1 Introduction

Urban street canyons (“canyons” hereafter) are the spaces formed between buildings. They are often simplified in terms of geometry with height H, width W, and length L. When L is much larger than W, a two-dimensional (2D) canyon is formed. Due to their poor self-ventilation capability, 2D canyons tend to trap pollutants or heat [5, 27, 35]. For perpendicular wind (i.e., wind blowing in the direction normal to the canyon axis), Oke [31] has categorized flow patterns in 2D canyons in three regimes based on the canyon height-to-width aspect ratio (H/W). At H/W < 0.3, the isolated flow regime is expected, where the effects of the upwind building are isolated from the downwind building. At 0.3 < H/W < 0.7, the wake interference regime is expected, as the wake from the upwind building begins to interfere with the downwind building. At H/W > 0.7, the skimming flow regime is expected, with one or multiple wind-induced vortices formed in the canyons. Multiple vortices have been observed in deep canyons, i.e., canyons with high aspect ratios (H/W ≥ 1.5). The skimming flow, also known as cavity flow [16], is the most severe in terms of pollutant or heat trapping inside canyons [5, 22]. Therefore, this paper investigates perpendicular winds across 2D canyons in the skimming flow regime.

Three common approaches to study flow fields in canyons are reduced-scale experiments, in situ field measurements, and numerical modeling. Comprehensive reviews of street canyon modeling are available in Vardoulakis et al. [35] and Ahmad et al. [1]. They highlight that a major limitation of field measurements is the difficulty of decoupling the effects from simultaneously operating parameters in a real built environment. For example, it is difficult to decouple the buoyancy induced by a heated building surface from the measured mean flow [29, 30]. In contrast, reduced-scale experiments allow independent control of each parameter, such as temperature and wind speed. However, reduced-scale experiments cannot meet all the requirements of similitude, including geometric, kinematic, and dynamic similarities of full-scale measurements [2, 16]. The dynamic similarity in momentum requires matching the Reynolds numbers between two experiments. The Reynolds number, Re = UrefH/ν, is an important parameter to ensure fully turbulent flow, where Uref is a reference velocity and ν the kinematic viscosity of the fluid. In reduced-scale experiments, scaled-down model blocks with idealized geometry are used to represent buildings in a wind tunnel or water channel. These model blocks often have a geometrical length scale on the order of 10−1 m. The corresponding Re in reduced-scale experiments is on the order of 103–104. On the other hand, real-size buildings have H on the order of 101–102 m. With a reference wind speed of 2 m s−1, the corresponding Re is on the order of 105–106. In order to match this Re in a wind tunnel with a 100 × scale reduction, the wind speed required is 200 m s−1, which is no longer in the incompressible regime. The mismatch of Re in reduced-scale experiments is justified by the “Reynolds number independence” hypothesis, stating that as long as the Re in reduced-scale experiments exceeds a critical value, the normalized flow field remains invariant with increasing Re. This means that reduced-scale experiments that exceed a critical Reynolds number, Rec, could reproduce the flow fields at full scale, since increasing the Re does not alter the overall normalized flow fields. Assuming this hypothesis is true, the challenging question is to quantify Rec.

Next, we outline a few studies that attempted to quantify Rec. Castro and Robins [7] conducted wind tunnel tests to study flows across a single cube at different Re. They found that in a uniform flow, Re independence is achieved at Rec ~ 30,000. In a simulated atmospheric boundary layer flow, Re independence is achieved at Rec ~ 4000. Uehara et al. [34] obtained an Rec ~ 3500 for Re independence for a single cube, close to that of Castro and Robins [7]. Note that the freestream velocity ten cube lengths above the cube was taken as the reference velocity for Rec in Castro and Robins [7], while the velocity in an undisturbed flow (i.e., without the cube) at the height of the cube was taken as the reference velocity for Rec in Uehara et al. [34]. The Rec from both experiments are close to each other, meaning both the above-roof velocity and the velocity in an undisturbed flow are suitable as a reference velocity to calculate Rec. Lim et al. [23] conducted wind tunnel experiments and field measurements on a single cube to cover an Re range spanning two orders of magnitude, from 2 × 104 to 3 × 106. They concluded that the normalized mean velocity is independent of Re within the range of Re studied. As opposed to a single cube (height-to-width ratio of 1) with an Rec ~ 3500, a higher Rec ~ 8000 was required for a rectangular block with height-to-width ratio of 2 [34], suggesting Re independence criteria depend on the aspect ratio. Based on a uniform flow across a single cube, Snyder [33] proposed Rec = 11,000 as a conservative value. Nonetheless, we still lack strong evidence to support the similarity assumption [39]. Although Snyder [33] has warned that experiments with different geometries should derive their own Re independence criteria, Rec = 11,000 (or lower, since 11,000 is a “conservative” value) has been widely adopted without questioning its validity for other geometries. In Sect. 2, we will demonstrate the drawback of this assumption after a literature review of reduced-scale experiments and field measurements. The experimental setup is described in Sect. 3. Section 4 discusses the experimental result, followed by conclusions in Sect. 5.

2 Literature review

We first review reduced-scale experiments in wind tunnels and water channels (a summary will be provided in Table 1). Gerdes and Olivari [14] visualized the flow fields in canyons with H/W of 0.7 and 1.0 using Particle Image Velocimetry at an Re around 4000. They concluded that both values of H/W produced the same flow pattern, where a single vortex induced by the mean flow was observed. Using Particle Tracking Velocimetry in a water channel, Caton et al. [8] visualized the flow in a canyon with H/W of 1.0. The Re was relatively low, between 1000 and 3000. They observed a stable vortex in the canyon, too. Brown et al. [6] conducted a wind tunnel experiment using a 2D array of six canyons with H/W of 1.0. At an Re around 30,000, they observed a single vortex in each canyon. Two sets of wind tunnel experiments at Re between 8000 and 20,000 [28] and between 9500 and 19,000 [20] also revealed one vortex in canyons with H/W of 1.0. Water channel experiments conducted at an Re around 10,000 observed one vortex in canyons with H/W of 1.0 [4, 21], consistent with the wind tunnel experiments. Baik et al. [4] extended their experiments to canyons with H/W of 1.5, 2.0 and 2.4. In these deep canyons where H/W ≥ 1.5, they observed two vertically aligned, counter-rotating vortices. Experiments by Li et al. [21] on canyons with H/W of 2.0 at Re = 12,000 also showed two vortices. The top vortex driven by the freestream was stronger than the bottom vortex driven by the top vortex.

Table 1 Summary of reduced-scale experiments (E) and field measurements (F) arranged by canyon height-to-width aspect ratio (H/W)
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Next, we briefly review in situ field measurements in full-scale canyons. There are two major distinctions between reduced-scale experiments and full-scale field measurements. First, Re in reduced-scale experiments is on the order of 103–104, whereas Re in field measurements is on the order of 105–106. Second, the canyons in reduced-scale experiments are often ideal, whereas real canyons possess three-dimensional features [13, 18, 38] and small-scale structures such as roof shape [15, 36] and roof roughness [3]. Despite the non-idealities in real canyons, many field measurements still observed in-canyon vortices under perpendicular winds. Louka et al. [24] reported a single vortex in a canyon with H/W of 0.7. Yamartino and Wiegand [37] reported a single vortex in a canyon with H/W of 0.9 in Cologne, Germany. Nakamura and Oke [29] conducted a field study in a canyon with H/W of about 1.1 in Kyoto, Japan. They reported a single vortex with the highest strength at local standard time 1700 h. These field measurements with H/W near unity or lower are consistent with reduced-scale experiments, which also observed a single vortex. At higher H/W, however, disagreement between them arises. Recall that two vortices were observed in the reduced-scale experiments with H/W of 1.5 and 2.0 [4, 21]. However, in a full-scale canyon with H/W around 1.4, Louka et al. [25] observed only one vortex in the canyon. DePaul and Sheih [11] observed a single vortex in a canyon with H/W around 1.5 by tracking the trajectories of simultaneously released balloons. The core of the vortex was not at the mid-canyon height but shifted upward at about 0.7H above the ground. The single-vortex flow pattern was also observed in field measurements in canyons with H/W ≥ 2.0. In a canyon with H/W around 2.1, Eliasson et al. [12] observed one vortex under quasi-steady perpendicular wind. They also found the core of the vortex at about 0.7H above the ground, similar to DePaul and Sheih [11]. Measuring both wind and temperature in a canyon with H/W about 2.4, Santamouris et al. [32] concluded that a secondary vortex might be induced only when thermal effects (e.g., heated canyon wall) are important. Nonetheless, this does not explain the formation of the secondary vortices in the reduced-scale experiments, as they were run with isothermal flows with no heated surfaces. Overall, only one vortex is induced in full-scale canyons regardless of the H/W (in the range of H/W between 0.7 and 2.4).

This review is summarized in Table 1. For reduced-scale experiments (“E” in the second column), the above-roof-level velocities are well defined, as they are steady and controllable. Correspondingly, the Re is well defined. In field measurements, however, the above-roof-level winds are often unsteady. We take a wind speed of 2 m s−1 to calculate the Re, as some researchers show that a threshold wind speed of 2 m s−1 is required to sustain the vortex in full-scale canyons [11, 12]. For canyons with H/W around 1.0 or smaller, the flow patterns seem to be independent of Re: both reduced-scale experiments (Re on the order of 103–104) and field measurements (Re on the order of 105–106) observed a similar flow pattern with a single vortex. Even at an Re as low as 3000 [8], the single vortex flow field was still observed. In contrast, for canyons with H/W ≥ 1.5, two vortices were observed in reduced-scale experiments, whereas only one vortex was observed in field measurements. This suggests that flows in canyons with H/W ≥ 1.5 have two regimes that are Re dependent. Referring to Table 1, the transition from the double-vortex regime to the single-vortex regime occurs in the range 104 < Re < 106. Clearly, Rec = 11,000 does not apply to canyons with H/W ≥ 1.5, since increasing Re from 104 to 106 changes the flow fields from the double-vortex regime to the single-vortex regime. Unfortunately, if it exists, we are missing the link for the Rec between 104 and 106.

The third approach, numerical modeling, covers a wide range of Re up to 107, but numerical models must be validated by experiments or field measurements. Since the inputs to numerical models are controllable, reduced-scale experiments with similar controllable parameters are used for model validation [35]. Such numerical studies therefore make an a priori assumption that the numerical models must reproduce the double-vortex regime in reduced-scale experiments to be considered validated. For example, Baik et al. [4] validated their Computational Fluid Dynamics (CFD) model with their water channel experiments (H = 0.24 m) and concluded from the CFD simulations that the double-vortex flow exists in a full-scale canyon with H = 96 m. Another example is CFD models that predicted one vortex in full-scale deep canyons were “corrected” to produce the double-vortex regime by employing rough wall functions [19]. Although heated canyon walls could induce a single vortex in a deep canyon [26], under isothermal conditions, the review by Li et al. [22] summarizes that CFD models would predict a single vortex in canyons with H/W ≤ 1.0 and two vortices in canyons with H/W ≥ 1.5. Many CFD models have been validated to reproduce the double-vortex flow regime, which may be incorrect should this regime not apply at high Re. Therefore, we exclude numerical studies from Table 1.

With an increasing number of dense cities expanding vertically, more deep canyons will be created. Therefore, fundamental understanding of flow fields in deep canyons is urgently needed. To understand the fundamentals correctly, we should first establish the criteria for Re independence in deep canyons, instead of generalizing the criteria from experiments with a single cube. In fact, when providing the Rec = 11,000 criterion based on flow across a cube, Snyder [33] warned that “tests to establish Reynolds number independence should be an integral part of any model study.” Table 1 shows that reduced-scale experiments of deep canyons were performed at Re on the order of 104, while field measurements had Re on the order of 106. We are missing the link of Re between 104 and 106, where the transition from two vortices to one vortex occurs. Therefore, we conducted water channel experiments for deep canyons with Re up to 105 to fill the gap.

3 Experimental setup

All experiments were conducted in a recirculating water channel in the Hydraulic Engineering Laboratory at the Department of Civil and Environmental Engineering, National University of Singapore. Figure 1a shows the side-view sketch of the recirculating water channel, which is 15 m long, 0.6 m tall and 0.6 m wide. A valve controls the flow rate up to a maximum 50 L s−1. High flow velocity at the test section was needed to achieve Re of 105, so a partition was installed to reduce the effective width from 0.6 to 0.3 m at the test section. The test section measures 3.6 m long, 0.6 m tall and 0.3 m wide. Flow straighteners made with a combination of plastic tubes, wire mesh and honeycombs were arranged at the inlet to minimize span-wise and vertical velocities. Two layers of ceramic marbles (1.27 cm diameter) accelerated the flow development so the flow profile at the test section was fully developed. The dimensions of the honeycomb and ceramic marbles are shown in Fig. 1b. For the layout of the components upstream (plastic tubes, wire mesh, honeycombs and ceramic marbles), the reader is referred to [9]. The water depth was controlled by the floodgate. The scaled-down model blocks of buildings spanned the full test section width to simulate 2D canyons. Acoustic Doppler Velocimeters (ADV, Vectrino by Nortek AS), were used throughout the experiments to measure all three velocity components (stream-wise, vertical, and span-wise) at the centerlines of canyons 3–6, as shown in Fig. 1c. The ADV were mounted on metal frames with adjustable vertical position. The diameter of the probe is 8 mm. The sampling volume diameter is 6 mm. The accuracy is ± 1 mm s−1 ± 0.5% of measured value (e.g., if the measured value is 200 mm s−1, the accuracy is ± 1 mm s−1 ± 0.5% × 200 mm s−1 = ± 2 mm s−1).

Fig. 1
figure 1

a Side view of the recirculating water channel with an adjustable floodgate and a control valve to vary the flow rate and water level. The models of buildings were placed in the middle of the test section. b Dimensions of the honeycomb and ceramic marbles (top scale in inches, bottom scale in centimeters). c Centerlines of canyons 3–6, where the velocity profiles were measured

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The scaled-down model blocks of buildings were made of marine plywood coated with epoxy. Three sets of experiments were conducted at H/W of 1.0, 1.5 and 2.0. The dimensions of model blocks and canyons are summarized in Table 2. For canyons with H/W of 1.0, seven canyons of 0.12 m by 0.12 m were formed between eight model blocks of 0.12 m by 0.10 m. For canyons with H/W of 1.5, six canyons of 0.20 m by 0.13 m were formed between seven model blocks of 0.20 m by 0.12 m. For canyons with H/W of 2.0, six canyons of 0.20 m by 0.10 m were formed between seven model blocks of 0.20 m by 0.12 m. The measurement frequency was at least 50 Hz and the measurement period was at least 60 s, which have been verified to be sufficient to obtain both the mean velocities and turbulence based on the advection time scale [10]. The advection time scale can be estimated as H/Uref. A small advection time scale requires higher measurement frequency, while a large advection time scale requires longer averaging period. The measurement frequency was increased to 100 Hz for the case with the smallest advection time scale, while the measurement period was increased to 360 s for the case with the largest advection time scale.

Table 2 Dimensions of model blocks and canyons in the experiments at three aspect ratios, H/W
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Due to the limitation of the ADV probes, the spacing between two model blocks must exceed 0.06 m (i.e., W > 0.06 m). We modeled canyons with H/W of 1.0 with H = W = 0.12 m. For canyons with H/W of 2.0, we could not reduce W to 0.06 m (while maintaining H = 0.12 m) due to the probe limitation. Therefore, we modeled them with H = 0.20 m and W = 0.10 m. The water channel has a maximum water depth of 0.55 m. In terms of H, the maximum water depths for experiments with H/W of 1.0 (where H = 0.12 m) and H/W of 1.5 and 2.0 (where H = 0.20 m) are 4.58H and 2.75H, respectively. The water channel experiments in Li et al. [21] and Baik et al. [4] had water depths of 4H and 1.4H. Therefore, the water depth of 2.75H in our experiments is lower than that in Li et al. [21] but double that in Baik et al. [4]. The Froude number (Fr) based on the water depth and reference velocity was between 0.025 and 0.26. We conducted a sensitivity study on the effects of water depth, which will be discussed in Sect. 4.1.

4 Results and discussion

This section discusses the experimental results. All three components of velocity were measured. The mean velocities were calculated by time averaging the velocities over the whole measurement periods. The mean stream-wise velocity, u, and the mean vertical velocity, w, are reported. The mean span-wise velocity, v, was zero due to span-wise symmetry and is not plotted. We also report the Reynolds stress, u′w′, and the turbulence kinetic energy, TKE = 0.5(u′2 + v′2 + w′2), where u′, v′, and w′ are the root-mean-squared stream-wise, span-wise, and vertical velocity fluctuations, respectively. The mean velocities are normalized by a reference velocity, Uref, which is the freestream velocity at least 2H above the canyon ground, as recommended by Kastner-Klein et al. [17] and Li et al. [21]. The Reynolds stress and TKE are normalized by \({\text{U}}_{\text{ref}}^{2}\).

4.1 Canyon with aspect ratio of 1.0

Figure 2 summarizes the experimental results for canyons with H/W of 1.0. The water channel experiment in Li et al. [21] with H = W = 0.1 m and Re = 12,000 is included for comparison. The experiment in Li et al. [21] used Laser Doppler Velocimetry (LDV), which is non-intrusive; while our experiments used ADV, which is categorized as an intrusive measuring device. The agreement between their LDV and our ADV measurements verified that the presence of the ADV probes in our experiments did not affect the results. The runs with Re = 19,000 and Re = 62,000 had a water depth of 0.4 m = 3.3H, with corresponding Fr = 0.083 and Fr = 0.26. A sensitivity study on the effects of water depth was conducted with a water depth of 0.24 m = 2.0H with a corresponding Fr = 0.15 for the run with Re = 30,000. The case with Re = 12,000 in Li et al. [21] had a water depth of 4.0H. To ensure repeatability of the experimental setup, the case with Re = 19,000 was repeated two times on two separate days. All three runs produced identical results, with negligibly small run-to-run standard deviations (0.6% of Uref for mean velocities and 0.3% of Uref for velocity fluctuations), confirming the repeatability of the experimental setup. Brown et al. [6] and Meroney et al. [28] showed that in a series of canyons, the first two canyons exhibit different flow fields compared to the third and downstream canyons. We measured the velocity profiles in the third to sixth canyons, and verified that they are similar. For brevity, only the profiles in the fourth canyon are shown.

Fig. 2
figure 2

Normalized profiles for canyons with aspect ratio of 1.0. a Mean stream-wise velocity, b mean vertical velocity, c turbulence kinetic energy, and d Reynolds stress. Water channel data from Li et al. [21] are included in a and b. All profiles were measured at the centerline of the fourth canyon. The Re 12,000 case has a water depth of 4.0H, the Re 19,000 and 62,000 cases have a water depth of 3.3H, the Re 30,000 case has a water depth of 2.0H

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Figure 2a shows that u/Uref is Re independent in the range of Re between 12,000 and 62,000. The freestream flow in the positive x-direction (i.e., positive Uref) drives a clockwise rotating vortex, which induces negative u/Uref near the bottom. Reducing the water depth from 3.3H (the cases with Re = 19,000 and Re = 62,000) to 2H (the case with Re = 30,000) does not affect u/Uref profiles. Increasing the Fr from 0.083 to 0.26 does not affect u/Uref profiles, confirming the free surface effect is negligible even at the largest Fr = 0.26. Figure 2b shows w/Uref is positive inside the canyon (z/H < 1). This suggests that the vortex is asymmetric, with its core slightly offset from the center of the canyon. Both u/Uref and w/Uref are zero at the bottom, satisfying the no-slip condition. The mean span-wise velocity is zero (not shown).

Figure 2c shows that the normalized TKE remains almost constant inside the canyon (z/H < 1) except near the bottom, where TKE is slightly higher. At the bottom, TKE decays to zero, as expected. The TKE peaks at the roof level (z/H = 1) and decays with increasing z/H. The water depth has no significant effect on TKE inside the canyon, but the Re = 30,000 case with 2H water depth had lower normalized TKE above the roof level. The ADV probe could measure only 5 cm or deeper below the water surface, so no data was acquired near the water surface (above z/H = 1.6) for this case. Figure 2d shows that the normalized Reynolds stress, \({\text{u}}^{{\prime }} {\text{w}}^{{\prime }} /{\text{U}}_{\text{ref}}^{2}\), remains zero inside the canyon and above z/H = 2, but peaks at the roof level. The case with 2H water depth (Re = 30,000) shows a faster decay of Reynolds stress with increasing z/H, possibly due to the smaller water depth. Note that the case with Re = 30,000 and 2H water depth (which shows a faster decay of TKE and Reynolds stress above the roof level) had Fr = 0.15. The case with Re = 62,000 had a larger Fr = 0.26, but does not show such decay. This suggests that the water depth (relative to the canyon height) plays a more important role on this decay when Fr is smaller than 0.26.

In this study, we consider Re independence is achieved when the mean velocity profiles do not change with increasing Re. Overall, all profiles in Fig. 2a–d show no apparent change in the flow pattern from Re = 12,000 to Re = 62,000, confirming that flows in canyons with H/W of 1.0 achieve Re independence at Re > 11,000. This criterion for Re independence is widely adopted in the literature [4, 21, 28, 33]. On the effects of water depth, reducing the water depth from 3.3H to 2H did not change the mean velocity profiles, but the normalized TKE and Reynolds stress are slightly reduced above the roof level. Since this study investigates Re independence based on the mean velocities and the number of vortices inside a canyon, lower normalized TKE and Reynolds stress above the roof level are not a concern. However, for studies where above-roof turbulence is important, a water depth larger than 2H and Fr smaller than 0.15 are recommended.

4.2 Canyon with aspect ratio of 1.5

Figure 3 summarizes the experimental results for canyons with H/W of 1.5. All runs had a water depth of 2.75H and Fr ≤ 0.22. The field measurement from DePaul and Sheih [11] is included in the mean profiles (a) and (b) for comparison. Note that DePaul and Sheih [11] adopted the wind speed at 1.3H as their reference velocity. To directly compare their data with ours (we used the freestream velocity above 2H as Uref), we scaled the normalized velocity in DePaul and Sheih [11] by a factor of 0.6, since the velocity at 1.3H is about 60% of the velocity at 2H. Figure 3a shows that above z/H = 0.3, the profiles of u/Uref do not change significantly with increasing Re. On the other hand, below z/H = 0.3, u/Uref is the lowest at the smallest Re = 12,000. When Re is increased to 31,000, the magnitude of u/Uref increases. A further increase of Re to 58,000 results in a further increase of u/Uref in magnitude. At the highest Re = 106,000, the profile is similar to that of Re = 58,000. This suggests that the Rec for the flow to be Re independent is between 31,000 and 58,000 for canyons with H/W of 1.5. Our u/Uref profiles for the two largest Re show good agreement with the field measurement, which has an Re around 1.4 × 107. In addition, all u/Uref profiles show negative u/Uref near the bottom, indicating that there is only one vortex in the canyon. If there were a second vortex, the velocity near the bottom would be positive, as the bottom vortex would rotate counterclockwise. Figure 3b shows that in general, w/Uref is very small and does not depend on Re. The larger mismatch between our experimental profiles with the field measurement is due to the asymmetric canyon in the latter. The canyon studied by DePaul and Sheih [11] is a “step-up” canyon [35] with a shorter upwind wall (33.5 m) and a taller downwind wall (40 m). The vertical velocity is more sensitive to the asymmetry of a canyon than the stream-wise velocity [10]. Figure 3c, d show that the normalized TKE and Reynolds stress profiles do not depend on Re, with their peaks recorded near the roof level. Overall, the canyon with H/W of 1.5 at Re ≥ 58,000 shows very similar flow profiles to that of a canyon with H/W of 1.0. Since further increasing the Re to 106,000 did not result in a significant change in the u/Uref profile, our experiments suggest that Rec is between 31,000 and 58,000 for canyons with H/W of 1.5.

Fig. 3
figure 3

Normalized profiles for canyons with aspect ratio of 1.5. a Mean stream-wise velocity, b mean vertical velocity, c turbulence kinetic energy, and d Reynolds stress. The field study in a and b had an Re about 1.4 × 107 [11]. All profiles (except the field study) were measured at the centerline of the fourth canyon

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Next, we will demonstrate that a higher canyon height-to-length ratio (H/L) in our experiments does not affect the outcome. Recall that canyons with H/W of 1.0 have H = 0.12 m, while canyons with H/W of 1.5 have H = 0.20 m (see Table 2). The canyon length, L, is 0.3 m, so increasing H from 0.12 to 0.20 m increases H/L from 0.40 to 0.67. To ensure that this increase in H/L does not affect the results, we repeated the experiments with Re at 12,000, 31,000 and 58,000 with canyons of H = 0.12 m and W = 0.08 m (i.e., H/L of 0.40, with H/W maintained at 1.5). All three sets of experiments produced similar results at H/L of 0.40 and 0.67, verifying that the increase in H/L has negligible effects on the experiments.

4.3 Canyon with aspect ratio of 2.0

Figure 4 summarizes the experimental results for canyons with H/W of 2.0. All runs had a water depth of 2.75H and Fr ≤ 0.22. The field measurement of a full-scale canyon with H/W of 2.1 in Eliasson et al. [12] is included for the mean velocity profiles (only the start hour 16 of Fig. 4 in Eliasson et al. [12] was selected, as it represents a perpendicular wind with negligible along-canyon wind). Figure 4a shows that u/Uref is independent of Re above z/H = 0.4. Below z/H = 0.4, the case with the lowest Re = 12,000 has almost zero velocity while the high Re cases (≥ 57,000) show u/Uref of about − 0.1. As u/Uref is the most important parameter to determine the number of vortices in the canyon, we will discuss u/Uref in detailed in the next paragraph. Figure 4b shows that there is no apparent change in w/Uref with increasing Re. Our experiments show smaller magnitudes of w/Uref compared to the field measurement but the difference is small. Figure 4c shows that the normalized TKE is the highest between z/H = 1 and z/H = 2. Above z/H = 2, TKE decays rapidly. Above z/H = 0.4, the normalized TKE is independent of Re. Below z/H = 0.4, the lowest Re case has the smallest normalized TKE. Increasing Re to 28,000 increases the normalized TKE below z/H = 0.4. Further increasing the Re to 57,000 and higher did not change the overall profiles, suggesting that only the case with the lowest Re = 12,000 has a different flow structure in the canyon. Figure 4d shows that the normalized Reynolds stress is near zero inside the canyon (z/H < 1) and increases rapidly at the roof level, with peaks between z/H = 1 and z/H = 1.5. There is no apparent change of normalized Reynolds stress with increasing Re.

Fig. 4
figure 4

Normalized profiles for canyons with aspect ratio of 2.0. a Mean stream-wise velocity, b mean vertical velocity, c turbulence kinetic energy, and d Reynolds stress. The field study in a and b had an Re about 2.7 × 106 [12]. All profiles (except the field study) were measured at the centerline of the fourth canyon

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Figure 5 replots u/Uref in Fig. 4a by zooming into the in-canyon region (z/H < 1) and using a smaller x-axis range to emphasize the trend with increasing Re. The field measurement in a real canyon of H/W of 2.1 has an Re about 2.7 × 106 [12]. From Table 1, there is only one vortex in the real canyon with Re on the order of 106, whereas there are two vortices in the reduced-scale canyons with Re on the order of 104. Our experiments cover a range of Re between 104 and 105. Two interesting features can be observed in Fig. 5. First, at the lowest Re = 12,000, the peak negative u/Uref is near z/H = 0.6. The magnitude of u/Uref decreases with decreasing z/H and approaches zero near z/H = 0.3. Between z/H = 0.1 and z/H = 0.2, there is very small positive u/Uref recorded. If there is a single vortex in a canyon (with positive freestream velocity), the near-ground velocity would be negative, similar to the cases with H/W of 1.0 discussed in Sect. 4.1 (see Fig. 2a). The u/Uref profile at the lowest Re = 12,000 suggests that two vortices are induced, but the bottom vortex is significantly weaker. The top vortex occupies the top part of the canyon between z/H = 0.4 and z/H = 1, while the bottom vortex occupies the bottom part of the canyon below z/H = 0.4. The core of the top vortex (where u/Uref approaches zero) is located near z/H = 0.75, close to the location in the reduced-scale experiments (z/H = 0.78) in Baik et al. [4].

Fig. 5
figure 5

Normalized mean stream-wise velocity for canyons with aspect ratio of 2.0. The field study is from Eliasson et al. [12] and has an Re about 2.7 × 106. At the lowest Re = 12,000, flow below z/H = 0.3 is very weak and could be positive, indicating a bottom vortex. Re independence is achieved between Re = 57,000 and Re = 87,000, since increasing Re from 87,000 to 105,000 does not result in a change in u/Uref profiles

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The second feature is the sign of near-ground u/Uref, which helps to determine the number of vortices in a canyon. Negative near-ground u/Uref indicates that there is only one vortex in the canyon. The transition to the single vortex flow regime is observed at Re = 28,000. The profile at Re = 28,000 shows a higher scatter, suggesting the transition could be unstable with intermittent switching between single vortex and double vortex. The profiles with Re = 57,000, Re = 87,000, and Re = 105,000 show negative near-ground u/Uref, indicating that only one vortex was induced in the canyon in each case. The cores of these single vortices are located near z/H = 0.75, within the range of the field measurement (between z/H = 0.63 and z/H = 0.85) [12]. The case with Re = 28,000 has its peak negative u/Uref near z/H = 0.4. The case with a higher Re = 57,000 has its peak negative u/Uref shifted downward near z/H = 0.2. Further increasing Re to 87,000 shifts the peak negative u/Uref to near the ground level. The case with the highest Re = 105,000 also has its peak negative u/Uref near the ground. Overall, the lowest Re case shows u/Uref profile with a top vortex and a very weak bottom vortex, while all other cases show u/Uref profiles with a single vortex. Below z/H = 0.4, increasing Re from 12,000 to 87,000 changes u/Uref profiles in terms of the locations of their peak magnitude. Since further increasing Re to 105,000 did not result in a significant change in the u/Uref profile, our experiments suggest that Rec is between 57,000 and 87,000 for canyons with H/W of 2.0.

4.4 Comparison of canyons with different aspect ratios

The previous subsections discuss the results of canyons with H/W of 1.0, 1.5 and 2.0 separately. At high Re, there is only one vortex in canyons with H/W of 1.5 and 2.0, similar to canyons with H/W of 1.0. This means the flow fields in deep canyons could be similar to that of the well-studied canyons with H/W of 1.0, provided the Re exceeds a critical value. To make a direct comparison at different H/W, Fig. 6 plots the normalized profiles at the lowest and highest Re in canyons with H/W of 1.0, 1.5 and 2.0. The first points nearest to the ground were removed to ease comparisons, as all cases have these points very close to zero (satisfying the no-slip condition at the ground). For H/W of 1.0, the low Re and high Re cases have Re = 19,000 and Re = 62,000. For H/W of 1.5, the low Re and high Re cases have Re = 12,000 and Re = 106,000. For H/W of 2.0, the low Re and high Re cases have Re = 12,000 and Re = 105,000. Open symbols represent low Re cases, while filled symbols represent high Re cases.

Fig. 6
figure 6

Normalized profiles at aspect ratios (H/W) of 1.0, 1.5 and 2.0, comparing cases with the lowest and highest Reynolds numbers at each aspect ratio. a Mean stream-wise velocity, b mean vertical velocity, c turbulence kinetic energy, and d Reynolds stress

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Figure 6a shows that for canyons with H/W of 1.0, the overall profiles remain unchanged from low Re to high Re, although the near-ground u/Uref magnitude is slightly higher in the high Re case. There is only one vortex in the canyon, and the peaks of u/Uref (in magnitude) are near the ground at both low Re and high Re. In contrast, for canyons with H/W of 2.0, the low Re case has a different profile than the high Re case. At low Re, the top vortex occupies the canyon space between z/H = 0.4 and z/H = 1.0. Below z/H = 0.4, u/Uref is near zero, so there is either no flow or a very weak bottom vortex in this region. On the other hand, at high Re, the u/Uref profile shows negative near-ground u/Uref, with its peak (in magnitude) near the ground. A similar trend is observed for the canyons with H/W of 1.5. Comparing u/Uref profiles at high Re, although all H/W cases exhibit profiles resembling a single vortex, there is a major difference: the location of the core of the vortex. For canyons with H/W of 1.0, the vortex core is near the mid-canyon height at z/H = 0.55, whereas for canyons with H/W of 1.5 and 2.0, the cores are located near z/H = 0.70 and z/H = 0.75. Figure 6b shows that w/Uref depends on H/W but not Re. Figure 6c shows that the normalized TKE profiles are almost the same, except the case with H/W of 2.0 at low Re. Below z/H = 0.4, the normalized TKE is lower at low Re. This is again due to a very weak flow in the lower region at low Re. Lastly, the normalized Reynolds stress profiles in Fig. 6d do not show any relationship with H/W or Re. All profiles show negligible Reynolds stress inside the canyon (z/H < 1) and peak between z/H = 1 and z/H = 1.5.

5 Conclusions

We revisited the critical Reynolds number (Rec) required to satisfy Reynolds number (Re) independence with 2D idealized canyons at aspect ratios (H/W) of 1.0, 1.5 and 2.0. From the literature, only one vortex is induced by the mean flow in canyons with H/W near unity or lower for both reduced-scale experiments (Re on the order of 103–104) and field measurements (Re on the order of 105–106). In contrast, for deep canyons with H/W ≥ 1.5, reduced-scale experiments (Re on the order of 104) observe two vortices while field measurements (Re on the order of 106) report only one vortex. There were no data available for deep canyons with Re on the order of 105. We conducted water channel experiments up to Re ~ 105 to fill this gap. The key observations are summarized in Table 3. For canyons with H/W of 1.0, the in-canyon flows are independent of Re at Re above 12,000, confirming the widely adopted Re independence criterion (Rec = 11,000). For canyons with H/W of 1.5, Re independence is achieved between Re = 31,000 and Re = 58,000 in our experiments. For canyons with H/W of 2.0, Re independence is achieved between Re = 57,000 and Re = 87,000. At the highest Re, only one vortex was observed in each case, regardless of the H/W. A direct comparison of the mean stream-wise velocity profiles among all cases shows that the flow structures in deep canyons at Re above 105 are similar to that of canyons with H/W of 1.0, but the cores of the vortices in deep canyons are shifted upward.

Table 3 The critical Reynolds number (Rec), the highest Reynolds number (Re) in our experiments, the number of vortices observed at the highest Re, and the location of the cores of these vortices at aspect ratios (H/W) 1.0, 1.5, and 2.0
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From the above observations, we made two conclusions. First, Rec = 11,000 for Re independence is satisfied only for canyons with H/W ≤ 1.0. For canyons with H/W of 1.5, our experiments show that Re = 58,000 is sufficient for Re independence. For canyons with H/W of 2.0, Re = 87,000 is sufficient for Re independence. This suggests that the Rec required for Re independence is a function of H/W. A more appropriate length scale should include both H and W in the definition of Re. For example, the hydraulic diameter HW/(W + 2H) = H/(1 + 2H/W) could be a suitable length scale. The second conclusion from our experiments is that at high Re, there is only one vortex in deep canyons, consistent with full-scale field measurements. This is contrary to reduced-scale experiments that reported two vortices (at low Re). Since full-scale deep canyons exhibit a single-vortex flow pattern, we should revisit studies of deep canyons that were based on the double-vortex flow regime. CFD models are often validated with low Re experiments that exhibit the double-vortex flow regime, and thus some “validated” CFD models predict two vortices in deep canyons. We provide experimental data for deep canyons at a range of Re up to 105 and encourage the revalidation of CFD models to correctly predict a single vortex in deep canyons at high Re.