Characterizing the turbulent drag properties of rough surfaces with a Taylor--Couette setup
Pieter Berghout, Pim A. Bullee, Thomas Fuchs, Sven Scharnowski, Christian J. Kähler, Daniel Chung, Detlef Lohse, Sander G. Huisman
TThis draft was prepared using the LaTeX style file belonging to the Journal of Fluid Mechanics Characterizing the turbulent drag propertiesof rough surfaces with a Taylor–Couettesetup
Pieter Berghout † , Pim A. Bullee , Thomas Fuchs ,Sven Scharnowski , Christian J. K¨ahler , Daniel Chung ,Detlef Lohse and Sander G. Huisman ‡ Physics of Fluids Group, Max Planck UT Center for Complex Fluid Dynamics, MESA+Institute and J.M. Burgers Centre for Fluid Dynamics, University of Twente, P.O. Box 217,7500 AE Enschede, The Netherlands Soft Matter, Fluidics and Interfaces, MESA+ Research Institute, University of Twente, P.O.Box 217, 7500AE Enschede, The Netherlands, Institut f¨ur Str¨omungsmechanik und Aerodynamik, Universit¨at der Bundeswehr M¨unchen,Werner-Heisenberg-Weg 39, 85577 Neubiberg, Germany, Department of Mechanical Engineering, University of Melbourne, Victoria 3010, Australia, Max Planck Institute for Dynamics and Self-Organization, Am Faßberg 17, 37077 G¨ottingen,Germany(Received xx; revised xx; accepted xx)
Wall-roughness induces extra drag in wall-bounded turbulent flows. Mapping any givenroughness geometry to its fluid dynamic behaviour has been hampered by the lack ofaccurate and direct measurements of skin-friction drag. Here the Taylor–Couette (TC)system provides an opportunity as it is a closed system and allows to directly andreliably measure the skin-friction. However, the wall-curvature potentially complicatesthe connection between the wall friction and the wall roughness characteristics. Here weinvestigate the effects of a hydrodynamically fully rough surface on highly turbulent, innercylinder rotating, TC flow. We carry out particle image velocimetry (PIV) measurementsin the Twente Turbulent Taylor–Couette (T C) facility with radius ratio η = 0 . . × < Re i < . × , with water asworking fluid. The inner cylinder is covered with P36 grit sandpaper, and the outercylinder remains smooth and stationary. We find that the effects of a hydrodynamicallyfully rough surface on TC turbulence, where the roughness height k is three ordersof magnitude smaller than the Obukhov curvature length L c (which characterizes theeffects of curvature on the turbulent flow, see Berghout et al. arXiv: 2003.03294, 2020),are similar to those effects of a fully rough surface on a flat plate turbulent boundary layer(BL). Hence, the value of the equivalent sand grain height k s , that characterizes the dragproperties of a rough surface, is similar to those found for comparable sandpaper surfacesin a flat plate BL. Next, we obtain the dependence of the torque (skin-friction drag) onthe Reynolds number for given wall roughness, characterized by k s , and find agreementwith the experimental results within 5%. Our findings demonstrate that global torquemeasurements in the TC facility are well suited to reliably deduce wall drag propertiesfor any rough surface. † Email address for correspondence: [email protected] ‡ Email address for correspondence: [email protected] a r X i v : . [ phy s i c s . f l u - dyn ] S e p P. Berghout et al.
1. Introduction
Turbulent boundary layers over fully rough walls
The transport of a fluid over a solid body or the transport of a solid body through afluid is always hindered by friction forces acting on the interface between the solid andthe fluid. Ideally, the solid surface is smooth, and the drag force is a purely viscous force.In nature and engineering applications, however, solid surfaces are nearly always rough.This means that in addition to a modified viscous force, the roughness also results in apressure contribution to the drag force (‘pressure drag’), and consequently, an increasein the total drag force (the so-called ‘drag penalty’). The contribution of the pressuredrag to the total friction drag at the surface grows with increasing roughness height.Ultimately, when the pressure drag dominates, the surface is called hydrodynamicallyfully rough .Due to the obvious interest in reducing the drag penalty, substantial research hasbeen carried out to investigate the effects of rough surfaces on wall-bounded turbulentflows (Jim´enez 2004; Flack & Schultz 2010; Chung et al. ∆u + ) of the mean streamwise velocity ( u + ) in the overlap (orlogarithmic) region of the turbulent BL (Clauser 1954; Hama 1954). This shift can beconsidered as a direct measure of the drag penalty. The mean velocity profile for a roughwall in the overlap region is given by the Prandtl–von K´arm´an profile for smooth walls,minus this shift (Pope 2000) u + = 1 κ log y + + A − ∆u + , (1.1)where y + is the wall-normal distance and the von K´arm´an constant κ ≈ .
40 and A ≈ . u τ = (cid:112) τ w /ρ and the viscous length scale δ ν = ν/u τ , where τ w is the wall shear stress, ρ the fluid density and ν is the kinematicviscosity. For a fully rough surface, it can be derived from dimensional arguments that thevelocity shift ∆u + depends logarithmically on the roughness height k + , see e.g. (Raupach et al. ∆u + = 1 κ log k + s + A − B, (1.2)where B ≈ . k + s is obtained by fitting, such that the velocity shift of any fully rough surface collapseswith the velocity shift of sand grains in turbulent pipe flow, that historically grew to bethe reference case (Nikuradse 1933). Hence, the key objective in research of wall boundedturbulent flows over rough surfaces is to relate the statistics of a rough surface to thevalue of k s , which characterizes the roughness (Forooghi et al. Taylor–Couette flow
TC flow — the flow between two coaxial, independently, rotating cylinders — is acanonical system in turbulence (Taylor 1923; Grossmann et al. et al. et al.
C flow with rough surfaces
T a = 14 (cid:18) η √ η (cid:19) ( r o − r i ) ( r i + r o ) ( ω i − ω o ) ν . (1.3)Here η is the geometric measure of curvature, namely the ratio r i /r o of the radii of thecylinders. The subscripts i and o indicate inner cylinder and outer cylinder, respectively.The angular velocity is denoted by ω , and ν is the kinematic viscosity.The global response of the system is expressed as the Nusselt number N u ω , which is theratio between the angular velocity flux J ω in radial direction and its laminar counterpart J ω lam (Eckhardt et al. N u ω = J ω J ω lam = r ( (cid:104) u r ω (cid:105) A ( r ) ,t − ν∂ r (cid:104) ω (cid:105) A ( r ) ,t )2 νr i r o ( ω i − ω o ) / ( r o − r i ) . (1.4)Here, (cid:104)·(cid:105) A ( r ) ,t denotes averaging over the cylinder surface A ( r ) and over time t . TheNusselt number N u ω is related to the torque T required to drive the inner cylinder. Innon-dimensional form the torque can be expressed as G = T πLρν = N u ω J ω lam ν . (1.5)From here onwards, we assume inner cylinder rotation only, hence ω o = 0, as thiscorresponds to our experiments where we kept the (smooth) outer cylinder stationary atall times.The torque is directly related to the wall shear stress τ w = T / (2 πr i L ). As commonlyused in other canonical systems (e.g. the flat plate BL), we define the friction factor C f as (Lathrop et al. C f = 2 πτ w,i ρd ω i = 2 πN u ω J ωlam ( νRe i ) − , (1.6)where Re i = r i ω i d/ν and d = r o − r i . This relation allows for straightforward comparisonwith other canonical wall-bounded flows like pipe flow, channel flow, flow over a flat plate,etc.The turbulent flow in the TC setup is strongly influenced by the curvature of itsbounding walls, i.e. the cylinders that drive the flow. This distinguishes turbulent TC flowfrom turbulent flows in other canonical systems. Bradshaw (1969) realized that the effectsof curvature on a turbulent BL are very similar to the effects of buoyancy stratificationon a turbulent BL (Obukhov 1971). In analogy to the Obukhov length (Obukhov 1971;Monin & Yaglom 1975), he derived a length scale that separates the curved BL in aregion where the effects of shear dominate (i.e. production of turbulence is dominatedby shear production), and a region further away from the wall where curvature effectsdominate (i.e. the production of turbulence is dominated by curvature). For smooth wallTC turbulence, this ‘curvature Obukhov length’ is well approximated by (Berghout et al. L c,s = u τ κω i , (1.7)where shear dominates for 0 . L c,s (cid:46) y , shear and curvature effects are both significantfor 0 . L c,s (cid:46) y (cid:46) .
65, and curvature effects dominate at 0 . L c,s (cid:46) y . Using data ofthe mean velocity profiles from PIV in turbulent TC flow (Huisman et al. et al. et al. P. Berghout et al. curvature effects are important was recently obtained for smooth wall TC flow (Berghout et al. et al. (2020), an analyticalexpression for
N u ( T a ) was derived (Berghout et al. et al. et al. et al. et al. et al. k/d between the height of thebars and the gap width d = r o − r i was as large as k/d = 0 .
05 or even 0 .
1. Later Berghout et al. (2019) numerically studied the effects of sand grain roughness ( k/d = 0 . . k and the gap width d .In this paper, we study the effects of a hydrodynamically fully rough inner cylinder onthe turbulent wall-bounded flow, with small roughness k/d = 0 . k ≡ k σ , and k σ is the standard deviation of the roughness elevation. In particular, we keep k smallerthan the curvature Obukhov length L c (see (3.2)), namely k/L c = 0 . . k (cid:28) d is not enough. Rather, k (cid:28) L c must also hold, to ensure that effects related tothe streamwise curved geometry are not influencing the effects of the roughness.Hence, we hypothesize that effects of roughness in TC turbulence (where k (cid:28) L c )are similar to the effects of roughness in other canonical systems without streamwisecurvature. Thus global measurements in the (closed) TC facility can be employed tocharacterize drag properties of the rough surface. The outer cylinder remains smooth, toallow for optical access of the velocity profiles.The paper is organised as follows: In §
2, we describe the experimental methods. Wethen ( §
3) discuss the relevant dynamical length scales in the experiment, and elaborateon the different regions in the BL where turbulent production is dominated by sheareffects, and where effects related to the streamwise curvature of the setup play a role.We also comment on the scale separation and show that the roughness mainly affects theinertial shear dominated regime, and hence, effects from the streamwise curved geometryof the TC flow do not modify the velocity shift. In § § ∆ω + , from which the equivalent sand grain roughness height isdetermined in §
6. In § § k s , in agreement withour experimental results. The paper ends with a summary, conclusions, and an outlook( §
2. Experimental setup and methods
Experimental setup
The experiments were performed in the Twente Turbulent Taylor–Couette (T C)facility (van Gils et al. a ), with water as working liquid. We used a fully roughinner cylinder with an outer radius of r i = 201 . r o = 279 . η = 0 .
720 and a gap
C flow with rough surfaces d = 78 . L = 927 mm and an aspect ratio of Γ = L/d = 11 .
9. For inner cylinder rotation only (the outer cylinder is stationary), theReynolds number is defined with the velocity of the inner cylinder and the gap width d as Re i = ω i r i dν . (2.1)Using the viscous velocity u τ obtained from measurements of the torque, the frictionReynolds number is defined as Re τ = u τ ( d/ ν . (2.2)The roughness used was P36 grit sandpaper ( VSM , ceramic industrial-grade), that wasfixed to the entire surface of the inner cylinder using double-sided adhesive tape ( tesa51970 ). We define the characteristic length scale of the roughness as k ≡ k σ ≈ .
07 mm(corresponding to the 99 .
8% interval of the height), where k σ is the standard deviation ofthe local roughness height h ( x, y ) (quantified using confocal microscopy (Bakhuis et al. k/d = 0 . Experimental procedure
We performed seven experiments with different rotation rates of the inner cylinder,see table 1. During all these experiments, the torque T that is required to drive theinner cylinder at fixed rotational velocity was measured constantly. The hollow reactiontorque sensor that connects the drive shaft to the middle section of the inner cylinderis indicated in figure 1(a). By only measuring the torque on the middle section, possible P. Berghout et al.
T a [ × ] Re i [ × ] Nu ω C f [ × − ] Re τ [ × ] L + c [ × ] k + σ .
31 0 .
46 312 2 .
21 7 . . .
57 0 .
62 403 2 .
13 10 . . .
92 0 .
78 513 2 .
14 12 . . .
47 0 .
99 643 2 .
12 16 . . .
18 1 .
20 784 2 .
12 19 . . .
55 1 .
54 998 2 .
11 24 . . .
71 1 .
77 1137 2 .
09 28 . . .
15 2 .
00 653 1 .
07 23 . . Table 1: Control parameters, global response and relevant length scales, measured duringthe PIV measurements.
T a or Re i characterize the driving of the system. N u ω is thedimensionless angular velocity flux, C f the friction factor, L + c the curvature Obukhovlength as defined in equation (3.2) and k + σ is the standard deviation of the sandpaperroughness, see Bakhuis et al. (2020). The final row presents the values correponding tothe smooth wall measurement of Huisman et al. (2013)end-plate effects are eliminated (van Gils et al. T smaller than 4% for all cases. These direct and reproducible measurements of thetorque (friction) have an accuracy that is comparable to the measurement accuracy ofwall shear stress in flat plate BLs, by means of a drag balance (Baars et al. Dantec FPP-RhB-10 with diameters from 1 µ m to 20 µ m) were added to the working fluid. A horizontal lasersheet of about 1 mm in thickness illuminated tracer particles in the working liquid atmid-height, through the transparent outer cylinder. The laser sheet was created usinga frequency doubled Quantel EverGreen
200 mJ laser. The fluorescent light emergingfrom the tracer particles was imaged from below, through a window placed in the bottomplate of the apparatus. For this, a 45 ◦ mirror was positioned under the bottom plateas drawn schematically in figure 1. The camera was a high-resolution sCMOS camera( LaVision PCO.edge ), with a resolution of 2560 px × . µ m.A 100 mm focal length objective ( Zeiss Makro Planar , 100 mm) was used, giving anoptical magnification of 0 . image pairs were acquired ata recording frequency equal to the rotation rate. The mean velocity distribution in thehorizontal plane was computed using single-pixel ensemble correlation (K¨ahler et al. µ m, leading to about 1600 independent measurementpoints in the radial direction, evenly spread over the entire gap. From the correlationfunction (obtained for every pixel) one can directly extract the standard deviation of thevelocity, by integrating the probability density function σ ( u ) = (cid:82) ∞−∞ ( u −(cid:104) u (cid:105) ) PDF( u ) du ,see Scharnowski et al. (2012). This ensures that all turbulent scales are included inthe standard deviation, as opposed to regular PIV analysis. The velocity profiles weresmoothed using a Gaussian filter with a standard deviation of σ ≈ . C flow with rough surfaces S between the turbulence production by shear and that by curvature, see equation (3.1).Dotted and dashed lines represent the slope of the logarithmic velocity profile of the shearand the curvature dominated regimes, κ − and λ − , respectively. (b) Ratio S versus thewall-normal distance shifted with the wall offset, y/L c = ( r − r i − k σ ) /L c , where 2 k σ is the approximated wall offset of the sandpaper. Coloured lines are calculated from thePIV data of the rough wall cases. The grey line ( k + σ = 0) is the smooth wall profile at T a = 6 . × ( Re τ = 23093), obtained from Huisman et al. (2013).
3. Curvature effects, the mean velocity profile and scale separation
The relative effects of curvature and shear
To characterize and quantify the relative effects of shear and curvature in TC tur-bulence, we study the ratio S of turbulence production by shear and curvature relatedeffects (Bradshaw 1969; Townsend 1976; Berghout et al. S − = u (cid:48) θ u (cid:48) r ddr U r u (cid:48) θ u (cid:48) r U = 1 ω dUdr , (3.1)where u (cid:48) θ and u (cid:48) r are the azimuthal and radial velocity fluctuations, respectively, and u (cid:48) θ u (cid:48) r is the Reynolds stress. The mean azimuthal velocity is denoted by U , and ω = U/r is the mean angular velocity. The curvature Obukhov length L c defined in equation (1.7)for a smooth wall marks the transition from a region where the production of turbulenceis dominated by shear ( y < . L c ), to a region where it is affected by curvature ( y > . L c ). Hence, by this definition, for S = 1 we have y + = L + c . The definition fromequation (1.7) builds on the existence of a shear logarithmic region, where the gradientof the mean angular velocity is ddr U = u τ / ( κy ). The angular velocity scale for roughwalls is approximated as ω = ω i + ∆ω . Thus the generic curvature Obukhov length L c for smooth and rough walls can be defined with the inner cylinder rotation rate ω i , andthe wall-shear stress τ w only, similar to equation 1.7, but now for a rough wall, L c = u τ κ ( ω i + ∆ω ) , (3.2)so that L + c ( ∆ω + = 0) = L + c,s .Figure 2(a) presents the gradient of the mean angular velocity profile versus S ,calculated from the PIV results. We find fair collapse of the velocity gradients of smooth(grey) and rough (coloured) wall profiles. When the effects of curvature are negligible S (cid:62) O (10), the gradient of the velocity profile approaches κ − ≈ .
5. This occurs in a
P. Berghout et al.
Figure 3: Schematic of the various regions in smooth and IC-rough turbulent TC flow atmatched L + c . The height y + = 0 . L + c is defined as the location where the logarithmicprofile with slope λ − ends and the constant angular momentum region of the bulkvelocity starts.very small region close to the wall, where we cannot measure due to the presence of thesandpaper roughness. For the rough and smooth wall velocity profiles, we find that thegradient approaches λ − in the region where curvature and shear affect the flow. For S (cid:54)
1, curvature effects dominate the flow, and a constant angular momentum region(i.e. the bulk flow) sets in (Berghout et al.
The mean angular velocity profile
Figure 2(a) shows that the curvature and shear affected region of the BL contains aconstant gradient (= λ − ) of the mean angular velocity. From this observation, Berghout et al. (2020) obtained an equation of the mean angular velocity in the shear and curvatureaffected region in the BL (in short ‘curvature log’).The offset of the logarithmic velocity profile (with slope λ − ) in the curvature and shearaffected region, as indicated in figure 3, is a function of the wall normal location wherecurvature related effects impact the flow. From PIV results, the exact location was foundto be y + = 0 . L + c , with L + c defined in (3.2). Therefore, the offset is κ − log 0 . L + c + A ,where A = 5 . y + = 0 . L + c is not sharp but gradual.To account for this, we introduce a constant C bl that connects the logarithmic velocityprofiles of both regions. Berghout et al. (2020) found that A + C bl + (cid:0) κ − λ (cid:1) log(0 .
2) = 1 . y + = 0 . L + c as ω + = 1 κ log 0 . L + c + A + C bl + 1 λ log y + . L + c = 1 λ log y + + (cid:18) κ − λ (cid:19) log L + c + 1 . , (3.3)with C bl = − .
30. The transition from the curvature and shear affected region to theconstant angular momentum region occurs at y + = 0 . L + c . This height we take as our C flow with rough surfaces
Scale separation
Key to the understanding of the effects of roughness in TC turbulence is the conceptof scale separation. To illustrate this, in figure 2(b) we plot S versus the wall-normaldistance y/L c = ( r − r i − k σ ) /L c . We note that the wall offset 2 k + σ of the roughwall is an approximation. As a reference, we also plot the smooth wall profile (grey)at T a = 6 . × (Huisman et al. Re τ , L + c and k + σ . The friction Reynolds number Re τ from equation (2.2) gives the ratio of thelargest dynamical length scale in the TC setup to the viscous length scale δ ν . Re τ is ofthe same order as in the smooth wall TC experiments by Huisman et al. (2013), where itwas Re τ = 488–23 093, comparable to the rough BL experiments by Squire et al. (2016),where Re τ = 2890–29 900.The roughness scale in our experiments is much larger than the viscous length scale δ ν , i.e. k + σ = 34–127 (cid:29)
1, and thus pressure drag dominates over viscous drag. For theflat plate BL experiments of Squire et al. (2016) in the fully rough regime, we estimatethat k + σ = 9–12 is required, based on the data for which ∆U + > . et al. (2016). Hence, we are confident that we are indeed far in the fully rough regime. Wealso find that the roughness sublayer height ≈ k + σ is smaller than the outer boundof the shear dominated logarithmic region, ≈ . L + c . For the lowest roughness we have3 k + σ / . L + c = 0 .
40, and for the highest roughness it is 3 k + σ / . L + c = 0 .
20. This separationof length scales allows for a region where the logarithmic velocity profile can form. Forexample in the smooth wall experiments of Huisman et al. (2013) such a profile was foundbetween 50 (cid:54) y + (cid:54)
600 for comparable
T a . We finally find that the outer bound of thecurvature dominated logarithmic region L + c is smaller than the outer length scale Re τ ,so that 0 . L + c /Re τ ≈ .
33. For y + > . L + c the curvature dominated bulk, constantangular momentum region, forms. The occurrence and extent of this constant angularmomentum region depends on the radius ratio η , i.e. L + c /Re τ depends on η .Table 1 suggests that the roughness only affects the inertial region where curvatureeffects are negligible. Hence, we expect that the velocity shift of that region is similar tothat of identical sandpaper in a flat plate turbulent BL. In other words, we would expecta fully rough asymptote with slope κ − and a similar value of k s as we would measurefor identical sandpaper in a flat plate turbulent BL.
4. Mean velocity profiles of the inner cylinder boundary layer
In this paper we will show the angular velocity profile ω + ( y + ) rather than the azimuthalvelocity profile u + ( y + ), as it is ω + ( y + ) which is expected, given the arguments based onthe Navier–Stokes equations, to have a logarithmic profile (Grossmann et al. ω + = (cid:104) ω i − ω ( r ) (cid:105) t /ω τ , with ω τ = u τ /r i , versus the wall-normal coordinate y + . In this and thenext section, we focus our analysis on the mean velocity profiles of the inner cylinderBL, hence u τ = u τ,i throughout. In § et al. (2020) for an analysis of the smooth velocity profiles of the outer cylinderBL.Figure 4(a) shows that with increasing roughness the rough wall profiles are increas-ingly shifted downwards, as expected. More importantly, we find from the diagnostic0 P. Berghout et al.
Figure 4: (a) Mean angular velocity ω + versus the wall-normal distance y + . The solidlines are the measured rough wall profiles. The dashed lines represent the theoreticalsmooth wall reference profiles (colors are the same), calculated from equation (3.3) andat matching L + c and Re τ with the rough wall profiles. (b) The compensated gradientof the rough wall profiles in (a), where the wall-normal distance is normalized withthe curvature length L + c . The colors are the same in both figures. The grey line is thesmooth wall profile at T a = 6 . × , obtained from Huisman et al. (2013). The dashedhorizontal line represents the slope λ − of the logarithmic velocity profile in the regionwhere turbulence production is dominated by curvature effects. The dotted horizontal linerepresents the slope κ − of the logarithmic velocity profile in the region where turbulenceproduction is dominated by shear.function y + dω + dy + (a useful representation of the gradients (Pope 2000)) in figure 4(b),that the slope λ − of the curvature dominated logarithmic region is the same for roughwall TC turbulence as for smooth wall TC turbulence (grey line). Unfortunately, wecould not resolve the very thin spatial region where a shear dominated logarithmic wasfound by Huisman et al. (2013), as the roughness peaks obstruct the view for the PIVvery close to the wall.For a rough wall, ∆ω + is a function of the equivalent sand grain roughness height k + s and the curvature length L c , so that ∆ω + ( k + s , k s L c ). When k s (cid:28) L c , the angular velocityshift only depends on k s , and the shift becomes ∆ω + ( k + s ). Since the inner cylinderrotates, the plus sign in the denominator of equation (3.2) is connected with the increaseof angular fluid velocity in the inner cylinder BL due to the roughness. When we normalizethe wall-normal distance with L + c , we expect the transition from a curvature logarithmicvelocity profile to the constant angular momentum bulk velocity profile to occur at y + /L + c = 0 .
65. In figure 4(b) we find a fair collapse of both smooth and rough wallprofiles in wall-normal direction, when normalized with curvature length L + c .
5. The fully rough asymptote
From the observation that both smooth and rough wall velocity profiles possess thesame slope λ − of the curvature logarithmic region (figure 4b) we proceed to calculatethe angular velocity shift ∆ω + . Due to the roughness the angular velocity profiles in theshear logarithmic region are shifted, as discussed in § C flow with rough surfaces ∆ω + of the rough wall profiles with respect to the reference smooth wall profiles.(a) Velocity shift versus the wall-normal distance y + /L + c (colors the same as figure 4). (b)The velocity shift ∆ω + , crosses in (a), versus the equivalent sand grain height k + s . Blacksymbols are the experimental values. The solid black line is the fully rough asymptote ofNikuradse (1933), equation (5.1). The solid blue line is an illustration of the curvaturefully rough asymptote, with slope λ − and arbitrary vertical shift.offset of that region scales with κ log(0 . L + c ) + A (figure 3). Hence, it is imperative tocalculate the angular velocity shift from the smooth wall velocity profile at matching L + c .Figure 4(a) shows these smooth wall profiles (dashed), where the colors match therespective rough wall cases, and both L + c and Re τ are matched. The velocity shift ∆ω + ( y + ) from the theoretical smooth wall profile, equation (3.3), is plotted in figure5(a). The horizontal plateaus confirm the similarity of the slopes of the velocity profiles.We extract ∆ω + at y + ≈ . L + c and plot the shift versus the roughness height in figure5(b). When we fit a function of the form ∆ω + = a log k + + b through all seven datapoints, we obtain a = 0 . ± .
02, to within 15% of the von K´arm´an constant κ ≈ . δ ν (cid:28) k < L + c has slope κ − .For reference, this is much higher than λ − ≈ . − , blue line in figure 5(b).To obtain a measure of the equivalent sandgrain roughness height k s , we fit the datapoints to the fully rough asymptote of Nikuradse (1933) ∆ω + ( k + s ) = 1 κ log k + s + 5 . − . , (5.1)and obtain k s = 5 . k σ = 0 .
97 mm, for κ ≈ .
40. For reference, the typical grain size isestimated by 6 k σ = 1 .
05 mm Bakhuis et al. (2020).
6. The equivalent sand grain roughness height
The hypothesis in this research, postulated in § δ ν (cid:28) k < L c is the same (or very similar) to the fully roughasymptote in flows without streamwise curvature. We have already demonstrated in § κ − of the fully rough asymptote is indeed (almost) the same. This leavesus with a comparison of the value of k s , between TC turbulence and canonical systemswithout streamwise curvature.In literature, we have found two reports on turbulent flows over sandpaper roughness:the work of Squire et al. (2016), employing 36 grit sandpaper in a turbulent BL, and2 P. Berghout et al.
Figure 6: Relationship between the equivalent sand grain roughness height divided by theroot-mean-square height k s /k σ , and the skewness parameter Sk of different sandpapersurfaces. The solid black line is the empirical correlation for Sk > et al. (2020). Data from turbulent boundary layer flow using grit (12, 24, and 80) sandpaper(Flack et al. et al. | dkd ( rθ ) | (Napoli et al. et al. (2007) who employed (12-, 24-, and 80-) grit sandpaper in turbulent BL flow.In the rough wall TC experiments reported here we used grit 36 sandpaper. However, itis essential to realize that sandpaper is not only defined by the grit size. Other statistics,like the skewness (an important parameter (Forooghi et al. et al. (2016)), do vary with manufacturing methods. We have triedto use the very same sandpaper type (SP40F, Awuko Abrasives ) as Squire et al. (2016).Unfortunately, the sandpaper turned out to be not waterproof, and detached from theinner cylinder. We then applied new water resistant sandpaper (
VSM , P36 grit ceramicindustrial grade), with different surface roughness statistics.To compare the drag property of the sandpaper surfaces in TC, to the respectivesandpaper surfaces in literature, we plot the relationship between k s and the root-mean-square and skewness in figure 6. The surface properties of the sandpaper surface fromFlack et al. (2007) are taken from Flack & Schultz (2010). The solid black line is theempirical correlation from Flack et al. (2020). We find that the relation between k s andthe Skewness Sk and the root-mean-square height k rms of sandpaper used in our roughwall TC experiments is consistent with the empirical trend given for the sandpaper usedin rough wall turbulent BL flow analysis. Whether, the deviation originates from thedifference between TC and canonical systems without curvature, or originates from thedifferent surface statistics (e.g. the ES for the present surface is higher, indicating adenser surface), remains to be resolved.
7. The constant angular momentum region in the bulk
Thus far, we have discussed the velocity profiles of the inner cylinder boundary layer,i.e. y + < . L + c . By means of matching this profile to the bulk velocity profile atboundary layer height, one can derive the relationship between the torque N u ω ( T a ) C flow with rough surfaces ω/ω i , versus the radius ( r − r i ) /d normalized with the gap width d . The profilesfor different roughness heights k + σ are compared. The bulk profile is strongly shiftedtowards the rough inner cylinder, as the roughness there enhances the coupling betweenthe inner BL and the bulk, similarly as the ribs have done in Zhu et al. (2018). (b) Theangular momentum M , normalized with the inner cylinder angular momentum M i . Solidlines are the PIV results and dashed lines ( M b /M i ) are calculated from equation (7.1).The colors are the same in both figures. The grey line is the smooth wall profile at T a = 6 . × , obtained from Huisman et al. (2013).and the velocity of the inner cylinder (Cheng et al. et al. M b ) is constant (Wendt 1933; Townsend 1976), and, in fact,very close to half the inner cylinder angular momentum ( M i = ω i r i ), M b = 0 . M i , forstationary outer cylinder. For rough wall TC flow however, and especially for asymmetricroughness when the inner cylinder is of a different roughness height than the outercylinder, the exact value of M b is a priori unknown. However, what was shown is that forvery rough walls the bulk azimuthal velocity profile is shifted towards the rough cylinder,due to the stronger coupling to that side thanks to the roughness (Zhu et al. et al. r = r i + δ r , where δ r = 0 . L c .The momentum ratio ( M b /M i ) is the angular momentum in the bulk over the angularmomentum of the inner cylinder M b M i = ω | y = δ r ( r i + δ r ) ω i r i , (7.1)where ω | y = δ r = ω τ,i ( ω + i − ω + r ( y + = δ + r )), and we use the velocity profile of the roughinner cylinder BL, figure 3 and (3.3)- ∆ω + , ω + r ( y + = δ + r ) = λ log δ + r + (cid:0) κ − λ (cid:1) log L + c,r +1 . − ∆ω + . Figure 7 compares the result from equation (7.1) (dashed line) with theexperimentally obtained velocity profiles (solid lines), demonstrating agreement betweenthe calculated and the measured profiles. This supports the assumption that also therough-wall velocity profiles conform to a constant angular momentum in the bulk. Finally,we point out that the ‘overshooting’ of the profiles in the bulk, i.e. the slight increase in M with increasing r , is likely an effect of the turbulent Taylor vortices, and is thereforeexpected to depend on the height coordinate z (Huisman et al. P. Berghout et al.
Similar overshooting is well known from temperature profiles in turbulent Rayleigh–B´enard flow (Tilgner et al. et al.
8. Calculation of
N u ω ( T a ) and C f ( Re ) Since the angular momentum in the bulk is to a good approximation constant, we canmatch the angular momentum of the inner cylinder BL at BL height with the angularmomentum of the outer cylinder BL at BL height, i.e. M ( δ i,r ) = M ( δ o,s ). This approach isbased on the matching of BL and bulk velocity profiles in the recent CPS model (Cheng et al. i , o ) refer to inner cylinder and outer cylinder BL quantities,where subscripts ( s , r ) refer to smooth and rough wall quantities, and δ = 0 . L c for innercylinder and outer cylinder (rough and smooth) so that δ + i,r = αL + c,i,r , δ + o,s = αL + c,o,s with α = 0 .
65. The matching argument becomes( r i + δ i,r ) ω τ,i ω + IC ( δ + i,r ) = ( r o − δ o,s ) ω τ,o ω + OC ( δ + o,s ) , (8.1)where we realize that ω τ,o = η ω τ,i . We substitute the BL equations for respectivelyrough and smooth walls into equation (8.1) and obtain( r i + δ i,r ) ω τ,i (cid:18) ω + i − λ log( δ + i,r ) − (cid:18) κ − λ (cid:19) log( L + c,i,r ) − C i + ∆ω + (cid:19) =( r o − δ o,s ) ω τ,o (cid:18) λ log( δ + o,s ) + (cid:18) κ − λ (cid:19) log( L + c,o,s ) + C o (cid:19) . (8.2)The rough wall, inner cylinder BL height δ + i,r , and the velocity shift ∆ω + are functions ofthe sand grain size k + s . This makes the matching equation more involved, in comparisonto the smooth wall case (Cheng et al. et al. et al. (2020), we now rewrite the equation in terms of Re τ,i and Re i .The inner cylinder angular velocity becomes ω + i = Re i Re τ,i . (8.3)The equivalent sand grand size is k + s = 2 k s d Re τ,i = (cid:15)Re τ,i . (8.4)The fully rough asymptote from equation (5.1) can now be rewritten as ∆ω + = 1 κ log( (cid:15)Re τ,i ) + A − B. (8.5)The inner cylinder, rough wall, BL height δ + i,r is rewritten from δ + i,r = αL + c,i,r as δ + i,r = 2 αηRe τ,i κ (1 − η ) Z ; with Z = (cid:18) Re i Re τ,i + 1 κ log( (cid:15)Re τ,i ) + A − B (cid:19) . (8.6)The outer cylinder, smooth wall, BL height δ + o,s is rewritten from δ + o,r = αL + c,o,r as δ + o,s = 4 αη Re τ,i κ (1 − η ) Re i , (8.7) C flow with rough surfaces k s /d = 0 .
012 and smooth wall TC turbulence. (a)
N u ω = Re τ η (1+ η ) Re i versus T a . (b) Friction factor C f = Re τ Re i versus Re i . Black diamondsare the smooth wall experiments of van Gils et al. (2011 b ), where the solid black lineshows the theory of Berghout et al. (2020) for smooth wall TC turbulence. Blue squaresare the rough inner cylinder measurements from the present work. The solid blue line isequation (8.8).We can now substitute equations (8.3)–(8.7) into (8.2), and obtain (cid:16) ακ Z (cid:17) (cid:18) Re i Re τ,i − λ log (cid:18) αηRe τ,i κ (1 − η ) Z (cid:19) − (cid:18) κ − λ (cid:19) log (cid:18) ηRe τ,i κ (1 − η ) Z (cid:19) + 1 κ log( (cid:15)Re τ,i ) + A − B − C i (cid:19) = (cid:18) − αηRe τ,i κRe i (cid:19) (cid:32) λ log (cid:32) αη Re τ,i κ (1 − η ) Re i (cid:33) + (cid:18) κ − λ (cid:19) log (cid:32) η Re τ,i κ (1 − η ) Re i (cid:33) + C o (cid:33) . (8.8)This implicit equation can be solved numerically to obtain Re τ,i ( Re i ) with parameters C i = 1 . , C o = 2 . , A = 5 . , B = 8 . , κ = 0 . , λ = 0 . , α = 0 .
65 for these experiments, η = 0 .
714 and (cid:15) = 0 . /
80. Finally, by means of equations (1.3) – (2.1), we express theresult Re τ,i ( Re i ) into N u ω ( T a ) and C f ( Re i ) respectively.Figure 8 presents the final result, together with the experimental data from smoothwalls (van Gils et al. b ) and with the equation for smooth wall TC (Berghout et al. κ − , λ − ), the offset of the smooth velocity profile ( A, C i , C o ) or the BL thickness fitfor smooth walls α . This reflects that all parameters are universal for all radius ratios,and cannot and need not be ‘tuned’.The agreement between equation (8.8) and the experimental data (the maximum erroris only ≈ k s with areasonable accuracy. This means that the TC facility can potentially be used for direct,fast, measurements of surface drag properties, as characterized by k s .6 P. Berghout et al.
9. Summary, conclusions, and outlook
We carried out experiments of inner cylinder rotating (and stationary outer cylinder)Taylor–Couette (TC) turbulence with a rough inner cylinder and a smooth outer cylinder.We measured the torque, and, by means of PIV, the mean angular velocity profiles. Therough surface consisted of P36 industrial grade sandpaper, where the roughness height k = 6 k σ , with k σ the standard deviation of the roughness height, over the gap width d was k/d = 0 . k was much larger than the viscous lengthscale δ ν , such that k/δ ν = 204–762. The velocity shift of the rough wall azimuthalvelocity profiles was, compared to the reference smooth wall, in the log-law region ∆ω + > . × < Re i < . × . Hence the sandpaperwas hydrodynamically fully rough. Furthermore, the roughness height k σ < . L c ,where 0 . L c is the height (Berghout et al. κ = 0 . ± .
02, was similar to previous findings in flat plateBLs κ ≈ .
38. Also, the value of the equivalent sand grain roughness height k s comparedreasonably well with those found for sandpaper in flat plate BLs (Flack et al. et al. N u ω ( T a ), we employed a matching argument between the innercylinder BL rough mean angular momentum profile at the inner cylinder BL height, andthe smooth outer cylinder BL mean angular momentum profile, at the outer cylinderBL height, based on the CPS model of Cheng et al. (2020), see also Berghout et al. (2020). To justify this, we first showed that for a rough wall inner cylinder, a region ofconstant angular momentum exists in the bulk. We find a convincing overlap between thecalculated value of the torque (or wall shear stress), and the experimentally measuredvalues of the torque, with a maximum error of ≈ k s . It seems that the value of k s found in TC is similar to the value of k s found in flat plate BLs.As an outlook to future work, we propose that more studies in both turbulent flatplate BLs and turbulent TC flow, with identical rough surfaces, are carried out to furthercompare the drag properties of these surfaces. Further unanswered questions include theeffects of even more considerable roughness penetrating the curvature affected logarithmicregime of the BL, which is related to finding the slope of the fully rough asymptote inthat region. This could also be achieved by employing a TC setup with a lower radiusratio η , thus increasing curvature effects. Acknowledgements
We would like to thank B. Benschop, M. Bos, and G.W. Bruggert for their technicalsupport, Y.A. Lee for his support in the lab and M.A. Bruning for discussions.This study was funded by the Netherlands Organisation for Scientific Research (NWO)through the Multiscale Catalytic Energy Conversion (MCEC) research center and theGasDrive project 14504, by the European Research Council (ERC) Advanced Grant“Droplet Diffusive Dynamics”, and by the Priority Programme SPP 1881 Turbulent
C flow with rough surfaces
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