# Water Tank Experiments on Stratified Flow over Double Mountain-Shaped Obstacles at High-Reynolds Number

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## Abstract

**:**

## 1. Introduction

- Test the applicability of wave interference theory to waves trapped on the inversion in dependence on the ratio of ridge separation distance (V) to horizontal lee-wave wavelength (λ);
- Examine the influence of the secondary obstacle height and reproduce conditions under which waves in the lee of the obstacle are totally cancelled;
- Examine the influence of lee-wave interference on rotors;
- Reproduce hydraulic jump rotors and examine the influence of secondary obstacles on them;
- Examine the inner structure of rotor turbulence.

## 2. Methods

#### 2.1. Experimental Design

^{−1}for the different experiments.

_{0i}and x = L

_{0i}+ 2 cm. Here, subscript i denotes the obstacle (first or second), L is the characteristic length scale of the obstacle shape and L

_{0i}is the actual width of each obstacle. In total, four different obstacles were used in laboratory experiments, as indicated in Table 1. Unlike the quasi two-dimensional obstacles in Knigge et al. [32], the obstacles used in these experiments are fully two-dimensional (2D) and fill the lateral extent of the part of the tank in which they were pulled (Figure 1d). Still, the tallest obstacle is the same height and length as in Knigge et al. [32], and it was used for all the single obstacle experiments and as the upstream obstacle in all the double obstacle experiments. The other obstacles were used in experiments examining the influence of secondary obstacle height, and they correspond to three-thirds, two-thirds and one-third of the primary obstacle height.

_{0}is the density of the lower neutrally stratified layer capped by a density jump Δρ, Z

_{i}is the mean height of the density jump (i.e., inversion), and g is the acceleration due to gravity. The density increases linearly above the density jump. Use of scaled variable (e.g., Froude number F

_{r}, non-dimensional inversion height H

_{1}/Z

_{i}) allows for transferability of the results of water tank measurements to the atmosphere [32].

_{2}/H

_{1}ranging from 0 to 1, cf. Table 1) for a range of Froude numbers F

_{r}extending from 0.17 to 1.4, and first obstacle height to inversion height ratio H

_{1}/Z

_{i}from 0.29 to 1.31. The non-linearity parameter NH

_{1}/U in the upper stably stratified layer ranged between 0.3 and 2, with a median value of 0.65. For most of the experiments (85%) the flow in upper layer was linear (NH

_{1}/U < 1) and non-hydrostatic since NL/U was below 0.1 for all experiments.

_{2}/H

_{1}, H

_{1}/Z

_{i}and F

_{r}were conducted; not all of the 395 experiments are unique. The full list of all the experiments is given in Table S1 (Supplementary Materials).

#### 2.2. Data Processing

_{1}) and downstream of the second peak (A

_{2}) were calculated as the difference between minimum and maximum interface displacement within the relevant horizontal region (Figure 3b). The wavelength (λ

_{1}and λ

_{2}) was determined as a physical distance between the consecutive minima (or maxima) of the first wave downstream of each obstacle. This definition was sometimes ambiguous, since the wavelength in some experiments evolved further downstream. From these variables, dimensionless ratios, governing lee-wave interference, were calculated following Stiperski and Grubišić [29]. These are:

- Amplitude ratios A
_{2}/A_{1}or A_{2}/A_{s}, where the numbers denote the obstacle downstream of which the amplitude was calculated (1, 2 and s for first, second and single, respectively). - Mountain height ratio H
_{n}, defined as the ratio of second to first obstacle height H_{2}/H_{1}. - Dimensionless inversion height H
_{1}/Z_{i}, defined as the ratio between first obstacle height and the inversion height. - Dimensionless wavelength (V/λ), defined as ratio between valley width V, taken as the distance between the ridges of the obstacles, and the lee-wave wavelength.

#### 2.3. Flow Classification

_{max}are able to propagate vertically through the inversion and are then reflected at the free surface of the water tank. Since their trapping is not directly linked to the inversion, their characteristics are not governed by Froude number and dimensionless inversion height but by the stratification of the upper water layer and the water depth. These long waves were classified as trapped lee waves in [32], where they were also shown to be able to produce lee wave rotors.

## 3. Results

#### 3.1. Flow over an Isolated Obstacle

_{1}/U in the upper layer was not perfectly constant for all the experiments and therefore does not correspond exactly to the one used in [35]. This might allow some trapped waves to exist beyond the limiting line. The separation between trapped lee waves and hydraulic jumps fits qualitatively well to that from regime diagrams of both Knigge et al. [32] and Vosper [35], but is best described by the limiting line of Sachperger et al. [41]. We can therefore conclude that the differences in obstacle shapes and flow characteristics do not have important repercussions for our results, providing confidence in analyzing experimental results for double obstacles.

#### 3.2. Flow over Double Obstacles

_{1}/U (and therefore also the limiting line separating the two) is similar. In addition, flushing of the valley atmosphere (cf. [16]) is observed and is associated with waves whose wavelength is similar to or larger than the ridge separation distance and these are shown in Figure 6 with cross symbols.

_{r}and H

_{1}/Z

_{i}) corresponds to long waves for a single obstacle and waves that propagate through the interface in the atmosphere according to [35]. The presence of a secondary obstacle thus forces a decrease of lee-wave wavelength that corresponds to a flow regime transition. An example of such a regime transition for H

_{1}/Z

_{i}= 0.9, Fr = 1.1 is given in Figure 7. Downstream of a single obstacle, this combination of Froude number and dimensionless inversion height results in long waves caused by wave reflection off the free water surface and is not governed by the trapping on the inversion. On the other hand, this same profile over double obstacles, leads either to valley flushing in between the obstacles and long waves downstream of them over a narrow valley or to large-amplitude trapped lee waves both in between the obstacles and downstream of them over a wide valley. It is important to note that valley flushing (cf. [16]) is not exclusive to long waves that do not undergo regime transition, but occurs also for trapped lee waves.

_{s}), but the regime transition is also strongly modulated by the second obstacle height. For the case where flow transitions to trapped lee waves, these large-amplitude trapped waves persist in between the obstacles for all mountain height ratios examined. Downstream of the second obstacle, however, the transition to trapped lee waves depends on mountain height ratio. When the second obstacle is low (e.g., H

_{n}= 1:3), the regime transition does not occur downstream of it and flow there again has the form of long waves. It is possible that this regime transition occurs due to a decrease of lee-wave wavelength over the valley, caused by the presence of the second obstacle. These new shorter waves over the valley are no longer able to propagate through the inversion and are therefore trapped there causing their amplitude to increase. Stiperski and Grubišić [29] have observed the modulation of lee-wave wavelength by double obstacles caused by the fact that the Fourier transform of double obstacles has distinct maxima. A detailed study of this phenomenon is, however, beyond the scope of this paper but will be addressed in a future publication.

_{r}and H

_{1}/Z

_{i}[41] and complex topography through the relation between lee-wave wavelength and ridge separation distance. This non-linearity in flow response and differences in the flow character are also clear from the difference in interface height between single obstacle and double obstacle experiments in Figure 7. Flow over single obstacle is blocked upstream so that the interface rises high over the obstacle, whereas for double obstacle experiments flow is not blocked but undergoes supercritical transition leading to a lower interface height.

_{r}and H

_{1}/Z

_{i}, different for the two experiments) that causes the waves downstream of the first obstacle to be in or out of phase with those generated by the second obstacle, over the same ridge separation distance. The second obstacle height also considerably modifies lee-wave amplitude and wavelength (Figure 9b). In this example, the effect of the second obstacle is maximized for larger H

_{2}as the higher second obstacle causes a larger reduction of wave amplitude. This, however, is not always the case as will be discussed later in this section.

_{2}/A

_{s}increases beyond 1 for dimensionless wavelength (V/λ

_{s}) close to integer, and is smaller than 1 for V/λ

_{s}close to half-integer (Figure 10a) which is a clear evidence of an interference pattern suggested by Scorer [15] and Stiperski and Grubišić [29]. The exact amount of amplitude increase or decrease (constructive or destructive interference) is not as clearly delineated as in [29], but depends on both the inversion height and Froude number, since the combination of these parameters controls flow non-linearity (cf. Equation (11) in Sachsperger et al. [41]). Still, there appears to be no systematic separation of results according to either F

_{r}or H

_{1}/Z

_{i}, or their combination, that would identify one of those parameters as the dominant influence on the interference. Interestingly, amplitude-increase for constructive interference (e.g., experiment 268) and decrease for destructive interference (experiment 223) is larger than observed by Stiperski and Grubišić [29] for both free- and no-slip simulations, in line with the hypothesis that non-linearity plays a more important role in the interference of waves trapped on the inversion than those trapped by positive wind shear. The difference to [29] could also stem from the fact that the turbulence in laboratory experiments is three-dimensional (3D) and the flow is allowed to develop in the cross-stream direction (albeit both the obstacles and the incoming flow itself are 2D), whereas the numerical simulations were fully 2D.

_{2}/A

_{1}ratio occurred for mountain height ratio approximately equal to two-thirds and was associated with maximum destructive interference, mirroring the fact that the lee-wave amplitude was reduced to approximately two-thirds of its original value at the location of the second obstacle height. Since the particular value of H

_{n}for which maximum destructive interference occurs will be a function of both surface friction (decreasing the wave amplitude and wavelength) and V/λ, the 2:3 ratio cannot be taken as a universal value. It is therefore not surprising that only two sets of experiments (Figure 10b) show a minimum in A

_{2}/A

_{1}ratio at H

_{n}= 2:3. Three other experiments show a smaller increase of A

_{2}/A

_{1}with increasing H

_{n}. The value of the amplitude ratio A

_{2}/A

_{1}= 0.2, found by Stiperski and Grubišić [29], for maximum destructive interference is observed only for experiment 277. The coarse resolution of H

_{n}at which the experiments were performed does not allow us to examine the actual value of H

_{n}for which destructive interference reaches a maximum for all experimental sets.

## 4. Discussion

_{r}< 1 for these flows. On the other hand, the wavelength of trapped lee waves developing within the valley as a consequence of regime transition (see Section 3.2) does not show such a strong dependence on the second obstacle height, since long waves develop for supercritical flow with F

_{r}≈ 1, where information propagates only downstream.

_{1}/U did not vary much between the relevant single and double obstacle experiments. As shown in [37], the stratification above has additionally relatively little influence on the non-linearity of the waves on the inversion.

^{3}and 48 × 10

^{3}. This value is larger than the critical value usually assumed for transition into turbulence in geophysical flows, although it is considerably smaller than in the atmosphere. Still, due to Reynolds number similarity [43], we can assume the structure of the flow to be similar once the flow is fully turbulent, irrespective of the differences in the actual magnitude of the Reynolds number. This was shown by Eiff and Bonneton [38] who examined lee-wave breaking in tanks of three different sizes (corresponding different Reynolds number ranges) in the same facility where our experiments were conducted, and showed that their results were Reynolds number independent. The fact that the transition between laminar trapped lee waves and turbulent hydraulic jumps in our experiments corresponds to that predicted by Vosper [32] gives us additional confidence that our experiments realistically reproduce geophysical turbulence.

^{5}, [44]), calculated based on the distance along the plate, would still not have been reached until the lee of the first obstacle (Re ~ 3 × 10

^{5}). However, the fact that the plate was neither smooth nor flat and that this critical Reynolds number does not account for pressure perturbations that are present in flows over obstacles, we can assume that the laminar to turbulent transition occurred already before reaching the lee of the first obstacle. The same appears to be confirmed by Knigge et al. [32] who show positive near-surface cross-stream vorticity on the lee side of the obstacle, due to friction within the boundary layer. They have not only successfully reproduced rotor flow over a range of Reynolds numbers, but the onset of rotors coincided with that predicted by Vosper [32] and also matched with the results of Large-Eddy Simulations [45]. Still, the actual value of surface roughness will have an impact on the results, because rougher surfaces cause larger reduction of wave amplitude and wavelength. Given that several experiments were conducted with a 1-m long plate covered by lego elements (having larger surface roughness), we can examine this effect. Figure 11 shows the interfaces for experiments with same H

_{1}/Z

_{i}and F

_{r}but different upstream roughness. It shows that for a rougher surface, lee-wave wavelength is reduced, however, the amplitude is not significantly affected. Since no experiments were conducted with the extra rough plate and double obstacles, we are unable to say if the results would be the same given the more nonlinear interactions found for flow over double obstacles. Still, we do not expect the differences in surface roughness to alter the general conclusions of this paper in as much as lee-wave interference is expected to occur even for free-slip simulations (cf. [29]).

_{1}/Z

_{i}larger than 0.3 and F

_{r}larger than 0.5. This means that trapped lee wave rotors are expected over the entire trapped lee-wave regime (Figure 6) within our regime diagram. Furthermore, average obstacle steepness of 34% (calculated as the derivative of the Gaussian obstacle at the obstacle half-width) and Froude number range, places our experiments in the part of the regime diagram of Baines [46] that corresponds to post-wave separation. Whether rotors are actually observed can only be confirmed via PIV analysis and preliminary results do indeed show that our water tank experiments reproduce rotors. Still, a detailed study of rotor flow and the effect of secondary obstacles on the rotors is beyond the scope of this paper and will be a topic of a future analysis.

## 5. Conclusions

_{s}), Froude number (F

_{r}), dimensionless inversion height (H

_{1}/Z

_{i}) and mountain height ratio (H

_{2}/H

_{1}). In general, the amplitude increase for constructive interference and decrease for destructive interference was found to be larger than in the numerical simulations of Stiperski and Grubišić [29], as was the influence of the second obstacle on the flow within the valley. These differences could be due to flow non-linearity induced by the close proximity of the density jump to the obstacle top, as well as due to the fact that, unlike in the numerical simulations of Stiperski and Grubišić [29] and Vosper [35], the turbulence in HyIV-CNRS-SecORo experiments was three-dimensional. The performed experiments for different second obstacle heights did not reproduce maximum destructive interference as defined by Stiperski and Grubišić [29], except for one experiment.

## Supplementary Materials

_{1}/Z

_{i}), mountain height ratio (H

_{2}/H

_{1}), ridge separation distance (V), height of the first obstacle (H

_{1}), height of the second obstacle (H

_{2}), strength of the density jump (Δρ), height of the density jump (Zi), Brunt-Vaisala frequency in the upper layer (N), towing speed (U), subjectively classified flow response (type: T—trapped lee waves, H—hydraulic jump, L—long waves).

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Abbreviations

2D | two-dimensional |

3D | three-dimensional |

A_{s} | Amplitude downstream of a single obstacle |

A_{1} | Amplitude downstream of the first obstacle |

A_{2} | Amplitude downstream of the second obstacle |

CNRM | Centre National de Recherches Météorologiques |

CNRS | Centre National de Recherches Scientifique |

H_{1} | Height of the first obstacle |

H_{2} | Height of the second obstacle |

HyIV-CNRS-SecORo | Hydralab IV–CNRS–Secondary Orography and Rotors Experiments |

λ_{s} | Lee-wave wavelength downstream of a single obstacle |

λ_{1} | Lee-wave wavelength downstream of the first obstacle |

λ_{2} | Lee-wave wavelength downstream of the second obstacle |

LES | Large Eddy Simulation |

PIV | Particle Image Velocimetry |

T-REX | Terrain-Induced Rotor Experiment |

V | ridge separation distance |

Z_{i} | height of the density jump (i.e., inversion) |

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**Figure 1.**From top to bottom: (

**a**) cross-section along the longer side of the water tank and the view from the top; (

**b**) the cross-section along the shorter side of the CNRM large stratified water flume set-up used for the experiments; (

**c**) photograph of the water tank; (

**d**) photograph of the narrow part of the tank with the obstacles.

**Figure 2.**Example of a density profile for experiment 242. The depth of the mounting plate for the obstacles is indicated with a shaded area at the bottom of the plot.

**Figure 3.**(

**a**) Example of interface detection for experiment 240; (

**b**) Schematic diagram showing the definition of variables used in the text.

**Figure 4.**Examples of prototype flow regimes: (

**a**) trapped lee wave; (

**b**) long waves; (

**c**) hydraulic jump.

**Figure 5.**Interface displacement for a set of single obstacle runs with fixed H

_{1}/Z

_{i}= 0.5 and Fr ranging from 0.41 to 0.88, corresponding to flow regimes of hydraulic jump (F

_{r}= 0.41), trapped lee waves (F

_{r}= 0.51–0.78) and long waves (F

_{r}= 0.83–0.88).

**Figure 6.**Regime diagram of HyIV-CNRS-SecORo experiments showing the dependence of flow type on Froude number Fr and dimensionless inversion height (H

_{1}/Z

_{i}) for the: (

**a**) single obstacle runs; (

**b**) runs where the secondary obstacle is one-third of the height of the first one; (

**c**) the secondary obstacle is two-thirds of the height of the first one; and (

**d**) for obstacles of equal height. The long-dashed line separates trapped lee waves from vertically propagating waves according to Vosper [35]. The short-dashed line separates hydraulic jumps from trapped lee waves according to Sachsperger et al. [41] with NH

_{1}/U equal to 0.5. Crosses correspond to cases of valley flushing.

**Figure 7.**Interface displacement for a set of double obstacle runs with fixed H

_{1}/Z

_{i}= 0.9 and median F

_{r}equal to 1.095, for (

**a**) different valley widths; (

**b**) different heights of the second obstacle.

**Figure 8.**Examples of (

**a**) destructive and (

**b**) constructive lee-wave interference for (

**a**) shorter 1.3 m and (

**b**) longer 1.8 m valley width.

**Figure 10.**Interference pattern for the trapped lee-wave experiments; (

**a**) Amplitude ratio A

_{2}/A

_{s}as a function of dimensionless valley width V/λ

_{s}for twin obstacles (H

_{n}= 1); (

**b**) Amplitude ratio A

_{2}/A

_{1}as a function of mountain height ratio H

_{n}for experiments with equal valley width (V = 1.8 m). The color of the points corresponds to dimensionless inversion height H

_{1}/Z

_{i}, and the symbol to the Froude number of the experiment F

_{r}. Vertical lines in panel (

**a**) show full and half-integer values of V/λ

_{s}indicating where constructive and destructive interference is expected. Dotted wavy line indicates possible interference pattern inspired by results of Stiperski and Grubišić [29] with the values of minima and maxima obtained from linear no-slip simulations in [29]. Colored lines in panel (

**b**) connect experiments with the same set-up (F

_{r}and H

_{1}/Z

_{i}).

**Figure 11.**Interface displacement for two single obstacle runs with fixed H

_{1}/Z

_{i}= 0.91 and Fr = 0.79 but for a run with just the baseplate (smooth) and run with additional lego elements to increase the surface roughness (rough).

H_{i} (m) | 2L^{2} (cm^{2}) | L_{0i} (cm) | |
---|---|---|---|

Primary or Single (H_{1}) | 13.2 | 1060 | 66.3 |

Secondary (H_{2}) | 13.2 | 1060 | 66.3 |

Secondary (H_{2}) | 8.8 | 1060 | 62.9 |

Secondary (H_{2}) | 4.4 | 1060 | 56.8 |

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**MDPI and ACS Style**

Stiperski, I.; Serafin, S.; Paci, A.; Ágústsson, H.; Belleudy, A.; Calmer, R.; Horvath, K.; Knigge, C.; Sachsperger, J.; Strauss, L.;
et al. Water Tank Experiments on Stratified Flow over Double Mountain-Shaped Obstacles at High-Reynolds Number. *Atmosphere* **2017**, *8*, 13.
https://doi.org/10.3390/atmos8010013

**AMA Style**

Stiperski I, Serafin S, Paci A, Ágústsson H, Belleudy A, Calmer R, Horvath K, Knigge C, Sachsperger J, Strauss L,
et al. Water Tank Experiments on Stratified Flow over Double Mountain-Shaped Obstacles at High-Reynolds Number. *Atmosphere*. 2017; 8(1):13.
https://doi.org/10.3390/atmos8010013

**Chicago/Turabian Style**

Stiperski, Ivana, Stefano Serafin, Alexandre Paci, Hálfdán Ágústsson, Anne Belleudy, Radiance Calmer, Kristian Horvath, Christoph Knigge, Johannes Sachsperger, Lukas Strauss,
and et al. 2017. "Water Tank Experiments on Stratified Flow over Double Mountain-Shaped Obstacles at High-Reynolds Number" *Atmosphere* 8, no. 1: 13.
https://doi.org/10.3390/atmos8010013