A new equivalent sand grain roughness relation for two-dimensional rough wall turbulent boundary layers
5.3 INTRODUCTION
A new equivalent sand grain roughness relation for two-dimensional
than flows over smooth surfaces. The increase of Cf is manifested in the downward shift of wall−unit normalised mean velocity profile when compared with the smooth wall profile (Hama 1954). The shift, known as the roughness function, is defined as ∆U+ = ∆U/Uτ, where U is the mean streamwise velocity and Uτ is the friction velocity Uτ = √
τw/ρ; the superscript, (+) means normalisation by the wall units Uτ or ν/Uτ, where ν is the kinematic viscosity.
The roughness function is a function of the equivalent sand grain roughness ks (Schlichting &
Kestin 1961). Ideally, one would like to determine ∆U+based solely on the roughness topology.
This would provide a way of predicting Cf without the need to perform lengthy measurements.
Unfortunately, ks is not a physical roughness parameter and its determination relies on an empirical method based on the mean velocity profile (Nikuradse 1933).
When ks+ is large enough the flow is considered to be in the ‘fully rough’ regime and the drag is mainly composed of the pressure (or form) drag (Squire et al. 2016). This regime is characterised by a log−linear relationship between ∆U+ and k+s. On the other hand, when ks+ is too small, the roughness does not have a noticeable effect on the viscous sublayer of the flow. This flow regime is called ‘dynamically smooth’ and can be treated as a smooth wall flow.
At moderate values of k+s, both the pressure drag and the viscous drag contribute to the total drag. This flow regime is called ‘transitionally rough’ (Nikuradse 1933). So far there are no clear thresholds that delimit these flow regimes. However, it is commonly agreed that the flow is hydraulically smooth when ks+<4, and fully rough whenk+s >70 (Jim´enez 2004).
The roughness function ∆U+ is determined by ‘measuring’ the vertical shift between the log regions of the rough wall and smooth wall wall-unit normalised velocity profiles. This requires that experiments be conducted for each rough surface to obtain the velocity profile and friction velocity. A more practical way to proceed, but one that also presents a big challenge, is to predict ∆U+ directly from surface topology or by other means, correlating ks to one or more parameters characterising the roughness. So far, there has been no definite consensus on which length scale or roughness parameter best characterises a surface that can correlate with the friction drag. Over the last decade, a large body of work has been undertaken to address this issue (Musker 1980; Napoli, Armenio & De Marchis 2008;Schultz & Flack 2009; Forooghi et al. 2017; Flack, Schultz & Barros 2020, among many others). A wide range of different surface parameters has been studied to predict which of these parameters dominantly affects the friction drag. Some of these parameters are the mean roughness height ka, root-mean-square heightkrms, maximum peak to valley heightk, average peak to valley height kz, effective slope ES, solidarityλf, skewnessksk, and flatnesskku. However, none of these parameters is sufficient to be generalised for all kinds of roughness.
Musker(1980) developed a modified roughness Reynolds number as a function of simple geometric statistics (average absolute slope, standard deviation, skewness, and flatness) for naturally occurring surfaces as follows:
ks+= σUτ
ν (1+aES)(1+bkskkku), (5.1) where a and b are constants to give the best fit of roughness functions related to naturally occurring surfaces, andσ is the standard deviation of the surface roughness.
Waigh & Kind (1998) studied a regular three-dimensional (3-D) roughness and found that the element bluntness and spanwise aspect ratio are the main parameters characterising
the roughness function as follows:
∆U+=
⎧⎪⎪⎪⎪⎪⎪⎪
⎪⎪⎨⎪⎪⎪⎪
⎪⎪⎪⎪⎪⎩
10.56log[( 1 λf
k bm)
0.87
(Aw Af)
0.44
]−7.59 Λ<6,
−5.57log[( 1 λf
k bm)
0.55
(Aw Af)
1.38
]+5.78 Λ>6,
(5.2)
where Λ = λfk/sm, sm is the streamwise width of the roughness element, bm is the spanwise width of the roughness element,λf =Af/As,As is the surface area, Af is the projected frontal area, and Aw is the wetted area.
Bons(2002) derived a newkscorrelation, based on the surface slope angle for real turbine roughness; the roughness panels were scaled models of actual turbine surfaces instead of using the traditional simulated roughness
ks=−0.0261ES+0.0138ES2. (5.3) Schultz & Flack (2009) studied close-packed pyramid roughnesses to study the effect of the height and slope of roughness on the flow. They found that ∆U+ mainly depends on the roughness height. They also found a range of ES such that, below this level, the surface is considered to be wavy, not rough, and the roughness function strongly depends on the ES. If ES<0.35, the surface is considered to be wavy.
Forooghiet al.(2017) investigated numerically the effect of various roughness parameters on ks in the fully rough regime of channel flow at friction Reynolds number Reτ ≈ 500. Their roughness geometries were generated by systematically changing the moments of the surface height probability density function, the effective slope of the random roughness and the size distribution of the roughness peaks. Forooghiet al. (2017) correlateks based onk,ksk and Es. They found that these roughness parameters can predictksaccurately for randomly distributed roughness in the fully rough regime of channel flows
ks/k=F(ksk).G(ES),
F(ksk)=0.67ksk2 +0.93ksk+1.3, G(ES)=1.07.(1−e−3.5ES).
⎫⎪⎪⎪⎪⎪⎪⎪
⎬⎪⎪⎪⎪⎪⎪⎪
⎭
(5.4)
Flacket al.(2020) investigated the importance of roughness height and skewness on the coefficient of friction. Aks correlation was derived as:
ks=A1krms(1+ksk)B1, (5.5) withA1 =4.43, andB1=1.37.
While many roughness function formulations in terms of one or more roughness param- eters have been proposed recently, they are limited to the particular roughness investigated. In other words, they lack universality. This is certainly associated with the fact that, so far, a ‘uni- versal’ critical parameter for all roughnesses is yet to be determined. Whether such a ‘universal’
parameter exists or not is still an open issue. It is nevertheless conceivable that such a parameter can be identified for families of rough surfaces, such as 2-D roughness, and 3-D roughness, in the fully rough regime. The present work is an attempt to determine such a parameter with the view
to develop an expression forks. The roughness family considered here is that of 2-D roughness.
One reason to focus on this roughness family is that it was shown to produce a fully rough regime at relatively low Reynolds number, which can be easily achieved in wind tunnels. For example the use of 2-D rods attached to a smooth wall where the spacing between two consecutive rods, sx, is approximately 8kleads to a fully rough regime (Djenidi, Talluru & Antonia 2018). In their direct numerical simulations of a turbulent rough wall channel flow,Leonardiet al.(2003) used transverse square bars as a roughness, with sx varying from 2k to 32k. They showed that the form drag was the main contributor to the total drag when 8k≤sx ≤16k. Later,Leonardiet al.
(2015) used circular rods and investigated friction and pressure drags numerically, confirming their previous observations. Kamruzzaman et al. (2015) investigated the turbulent boundary layer developed over a rod roughened wall that has a spacing between two consecutive rods of 8k. They calculated the pressure drag directly from the distribution of static pressure around one rod. They revealed that theCf is constant and independent of the Reynolds number in the fully rough regime. This was further confirmed by Djenidiet al. (2018).
In this present work, we carry out turbulent boundary layer measurements over various 2-D rough surfaces with a height ratio δ/k ranging from 23 to 41. Two sets of experiments are conducted. The first one is to validate our rough wall measurements and estimation of Uτ. Also, the experiments are used to validate the Reynolds number independence when the flow is fully rough. The second set of experiments is exploited to determine the most critical dominant roughness parameters impacting Cf with the view to developing an expression for k+s valid for the 2-D rough wall family.