figure 28 shows the difference between the horizontal shear angle $\gamma _g$ and the horizontal turbulent stress angle $\gamma _\tau$ , defined in (4.16):

(4.16) \begin{equation} \gamma _g = \mathrm{tan}^{-1}\frac {\partial \overline {u}_{{\zeta }}/\partial n}{\partial \overline {u}_s/\partial n}, \quad \quad \gamma _\tau = \mathrm{tan}^{-1}\frac {\overline {u_{{\zeta }}'u_n'}}{\overline {u_s'u_n'}} .\end{equation}

The angle difference $\Delta \gamma$ is positive close to the wall ( $n^+ \lt 30$ ) and decreases quickly to reach a minimum at $n^+ \approx 50$ with $\Delta \gamma \approx -6^\circ$ for case 2110 and $\Delta \gamma \approx -11^\circ$ for case 2150. The positive $\Delta \gamma$ near the wall is attributed to the minimal impact of Reynolds stresses compared with shear within the viscous sublayer. The drop and minimum of $\Delta \gamma$ in the buffer region and at the bottom of the log layer are a consequence of the presence of the horizontal Reynolds stresses $\overline {u_s'u_{{\zeta '}}}$ as observed in figure 21. The lag in $\gamma _\tau$ with respect to $\gamma _g$ in the presence of crossflow ( $\overline {u_n'u_{{\zeta }}'}/\overline {u_s'u_n'} \lt ( {\partial {\overline {u}_{{\zeta }}}}/{\partial {n}})/({\partial {\overline {u}_s}}/{\partial {n}})$ ) is widely accepted and has been reported by Bradshaw & Pontikos (Reference Bradshaw and Pontikos1985), Chesnakas & Simpson (Reference Chesnakas and Simpson1996) and Hu et al. (Reference Hu, Hayat and Park2023). In other words, the shear stress vector changes direction more slowly than the velocity gradient vector. This observed angle difference between the shear and the turbulent stress is important to note in the context of turbulence modelling. Specifically, Reynolds-averaged Navier–Stokes models that rely on the Boussinesq hypothesis assume a collinear relationship between the deviatoric component of Reynolds stress and the mean rate of strain; however, figure 28 shows that this assumption may not be valid even for small curvature values such as those considered in the present study as the eddy viscosity must vary in the streamwise and spanwise directions (Bradshaw & Sendstad Reference Bradshaw and Sendstad1990).

Figure 28. Flow angle $\Delta \gamma = \gamma _g - \gamma _\tau$ for case 2000 (black), case 2110 (green) and case 2150 (red) at $s = 145h$ .

4.7. Turbulent structure

Coherent structures within the flow can be visualised with isocontours of $\lambda _2$ , defined as the second largest eigenvalue of the tensor $\epsilon _{ik}\epsilon _{kj} + \varOmega _{ik}\varOmega _{kj}$ (Jeong & Hussain Reference Jeong and Hussain1995). The rate-of-strain tensor $\epsilon _{\textit{ij}}= ({1}/{2})(\partial _iu_j + \partial _ju_i)$ describes fluid element deformation and the vorticity tensor $\varOmega _{\textit{ij}}= ({1}/{2})(\partial _iu_j - \partial _ju_i)$ describes fluid element rotation. Figure 29 shows instantaneous snapshots of vortical structures at $\textit{Re}_h=1000$ for the baseline case (panel a) and the case with spanwise curvature of radius $R=500h$ (panel b). Horseshoe vortices are visible around each trip and hairpin structures are found in the near wake. These coherent structures interact with each other and break down as the flow becomes turbulent. The location of this breakdown, $s_t$ , appears to be similar for case 1150 and case 1000, as noted in § 4.1. Farther downstream, the vortical structures are oriented with the direction of the mean free-stream flow in the presence of spanwise curvature. This is shown in greater detail in figures 30 and 31, which highlight the $x \in [100h, 140h]$ region. The top-down view shows remnants of horseshoe structures and streamwise vortex cores rotating both clockwise and counter-clockwise about the $x$ axis. The three-dimensionality of the boundary layer is apparent in figure 31; the vortical structures are aligned with the mean free-stream flow and have been deflected from the $x$ axis.

Figure 29. Vortical structure evolution by isocontours of $\lambda _2=0.15{U_\infty }^2/h^2$ coloured with streamwise velocity for (a) case 1000 (ZPG) and (b) case 1150 (spanwise curvature with radius $R = 500h$ ).

Figure 30. Visualisation of instantaneous vortical structures for case 1000 by isocontours of $\lambda _2=0.15{U_\infty }^2/h^2$ coloured with streamwise vorticity $\omega _x$ from $x \in [100h, 140h]$ across the entire span. Blue shading represents negative $\omega _x$ and red shading represents positive $\omega _x$ .

Figure 31. Visualisation of instantaneous vortical structures for case 1150 by isocontours of $\lambda _2=0.15{U_\infty }^2/h^2$ coloured with streamwise vorticity $\omega _x$ from $x \in [100h, 140h]$ across the entire span. Blue shading represents negative $\omega _x$ and red shading represents positive $\omega _x$ .

The effect of spanwise curvature on the shape of the vortical structures can be analysed with two-point spatial correlation, which, for a given $n$ location, is defined as

(4.17) \begin{equation} R_{\textit{AB}} = \dfrac {\overline {A(s, {\zeta })B(s+\Delta s, {\zeta }+\Delta {\zeta })}}{\sigma _A\sigma _B}, \end{equation}

where $A$ and $B$ are fluctuating parameters of interest at $(s, {\zeta })$ and $(s + \Delta s, {\zeta } + \Delta {\zeta })$ . Here $R_{\textit{AB}}$ is ensemble averaged for all locations with the same separation distances $\Delta s$ and $\Delta {\zeta }$ , time averaged and normalised by standard deviations $\sigma _A$ and $\sigma _B$ . Quantities of interest include fluctuating velocities $u', v', w' {\rm and }\,\lambda$ , where $\lambda$ is the swirl strength defined as the imaginary component of the complex eigenvalue of the local velocity gradient tensor (Zhou et al. Reference Zhou, Adrian, Balachandar and Kendall1999) used in a 2-D form (Chong, Perry & Cantwell Reference Chong, Perry and Cantwell1990; Hutchins, Hambleton & Marusic Reference Hutchins, Hambleton and Marusic2005; Volino, Schultz & Flack Reference Volino, Schultz and Flack2007; Volino Reference Volino2020b ). By definition, $\lambda \geqslant 0$ , but a sign is assigned to indicate the direction of rotation based on the local vorticity. The swirl strength can be used to locate vortices; it identifies hairpin vortices by their heads in $s{-}n$ planes and by their legs in $s{-}{\zeta }$ planes. Samples were collected at a frequency of $0.1h{U_\infty }^{-1}$ for a total time of $300h{U_\infty }^{-1}$ on a $10h \times 10h \,\,s-{\zeta }$ plane ranging the span at $n=0.25h$ , centred at $s=125h$ , with $\Delta s = \Delta {\zeta } = 0.2h$ .

Figure 32. Two-point spatial autocorrelation of the streamwise velocity component $R_{\textit{uu}}$ for (a) case 2000; (b) case 2150; (c) difference between $R_{\textit{uu}}$ in case 2150 and $R_{\textit{uu}}$ in case 2000.

Figure 33. Two-point spatial autocorrelation of $R_{\lambda \lambda }$ for (a) case 2000; (b) case 2150; (c) difference between $R_{\lambda \lambda }$ in case 2150 and $R_{\lambda \lambda }$ in case 2000.

Figure 34. Schematic of a hairpin vortex attached to the wall.

Figure 35. Two-point spatial cross-correlation of $u'$ and $v'$ $R_{uv}$ for (a) case 2000; (b) case 2150; (c) difference between $R_{uv}$ in case 2150 and $R_{uv}$ in case 2000.

Figure 36. Two-point spatial cross-correlation of $u'$ and $w'$ $R_{uw}$ for (a) case 2000; (b) case 2150; (c) difference between $R_{uw}$ in case 2150 and $R_{uw}$ in case 2000.

Figure 35 shows the coupling between streamwise and wall-normal velocity fluctuations. The correlation in the streamwise direction can be explained by considering the behaviour of legs of hairpin vortices, shown in figure 34. Each vortex leg is a tube of streamwise vorticity that induces spanwise motion. Fluid is drawn towards the centre of the hairpin near the wall, lifted upwards and ejected above the vortex tubes. The lifting of slowly moving fluid near the wall (negative $u'$ , positive $v'$ ) is consistent with Q2 events, and the downward motion of faster moving fluid outside the legs of the hairpins (positive $u'$ , negative $v'$ ) is consistent with Q4 events (Adrian et al. Reference Adrian, Meinhart and Tomkins2000). This behaviour is shown in figure 35. Negative $\overline {u'v'}$ correlation is exhibited near $\Delta {\zeta }/h= 0$ (Q2). The slightly positive correlations further away from the origin in the span are Q4 events that appear positive due to the reference points being in a low-speed streak ((4.17) considers $u'$ and $v'$ at different points in space, so a Q4 event will show a correlation of $-u'$ within the low-speed streak and $-v'$ at the appropriate distance in the span where the fluid is being swept, resulting in a positive $R_{uv}$ ). As seen in figure 31, the orientation of the hairpin vortices changes in the presence of spanwise curvature. Considering the change in structure in figure 35(c), the hairpins show an increase in correlation in the streamwise direction and a decrease in the spanwise direction, indicating weaker sweep correlation and a broader sweep structure in the presence of spanwise curvature. Figure 36 shows the correlation contour for $R_{uw}$ considering the coupling of streamwise and spanwise velocity fluctuations. A dipole pattern is present with a positive correlation in positive $\Delta {\zeta }/h$ and a negative correlation in negative $\Delta {\zeta }/h$ . This can again be understood by considering figure 34. Near the wall, in a low-speed streak (negative $u'$ ), fluid is drawn inward toward the centre of the vortex. This corresponds to negative $w'$ for the right leg and positive $w'$ for the left leg, creating the antisymmetric pattern present in figure 36(a). In the presence of spanwise curvature, the legs of the hairpin vortex are realigned. Additionally, these correlations are asymmetric. The minimum and maximum correlations seen in figure 36(a) are −0.29 and 0.27, whereas the minimum and maximum correlations in figure 36(b) are −0.23 and 0.42, indicating that one leg of the vortex is more coherent than the other. This is recognised by the sketch in figure 34 where any given eddy is rarely truly symmetric due to interactions with nearby structures (Adrian et al. Reference Adrian, Meinhart and Tomkins2000). Figure 36(c) shows stronger coupling at small $\Delta s/h$ with curvature as well as a stronger correlation in the direction of positive $\Delta s/h$ and negative $\Delta {\zeta }/h$ . Conversely, there is a stronger correlation at all other $(\Delta s/h, \Delta {\zeta }/h)$ locations in the absence of curvature.

Contours of $R_{\textit{uu}}$ are shown in figure 32 for the ZPG case and spanwise curvature of radius $R = 500h$ . Figure 32(a) shows an elongated shape in the streamwise direction and compact symmetry in the span. For a given streamwise fluctuation $u'$ at some point, strong correlation is shown with streamwise fluctuations at nearby points. This coherence reveals the signature of low-speed streaks caused by vortices that transport low-momentum fluid away from the wall. The regions of negative correlation next to the low-speed streak correspond to high-speed streaks: a quasi-periodic arrangement in the spanwise and streamwise directions in the near-wall region of turbulent boundary layers (Kim, Moin & Moser Reference Kim, Moin and Moser1987; Adrian, Meinhart & Tomkins Reference Adrian, Meinhart and Tomkins2000; Pope Reference Pope2000). Extracting data in streamline coordinates was accomplished by rotation of an inclination angle $\theta$ , determined by Christensen & Wu (Reference Christensen and Wu2005) with a least-squares fit through points on the contours. The difference between the rotated correlation and the baseline correlation is shown in figure 32(c) and indicates an attenuation of $u'$ perturbations in the streamwise directions and an amplification in the tangential direction. Figure 33 shows contours of $R_{\lambda \lambda }$ in the same format as figure 32. The correlation is highly centred around the origin, meaning that the swirl is highly localised in both $\Delta s/h$ and $\Delta {\zeta }/h$ . The general agreement between contours of $R_{\lambda \lambda }$ and $R_{\textit{uu}}$ is because hairpin packets cause low-speed streaks, and the legs of these hairpins are oriented along the side of the low-speed regions (Volino et al. Reference Volino, Schultz and Flack2007).

The shape of these vortices can be compared with an isotropic state to determine the anisotropy within the boundary layer. A commonly used approach was introduced by Lumley & Newman (Reference Lumley and Newman1977) by extracting the invariants of the anisotropy tensor $b_{\textit{ij}}$ :

(4.18) \begin{equation} b_{\textit{ij}} = \frac {\overline {u_i'u_j'}}{u_k'u_k'} - \frac {1}{3}\delta _{\textit{ij}} ,\end{equation}

where $\delta _{\textit{ij}}$ is the Kronecker delta. Given the symmetry of this tensor, the first invariant ( $b_ib_i$ ) is zero, leaving two remaining invariants that represent any realisable Reynolds stress. Plotting the invariant pairs from a flow within the 2-D phase space yields the Lumley triangle. This anisotropy-invariant map presents itself nonlinearly such that the vertices defining the boundaries of the triangle do not have equal weight in determining the position of any single point within. To this end, Banerjee et al. (Reference Banerjee, Krahl, Durst and Zenger2007) suggested a barycentric representation such that the barycentric coordinates of any turbulent state within the map can be represented by a linear combination of each of the vertices in Euclidean space. In this equilateral triangular map, the top vertex represents three-component turbulence (isotropic) similar to a spherical structure, the bottom-left vertex represents two-component ‘disk-like’ turbulence and the bottom-right vertex represents one-component ‘rod-like’ turbulence. The identifying vertices of this map can be defined as $(x_{1c}, y_{1c}), (x_{2c}, y_{2c})$ and $(x_{3c}, y_{3c})$ , and along with weighting coefficients $C_{1c}, C_{2c}, C_{3c}$ , each barycentric coordinate pair ( $x_B$ , $y_B$ ) can be defined as

(4.19) \begin{equation} (x_B, y_B) = C_{1c}(x_{1c}, y_{1c}) + C_{2c}(x_{2c}, y_{2c}) + C_{3c}(x_{3c}, y_{3c}), \end{equation}

where $C_{1c} \geqslant 0; C_{2c} \geqslant 0; C_{3c} \geqslant 0$ and $C_{1c} + C_{2c} + C_{3c} = 1$ . At their extrema, $C_{1c} = 1$ represents the linear limit of anisotropy (fully one-component flow) and $C_{2c} = 1$ represents the planar limit (fully two-component flow). The coefficients are computed as

(4.20) \begin{equation} \begin{aligned} C_{1c} = \lambda _1 - \lambda _2, \qquad C_{2c} = 2(\lambda _2 - \lambda _3),\qquad C_{3c} = 3\lambda _3 + 1 ,\end{aligned} \end{equation}

where $\lambda _1, \lambda _2, \lambda _3$ are the eigenvalues of $b_{\textit{ij}}$ with $\lambda _1 \geqslant \lambda _2 \geqslant \lambda _3$ . These coefficients can be manipulated to create an anisotropy metric, i.e.

(4.21) \begin{equation} C_{\textit{ani}} = C_{1c} + C_{2c} = 1 - C_{3c} = -3\lambda _3, \end{equation}

where $C_{\textit{ani}} = 0$ represents perfectly isotropic flow. This barycentric anisotropy-invariant map has also been used by Emory & Iaccarino (Reference Emory and Iaccarino2014), Stiperski, Chamecki & Calaf (Reference Stiperski, Chamecki and Calaf2021) and Morse & Mahesh (Reference Morse and Mahesh2023).

Figure 37. (a) Barycentric anisotropy-invariant map coloured by wall-normal distance $n^+$ and (b) velocity profile in wall units coloured by anisotropy metric $C_{\textit{ani}}$ for case 2000 at $s=145h$ .

Figure 37 shows the average turbulent structure within the boundary layer ( $n^+\leqslant 400$ ) for case 2000 (ZPG at $\textit{Re}_h=2000$ ) coloured by $n^+$ , as well as the mean velocity profile coloured by $C_{\textit{ani}}$ . The near-wall behaviour is dominated by streamwise and wall-normal fluctuations, while spanwise fluctuations are negligible, as the flow is highly influenced by proximity to the flat plate. This two-component flow in the viscous sublayer is highly anisotropic with $C_{\textit{ani}} = 1$ . Outside the viscous sublayer, the streamwise fluctuations begin to dominate, shown by the move towards one-component flow as the anisotropy decreases. Wall effects diminish as the flow returns to isotropy closer to $\delta _{99}$ .

Figure 38. (a) Barycentric anisotropy-invariant map and (b) anisotropy coefficients $C_{\textit{ani}}$ (—), $C_{1c}$ (- - -), $C_{2c}$ ( $\boldsymbol{\cdot }\boldsymbol{\cdot }\boldsymbol{\cdot}$ ) comparing case 2000 (black) and case 2150 (red) at $s=145h$ .

Figure 38 shows a comparison between case 2000 and case 2150 (spanwise curvature of radius $R = 500h$ at $\textit{Re}_h=2000$ ). The general trend previously described still holds showing near-wall anisotropy and a return-to-isotropy in the outer layer. However, the presence of spanwise curvature shifts the entire barycentric curve towards the two-component vertex, a consequence of the increase of the spanwise fluctuations. This is visible by examining the $C_{2c}$ profile. The anisotropy coefficients are the same between the two cases within the viscous sublayer, but the structures are more two-component in the log layer until the boundary layer edge due to the presence of spanwise flow. The similarity between the Lumley triangles of case 2000 and 2150 despite the change in the structure of turbulence is due to the invariance of eigenvalues to a change in coordinate basis. The invariants that the Lumley triangle derive from are eigenvalues of $b_{\textit{ij}}$ , which by definition, remain unchanged under an orthogonal coordinate system transformation.

4.8. Tilting of turbulent eddies by spanwise curvature

Figure 39 shows the normalised helicity density $h$ , which is equal to the cosine of the angle between the mean velocity and mean vorticity vectors, normalised by their respective magnitudes, for case 2000, case 2110 and 2150 ( $\textit{Re}_h = 2000$ ZPG and spanwise curvature of radius $R = 1000h$ and $R = 500h$ , respectively). The vectors are perpendicular at the wall and then start to diverge before flattening at the top of the log layer.

Figure 39. Helicity density comparisons for case 2000 (black), case 2110 (green) and case 2150 (red) at $s = 145h$ .

Figure 40. Canonical eddy diagram.

Consider a steady inviscid eddy with axis of rotation $z$ , in a constant pressure gradient in $z$ as displayed in figure 40. For this case, the $z$ momentum can be written as

(4.22) \begin{align} \frac {\partial {\overline {p}}}{\partial {z}} = \frac {\rho u'}{r}. \end{align}

This shows that the pressure gradient curves the streamline, which is visible as a non-zero $w'$ component and a secondary flow. In addition, the sign of $r$ indicates the direction of curvature. On the top of the eddy and assuming (without loss of generality) that $\partial \overline {p}/\partial z \gt 0$ then $u' \gt 0 \Rightarrow r \gt 0 \Rightarrow w' \gt 0$ . Similarly, on the bottom of the eddy, $u' \lt 0 \Rightarrow r \lt 0 \Rightarrow w' \lt 0$ . This effect is manifested by a tilting of the eddy toward the streamwise direction and an increase of the normalised helicity density. It also explains the non-zero, positive correlation between the first and third components of velocity perturbation $\overline {u_s'u_{{\zeta }}'}^+$ seen in figure 21. The fact that the eddies have non-zero helicity on average also explains the observation that $\overline {u_n'u_{{\zeta }}'}^+$ is non-zero in that case. In the case without curvature, the mean axis of rotation can be written as $\boldsymbol{\omega } = \omega _z \hat {k}$ , i.e. the mean axis of rotation is along the span. In the presence of spanwise curvature, a tilting occurs such that the rotation vector is not aligned with the spanwise direction $\hat {k}$ : $\boldsymbol{\omega } = \omega _x \hat {i} + \omega _y \hat {j} + \omega _z \hat {k}$ . Thus, a streamwise-aligned component $\omega _y$ is induced. From this understanding, the magnitude of $\overline {u_n'u_{{\zeta }}'}^+$ can be estimated as $\overline {u_n'u_{{\zeta }}'}^+ \approx h \boldsymbol{\cdot }\overline {u_s'u_n'}^+$ , where $h$ is the normalised helicity density. The estimation of $\overline {u_n'u_{{\zeta }}'}^+$ is compared with its measured value in figure 41. Good agreement is shown in the wall and free-stream limits, and the approximation is accurate to the correct order of magnitude within the log layer. An alternative, though equivalent explanation is to consider the eddy in a non-uniform secondary flow such that $w_{top} \gt w_{bottom}$ , where $w_{top}$ and $w_{bottom}$ are the mean spanwise velocity at the top and bottom of the eddy. The spanwise shear applied by the mean flow on the eddy is stronger at the bottom than at the top, thus, a tilting torque is applied on it.

Figure 41. Comparison between measured $\overline {u_n'u_{{\zeta }}'}^+$ (—) and analytical approximation $\overline {u_n'u_{{\zeta }}'}^+ = h \boldsymbol{\cdot }\overline {u_s'u_n'}^+$ (- - -) for case 2000 (black), case 2110 (green) and case 2150 (red) at $s=145h$ .