2.1 Computational set-up

A schematic of the flow configuration is shown in figure 1. In order to generate a realistic inflow TBL for the main computations, a precursor simulation of a transitional boundary layer was performed. A spatially and temporally resolved cross-flow plane was stored in the fully turbulent regime and used as inflow in the main simulation. A similar approach was adopted by Lee et al. (Reference Lee, Jung, Sung and Zaki2013) and Lee et al. (Reference Lee, Sung and Zaki2017). The inflow condition in the auxiliary transitional computation is a Blasius boundary layer and a superposition of vortical perturbations which were prescribed inside the mean shear only, such that the transitional boundary layer develops below a quiescent free stream (in contrast to bypass transition). The domain length spanned $108\leqslant Re_{\unicode[STIX]{x1D703}}\leqslant 1379$ which overlaps with the main simulation domain. Instantaneous TBL data were extracted in the precursor simulation at $Re_{\unicode[STIX]{x1D703}}=1200$ . The time series was subsequently applied at the inlet in the main simulation. Even though the inflow Reynolds number is lower than the recommendation by Schlatter & Örlü (Reference Schlatter and Örlü2012), the streamwise extent of the domain ensures that their criterion is satisfied where results are examined. In addition, the accuracy of this inflow condition was extensively validated. Figure 2 compares the mean streamwise velocity profile $\overline{u}$ and the root-mean-square fluctuations $u_{i,rms}^{\prime }$ in inner scaling with the data from Schlatter & Örlü (Reference Schlatter and Örlü2010). The profiles are scaled by the friction velocity $u_{\unicode[STIX]{x1D70F}}\equiv \sqrt{\unicode[STIX]{x1D708}(\unicode[STIX]{x2202}\overline{u}/\unicode[STIX]{x2202}y)|_{y=0}}$ , where $\unicode[STIX]{x1D708}$ is the kinematic viscosity and $y^{+}\equiv (yu_{\unicode[STIX]{x1D70F}})/\unicode[STIX]{x1D708}$ .

In addition to the inflow boundary layer, free-stream vortical perturbations are needed at the inlet. Decaying, HIT was simulated in a rectangular box with dimensions $\{L_{x},L_{y},{L_{z}\}}_{HIT}=$ $\{80,80,160\}\,\unicode[STIX]{x1D703}_{in}$ using a pseudo-spectral algorithm, with periodic boundary conditions in all three coordinate directions. Both $x$ and $y$ were discretized using 640 Fourier modes, and 1280 modes were used in the spanwise direction. Figure 3 shows the time history of the turbulent intensity, skewness of the velocity derivative, integral length scale and Taylor microscale:

(2.3a-d ) $$\begin{eqnarray}Tu\equiv \sqrt{\frac{u^{\prime 2}+v^{\prime 2}+w^{\prime 2}}{3}},\quad S^{\prime }(u^{\prime })\equiv \frac{\overline{(\unicode[STIX]{x2202}u^{\prime }/\unicode[STIX]{x2202}t)^{3}}}{(\overline{(\unicode[STIX]{x2202}u^{\prime }/\unicode[STIX]{x2202}t)^{2}})^{3/2}},\quad L_{k}\equiv \frac{k^{3/2}}{\unicode[STIX]{x1D716}},\quad \unicode[STIX]{x1D706}\equiv \sqrt{\frac{15\unicode[STIX]{x1D708}\overline{u^{\prime 2}}}{\unicode[STIX]{x1D716}}}.\end{eqnarray}$$

In the above expressions, $k$ is the turbulence kinetic energy and $\unicode[STIX]{x1D716}$ is the dissipation rate. The time instant when the data are extracted is identified by a dashed-dotted line in the figure ( $Tu=0.1$ and $L_{k}\approx 10.8\unicode[STIX]{x1D703}_{in}$ ). Note that $S^{\prime }$ is approaching its asymptotic value (Batchelor Reference Batchelor1953). At the time when the data are extracted, the Reynolds number based on the Taylor microscale is $Re_{\unicode[STIX]{x1D706}}$ $\equiv u_{rms}^{\prime }\unicode[STIX]{x1D706}/\unicode[STIX]{x1D708}=105$ . The turbulence intensity follows the decay law $Tu\propto t^{-5/7}$ , and the length scale shows a consistent power-law dependence $L_{k}\propto t^{2/7}$ .

The three-dimensional energy spectrum for the inflow FST is provided in figure 4(a). The height of the plateau in $E(\unicode[STIX]{x1D705}\unicode[STIX]{x1D702})$ , or the Kolmogorov constant $C_{k}$ , is approximately 1.96. The spatial resolution of a spectral simulation is adequate since $\unicode[STIX]{x1D705}_{max}\unicode[STIX]{x1D702}>1$ , where $\unicode[STIX]{x1D705}_{max}$ is the maximum wavenumber and $\unicode[STIX]{x1D702}$ is the Kolmogorov scale.

The free-stream disturbances at the inflow of the main simulations are extracted from the HIT data using Taylor’s hypothesis, where space is converted to time using the free-stream velocity of the boundary layer, $(U_{\infty },V_{\infty })$ at $Re_{\unicode[STIX]{x1D703},in}=1200$ . Note that periodicity of the free-stream forcing is not a concern because the underlying TBL is time dependent and not periodic. We have also exploited isotropy of the free stream and rotated the HIT volume about its $z$ -axis by $\unicode[STIX]{x1D709}=7.829^{\circ }$ . In this manner, every $L_{x,HIT}/(U_{\infty }\cos \unicode[STIX]{x1D709})$ time units, the HIT plane prescribed as inflow to the direct numerical simulation is shifted vertically by $L_{x,HIT}\tan (\unicode[STIX]{x1D709})=11\unicode[STIX]{x1D703}_{in}$ , which is of the same order as the integral length scale.

Figure 3. Evolution of HIT in time. The dashed-dotted line marks the time instant when data are extracted to apply as inflow condition in the main simulation. (a) (——) $Tu$ , (○) $Tu\propto t^{-5/7}$ and (– – –) $-S^{\prime }$ ; (b) (——) $L_{k}$ , (▫) $L_{k}\propto t^{2/7}$ , (– – –) $\unicode[STIX]{x1D706}$ and (○) $\unicode[STIX]{x1D706}\propto t^{0.5}$ .

Figure 4. (a) Three-dimensional energy spectrum of the FST at the inlet. (b) Comparison of downstream evolution of FST in the main simulation: (——) $u_{rms}^{\prime }$ , (– – –) $v_{rms}^{\prime }$ , (— ⋅ —) $w_{rms}^{\prime }$ and (○) $Tu$ of the decaying turbulence (reproduced from figure 3 a). (c) Downstream evolution of the integral length scale, $L_{k}/\unicode[STIX]{x1D6FF}_{99}$ . Thin dashed-dotted lines mark streamwise positions where $Re_{\unicode[STIX]{x1D703}}=\{1900,3000\}$ in FRC computation.

The dimensions of the computational domains, the number of grid points and resolutions are provided in table 1. Two simulations are contrasted: one of a conventional TBL beneath a quiescent free stream and a second case with free-stream vortical forcing. They are designated the reference (REF) and forced (FRC) cases (an additional forced case with smaller-scale HIT is reported in appendix B). In the forced configuration, the FST decays with downstream distance, and figure 4(b,c) shows the evolution of its normal stresses and length scale. The Reynolds normal stresses are isotropic, and their decay in space in the main simulation matches the temporal evolution of $Tu$ shown in figure 3(a), and reproduced in figure 4(b) (circle symbols). The length scale of FST is now redefined as $L_{k}\equiv -k^{3/2}/U_{\infty }(\text{d}k/\text{d}x)$ , and remains of the order of the $99\,\%$ boundary-layer thickness, $\unicode[STIX]{x1D6FF}_{99}$ (figure 4 c). This choice is motivated by our understanding of the interaction of vortical perturbations with mean shear: disturbances with spanwise and wall-normal length scales of the order of the boundary-layer thickness, and elongated in the streamwise direction, are most effective at permeating the mean shear and inducing an energetic response (Zaki & Durbin Reference Zaki and Durbin2005; Zaki & Saha Reference Zaki and Saha2009). For the present FST, the first two criteria are satisfied and the low-frequency components of the streamwise wavenumbers satisfy the third one.

Table 1. Summary of simulation parameters, and the spatial and temporal resolutions.

Both the REF and FRC boundary-layer simulations start at the same streamwise position, $Re_{\unicode[STIX]{x1D703},in}=1200$ . The differences in lengths and widths of the domains were guided by results from preliminary simulations. The domain of the FRC case is slightly shorter in the streamwise direction, which is partially compensated by a faster increase in the momentum-thickness Reynolds number (see figure 5). Its larger spanwise extent is required to accommodate the formation of wider structures. In both cases, the grid is uniformly spaced in the streamwise and spanwise directions, whereas a non-uniform grid distribution is used in the wall-normal direction. The grid spacings reported in table 1 in wall units are normalized by the friction velocity at the inlet to the simulation domain.

Figure 5. Downstream dependence of (a) momentum-thickness Reynolds number $Re_{\unicode[STIX]{x1D703}}\equiv U_{\infty }\unicode[STIX]{x1D703}/\unicode[STIX]{x1D708}$ and (b) mean pressure coefficient at the wall $C_{pw}$ . Grey: reference simulation; black: forced case.

The convective outflow condition $\unicode[STIX]{x2202}u_{i}/\unicode[STIX]{x2202}t+c\unicode[STIX]{x2202}u_{i}/\unicode[STIX]{x2202}x=0$ is applied at the outlet of the simulation domain, where $c$ is the local bulk velocity. The no-slip condition is imposed at the bottom wall. Periodic boundary conditions are applied in the spanwise direction. At the top boundary, appropriate treatment is required to maintain a ZPG boundary layer, even in the presence of high levels of FST.

2.2 Top boundary condition

A distribution of suction velocity is applied at the top boundary in order to maintain ZPG in the streamwise direction. The suction velocity is evaluated using an active controller. First, a measure of the streamwise free-stream velocity is evaluated and used as a sensor:

(2.4) $$\begin{eqnarray}\displaystyle u_{s}(x;t)=\frac{1}{\unicode[STIX]{x1D70F}}\int _{t-\unicode[STIX]{x1D70F}}^{t}\frac{\displaystyle \int _{0}^{L_{y}}\int _{0}^{L_{z}}(1-\unicode[STIX]{x1D6E4}(\boldsymbol{x},t))u(\boldsymbol{x},t)\,\text{d}z\,\text{d}y}{\displaystyle \int _{0}^{L_{y}}\int _{0}^{L_{z}}(1-\unicode[STIX]{x1D6E4}(\boldsymbol{x},t))\,\text{d}z\,\text{d}y}\,\text{d}t, & & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D70F}$ is an averaging time scale for the controller and $\unicode[STIX]{x1D6E4}$ is an indicator function which is zero in the free stream and unity in the boundary layer. The method for defining $\unicode[STIX]{x1D6E4}$ is covered in the next section. The averaged streamwise velocity $u_{\unicode[STIX]{x1D70F}}$ during $\unicode[STIX]{x1D70F}$ is

(2.5) $$\begin{eqnarray}\displaystyle u_{\unicode[STIX]{x1D70F}}(x,y;t)=\frac{1}{\unicode[STIX]{x1D70F}}\frac{1}{L_{z}}\int _{t-\unicode[STIX]{x1D70F}}^{t}\int _{0}^{L_{z}}u(\boldsymbol{x},t)\,\text{d}z\,\text{d}t. & & \displaystyle\end{eqnarray}$$

The suction velocity at the top boundary is computed as a superposition of continuity constraint over time $\unicode[STIX]{x1D70F}$ and the action of the controller:

(2.6) $$\begin{eqnarray}\displaystyle v_{top}(x;t)=-\frac{\text{d}}{\text{d}x}\int _{0}^{L_{y}}u_{\unicode[STIX]{x1D70F}}(x,y;t)\,\text{d}y+\unicode[STIX]{x1D70E}\frac{V_{\infty }}{U_{\infty }}(u_{s}(x;t)-u_{T}(x)), & & \displaystyle\end{eqnarray}$$

where $u_{T}$ is the target streamwise free-stream velocity which is unity in ZPG. The control factor is $\unicode[STIX]{x1D70E}$ , and $V_{\infty }/U_{\infty }$ is a reference rate of free-stream vertical to streamwise velocities for the canonical mean boundary-layer profile. The boundary conditions on the other directions are $u_{top}=1$ and $(\unicode[STIX]{x2202}w/\unicode[STIX]{x2202}y)_{top}=0$ . In the present study, the time scale $\unicode[STIX]{x1D70F}$ is comparable to the integral time scale of FST, $t_{k}=L_{k}/U_{\infty }$ .

The downstream dependence of pressure coefficient at the wall,

(2.7) $$\begin{eqnarray}C_{pw}\equiv \frac{\overline{p}_{(x,y=0)}-\overline{p}_{(x_{0},y=0)}}{\frac{1}{2}\unicode[STIX]{x1D70C}U_{\infty }^{2}},\end{eqnarray}$$

is shown in figure 5. It remains well within the range $-0.005<C_{pw}<0.002$ , which satisfies the condition for ZPG.

2.3 Conditional sampling

The starting point for conditional sampling is to define an indicator function, $\unicode[STIX]{x1D6E4}(\boldsymbol{x},t)$ , which is unity in the boundary layer and zero in the free stream. Its spanwise and time average is the intermittency,

(2.8) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FE}=\frac{1}{L_{z}\unicode[STIX]{x0394}T}\int _{T_{0}}^{T_{0}+\unicode[STIX]{x0394}T}\int _{0}^{L_{z}}\unicode[STIX]{x1D6E4}(\boldsymbol{x},t)\,\text{d}z\,\text{d}t, & & \displaystyle\end{eqnarray}$$

which is the probability that the fluid is part of the boundary-layer flow. A general quantity $\unicode[STIX]{x1D719}$ can then be separated into its boundary-layer $\unicode[STIX]{x1D719}^{B}$ and free-stream $\unicode[STIX]{x1D719}^{F}$ constituents:

(2.9) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D719}^{B}=\unicode[STIX]{x1D6E4}(\boldsymbol{x},t)\unicode[STIX]{x1D719}(\boldsymbol{x},t), & \displaystyle\end{eqnarray}$$

(2.10) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D719}^{F}=(1-\unicode[STIX]{x1D6E4}(\boldsymbol{x},t))\unicode[STIX]{x1D719}(\boldsymbol{x},t). & \displaystyle\end{eqnarray}$$

Using this decomposition, the conventional and conditional means are

(2.11) $$\begin{eqnarray}\displaystyle & \displaystyle \overline{\unicode[STIX]{x1D719}}=\frac{1}{L_{z}\unicode[STIX]{x0394}T}\int _{T_{0}}^{T_{0}+\unicode[STIX]{x0394}T}\int _{0}^{L_{z}}\unicode[STIX]{x1D719}(\boldsymbol{x},t)\,\text{d}z\,\text{d}t, & \displaystyle\end{eqnarray}$$

(2.12) $$\begin{eqnarray}\displaystyle & \displaystyle \overline{\unicode[STIX]{x1D719}}^{B}=\frac{\displaystyle \int _{T_{0}}^{T_{0}+\unicode[STIX]{x0394}T}\int _{0}^{L_{z}}\unicode[STIX]{x1D719}(\boldsymbol{x},t)\unicode[STIX]{x1D6E4}(\boldsymbol{x},t)\,\text{d}z\,\text{d}t}{\displaystyle \int _{T_{0}}^{T_{0}+\unicode[STIX]{x0394}T}\int _{0}^{L_{z}}\unicode[STIX]{x1D6E4}(\boldsymbol{x},t)\,\text{d}z\,\text{d}t}, & \displaystyle\end{eqnarray}$$

(2.13) $$\begin{eqnarray}\displaystyle & \displaystyle \overline{\unicode[STIX]{x1D719}}^{F}=\frac{\displaystyle \int _{T_{0}}^{T_{0}+\unicode[STIX]{x0394}T}\int _{0}^{L_{z}}\unicode[STIX]{x1D719}(\boldsymbol{x},t)(1-\unicode[STIX]{x1D6E4}(\boldsymbol{x},t))\,\text{d}z\,\text{d}t}{\displaystyle \int _{T_{0}}^{T_{0}+\unicode[STIX]{x0394}T}\int _{0}^{L_{z}}(1-\unicode[STIX]{x1D6E4}(\boldsymbol{x},t))\,\text{d}z\,\text{d}t}, & \displaystyle\end{eqnarray}$$

where superscripts $B$ and $F$ identify boundary-layer and free-stream quantities. The conventional mean can be related to boundary-layer and free-stream components, weighted by the local intermittency factor,

(2.14) $$\begin{eqnarray}\displaystyle \overline{\unicode[STIX]{x1D719}}=\unicode[STIX]{x1D6FE}\overline{\unicode[STIX]{x1D719}}^{B}+(1-\unicode[STIX]{x1D6FE})\overline{\unicode[STIX]{x1D719}}^{F}. & & \displaystyle\end{eqnarray}$$

Reynolds decomposition can be performed relative to each of the three averages:

(2.15) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D719}=\overline{\unicode[STIX]{x1D719}}+\unicode[STIX]{x1D719}^{\prime }, & \displaystyle\end{eqnarray}$$

(2.16) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D719}^{B}=\overline{\unicode[STIX]{x1D719}}^{B}+\unicode[STIX]{x1D719}^{\ast }\quad \text{for boundary-layer flow,} & \displaystyle\end{eqnarray}$$

(2.17) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D719}^{F}=\overline{\unicode[STIX]{x1D719}}^{F}+\unicode[STIX]{x1D719}^{\prime \prime }\quad \text{for free-stream flow.} & \displaystyle\end{eqnarray}$$

Analysis of the perturbations relative to the conditional means is intended to examine the turbulence dynamics within each region of the flow, separately. In contrast, Hancock & Bradshaw (Reference Hancock and Bradshaw1989) evaluated conditional statistics of perturbation quantities relative to the conventional mean, in order to measure the contribution from each fluid to overall statistics.

Unlike the intermittency weighted average (2.14), higher-order statistics evaluated relative to conditional and conventional means are related by more elaborate expressions. For example, the Reynolds stress is given by

(2.18) $$\begin{eqnarray}\overline{u_{i}^{\prime }u_{j}^{\prime }}=\unicode[STIX]{x1D6FE}\overline{u_{i}^{\ast }u_{j}^{\ast }}^{B}+(1-\unicode[STIX]{x1D6FE})\overline{u_{i}^{\prime \prime }u_{j}^{\prime \prime }}^{F}+\unicode[STIX]{x1D6FE}(1-\unicode[STIX]{x1D6FE})(\overline{u_{i}}^{F}-\overline{u_{i}}^{B})(\overline{u_{j}}^{F}-\overline{u_{j}}^{B}).\end{eqnarray}$$

Also, commutation of conditional averaging and the derivative operator is not always possible, for example,

(2.19a,b ) $$\begin{eqnarray}\displaystyle \overline{\frac{\unicode[STIX]{x2202}\overline{\unicode[STIX]{x1D719}}^{B}}{\unicode[STIX]{x2202}x_{i}}}^{B}=\frac{\unicode[STIX]{x2202}\overline{\unicode[STIX]{x1D719}}^{B}}{\unicode[STIX]{x2202}x_{i}};\quad \overline{\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D719}}{\unicode[STIX]{x2202}x_{i}}}^{B}=\frac{\unicode[STIX]{x2202}\overline{\unicode[STIX]{x1D719}}^{B}}{\unicode[STIX]{x2202}x_{i}}+\overline{\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D719}^{\ast }}{\unicode[STIX]{x2202}x_{i}}}^{B}\neq \frac{\unicode[STIX]{x2202}\overline{\unicode[STIX]{x1D719}}^{B}}{\unicode[STIX]{x2202}x_{i}}, & & \displaystyle\end{eqnarray}$$

and similarly for $F$ . This affects various terms in the turbulence kinetic energy budget (see appendix A), for example the pseudo-dissipation:

(2.20) $$\begin{eqnarray}\displaystyle \frac{1}{Re}\overline{\frac{\unicode[STIX]{x2202}u_{i}^{\prime }}{\unicode[STIX]{x2202}x_{k}}\frac{\unicode[STIX]{x2202}u_{i}^{\prime }}{\unicode[STIX]{x2202}x_{k}}} & = & \displaystyle \frac{1}{Re}\left\{\unicode[STIX]{x1D6FE}\overline{\frac{\unicode[STIX]{x2202}u_{i}^{\ast }}{\unicode[STIX]{x2202}x_{k}}\frac{\unicode[STIX]{x2202}u_{i}^{\ast }}{\unicode[STIX]{x2202}x_{k}}}^{B}+(1-\unicode[STIX]{x1D6FE})\overline{\frac{\unicode[STIX]{x2202}u_{i}^{\prime \prime }}{\unicode[STIX]{x2202}x_{k}}\frac{\unicode[STIX]{x2202}u_{i}^{\prime \prime }}{\unicode[STIX]{x2202}x_{k}}}^{F}\right.\nonumber\\ \displaystyle & & \displaystyle +\,\unicode[STIX]{x1D6FE}\left(2\frac{\unicode[STIX]{x2202}\overline{u_{i}}^{B}}{\unicode[STIX]{x2202}x_{k}}\overline{\frac{\unicode[STIX]{x2202}u_{i}}{\unicode[STIX]{x2202}x_{k}}}^{B}-\frac{\unicode[STIX]{x2202}\overline{u_{i}}^{B}}{\unicode[STIX]{x2202}x_{k}}\frac{\unicode[STIX]{x2202}\overline{u_{i}}^{B}}{\unicode[STIX]{x2202}x_{k}}\right)+(1-\unicode[STIX]{x1D6FE})\left(2\frac{\unicode[STIX]{x2202}\overline{u_{i}}^{F}}{\unicode[STIX]{x2202}x_{k}}\overline{\frac{\unicode[STIX]{x2202}u_{i}}{\unicode[STIX]{x2202}x_{k}}}^{F}-\frac{\unicode[STIX]{x2202}\overline{u_{i}}^{F}}{\unicode[STIX]{x2202}x_{k}}\frac{\unicode[STIX]{x2202}\overline{u_{i}}^{F}}{\unicode[STIX]{x2202}x_{k}}\right)\nonumber\\ \displaystyle & & \displaystyle -\left.\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FE}\overline{u_{i}}^{B}}{\unicode[STIX]{x2202}x_{k}}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FE}\overline{u_{i}}^{B}}{\unicode[STIX]{x2202}x_{k}}-\frac{\unicode[STIX]{x2202}(1-\unicode[STIX]{x1D6FE})\overline{u_{i}}^{F}}{\unicode[STIX]{x2202}x_{k}}\frac{\unicode[STIX]{x2202}(1-\unicode[STIX]{x1D6FE})\overline{u_{i}}^{F}}{\unicode[STIX]{x2202}x_{k}}-2\left(\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FE}\overline{u_{i}}^{B}}{\unicode[STIX]{x2202}x_{k}}\frac{\unicode[STIX]{x2202}(1-\unicode[STIX]{x1D6FE})\overline{u_{i}}^{F}}{\unicode[STIX]{x2202}x_{k}}\right)\!\right\}.\nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

The averaging durations in the present simulations are $\unicode[STIX]{x0394}T=1200\unicode[STIX]{x1D703}_{in}/U_{\infty }$ and $810\unicode[STIX]{x1D703}_{in}/U_{\infty }$ for the reference and forced cases, respectively. Since the forced configuration is twice as wide in the homogeneous spanwise direction, the shorter averaging time is sufficient for statistical convergence.

3.1 Comparison of conditional sampling using $|\unicode[STIX]{x1D714}|_{thres}^{\ast }$ and $\unicode[STIX]{x1D713}$

The vorticity threshold and the level-set approaches identify two different interfaces. The former marks the outer diffusive edge of the boundary layer, while the latter marks the material fluid that belongs to the boundary layer at the inlet. Previous studies of conventional boundary layers, with quiescent free streams, have used the vorticity threshold. Since it is not applicable in the presence of FST, we will adopt the level-set method throughout this work. A comparison of both techniques is, nonetheless, possible in the reference case without FST.

Figure 7. Contours of streamwise velocity $u$ , the line identifying the level set $\unicode[STIX]{x1D713}=0.5$ , in TBL (a) without and (b) with FST.

Figure 8. Mean streamwise velocity profiles in inner scaling at (a) $Re_{\unicode[STIX]{x1D703}}=1900$ and (b) $Re_{\unicode[STIX]{x1D703}}=3000$ in REF case. (——) Conventional mean; (– – –) boundary-layer contribution; (— ⋅ —) free-stream contribution. Conditional statistics are evaluated using (black) vorticity threshold $|\unicode[STIX]{x1D714}|$ and (grey) level set function $\unicode[STIX]{x1D713}$ . The thin dashed line is given by $\overline{u}^{+}=2.44\ln (y^{+})+5.2$ .

Figure 9. Reynolds stress profiles at (a) $Re_{\unicode[STIX]{x1D703}}=1900$ and (b) $Re_{\unicode[STIX]{x1D703}}=3000$ in REF case; (i) $\overline{u^{\prime }u^{\prime }}$ and (ii) $-\overline{u^{\prime }v^{\prime }}$ , both normalized by $U_{\infty }^{2}$ . Other details as in figure 8.

Figure 8 shows the mean velocity profiles at two different streamwise locations in the reference simulation. In addition to the conventional mean, the figure also shows the conditional averages in the boundary layer, $\unicode[STIX]{x1D6FE}\overline{u}^{+B}$ , and in the free stream, $(1-\unicode[STIX]{x1D6FE})\overline{u}^{+F}$ . The conditional curves are plotted twice, using the vorticity threshold (black) and the level-set approach (grey). Considering the free-stream contribution, it decays faster into the boundary layer when evaluated using the vorticity threshold relative to the level-set approach. In the former case, free-stream fluid that becomes part of the boundary layer as it diffuses only contributes to the boundary-layer statistics. In the level-set case, however, fluid that is assigned to the free stream at the inlet continues to contribute to the free-stream statistics even if it became part of the TBL. It is important to note that, at both downstream locations considered, the log-law region is fully captured by the boundary-layer conditional average, independent of the sampling technique.

The boundary-layer and free-stream contributions to the Reynolds stresses, evaluated using both the vorticity threshold and the level-set approach, are plotted in figure 9. The boundary-layer term $\unicode[STIX]{x1D6FE}\overline{u_{i}^{\ast }u_{j}^{\ast }}$ traces the conventional average from the wall through the inner peak and the log-law region. When $|\unicode[STIX]{x1D714}|_{thres}^{\ast }$ is used for conditional averaging, the free-stream contribution $(1-\unicode[STIX]{x1D6FE})\overline{u_{i}^{\prime \prime }u_{j}^{\prime \prime }}$ is essentially negligible even in the wake region. When $\unicode[STIX]{x1D713}$ is used, the free-stream contribution is small, and increases slightly downstream as more outer fluid becomes turbulent. Throughout the rest of this study, the level-set method will be adopted to evaluate conditional statistics, since it is equally applicable when the boundary layer is beneath a quiescent or turbulent free stream.

3.2 Statistics of the interface

The interface between the boundary layer and the free stream is defined as $y_{I}\equiv y(\unicode[STIX]{x1D713}=0.5)$ . Iso-surfaces of this quantity are plotted in figure 10, coloured by distance from the wall over a limited streamwise range. The figure qualitatively shows that the presence of the FST significantly enhances the undulation of the interface, relative to a canonical boundary layer.

Figure 10. Iso-surfaces of $y_{I}$ coloured by wall-normal height $y/\unicode[STIX]{x1D703}_{in}$ . (a) REF case and (b) FRC case.

Figure 11. (a) Downstream evolution of (——) $\overline{y_{I}}$ , (– $\times$ –) $\overline{y_{I}}^{+}$ and (– – –) $\unicode[STIX]{x1D6FF}_{99}$ . (b) Evolution of (——) $y_{I,rms}^{\prime }$ and (– – –) $y_{I,rms}^{\prime }/\overline{y_{I}}$ . Grey: REF case; black: FRC case.

Figure 11(a) contrasts the downstream development of the mean height of the interface, $\overline{y_{I}}$ , and the $99\,\%$ thickness of the boundary layer, $\unicode[STIX]{x1D6FF}_{99}$ . By construction, both quantities start at the same height at the inlet plane since $\unicode[STIX]{x1D713}=0.5$ was instantaneously assigned based on the vorticity threshold $|\unicode[STIX]{x1D714}|_{thres}^{\ast }=0.2$ (see figure 6). For the reference boundary layer (grey lines), both $\overline{y_{I}}$ and $\unicode[STIX]{x1D6FF}_{99}$ grow at similar rates, although the latter is slightly higher because the growing boundary layer entrains free-stream fluid. The results for the forced case show a steeper increase in $\unicode[STIX]{x1D6FF}_{99}$ , which was noted in previous studies (Hancock & Bradshaw Reference Hancock and Bradshaw1989). Note, however, that $\unicode[STIX]{x1D6FF}_{99}$ bears no physical significance, and is sensitive to the details of the mean velocity profile. On the other hand, $\overline{y_{I}}$ is the mean height of the material line separating the boundary layer from the free stream at the inlet. This quantity has a much smoother evolution with downstream distance, and shows only a moderate increase relative to the reference case.

The excursions of the interface relative to the mean location are measured by the root-mean-square of the fluctuations in its height, $y_{I,rms}^{\prime }$ , in figure 11(b). Lee et al. (Reference Lee, Sung and Zaki2017) showed that $y_{I,rms}^{\prime }$ increases with downstream distance, and is nearly linearly proportional to $\overline{y_{I}}$ in TBLs. The same trend is observed in the reference flow, where $y_{I,rms}^{\prime }/\overline{y_{I}}$ is nearly flat in figure 11(b). When the boundary layer is forced by FST, the root-mean-square of the interface excursions increases significantly. When normalized by $\overline{y_{I}}$ , it still shows an initial increase before it plateaus. The larger excursions in the interface height correspond to enhanced transport at the interface.

Figure 12. Probability density function of (i) $y_{I}^{\prime }$ and (ii) $y_{I}^{\prime }/\overline{y_{I}}$ for (a) REF case and (b) FRC case. Thin to thick lines correspond to the streamwise position, $(x-x_{0})/\unicode[STIX]{x1D703}_{in}=\{100,200,400,800\}$ .

The probability density function of the interface excursions is plotted in figure 12. The distributions for each flow nearly collapse when normalized by the mean value, $y_{I}^{\prime }/\overline{y_{I}}$ . In the reference boundary-layer simulation, the probability density function is practically symmetric, and its skewness at the shown locations is in the range $[0.12,0.15]$ . In the presence of free-stream forcing, the normalized probability density function curves are wider, and are more positively skewed in the direction of the free stream. Quantitatively, the skewness increases by as much as three-fold, and is in the range $[0.35,0.5]$ . Since $\overline{y_{I}}$ is larger in the forced case, the spread in the probability density function of $y^{\prime }$ is even more pronounced than in the reference case.

Due to the larger undulation of the interface height in the forced case, the intermittency $\unicode[STIX]{x1D6FE}$ spreads more rapidly as shown in the left-hand panel of figure 13. In the middle panels, wall-normal profiles for two streamwise positions are plotted versus $y/\overline{y_{I}}$ , and show good collapse in this outer scaling. When the boundary layer is buffeted by external disturbances, the profiles of $\unicode[STIX]{x1D6FE}$ clearly show its spread both towards the free stream and the wall. The FST is thus expected to influence the flow deep inside the mean shear. The extent of its penetration should, however, be viewed in inner scaling as shown in the rightmost panels. In the reference simulation, the contribution of the outer flow, or $(1-\unicode[STIX]{x1D6FE})$ , vanishes in the logarithmic layer. In the forced case, while the contribution of the free stream remains finite in the logarithmic layer, it is vanishingly small in the buffer layer.

4.1 The skin friction

When TBLs are exposed to free-stream vortical forcing, the wall shear stress is enhanced. This effect is shown in figure 14(a) where the downstream evolution of the skin-friction coefficient, $C_{f}\equiv \unicode[STIX]{x1D70F}_{w}/(0.5\unicode[STIX]{x1D70C}U_{\infty }^{2})$ , is plotted versus the momentum-thickness Reynolds number. The coefficient increases by approximately $15\,\%$ at $Re_{\unicode[STIX]{x1D703}}=1900$ . Many of the subsequent discussions will be supported by results at this location and also $Re_{\unicode[STIX]{x1D703}}=3000$ farther downstream. Note that these Reynolds numbers do not correspond to the same streamwise positions in the reference and forced boundary layers (cf. figure 5). Matching $Re_{\unicode[STIX]{x1D703}}$ is, however, the appropriate choice for comparing the two flows. Hereafter, grey and black lines always indicate the profile of the reference and forced TBLs, respectively.

In figure 14(a), the $C_{f}$ curve is shifted upwards in presence of free-stream forcing, and retains its dependence on $Re_{\unicode[STIX]{x1D703}}$ . Similar to Esteban et al. (Reference Esteban, Dogan, Rodríguez-López and Ganapathisubramani2017), the correlation $C_{f}=2[\log (Re_{\unicode[STIX]{x1D703}})/0.384+C]^{-2}$ is used to match the data by adjusting $C$ . For the reference flow, $C=4.127$ is anticipated (Nagib, Chauhan & Monkewitz Reference Nagib, Chauhan and Monkewitz2007) and yields good agreement for the present data. The constant must be adjusted to $C=2.77$ in order to accurately capture the dependence of $C_{f}$ on $Re_{\unicode[STIX]{x1D703}}$ in the presence of FST.

An interpretation of the skin friction in terms of three physical phenomena was recently proposed by Renard & Deck (Reference Renard and Deck2016). In a frame moving with the free-stream speed $U_{\infty }$ , the flat plate is pulled to the left at $-U_{\infty }$ . In that setting, the skin-friction coefficient can be interpreted as of the average normalized power imparted by the wall motion onto the fluid, and is the sum of three contributions:

(4.1) $$\begin{eqnarray}\displaystyle C_{f,RD} & = & \displaystyle \underbrace{\frac{2}{U_{\infty }^{3}}\int _{0}^{\infty }\unicode[STIX]{x1D708}\left(\frac{\unicode[STIX]{x2202}\overline{u}}{\unicode[STIX]{x2202}y}\right)^{2}\,\text{d}y}_{C_{f,a}}+\underbrace{\frac{2}{U_{\infty }^{3}}\int _{0}^{\infty }-\overline{u^{\prime }v^{\prime }}\frac{\unicode[STIX]{x2202}\overline{u}}{\unicode[STIX]{x2202}y}\,\text{d}y}_{C_{f,b}}\nonumber\\ \displaystyle & & \displaystyle +\,\underbrace{\frac{2}{U_{\infty }^{3}}\int _{0}^{\infty }(\overline{u}-U_{\infty })\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}y}\left(\frac{\overline{\unicode[STIX]{x1D70F}}}{\unicode[STIX]{x1D70C}}\right)\,\text{d}y}_{C_{f,c}},\end{eqnarray}$$

where $\overline{\unicode[STIX]{x1D70F}}/\unicode[STIX]{x1D70C}$ is the total shear stress. The term $C_{f,a}$ is the rate of dissipation of mean streamwise kinetic energy into heat, $C_{f,b}$ is the rate of production of turbulence kinetic energy and $C_{f,c}$ is the rate of change in the streamwise kinetic energy in the mean flow. The symbols in figure 14(a) show the reconstruction of the right-hand side of (4.1), which agrees with the direct evaluation of the skin friction from the gradient at the wall.

Figure 13. (i) Contours of intermittency $\unicode[STIX]{x1D6FE}$ near the inlet of the simulation domain. Profiles of $\unicode[STIX]{x1D6FE}$ (ii) in outer scaling and (iii) in inner scaling at (– – –) $Re_{\unicode[STIX]{x1D703}}=1900$ and (——) $Re_{\unicode[STIX]{x1D703}}=3000$ . (a) REF case and (b) FRC case.

Figure 14. (a) Skin-friction coefficients computed from (——) the wall shear stress $C_{f}=\unicode[STIX]{x1D70F}_{w}/(\unicode[STIX]{x1D70C}U_{\infty }^{2})/2$ and (○) the decomposition (4.1). Dashed lines are the correlation $C_{f}=2[\log (Re_{\unicode[STIX]{x1D703}})/0.384+C]^{-2}$ , where $C=4.127$ for REF and $C=2.77$ for FRC. (b) Contributions to the skin-friction coefficient normalized by $C_{f,REF}$ ; (——) $C_{f}$ , (— ⋅ —) $C_{f,a}$ , (– – –) $C_{f,b}$ and (— ⋅ ⋅ —) $C_{f,c}$ . Grey: REF; black: FRC.

All the terms in the decomposition (4.1) are plotted in figure 14(b), normalized by the total friction coefficient from the reference boundary-layer simulation, $C_{f,REF}(Re_{\unicode[STIX]{x1D703}})$ . The term $C_{f,c}$ , which accounts for the streamwise energy in the mean flow, is the smallest contributor to the friction and is initially reduced in response to the FST but subsequently recovers. The overall increase in the friction coefficient is therefore due to the augmentations of $C_{f,a}$ and $C_{f,b}$ , namely the dissipation in the mean profile and the production of turbulence kinetic energy. Both involve $-\overline{u^{\prime }v^{\prime }}$ , although indirectly in the first term through the mean-flow distortion by the stress.

The integral in $C_{f,a}$ converges to $95\,\%$ of its total below $y^{+}=20$ and to $99\,\%$ by $y^{+}=100$ . Within the region $y^{+}<100$ , the change in $\unicode[STIX]{x2202}\overline{u}/\unicode[STIX]{x2202}y$ from its wall value is nearly equal to the Reynolds shear stress, and therefore the increase in $C_{f,a}$ when the boundary layer is forced is related to the increase in $\overline{-u^{\prime }v^{\prime }}$ . This relation is demonstrated by integrating the mean-momentum equation for a ZPG boundary layer from the wall to a height $y$ :

(4.2) $$\begin{eqnarray}\displaystyle \frac{1}{Re}\left\{\frac{\unicode[STIX]{x2202}\overline{u}}{\unicode[STIX]{x2202}y}-\left[\frac{\unicode[STIX]{x2202}\overline{u}}{\unicode[STIX]{x2202}y}\right]_{y=0}\right\}=\int _{y}\frac{\unicode[STIX]{x2202}(\overline{u}\,\overline{u})}{\unicode[STIX]{x2202}x}\,\text{d}y+\overline{u}\,\overline{v}+\overline{u^{\prime }v^{\prime }}. & & \displaystyle\end{eqnarray}$$

The above expression was evaluated for both the reference and forced flows, and the difference $(\bullet )_{FRC}-(\bullet )_{REF}$ is plotted in figure 15, at $Re_{\unicode[STIX]{x1D703}}=1900$ and $3000$ . The results demonstrate how changes in $\overline{-u^{\prime }v^{\prime }}$ indirectly impact the dissipation in the mean-velocity profile, $C_{f,a}$ .

Figure 15. The differences in (——) $(1/Re)(\unicode[STIX]{x2202}\overline{u}/\unicode[STIX]{x2202}y-\unicode[STIX]{x2202}\overline{u}/\unicode[STIX]{x2202}y|_{y=0})$ and in (– – –) $\overline{u^{\prime }v^{\prime }}$ in the reference and forced boundary layers. (a) $Re_{\unicode[STIX]{x1D703}}=1900$ ; (b) $Re_{\unicode[STIX]{x1D703}}=3000$ .

Figure 16. Wall-normal distributions of turbulence kinetic energy budget terms normalized by $U_{\infty }^{3}/\unicode[STIX]{x1D703}_{in}$ at (a) $Re_{\unicode[STIX]{x1D703}}=1900$ and (b) $Re_{\unicode[STIX]{x1D703}}=3000$ ; (——) production ${\mathcal{P}}$ , (– – –) pseudo-dissipation $-\unicode[STIX]{x1D700}$ , (— ⋅ —) viscous diffusion ${\mathcal{D}}$ , ( $\cdots \cdots$ ) turbulent advection ${\mathcal{T}}$ , (— ⋅ ⋅ —) pressure diffusion ${\mathcal{R}}$ and (— —) mean advection ${\mathcal{A}}$ . Grey: REF; black: FRC.

The shear stress appears directly in $C_{f,b}$ whose increase signals a potential change in the energetics of the wall turbulence. Consider the kinetic energy budget

(4.3) $$\begin{eqnarray}\displaystyle \underbrace{\overline{u}_{j}\frac{\unicode[STIX]{x2202}k}{\unicode[STIX]{x2202}x_{j}}}_{{\mathcal{A}}}=-\underbrace{\overline{\frac{\unicode[STIX]{x2202}u_{i}^{\prime }p^{\prime }}{\unicode[STIX]{x2202}x_{i}}}}_{{\mathcal{R}}}\underbrace{-\frac{1}{2}\frac{\unicode[STIX]{x2202}\overline{u_{j}^{\prime }u_{i}^{\prime }u_{i}^{\prime }}}{\unicode[STIX]{x2202}x_{j}}}_{{\mathcal{T}}}\underbrace{-\overline{u_{i}^{\prime }u_{j}^{\prime }}\frac{\unicode[STIX]{x2202}\overline{u_{i}}}{\unicode[STIX]{x2202}x_{j}}}_{{\mathcal{P}}}-\underbrace{\frac{1}{Re}\overline{\frac{\unicode[STIX]{x2202}u_{i}^{\prime }}{\unicode[STIX]{x2202}x_{j}}\frac{\unicode[STIX]{x2202}u_{i}^{\prime }}{\unicode[STIX]{x2202}x_{j}}}}_{\unicode[STIX]{x1D700}}+\underbrace{\frac{1}{Re}\frac{\unicode[STIX]{x2202}^{2}k}{\unicode[STIX]{x2202}x_{j}\unicode[STIX]{x2202}x_{j}}}_{{\mathcal{D}}}, & & \displaystyle\end{eqnarray}$$

where $k\equiv (\overline{u_{i}^{\prime }u_{i}^{\prime }})/2$ is the turbulence kinetic energy per unit mass. In the above expression, ${\mathcal{A}}$ is mean advection, ${\mathcal{R}}$ is pressure-diffusion, ${\mathcal{T}}$ is turbulent advection, ${\mathcal{P}}$ is rate of production, $\unicode[STIX]{x1D700}$ is the pseudo-dissipation and ${\mathcal{D}}$ is viscous diffusion. These terms are plotted in the near-wall region in figure 16, and compared in the reference and forced boundary layers. The magnitude of every term is increased in the forced flow, especially the rate of dissipation and viscous diffusion at the wall. The production term is also appreciably increased in the buffer layer, which is consistent with the results of Péneau et al. (Reference Péneau, Boisson and Djilali2000). Based on the analysis by Renard & Deck (Reference Renard and Deck2016), in a frame moving with the free-stream speed $U_{\infty }$ , an increase of $\overline{u^{\prime }v^{\prime }}\unicode[STIX]{x2202}\overline{u}/\unicode[STIX]{x2202}y$ , and hence ${\mathcal{P}}$ , requires additional power input from the moving wall. This leads to an increase in wall shear stress, or drag. Further evidence of the connection between $\overline{u^{\prime }v^{\prime }}\unicode[STIX]{x2202}\overline{u}/\unicode[STIX]{x2202}y$ and drag is provided in figure 17. The wall-normal peak of the former term is plotted versus downstream Reynolds number, and its trend matches the change in the skin friction very well.

Figure 17. Evolutions of (——) $C_{f}$ and (▫) a peak value of $-\overline{u^{\prime }v^{\prime }}(\unicode[STIX]{x2202}\overline{u}/\unicode[STIX]{x2202}y)$ . Grey: REF; black: FRC.

A number of interesting lines of query arise from the above results. Firstly, the enhanced production in the buffer layer can be ascribed to either a change in the mean-flow profile or the Reynolds shear stresses. The two are not independent of one another, and which has a more pronounced effect is of interest. Secondly, the most pronounced change in the production term takes place in the buffer layer. Whether the FST penetrates this deep into the boundary layer or the near-wall turbulence dynamics are modified should be assessed. If the former, an increase in the shear stress is curious because the FST is isotropic; and the latter case would also warrant an explanation. These factors are examined by evaluating the contributions of the boundary-layer and free-stream fluids to the flow statistics.

Figure 18. Profiles of mean streamwise velocity at (a) $Re_{\unicode[STIX]{x1D703}}=1900$ and (b) $Re_{\unicode[STIX]{x1D703}}=3000$ , normalized by (i) $U_{\infty }$ and (ii, iii)  $u_{\unicode[STIX]{x1D70F}}$ . (i, ii) (– – –) Boundary-layer velocity $\overline{u}^{B}$ ; (— ⋅ —) free-stream velocity $\overline{u}^{F}$ . (iii) (– – –) Boundary-layer contribution $\unicode[STIX]{x1D6FE}\overline{u}^{B}$ ; (— ⋅ —) free-stream contribution $(1-\unicode[STIX]{x1D6FE})\overline{u}^{F}$ . The thin dashed line: $\overline{u}^{+}=2.44\ln (y^{+})+5.2$ . Grey: REF; black: FRC.

4.2 Conditional statistics

The mean streamwise velocity profiles are plotted in outer and inner scalings in figure 18, at $Re_{\unicode[STIX]{x1D703}}=1900$ and $3000$ . Note that the adopted outer length scale is $\overline{y_{I}}$ (left-hand panels), and not $\unicode[STIX]{x1D6FF}_{99}$ which does not bear a clear physical significance. The mean-velocity profiles are fuller in the FRC case, which is indicative of the enhanced mixing that was remarked upon by Hancock & Bradshaw (Reference Hancock and Bradshaw1989). It is also consistent with the increase in $C_{f,a}$ (4.1), which is the integral of $(\unicode[STIX]{x2202}\overline{u}/\unicode[STIX]{x2202}y)^{2}$ in the wall-normal direction, or dissipation of mean streamwise kinetic energy. The left-hand panels also show the conditional velocity profiles in the boundary layer, $\overline{u}^{B}$ , and in the free stream, $\overline{u}^{F}$ . These profiles, when weighted by their probabilities, make up the unconditional mean (2.14). Therefore they lie on either side of the mean, and agree with it near the wall and in the free stream, respectively. The free-stream conditional profile, $\overline{u}^{F}$ , is larger in the forced flow than in the reference case, but seems to only impact the mean profile near the outer edge of the boundary layer. It appears, at least based on outer scaling, that changes in the overall mean track the changes in the boundary-layer curve.

The middle panels of figure 18 show the same profiles in viscous scaling. Both the unforced and forced boundary layers have a logarithmic region, $\overline{u}^{+}=2.44\ln (y^{+})+5.2$ , at both Reynolds numbers. In the presence of FST, a significant depression of the boundary-layer profile occurs in the wake region, which is consistent with the increase in drag; the logarithmic, buffer and viscous regions are hardly affected. The unconditional mean profiles are thus consistent with earlier experiments (Hancock & Bradshaw Reference Hancock and Bradshaw1983; Sharp et al. Reference Sharp, Neuscamman and Warhaft2009) and simulations (Li et al. Reference Li, Schlatter and Henningson2010). The conditional profiles, $\overline{u}^{+B}$ and $\overline{u}^{+F}$ , are interesting. In both the reference and forced computations, $\overline{u}^{+B}$ faithfully follows the conventional mean up to the edge of the logarithmic layer, and is lower in the intermittent wake region. Conversely, the free-stream curves reproduce the outer uniform flow, retain a higher velocity than the conventional mean inside the boundary layer and collapse in viscous scaling. A deeper reach of $\overline{u}^{+F}$ towards the wall in the forced flow is evident in the figure. When weighted by their probabilities (right-hand panels), the conditional averages yield the contributions to the mean by the boundary layer, $\unicode[STIX]{x1D6FE}\overline{u}^{+B}$ , and the free stream, $(1-\unicode[STIX]{x1D6FE})\overline{u}^{+F}$ . In the unforced flow, the boundary-layer contributions still show a logarithmic-layer behaviour and decay sharply in the wake region due to the intermittency weighting. In contrast, when FST forcing is present, the boundary-layer contribution no longer traces the logarithmic law, although the logarithmic behaviour is re-established once the free-stream contribution is added to recover the conventional mean. In order to explain the ‘universality’, or robustness, of the logarithmic law in forced boundary layers, Hancock & Bradshaw (Reference Hancock and Bradshaw1989) verified that the departure from equilibrium is inappreciable, and that the rates of production and dissipation of turbulence kinetic energy are dominant and nearly balance in that region. The same dominant balance was verified in the present FRC case.

Figure 19. Reynolds stresses profiles at (a) $Re_{\unicode[STIX]{x1D703}}=1900$ and (b) $Re_{\unicode[STIX]{x1D703}}=3000$ : (i) $\overline{u^{\prime }u^{\prime }}$ ; (ii) $\overline{v^{\prime }v^{\prime }}$ ; (iii) $\overline{w^{\prime }w^{\prime }}$ ; (iv) $-\overline{u^{\prime }v^{\prime }}$ , all normalized by $U_{\infty }^{2}$ . (——) Conventional statistics; (– – –) boundary-layer contribution; (— ⋅ —) free-stream contribution. Grey: REF; black: FRC.

Wall-normal profiles of Reynolds stresses are presented in figure 19. In general, all the stresses are enhanced when the boundary layer is subjected to FST and, outside the mean shear, all the normal stresses match the outer turbulence levels and the shear stress is nearly zero. Two key observation are important to note, related to the stresses in the logarithmic and buffer layers, respectively. Firstly, the increase in $\overline{u^{\prime }u^{\prime }}$ in the logarithmic layer exceeds the free-stream value, and is therefore not a mere upward shift of the curve. Partial evidence of enhanced local production of $\overline{u^{\prime }u^{\prime }}$ is available from the Reynolds shear stress, $\overline{u^{\prime }v^{\prime }}$ , which is increased in the logarithmic layer. This trend is curious, and cannot be directly ascribed to the FST since it is itself void of $\overline{u^{\prime }v^{\prime }}$ . Instead, the increase in normal stresses $\overline{v^{\prime }v^{\prime }}$ due to the ingested FST acts against the mean shear to produce Reynolds shear stress that, in turn, enhances the production of $\overline{u^{\prime }u^{\prime }}$ . Profiles of the relevant production terms, ${\mathcal{P}}_{u_{i}^{\prime }u_{j}^{\prime }}\equiv -\overline{u_{j}^{\prime }u_{k}^{\prime }}\unicode[STIX]{x2202}\overline{u_{i}}/\unicode[STIX]{x2202}x_{k}-\overline{u_{i}^{\prime }u_{k}^{\prime }}\unicode[STIX]{x2202}\overline{u_{j}}/\unicode[STIX]{x2202}x_{k}$ , are provided in figure 20, pre-multiplied by $y^{+}$ . In this form, the area under the curve corresponds to the integral of production, which is clearly enhanced in the logarithmic layer. The symbols identify the dominant contributions, and confirm that $\overline{v^{\prime }v^{\prime }}$ leads to an increase in production of the shear stress and, in turn, production of $\overline{u^{\prime }u^{\prime }}$ in the outer layer.

Figure 20. Profiles of production of Reynolds shear and streamwise-normal stresses at $Re_{\unicode[STIX]{x1D703}}=1900$ : (a) $y^{+}{\mathcal{P}}_{-u^{\prime }v^{\prime }}$ ; (b) $y^{+}{\mathcal{P}}_{u^{\prime }u^{\prime }}$ . Symbols mark the leading contributions: (a) $y^{+}\overline{v^{\prime }v^{\prime }}\unicode[STIX]{x2202}\overline{u}/\unicode[STIX]{x2202}y$ ; (b) $-2y^{+}\overline{u^{\prime }v^{\prime }}\unicode[STIX]{x2202}\overline{u}/\unicode[STIX]{x2202}y$ . Production terms are normalized by $U_{\infty }^{3}/\unicode[STIX]{x1D703}_{in}$ . Grey: REF; black: FRC.

The second observation from figure 19 concerns the depth towards the wall over which the Reynolds stress profiles are altered in the presence of FST. The increase in $\overline{v^{\prime }v^{\prime }}$ diminishes as we approach the wall and nearly vanishes within the buffer layer, which is consistent with the extent of penetration of FST into the boundary layer and the decay of the intermittency curves (see figure 13). In contrast, the increases in the other stresses preserve their magnitudes deeper into the boundary layer, beyond the decay of the intermittency curves. For example, the increases in the streamwise and spanwise normal stresses are evident below the locations of their respective peaks, and even deeper than the buffer layer. Therefore, these changes cannot be caused by the FST directly, and the explanation for enhanced production in the logarithmic layer does not carry over to the buffer region. This point is further supported by the conditional free-stream contributions, $(1-\unicode[STIX]{x1D6FE})\overline{u_{i}^{\prime \prime }u_{j}^{\prime \prime }}^{F}$ , which vanish farther away from the wall than the increase in the stresses. In summary, while the FST has a direct contribution to the normal stresses in the outer intermittent region and enhances production of $\overline{u^{\prime }v^{\prime }}$ and $\overline{u^{\prime }u^{\prime }}$ in the logarithmic layer, it also indirectly modifies the turbulence deeper towards the wall in a manner that warrants further examination – a matter that we address in § 5.

Figure 21. Contributions to Reynolds shear stress, $-\overline{u^{\prime }v^{\prime }}$ normalized by $U_{\infty }^{2}$ , from each quadrant at (a) $Re_{\unicode[STIX]{x1D703}}=1900$ and (b) $Re_{\unicode[STIX]{x1D703}}=3000$ . (——) Conventional statistics; (– – –) boundary-layer contribution; (— ⋅ —) free-stream contribution. Grey: REF; black: FRC.

The increase in production is connected to the changes in Reynolds shear stress within the boundary layer. The cross-correlation coefficient, $\overline{u^{\prime }v^{\prime }}/(u_{rms}^{\prime }v_{rms}^{\prime })$ , is reduced in the presence of external forcing because the isotropy of the FST destroys the coherence of the boundary-layer turbulence (Hancock & Bradshaw Reference Hancock and Bradshaw1989). Nonetheless, the magnitude of Reynolds shear stress $\overline{u^{\prime }v^{\prime }}$ increases in the boundary layer as shown in figure 19, and as reported by others (Péneau et al. Reference Péneau, Boisson and Djilali2000). This increase is curious because the free-stream forcing is free of any mean shear stress, and because the extent of its penetration into the mean shear, towards the wall, is shallower than the region of increase in kinetic energy and production which persist deeper towards the wall.

A more detailed account of the increase in the Reynolds shear stress can be obtained from quadrant analysis (Wallace Reference Wallace2016). The analysis was performed for both the reference and forced flows, and is shown in figure 21 at the two downstream locations, $Re_{\unicode[STIX]{x1D703}}=1900$ and $3000$ . In the reference flow, ejection (Q2: $u^{\prime }<0$ and $v^{\prime }>0$ ) and sweep (Q4: $u^{\prime }>0$ and $v^{\prime }<0$ ) events are dominant (figure 21). In the forced case, the isotropy of FST leads to finite, nearly equal contribution to all four quadrants at and beyond the free-stream edge of the boundary layer, thus enhancing mixing in that region. Inside the boundary layer, the increases in Q2 and Q4 events far exceed those in the other two components. In addition, while the free-stream contribution decays within the boundary layer, a significant increase in the unconditional Q2 and Q4 events is observed near their respective peaks close to the wall. The peak of the boundary-layer contribution, which resides close to the wall, increases in magnitude but its location does not shift. The more pronounced ejections and sweeps are consistent with the higher rate of production, $P_{-u^{\prime }v^{\prime }}$ , in the forced case (figure 20 a).

Figure 22. Profiles of (i) pseudo-dissipation $\unicode[STIX]{x1D700}$ and (ii) production ${\mathcal{P}}$ terms, both normalized by $U_{\infty }^{3}/\unicode[STIX]{x1D703}_{in}$ . (a) $Re_{\unicode[STIX]{x1D703}}=1900$ ; (b) $Re_{\unicode[STIX]{x1D703}}=3000$ . (——) Conventional statistics; (– – –) boundary-layer contribution; (— ⋅ —) free-stream contribution. Grey: REF; black: FRC.

Figure 22 highlights the change in the near-wall turbulence kinetic energy production and dissipation rates and their conditional contributions, all plotted in logarithmic scale. The relationship between the conventional and conditional terms is provided in appendix A. In the reference simulation, the dissipation is zero in the free stream, and its high level near the wall is entirely due to the boundary-layer contribution. In the forced case, the dissipation is enhanced throughout the extent of the boundary layer and is finite, albeit small, in the free stream where it is entirely due to the contribution of the outer fluid. The free-stream conditional average decays inside the boundary layer and is more than two orders of magnitude smaller than the boundary-layer counterpart. Similar observations are applicable to the production terms. In this case, however, the free-stream contribution vanishes outside the mean shear in both the reference and forced configurations. While it is enhanced inside the boundary layer when the flow is forced, that contribution remains more than two orders of magnitude smaller than the peak value of the total production. The increase in production near the wall $(y^{+}\simeq 10)$ in the forced flow is essentially entirely due to the boundary-layer contribution. These results are consistent with the notion that the increase in the peak Reynolds stresses (figure 19), especially $\overline{u^{\prime }u^{\prime }}$ , is not an additive effect of injection of FST into the near-wall region. Instead, the turbulence kinetic energy and its production rate are enhanced near the wall, below the extent of penetration of free-stream perturbations. The associated changes in the structures and spectra of the wall turbulence are examined in § 5.