1. Introduction
Turbulent boundary layers (TBLs) developing over rough walls are found in many engineering applications. Studying this phenomenon is crucial for the performance evaluation of an engineering system. For example, in the aeronautical or automotive industry, the manipulation of a TBL using a surface treatment (i.e. roughness) may result in drag reduction (Whitmore & Naughton Reference Whitmore and Naughton2002). On the other hand, in the wind energy sector, an atmospheric boundary layer (ABL) in neutral conditions developing over a wind farm behaves like a large-scale TBL over roughness. Understanding the physics of this flow leads to more accurate wind power predictions and strategic site selections (Bou-Zeid et al. Reference Bou-Zeid, Anderson, Katul and Mahrt2020).
A realistic representation of a rough-wall TBL in these applications hardly ever involves a homogeneous rough wall. In some scenarios, it can be better approximated with a streamwise transition in roughness. For example, the roughness on a ship hull (due to biofouling and coating deterioration) forms at various roughness length scales and sites, resulting in multiple streamwise transitions in roughness that affect the development of the TBL. At the same time, when analysing sites for wind farm locations we might encounter areas of complex terrain where we see a combination of forests and plains or sea and coastline. These variations significantly affect the behaviour of the ABL and, consequently, the drag production near the surface.
The main change occurring in a TBL over a rough wall compared with one over a smooth wall is an increase in wall shear stress (WSS). This results in a momentum deficit
$\Delta U^+$
, characterised by a vertical shift in the logarithmic layer of the streamwise mean velocity profile, which, for fully rough flows, is defined as follows:
\begin{equation} U^+=\frac {1}{\kappa }\text {ln}\left (y^+\right )+A-\Delta U^+ = \frac {1}{\kappa }\text {ln}\left ( \frac {y}{k_s}\right )+B, \end{equation}
where
$\kappa \approx 0.39, A\approx 4.3 \text { and } B=8.5$
. Equation (1.1) shows that the two main parameters used to scale TBLs over rough walls are
$k_s$
as the length scale, and the friction velocity
$u_\tau$
(see Jiménez (Reference Jiménez2004) or other similar works for the details on the scaling arguments). A surface with arbitrary representative roughness height
$k$
is associated with a length scale
$k_s$
, as shown in figure 1, which affects the logarithmic layer of the mean velocity profile in the same way as a surface covered by an ideally uniform sand-grain type of roughness with physical height
$k_s$
. Its definition is given in Colebrook et al. (Reference Colebrook, White and Taylor1937) and Nikuradse (Reference Nikuradse1933) and some examples of its usage can be found in Flack & Schultz (Reference Flack and Schultz2014) and Schultz & Flack (Reference Schultz and Flack2009). This height is usually calculated by taking a point measurement in the logarithmic layer of a TBL and using (1.1), with the main assumption being that the flow is within the fully rough regime. Another method of calculating
$k_s$
is given by Monty et al. (Reference Monty, Dogan, Hanson, Scardino, Ganapathisubramani and Hutchins2016), which consists of an iterative procedure to obtain a direct relation between the surface friction and
$k_s$
. This method assumes that the TBL adheres to the outer layer similarity (see Townsend Reference Townsend1965). As such, the TBL is in equilibrium with the surface texture.
A TBL past a step change in roughness undergoes a well-documented change in its internal structure (Elliott Reference Elliott1958; Townsend Reference Townsend1965). Specifically, the logarithmic region of the TBL is split into two parts: an internal layer (IL), mostly in equilibrium with the downstream roughness, and an outer layer, still in equilibrium with the upstream surface. Whilst
$k_s$
has traditionally been defined as a constant, it can be argued that this assumption may overlook significant opportunities to account for deviations from idealised roughness effects. In fact, the mean velocity in the two log regions of the TBL past a step change in roughness can be described using (1.1) by allowing
$k_s$
to vary and reflect the surface with which the region is in equilibrium. Furthermore, varying
$k_s$
as a function of fetch length might provide a straightforward way to describe the onset of equilibrium in the IL and its development within the TBL until equilibrium is achieved throughout the entire structure. This will be discussed in more detail in § 3.
Figure 1. Schematic of physical roughness height
$k$
vs. the equivalent sand-grain roughness height
$k_s$
.
Another effect of step changes in roughness is the immediate impact on WSS and its recovery to an equilibrium state. This phenomenon has been extensively studied using both experimental and numerical approaches. The WSS either increases or decreases abruptly overshooting or undershooting the expected value for the downstream surface in smooth-to-rough (Bradley Reference Bradley1968; Antonia & Luxton Reference Antonia and Luxton1971a
) and rough-to-smooth transitions (Antonia & Luxton Reference Antonia and Luxton1972; Efros & Krogstad Reference Efros and Krogstad2011; Hanson & Ganapathisubramani Reference Hanson and Ganapathisubramani2016), respectively. Experimentally, this has been researched with direct WSS measurements immediately downstream of the step change by using floating element balances (Bradley Reference Bradley1968; Efros & Krogstad Reference Efros and Krogstad2011), near-wall hot-wires (Chamorro & Porté-Angel Reference Chamorro and Porté-Angel2009), Preston tubes (Loureiro et al. Reference Loureiro, Sousa, Zotin and Freire2010) and pressure taps (Antonia & Luxton Reference Antonia and Luxton1971a
, Reference Antonia and Luxton1972), coupled with indirect methods to obtain the development of the WSS with distance from the step change. This was mainly done using a logarithmic fit to match the measured value and the expected one for the downstream surface if there were no surface changes upstream. Numerically, the WSS recovery after a step change in roughness has been mainly investigated with direct numerical simulations (Lee Reference Lee2015; Ismail et al. Reference Ismail, Zaki and Durbin2018a
; Rouhi et al. Reference Rouhi, Chung and Hutchins2019b
) and large eddy simulations (Saito & Pullin Reference Saito and Pullin2014; Sridhar Reference Sridhar2018). The results of all these studies were conducted at friction Reynolds numbers in the range
$10^2\leqslant Re_\tau \leqslant 10^6$
, and a variety of downstream-to-upstream roughness height ratios,
$-6 \leqslant \text {ln}(k_{s,2}/k_{s,1})\leqslant 6$
(where the subscripts 1 and 2 indicate the values for the upstream and downstream surfaces, respectively). Here, negative ratios correspond to rough-to-smooth transitions, and positive values correspond to smooth-to-rough changes.
Previous studies highlighted some remaining questions regarding the TBL recovery to an equilibrium condition after being subjected to a streamwise step change in roughness. Firstly, as mentioned above, the characteristic overshoot or undershoot of the WSS just after a step change in roughness renders the characterisation methods developed for the homogeneous rough wall inaccurate, since both scaling parameters depend on WSS and are calculated assuming fully rough homogeneous roughness. This leads to a need to define a minimum recovery length in which the flow recovers to the homogeneous rough-wall TBL. Secondly, the use of experimental indirect methods and numerical methods to obtain the WSS recovery after a step change resulted in a wide range of recovery fetch lengths between
$1\delta$
and
$10\delta$
, making it difficult to draw specific conclusions from these predictions. Moreover, some studies such as Saito & Pullin (Reference Saito and Pullin2014) and Sridhar (Reference Sridhar2018) showed an increase in recovery fetch length with Reynolds number which is inconsistent with other studies, highlighting the necessity of a direct WSS measurement method for a more accurate prediction. An extensive review and comparison between existing studies can be found in Li et al. (Reference Li, de Silva, Chung, Pullin, Marusic and Hutchins2019).
In this study, we consider a TBL developing from a baseline smooth wall and subjected to a streamwise transition to a rough wall. We aim to investigate and achieve a reliable value for the minimum roughness fetch length that allows a TBL developing past such step change in roughness to recover to an equilibrium condition, i.e. fully adjust to the rough wall downstream of the transition. This is essential since all of the scaling arguments used in rough-wall TBLs depend on the WSS and the latter changes drastically after a step change in wall roughness. Secondly, we aim to quantify the error in choosing a shorter fetch to conduct experiments/simulations on presumably homogeneous fully rough flows. This would be helpful to quantify the uncertainty of the data if, for instance, a study needed to be conducted in a facility with a shorter test section, or if there were limitations on the domain size for a numerical investigation dictated by the available computational power. Finally, we aim to develop a relationship between the
$k_s$
value of a surface with short fetch (where the flow is not in equilibrium) in terms of the equilibrium value of
$k_s$
. We designed an experiment to directly measure the change in WSS to sequential increases in roughness fetch, covering a wide range of Reynolds numbers,
$4\times 10^3\leqslant Re_\tau \leqslant 2\times 10^4$
, to ensure all or most common conditions are covered. The set-up of the experiment is covered in § 2, followed by a detailed discussion of the results in § 3 leading to the conclusions and future work in § 4.
2. Experimental set-up and methodology
The experimental campaign is conducted inside the closed return boundary layer wind tunnel (BLWT) at the University of Southampton. The TBL is tripped by a turbulator tape located at the inlet of the test section, marking the streamwise datum (x = 0) and further developed along the floor of the 12 m-long test section, which has a width and height of 1.2 m and 1 m, respectively. Figure 2 illustrates the tunnel and coordinate system where x, y and z denote the streamwise, wall-normal and spanwise directions, respectively. The tunnel is equipped with a ‘cooling unit’ comprising two heat exchangers and a temperature controller such that the air temperature remains constant during measurements (
$21^{\circ }$
C
$\pm 0.5^{\circ }$
C). The free stream has a turbulence intensity of less than 0.1 % of the free-stream velocity
$U_\infty$
, which is measured with hot-wire anemometry before the experimental campaign. The tunnel is equipped with a closed-loop feedback controller to set
$U_\infty$
, and air properties are measured with a Pitot-static tube and a thermistor inside the BLWT.
Figure 2. Schematic of the experiment with the fetch length,
$L$
, measured from the centreline of the balance: (a)
$L=1\delta _2$
and (b)
$L=39\delta _2$
.
As seen in figure 2, the experiment consisted of a roll of P24 sandpaper cut in patches of size
$2\delta _2\times 8\delta _2$
, where
$\delta _2$
refers to the TBL thickness of the case with the longest fetch measured at the balance location. This length scale was chosen instead of the more commonly used
$\delta _1$
(where
$\delta _1$
is the TBL thickness over the smooth surface measured at the measurement point) for two reasons. Firstly, having the recovery length as a function of the downstream TBL thickness removes all dependency on the type of surface upstream of the step change, making it applicable to more cases. Secondly, the TBL thickness was measured using particle image velocimetry (PIV) above the balance to ensure consistency between the balance readings and the flow field above while no PIV measurement was taken upstream of the step change in any of the cases. For the sake of clarity, the relationship between
$\delta _1$
and
$\delta _2$
in the current experimental campaign is quantified. This analysis is based on PIV measurements of the smooth wall at the measurement location, combined with the equation for the evolution of
$\delta$
over a smooth surface (
$\delta = 0.37x/Re_x^{1/5}$
, White Reference White2011) to estimate the value at the step-change location for the case where the TBL achieves equilibrium. The resulting ratio,
$\delta _1/\delta _2 \approx 0.8$
, provides a basis for direct comparison with previous studies.
Figure 3. Laser scan of the P24 sandpaper used in the campaign with probability density function (PDF) of the surface height variation,
$h'$
from the mean physical height
$\overline {h}$
. Key surface statistics are as follows:
$k=6\sqrt {\overline {h^{\prime 2}}}=1.6953$
,
$k_a=\overline {|h'|}=0.2257$
,
$k_p=\text {max}(h')-\text {min}( h')=2.0227$
,
$k_{rms}=\sqrt {\overline {h^{\prime 2}}}=0.2825$
.
The patches are sequentially taped on the floor of the wind tunnel’s test section starting at the measurement point, which is located approximately
$59\delta _2$
downstream from the test section’s inlet. The patches are then added upstream, so that the roughness fetch measured from the centreline of the balance is systematically increased. The shortest fetch is
$1\delta _2$
, and the longest is
$39\delta _2$
, which corresponds to the distance between the measurement point and the step change in roughness. This means that, for the longest fetch configuration, the distance between the step change in roughness and the test section inlet is approximately
$20\delta _2$
, while the distance between the measurement point and the step change in roughness is approximately
$39\delta _2$
, as shown in figure 2. A laser scan of the sandpaper used in the experimental campaign can be found in figure 3. The mean physical height,
$k$
of the sandpaper was computed as
$k=6\sqrt {\overline {h^{\prime 2}}}$
, similarly to Gul & Ganapathisubramani (Reference Gul and Ganapathisubramani2021), with
$h'$
being the variation from the mean surface height showed in figure 3 and
$\sqrt {\overline {h^{\prime 2}}}$
being the surface variance.
All cases tested in the experimental campaign are listed in table 1. The longest fetch tested was chosen as a threshold between having as long of a roughness fetch as achievable in our facility and ensuring the TBL on the smooth surface upstream of the step change would also have enough development length to be in equilibrium conditions (
$\approx 25\delta _1$
or
$\approx 20\delta _2$
). As the sandpaper sheets are taped on the smooth wall, there is a step change of physical height
$k$
at the smooth-to-rough transition. The effect of the physical height difference between the smooth wall and the sandpaper taped onto it has been previously explored in Antonia & Luxton (Reference Antonia and Luxton1971a
,
Reference Antonia and Luxtonb
). Both experiments yielded the same results in terms of
$\overline {u'v'}$
distribution and WSS. Therefore, the physical height difference plays a negligible role on the flow development. Thus, we did not implement a system to isolate the effects of the superposition of the physical height of the two surfaces from the step change in roughness.
Table 1. Colour legend for different roughness fetches applied to all plots in § 3.
The experimental campaign was designed to take direct WSS measurements at different Reynolds numbers and with sequentially increased roughness fetch (the distance between the step change in roughness and the measurement point). This was possible by employing a floating element drag balance (located at the previously mentioned measurement point), designed and manufactured by Ferreira et al. (Reference Ferreira, Costa and Ganapathisubramani2024). With this tool, the friction on the wall was monitored during velocity sweeps (0–40 ms
$^{-1}$
) while systematically increasing the length of the roughness fetch. The velocity sweeps were run three times per case to ensure the repeatability of the results. The measurement uncertainty of skin friction from the balance is estimated to be less than 1 %. A detailed description of the balance, its specifications as well as uncertainty quantification can be found in Ferreira et al. (Reference Ferreira, Costa and Ganapathisubramani2024).
For each fetch length, measurements are conducted within a range of free-stream velocities 10 ms
$^{-1}$
$\leqslant U_\infty \leqslant 40$
ms
$^{-1}$
, allowing 10 s for the flow to adjust after each velocity increase and 60 s for the flow to come to rest completely before restarting the sweep. The sampling rate was set to
$f_s = 256$
Hz, while the sampling time was set to 60 s. Once the friction force,
$F$
, has been measured, the WSS,
$\tau _w$
, and friction velocity,
$u_\tau$
, can be directly computed
\begin{equation} \tau _w = \frac {F}{A} = \frac {1}{2}\rho U_\infty ^2 C_{\! f} = \rho u_\tau ^2 , \end{equation}
where
$A$
is the surface area of the balance plate and
$C_{\! f}$
is the friction coefficient.
Planar PIV was also performed in the streamwise wall-normal plane above the floating element location. This was done to check whether the outer layer similarity (OLS) used in Monty et al. (Reference Monty, Dogan, Hanson, Scardino, Ganapathisubramani and Hutchins2016) to calculate
$k_s$
holds for some or all of our cases and to calculate the boundary layer thickness for all cases. The additional PIV measurements were only conducted at a free-stream velocity of
$20$
ms
$^{-1}$
and only for the cases with fetch length
$1\delta _2,3\delta _2,5\delta _2,7\delta _2,9\delta _2,19\delta _2 \text { and }39\delta _2$
. The selection of free-stream velocity and fetches to study with PIV was dictated by the trends obtained in the drag balance measurements, as seen in § 3. The data were sampled at
$f_s =1$
Hz (
$t_r=N \times U_\infty /(f_s\times \delta _2)\approx 267\times 10^3$
, where
$t_r$
is the TBL turnover rate,
$U_\infty =20$
ms
$^{-1}$
and
$N$
is the number of samples taken, Marusic et al. (Reference Marusic, Chauhan, Kulandaivelu and Hutchins2015)) with Lavision Imager CMOS 25 MP cameras (resolution of 17 pixels mm−1), using a Bernoulli 200 mJ, 532 nm Nd:YAG laser and the Lavision software Davis 10 for acquisition. The data were processed using an in-house code for cross-correlation, with a final window size of 16
$\times$
16 px with 75 % overlap, and a viscous-scaled final window size
$\Delta x^+ \times \Delta y^+$
of 30
$\times$
30. The uncertainty in the mean flow is approximately 2 %, estimated following standard uncertainty propagation methods Bendat & Piersol (Reference Bendat and Piersol2010).
3. Results
The colour legend for all the plots in § 3 is shown in table 1.
The evolution of the friction coefficient obtained with the drag balance at different fetch lengths and increasing Reynolds number (
$Re_x=\rho U_\infty x/\mu$
, where
$x$
is the streamwise distance between the wind tunnel’s test section inlet and the balance centreline) can be seen in figure 4(a). This plot shows that, for a fixed fetch length,
$C_{\! f}$
is independent of
$Re_x$
, which is a sign that the flow is within the fully rough regime bounds mentioned in § 1. However, it is not fetch-length independent since the fetch length is inversely proportional to
$C_{\! f}$
, consistent with the overshoot downstream of the transition observed by multiple studies listed in Li et al. (Reference Li, de Silva, Chung, Pullin, Marusic and Hutchins2019), and the slow recovery with increasing distance from the step change.
Figure 4. Panel (a) shows
$C_{\! f}$
plotted against
$Re_x=U_\infty x/\nu$
, where
$x$
is the distance of the balance centreline from the test section’s inlet, with (
3) being lines of constant unit
$Re=U_\infty /\nu$
(while fetch length
$x$
varies), and (
4) lines of constant
$k_s/x$
(while unit
$Re$
varies) as described by Monty et al. (Reference Monty, Dogan, Hanson, Scardino, Ganapathisubramani and Hutchins2016). Panel (b) shows
$C_{\! f}$
at
$Re_x\approx 10^7$
plotted against the fetch length
$L$
normalised by the downstream TBL thickness
$\delta _2$
.
The recovery of WSS with fetch length is shown more clearly in figure 4(b). This plot shows the recovery of the friction coefficient measured at around 20 ms
$^{-1}$
with fetch length. This is the lowest flow speed at which the TBL seems to reach fully rough conditions and is thus used for the PIV measurements as well. The friction coefficient is plotted against the normalised fetch length, where
$\delta _2$
is the boundary layer thickness at the balance location of the fully rough case with a fetch length of
$\approx 39\delta _2$
. For clarity, table 2 lists the TBL thickness measured above the balance for all the different fetches.
Table 2. The TBL thickness at the drag balance location of the cases tested with PIV, fetch length defined as a function of
$\delta _2$
(the TBL thickness of the case with longest fetch).
Figure 1(b) in Li et al. (Reference Li, de Silva, Chung, Pullin, Marusic and Hutchins2019) presents a comparison of recovery lengths compiled from previous studies, indicating a wide range of recovery lengths, from 1to 10
$\delta _1$
, for both smooth-to-rough and rough-to-smooth transitions. The criteria for defining convergence in these studies remain unclear, with Li et al. (Reference Li, de Silva, Chung, Pullin, Marusic and Hutchins2019) suggesting, as an example, defining the recovery length
$L$
as the downstream fetch where the local
$C_{\! f}$
reaches approximately 80 % of the full-recovery value
$C_{\! f0}$
. Applying this criterion to our data (shown in figure 4
b) suggests a recovery length of approximately
$6\delta _2$
, where
$C_{\! f}$
reaches 80 % of the equilibrium value. However, we propose an alternative criterion for convergence, defining it as the fetch where
$C_{\! f}$
plateaus within 5 % of the full-recovery value. Using this definition, our results indicate a longer recovery length of at least
$20\delta _2$
. Secondly, although we expect the overshoot in
$C_{\! f}$
immediately after transition (i.e.
$C_{\! f}$
measured in shorter patch lengths,
$1\delta _2-5\delta _2$
), we observe that, for
$L\gt 10\delta _2$
, the error in
$C_{\! f}$
is within
$\approx 10\,\%$
of the converged value. This type of error is to be expected when a shorter development length or computational domain is used. Figure 4(a) can also be used to obtain the equivalent sand-grain height following the method proposed by Monty et al. (Reference Monty, Dogan, Hanson, Scardino, Ganapathisubramani and Hutchins2016). As briefly mentioned in § 1, this method assumes that the flow has already reached an equilibrium state and therefore employs OLS from Townsend (Reference Townsend1965), to obtain a relationship between
$C_{\! f}$
at constant unit Reynolds number and
$k_s$
, via what the authors refer to as lines of constant length,
$k_s/x$
. These are shown in figure 4(a) as pink horizontal solid lines. The intersection of these and the
$C_{\! f}$
values at given
$Re_x$
, give us a way of calculating
$k_s$
for different fetch lengths.
Figure 5. Panel (a) shows velocity defect plotted against
$y$
normalised by
$\delta _{99}$
as listed in table 2 for each case. Panel (b) shows
$k_s$
evolution, normalised by
$k_{s,2}$
, with fetch length
$L$
scaled with
$\delta _2$
.
$k_{s,IL}$
and
$k_{s,OLS}$
calculated at https://www.cambridge.org/S0022112025003118/files/figure5B.
Before discussing the result of this operation, the OLS hypothesis from Townsend (Reference Townsend1965) was reproduced and is shown in figure 5(a). From this plot, it can be seen that, for shorter fetch lengths, velocity defect profiles do not collapse and hence do not conform to OLS. This is to be expected as OLS is a measure of equilibrium with the boundary layer and for fetches lower than
$\approx 10\delta _2$
equilibrium cannot be achieved due to the development of the IL. On the other hand, for the longer fetches, OLS can be observed as the profiles perfectly collapse onto smooth-wall TBLs from
$\approx 0.4\delta$
. The latter is the main conclusion from Townsend (Reference Townsend1965), which means that the near-wall region and anything that is associated with it should not affect the outer portion of the TBL. From our results, we can conclude that this indeed holds for the longest fetches tested. In the following analysis, we will see more in detail how the non-equilibrium conditions affect the prediction of
$k_s$
values based on OLS and how this compares with the standard practice of calculating it from the roughness function
$\Delta U^+$
where fully rough as well as equilibrium conditions are assumed.
In figure 5(b) we show the
$k_s$
development with fetch length obtained using two methods. Firstly, the method from Monty et al. (Reference Monty, Dogan, Hanson, Scardino, Ganapathisubramani and Hutchins2016) defined by the symbol 
5; secondly, we used an IL curve fitting method to obtain
$k_s$
. In order to only fit the IL region, we used the first derivative of the Reynolds shear stress, which will be discussed later on in figure 7, with respect to the natural log of
$y/\delta _2$
. This highlighted a region of blending between two distinct profiles that we identified as the IL thickness. Finally, below this point we used (1.1) to curve fit the data. This is represented by the symbol 
6. Starting with the method from Monty et al. (Reference Monty, Dogan, Hanson, Scardino, Ganapathisubramani and Hutchins2016) we observe that
$k_s$
follows the same trend as
$C_{\! f}$
, i.e. overshooting its ‘real’ value for a certain surface at fixed Reynolds number and logarithmically approaching its true value with increasing fetch length. Figure 5(b) shows how crucial it is to ensure sufficient fetch length for the WSS to recover to be able to treat
$k_s$
as universal and use it as a length scale/modelling constant for rough-wall TBLs. It can also be noted that the minimum fetch length for full WSS recovery is around
$L\geqslant 20\delta _2$
, where
$C_{\! f}$
becomes both Reynolds number and fetch-length independent. We note here that this streamwise fetch might depend on the type of roughness and the extent of change in
$k_s$
(from upstream to downstream). Regardless, the results suggest that TBLs flowing over a change in wall texture with fetch lengths shorter than at least
$10\delta _2$
(
$\text {error} \geqslant 10\, \%$
) will inevitably result in a significant overestimation of the roughness function and corresponding mean flow.
In figure 5(b), we also see the trend of
$k_s$
when calculated by fitting a logarithmic profile to the velocity profile in the near-wall region (i.e. below the inflection point), which is the point where the IL blends into the outer layer. As shown in this figure, the trend captured by this method is opposite to the one given by the method in Monty et al. (Reference Monty, Dogan, Hanson, Scardino, Ganapathisubramani and Hutchins2016) used previously. Nonetheless, the fetch length at which we can infer equilibrium conditions after a step change does not change and is in full agreement between the two methods. Moreover, the converged value of
$k_s$
for the longer fetch cases appears to be in perfect agreement as well. The challenge in using this method lies in accessing velocity measurements in the region close to the wall with enough resolution to fit a logarithmic profile and achieving a Reynolds number large enough to be able to distinguish the inflection point.
In this article, we propose using the equivalent sand-grain roughness height,
$k_s$
, as a parameter to describe and model TBLs over streamwise variations in surface roughness. This approach is motivated by the challenges associated with using the IL growth rate, which is highly dependent on the criteria used to identify the inflection point where the log region governed by the downstream roughness meets that in equilibrium with the upstream surface. As highlighted in the literature (e.g. Antonia & Luxton Reference Antonia and Luxton1971a
,
Reference Antonia and Luxtonb
, Reference Antonia and Luxton1972), the same case can often be interpreted with markedly different power laws depending on the chosen criteria, making it difficult to draw generic conclusions. In contrast,
$k_s$
offers a simpler and more consistent parameter for characterising the effects of step changes in surface roughness.
We propose redefining
$k_s$
as a function of fetch length with the form
\begin{equation} \lim _{x \to \infty } k_s(x) = k_{s,}\text {homogeneous}. \end{equation}
This approach aligns with the physical processes governing the development of equilibrium and is consistent with existing wall models for low-fidelity simulations. In Reynolds-averaged Navier–Stokes models, roughness is typically applied using the law of the wall with
$k_s$
specified a priori. By allowing
$k_s$
to vary with fetch length, this approach improves the reliability of low-fidelity simulations in TBL studies while maintaining computational efficiency and simplicity.
7 in 
8 in Figure 6 shows the streamwise-averaged mean flow profiles measured by PIV, taken above the drag balance and scaled by the friction velocity given by the balance measurement, where the black dashed line represents the log profile. In figure 6(a), the wall-normal coordinate used to plot all the profiles is normalised by the fully rough, equilibrium value of
$k_s = k_{s,2}$
, which is computed for the longest fetch case. Here, we can see that, although the two longest fetches collapse onto the dashed line perfectly in the log region, the rest of the cases slowly diverge from it with the shortest fetch displaying a change in slope across the logarithmic domain of the TBL. This is clearly explained by the blending of the logarithmic regions from the upstream and downstream surface near the step change in roughness. In the next plot, figure 6(b), we used a
$k_s$
value for each fetch case that attempts to include the effect of the IL development by computing it using the local
$C_{\! f}$
value as described in Monty et al. (Reference Monty, Dogan, Hanson, Scardino, Ganapathisubramani and Hutchins2016) and shown in figure 5(b) – ‘OLS’. However, when using this method, the profiles seem to diverge to a greater extent than using the
$k_{s,2}$
value for all the cases. This can be explained by the equilibrium assumption made when employing the method described in Monty et al. (Reference Monty, Dogan, Hanson, Scardino, Ganapathisubramani and Hutchins2016). Lastly, in figure 6(c), we used the
$k_s$
value for each individual case given by fitting a logarithmic profile to the IL only as shown in figure 5(b) – ‘IL fit’. Using this method we achieved a perfect match for all fetches below the inflection point, while above this point the shorter fetch profiles do not collapse onto the longer ones. This is because the
$k_s$
value that models the IL region would inevitably fail in the outer region in cases of non-equilibrium such as a TBL after a step change in roughness. Therefore, in order to achieve a universal scaling we would have to make
$k_s$
a function of
$x$
, by employing a different value for different fetches, and
$y$
, by using a different value below and above the inflection points where the IL is still developing. Finally, this method is only possible when a direct way of measuring drag is available as the drag given by the slope of the IL is not correct for short fetches.
Figure 7. Turbulent fluxes (a,b), and Reynolds shear stress (c), in viscous units against wall-normal distance in outer scaling.
Finally, we looked at the second-order statistics for all cases and compared the patterns with the mean fields that we discussed above. Figure 7 illustrates the turbulent intensities (panels a and b) and Reynolds shear stress (panel c) in viscous units, plotted against the wall-normal distance using outer scaling. These plots allow for a detailed examination of the onset and development of equilibrium within the TBL downstream of the step change in roughness.
From these profiles, it is evident that equilibrium is achieved at approximately
$20\delta _2$
, as the curves align closely with those of the longest fetch case. This observation aligns with the mean velocity analysis presented in figure 6. Additionally, the growth of the IL within the TBL is particularly pronounced in all the second-order statistics plots, especially for the shortest fetches of
$1\delta _2$
and
$3\delta _2$
. For these cases, a noticeable change in the curvature of the turbulent profiles is apparent, with the transition occurring closer to the wall for the shortest fetch and gradually extending outward with increasing fetch length.
4. Conclusions
The current paper aims to describe the outcome of an experimental campaign involving direct measurements of the WSS recovery after a step change in wall roughness with systematically increasing fetch length. The results show that full WSS recovery is achieved
$20\delta _2$
downstream of the step change, while previous studies employing indirect ways of measuring the WSS recovery predicted a full recovery between
$1\delta _1$
and
$10\delta _1$
. This difference is most likely due to the logarithmic nature of the WSS recovery. Therefore, even the smallest difference in WSS results in a significant difference in fetch length. We also show that the greatest change in WSS appears for fetch lengths between
$1\delta _2$
and
$10\delta _2$
, resulting in an error of
$\leqslant 10\, \%$
of the converged WSS value when fetches
$\geqslant 10\delta _2$
are used.
Moreover, we have shown that the equivalent sand-grain height,
$k_s$
, given by the method adopted in Monty et al. (Reference Monty, Dogan, Hanson, Scardino, Ganapathisubramani and Hutchins2016) cannot be used to scale or model the mean velocity profile of a TBL for fetches measuring less than
$10\delta _2$
, as this would inevitably result in a significant overprediction of the roughness effects and erroneous velocity profiles. This is due to the assumptions employed when deriving
$k_s$
, including fully rough regime and equilibrium conditions in the TBL, which do not apply in the case of a TBL flowing past a step change with fetch length measuring less than
$10\delta _2$
. On the other hand, when fitting a logarithmic profile to the IL region, we can achieve a unique
$k_s$
value for finite fetches that is able to scale/model the velocity profile below the inflection point and, by making
$k_s$
vary in the wall-normal direction, we could be able to model TBLs past step changes in roughness and their development to a greater extent.
A new way of modelling
$k_s$
, which takes into account both log regions of the internal boundary layer downstream of the transition and the outer layer (containing the flow history prior to the transition), could help with modelling streamwise-varying rough-wall TBLs. A correction factor between the
$k_s$
trend with increasing fetch given by Monty et al. (Reference Monty, Dogan, Hanson, Scardino, Ganapathisubramani and Hutchins2016) and the one given by fitting should also be developed in cases where high-resolution PIV at the right Reynolds number near the wall is not viable.
Supplementary material.
Supplementary material and Computational Notebook files are available at https://doi.org/10.1017/jfm.2025.311. Computational Notebooks can also be found online at https://www.cambridge.org/S0022112025003118/JFM-Notebooks.
Acknowledgements.
The authors acknowledge funding from the Leverhulme Early Career Fellowship (Grant ref: ECF-2022-295), the European Office for Airforce Research and Development (Grant ref: FA 8655-23-1-7005) and EPSRC (Grant ref no: EP/W026090/1).
Declaration of interests.
The authors report no conflict of interest.
Data availability statement.
All data published in this article are publicly available from the University of Southampton repository at https://doi.org/10.5258/SOTON/D3426.
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