3.1 Approaches based on nozzle ICs

As noted by Reference Narasimha, Narayan and ParthasarathyNYP, early approaches have relied upon $U_{j}$ and $b$ as the relevant scaling parameters, i.e. relations of the form $U_{max}/U_{j}=F(x/b)$, where $F$ is a universal function, should hold. However, after compiling a large amount of experimental data on wall jets in still air, Reference Narasimha, Narayan and ParthasarathyNYP have found that the data do not scale simply on $U_{j}$ and $b$. Based on the dynamical importance of the nozzle kinematic momentum rate $M_{j}$, Reference Narasimha, Narayan and ParthasarathyNYP have proposed the relevant scaling parameters to be $M_{j}$ (instead of $U_{j}$ and $b$ separately) and $\unicode[STIX]{x1D708}$. Reference Narasimha, Narayan and ParthasarathyNYP have showed that this $M_{j}$–$\unicode[STIX]{x1D708}$ scaling leads to better collapse of the data with relationships of the form $U_{max}\unicode[STIX]{x1D708}/M_{j}=f_{1}(xM_{j}/\unicode[STIX]{x1D708}^{2})$, $z_{T}M_{j}/\unicode[STIX]{x1D708}^{2}=f_{2}(xM_{j}/\unicode[STIX]{x1D708}^{2})$ etc. where $f_{1}$ and $f_{2}$ are supposed to be universal functions. Several further studies (Wygnanski et al. Reference Wygnanski, Katz and Horev1992; George et al. Reference George, Abrahamsson, Eriksson, Karlsson, Löfdahl and Wosnik2000) have used the Reference Narasimha, Narayan and ParthasarathyNYP scaling to present their data and found it to be reasonably robust.

More recently, Barenblatt et al. (Reference Barenblatt, Chorin and Prostokishin2005) – henceforth Reference Barenblatt, Chorin and ProstokishinBCP – have argued that the strong dependence on slot height $b$ remains important and this leads to the so-called incomplete similarity in the dimensionless independent variables $\unicode[STIX]{x1D6F1}_{1}=z/b$, $\unicode[STIX]{x1D6F1}_{2}=x/b$ and $\unicode[STIX]{x1D6F1}_{3}=\sqrt{M_{j}b/\unicode[STIX]{x1D708}^{2}}$. Note that, in the Reference Barenblatt, Chorin and ProstokishinBCP framework, the governing (independent) parameters are taken to be $x$, $z$, $b$, $M_{j}$ and $\unicode[STIX]{x1D708}$, and $b$ and $\unicode[STIX]{x1D708}$ are used to make $x$, $z$ and $M_{j}$ dimensionless in accordance with Buckingham’s Pi theorem. Since $\unicode[STIX]{x1D6F1}_{3}$ is a measure of the nozzle Reynolds number $Re_{j}$, streamwise variations of all the length and velocity scales may be expressed as $U_{max}b/\unicode[STIX]{x1D708}=f_{3}(x/b,Re_{j})$, $z_{T}/b=f_{4}(x/b,Re_{j})$ etc. For complete similarity according to Reference Barenblatt, Chorin and ProstokishinBCP, this implies that the data from various experiments should collapse to universal curves in plots of $U_{max}b/\unicode[STIX]{x1D708}$ versus $x/b$, $z_{T}/b$ versus $x/b$ etc. as $Re_{j}\rightarrow \infty$. However, the results of Reference Narasimha, Narayan and ParthasarathyNYP indicate that the $z_{T}/b$ versus $x/b$ plot does not show such collapse. This observation could be taken to support the incomplete similarity hypothesis of Reference Barenblatt, Chorin and ProstokishinBCP.

As a side line, one may ask if there is any connection at all between these seemingly different approaches of Reference Narasimha, Narayan and ParthasarathyNYP and Reference Barenblatt, Chorin and ProstokishinBCP. To see this, consider choosing $M_{j}$ and $\unicode[STIX]{x1D708}$ (instead of $b$ and $\unicode[STIX]{x1D708}$ but still remaining within the Reference Barenblatt, Chorin and ProstokishinBCP framework) to form the dimensionless groups. One then obtains $\unicode[STIX]{x1D6F1}_{1}^{\ast }=zM_{j}/\unicode[STIX]{x1D708}^{2}$, $\unicode[STIX]{x1D6F1}_{2}^{\ast }=xM_{j}/\unicode[STIX]{x1D708}^{2}$ and $\unicode[STIX]{x1D6F1}_{3}^{\ast }=\unicode[STIX]{x1D6F1}_{3}=\sqrt{M_{j}b/\unicode[STIX]{x1D708}^{2}}$. This choice leads to the scaling relations $U_{max}\unicode[STIX]{x1D708}/M_{j}=f_{3}^{\ast }(xM_{j}/\unicode[STIX]{x1D708}^{2},Re_{j})$, $z_{T}M_{j}/\unicode[STIX]{x1D708}^{2}=f_{4}^{\ast }(xM_{j}/\unicode[STIX]{x1D708}^{2},Re_{j})$ etc. Interestingly, Reference Narasimha, Narayan and ParthasarathyNYP have already shown that the data (accessible to them) collapse fairly well to near-universal curves, i.e. no $Re_{j}$-dependence, in plots of $U_{max}\unicode[STIX]{x1D708}/M_{j}$ versus $xM_{j}/\unicode[STIX]{x1D708}^{2}$, $z_{T}M_{j}/\unicode[STIX]{x1D708}^{2}$ versus $xM_{j}/\unicode[STIX]{x1D708}^{2}$, etc. This suggests complete similarity! In other words, simple change of repeating variables in the Reference Barenblatt, Chorin and ProstokishinBCP approach leads to the approach of Reference Narasimha, Narayan and ParthasarathyNYP which in turn supports complete similarity, i.e. the direct effect of slot height $b$, if any, appears to be negligibly small.

To summarize, the approach of Reference Narasimha, Narayan and ParthasarathyNYP appears to be the most appropriate amongst those that use nozzle ICs as scaling parameters.

3.2 Approach based on FCs

Recently, Gersten (Reference Gersten2015) has pointed out the need for a distinction between different wall jet configurations based on the geometry of the experimental set-up, i.e. whether (or not) the nozzle is located in a large wall perpendicular to the test surface. This distinction is motivated by the theoretical results for turbulent plane free jets by Schneider (Reference Schneider1985), wherein it is shown that the kinematic momentum flux $M$ at large distances ($x\rightarrow \infty$) from the nozzle – (i) asymptotically vanishes ($M\rightarrow 0$) if the nozzle is located in a large plane wall perpendicular to the jet exit velocity vector and (ii) remains constant ($M=M_{j}$) for nozzles without any such wall. Extending this idea to wall jets, Gersten (Reference Gersten2015) has hypothesized that the asymptotic state far downstream (as Reynolds number tends to infinity) would be a half-free jet with finite momentum rate $M_{\infty }$ (we denote this by FC; see § 1) if the nozzle in the wall jet set-up is not located in a wall perpendicular to the test surface. With this, the wall jet flow could be considered as a perturbation of the limiting half-free jet state and therefore Gersten (Reference Gersten2015) has proposed $M_{\infty }$ as the scaling parameter for the streamwise coordinate $x$. This leads to the definition of a local Reynolds number $Re_{x}=\sqrt{(x-x_{0})M_{\infty }}/\unicode[STIX]{x1D708}$ ($x_{0}$ is the adjustment for the virtual origin effect) and all the other dimensionless parameters are then expected to be universal functions of $Re_{x}$. Consistent with the current presentation, the expected universal relationships involving $M_{\infty }$ and $\unicode[STIX]{x1D708}$ may be written as $U_{max}\unicode[STIX]{x1D708}/M_{\infty }=f_{5}(xM_{\infty }/\unicode[STIX]{x1D708}^{2})$, $z_{T}M_{\infty }/\unicode[STIX]{x1D708}^{2}=f_{6}(xM_{\infty }/\unicode[STIX]{x1D708}^{2})$, etc.

3.3 The present approach based on LCs

As outlined in § 1, if indeed wall jet flow evolves downstream in a self-similar fashion, then the local kinematic momentum rate $M=\int _{0}^{\infty }U^{2}\,\text{d}z$ and $\unicode[STIX]{x1D708}$ (i.e. LCs) could very well be the candidate scaling parameters. Some support for this proposal may be derived from the work of Reference Narasimha, Narayan and ParthasarathyNYP who have suggested ‘A similar correlation using local momentum flux $M$ is implied by (3) $\ldots$ but clearly $M$ is less convenient than $M_{j}$’. In case of self-similar development, data should show better collapse with LCs than with ICs or FCs. Therefore, according to our present proposal of ‘local’ scaling, we expect universal relationships of the form $U_{max}\unicode[STIX]{x1D708}/M=f_{7}(xM/\unicode[STIX]{x1D708}^{2})$, $z_{T}M/\unicode[STIX]{x1D708}^{2}=f_{8}(xM/\unicode[STIX]{x1D708}^{2})$, etc.

3.4 Wall jet data sets from the literature

In order to evaluate the scaling approaches mentioned above, we have selected, apart from our own experimental data (§ 2), five well-documented studies from the literature. These are briefly described below with necessary parameters given in table 1.

The main constraints on the selection of the data sets are the availability of measured mean velocity profiles and an accurate estimate of the wall shear stress $\unicode[STIX]{x1D70F}_{w}$. The former is necessary to evaluate the local momentum rate $M$ and the latter is crucial for testing the scaling of skin friction. Accurate measurement of $\unicode[STIX]{x1D70F}_{w}$ poses serious challenges in wall jets (Launder & Rodi Reference Launder and Rodi1979). Computation of $\unicode[STIX]{x1D70F}_{w}$ from the mean velocity gradient in the linear viscous sublayer is accurate only with non-heat transfer techniques such as laser Doppler velocimetry (LDV). Hot-wire data are known to yield spurious large values of velocity in the near-wall region due to strong conduction heat transfer to the wall (Durst, Zanoun & Pashtrapanska Reference Durst, Zanoun and Pashtrapanska2001), the so-called wall proximity effect. This could result in reduction in the estimated values of the velocity gradient at the wall, leading to lower-than-actual values of $\unicode[STIX]{x1D70F}_{w}$ (and hence $U_{\unicode[STIX]{x1D70F}}$). Use of impact tube devices such as the Stanton tube relies on the universality of the mean velocity profile over the wall-normal extent of the device. Usually such devices are calibrated in a canonical flow such as a fully developed channel or a zero-pressure-gradient TBL and then used in wall jets to measure skin friction (Bradshaw & Gee Reference Bradshaw and Gee1962). However, it has been reported that the linear velocity profile in the sublayer of wall jets extends only up to $z_{+}\approx 3$ in contrast to the celebrated $z_{+}\approx 5$ in canonical TBLs, pipes and channels (Eriksson et al. Reference Eriksson, Karlsson and Persson1998). As we shall see later, such a reduction in the thickness of the linear sublayer is consistent with the strong influence of the outer jet flow on the inner wall flow. In view of these difficulties, one may consider applying some judiciously estimated corrections to $\unicode[STIX]{x1D70F}_{w}$ values measured in wall jets using the hot-wire mean velocity gradient and impact tubes. These corrections are discussed further in the relevant places. In our experiments described in § 2, we have used OFI that does not suffer from these drawbacks and allows direct, reliable measurement of $\unicode[STIX]{x1D70F}_{w}$.

The first data set is from Reference Eriksson, Karlsson and PerssonEKP and the data measured for $Re_{j}=9600$ at four different streamwise locations are selected for the present purpose. These are available in the ERCOFTAC Classic Collection Database. The working fluid is water and the nozzle is located in a vertical wall perpendicular to the test surface. Mean velocity has been measured using high-resolution LDV and $\unicode[STIX]{x1D70F}_{w}$ has been computed from the velocity gradient in the linear viscous sublayer. There is reverse flow above the wall jet in this set-up due to the vertical wall above the nozzle exit extending all the way to the free surface of water.

The second set of data come from Schwarz & Cosart (Reference Schwarz and Cosart1961) – henceforth Reference Schwarz and CosartSC – and have been measured at different nozzle Reynolds numbers of which the $Re_{j}=42\,839$ data have been digitized from figure 2 given in Reference Schwarz and CosartSC. The working fluid is air and nozzle is not located in a wall perpendicular to the test surface. Mean velocity has been measured using hot-wire anemometry and $\unicode[STIX]{x1D70F}_{w}$ has not been measured in these experiments.

The third data set is from the extensive measurements by Bradshaw & Gee (Reference Bradshaw and Gee1962) – hereafter Reference Bradshaw and GeeBG. These are also air-based experiments with $Re_{j}=6113$ (see table 1) and four representative hot-wire mean velocity profiles have been digitized from the figures 2, 3 and 6 in Reference Bradshaw and GeeBG. The nozzle is not located in a wall perpendicular to the test surface and $\unicode[STIX]{x1D70F}_{w}$ has been measured using a Stanton tube (Bradshaw & Gregory Reference Bradshaw and Gregory1961; Bradshaw & Gee Reference Bradshaw and Gee1962) made typically from a 0.002 inch ($z_{+}\approx 3.5$ to 6 for Reference Bradshaw and GeeBG data) thick steel blade; the Stanton tube has been calibrated in a channel flow and used to measure $\unicode[STIX]{x1D70F}_{w}$ in the wall jets. However, as mentioned before, the linear sublayer extends only up to $z_{+}\approx 3$ in wall jets (see figure 10 of Reference Eriksson, Karlsson and PerssonEKP) as against $z_{+}\approx 5$ in channel flows. Therefore, the Stanton tube $U_{\unicode[STIX]{x1D70F}}$ values in the Reference Bradshaw and GeeBG data sets require corrections that may be estimated using the Reference Eriksson, Karlsson and PerssonEKP near-wall profile as follows. Figure 10 of Reference Eriksson, Karlsson and PerssonEKP shows that the near-wall profile in wall jets is described well by $U_{+}=z_{+}+C_{4}z_{+}^{4}$, where $C_{4}=-0.0004$ (Eriksson et al. Reference Eriksson, Karlsson and Persson1998) represents the maximum departure of the profile from linear variation. Now, a Stanton tube is expected to respond to the dynamic pressure exerted by the average flow velocity in the gap between the test surface and the chamfered sharp leading edge of the steel blade. Therefore, one may compute the inner-scaled average velocity $U_{avg+}$ in a wall jet over the thickness of the blade using the above-mentioned wall jet inner layer profile. Similarly, $U_{avg+}$ may be computed using the linear velocity profile, which is typical of the channel flow calibration of a Stanton tube. For the same value of $U_{avg}$, one may then compute the ratio of these two inner-scaled average velocities to obtain the ratio of actual $U_{\unicode[STIX]{x1D70F}}$ value in the wall jet to that read by the Stanton tube as per its channel flow calibration. For the four flows of Reference Bradshaw and GeeBG (BG1-1 to BG1-4), these ratios respectively turn out to be 1.0172, 1.0102, 1.007 and 1.005. Therefore, the $U_{\unicode[STIX]{x1D70F}}$ values for Reference Bradshaw and GeeBG data listed in table 1 have been corrected by multiplying the original $U_{\unicode[STIX]{x1D70F}}$ values by these enhancement factors.

The fourth data set belongs to the well-cited experiments of Tailland & Mathieu (Reference Tailland and Mathieu1967) – henceforth Reference Tailland and MathieuTM. These are air-based experiments with three different nozzle Reynolds numbers ($Re_{j}=11\,000$, 18 000 and 25 000) as seen in table 1. Four representative hot-wire mean velocity profiles have been digitized from figure 2 in Reference Tailland and MathieuTM. The nozzle is not located in a wall perpendicular to the test surface and $\unicode[STIX]{x1D70F}_{w}$ has been measured using the velocity gradient in the near-wall region. It is known that the $\unicode[STIX]{x1D70F}_{w}$ values in Reference Tailland and MathieuTM are 20 % to 35 % lower than the consensus impact tube data (Launder & Rodi Reference Launder and Rodi1979), which is most likely due to the wall proximity effect of hot-wires described above. In view of this, we have corrected the skin friction values in the Reference Tailland and MathieuTM data sets using an enhancement of 10 % in $U_{\unicode[STIX]{x1D70F}}$ (i.e. 20 % in $\unicode[STIX]{x1D70F}_{w}$).

Finally, the fifth data set is taken from the recent DNS of Naqavi et al. (Reference Naqavi, Tyacke and Tucker2018) – henceforth Reference Naqavi, Tyacke and TuckerNTT – having $Re_{j}=7500$. The nozzle is not located in a wall perpendicular to the test surface because a co-flow velocity (6 % of $U_{j}$) has been used as the inlet boundary condition above the nozzle as well as the outer boundary condition. The flow used in the present analysis corresponds to $x/b=35$, which is the farthest downstream station from the nozzle exit free from the domain end effects. Since the wall jet flow attains fully developed state beyond $x/b=30$ as mentioned before, we have not used the data for $x/b<35$ from this DNS study. In addition to the mean velocity profile data at $x/b=35$, we have also used the Reynolds shear stress profile to investigate the layered structure of wall jet flow based on the mean momentum balance (§ 4.4).

There have been some recent water-based wall jet experiments (Tachie et al. Reference Tachie, Balachandar and Bergström2002; Rostamy et al. Reference Rostamy, Bergstrom, Sumner and Bugg2011; Tang et al. Reference Tang, Rostamy, Bergstrom, Bugg and Sumner2015) that have used a facility similar to the Reference Eriksson, Karlsson and PerssonEKP facility and covered a similar range of Reynolds numbers. However, we have noticed a certain peculiar behaviour of this facility. Upon coming out of the nozzle, the kinematic momentum rate $M$ of the wall jet has been initially found to increase downstream up to $x/b\approx 50$ and to reach a maximum value of $M/M_{j}\approx 1.25$ (see Tachie Reference Tachie2001). Beyond this point, the momentum rate starts reducing in the downstream direction. Although there has been some explanation provided for this anomalous increase of momentum rate in this facility (Tachie Reference Tachie2001), no other wall jet studies have reported such a behaviour. Our contention is that this mean flow acceleration could be related to the secondary flow set up in the tank above the main wall jet flow. Since our scaling approach crucially hinges on the correctness of the local kinematic momentum rate $M$, these data sets had to be excluded from our analysis.

3.5 Scaling results

Figure 3(a) shows the plot of $U_{max}$ as a function of $x$ for the data of table 1; note that $x$ is measured from the nozzle exit in each case and no adjustment for the virtual origin has been made. Three scaling approaches using $M_{j}$, $M$ and $M_{\infty }$ (ICs, LCs and FCs of §§ 3.1, 3.3 and 3.2, respectively) are shown. Note that $M_{\infty }$ has been computed using the procedure given in §8 of Gersten (Reference Gersten2015) which is further elaborated here in appendix B. Also note that $xM_{\infty }/\unicode[STIX]{x1D708}^{2}$ is equivalent to the $Re_{x}^{2}$ recommended by Gersten (Reference Gersten2015). Solid line in each case, shows the least-squares power-law curve fit to our experimental data (WJ1, WJ2 and WJ3 in table 1) and the shading shows the $\pm 5\,\%$ band around the curve fit (and also for subsequent curve fits) to enable visual aid for assessing data collapse. To quantify the quality (goodness) of scaling with each approach, the normalized root-mean-squared error (RMSE) of all the data points is computed with respect to the corresponding curve fit. This RMSE value is mentioned alongside each curve fit; the lower the RMSE value the better is the scaling quality. It is clear that the a much better data collapse is seen with our local scaling approach ($M$–$\unicode[STIX]{x1D708}$), i.e. with the LCs. Scaling using ICs ($M_{j}$–$\unicode[STIX]{x1D708}$) clearly shows larger data scatter. Scaling with FCs ($M_{\infty }$–$\unicode[STIX]{x1D708}$) also shows some data scatter but, more importantly, this approach has limitations since it does not apply to all flows due to the nozzle being located in a perpendicular wall (Gersten Reference Gersten2015) in some cases, such as the Reference Eriksson, Karlsson and PerssonEKP data; these data are, however, included in the plot for completeness. Our approach (using LCs) does not suffer from this limitation and is applicable to all flows. Figures 3(b) and 3(c) respectively show plots of $z_{T}$ and $U_{\unicode[STIX]{x1D70F}}$ as functions of $x$. While $M_{j}$, $M$ and $M_{\infty }$ all give almost similar data collapse for $z_{T}$ (almost identical RMSE values in figure 3b), $U_{\unicode[STIX]{x1D70F}}$ clearly favours the $M$–$\unicode[STIX]{x1D708}$ scaling (figure 3c). Also, contrary to the expectation of Reference Narasimha, Narayan and ParthasarathyNYP, our data indicate that $M/M_{j}$ is not a universal function of $xM_{j}/\unicode[STIX]{x1D708}^{2}$ (not shown). Further, $M_{\infty }/M_{j}$ is also not a universal constant (Gersten Reference Gersten2015). Therefore $M$ cannot be expressed in terms of either $M_{j}$ or $M_{\infty }$, and hence may be considered as an independent scaling parameter. Figure 4 shows the data of figure 3 cross-plotted by eliminating the explicit dependence on the streamwise distance $x$ (similar to George et al. Reference George, Abrahamsson, Eriksson, Karlsson, Löfdahl and Wosnik2000). These plots also support the broad conclusion that the LCs ($M$–$\unicode[STIX]{x1D708}$) lead to better data collapse than the ICs or FCs.

Figure 3. Streamwise variations of (a) $U_{max}$, (b) $z_{T}$ and (c) $U_{\unicode[STIX]{x1D70F}}$ in the $M_{j}$–$\unicode[STIX]{x1D708}$ scaling of Reference Narasimha, Narayan and ParthasarathyNYP, the presently proposed ‘local’ $M$–$\unicode[STIX]{x1D708}$ scaling and the $M_{\infty }$–$\unicode[STIX]{x1D708}$ scaling of Gersten (Reference Gersten2015). Data come from the experiments and simulations listed in table 1. Each solid line shows the least-squares power-law fit to our data (WJ1, WJ2 and WJ3 in table 1). To avoid clutter in each plot, data points and the fitted curve in categories $M_{i}=M$ and $M_{i}=M_{\infty }$ have been shifted upward using suitable arbitrary multiplying factors. Shading shows the $\pm 5\,\%$ band around each curve fit. For each scaling alternative, the RMSE of all data points with respect to the curve fit is also shown.

Figure 4. Cross-plots of (a) $U_{\unicode[STIX]{x1D70F}}$ and $z_{T}$, (b) $U_{max}$ and $z_{T}$, and (c) $U_{max}$ and $U_{\unicode[STIX]{x1D70F}}$ for the data of figure 3. Each solid line shows the least-squares power-law fit to our data (WJ1, WJ2 and WJ3 in table 1). To avoid clutter in each plot, data points and fitted curve in categories $M_{i}=M$ and $M_{i}=M_{\infty }$ have been shifted upward using suitable arbitrary multiplying factors. Shading shows the $\pm 5\,\%$ band around each curve fit. For each scaling alternative, the RMSE of all data points with respect to the curve fit is also shown.

In summary, the streamwise variations of $U_{max}$, $z_{T}$ and $U_{\unicode[STIX]{x1D70F}}$ exhibit robust scaling behaviour in the local $M$–$\unicode[STIX]{x1D708}$ framework irrespective of the facility, working fluid and whether the nozzle is located in a wall perpendicular to the test surface or not. Thus, the basic condition for self-similarity, i.e. lack of any imposed length or velocity scales, is met.

4.1 Self-similar inner and outer regions

We first plot mean velocity profiles in classical outer or jet ($U/U_{max}$ versus $\unicode[STIX]{x1D702}$, figure 5a) and inner or wall ($U_{+}$ versus $z_{+}$, figure 5b) coordinates to look for any meaningful scaling; $\unicode[STIX]{x1D702}=z/z_{T}$, $U_{+}=U/U_{\unicode[STIX]{x1D70F}}$ and $z_{+}=zU_{\unicode[STIX]{x1D70F}}/\unicode[STIX]{x1D708}$. For the entire range of Reynolds numbers covered here, figure 5(a) shows that all the data exhibit excellent scaling (self-similarity) in the outer region, including the velocity maximum and a small extent below it. The observation that the outer (jet) scaling holds even below the velocity maximum, suggests that the outer flow is in fact akin to a full-free jet rather than a half-free jet as contended by many other studies; this will be discussed later in § 4.4 in some detail. Note that the outer-scaled profiles show some scatter in the region close to the edge of the wall jet (see figure 7 of George et al. (Reference George, Abrahamsson, Eriksson, Karlsson, Löfdahl and Wosnik2000), also). This could be due to the differences in the outer boundary condition such as reverse flow in the Reference Eriksson, Karlsson and PerssonEKP set-up or co-flow in the Reference Naqavi, Tyacke and TuckerNTT data (see § 3.4) as well as larger uncertainties in hot-wire measurements due to low velocities near the edge (Deo, Mi & Nathan Reference Deo, Mi and Nathan2008). Figure 5(b) shows the same data in the inner (wall) coordinates. Reference Bradshaw and GeeBG and Reference Tailland and MathieuTM data clearly show the wall proximity effect affecting the first few near-wall points in the hot-wire profiles; the effect is very severe in the Reference Tailland and MathieuTM profiles, leading to spuriously high values of $U_{+}$. Reference Schwarz and CosartSC data are not plotted because $U_{\unicode[STIX]{x1D70F}}$ has not been measured in their experiments. Notwithstanding these difficulties, remarkably robust scaling is evident for all the other data in the near-wall region (figure 5b). Thus, the mean velocity data support classical inner and outer scalings (self-similarity) respectively of the form

(4.1)$$\begin{eqnarray}\displaystyle & \displaystyle U_{+}=f(z_{+}), & \displaystyle\end{eqnarray}$$

(4.2)$$\begin{eqnarray}\displaystyle & \displaystyle U/U_{max}=g(\unicode[STIX]{x1D702}), & \displaystyle\end{eqnarray}$$

where $f$ and $g$ are universal functions independent of $Re_{\unicode[STIX]{x1D70F}}$ (at least to the lowest order).

4.2 Overlap layer analysis

Figures 5(a) and 5(b) indicate that the overlap of outer and inner scalings in wall jets is expected to occur below the velocity maximum. Matching $U$ and $\unicode[STIX]{x2202}U/\unicode[STIX]{x2202}z$ from the inner (4.1) and outer (4.2) descriptions in the overlap layer leads to

(4.3)$$\begin{eqnarray}\frac{\text{d}\ln f}{\text{d}\ln z_{+}}=\frac{\text{d}\ln g}{\text{d}\ln \unicode[STIX]{x1D702}}.\end{eqnarray}$$

Note that (4.3) is only superficially similar to (8.3) from George et al. (Reference George, Abrahamsson, Eriksson, Karlsson, Löfdahl and Wosnik2000); the main difference is that, within the framework of George et al., $f$ and $g$ essentially depend on $Re_{\unicode[STIX]{x1D70F}}$ whereas in (4.3) they are independent of $Re_{\unicode[STIX]{x1D70F}}$ (due to (4.1) and (4.2)). Next, it is known that the solution to (4.3) is invariant under the transformation $z\rightarrow z+\unicode[STIX]{x1D6FC}$, where $\unicode[STIX]{x1D6FC}$ is an arbitrary shift in the origin for $z$ (see George et al. Reference George, Abrahamsson, Eriksson, Karlsson, Löfdahl and Wosnik2000). Alternatively, we note that (4.3) also permits the transformation $U\rightarrow U+\unicode[STIX]{x1D6FD}$, where $\unicode[STIX]{x1D6FD}$ is an arbitrary shift in the origin for $U$. Physically, either of these shifts accounts for a finite slip velocity at the wall since the overlap layer profile cannot satisfy the inner (no-slip) boundary condition due to matching of the inner layer with the outer layer. In what follows, we replace $U$ by $\widetilde{U}=U+\unicode[STIX]{x1D6FD}$, so that $\widetilde{f}=\widetilde{U}/U_{\unicode[STIX]{x1D70F}}=f+\unicode[STIX]{x1D6FD}_{+}$ and $\widetilde{g}=\widetilde{U}/U_{max}=g+\left(\unicode[STIX]{x1D6FD}/U_{max}\right)$. Since the left and right sides of (4.3) are purely functions of $z_{+}$ and $\unicode[STIX]{x1D702}$, respectively, each side must be equal to a universal constant, say $A$. Integrating $\text{d}\ln \widetilde{f}/\text{d}\ln z_{+}=A$ leads to $\ln \widetilde{f}=A\ln z_{+}+B^{\prime }$ in the overlap layer. Note that while $B^{\prime }$ must be constant with respect to $z_{+}$ in the overlap layer, mathematically, nothing precludes it from being a function of the local Reynolds number $Re_{\unicode[STIX]{x1D70F}}$. Physically, this is closely related to the mesolayer discussed in detail in § 4.4. Rewriting $B^{\prime }=A\ln B+\ln D$ yields

(4.4)$$\begin{eqnarray}\widetilde{f}=Dz_{+}^{A}B^{A},\end{eqnarray}$$

where $B(Re_{\unicode[STIX]{x1D70F}})$ and $D(Re_{\unicode[STIX]{x1D70F}})$ are unknown functions still to be determined. Similarly, $\text{d}\ln \widetilde{g}/\text{d}\ln \unicode[STIX]{x1D702}=A$ leads to

(4.5)$$\begin{eqnarray}\widetilde{g}=E\unicode[STIX]{x1D702}^{A}C^{A},\end{eqnarray}$$

where $E(Re_{\unicode[STIX]{x1D70F}})$ and $C(Re_{\unicode[STIX]{x1D70F}})$ are functions unknown as yet. Clearly, (4.4) and (4.5) do not scale the mean velocity profiles at different $Re_{\unicode[STIX]{x1D70F}}$ in the overlap layer in classical inner and outer coordinates (figures 5a and 5b). However, if one rearranges (4.4) and (4.5) as

(4.6)$$\begin{eqnarray}\displaystyle & \displaystyle (\widetilde{f}/D)=(z_{+}B)^{A}, & \displaystyle\end{eqnarray}$$

(4.7)$$\begin{eqnarray}\displaystyle & \displaystyle (\widetilde{g}/E)=(\unicode[STIX]{x1D702}C)^{A}, & \displaystyle\end{eqnarray}$$

then $\widetilde{f}/D$ and $\widetilde{g}/E$ could become universal functions of $z_{+}B$ and $\unicode[STIX]{x1D702}C$, respectively. In order to proceed further with this contention, the functional forms of $B,C,D$ and $E$ are required to be determined.

Note that $z_{+}B$ and $\unicode[STIX]{x1D702}C$ are both akin to the so-called intermediate variable that is commonly used in the asymptotic analysis of multiscale problems (Kevorkian & Cole Reference Kevorkian and Cole2013). Physically, the role of function $B$($C$) is to ‘rescale’ $z_{+}(\unicode[STIX]{x1D702})$ appropriately so that the overlap region remains in focus, i.e. $z_{+}B(\unicode[STIX]{x1D702}C)$ remains of $O(1)$ although $z_{+}\rightarrow \infty (\unicode[STIX]{x1D702}\rightarrow 0)$ as $Re_{\unicode[STIX]{x1D70F}}\rightarrow \infty$. Therefore, by the definition of an intermediate variable, $z_{+}B\sim \unicode[STIX]{x1D702}C$. Next, if one assumes power-law functional forms $B\sim Re_{\unicode[STIX]{x1D70F}}^{m}$ and $C\sim Re_{\unicode[STIX]{x1D70F}}^{n}$, then the condition $z_{+}B\sim \unicode[STIX]{x1D702}C$ leads to the constraint $1+m=n$ on the constants $m$ and $n$. In order to estimate the individual values of $m$ and $n$, we need additional information. Towards this, we note that since $Re_{\unicode[STIX]{x1D70F}}$ assumes importance in the mean-velocity overlap layer in wall jets (through functions $B$ and $C$ in (4.6) and (4.7), respectively), this layer could correspond to the mesolayer that is studied extensively in the literature on mean momentum balance in wall-bounded flows (George et al. Reference George, Abrahamsson, Eriksson, Karlsson, Löfdahl and Wosnik2000; Wei et al. Reference Wei, Fife, Klewicki and McMurtry2005); it will be demonstrated later (§ 4.4) from DNS data that this indeed is the case. The momentum-balance mesolayer in canonical wall-bounded flows occurs around the location of maximum Reynolds shear stress where the Reynolds stress gradient term in the mean momentum equation loses its dynamical significance (Afzal Reference Afzal1982; Sreenivasan & Sahay Reference Sreenivasan, Sahay and Panton1997; Wei et al. Reference Wei, Fife, Klewicki and McMurtry2005) and this location scales as $z_{+}\sim \sqrt{Re_{\unicode[STIX]{x1D70F}}}$. As we shall see later, DNS data show the existence (very similar to other wall-bounded flows) of a momentum-balance mesolayer in wall jets as well. Therefore, assuming that the mean-velocity overlap layer in the present analysis coincides with the momentum-balance mesolayer in wall jets, one has $z_{+}\sim \sqrt{Re_{\unicode[STIX]{x1D70F}}}$ in the mean-velocity overlap layer. Since $z_{+}B\sim O(1)$ in the mean-velocity overlap layer, it follows that $B\sim 1/\sqrt{Re_{\unicode[STIX]{x1D70F}}}$, i.e. $m=-1/2$ and, consequently, the constraint $1+m=n$ yields $n=1/2$, i.e. $C\sim \sqrt{Re_{\unicode[STIX]{x1D70F}}}$.

Since $z_{+}B\sim \unicode[STIX]{x1D702}C$ and $A$ is a universal constant, (4.6) and (4.7) imply $(\widetilde{f}/D)\sim (\widetilde{g}/E)$. Physically, functions $D$ and $E$ represent $Re_{\unicode[STIX]{x1D70F}}$-dependent modulation of the inner and outer velocity scales, respectively. For wall jets without an external stream, the jet momentum is expected to overwhelm the drag at the surface as $Re_{\unicode[STIX]{x1D70F}}\rightarrow \infty$ (Gersten Reference Gersten2015) with the asymptotic state being a half-free jet (see § 4.4). Therefore, the effect of the wall in the overlap region on the outer velocity scale ($Re_{\unicode[STIX]{x1D70F}}$ dependence) may be expected to be very weak and enter only in the length scale. Therefore, to the lowest order, it is reasonable to assume $E(Re_{\unicode[STIX]{x1D70F}})\approx \text{const.}$ in the overlap layer. With this, $D(Re_{\unicode[STIX]{x1D70F}})\sim \widetilde{f}/\widetilde{g}\sim U_{max}/U_{\unicode[STIX]{x1D70F}}$ and the velocity profile ((4.6) and (4.7)) in the overlap layer scales according to either of the two equivalent relations

(4.8)$$\begin{eqnarray}\displaystyle & \displaystyle \widetilde{f}/D=K_{i}(z_{+}/\sqrt{Re_{\unicode[STIX]{x1D70F}}})^{A}, & \displaystyle\end{eqnarray}$$

(4.9)$$\begin{eqnarray}\displaystyle & \displaystyle \widetilde{g}=K_{o}(\unicode[STIX]{x1D702}\sqrt{Re_{\unicode[STIX]{x1D70F}}})^{A}, & \displaystyle\end{eqnarray}$$

where $K_{i}$, $K_{o}$ and $A$ are universal constants. We shall now focus on (4.9). Note that $\widetilde{g}=g+(\unicode[STIX]{x1D6FD}/U_{max})$, where $\unicode[STIX]{x1D6FD}$ is constant for a given profile. Therefore $\unicode[STIX]{x1D6FD}/U_{max}$ could be a function of $Re_{\unicode[STIX]{x1D70F}}$. However, the same arguments that led to $E\left(Re_{\unicode[STIX]{x1D70F}}\right)\approx \text{const.}$, indicate that, to the lowest order, one may expect $\unicode[STIX]{x1D6FD}/U_{max}\approx \text{const}$. With this, (4.9) simplifies to

(4.10)$$\begin{eqnarray}\frac{U}{U_{max}}=K_{o}(\unicode[STIX]{x1D702}\sqrt{Re_{\unicode[STIX]{x1D70F}}})^{A}-\frac{\unicode[STIX]{x1D6FD}}{U_{max}},\end{eqnarray}$$

indicating that the mean velocity profiles in the overlap region scale (collapse) in the coordinates $U/U_{max}$ versus $\unicode[STIX]{x1D702}\sqrt{Re_{\unicode[STIX]{x1D70F}}}$.

4.3 Overlap scaling in the experimental data

Figure 6 shows all the data in table 1 (except the Reference Schwarz and CosartSC data) plotted in coordinates $U/U_{max}$ versus $\unicode[STIX]{x1D702}\sqrt{Re_{\unicode[STIX]{x1D70F}}}$ as suggested by (4.10). Indeed, all the data collapse remarkably well in the region where overlap is expected to occur. The collapse extends over the range $0.7\leqslant \unicode[STIX]{x1D702}\sqrt{Re_{\unicode[STIX]{x1D70F}}}\leqslant 3$, i.e. 42 % of a decade in $\unicode[STIX]{x1D702}\sqrt{Re_{\unicode[STIX]{x1D70F}}}$; the extent of this region is decided empirically by observation. Least-squares fit of (4.10) to all of the data of figure 6 over this range, yields the values for the universal constants as $K_{o}=-0.2879$, $A=-0.5346$ and $\unicode[STIX]{x1D6FD}/U_{max}=-1.1145$. More data from future high-Reynolds-number experiments on wall jets would help accurate determination of these universal constants.

Figure 6. Mean velocity profiles for the data sets listed in table 1 (except for the Reference Schwarz and CosartSC data) in the overlap layer scaling. Ordinate is the intermediate variable $\unicode[STIX]{x1D702}\sqrt{Re_{\unicode[STIX]{x1D70F}}}$ or $z_{+}/\sqrt{Re_{\unicode[STIX]{x1D70F}}}$. Solid line is the least-squares curve fit of (4.10) to all data between the dashed line and the dashed-dotted line ($0.7\leqslant \unicode[STIX]{x1D702}\sqrt{Re_{\unicode[STIX]{x1D70F}}}\leqslant 3$). The curve is extended beyond these limits for visual aid. For symbols, refer to the legend of figure 5(a).

Note that the position and extent of the mean-velocity overlap are fixed in $\unicode[STIX]{x1D702}\sqrt{Re_{\unicode[STIX]{x1D70F}}}$ or $z_{+}/\sqrt{Re_{\unicode[STIX]{x1D70F}}}$ and not in $\unicode[STIX]{x1D702}$ or $z_{+}$. In fact, with increasing Reynolds number, the position of the overlap layer is expected to shift towards the wall and its extent is expected to reduce in the $\unicode[STIX]{x1D702}$ coordinate; the opposite is expected to be true for the $z_{+}$ coordinate. We shall come back to this point in the next section.

4.4 Layered structure

Figure 7. Correspondence between the mean-velocity overlap layer and the momentum-balance mesolayer in wall jets. Inner-scaled profiles are shown for the mean velocity and the ratio of viscous stress gradient to the Reynolds shear stress gradient for the flow NTT1-1 (DNS data) of table 1. Dashed and dashed-dotted lines respectively denote the outer and inner limits $\unicode[STIX]{x1D702}\sqrt{Re_{\unicode[STIX]{x1D70F}}}=3$ and $\unicode[STIX]{x1D702}\sqrt{Re_{\unicode[STIX]{x1D70F}}}=0.7$ of the mean-velocity overlap layer for $Re_{\unicode[STIX]{x1D70F}}=865$. These limits have been fixed empirically in figure 6. Sharp flip in the stress gradient ratio is characteristic of the momentum-balance mesolayer.

In order to gain insight into the layered structure of wall jet flow, we plot in figure 7 the profile of the ratio of viscous stress gradient ($\unicode[STIX]{x2202}\unicode[STIX]{x1D70F}_{v+}/\unicode[STIX]{x2202}z_{+}=\unicode[STIX]{x2202}^{2}U_{+}/\unicode[STIX]{x2202}z_{+}^{2}$) to the Reynolds shear stress gradient ($\unicode[STIX]{x2202}\unicode[STIX]{x1D70F}_{t+}/\unicode[STIX]{x2202}z_{+}=\unicode[STIX]{x2202}(-\overline{u_{+}^{\prime }w_{+}^{\prime }})/\unicode[STIX]{x2202}z_{+}$) for the DNS data (flow NTT1-1 in table 1) of Naqavi et al. (Reference Naqavi, Tyacke and Tucker2018); $u^{\prime }$ and $w^{\prime }$ are velocity fluctuations in the streamwise and wall-normal directions, respectively, and overbar denotes time average. Also plotted alongside are the mean velocity profile and the overlap layer profile (4.10). We observe that, as far as the mean momentum balance is concerned, the layered structure of a wall jet is qualitatively very similar to that of channels and TBLs (Wei et al. Reference Wei, Fife, Klewicki and McMurtry2005). This suggests that the momentum-balance approach does not distinguish between the jet or wake character of the outer layer in these flows, a fact that is not surprising given the nonlinear character of the Navier–Stokes equations and differences in boundary conditions in these flows. The innermost region is characterized by the balance of viscous and Reynolds shear stress gradients (ratio ${\approx}-1$) with advection being negligibly small. In the outer region, advection balances the Reynolds shear stress gradient with the viscous stress gradient being negligibly small (ratio ${\approx}0$). The sharp flip in the ratio $(\unicode[STIX]{x2202}\unicode[STIX]{x1D70F}_{v+}/\unicode[STIX]{x2202}z_{+})/(\unicode[STIX]{x2202}\unicode[STIX]{x1D70F}_{t+}/\unicode[STIX]{x2202}z_{+})$ from large negative to large positive values indicates the mesolayer where the Reynolds shear stress reaches a maximum and viscous effects dominate the dynamics even at distances much further away from the viscous sublayer. The extent of the mean-velocity overlap layer for this flow is shown by horizontal dashed and dashed-dotted lines in figure 7. It is clear that the mean-velocity overlap layer, whose extent is decided empirically from the mean-velocity data (figure 6), matches quite well with the momentum-balance mesolayer. This a posteriori justifies the assumption $z_{+}\sim \sqrt{Re_{\unicode[STIX]{x1D70F}}}$ for the mean-velocity overlap layer in wall jets and provides a dynamical basis for the matching arguments of § 4.2. Despite the fact that the DNS flow might still not have reached the fully developed state (largest $x/b$ is $35$ only) and the flow Reynolds number is also relatively low ($Re_{\unicode[STIX]{x1D70F}}=865$ only), the mean velocity profile in the overlap layer in figure 7 is quite close to the curve of (4.10).

Two important quantitative differences, however, exist when one compares wall jets with canonical channel and TBL flows. First, it may be noted that, for channel and TBL flows, the location of the Reynolds shear stress maximum (or the location of the flip in the stress gradient ratio and hence the mesolayer) moves outwards from the wall in inner coordinates with increasing Reynolds number. For $Re_{\unicode[STIX]{x1D70F}}\approx 590$, the maximum is located at $z_{+}\approx 45$ for channel flow DNS data (see figure 1 of Wei et al. (Reference Wei, Fife, Klewicki and McMurtry2005)). Similarly for $Re_{\unicode[STIX]{x1D70F}}\approx 650$, the maximum is located at $z_{+}\approx 55$ for TBL DNS data (see figure 3 of Wei et al. (Reference Wei, Fife, Klewicki and McMurtry2005)). For the wall jet data NTT1-1 of table 1 and figure 7, however, the Reynolds shear stress maximum occurs much closer to the wall, i.e. below $z_{+}=30$ at $Re_{\unicode[STIX]{x1D70F}}=865$. This observation is consistent with a thinner linear sublayer in wall jets compared to canonical flows (see § 3.4) and clearly shows that, for the same Reynolds number, the influence of the outer full-jet flow penetrates deeper into the near-wall region in wall jets than the influence of the outer wake flow in the case of channels and TBLs. Second, the width of the mesolayer in channels and TBLs in the $z_{+}$ coordinate is approximately equal to $\sqrt{Re_{\unicode[STIX]{x1D70F}}}$ (Wei et al. Reference Wei, Fife, Klewicki and McMurtry2005). For wall jets, on the other hand, the width of the mean-velocity overlap layer is approximately $2.3\sqrt{Re_{\unicode[STIX]{x1D70F}}}$ in the $z_{+}$ coordinate (see figure 6). Thus the mean-velocity overlap layer in wall jets appears to be wider than the mesolayer in channels and TBLs.

Figure 8. (a) Outer and (b) inner scaling for the low-Reynolds-number wall jets of table 1. Dashed line in panel (a) denotes the outer limit $\unicode[STIX]{x1D702}=0.077$ of the overlap layer computed from $\unicode[STIX]{x1D702}\sqrt{Re_{\unicode[STIX]{x1D70F}}}=3$ for the average $Re_{\unicode[STIX]{x1D70F}}=1487$. Dashed-dotted line in panel (b) denotes the inner limit $z_{+}=27$ of the overlap layer computed from $\unicode[STIX]{x1D702}\sqrt{Re_{\unicode[STIX]{x1D70F}}}=0.7$ for the same average $Re_{\unicode[STIX]{x1D70F}}$.

Figure 9. (a) Outer and (b) inner scaling for high-Reynolds-number wall jets of table 1. Dashed line in panel (a) denotes the outer limit $\unicode[STIX]{x1D702}=0.056$ of the overlap layer computed from $\unicode[STIX]{x1D702}\sqrt{Re_{\unicode[STIX]{x1D70F}}}=3$ for the average $Re_{\unicode[STIX]{x1D70F}}=2868$. Dashed-dotted line in panel (b) denotes the inner limit $z_{+}=37.5$ of the overlap layer computed from $\unicode[STIX]{x1D702}\sqrt{Re_{\unicode[STIX]{x1D70F}}}=0.7$ for the same average $Re_{\unicode[STIX]{x1D70F}}$. Reference Tailland and MathieuTM and Reference Bradshaw and GeeBG data are not plotted in panel (b) since these suffer from the errors due to wall proximity effect as mentioned in § 4.1.

A further question that now arises is whether there exists any inertial overlap layer between the inner and outer scaling regions. Usually the mesolayer overlap is considered to be the near-wall part (with $Re_{\unicode[STIX]{x1D70F}}$-dependence) of the entire overlap layer and is followed by the inertial overlap (independent of $Re_{\unicode[STIX]{x1D70F}}$) as one moves further outwards from the wall (George et al. Reference George, Abrahamsson, Eriksson, Karlsson, Löfdahl and Wosnik2000). To examine this, we plot the mean velocity profile data (table 1) by dividing them into two broad categories, low-$Re_{\unicode[STIX]{x1D70F}}$ and high-$Re_{\unicode[STIX]{x1D70F}}$. We plot the low-$Re_{\unicode[STIX]{x1D70F}}$ data in figure 8 with the average $Re_{\unicode[STIX]{x1D70F}}=1487$ and the range $1140\leqslant Re_{\unicode[STIX]{x1D70F}}\leqslant 1757$. Outer and inner limits ($\unicode[STIX]{x1D702}\sqrt{Re_{\unicode[STIX]{x1D70F}}}=3$ and $\unicode[STIX]{x1D702}\sqrt{Re_{\unicode[STIX]{x1D70F}}}=0.7$) of the mean-velocity overlap layer translate to $\unicode[STIX]{x1D702}=0.077$ and $z_{+}=27$, respectively, for the average $Re_{\unicode[STIX]{x1D70F}}=1487$. These are shown in figures 8(a) and 8(b), respectively, by dashed and dashed-dotted lines. Figure 9 shows the high-$Re_{\unicode[STIX]{x1D70F}}$ data with average $Re_{\unicode[STIX]{x1D70F}}=2868$ and the range $2450\leqslant Re_{\unicode[STIX]{x1D70F}}\leqslant 3742$ plotted in a fashion similar to figure 8. Figures 8(a) and 9(a) clearly show that the outer scaling extends well below the velocity maximum supporting the present contention that the outer flow is akin to a full-free jet. The most striking observation from figures 8 and 9, however, is that beyond the outer limit (in $\unicode[STIX]{x1D702}$ coordinate) of the mean-velocity overlap layer, data show remarkable collapse in outer scaling, and below the inner limit (in $z_{+}$ coordinate) of the overlap layer, data collapse quite well in the inner scaling. This implies that only the $Re_{\unicode[STIX]{x1D70F}}$-dependent mean-velocity overlap (mesolayer) of figure 6 exists in wall jets and the inertial overlap layer does not appear to exist, at least for the range of Reynolds numbers considered here. Also, it may be noted that, with increase in Reynolds number, the outer edge of the overlap layer in figure 9(a) shifts towards the wall in outer ($\unicode[STIX]{x1D702}$) coordinate as compared to figure 8(a). This indicates that the outer full-free jet flow in wall jets progressively dominates the inner wall flow with increasing Reynolds number. This is to be contrasted with the outer edge of the mean-velocity inertial sublayer (logarithmic overlap layer) being fixed at $\unicode[STIX]{x1D702}\approx 0.15$ in channels and TBLs (Wei et al. Reference Wei, Fife, Klewicki and McMurtry2005; Smits, McKeon & Marusic Reference Smits, McKeon and Marusic2011).

Thus, the layered structure of wall jets that emerges from the data and our analysis in the preceding may be summarized schematically as shown in figure 10(a). The outer flow is a self-similar turbulent full-free jet centred on the velocity maximum and described by the outer layer scaling (4.2). The outer jet layer does not satisfy the no-slip boundary condition at the wall as seen in figure 10(a). An inner or wall region, governed by (4.1), is therefore required to satisfy the no-slip condition at the wall and provide inner boundary condition for the outer flow. These inner and outer regions overlap in the form of an $Re_{\unicode[STIX]{x1D70F}}$-dependent mean-velocity overlap layer that coincides with the momentum-balance mesolayer. There appears to be no inertial overlap between inner and outer scaling regions.

Figure 10. (a) Schematic of the layered structure of wall jet flow as per data and our analysis. Outer layer is a full-free jet centred on the velocity maximum having non-zero slip velocity at the wall. The part of the outer layer velocity profile shown by the thick dashed line gets modified due to overlap with the inner wall layer. The overlap layer is $Re_{\unicode[STIX]{x1D70F}}$-dependent and coincides with the momentum-balance mesolayer. There is no inertial overlap between inner and outer scalings. (b) Schematic of the prevalent proposal of the layered structure of wall jets from the literature (Afzal Reference Afzal2005; Gersten Reference Gersten2015). Part below $U_{max}$ is akin to a TBL, the part above is a half-free jet and these parts merge smoothly at the velocity maximum. Overlap occurs between the defect and inner regions of the TBL part of the flow.

Note that the structure depicted in figure 10(a) supports the asymptotic half-free jet state proposed by Afzal (Reference Afzal2005) and Gersten (Reference Gersten2015), as will be discussed shortly. However, in terms of the spirit and the details, the present proposal is distinct and minimal compared to the alternatives presented in other theoretical studies in the literature (George et al. Reference George, Abrahamsson, Eriksson, Karlsson, Löfdahl and Wosnik2000; Afzal Reference Afzal2005; Barenblatt et al. Reference Barenblatt, Chorin and Prostokishin2005; Gersten Reference Gersten2015). The most prevalent alternative (Afzal Reference Afzal2005; Gersten Reference Gersten2015) of these is the structure depicted in figure 10(b). Here, the part of the flow below the velocity maximum $U_{max}$ behaves similar to a TBL with its own inner and defect scaling regions and a logarithmic (or power-law) overlap. The part above $U_{max}$ is a half-free jet and these parts patch up at the velocity maximum. The contention in this approach is that the asymptotic state far downstream would be a half-free jet, i.e. with increasing $Re_{\unicode[STIX]{x1D70F}}$, the TBL part occupies an increasingly smaller fraction of the overall flow thickness. Note that the proposal of an asymptotic half-free jet state is quite plausible and consistent with our approach as well. Indeed, our data for WJ1, WJ2 and WJ3 experiments in table 1 show that $U_{max}/U_{\unicode[STIX]{x1D70F}}$ and $Re_{\unicode[STIX]{x1D70F}}=z_{T}/(\unicode[STIX]{x1D708}/U_{\unicode[STIX]{x1D70F}})$ for each flow increase continually with $x/b$, implying more rapid decay of inner scales than the outer scales, and this is consistent with the asymptotic half-free jet state. In fact, contrary to the expectation of Gersten (Reference Gersten2015), Reference Eriksson, Karlsson and PerssonEKP data from table 1 also show a similar tendency towards a half-free jet state. However, the contention that the entire flow below $U_{max}$ behaves like a TBL appears to be rather untenable (Bradshaw Reference Bradshaw1963). The occurrence of a region of counter-gradient momentum diffusion below $U_{max}$ in wall jets (Narasimha Reference Narasimha1990; Ahlman et al. Reference Ahlman, Brethouwer and Johansson2007; Naqavi et al. Reference Naqavi, Tyacke and Tucker2018) is in stark contrast with the always down-the-gradient diffusion in the outer region of a TBL flow. A relatively recent LES study (Dejoan & Leschziner Reference Dejoan and Leschziner2005) shows that the anisotropy parameters, profiles of production–dissipation ratio etc. in wall jets differ radically from the corresponding turbulent channel flow results. Given the known similarity between channel flow and TBL structure, it is highly improbable that the part of wall jet flow just below $U_{max}$ has a structure similar to the outer region of a TBL. The innermost region could still have similarities due to dominant role of the wall. Further detailed analyses of the experimental data of table 1 following the approaches of Afzal (Reference Afzal2005) and Gersten (Reference Gersten2015) are presented in appendices A and B, respectively.

The layered structure of wall jets proposed by George et al. (Reference George, Abrahamsson, Eriksson, Karlsson, Löfdahl and Wosnik2000) is summarized in figure 2 therein. While the structure we have proposed is very similar to their proposal, there are distinct differences such as $Re_{\unicode[STIX]{x1D70F}}$-independent inner and outer scalings, absence of inertial overlap, arbitrary shift in $U$ instead of $z$ in the overlap analysis, etc. in our proposal. The outer analysis of George et al. rests on the contention that the Reynolds shear stress in the outer layer scales on $U_{\unicode[STIX]{x1D70F}}$ rather than $U_{max}$. The experimental evidence (figure 10 of their paper) for this proposal is, however, very limited; there have been some recent studies that challenge this view (see Ahlman et al. Reference Ahlman, Brethouwer and Johansson2007; Rostamy et al. Reference Rostamy, Bergstrom, Sumner and Bugg2011). Theoretical argument in favour of this contention is presented in appendix B of their paper, where the outer equations reduce to a constant-stress layer in the overlap region. Again, the experimental evidence in wall jets for the occurrence of a constant-stress layer, or even a tendency towards it, is plainly absent (see figure 17 from their paper); this is in fact consistent with the absence of inertial overlap in wall jets. Our perception is that part of this confusion arises because the outer flow structure has not been identified correctly. With our structural model, the outer full-free jet flow does not lead to a constant-stress layer in the near-wall region due to non-negligible advection on account of a large, finite slip velocity. More systematic experiments on wall jets at higher Reynolds numbers are required to shed more light on the correct scaling behaviour of the Reynolds shear stress. Appendix C presents further analysis of the experimental data of table 1 following the approach of George et al. (Reference George, Abrahamsson, Eriksson, Karlsson, Löfdahl and Wosnik2000).

Barenblatt et al. (Reference Barenblatt, Chorin and Prostokishin2005) also have proposed a layered structure for wall jets based on rather limited data of Reference Eriksson, Karlsson and PerssonEKP experiments alone. In this proposal, the inner and outer layer length scales are respectively taken to be the heights of the half-velocity points ($U_{max}/2$) below and above the velocity maximum. Our view is that, whilst the height of the upper half-velocity point could be justified as a measure of the overall thickness of the flow (as in turbulent free jets, see Deo et al. Reference Deo, Mi and Nathan2008), the height of the lower half-velocity point seems a rather ad hoc and arbitrary choice without an immediately apparent physical significance (the viscous length scale would be more appropriate in the near-wall region). The velocity scale $U_{max}$ essentially remains the same for both the layers. Furthermore, Barenblatt et al. do not discuss any overlap analysis.