1. Introduction
Favourable pressure gradients (FPGs) can have a strong effect on turbulent boundary layers, and are seen in practice in numerous applications. Some examples include the leading regions of airfoil surfaces and the hulls of marine vessels. An FPG increases velocity in the streamwise direction and reduces the boundary-layer thickness, resulting in increasing shear. The increasing shear can, however, be mitigated or even reversed by a second effect. An FPG is stabilising, and tends to reduce the turbulence in a boundary layer. If the FPG is strong enough and sustained for sufficient distance, a boundary layer can relaminarise. Reduced turbulent mixing tends to reduce skin friction. Many studies have considered FPG boundary layers on smooth walls. Narasimha & Sreenivasan (Reference Narasimha and Sreenivasan1973), for example, considered the mechanisms by which relaminarisation occurs. Narasimha & Sreenivasan (Reference Narasimha and Sreenivasan1979) and Sreenivasan (Reference Sreenivasen1982) provided discussions and reviews of relaminarisation.
An interesting type of FPG boundary layer is the sink flow that develops when the acceleration parameter, K, is held constant:
\begin{equation}K=\frac{\nu }{U_{e}^{2}}\frac{{\rm d}U_{e}}{{\rm d}x},\end{equation}
where ν is the kinematic viscosity, U e is the local free-stream velocity and x is the streamwise coordinate. In the limit of zero boundary-layer thickness, a constant K is produced in the flow between two converging flat plates. A close approximation to a constant K also occurs in real boundary-layer flow between converging flat plates because the FPG keeps the boundary layers thin. If a sink flow could be continued for a sufficient streamwise distance, a singularity would be reached where the converging plates would meet, and the velocity would go to infinity. This distance is
\begin{equation}x_{\textit{sink}}=x-x_{o}=\frac{\nu }{\textit{KU}_{\textit{eo}}},\end{equation}
where x o and U eo are taken at some upstream location, e.g. the start of the FPG. A sink flow boundary layer on a smooth wall will approach complete equilibrium, meaning that all dimensionless quantities reach asymptotic values and stop changing with x. This includes Reynolds numbers of all types, the skin friction coefficient and profiles of mean velocity and turbulence quantities expressed in inner or outer coordinates.
For a laminar boundary layer, the sink flow has an exact, closed-form solution, obtained by Pohlhausen (Reference Pohlhausen1921) and presented by Schlichting (Reference Schlichting1968). The turbulent, smooth-wall case has been considered in several studies, both experimentally and numerically. Jones & Launder (Reference Jones and Launder1972), for example, conducted experiments with K values between 1.5 × 10–6 and 3 × 10–6. Spalart (Reference Spalart1986) performed direct numerical simulations (DNS) for the same range of K. The studies found that relaminarisation occurred for K larger than about 3 × 10–6. In cases where the boundary layer relaminarised, it was noted that the flow became ‘quasi-laminar’ as described by Narasimha & Sreenivasan (Reference Narasimha and Sreenivasan1973). Turbulent fluctuations remained in the boundary layer, but their dimensional magnitude remained approximately constant even as the flow accelerated. Similar behaviour has been noted in several studies (e.g. Blackwelder & Kovasznay Reference Blackwelder and Kovasznay1972). Escudier et al. (Reference Escudier, Abdel-Hameed, Johnson and Sutcliffe1998) noted that the frequencies associated with the fluctuating velocities also remained constant instead of scaling with the increasing free-stream velocity. Narasimha & Sreenivasan (Reference Narasimha and Sreenivasan1973) referred to this fixed dimensional magnitude as ‘frozen’ turbulence, and ‘frozen’ is used to refer to this behaviour below. Piomelli & Yuan (Reference Piomelli and Yuan2013) also noted that the streamwise Reynolds stress remained frozen at upstream values. They saw that the decrease of wall-normal and spanwise fluctuations was the main cause of inner-layer stabilisation, and that there were few ejections from the inner layer to the outer layer. As the fluctuations became increasingly small relative to the free-stream velocity, the mean flow approached a quasi-laminar state. While the turbulent fluctuations remain, their structure can change. Bourassa & Thomas (Reference Bourassa and Thomas2009) used quadrant analysis to show that an FPG reduces sweeps of high-speed fluid towards the wall relative to ejections of low-speed fluid moving away from the wall. They also noted an aliasing of quadrant-four events into quadrant three due to the streamwise pressure gradient, and found that this reduced the near-wall Reynolds shear stress.
At lower K the boundary layer remains turbulent with KRe
θ
≈ 0.001, and skin friction coefficient, C
f
= 2u
τ
2/U
e
2 ≈ 0.005, where u
τ
is the friction velocity and θ is the momentum thickness. The Clauser pressure gradient parameter,
$\beta = Re_{\delta^{*}}K/(C_{f}/2)\approx -0.6$
, where δ
* is the displacement thickness. These values appear to hold regardless of the K value. There is not a gradual change from the equilibrium turbulent condition to the laminar as K increases. Although turbulent, the Reynolds stresses are greatly reduced below those in a canonical zero-pressure-gradient (ZPG) boundary layer. Spalart (Reference Spalart1986) noted that turbulent near-wall streaks remained, but that patches of non-turbulent flow were present in the instantaneous flow field. Other studies have reported similar results. Castillo & George (Reference Castillo and George2001) examined cases from multiple studies and found that the mean velocity profile in defect coordinates was essentially the same for all equilibrium turbulent FPG boundary layers, regardless of the strength of the FPG. The Spalart (Reference Spalart1986) DNS results indicate that some differences in the defect profiles are apparent as K is varied, particularly in the inner region, but that the differences are vanishingly small for y/δ > 0.2, where y is the wall-normal distance and δ is the 99 % boundary-layer thickness. In inner coordinates the variation with K is clearer, with the mean profile rising above the ZPG law of the wall in the log region, as shown in studies such as Jones & Launder (Reference Jones and Launder1972), Spalart (Reference Spalart1986) and Dixit & Ramesh (Reference Dixit and Ramesh2008). The non-equilibrium region between the beginning of a sink flow FPG and the attainment of equilibrium depends on the upstream condition and the strength of the pressure gradient. In mild to moderate pressure gradients, studies such as Jones, Marusic & Perry (Reference Jones, Marusic and Perry2001) and Volino (Reference Volino2020) showed a monotonic progression from ZPG conditions towards an equilibrium sink flow. The mean velocity deficit gradually decreased while
$\beta$
increased in magnitude or remained nearly constant at the equilibrium value. In stronger FPG cases, Volino (Reference Volino2020) saw
$\beta$
dropping from higher magnitudes toward the equilibrium
$\beta$
= −0.6, and the mean velocity deficit decreasing from the ZPG value past the sink flow equilibrium and then increasing back towards the equilibrium value.
Many flows of interest occur on rough surfaces, and several studies have considered the combined effects of roughness and FPGs. Coleman, Moffat & Kays (Reference Coleman, Moffat and Kays1977) proposed conditions for equilibrium on rough walls, which required a variable K. Tachie & Shah (Reference Tachie and Shah2008) considered mild FPG cases and saw that in the near-wall region differences between the ZPG and FPG cases were small, with the roughness effect dominating over the pressure gradient effect. In the outer part of the flow, the FPG reduced the Reynolds stresses. Cal et al. (Reference Cal, Brzek, Johansson and Castillo2009) considered cases with K below 0.5 × 10–6. As with a smooth wall, the pressure gradient decreased the mean velocity deficit and the Reynolds stresses. Volino & Schultz (Reference Volino and Schultz2023) investigated rough-wall cases with constant K ranging from 0.125 × 10–6 to 2 × 10–6. When normalised using δ and u τ , the mean velocity and Reynolds stress results agreed with those from comparable smooth-wall cases, indicating that the outer-layer similarity between rough- and smooth-wall cases described by Townsend (Reference Townsend1976) for ZPG flows continues to apply for FPG conditions.
Only a few studies have considered rough-wall boundary layers with K> 3 × 10–6 (i.e. large enough to cause relaminarisation in a smooth-wall case). Yuan & Piomelli (Reference Yuan and Piomelli2015) reported DNS results for spatially developing boundary layers. The pressure gradient matched the smooth-wall experiment of Warnack & Fernholz (Reference Warnack and Fernholz1998), with K increasing from 0 to 4 × 10–6 and then decreasing back to zero. In the smooth-wall case, relaminarisation was observed. For the rough wall, the roughness Reynolds number, k + = ku τ /ν, with k the average roughness height, rose from 23 to 80, going from the transitionally rough regime to fully rough. For the surface simulated, the equivalent sandgrain roughness, k s , was equal to 1.6k. The acceleration increased the shear rate at the wall and caused the roughness effect to increase. The skin friction coefficient increased continuously as K increased, while k/δ increased as the boundary layer became thinner. The boundary layer did not relaminarise. The Reynolds stresses decreased relative to the increasing u τ 2, but still remained significant. The roughness resulted in turbulence production in the near-wall region, in spite of the strong FPG. Relaminarisation was only expected to occur if the roughness was small enough for the wall to be considered hydraulically smooth.
The rough-wall equilibrium sink flow case was considered by Yuan & Piomelli (Reference Yuan and Piomelli2014). In equilibrium, all dimensionless quantities, including k/δ, are constant. The DNS and large-eddy simulations, therefore, were equivalent to a spatially evolving boundary layer with k decreasing linearly in the streamwise direction to match the decreasing boundary-layer thickness. They considered K values from 0.45 × 10–6 to 2.50 × 10–6. The roughness modelled was sandgrain with k/δ between 0.015 and 0.04 and k + between 10 and 42, indicating transitionally rough conditions. Full reverse transition did not occur in any case, but a case with the strongest K and k + = 13 was close to hydraulically smooth conditions and showed some signs of relaminarisation.
While the equilibrium sink flow case is interesting, a more physically likely surface would have uniform roughness. When subject to a constant K acceleration, the ratio k/δ would continuously increase as δ decreased in the streamwise direction. Such a surface might approach a quasi-equilibrium as long as k/δ remained sufficiently small. If hydraulically smooth, relaminarisation could potentially occur. Eventually, however, as k/δ became continuously larger, any quasi-equilibrium would presumably be broken. The present study considers a spatially developing sink flow with uniform random roughness. The acceleration parameter is set to K = 3.2 × 10–6, which is large enough to cause relaminarisation on a smooth wall. It appears that such a case has not been reported in the literature. Two sizes of roughness are used, one intended to produce transitionally rough conditions and the other fully rough. A smooth-wall case is also included for comparison. The spatial development of the boundary layers is documented to show the progression from an initial ZPG condition.
2. Experiments
Experiments were conducted in the recirculating water tunnel described in Volino (Reference Volino2020). Water was supplied to the test section from a 4000 l cylindrical tank via two variable-speed pumps operating in parallel and a flow conditioning section consisting of a honeycomb and three screens followed by a three-dimensional contraction and a second honeycomb. Following the test section the water returned through a perforated plate into the cylindrical tank. The free-stream turbulence intensity was 0.3 % in the test section. The test fluid was filtered and deaerated water. A chiller was used to keep the water temperature constant to within 0.5 °C. The test section was 0.2 m wide, 2 m long and 0.1 m tall at the inlet. The bottom surface of the test section was the flat test wall and included a trip at the leading edge, as shown in figure 1. The upper surface consisted of four flat plates that could be independently adjusted to set the pressure gradient. The sidewalls and upper wall were transparent for optical access.
Roughness surface statistics. Parameters: ka , mean amplitude; k rms , root-mean-square height; kt , average peak-to-trough height; Sk, skewness; Fl, flatness; ES, effective slope; ks , equivalent sandgrain roughness height.
Cross-section of the test section in the streamwise–wall-normal plane. Numbers in the test section indicate streamwise measurement stations. Shown approximately to scale.
The inlet velocity to the test section was set to 0.5 m s–1 in all cases. A ZPG was set for the first 0.6 m downstream of the trip to establish a canonical ZPG boundary layer. This was followed by a 0.5 m long FPG section set for K = 3.2 × 10–6. The FPG was followed by an adverse-pressure-gradient section, as shown in figure 1, that was used as a diffuser before the flow was returned to the cylindrical tank.
The test wall consisted of a 0.23 m long smooth section followed by a replaceable downstream section where roughness could be included. An acrylic plate was used for the smooth-wall case. For the two rough-wall cases, additively manufactured plates with the mathematically generated three-dimensional random roughness described in Flack, Schultz & Volino (Reference Flack, Schultz and Volino2020) were used. The statistics for these surfaces are given in table 1, and the geometry is shown in figure 2. The two roughnesses differed in their skewness, Sk, with one having Sk = +1 and the other Sk = −1. The average equivalent sandgrain roughness was estimated based on the ZPG velocity profiles of Flack et al. (Reference Flack, Schultz and Volino2020), Volino & Schultz (Reference Volino and Schultz2022) and the present study using the relation between the roughness function, ΔU +, and k s :
\begin{equation} \Delta U^{+}=\frac{1}{\kappa }\ln\left(k_{s}^{+}\right)+B-8.5.\end{equation}
Elevation maps for sections of rough surfaces: (a) Sk = −1; (b) Sk = + 1. Height indicated by colour bar in mm.
The roughness function is the downward displacement of the log region of the mean velocity profile in inner coordinates below the smooth-wall law of the wall. The values used for κ and B are 0.384 and 4.17, respectively. The average k s values used below are 2 mm for the Sk = +1 surface and 0.6 mm for the Sk = −1 surface.
Velocity profiles were acquired at 13 streamwise stations along the spanwise centreline of the test section using a TSI FSA3500 two-component laser Doppler velocimeter. A four-beam fibre optic probe was used to collect data in backscatter mode. The beams entered the test section through one sidewall. A custom-designed beam displacer was used to shift one of the four beams, resulting in three co-planar beams that were aligned parallel to the test wall. A 2.6 : 1 beam displacer was located at the exit of the probe to reduce the size of the measurement volume to 45 μm in diameter and 340 μm long. The flow was seeded with 2 μm diameter silver-coated glass spheres. Data were acquired at 43 locations within each velocity profile using a traverse with 6.25 μm resolution to move the probe from the wall to the free stream. At each location, data were acquired for 10 000 large-eddy turnover times, δ/U e . The uncertainty in the mean streamwise velocity was 0.5 % of U e . The 95 % confidence interval uncertainty in the Reynolds stresses ranged from 1 % to 4 %. More details of the uncertainty analysis are available in Volino & Schultz (Reference Volino and Schultz2022).
Because of the pressure gradient and test section geometry, the free-stream velocity varied slightly in the wall-normal direction. The variation was very nearly linear. To determine the 99 % boundary-layer thickness, a straight line was fitted through the measured velocity points in the free stream for each profile, and a line at 99 % of the fit line was extrapolated to the wall. The location where the measured velocity intersected the 99 % line was taken as the boundary-layer thickness, δ.
The friction velocity was found from each velocity profile using the method described by Volino & Schultz (Reference Volino and Schultz2018) with an uncertainty of 5 %. The method is based on the streamwise momentum equation and uses the measured mean streamwise velocity and Reynolds shear stress profiles. The effective location of y = 0 for each profile was found by shifting the data in the wall-normal direction until the near-wall data agreed with a log law as closely as possible. The data were not forced to agree with the standard ZPG log-law slope, since the slope could be different in an FPG. The average shift was 0.09 mm for the smooth-wall case, 0.18 mm for the Sk = −1 rough-wall case and 0.85 mm for the Sk = +1 case.
3. Results
3.1. Comparison cases
To assess the state of the boundary layer with respect to equilibrium and relaminarisation, it is useful to have equilibrium sink flow comparison cases. As noted above, an exact solution exists for the mean velocity profile in the laminar case, and various quantities such as the skin friction coefficient can be found as functions of K. For turbulent cases, DNS solutions are available up to K = 2.75 × 10–6, but above this, turbulence cannot be sustained on a smooth wall. On a rough wall, however, the boundary layer can remain turbulent, so it is useful to know how dependent the solution is on K, and whether the DNS result at K = 2.75 × 10–6 might be useful for comparison with experiments at higher K. The results of Spalart (Reference Spalart1986) and Castillo & George (Reference Castillo and George2001), mentioned above, indicate that the mean profile for an FPG boundary layer is largely independent of K. This is illustrated and checked for other quantities in figure 3, which compares the equilibrium DNS results of Spalart (Reference Spalart1986) at K = 1.5 × 10–6, 2.5 × 10–6 and 2.75 × 10–6 with experimental data from the literature at various K values. Mean velocity profiles in defect coordinates are shown in figure 3(a). There is little difference between the DNS profiles. The experiments of Jones et al. (Reference Jones, Marusic and Perry2001) agree closely with the DNS, in spite of an order-of-magnitude difference in K. The smooth-wall results of Volino (Reference Volino2020) fall below those of the DNS. As discussed in Volino (Reference Volino2020), the Jones et al. (Reference Jones, Marusic and Perry2001) cases approached equilibrium monotonically, with Re θ increasing to the equilibrium value and the velocity defect dropping from the upstream ZPG condition to equilibrium at the location of the profiles shown. In the Volino (Reference Volino2020) cases, Re θ was dropping and the boundary layer had not yet reached equilibrium. Computations suggested that this boundary layer had overshot the equilibrium profile and would have risen back to match it if given further streamwise distance to develop. The rough-wall cases of Volino & Schultz (Reference Volino and Schultz2023) behaved similarly.
Comparison of experimental turbulent sink flow data of Jones et al. (Reference Jones, Marusic and Perry2001) (JMP, smooth wall), Volino (Reference Volino2020) (V, smooth wall) and Volino & Schultz (Reference Volino and Schultz2023) (VS, rough wall) with equilibrium sink flow DNS of Spalart (Reference Spalart1986). (a) Mean streamwise velocity, (b) streamwise Reynolds normal stress, (c) wall-normal Reynolds normal stress and (d) Reynolds shear stress.
The streamwise component of the Reynolds normal stress,
$\overline{u'^{2}}^{+}$
, is shown in figure 3(b). All of the experimental cases collapse for y/δ> 0.15 and agree best with the lowest-K DNS result. Close to the wall there is more variation between the experimental cases and K dependence for the DNS. The wall-normal component of the Reynolds stress,
$\overline{v'^{2}}^{+}$
, is shown in figure 3(c). There is clear K dependence for the DNS. The Reynolds shear stress,
$-\overline{u'v'}$
, in figure 3(d) shows less K dependence for y/δ> 0.2 in both the experimental and DNS results. The smooth-wall cases from Volino (Reference Volino2020) lie below the others, showing the same overshoot of equilibrium noted in the mean velocity profiles, but the other cases are in good agreement with each other and the DNS.
The results indicate that at least for the mean profile and the Reynolds shear stress, the equilibrium turbulent sink flow profiles do not depend strongly on K. The K = 2.75 × 10–6 DNS results are shown for reference in the figures below.
3.2. Present results with K = 3.2 × 10–6
Boundary-layer parameters for the three experimental cases are given in table 2. The momentum thickness Reynolds number and friction Reynolds numbers are shown in figure 4. In the smooth-wall case, Re θ drops from a ZPG value of 1100 to 600 at x/x sink = 0.4. The origin for x is taken as the start of the FPG. The expected equilibrium value for a turbulent boundary layer at K = 3.2 × 10–6 is about Re θ = 310. Downstream there is a more sudden drop to Re θ = 264 at the last station. The laminar equilibrium sink flow solution is Re θ = 0.3754K −0.5 = 210. The behaviour is similar in Re τ , dropping after x/x sink = 0.4 towards a value of about 184. The laminar equilibrium sink flow value is Re τ = 3.635K −0.25 = 86. This suggests the smooth-wall case is approaching but has not yet reached the laminar equilibrium state. In the rough-wall cases, Re θ drops from the ZPG value to about 600 in the Sk = −1 case and to about 900 in the Sk = +1 case. Both show a slight increase at the most downstream stations. The Re τ values for both cases remain fairly constant, with an increase in the Sk = +1 case at the downstream stations. The shape factor, H = δ */θ, provides another view of the state of the boundary layer, and is shown in figure 5. The H values are about the same for the smooth-wall and Sk = −1 cases from the ZPG region to x/x sink = 0.4. This suggests that roughness is small enough that it is not having a large effect on the boundary layer, which is consistent with the Reynolds numbers of figure 4, which are not very much higher for the Sk = −1 case compared with the smooth-wall case. The streamwise drop in H is consistent with expectations for a turbulent FPG boundary layer. The Sk = +1 roughness causes higher H but the behaviour is otherwise similar, with a drop from the upstream ZPG value. Downstream of x/x sink = 0.4, H rises in all cases. In the smooth-wall case it rises to H = 1.71, which compares with the laminar equilibrium sink flow value of H = 2.07. It appears that H rises in the rough-wall cases because the roughness effect is increasing as the boundary layer becomes thinner, while in the smooth-wall case it increases because the boundary layer is approaching a laminar-like state.
Boundary-layer parameters.
The results in figures 4 and 5 suggest that the boundary layer is relaminarising in the smooth-wall case and is approaching the expected equilibrium. The onset of relaminarisation appears to be somewhat abrupt, occurring near x/x sink = 0.4. In the rough-wall cases, the boundary layer appears to remain turbulent. This is supported by the skin friction coefficient, shown in figure 6. In the smooth-wall case, the upstream value of C f is 0.0043, which is consistent with expectations for a turbulent ZPG boundary layer at the corresponding Re θ . The value remains essentially constant, and at the downstream stations matches the expected laminar sink flow value of C f = 2.30K 0.5 = 0.0041. It is coincidental that the upstream and downstream values are equal. In the rough-wall cases, C f rises in the streamwise direction. It appears to be levelling off towards an equilibrium in the Sk = −1 case, but is still rising in the Sk = +1 case. Figure 7, which compares the roughness height with the boundary-layer thickness, may provide some explanation. In the Sk = −1 case, k s + rises from a transitionally rough value of 16 to a fully rough value of 100. The low upstream value is consistent with the similarity observed in figure 6 between this case and the smooth-wall case, while the downstream value is consistent with the boundary layer remaining turbulent while the smooth-wall case relaminarised. The ratio of δ/k s continuously drops, but is still above five at the last measurement station. Although the ratio is well below the recommended value of about 40 needed to ensure outer and inner scale separation (Jiménez Reference Jiménez2004), it may still be high enough that the boundary layer is able to reach a quasi-equilibrium sink flow condition. In the Sk = +1 case, k s + > 65, so fully rough behaviour is anticipated. The δ/k s ratio drops as low as two, which appears to be too low for establishment of a quasi-equilibrium. Hence the continuously rising C f in this case.
Momentum thickness Reynolds number (filled symbols) and friction Reynolds number (open symbols) for the present study.
Shape factor. Legend as in figure 4.
Skin friction coefficient. Legend as in figure 4.
Roughness height: (a) δ/k s ; (b) k s + . Legend as in figure 4.
3.3. Mean velocity
To show how the state of the boundary-layer changes, mean velocity profiles are considered next. Figure 8(a) shows the profiles for the smooth-wall case in inner coordinates. The streamwise stations correspond to the streamwise locations shown in figure 1 and to the points in figures 4–7. The profiles agree well with the turbulent law of the wall at the upstream ZPG stations. At stations 4–6, the profiles rise above the standard log law and appear to reach a new equilibrium, as expected for a turbulent sink flow (e.g. Jones & Launder Reference Jones and Launder1972). The sink flow profile for K = 2.75 × 10–6 from Spalart (Reference Spalart1986) is shown for comparison. In these coordinates the equilibrium profile depends on K, so it is reasonable that the experimental profiles should lie above the K = 2.75 × 10–6 result. Between stations 7 and 8, a change occurs. This location corresponds to the drop in Reynolds number at x/x sink = 0.4 seen in figure 4. The profiles continue to rise and appear to reach a new equilibrium at stations 12 and 13 that agrees with the laminar equilibrium sink flow solution.
Mean streamwise velocity profiles for smooth-wall case: (a) inner coordinates; (b) defect coordinates for stations 1–9; (c) defect coordinates for stations 9–13. Legend indicates streamwise station. DNS is K = 2.75 × 10−6 equilibrium sink flow simulation from Spalart (Reference Spalart1986). Here LS is laminar equilibrium sink flow solution. Arrows indicate progression of profiles in the streamwise direction.
The profiles are shown in defect coordinates in figures 8(b) and 8(c). The velocity defect decreases, as expected for an FPG, and appears to reach a temporary equilibrium at stations 7 to 11. These profiles lie below the DNS solution. The overshoot of the turbulent equilibrium result agrees with the cases of Volino & Schultz (Reference Volino and Schultz2023) in figure 3. Downstream of station 11 the profiles begin to rise but do not reach the laminar sink flow solution, consistent with the results of figures 4 and 5. Cal & Castillo (Reference Cal and Castillo2008) describe similar behaviour for smooth-wall cases.
Results for the Sk = −1 case are shown in figure 9. In inner coordinates the profiles are shifted upward by the roughness function given by (2.1). Since (2.1) is based on fully rough ZPG results, there is no reason to presume it must apply in the present transitionally rough FPG boundary layer, but it does collapse the results to within the experimental uncertainty. The FPG suppresses the wake seen at the ZPG stations. The profiles at most stations lie above the standard log law, in agreement with the DNS sink flow result. At stations 4–8 the near-wall data show signs of a viscous sublayer, which is possible since k s + < 32 at these stations. The outer region shows no sign of relaminarising at any location. In defect coordinates the profiles shift downward through station 9, agreeing with the smooth-wall results of figure 8. At stations 10–13 there is a very slight upward shift of the profiles. Since this shift is within the uncertainty, it may not be significant, but it could indicate a reversal of the overshoot and return towards the turbulent sink flow result. Another possibility is that the rise is caused by an increased roughness effect as δ/k s decreases. The mean profiles for the Sk = +1 case are shown in figure 10. The results are essentially the same as those of figure 9, with the exception that in inner coordinates the scatter between stations is larger, presumably because ΔU + is significantly larger.
Mean streamwise velocity profiles for Sk = −1 case: (a) inner coordinates; (b) defect coordinates. Legend as in figure 8. Arrows indicate progression of profiles in the streamwise direction through station numbers indicated.
Mean streamwise velocity profiles for Sk = +1 case: (a) inner coordinates; (b) defect coordinates. Legend as in figure 8. Arrows indicate progression of profiles in the streamwise direction through station numbers indicated.
3.4. Reynolds stresses
Profiles of
$\overline{u'^{2}}^{+}$
are shown for the smooth-wall case in figure 11. For stations 1–8, the inner peak is unchanged while the turbulence is suppressed in the outer region. These are expected results for a turbulent FPG boundary layer, and are consistent with the mean velocity profiles. For y/δ< 0.2 the data show the same drop below the DNS solution as in the fully turbulent cases of figure 3(b). Farther from the wall the data are above the DNS result. This is believed to be an artefact of the measurement technique. The actual free-stream turbulence level in this facility, as measured with a particle image velocimetry system, was about 0.3 %. As explained by DeGraaff (Reference DeGraaff1999), laser Doppler velocimetry systems with small probe volumes produce artificially high turbulence readings that scale with the local mean velocity. The resulting error is insignificant in comparison with the actual turbulence in most of the boundary layer, but is significant near the edge of the boundary layer, where the actual turbulence is low. The noise is uncorrelated in the two velocity components, so it does not affect the Reynolds shear stress.
Streamwise Reynolds normal stress profiles for smooth-wall case: (a) stations 1–9; (b) stations 9–13. Legend as in figure 8. Arrows indicate progression of profiles in the streamwise direction.
Wall-normal Reynolds normal stress profiles for smooth-wall case. Legend as in figure 8. Arrow indicates progression of profiles in the streamwise direction.
Downstream of station 8, the behaviour changes, as shown in figure 11(b). The peak drops, while the profiles rise somewhat in the outer region. This is the region where the mean profiles of figure 8 indicate relaminarisation. The change in the
$\overline{u'^{2}}^{+}$
profiles at the downstream stations is due mainly to the increase in u
τ
. In dimensional coordinates the profiles at the downstream stations do not change much. This is the quasi-laminar state described by Piomelli & Yuan (Reference Piomelli and Yuan2013) and others. The turbulence is essentially frozen, and the fluctuations acting across an increasingly large wall-normal mean streamwise velocity gradient result in significant
$u^{\prime}$
fluctuations. Bradshaw (Reference Bradshaw1967) referred to turbulent motions that do not produce shear stress as inactive, and this applies to much of the present
$u^{\prime}$
fluctuations, as indicated by the other components of the Reynolds stress. Since the motions are inactive, the mean profile can still relaminarise.
Figure 12 shows the wall-normal component of the Reynolds normal stress. The profile peaks decrease steadily in the streamwise direction. Figure 13 shows profiles of the ratio of
$\overline{u'^{2}}^{+}$
to
$\overline{v'^{2}}^{+}$
, illustrating the changing anisotropy of the turbulence. From stations 1 to 7 the ratio remains between 2 and 2.5 for y/δ> 0.2 and rises to a peak at y/δ ≈ 0.02, with the magnitude of the peak increasing from about 30 in the upstream ZPG to about 70 at station 7. Beginning at station 6 the near-wall peak begins to widen, and the widening becomes more rapid beyond station 8 and extends into the outer part of the boundary layer. This corresponds to the beginning of the relaminarisation, in which the inactive
$u^{\prime}$
fluctuations continue to scale with the increasing velocity gradient, while the
$v^{\prime}$
fluctuations remain frozen at the upstream magnitude. Since the active motions are of most importance for turbulent mixing and the development of the boundary layer, they are the focus of much of the discussion below.
Profiles of
$\overline{u'^{2}}/\overline{v'^{2}}$
ratio for smooth-wall case. Legend as in figure 8.
Reynolds shear stress profiles for smooth-wall case: (a) dimensionless; (b) dimensional. Legend as in figure 8. Arrow indicates progression of profiles in the streamwise direction.
Reynolds stress profiles for Sk = −1 case: (a)
$\overline{u'^{2}}^{+}$
; (b)
$\overline{u'^{2}}/\overline{v'^{2}}$
ratio; (c)
$-\overline{u'v'^{+}}$
; (d) dimensional
$-\overline{u'v'}$
. Legend as in figure 8. Arrows indicate progression of profiles in the streamwise direction through station numbers indicated.
Reynolds stress profiles for Sk = +1 case: (a)
$\overline{u'^{2}}^{+}$
; (b)
$-\overline{u'v'^{+}}$
. Legend as in figure 8. Arrows indicate progression of profiles in the streamwise direction through station numbers indicated.
The Reynolds shear stress, shown in figure 14, exhibits the same behaviour as
$\overline{v'^{2}}^{+}$
. Of note is the agreement of the profiles at stations 7 and 8, followed by the significant drop, particularly in the near-wall region, and the outward shift of the peaks farther downstream. The agreement at stations 7 and 8 may indicate an approach to a turbulent quasi-equilibrium, and the downstream drop corresponds to the beginning of relaminarisation. Figure 14(b) shows
$-\overline{u'v'}$
dimensionally. The profiles at the first 10 stations collapse. The FPG prevents
$-\overline{u'v'}$
from growing with the increasing wall shear, but it does not go away. It is large enough to maintain a turbulent boundary layer for some distance, but it becomes increasingly insignificant as the mean velocity increases. Eventually the level, particularly near the wall, becomes too small, and the boundary layer relaminarises. There is never agreement with the equilibrium turbulent DNS solution. At the upstream stations the boundary layer is still turbulent, but has not reached equilibrium, so the data lie above the DNS curve. Downstream, after relaminarisation begins, the data lie below the turbulent solution.
The Sk = −1 rough-wall Reynolds stress profiles are shown in figure 15. The
$\overline{u'^{2}}^{+}$
results are similar to those of the smooth-wall case in figure 11(a) and the fully turbulent cases of figure 3(b). The near-wall peak would be suppressed for fully rough conditions, but since the present case is transitionally rough, the peak is clear at some stations. This is in agreement with the appearance of a sublayer in the mean profiles of figure 8. At the downstream stations, the rough- and smooth-wall cases are different. In the near-wall region,
$\overline{u'^{2}}^{+}$
is higher for the rough-wall case, consistent with the boundary layer remaining turbulent. There is a slight rise in
$\overline{u'^{2}}^{+}$
between stations 9 and 13 in figure 15(a). This may be due to a closer approach to a turbulent quasi-equilibrium, or it may be due to an increased roughness effect as δ and δ/k
s
decrease.
The ratio of
$\overline{u'^{2}}$
to
$\overline{v'^{2}}$
is shown in figure 15(b). Similar to the upstream stations of the smooth-wall case, the ratio is about 2.5 for y/δ> 0.2 and rises to a peak near the wall. Unlike the smooth-wall case, the ratio does not rise at the downstream stations, because the boundary layer remains turbulent in the rough-wall case. The
$u^{\prime}$
and
$v^{\prime}$
fluctuations remain correlated and related to active motions.
The Reynolds shear stress for the Sk = −1 case is shown in figure 15(c). The
$\overline{v'^{2}}^{+}$
normal stress profiles are very similar to the
$-\overline{u'v'^{+}}$
profiles and are not presented. The behaviour is essentially the same as in
$\overline{u'^{2}}^{+}$
, again showing that
$u^{\prime}$
remains related to active motions in this case. The profiles drop below the equilibrium DNS at stations 7–9, and then rise slightly above it at stations 10–13. The dimensional
$-\overline{u'v'}$
is shown in figure 15(d), and it rises rapidly in response to the rising wall shear. Reynolds shear stress is clearly being produced in the turbulent boundary layer, unlike the frozen turbulence of the smooth-wall case of figure 14(b).
Results for the Sk = +1 case are shown in figure 16. Comparing with figure 15, the inner peak is more suppressed because of the larger roughness, and in the outer region there is less of a drop below the sink flow DNS at stations 7–9, perhaps because the larger roughness is mitigating some of the effect of the FPG and limiting the overshoot of the smooth-wall equilibrium. At the most downstream station, the larger roughness causes a higher
$-\overline{u'v'^{+}}$
peak in the Sk = +1 case. This may indicate a departure from quasi-equilibrium as δ/k
s
approaches two. The
$\overline{u'^{2}}$
to
$\overline{v'^{2}}$
ratio is essentially the same as for the Sk = −1 case and is not shown.
Primary production term for
$\overline{u'^{2}}$
: (a) smooth wall; (b) Sk = −1; (c) Sk = +1. Legend as in figure 8. Arrows indicate progression of profiles in the streamwise direction.
3.5. Production terms
The primary production term for the
$\overline{u'^{2}}$
Reynolds stress is shown in figure 17. In the smooth-wall case, there is measurable production to the edge of the boundary layer in the ZPG region, but it is suppressed to near zero in the outer half of the boundary layer by station 7, matching the equilibrium DNS. Bader et al. (Reference Bader, Pschernig, Sanz, Woisetschläger, Heitmeir, Meile and Brenn2018) also reported a drop in production for a smooth-wall case along with the frozen turbulence described above. The near-wall peak in figure 17 drops steadily, approaching the level of the DNS by station 7. This is followed by a drop below the DNS, and by station 12 the production is essentially zero everywhere in the boundary layer. This result is consistent with the relaminarisation observed in the mean flow and Reynolds stresses. In the two rough-wall cases, the same drop is observed in the outer region, but in the near-wall region the production appears to reach an equilibrium by about station 4. The data lie below the DNS for y/δ> 0.06, but the roughness results in increasing production as the wall is approached, while the smooth-wall DNS shows a drop to zero.
Figure 18 shows the primary production term for
$-\overline{u'v'}$
. As in figure 17, the production drops steadily in the smooth-wall case, dropping below the equilibrium DNS at 9 for y/δ> 0.1, and to near-zero values at the last station. With the rough wall, there is a rapid drop to an apparent equilibrium when the FPG is imposed. Values in the experiments are just above the equilibrium DNS for y/δ> 0.15, but are several times higher closer to the wall.
Primary production term for
$-\overline{u'v'}$
: (a) smooth wall; (b) Sk = −1; (c) Sk = +1. Legend as in figure 8. Arrows indicate progression of profiles in the streamwise direction.
3.6. Higher-order moments
The higher-order moments of the fluctuating velocity show essentially the same trends as the Reynolds stresses. Two examples that provide additional insight are shown in figures 19 and 20. The skewness of the streamwise component of the velocity,
$\overline{u'^{3}}/(\overline{u'^{2}})^{3/2}$
, is shown in figure 19. The
$u^{\prime}$
skewness is negative across most of the boundary layer, which indicates that strong negative
$u^{\prime}$
motions, which are associated with ejections of low-speed fluid away from the wall, are more significant than strong positive
$u^{\prime}$
motions, which are associated with higher-speed fluid moving towards the wall. In all cases the negative peak located at about y/δ = 0.9 decreases in magnitude in the streamwise direction, suggesting that the FPG suppresses the larger fluctuations reaching the outer region. In the near-wall region, a second negative peak emerges at the downstream stations. This peak suggests that as the FPG suppresses all of the turbulent fluctuations, the outward motions generated closer to the wall are possibly more persistent. In the smooth-wall case the near-wall peak is confined in the inner half of the boundary layer, while in the rough-wall cases it extends farther out. The
$v^{\prime}$
skewness (not shown) has the same behaviour as the
$u^{\prime}$
skewness, but has positive sign, again associated with motion of low-speed fluid away from the wall.
Skewness of
$u^{\prime}$
: (a) smooth wall; (b) Sk = −1; (c) Sk = +1. Legend as in figure 8. Arrows indicate progression of profiles in the streamwise direction.
Kurtosis of
$v^{\prime}$
: (a) smooth wall; (b) Sk = −1; (c) Sk = +1. Legend as in figure 8. Arrows indicate progression of profiles in the streamwise direction.
Figure 20 shows profiles of the kurtosis of the
$v^{\prime}$
fluctuations,
$\overline{v'^{4}}/(\overline{v'^{2}})^{2}$
. High kurtosis indicates the importance of strong fluctuations relative to the average. Beyond the edge of the boundary layer (y/δ> 1) the FPG suppresses the kurtosis, indicating less of an influence of the strong fluctuations extending into this region in all cases. Near that wall there is an increase at the downstream stations, particularly in the smooth-wall case. As the boundary layer begins to relaminarise in the smooth-wall case, the turbulence level drops, so the occasional stronger fluctuation stands out more and produces a higher kurtosis. There is also some sign of this rise in the rough-wall cases between y/δ = 0.5 and 1, but it less pronounced, possibly because the higher turbulence level in these cases makes the stronger fluctuations less pronounced.
3.7. Quadrant analysis
The boundary layer data were examined using quadrant analysis, as presented by Willmarth & Lu (Reference Willmarth and Lu1972). Quadrants 2 (Q2; ejections, u
′ < 0, v
′
> 0) and 4 (Q4; sweeps, u
′
> 0, v
′
< 0) contribute to negative
$\overline{u'v'}$
, and are most significant in turbulent boundary layers. Quadrants 1 (Q1; u′
> 0, v′
> 0) and 3 (Q3; u
′ < 0, v
′ < 0) have the opposite sign. Examination of the relative contributions from the quadrants can provide insights into the response of the boundary layer to pressure gradients and the behaviour during relaminarisation. Figure 21 shows profiles of the ratio of the combined contributions to
$-\overline{u'v'}$
from Q2 and Q4 to the contributions from Q1 and Q3. Under ZPG conditions, for both the smooth- and rough-wall cases, the ratio rises from one at the wall to a peak of about four at y/δ = 0.3. It then drops towards one as the free stream is approached. When the ratio equals one, the turbulent motions are uncorrelated, and the contributions from the four quadrants cancel each other, resulting in zero Reynolds shear stress. On the smooth wall, the peak starts to drop between y/δ = 0.2 and 0.7 at stations 6–8 in response to the FPG. The drop becomes more rapid and extends into the inner region as relaminarisation begins at stations 9–13. The drop in the ratio is consistent with the fluctuating velocities becoming less correlated as the boundary layer becomes more laminar-like. For the Sk = −1 case, a similar drop occurs in the centre of the boundary layer at stations 7–9, but the near-wall drop is not observed. It appears that when k
s
+ is sufficiently small, the boundary layer begins to respond as in the smooth-wall case, but the roughness sustains the turbulence near the wall. As ks
+ rises, the profile returns to the ZPG shape at the downstream stations. For the Sk = +1 case, conditions are fully rough, and there is no clear drop in the peak.
Profiles of ratio of contributions to
$\overline{u'v'}$
from (Q2 + Q4) and (Q1 + Q3) with hole size H = 0: (a) smooth wall; (b) Sk = −1; (c) Sk = +1. Legend as in figure 8. Arrow indicates progression of profiles in the streamwise direction.
Figure 22 shows the ratio of the contributions to
$-\overline{u'v'}$
of Q2 and Q4. With a ZPG, in all cases, the ratio rises from the wall to a local peak of about 1.4 at y/δ = 0.07. It then drops to a local minimum of about 1.2 at y/δ = 0.3 before rising to 2 at the edge of the boundary layer. It then drops to one as the turbulence becomes uncorrelated in the free stream. The FPG has a strong effect in suppressing the turbulence in the outer region, which reduces the influence of sweeps moving towards the wall. Ichimiya, Nakamura & Yamashita (Reference Ichimiya, Nakamura and Yamashita1998) also reported changes in sweeps and ejections for a smooth-wall case. In figure 22, this causes the Q2/Q4 ratio to rise in all cases between y/δ = 0.1 and 1 at stations 6–13. Near the wall there is a drop in the ratio at the most downstream stations of the smooth-wall case. As relaminarisation begins, it appears that ejections are suppressed near the wall, causing the Q2/Q4 ratio to drop below ZPG values at stations 11–13. In all cases the Q2/Q4 ratio is also high at the downstream stations for δ > 1. The contribution from all quadrants is small in this region, but it again appears that the FPG is suppressing the sweeps, as in the centre of the boundary layer. The observations appear to be consistent with the quadrant analysis results of Bourassa & Thomas (Reference Bourassa and Thomas2009).
Profiles of ratio of contributions to
$\overline{u'v'}$
from Q2 and Q4 with hole size H = 0: (a) smooth wall; (b) Sk = −1; (c) Sk = +1. Legend as in figure 8. Arrows indicate progression of profiles in the streamwise direction.
The hole size, H, can be used to investigate the effect of strong instantaneous fluctuating velocities and is defined such that
\begin{equation}\overline{u'v'}_{i}=\frac{\sum _{1}^{N}Iu'v'_{i}}{N},\quad \textit{where}\ I=\left[\!\!\begin{array}{c} 1\, \textit{if}\, u'v'\;\textit{in}\, \textit{quadrant}\, i\,\textit{and}\ | u'v'| \gt \big| H\overline{u'v'}\big| \\ 0\, \textit{otherwise}, \end{array}\right.\end{equation}
where i is the quadrant number and N is the number of instantaneous samples. When H = 0, all the data are used, and the results are as in figures 21 and 22. Figures 23 and 24 show the results of figures 21 and 22 with H = 5. Qualitatively the two figures are the same, but with the larger hole size the magnitudes are increased by a factor of about five. The same qualitative behaviour is seen with other hole sizes. The stronger events are more biased towards Q2 and Q4, which is expected since these quadrants are associated with the active motions that promote mixing.
Profiles of ratio of contributions to
$\overline{u'v'}$
from (Q2 + Q4) and (Q1 + Q3) with hole size H = 5: (a) smooth wall; (b) Sk = −1; (c) Sk = +1. Legend as in figure 8.
Profiles of ratio of contributions to
$\overline{u'v'}$
from Q2 and Q4 with hole size H = 5: (a) smooth wall; (b) Sk = −1; (c) Sk = +1. Legend as in figure 8.
3.8. Spectra
To examine the scales of the turbulence, spectra of
$-\overline{u'v'}$
are shown in figure 25. Results are shown for y/δ = 0.2, which is near the peak in the
$-\overline{u'v'}$
profiles at most stations. Frequencies were converted to wavenumber, k, using Taylor’s hypothesis with the local mean streamwise velocity. The wavenumbers and premultiplied spectra were normalised using δ and uτ
. In the smooth-wall case, the magnitudes of the spectra drop continuously in the streamwise direction, and the peak moves to lower dimensionless wavenumber. At the most downstream stations, y/δ = 0.2 is in the region where the u
′ and v
′ fluctuations are becoming uncorrelated, as shown in figure 21, so the spectral magnitudes drop towards zero. For a different perspective, figure 26 shows the same spectra dimensionally. The smooth-wall case spectra collapse, with peaks at a frequency, f, of about 8 Hz. As noted above, the turbulence in this case is essentially frozen, remaining at the same magnitude as the mean velocity increases.
Dimensionless spectra of
$-\overline{u'v'}$
at y/δ = 0.2: (a) smooth wall; (b) Sk = −1; (c) Sk = +1. Legend indicates station number.
Dimensional spectra of
$-\overline{u'v'}$
at y/δ = 0.2: (a) smooth wall; (b) Sk = −1; (c) Sk = +1. Legend indicates station number.
Spectra for the Sk = −1 case are shown in figures 25(b) and 26(b). Results are similar to the smooth-wall case at stations 1–7. Farther downstream, the relaminarisation seen in the smooth-wall case is not observed. The dimensionless spectra stop changing at stations 7–9, while the dimensional magnitudes and frequencies begin to increase. The turbulence appears to be frozen at first, similar to the smooth-wall case, but eventually the roughness causes the Reynolds shear stress to increase in proportion to the increasing wall shear. The spectra for stations 10–13 are not shown due to limitations of the measurements. The average laser Doppler velocimetry sampling rate was about 180 Hz, which was sufficient to resolve the energy-containing fluctuations through about station 8. At station 9, some attenuation is clear at the higher frequencies in figure 26(b), and farther downstream the attenuation increased, causing aliasing errors.
The Sk = +1 case shows a rapid drop of the dimensionless spectra at the beginning of the FPG, and no subsequent change. It appears that a turbulent equilibrium has been reached. The dimensional values show an increase in magnitude and frequency after station 5, similar to the Sk = −1 case, but starting farther upstream, presumably due to the larger roughness effect.
4. Conclusions
Boundary layers were subject to strong FPGs with K = 3.2 × 10–6. In all cases the acceleration caused a drop in the turbulence relative to the wall shear, particularly in the outer part of the boundary layer. Sweeps were suppressed relative to ejections originating near the wall. The sink flow boundary layer appeared to approach a turbulent equilibrium, which was similar to that observed in flows with more moderate K. At some locations the mean velocity and turbulence profiles dropped below the lower K equilibrium and then rose towards it from below.
After the initial adjustment to the FPG, the smooth- and rough-wall cases behaved differently. On the smooth wall, after Reθ had dropped to about 600 in the turbulent boundary layer, relaminarisation began. The turbulence level in the near-wall region dropped, and ejections contributed less relative to sweeps. The fluctuating velocities also lost correlation, causing the Reynolds shear stress to drop more than the normal stresses. In the outer flow the turbulence was essentially frozen, remaining at the values of the upstream ZPG region in both magnitude and frequency instead of rising with Ue and uτ . The turbulence eventually became insignificant, which resulted in a quasi-laminar boundary layer. The mean velocity profile, shape factor, skin friction coefficient and Reynolds numbers approached those of the analytical laminar sink flow solution.
The rough-wall cases showed no signs of relaminarisation, but the case with the smaller effective roughness did exhibit considerable similarity to the smooth-wall case at the upstream stations. This case had transitionally rough conditions at the upstream stations, with ks
+ < 20, and showed some signs of a viscous sublayer. The roughness sustained the turbulence near the wall, and the contribution to
$-\overline{u'v'}$
from ejections remained large relative to sweeps, in contrast to the smooth-wall case. The peak in the
$-\overline{u'v'}$
spectra remained at a dimensionless wavenumber, kδ, of about 0.3, indicating that smaller-scale eddies were being generated as the velocity increased. The case appeared to have reached a temporary equilibrium.
The case with larger roughness had similar behaviour, but conditions were fully rough, so no signs of a viscous sublayer appeared. At the downstream stations, the profiles were similar to those in the case with smaller roughness, but the Reynolds numbers, shape factor and skin friction coefficient were rising, indicating that this case was not in equilibrium. This was likely due to the drop of δ/ks to values below five. Since δ decreases continuously in a boundary layer with constant K, any surface with uniform roughness will eventually depart from equilibrium as δ/k s becomes small. Still, it is interesting that this departure was not large in the present rough-wall cases, even with δ/k s < 10 in both of them.
Funding
The authors thank the Office of Naval Research for providing financial support under Grant N0001425GI00573, and the United States Naval Academy Hydromechanics Laboratory and Project Support Branch for providing technical support.
Disclaimer. The views expressed in this article are those of the authors and do not reflect the official policy or position of the US Naval Academy, the Department of the Navy, the Department of Defense or the US Government.
Declaration of interests
The authors report no conflict of interest.
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