Abstract
Turbulent flows over wavy surfaces give rise to the formation of ripples, dunes and other natural bedforms. To predict how much sediment these flows transport, research has focused mainly on basal shear stress, which peaks upstream of the highest topography, and has largely ignored the corresponding pressure variations. In this article, we reanalyze old literature data, as well as more recent wind tunnel results, to shed a new light on pressure induced by a turbulent flow on a sinusoidal surface. While the Bernoulli effect increases the velocity above crests and reduces it in troughs, pressure exhibits variations that lag behind the topography. We extract the in-phase and in-quadrature components from streamwise pressure profiles and compare them to hydrodynamic predictions calibrated on shear stress data.
1 Introduction
Most natural flows occur on evolving topography. The resulting hydrodynamic variations are described by a linear theory that Jackson and Hunt [1] developed for wind profiles over low hills. Their work inspired analyses of laminar [2–6] and turbulent [7–18] flows on shallow bedforms, as recently reviewed by Finnigan et al. [19]. Flow modulation associated with fluid-structure interactions also drives the dynamics of wind-driven wave generation at a liquid surface [20, 21], or on compliant thin sheets [22–24], leading to the flag instability when a free end is allowed to flap [25].
Most studies of fluid motion on wavy surfaces have focused on basal shear stress, which drives sediment transport [26]. As Charru et al [27] reviewed, coupling the latter to the Jackson and Hunt theory or its variants [10, 13] explains the formation of erodible objects like sand ripples and dunes, which owe their initial growth to a basal shear stress peaking upstream of the highest elevation.
However, basal pressure is also affected by evolving topography. In porous sand beds, streamwise pressure variations produce an internal flow that drives humidity and microscopic particles below the surface [28]. With strong enough winds, the resulting pore pressure can also relieve part of the bed weight, thereby facilitating the onset of its erosion [29]. At much larger scale, topography-induced pressure variations are important to atmospheric science, especially mountain meteorology [30, 31].
Relatively few experiments conducted in air [29, 32–39], water or other liquids [40–48] flowing over wavy surfaces staged harmonic bedforms with low enough ratio of amplitude ζ and wavelength λ to avoid flow detachment. This has made it difficult to compare data with linear theories predicated on small . As the experiments of Hanratty, et al. [47] have shown, a hydrodynamic anomaly occurs when the flow response to topographical variations transitions from laminar to turbulent behavior. This phenomenon is at the origin of an instability that carves rippled or scalloped features on surfaces able to sublimize or dissolve into the fluid [49, 50], but that disappears when the bed becomes hydrodynamically rough [51]. Recently, such sublimation ripples have been found at the surface of the Martian north polar cap [52], with a typical wavelength much larger than blue ice ripples found in Antarctica [53], but still proportional to the viscous length [54], based on kinematic viscosity ν and shear velocity . This anomalous hydrodynamic response is also essential to understand how subaqueous or Martian ripples are superimposed on dunes [55], acting as a separation of small and large scale bedforms. Therefore, a question is whether streamwise pressure variations are also subject to such anomalous transition.
When fluid flowing on a flat surface reaches an ascending bedform, the narrowing of streamlines raises speed and decreases static pressure, as predicted by energy conservation in the Bernoulli equation. To leading order, this effect is captured by dimensionless coefficients and , which respectively represent the bedform’s role on speed and pressure. Because speed rises when pressure decreases, these coefficients have opposite signs, and . In the “outer region” far above the surface, such energy conservation holds. However, in the closer “inner region”, inertia causes fluid flow to lag changes in the bedform, a phenomenon that is captured by dimensionless coefficients and that are both positive, and respectively represent effects on shear and normal stress. Overall, fluid inertia causes surface shear stress to lead topographical variations with a positive phase , whereas static pressure lags those changes with . While others have addressed and [27, 32, 39], this paper focuses on and . We begin with a summary of the theory, which predicts how and depend on the wavenumber of bed oscillations.
Our objective is to review articles reporting pressure measurements on wavy surfaces subject to a turbulent flow. As we will discuss, existing data [29, 36, 37, 47, 56] suggest that the anomalous transition in shear stress may also arise in the pressure response. However, we recognize that the corresponding experiments, which were not designed to address this question, do not support a definitive conclusion. In the context of the anomalous transition, a crucial shortcoming of these experiments is their determination of , which may have been approximate. Because the coupling of surface pressure with porous media is relevant to industrial applications and the formation of geophysical bedforms, we hope that this article will inspire future experiments in the dimensionless wavenumber range , where our theory predicts distinct behavior of for rough and smooth walls.
2 Turbulent Flow Over A Wavy Bed
Because our main objective is to reanalyze existing data for turbulent flows over wavy beds, this section does not repeat our own derivations of the underlying theory, but rather provides a summary of key quantities and concepts. To account for the hydrodynamic anomaly, the framework of Fourrière, et al. [9, 10] was recently extended, as detailed in [27, 51]. We examine a turbulent fluid flow along the x direction, unbounded vertically and driven by a shear stress imposed far above the bed. Restricting attention to the linear flow response to bed relief, the elevation can be decomposed in Fourier modes. Therefore, without loss of generality, we consider a bed profile of the formwhere is a parameter . From Eq. 1, troughs reside at . z is the crosswise distance normal to the reference mean bed elevation. We assume invariance in the spanwise direction y that completes the Cartesian coordinate system.
In this framework, hydrodynamics is described by Reynolds-averaged Navier-Stokes equations governing the mean velocity field and pressure p. A first order turbulence closure relates the stress tensor to the velocity gradient. This closure involves a turbulent kinematic viscosity associated with a mixing length and a mixing frequency representing typical eddy length and time scales [57]. The mixing frequency is given by the strain rate, and the mixing length depends explicitly on distance from the bed. To account for both the smooth and rough regimes, we adopt a mixing length inspired from van Driest [58].where is von Kármán’s constant, is the bed shear stress, and ρ is the constant fluid density. d is the sand-equivalent roughness size, from which we define the Reynolds number . The exponential term in Eq. 2 suppresses turbulent mixing within the viscous sub-layer close to the bed. The term corresponds to the standard Prandtl hydrodynamic roughness extracted by extrapolating the logarithmic law of the wall at vanishing velocity, . The term controls the reduction of the viscous layer thickness upon increasing bed roughness. In the rough limit where , the exponential term vanishes and the hydrodynamic anomaly is suppressed altogether. The dimensionless parameters and are calibrated from measurements of velocity profiles over various rough walls [59, 60].
In Eq. 2, is the van Driest transitional Reynolds number. Following Hanratty [49], the hydrodynamic anomaly is captured by a spatial relaxation of . In the homogeneous case of a flat bed, it is . However, in general, is not constant but instead trails behind the pressure gradient by a space lag on the order of that is associated with a thickening of the boundary layer,
Charru, et al. [27] calibrated this additional equation with and by matching theoretical predictions to basal shear stress measurements [32, 33, 44, 47, 61]. These predictions were obtained in the regime where the hydrodynamic equations could be linearized with respect to , and then solved for boundary conditions [9, 10, 51]. In this regime, we write the basal shear stress response to the bed perturbation (1) as
Here, is reference shear stress in the flat base state. In Eq. 4, the two terms in straight brackets respectively quantify the in-phase and in-quadrature contributions of the response. Both and have values of order unity [32, 39] and are positive, thereby producing a shear stress leading the bed elevation. They are weak logarithmic functions of the bed wavenumber, except in the range where strong variations arise around the hydrodynamic anomaly [27, 51].
Similarly, we write the basal pressure response in the linear regime as
Figure 1 shows how and vary with the dimensionless wavenumber in the rough and smooth limits [9, 10, 51]. The opposite signs and mean that, in contrast with shear stress, the phase of pressure modulations are delayed with respect to the bed profile. In addition, because , the pressure response is dominated by the Bernoulli effect, in that it is nearly out-of-phase with the topography. Therefore, variations of with wavenumber are well captured by approximating the flow as inviscid and irrotational. In this case, the pressure varies as the square of the velocity at a height on the order of the wavelength, which is the only macroscopic scale over which a pressure disturbance is expected to penetrate the flow. From the logarithmic law of the wall, the velocity therefore scales as , where b is . This argument suggests that, in the rough case where complications associated with the hydrodynamic anomaly do not arise, should scale as the square of . As the parabola in Figure 1A shows, this approximation indeed conforms well to the theory for the rough case [9, 10]. The smooth case differs from this log-parabolic behavior above where the anomaly comes into play. While the dependence of flattens somewhat at the larger wavenumbers, the anomaly has a more pronounced effect on , with a distinctive non-monotonic behavior spanning a decade around (Figure 1B).
FIGURE 1
3 Pressure Measurements Over Wavy Surfaces
In this section, we compare theoretical predictions to available experimental data. We first outline how to fit the recorded pressure profiles. Then, for each set, we discuss how this procedure yields , , and their respective uncertainties.
3.1 Fitting Procedure
Because the theory is built on a linear analysis of hydrodynamic equations, we restrict attention to data sets with a harmonic pressure response to topographical variations at low (Table 1). However, as Figure 2 and graphs in Supplementary Appendix S1 indicate, relatively weak non-linearities are apparent. Accordingly, we fit dimensionless pressure response profiles to third-order expansions of the formbut we infer , from the leading order
TABLE 1
| Experiment | Data symbol, bed and fluid properties | |||||
|---|---|---|---|---|---|---|
| Profiles | (m/s) | (%) | ||||
| Zilker et al. [47] | , m, , m2/s, kg/m3 | |||||
| Figure 2A | 0.028 | 0.0038 | ||||
| Cook [56] | , m, , m2/s, kg/m3 | |||||
| Supplementary Figure S1A | 0.030 | 0.0045 | ||||
| Supplementary Figure S1B | 0.062 | 0.0022 | ||||
| Supplementary Figure S1C | 0.076 | 0.0018 | ||||
| Supplementary Figure S1D | 0.079 | 0.0017 | ||||
| Motzfeld [37] | , m, , m2/s, kg/m3 | |||||
| Supplementary Figure S2 | 0.69 | 0.0005 | ||||
| Kendall [36] | , m, , m2/s, kg/m3 | |||||
| Supplementary Figure S3A | 0.13 | 0.0072 | ||||
| Supplementary Figure S3B | 0.21 | 0.0044 | ||||
| Supplementary Figure S3C | 0.30 | 0.0031 | ||||
| Supplementary Figure S3D | 0.39 | 0.0024 | ||||
| Musa et al. [29] | , m, , m2/s, kg/m3 | |||||
| Supplementary Figure S4A | 0.16 | 0.0058 | ||||
| Supplementary Figure S4B | 0.33 | 0.0028 | ||||
| Supplementary Figure S4C | 0.55 | 0.0017 | ||||
| Supplementary Figure S4D | 0.76 | 0.0012 | ||||
| Supplementary Figure S4E | 0.95 | 0.0010 | ||||
| Supplementary Figure S4F | 1.21 | 0.0008 | ||||
FIGURE 2
For the data sets under consideration, the second and third terms have amplitudes and . As expected, fitting them to first-order alone (, ) does not significantly affect the resulting and .
Uncertainties in and depend not only on experimental scatter in and , but also on how was inferred. Using Eqs 7, 8, we estimate uncertainties due to scatter by carrying out a least-squares regression of data to and , while assuming that and are uncorrelated and normally distributed. Unfortunately, too little information is available to gauge how accurately was established.
3.2 Zilker et al. and Cook
We first review experiments reported in [47], which were used to compare predictions for basal shear stress and related behavior of and in [27]. Experiments were conducted in a rectangular channel circulating an electrolyte of density kg/m3 and kinematic viscosity m2/s. The bottom of the test section featured interchangeable smooth Plexiglas wavy surfaces of fixed cm but different amplitudes ζ. Basal shear stress and static wall pressure were respectively measured with an electrochemical method and small taps. Shear velocity was inferred from measurements of flow rate by integrating the logarithmic law of the wall recorded on a flat surface.
As Figure 2 shows for , the pressure profiles exhibit a minimum shortly after the crest. From Eq. 6, we find and . Cook [56] measured additional pressure profiles in a similar setup, albeit using another electrolyte solution with kg/m3 and m2/s. Data at four different flow rates are shown in Supplementary Figure S1. Pressure amplitude increases with flow velocity, whereas phase shift with respect to bed elevation decreases. remains nearly constant , while rises with (Table 1).
3.3 Motzfeld and Kendall
We analyzed experiments performed in wind tunnels over smooth solid sinusoidal waves. The oldest work is Motzfeld’s [37], who staged four different bed profiles carved in plaster and varnished. To stay within reach of the linear assumption, we only exploited his data for the smallest amplitude (his ‘model I’ with and m). The corresponding pressure profile and fits are shown in Supplementary Figure S2. Eq. 6 yields a relatively precise , but a less accurate , which hints at pressure variations almost in phase with the bed.
The other wind tunnel data are Kendall’s [36], who studied turbulent flows over mobile and immobile waves on a rubber surface. We only considered his immobile sinusoidal bed (, m) at different wind velocities. Profiles are shown in Supplementary Figure S3. Consistent with the results of [56], Kendall’s [36] pressure amplitude increases with flow velocity, while phase with respect to bed elevation decreases.
Motzfeld [37] and Kendall [36] both inferred shear velocity from logarithmic fits of vertical velocity profiles. However, Kendall fitted and separately. Because hydrodynamic roughness on a smooth wall is correlated with shear velocity, we recalculated Kendall’s using .
3.4 Musa et al
Musa et al. [29] also acquired data on sinusoidal, smooth, rigid walls in the wind tunnel ( m), but their objective was to record pore pressure within a permeable rigid material mimicking a sand bed. We only consider their ripple of smallest amplitude (), which they deployed at six different wind speeds. From a solution of the Laplace equation governing pore pressure, Musa et al. fitted at the surface to their pore pressure measurements at 45 locations within the ripple [In the nomenclature of their Eq. 9, is equivalent to our ]. They then inferred shear velocities by fitting vertical wind profiles to the logarithmic law of the wall using an aerodynamic roughness proportional to the viscous length. For consistency with other data reviewed here, we adopt , which differs slightly from their fit of in the smooth limit. Surface pressure profiles and fits to Eq. 6 are shown in Supplementary Figure S4 for their six values of . As the open diamonds in Figure 3 suggest, rose slowly with and was non-monotonic.
FIGURE 3
4 Discussion and Concluding Remarks
Table 1 summarizes conditions of all available experiments on smooth walls with nearly harmonic response, and the resulting and . First- and third-order fits of experimental pressure profiles yield similar values for these quantities, thereby demonstrating robustness of the fitting procedure. Uncertainties are relatively small, except for from Motzfeld and from Musa, et al.
Figure 3 shows corresponding variations with the rescaled wavenumber and, consistently with experimental conditions, compares them with theoretical predictions in the smooth case [9, 10, 51], which rely on a calibration of the hydrodynamic equations and the relaxation framework of Hanratty [49] on the streamwise evolution of basal shear stress [27]. While these preditions capture the correct trend, they clearly underpredict . A Bernoulli-like approximation (dotted line in Figure 3A) is more faithful to the data, but it requires , which is lower than the value used in the inset of Figure 1 by a factor of 2. We attribute the discrepancy to challenges in extracting from experiments.
At first glance, Figure 3 suggests that is nearly constant within experimental error. However, its trend vs hints at the presence of a local minimum from the hydrodynamic anomaly for a smooth wall (black curve in Figure 1). Nonetheless, the precision in is not yet sufficient to distinguish this behavior from that of a rough wall (blue curve).
Overall, and exhibit dispersion among different experiments. One reason is inconsistent ways to evaluate , which affects both axes in Figure 3 by rescaling stresses with and lengths with . Because velocity fluctuations or Reynolds stresses close to the bed are difficult to measure, was inferred from velocity profiles or flow rates using the law of the wall. In addition, it is open to question how flows driven by a pressure gradient, such as those in wind tunnels or pipes, can be quantitatively compared to a theoretical framework where shear stress is imposed. In this context, the Bernoulli-like approximation suggests that the velocity very close to the bed at an altitude is a better proxy for an effective than the average flow velocity.
A second issue is non-linear effects. Weakly non-linear developments [9, 62] and measurements [10] suggest that is an upper bound for validity of the linear theory. In recorded pressure profiles, we clearly discern weakly non-linear effects, especially in Musa et al. (Supplementary Figure S4), whose pressure is lower than expected on bed crests and troughs, although this effect may be due to an interaction with the porous bed underneath [29]. In addition, non-linearities also raise the effective bed roughness on a scale comparable to λ, with a first corrective term in [9]. This further complicates an estimation of the relevant experimental shear velocity.
These observations call for more measurements, particularly in the range that resolves the peak of the hydrodynamic anomaly. For air at ordinary wind speed, for example m/s, this implies a wavy bed with 2 mm cm. For more gentle winds, the smaller wavelength could rise to 1 cm. Larger λ could be staged with oils of larger viscosity. Data at significantly smaller wavenumbers would require a natural wavy surface such as a sand dune. An example is the hump studied in [32], where m and in the rough limit. Here, the theory predicts and , i.e. a phase shift rad, corresponding to a distance m downwind of the crest. If small pressure differences could be reliably recorded over relatively long distances, such spatial phase lag could also be measured.
Finally, DNS or LES simulations would also constitute another source of data, since runs could be performed with strictly imposed values of [63, 64], thereby mirroring the theoretical approach. Unfortunately, simulations of Maaß and Schumann [65, 66] or those of Salvetti et al. [67] involve amplitudes too large to avoid non-linear effects arising at . In both experimental and numerical investigations, is typically adjusted at fixed wavenumber using different winds. However, investigating the role of k under constant flow is equally valuable, perhaps again with DNS, to gain a deeper understanding of the hydrodynamic anomaly. At present, the relaxation closure inspired from Hanratty [49] is convenient. However, the interplay between a wavy bed and modulation of the viscous sublayer remains an open problem.
The evolution of pressure on geophysical bedforms such as sand ripples creates an internal seepage flow that brings nutrients to the liveforms they shelter [68], and it provides a mechanism for the accumulation of moisture or dust within them [28]. The phase lag that is proportional to also induces surface variations that future research could relate to complex drying patterns that are observed in sand seas [69]. More generally, pressure variations affect phase change and thermodynamics. Pressure is also a stress scale that may alter the rheology of dense granular flows and suspensions in subtle ways, for example by altering the opening and closing of contacts among grains [70], especially when the continuous phase is a liquid [72]. Although initially motivated by wind flows over ripple-like bed oscillations, the pressure effects that we have reviewed could then be relevant to subjects as diverse as dissolution and sublimation patterns [50, 51, 72], granular soil liquefaction [71], droplets and aerosols production [73] and cloud formation over larger scale topography [30, 31].
We thank F. Charru, O. Durán and A. Fourrière for fruitful discussions. This research was supported in part by the National Science Foundation under Grant No. NSF PHY-1748958, since it was initiated at KITP (Santa Barbara, United States) during the conference on Particle-Laden Flows in Nature (16–22 December, 2013), as part of the program “Fluid-mediated particle transport in geophysical flows.” This work has benefited from the financial support of the Agence Nationale de la Recherche, grant “Zephyr” No. ERCS07 18. MYL’s contribution was made possible by the support of NPRP grant 6–059-2-023 from the Qatar National Research Fund.
Statements
Data availability statement
Publicly available datasets were analyzed in this study. This data can be found here: The data are available in the cited references.
Author contributions
All authors contributed equally to this work.
Conflict of interest
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
Supplementary material
The Supplementary Material for this article can be found online at: https://www.frontiersin.org/articles/10.3389/fphy.2021.682564/full#supplementary-material
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Summary
Keywords
pressure modulation, bedforms, sinusoidal topography, linear response, laminar-turbulent transition
Citation
Claudin P, Louge M and Andreotti B (2021) Basal Pressure Variations Induced by a Turbulent Flow Over a Wavy Surface. Front. Phys. 9:682564. doi: 10.3389/fphy.2021.682564
Received
18 March 2021
Accepted
30 April 2021
Published
14 May 2021
Volume
9 - 2021
Edited by
Hezi Yizhaq, Ben-Gurion University of the Negev, Israel
Reviewed by
Anna Carbone, Politecnico di Torino, Italy
Dong Chen, Zhejiang University, China
Updates
Copyright
© 2021 Claudin, Louge and Andreotti.
This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.
*Correspondence: Philippe Claudin, philippe.claudin@espci.fr
This article was submitted to Interdisciplinary Physics, a section of the journal Frontiers in Physics
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