1.1. Generalised description of the surface layer: aims

Consider the general case of fully developed high Reynolds number wall-bounded turbulent flows over solid surfaces. In such flows, there exists a turbulence-filled region near the surface where turbulence structure is directly modified by the interactions between the turbulence eddies within the region and the impermeable surface. This ‘surface-modified layer’ (SML) has a characteristic thickness, $\delta _{SML}$ , that will depend on the scale, structure and source of the turbulence. Within the surface-modified turbulence region there exist sub-layers with definable characteristics. Directly adjacent to hydrodynamically smooth surfaces, for example, a frictional layer is created characterised by high viscous stresses and frictional impacts on the turbulence dynamics. Similarly, over hydrodynamically rough surfaces a roughness layer is created by the amalgamation of the local separated flows over the roughness elements. The turbulence within this ‘surface-adjacent sub-layer’ (SAS) of thickness $\delta _{SAS}$ is characterised by a viscous scale $\delta _\nu$ over smooth surfaces, or by a roughness scale $z_0$ over rough surfaces. We consider high Reynolds numbers and small roughness elements where the turbulence scales within the SAS are confined to a thin layer adjacent to the surface ( $\delta _{SAS}\lt \lt \delta _{SML}$ ). On the other extreme, the largest surface-modulated turbulence eddies have coherence lengths of order $\delta _{SML}$ .

However, there exists the potential for an inertia-dominated near-surface sub-layer to exist just outside the SAS with turbulence eddies that are strongly impacted by the surface. If this inertial sub-layer is sufficiently thin relative to $\delta _{SML}$ , and if aspects that impact coherence of the inertia-dominant turbulence motions are not significantly influenced by either $\delta _{SML}$ -scale motions or the SAS below, we anticipate that specific coherence lengths related to these inertia-dominated aspects of turbulence motions will scale on the distance from the surface. This could occur only in the absence of external forcing sufficiently strong to interfere with linear scaling, such as strong horizontal pressure gradients induced by flow over non-planar surfaces. We use the term ‘surface layer’ to refer to an inertia-dominated sub-layer as described above which displays linear increases in specific integral scales with distance $z$ from the surface. Whereas it is not a priori obvious that surface layers so defined exist, this concept of surface layer originated with the law-of-the-wall phenomenology for the canonical shear-driven smooth-wall turbulent boundary layer (Tennekes & Lumley Reference Tennekes and Lumley1972), and was generalised by Monin & Obukhov (Reference Monin and Obukhov1954) to the canonical daytime rough-surface atmospheric boundary layer driven by both shear and buoyancy, further discussed in § 1.1.1. The current study aims to give substance to this concept on the basis of experimental evidence and considers how generally applicable it may be.

In context with our experimental study, consider figure 1, which contrasts the concept of the surface layer in two classes of high Reynolds number wall-bounded turbulent flows. Figure 1(a) illustrates the classical surface layer in a high Reynolds number turbulent boundary layer (TBL) that contains a surface layer. In this case, turbulence generated very near the surface in the buffer layer spreads from the surface to fill the TBL. As a result, the TBL is, itself, the ‘SML’ since it is the region of turbulence that has been directly modified by the solid surface. The SAS, in this case, is the viscous sub-layer with thickness $\delta _{SAS}$ that scales on $\delta _\nu = \nu /u_\tau$ when the surface is hydrodynamically smooth, or a roughness layer with thickness $\delta _{SAS}$ that scales on $z_0\lt \lt \delta _{TBL}$ when the surface is hydrodynamically rough ( $\nu$ is kinematic viscosity and $u_\tau$ is friction velocity). The lower boundary of the surface layer, $\delta _L$ , is outside the buffer or roughness layers, while the upper margin typically extends to $\delta _U\sim 0.15\delta _{TBL}$ .

Figure 1. Two examples of wall-bounded turbulent flows: (a) a high Reynolds number turbulent boundary layer over a solid surface within an irrotational external flow; (b) the transport of externally generated turbulence over an impermeable surface. In both flows, horizontal dimensions of surface curvature are large relative to the thickness of the surface-modified layer.

In contrast, figure 1(b) illustrates a SML within a region of turbulence that is in a fully developed state of interaction with the surface as turbulence eddies are transported over within a roughly uniform mean flow. A fundamental distinction between the flows illustrated in figure 1(a) versus figure 1(b) is that, whereas in the TBL of figure 1(a) the SML is created by turbulence that was generated at the surface and is subsequently modified by the surface as it advects downstream and away from the surface, the SML in figure 1(b) is created from turbulence that originates upstream and is subsequently modified as it passes over and interacts with the impermeable surface outside a SAS. In this case the SAS is partly generated by friction or roughness with an upper boundary that confines turbulence fluctuations that are generated adjacent to the surface by strong shear; the SAS is effectively a non-canonical highly turbulent boundary layer. We consider cases where $\delta _{SAS} \lt \lt \delta _{SML}$ and hypothesise that the externally generated turbulence interacts with the impermeable surface through a SAS that is sufficiently thin to have negligible impact on the surface-modified turbulence above. Thus, like the classical law-of-the-wall layers in the TBL (§ 1.1.1), there might exist a near-surface inertia-dominated turbulence layer that is strongly modified by interactions with the impermeable surface without being significantly impacted by the much larger $\delta _{SML}$ eddies or much smaller $\delta _{SAS}$ eddies within the SAS. If that is the case, and if the mechanisms that cause linear scaling between some integral scales and the distance from the surface within the surface layer of the TBL are the same, we anticipate the potential development of a surface layer in the flow of figure 1(b). This surface layer would extend from a lower bound $\delta _L$ outside the SAS to an upper bound $\delta _U$ that is sufficiently far below the upper boundary of the SML, $\delta _{SML}$ , that the eddies of order $\delta _{SML}$ in scale do not significantly impact the turbulence within, as illustrated in figure 1(b).

1.2. Previous studies: the shear-free surface layer

Prior works have examined shear-free turbulence–surface interactions within what is sometimes described as a ‘shear-free TBL.’ Hunt & Graham (Reference Hunt and Graham1978) developed a theory to characterise the interaction between initially homogeneous and isotropic turbulence within a uniform flow and a flat surface with the surface velocity equal to the mean flow velocity. The theory applies as the turbulence initiates distortion near the leading edge of the flat plate where linearisation akin to ‘rapid distortion theory’ (RDT) is applicable. Three distinct regions in the flow were developed within the linearisation: (i) a very thin wall-adjacent viscous laminar sub-layer where no slip is applied, (ii) an inertia-dominated ‘source layer’ (using the terminology of Hunt & Graham (Reference Hunt and Graham1978)) where the RDT-like model predicts an irrotational modification to the flow from blockage and (iii) an outer region where the flow returns to the free-stream state. A lower boundary condition that enforces blockage of vertical motions at the flat plate defines the source of fluctuations that drive the ‘source layer.’ This is the vertical velocity within the isotropic free-stream turbulence at the location of the surface. In effect, the turbulence flowing over the flat plate is blocked at the lower boundary and turbulence structure within the source region is adjusted through an irrotational response to blockage. It was shown that the model predicts linear growth in integral scales in the limit $z\rightarrow 0$ (i.e. to the thin viscous layer), but the relationship to surface layer was not analysed. An important element in the theoretical model is that the viscous layer is so thin that that blockage directly modulates the turbulence in their source layer, similar to an extended surface layer.

Perot & Moin (Reference Perot and Moin1995) note that the Hunt & Graham (Reference Hunt and Graham1978) model of turbulence response to a moving wall is mathematically equivalent to an RDT model of a wall instantaneously placed within a field of isotropic turbulence. Using this approach, Hunt & Carlotti (Reference Hunt and Carlotti2001) extended the Hunt & Graham analysis to show that $l_{ww,x}\sim z$ in the limit $z\rightarrow 0$ (with a thin viscous layer) when the wavenumber spectrum contains a high Reynolds number $k^{-5/3}$ inertial sub-range. However, with more rapid spectral roll-off consistent with lower Reynolds numbers, the integral scale has a nonlinear dependence on $z$ in the limit, suggesting that linear scaling requires inertia dominance. In the TBL, where both shear production and blockage are active, Hunt & Carlotti (Reference Hunt and Carlotti2001) represent vertical velocity variances as the sum of ‘shear’ and ‘blocking’ terms that are treated separately. Hunt (Reference Hunt1984) used the Hunt & Graham (Reference Hunt and Graham1978) concepts to develop a heuristic model of a fully convection-driven turbulent ABL that they found agrees well with measurements. They show that blockage in the near-surface region leads to linear growth in the streamwise integral length scale of vertical velocity fluctuations, $l_{ww,x}$ , in the limit $z\rightarrow 0$ . These studies suggest that both blockage and strong inertia dominance underlie the linear growth in integral scale that identifies the surface layer. However, linear scaling appears only in the limit. Furthermore, RDT restricts the dynamics to a short-time response with no turbulence–turbulence interactions. Magnaudet (Reference Magnaudet2003) argued that the RDT theory represents the leading-order terms for longer-time expansions in the limit of high Reynolds numbers, potentially extending applicability to longer times.

Early experimental measurements by Uzkan & Reynolds (Reference Uzkan and Reynolds1967) of grid-generated turbulence passed over a moving belt in a water channel facility, concluded that surface blockage acted to attenuate turbulence intensity near the surface in the absence of mean velocity gradients. Thomas & Hancock (Reference Thomas and Hancock1977) improved on these experiments with similar wind tunnel measurements of grid turbulence over a moving belt designed to compare with the theory of Hunt & Graham (Reference Hunt and Graham1978). These experiments agreed overall with the Hunt & Graham (Reference Hunt and Graham1978) predictions. Specifically, they found that vertical velocity variance decreases towards the (moving) surface like $\langle {w'}^2\rangle \sim z^{2/3}$ in the limit $z\rightarrow 0$ . Consistent with arguments from continuity (§ 4.3), they find that horizontal variances, $\langle {u'}^2\rangle$ and $\langle {v'}^2\rangle$ , increase towards the surface.

McCorquodale & Munro (Reference McCorquodale and Munro2017, Reference McCorquodale and Munro2018) studied experimentally shear-free turbulence–surface interactions with oscillating grid turbulence to analyse pressure–strain-rate correlations and intercomponent energy transfer. They concluded that the ‘splat–antisplat disequilibrium’ mechanism proposed by Perot & Moin (Reference Perot and Moin1995) is a viscous effect in the thin wall-adjacent sub-layer and that inter-component energy transfer acts as a ‘return-to-isotropy’ mechanism outside the viscous sub-layer, as originally proposed by Walker et al. (Reference Walker, Leighton and Garza-Rios1996). Measured increases in horizontal integral length scale of vertical velocity fluctuations were found to agree qualitatively with Thomas & Hancock (Reference Thomas and Hancock1977), but assessment of linearity is not possible due to significant scatter in the data points.

3.1. The existence of the shear-free surface layer

Figure 4. Wall-normal profile of the horizontal integral length scales of vertical velocity fluctuations in the streamwise direction ( $l_{ww,x}(z)$ ) for (a) the large grid and (b) the small grid. Using similar notation as in figure 1, the green dashed lines define the upper ( $\delta _U$ ) and lower ( $\delta _L$ ) margins of the linear region, the magenta dashed line is the upper margin of the SML ( $\delta _{SML}$ ) and the red dashed line is the upper margin of the SAS ( $\delta _{SAS}$ ) estimated as $\delta _{99}$ .

Figure 5. Wall-normal profiles of integral length scales of (a,c) streamwise velocity fluctuations in the streamwise direction ( $l_{uu,x}(z)$ ) and (b,d) cross-stream velocity fluctuations in the streamwise direction ( $l_{vv,x}(z)$ ). The top row (a,b) is for the large grid and the bottom (c,d) for the small. Using similar notation as in figure 4, the green dashed lines define the upper ( $\delta _U$ ) and lower ( $\delta _L$ ) margins of the linear region, the magenta dashed line is the upper margin of the SML ( $\delta _{SML}$ ) and the red dashed line is the upper margin of the SAS ( $\delta _{SAS}$ ) estimated using $\delta _{99}$ .

In figure 4, $l_{ww,x}$ is plotted against the distance from the plate, $z$ , for the grid-turbulence experiments with both grids, where integral scales are quantified according to § 2.3. In this figure the black solid line is $l_{ww,x}(z)$ with the plate installed in the test section and the dotted line is without the plate. The SML is the region within the grid-generated turbulence where the integral length scale, $l_{ww,x}$ , is significantly modified by the interactions between the grid turbulence and the flat plate. Therefore, $\delta _{SML}$ is defined as the height below which $l_{ww,x}$ differs significantly from $l_{ww,x}$ at the measurement location without the plate installed, where ‘significantly’ is defined as within one standard deviation of the variation of $l_{ww,x}(z)$ over the test section without the plate. The value of $\delta _{SML}$ is shown with the magenta dashed line in figure 4. As will be discussed below, two internal layers within the SML are objectively quantified: (i) the SAS (red dashed line), and (ii) the inertial SFSL (green dashed lines).

The SAS was discussed in § 1 in context with figure 1(b). It forms as the grid turbulence embedded within a quasi-uniform mean flow passes over the flat plate, creating a sub-layer adjacent to the surface with a mix of highly non-steady frictional stresses and turbulence production very close to the surface from the distortion of turbulence by a high mean shear rate. The surface-adjacent shear-generated turbulence and strong frictional stresses fill a region akin to a highly non-canonical TBL. We define the thickness of this SAS as the height at which the mean velocity profile is $99\%$ of the free-stream value, $U(\delta _{SAS}) = 0.99U_\infty$ , where $U_\infty$ is the mean velocity in the free stream (outside the surface-modified region). The SAS region, indicated by a dashed horizontal red line in figure 4, is discussed in more detail in § 4.

In figure 5 we plot the streamwise integral scales of the two horizontal components of fluctuating velocity over $z$ for the two grids. From figures 4 and 5 it is apparent that only the vertical velocity generates a horizontal integral scale that clearly increases with distance from the surface through the inertia-dominated region above, and separated from, the SAS (the red-dashed line). The SFSL is therefore defined from the regions of clear linear increase in $l_{ww,x}$ with $z$ in figure 4 for both grids. The upper and lower margins of the linear regions are objectively defined as described in Appendix B using an iterative procedure to identify the portion of the curve that is ‘most’ linear. This approach limits bias in quantifying $\delta _U$ and $\delta _L$ , the upper and lower margins of the SFSL shown with green horizontal dashed lines. The coefficient of determination of the linear fit over the SFSL is 0.997 and 0.992 for the large and small grids, respectively. As will be discussed in § 4, the mean shear rate is zero within and above this ‘shear-free’ surface-layer region.

Normalised thicknesses of the SAS, SFSL and SML for both grid cases are given in table 5. The upper boundary of the surface-modified region for the small and large grids is found to extend to approximately two free-stream streamwise integral length scales, $l_{uu,x\infty }$ , from the surface: $\delta _{SML}\approx 2l_{uu,x\infty }$ . Because the longitudinal integral length scale quantifies the coherence length of the energy-dominant eddies in the quasi-isotropic free-stream turbulence, it is not surprising that the thickness of the SML scales on the coherence length of the energy-containing turbulence motions that interact with the surface. This result is consistent with TH77, who measured the influence of the surface to extend to roughly twice the free-stream streamwise integral scale in their moving wall experiment, as well as with the theoretical predictions of HG78, where we use the acronyms HG78 for Hunt & Graham (Reference Hunt and Graham1978) and TH77 for Thomas & Hancock (Reference Thomas and Hancock1977).

Table 5. Surface-layer parameters for the grid-turbulence SDSL and TBL SDSL.

The consistencies with the HG78 theory and TH77 experiments are particularly significant because these were developed with a moving wall to minimise the thickness of the SAS. Specifically, the theory was developed in the limit $\delta _{visc}/l_{uu,x\infty }\lt \lt 1$ , while TH77 estimated this ratio to be approximately 0.05 in their experiments, with $\delta _{visc}$ defined as a viscous scale in a laminar SAS. As shown in table 5, in our fixed-wall grid-turbulence experiments $\delta _{SAS}/\delta _{SML}\approx 0.15$ , where $\delta _{SML}\approx 2l_{uu,x\infty }$ and $\delta _{SAS}$ is defined as $\delta _{99}$ , much thicker than the wall-adjacent viscous layer. Given the well-defined structure of the surface layers in figure 4, we conclude that the SASs in our fixed-wall experiments are sufficiently thin to maintain the nature of the turbulence-surface interactions that were modelled in the HG78 theory and measured in the TH77 experiments.

Interestingly, the upper margin of the surface layer is approximately one free-stream streamwise integral length scale from the surface ( $\delta _U \approx l_{uu,x\infty }$ ), and the thickness of the SFSL, $\Delta _{SL}=\delta _U-\delta _L$ , is approximately two thirds of the longitudinal integral scale in the free stream. These results are consistent between the two grids which create free-stream correlation lengths a factor of two different. As shown in figure 4, there exists a gap between the lower margin of the surface layer defined by linear growth in $l_{ww,x}$ , and the upper margin of the SAS as defined by $\delta _{99}$ . If the mechanism that creates linear growth in the integral length scale with $z$ is associated with blockage at the impermeable plate and the inertia dominance of the free-stream eddies that are modified, the linear SFSL region will not initiate until some distance above the SAS where length scales associated with frictional stresses and turbulence production dominate. Table 5 indicates that the gap between $\delta _{SAS}$ and $\delta _L$ is approximately 30 $\%$ of $\delta _L$ . This transition will be further discussed in § 4.

The streamwise integral length scales of the horizontal velocity components $l_{uu,x}$ and $l_{vv,x}$ plotted in figure 5 show consistency between the large and small grids. Importantly, none of these profiles increase with distance from the surface in the surface layer, much less linearly as is the case for $l_{ww,x}(z)$ . This significant result supports the hypothesis that the ‘special’ integral scales that display linear scaling with distance from the surface are those associated with vertical fluctuating velocity which feels directly the impacts of blockage at the impermeable surface. Contrasting figures 5(a), 5(b) (large grid) with figures 5(b), 5(c) (small grid), the variations in the streamwise integral scales of horizontal velocity fluctuations are consistent, in that for both grids all these integral scales increase from the upper margin towards the lower margin of the SFSL. With the exception of $l_{vv,x}$ for the large grid, all integral scales decrease from the lower margin of the surface layer into the SAS, although the $z$ -location where the decrease initiates varies in relation to the lower margin of the surface layer. Overall, one anticipates increasing measurement inaccuracy approaching the surface in the SAS. It is not clear if the increase in $l_{vv,x}$ at the lowest measurement location adjacent to the plate is physical, and the PIV field of view does not extend sufficiently close to the plate to capture the peak, or if the increase in $l_{vv,x}$ at the lowest measured point reflects measurement error.

3.2. The shear-dominated surface layer

Figure 6. (a) Profile of $l_{ww,x}$ in the wall-normal direction $z$ in the TBL, where $z$ and $l_{ww,x}$ are normalised with the boundary layer thickness ( $\delta _{99})$ . The green dashed lines identify the upper ( $\delta _U$ ) and lower ( $\delta _L$ ) margins of the linear region and the red dashed line is 40 viscous units from the surface, an estimate for the upper margin of the SAS; (b) mean velocity magnitude plotted against distance from the surface, both non-dimensionalised with wall units, plotted log linear to display the region with logarithmic dependence. The dashed green lines are the upper and lower margins of the linear region in (a).

Figure 7. Profiles of (a) $l_{uu,x}$ and (b) $l_{vv,x}$ in the wall-normal direction $z$ in the TBL, where $z$ is normalised with the boundary layer thickness ( $\delta _{99}$ ). The green dashed lines define the upper ( $\delta _U$ ) and lower ( $\delta _L$ ) margins of the linear region and the red dashed line is 40 viscous units from the wall, an estimate for the upper margin of the SAS.

A similar analysis of surface-layer statistics from the TBL experiment is presented in this section, where the variations with distance from the surface of the streamwise integral scales of the fluctuating vertical velocity, $l_{ww,x}$ , are plotted in figure 6(a), and of the horizontal fluctuating velocity components, $l_{uu,x}$ and $l_{vv,x}$ , in figure 7. Like the grid-turbulence experiments, only the vertical velocity integral scale displays clear linear increase over a region that we identify as the surface layer. As discussed in § 3.1, unlike grid-turbulence interactions where $l_{uu,x}$ and $l_{vv,x}$ decrease or remain roughly constant with increasing $z$ over the surface layer, in the TBL these integral scales increase with $z$ over the surface layer due to the existence of non-zero mean shear rate, with consequent turbulence production. It is for this reason that we refer to the surface layer in the TBL as ‘shear dominated’ (SDSL) in contrast with the ‘shear-free’ surface layer created by the interaction between grid turbulence and the surface where mean shear and production rates are zero over the SFSL (§ 4.1).

The linear region of $l_{ww,x}(z)$ is identified using the same procedure discussed above and in Appendix B. The coefficient of determination of the linear fit over the SDSL is 0.997. In figure 6(a) the objectively deduced boundaries of the surface layer are shown with green dashed lines, extending between $\delta _L=83\delta _\nu$ and $\delta _U=0.1\delta _{99}$ . As previously discussed, the inertial surface layer of the TBL is associated with the log region of the mean velocity profile

(3.1) \begin{equation} U^+=\kappa ^{-1} log(z^+)+B. \end{equation}

Equation 3.1 is plotted alongside the measured mean velocity profile in figure 6(b) as a red dotted line with $\kappa =0.41$ . The region where the measured mean velocity profile matches the log profile coincides with the surface layer determined from the linear region of $l_{ww,x}$ , shown with the green dashed lines. As discussed in § 1, the SML in this flow is the region filled by turbulence that was generated below the surface layer and transported to the upper boundary layer with thickness $\delta _{SML}=\delta _{99}$ in the mean.

In the TBL, the SDSL resides above a SAS with thickness that scales on the surface viscous scale $\delta _\nu$ , and which is dominated by high viscous stress adjacent to the surface and turbulence production that peaks at $z^+ \approx 11$ (Kim et al. Reference Kim, Moin and Moser1987; Spalart Reference Spalart1988). The turbulence production rate plotted with LMFL data from a TBL with the same flow variables as the current TBL (Foucaut et al. Reference Foucaut, Cuvier, Willert and Soria2018) is consistent with plots from the literature (Spalart Reference Spalart1988), all indicating a change in slope in the turbulence production curve variation when $z^+$ is near 40. At $z^+=40$ production rate is approximately 21 % of the peak value at $z^+ \approx 11$ followed by a slow decrease through the SDSL to the upper boundary layer. We use this approximate elbow as a sensible estimate of the upper margin of the SAS (i.e. $\delta _{SAS}^+=40$ ) as indicated by the horizontal dashed red lines in figures 6 and 7.

As in the SFSL, these figures show a transition region between the top of the SAS and the bottom of the SDSL where mixed scaling transitions to purely surface-layer scaling, $l_{ww,x}\sim z$ . Using $z^+=40$ as an estimate of $\delta _{SAS}$ , table 5 indicates that the transition occupies approximately 50 $\%$ of the distance to the lower margin of the SDSL, $\delta _L$ , in comparison with 30 $\%$ for the transition to the SFSL in the grid-turbulence experiment. This difference likely reflects the transition of turbulence production from below to within the SDSL in the TBL, in contrast to the SFSL within the grid turbulence where turbulence produced adjacent to the surface is confined to the SAS (§ 4).

3.3. Comparisons of integralscale variations in the shear-free and shear-dominated surface layers

Figure 8. Variations of $x{-}z$ integral length scales $\hat {l}$ , defined and normalised by (3.3), for each of the three measured velocity components and for the two grid turbulence–surface interaction and TBL flows. The normalised wall-normal direction $\hat {z}$ is defined by (3.2) so as to centre the variations around the surface-layer regions. The three cases are plotted with these line types: —–, large grid; $\cdots \cdots$ , small grid; - $\cdot$ - $\cdot$ -, TBL. As in figure 4, the green lines indicate the upper ( $\delta _U$ ) and lower ( $\delta _L$ ) margins of the linear regions, the magenta lines indicate the upper margins of the SMLs ( $\delta _{SML}$ ), and the red lines indicate the upper margins of the SASs ( $\delta _{SAS}$ ). Note that the TBL SML is well beyond the upper bounds of the figure at approximately $\hat {z} = 16$ .

We aim to identify characteristics of the shear-free and shear-dominated surface layers that are similar and dissimilar. To this end, in figure 8 we define a vertical axis $\hat {z}$ that centres on the surface layer and nearby regions to compare integral-scale variations in and near the surface layer in the two very different wall-bounded turbulent flows created in the grid turbulence and TBL experiments

(3.2) \begin{equation} \hat {z} = \frac {z-\delta _L}{\delta _U-\delta _L}. \end{equation}

This normalisation places the lower and upper bounds of the surface layer at $\hat {z} = 0$ and $1$ , respectively, for both flows. All integral scales are plotted as a function of $\hat {z}$ relative to the horizontal integral scales of vertical velocity fluctuation as follows:

(3.3) \begin{equation} \hat {l}_{\alpha \alpha ,\beta }=\frac {l_{\alpha \alpha ,\beta }-l_{ww,x}(\delta _L)}{l_{ww,x}(\delta _U)-l_{ww,x}(\delta _L)}. \end{equation}

Therefore, in figure 8 all horizontal integral scales of vertical velocity fluctuation, $\hat {l}_{ww,x}$ , pass through $0$ and $1$ at the lower and upper margins of the surface layer by construction. For vertical velocity fluctuations, the horizontal integral scales of the grid and TBL flows overlap at $\hat {z}=0$ and $1$ and throughout the surface layer due to linear scaling.

This is not the case, however, with the other integral scales. Figure 8 shows that the two grid-turbulence SFSL cases agree well for normalised streamwise length scales, while the correlations in the vertical direction show larger differences while still trending together. In the vertical what is plotted is the correlation length as a function of starting $z$ location. The vertical correlation of the vertical fluctuating velocity, $l_{ww,z}$ , in particular, appears to suggest overall larger vertical correlation lengths relative to the thickness of the surface layers with the smaller grid. Whether this difference reflects a different turbulence structure, a Reynolds number effect or is related to the measure used to quantify coherence in inhomogeneous directions (2.3), is unclear.

In context with the previous observation, note from figure 4 and table 5 that, while the thicknesses of the SML and SFSL are smaller with the smaller grid, the ratio of SFSL to SML thickness is the same. Interestingly, figure 8 shows that, while the $\hat {z}$ normalisation on the SFSL thickness collapses both the height of the surface-modulated layers for the large and small grids (SML, magenta lines), it also collapses the thickness of the surface-adjacent layer (SAS, red lines), suggesting a relative gap between $\delta _{SAS}$ and $\delta _L$ set by the largest surface-layer eddies in the vertical. Based on the definition $\delta _{SAS}^+=40$ , designed to capture most (but not all) of the turbulence production rate, the relative gap between the SAS and the SDSL in the TBL is wider than between the SAS and SFSL in the grid-turbulence flows, as previously discussed (§ 3.2). The separation between $\delta _{SML}$ and $\delta _U$ is much greater for the TBL than for the grid-turbulence cases, so that the surface-layer scaling does not collapse the SML as in the grid-turbulence cases.

As previously observed, figures 8(a)–8(f) as a group show clearly that only coherence lengths of vertical fluctuating velocity in the streamwise direction increase linearly with wall-normal distance from the surface in all flows. The only other coherence length that increases with distance from the surface in all three datasets is the correlation of vertical velocity fluctuations in the vertical. This observation is consistent with the hypothesis that the essential coherence character of the surface layer is associated with blockage over wall-normal fluctuations by the impermeable surface. As mentioned above, the lack of linearity in $l_{ww,z}(z)$ may reflect, in part, the impact of inhomogeniety in the vertical on the coherence measure (2.3).

The other global observation that is apparent from the group of figures is that, whereas for the SFSL flows some integral scales may increase, decrease, or remain constant with distance from the surface, in the TBL SDSL flow all integral scales increase with distance from the surface, with linearity in $l_{ww,x}(z)$ as a special case. As will be discussed in coming sections, this universal behaviour likely reflects the existence of turbulence production above the SAS and throughout the surface layer as a consequence of shear in the SDSL in comparison with the SFSL. Although the integral length scale of vertical velocity in the cross-stream direction, $l_{ww,y}$ , is not available in the present datasets, the RDT analysis of Hunt & Carlotti (Reference Hunt and Carlotti2001) suggests that $l_{ww,y}$ displays linear scaling with distance from the surface similar to that observed in $l_{ww,x}(z)$ .

In the grid-turbulence SFSL, the integral length scale of the streamwise velocity component in the streamwise direction, $l_{uu,x}$ , is relatively constant with $z$ in the SFSL, while the integral scale of the cross-stream velocity in the streamwise direction, $l_{vv,x}$ , shows a clear increase towards the surface. In figure 8 from TH77, predictions from the HG78 RDT theory are compared with their grid-turbulence experimental results for $l_{uu,x}(z)$ . Whereas our grid-turbulence experiments in figure 8 show roughly uniform $l_{uu,x}$ through the SFSL, there is an indication, with both grids, that $l_{uu,x}(z)$ decreases towards the surface beginning in the lower SFSL and into the SAS. The RDT theory also predicts a reduction, but it is a somewhat greater and more gradual reduction that initiates near the upper margin of a linear variation in $l_{ww,x}$ observed in the overlaid TH77 experimental data. However, the reduction in $l_{uu,x}$ towards the surface shown in TH77 data initiates in the lower margin of the linear region more consistent with our figure 8.

One might suspect that in the SFSL the reduction towards the surface in $l_{ww,x}$ together with increasing $l_{vv,x}$ may indirectly reflect continuity (§ 4.3). On the other hand, it is curious that, whereas we later show in figure 11 that the normalised variances of the streamwise and transverse velocity fluctuations increase equivalently towards the surface in the lower SFSL consistent with the HG78 theory for isotropic turbulence, the streamwise integral scale of transverse velocity in figure 8 clearly increases towards the surface in the lower SFSL while the integral scale of streamwise velocity clearly does not. The same is true in the TH77 data.

In stark contrast with the SFSL grid-turbulence variations just discussed, in the TBL $l_{uu,x}(z)$ increases strongly and consistently from the surface and through the SDSL, reaching a peak near the upper margin of the SDSL. This, we argue, is a consequence of the existence of mean shear rate and turbulence production, which peaks at $z^+\approx 11$ , within the SAS and well below the SDSL. Turbulence eddies created below the SDSL are distorted both by interactions with the impermeable surface and by mean shear as they move through the surface layer and into the upper regions of the TBL. It is well understood that mean shear rate causes coherent structure in the fluctuating streamwise velocity to elongate in the streamwise direction (Lee et al. Reference Lee, Kim and Moin1990) so that the $u'$ eddies elongate and the streamwise coherence length increases from the lower to upper SDSL where mean shear rate decreases like $1/z$ . Whereas $l_{uu,x}(z)$ increases strongly through the SDSL, $l_{vv,x}$ increases only gradually, through the SDSL and into the upper TBL.

What is particularly interesting in the observations above is that, independent of the existence of mean shear with corresponding turbulence production and the distortion/elongation of turbulence eddies, and independent of the increase or not of other integral scales, only the integral scales of vertical fluctuating velocity increase with distance from the surface in all three experiments. Furthermore, only the horizontal integral scales of vertical fluctuating velocity increase linearly with $z$ in a region we define as a surface layer. Whereas in the current experiments it was not possible to measure transverse correlations, linear growth of the integral length scales of vertical fluctuating velocity in both horizontal directions was identified by Apostolidis et al. (Reference Apostolidis, Laval and Vassilicos2022) in direct numeric simulation channel flow data. The observation that only the vertical fluctuating velocity has correlation lengths that scale on wall-normal distance strongly supports the hypothesis that the general mechanisms that create a surface layer are associated with blockage of vertical velocity at an impermeable surface.

4.1. Turbulence production rate

The two classes of wall-bounded turbulent flow considered here have fundamentally different mean shear rate and shear-generated turbulence production-rate characteristics as shown in figures 9 and 10, where normalised mean velocity, mean shear rate, Reynolds shear stress and turbulent production rate are plotted against the normalised wall-normal distance, $\hat {z}$ (3.2). In the grid-turbulence flows, mean shear, Reynolds shear stress and turbulence production are confined to the SAS, which in this case is a highly non-canonical boundary layer in which the turbulence fluctuations generated very near the surface are confined. The HG78 model predicts that if the SAS is sufficiently thin, it does not impact the suppression of vertical velocity by surface impermeability, which we argued in § 3.1 is also the case in both the TH77 and the current experiments.

Interestingly, figure 9 shows that mean shear rate, Reynolds shear stress and turbulence production rate all decrease in the vertical direction from the upper margin of the SAS to reach zero precisely at the lower margin of the SFSL, where $l_{ww,x}$ initiates a linear increase with $z$ . The SFSL, as indicated by dashed green lines, exists in the region of zero shear-induced turbulence production rate. The observation that the two grid flow profiles and the locations of $\delta _{SAS}$ and $\delta _{SML}$ collapse when scaled on the depth of the SFSL suggests that the initiation of a SFSL outside a SAS that confines near-wall turbulence production and Reynolds shear stress is a generalisable result for the influence of a surface on turbulence eddies that originated away from the surface. However, the fact that the canonical TBL contains a surface layer within a region in which mean shear rate, turbulence production rate and Reynolds shear stress are non-zero, as shown in figure 10, implies that the lack of mean shear and turbulence production is not a general requirement for the existence of a surface layer. This leads to the question of what is the fundamental mechanism that allows the surface layer to exist both in the presence of turbulence production in the TBL SDSL and in the absence of turbulence production in the grid-turbulence SFSL.

The fundamental difference between the two classes of flow is in the origin of the turbulence that is subsequently distorted in a similar manner in both flows so as to produce linear growth in $l_{ww,x}(z)$ in an inertia-dominated surface layer. In the grid-turbulence experiments the SFSL is created within turbulence eddies that were generated away from the surface and interact with the impermeable surface through a SAS. This SAS must be sufficiently thin, relative to the free-stream integral scale, that the eddies outside the SAS in the SFSL can respond directly to blockage at the surface (note that $\delta _L\approx 1.4\delta _{SAS}$ , table 5). This blockage impacts turbulence eddy structure over a sufficiently extended distance from the surface to create measurable linear growth in the horizontal integral scale of vertical fluctuations over a distance that scales on free-stream turbulence eddy size. In the TBL, however, the turbulence eddies that are modulated by the surface to create the SDSL originate very close to the surface below the lower margin of the SDSL in the vicinity of peak production rate at $z^+\approx 11$ . In the $\hat {z}$ coordinates of figure 10, peak production rate occurs at $\hat {z}\approx -0.51$ , well below $\delta _{SAS}$ and, especially, $\delta _L$ . Furthermore, we estimate the shear production-rate value at $\delta _{SAS}$ to be approximately one fifth of the peak value and note that $\delta _L\approx 2\delta _{SAS}$ (table 5). As the turbulence eddies that were generated very near the surface fill the boundary layer downstream, they pass through the surface layer where mean shear rate in the surface layer (figure 10 b) elongates the eddies to produce the continual increase in $l_{uu,x}$ with $z$ through the SDSL, as observed in figure 8(b) (and previously discussed in § 3.3).

Figure 11. Wall-normal variations in normalised component velocity variances in the grid-turbulence–surface experiments for (a) the large grid and (b) the small grid, plotted against $\hat {z}$ to centre on the SFSL. Also plotted are the normalised sum of the horizontal variances, $\langle u_h'^2 \rangle$ , and the normalised sum of the three variances, $\langle q^2 \rangle$ . All variances are normalised by their values in the free stream (see table 1 for relative variances). The horizontal lines indicate $\delta _U$ , $\delta _L$ , $\delta _{SML}$ and $\delta _{SAS}$ , as labelled in the figure. The location of the surface is: (a) $\hat {z} = -0.59$ and (b) $\hat {z} = -0.64$ .

Figure 12. Wall-normal variations in velocity variances in the TBL normalised by the corresponding component variances at the upper margin of the SDSL, $\delta _U$ , plotted against $\hat {z}$ to centre on the SDSL. Like figure 11, also plotted are the sum of the horizontal fluctuations, $\langle u_h'^2\rangle$ and the sum of all three variances, $\langle q^2\rangle$ . The horizontal lines indicate $\delta _U$ , $\delta _L$ and $\delta _{SAS}$ . At $\delta _U$ , $\langle u'^2\rangle /\langle w'^2\rangle = 3.39$ and $\langle u'^2\rangle /\langle v'^2\rangle = 2.28$ . The location of the surface is $\hat {z} = -0.58$ .

4.2. Impacts of the surface on the variations in velocity variances

The relative variations in the variances of the fluctuating velocity components through the SFSL and SDSL are presented in figures 11 and 12. Specifically, we compare the wall-normal variations of normalised velocity variances, $\langle u'^2\rangle$ , $\langle w'^2\rangle$ and $\langle v'^2\rangle$ , centred on the surface layer using the normalised vertical height, $\hat {z}$ in (3.2). Also plotted are the normalised variances of the ‘horizontal’ velocity vector in the plane parallel to the surface, $\langle u'^2_h\rangle =\langle u'^2\rangle +\langle v'^2\rangle$ , and twice the turbulent kinetic energy, $\langle q^2\rangle =\langle u'^2\rangle +\langle v'^2\rangle +\langle w'^2\rangle$ . Each variance is plotted normalised. For the grid-turbulence cases we normalise with values in the free stream, while in the TBL experiments we normalise with the values at the upper margin of the SDSL, $\delta _U$ .

In and surrounding the SFSL created by grid-turbulence–surface interactions, there is consistency in the $z$ variations of the normalised component variances between the large and small grids shown in figure 11. There is also consistency between the current results above $\delta _{SAS}$ , the experimental results obtained by TH77 and the theory of HG78, where the latter two are compared in figure 5 of HG78. As analysed in detail in § 4.3, data and theory all demonstrate a trade-off between the vertical and horizontal velocity variances in the SFSL. Whereas $\langle w'^2 \rangle$ decreases through the entire surface-modified region towards zero at the surface, it decreases more rapidly from the upper to lower margins of the SFSL. The more rapid decrease towards the lower margin of the SFSL coincides with an increase in horizontal velocity variance that continues into the SAS, a trade-off also clear in the HG78 theory and TH77 data.

Qualitatively consistent with the HG78 theory, figure 11 shows a decrease in turbulent kinetic energy ( $\langle q^2\rangle /2$ ) followed by an increase from the lower margins of the SFSL towards $\delta _{SAS}$ . However, because the HG78 theory applies in the limit of an infinitesimally thin SAS, the vertical velocity variance approaches zero due to surface impermeability at the upper margin of their viscous SAS. By contrast, in the current SFSL grid-turbulence experiments the SAS is relatively thick, 26 %–32 $\%$ of $l_{uu,x_\infty }$ (table 5). Consequently, the vertical velocity variance at the lower boundary of the SFSL is relatively large, approximately $50\%$ of the free-stream value. Figure 11 shows that, when normalised by the free-stream values, the streamwise, cross-stream and horizontal velocity variances coincide throughout the SML. These horizontal variances decrease from the upper margin of the SML to a minor minimum slightly below the free-stream values near the upper margin of the surface layer, consistent with the HG78 theory, before transitioning to increasing $\langle u'^2_h \rangle$ towards and below the lower margin of the SFSL.

In comparison with figure 11 for the SFSL, figure 12 shows both similarities and differences in the variations of the normalised variances surrounding the SDSL of the TBL, differences that can be understood in context with the discussions surrounding turbulence production in § 4.1 above. The vertical velocity variance, in blue, decreases relatively slowly in the surface-layer region compared with the transitional layer approaching the SAS below, where the variance decreases sharply towards zero at the impermeable surface. In contrast with the SFSL, vertical velocity variance decreases from the upper to lower margins of the SDSL (figure 12) much more slowly than through the SFSL (figure 11), while the streamwise and cross-stream velocity variances (in black) increase towards the surface within the SDSL more rapidly than within the SFSL. In the grid-turbulence flows, the two normalised horizontal variances vary similarly through the SFSL. In the TBL, whereas the horizontal velocity variance is dominated by the streamwise velocity fluctuations, the transverse variance has a somewhat different variation than the streamwise component, especially in the SAS region.

In comparison with the SFSL we attribute the more rapid increase in streamwise velocity variance from the upper to lower margins of the SDSL to turbulence production in this region (§ 4.1). Shear-driven turbulence production enters the streamwise fluctuating velocity directly before being transferred into the cross-stream and vertical components through pressure–strain-rate inter-component energy transfer. However, because both peak production rate of streamwise variance and streamwise velocity variance occur well below $\delta _L$ ( $z+\approx 11$ and $15$ , respectively), the variations in transverse and vertical velocity variances through the SDSL are likely driven both by inter-component energy transfer and the vertical transport of turbulence eddies into the SDSL region from below. In the SFSL where turbulence production does not exist, however, the increase in the horizontal velocity variance from the upper to lower boundary of the surface layer is likely associated with incompressibility, as discussed in the following section (§ 4.3). Also similar to the SFSL results in figure 11, whereas the HG78 theory predicts a decrease in $\langle w'^2\rangle$ towards zero together with an increase in $\langle u'^2\rangle$ without turbulence production, normalised $\langle w'^2\rangle$ in the TBL approaches a value at the upper margin of the SAS that is even larger than that for the fixed-wall SFSL of figure 11. This suggests that wall blockage of vertical velocity at the SAS of the TBL is partially masked by production and inter-component energy transfer from streamwise to vertical turbulence fluctuations in the region below the SDSL, where $\langle u'^2\rangle$ is most strongly produced by mean shear rate.

Related to these observations, we note that whereas vertical velocity variance decreases from the upper to lower margins of the surface layer in both the SFSL and SDSL, turbulent kinetic energy decreases in the SFSL but increases in the SDSL due to the existence of turbulence production. What is particularly interesting is that even with the generation of streamwise velocity fluctuations by shear production and the transfer of variance into vertical and transverse velocity fluctuations through correlations between pressure and strain-rate fluctuations, surface layers with linear increases with $z$ in the horizontal coherence length of vertical fluctuations are created in both flows.

An attempt to account for these observations may follow from a simple spectral model in a horizontally homogeneous channel flow. Perry et al. (Reference Perry, Henbest and Chong1986) argued that, because of blocking at the surface, primarily only eddies of size $O(z)$ contribute to $w'$ motions whereas eddies of size larger than $O(z)$ contribute primarily to $u'$ and $v'$ motions but not significantly to $w'$ motions for $z\ll \delta$ . The one-dimensional streamwise energy spectrum $E_w (k_{x}, z)$ of fluctuating vertical velocities $w'$ at distance $z$ from the wall is therefore modelled as approximately constant up to $k_{x}\sim 1/z$ , hence equal to $E_w (k_{x}=0 , z)$ for $k_{x} \lt O(1/z)$ . Assuming, following Perry et al. (Reference Perry, Henbest and Chong1986), that eddies of size smaller than distance $z$ from the wall are in approximate Kolmogorov 1941 (K41) equilibrium, we may write $E_w (k_{x},z) \sim \epsilon ^{2/3}k_{x}^{-5/3}$ for some range of wavenumbers $k_{x}$ larger than $O(1/z)$ . Matching at $k_{x} = O(1/z)$ gives $E_w(k_{x}=0 , z) \sim \epsilon ^{2/3} z^{5/3}$ . Integrating $E_w (k_{x},z)$ over all $k_x$ yields a contribution $\sim (\epsilon z)^{2/3}$ to $\langle w'^{2} \rangle$ from $k_{x}=0$ to $k_{x} \sim 1/z$ and a contribution to $\langle w'^{2} \rangle$ from larger $k_x$ which is bounded from above by a term $\sim (\epsilon z)^{2/3}$ and is in fact proportional to $(\epsilon z)^{2/3}$ rather than just bounded by it if the Reynolds number is high enough to support a significant K41 inertial range. We can therefore consider $\langle w'^{2} \rangle \sim (\epsilon z)^{2/3}$ to be a good approximation even at Reynolds numbers that are not so high. In the SDSL of the TBL, which coincides with the log layer (figure 6), we may expect $\epsilon$ to approximately balance production and therefore scale approximately as $u_{\tau }^{3}/z$ . Hence, $\langle w'^{2} \rangle$ is roughly constant with $z$ in the high Reynolds number TBL (more so at the upper part of the SDSL where Kolmogorov spectra will be better defined than at the lower part). In the SFSL, $\epsilon$ may be assumed approximately homogeneous, hence independent of $z$ , and therefore $\langle w'^{2} \rangle \sim z^{2/3}$ . The overall shape of $E_w (k_{x}, z)$ assumed by Perry et al. (Reference Perry, Henbest and Chong1986) can therefore imply, in agreement with our observations, that $\langle w'^{2}\rangle$ decreases from the upper to the lower margins of the surface layer much more slowly for the SDSL than for the SFSL. We note (Tennekes & Lumley Reference Tennekes and Lumley1972) that the $k_{x}=0$ limit of $\pi E_w (k_{x},z)$ is the integral of the non-normalised two-point correlation of $w'$ , that is, $C_{ww,x}(z)\equiv l_{ww,x} (z) \langle w'^{2} \rangle$ . The model above for $E_w (k_{x}=0,z)$ and $\langle w'^{2} \rangle$ therefore also implies that $l_{ww,x} (z)$ scales linearly with z for both the SFSL and the SDSL identically, as observed. Additionally, for the SFSL this model leads to ${d \langle w'^{2} \rangle / {\rm d} z}\sim \epsilon ^{2/3}z^{-1/3}$ where $\epsilon ^{2/3}$ estimated as $\langle u'^{2} \rangle /l_{uu,x}^{2/3}$ from table 1 is approximately 2.3 times larger for the large mesh grid than for the small mesh grid. This difference is consistent with the observation in figure 13 that $d \langle w'^{2}\rangle / {\rm d} z$ takes 2–2.5 times higher values, on average, in the SFSL with the large mesh grid than with the small mesh grid.

Figure 13. Wall-normal variations of the terms in (4.3) for (a) the large grid and (b) the small grid, plotted against $\hat {z}$ to centre on the SFSL. The dotted black line is the sum of the calculated terms on the right-hand side of (4.3). The horizontal lines indicate $\delta _U$ , $\delta _L$ , $\delta _{SML}$ and $\delta _{SAS}$ as labelled in the figure.

4.3. The interplay between vertical and horizontal velocity variances in the surface layer

The theory at the end of the previous section makes clear that linear growth in $l_{ww,x} = C_{ww,x}/\langle w'^2 \rangle$ implies coordinated variation between the vertical gradient ${\rm d}C_{ww,x}/{\rm d}z$ of the integral of the non-normalised two-point correlation function, $C_{ww,x}$ , and the vertical gradient $d\langle w'^2 \rangle /{\rm d}z$ of vertical velocity variance, $\langle w'^2 \rangle$ . As discussed earlier in § 4.2, our results in figure 11, the results of TH77 and the theory of HG78 all demonstrate the existence of a trade-off between a reduction in vertical velocity variance towards the surface that results from blockage of vertical velocity at the surface, and an increase in horizontal velocity variance towards the lower margin of the SFSL and into the SAS. Figures 11 vs. 12 show that the variation of vertical velocity variance with $z$ and the trade-off between vertical and horizontal velocity variances are different in the TBL and grid-turbulence flows. We speculate that this trade-off may be associated with continuity. Therefore, to interpret the blockage-induced vertical derivative of vertical velocity variance in relationship to horizontal velocity variance, and to compare the contributions with the gradient in vertical variance between the shear-free and shear-dominated surface layers, consider the incompressibility constraint

(4.1) \begin{equation} \frac {\partial w'}{\partial z}=-\left (\frac {\partial u'}{\partial x}+\frac {\partial v'}{\partial y}\right )=-\boldsymbol {\nabla }_h\cdot \boldsymbol {u}^{\prime}_h, \end{equation}

where $\boldsymbol {u}^{\prime}_h$ is wall-parallel fluctuating velocity vector and $\nabla _h$ is the wall-parallel gradient operator. This leads to

(4.2) \begin{equation} \frac {d\left \langle w'^{2}\right \rangle }{d z}=\left \langle \frac {\partial w'^{2}}{\partial z}\right \rangle =\left \langle 2w'\frac {\partial w'}{\partial z}\right \rangle =-2\left (\left \langle w'\frac {\partial u'}{\partial x}\right \rangle +\left \langle w'\frac {\partial v'}{\partial y}\right \rangle \right ), \end{equation}

which can be written equivalently as

(4.3) \begin{equation} \frac {d\langle w'^2\rangle }{d z}=\ 2\left (\left \langle u'\frac {\partial w'}{\partial x}\right \rangle +\left \langle v'\frac {\partial w'}{\partial y}\right \rangle \right )-2\ \left (\frac {\partial \left \langle u' w'\right \rangle }{\partial x}+\frac {\partial \left \langle v' w'\right \rangle }{\partial y}\right ). \end{equation}

If the turbulence is quasi-homogeneous in the horizontal, (4.3) is well approximated by

(4.4) \begin{equation} \frac {d\langle w'^2\rangle }{d z}\approx \ 2\left (\left \langle u'\frac {\partial w'}{\partial x}\right \rangle +\left \langle v'\frac {\partial w'}{\partial y}\right \rangle \right ) = 2\langle {\boldsymbol {u}^\prime}_h\cdot \nabla _hw'\rangle , \end{equation}

which implies that the vertical gradient of vertical velocity variance is largely driven by the horizontal advection of vertical velocity fluctuations. Therefore, the rate of decrease in vertical velocity variance that results from the blockage of vertical velocity fluctuations at an impermeable surface is expected to be accompanied by an overall increase in horizontal velocity fluctuations due to incompressibility. It is largely for this reason, we argue, that the more rapid decrease in vertical velocity variance toward the surface in the SFSL is associated with increasing horizontal velocity variance as observed in figure 11 and as predicted by the HG78 theory.

Here, we compare experimentally the individual contributions to $\partial \langle w'^2\rangle /\partial z$ in (4.3) between the SFSL and SDSL. To calculate the local derivatives in (4.3), we apply central finite differences to the PIV velocity vectors. Due to the vector resolution of the PIV being larger than the smallest scales of turbulence motion, the derivative calculations are likely underestimates of the true values (Adrian & Westerweel Reference Adrian and Westerweel2011).

In figure 13 the terms on the right-hand side of (4.3) are plotted against distance from the surface ( $z$ ) with data from the grid experiments. The sums of the calculated terms on the right-hand side of (4.3) are plotted with the dotted black lines in figure 13. These agree well with the derivatives of the vertical velocity variances plotted with solid black lines. Outside the SML, all terms fluctuate near zero. As discussed in the previous section, the vertical derivative of vertical velocity variance is shown in figure 13 to be significantly larger in the SFSL region than within the surface-modified region above and it grows from the upper to the lower margin of the SFSL, below which it decreases rapidly. We also note that the dominant terms in (4.3) are, as anticipated, those given by (4.4) since the Reynolds stress divergence terms (the green and cyan curves in figure 13) oscillate around zero. The deviation from zero is most dramatic in the SFSL regions. In the SFSL, the advection terms in the streamwise direction (the red curves) and cross-stream direction (the blue curves) are roughly the same with the small grid as anticipated, but appear to differ in the lower SFSL with the large grid. We anticipate that the differences are likely due to higher error in the local derivatives in the cross-stream direction.

Figure 14. Wall-normal variations of the terms in (4.3) for the TBL plotted against $\hat {z}$ to centre on the SDSL. The dotted black line is the sum of the calculated terms on the right-hand side of (4.3). The horizontal lines indicate $\delta _U$ , $\delta _L$ and $\delta _{SAS}$ as labelled in the figure.

Our analysis indicates that the correlation between a more rapid reduction of $\langle w'^2\rangle$ and a rapid increase in $\langle u'^2\rangle$ towards the surface in the SFSL, observed experimentally and within the HG78 theory, is consistent with (4.4) – in the sense that (4.4) suggests that an increased level of horizontal advection of $w'$ is required to reduce vertical velocity variance at the necessary rate in the SFSL. Within this relationship is embedded the solenoidal requirement for incompressibility that relates vertical derivatives of $w'$ to horizontal derivatives of $\boldsymbol {u_h}^{\prime}$ . Dynamical considerations suggest a potential role for pressure–strain-rate correlations in the exchange of component energy between horizontal and vertical variances near the surface (e.g. McCorquodale & Munro Reference McCorquodale and Munro2017).

To compare with figure 13 for the SFSL, we present in figure 14 the variations in the terms in (4.3) in the SDSL of the TBL. As previously observed, the sum of the right-hand side terms in the dotted black line agrees well with the derivative of the vertical velocity variance in the solid black line. Like the SFSL, $|\partial \langle w'^2\rangle /\partial z|$ increases from the upper to lower margins of the SDSL, albeit more slowly in the upper half of the SL and more rapidly in the lower half (in agreement with figure 12). Furthermore, because the stress divergence terms are close to zero, in both the shear-free and shear-dominated surface layers, the vertical gradient in $\langle w'^2\rangle$ corresponds to the horizontal advection of vertical velocity fluctuations (4.4).

However, an interesting difference is observed when comparing figure 14 for the SDSL with figure 13 for the SFSL. Whereas in the SFSL (figure 13) the streamwise and transverse advective derivative terms, the red and blue curves, respectively, are of the same sign, in the SDSL (figure 14) the streamwise component is positive while the transverse component is negative. The sum of the two components increases in magnitude from the upper to lower surface layer consistent with (4.4) and the $z$ variation in $d\langle w'^2\rangle /d z$ . The implication is that, whereas in the SFSL vertical velocity fluctuations are advected in the same direction in $x$ and $y$ , in the SDSL advection in $x$ and $y$ are in opposite directions and the dominance of advection in $x$ over that in $y$ is responsible for the magnitude of the vertical gradient of $\langle w'^2\rangle$ on the right-hand side of (4.4). Whereas the shear-free grid-turbulence flow and the shear-dominated TBL flow are fundamentally different in structure with corresponding differences in the balance between the streamwise and transverse directions of horizontal advection of vertical velocity fluctuations in (4.4), a surface layer is nevertheless generated in each.