Numerical observations of turbulence structure modification in channel flow over 2D and 3D rough walls

International Journal of Heat and Fluid Flow 56 (2015) 108–123

Contents lists available at ScienceDirect

International Journal of Heat and Fluid Flow journal homepage: www.elsevier.com/locate/ijhff

Numerical observations of turbulence structure modification in channel flow over 2D and 3D rough walls M. De Marchis a,⇑, B. Milici a, E. Napoli b a b

Facoltà di Ingegneria e Architettura, Università degli Studi di Enna ‘‘Kore’’, Cittadella Universitaria, Enna, Italy Dipartimento di Ingegneria Civile, Ambientale, Aerospaziale, dei Materiali, Università degli Studi di Palermo, Viale delle Scienze, Palermo, Italy

a r t i c l e

i n f o

Article history: Received 2 March 2015 Received in revised form 3 June 2015 Accepted 2 July 2015

Keywords: Turbulence simulation Channel flow LES Three-dimensional irregular roughness

a b s t r a c t The effects of wall roughness on turbulence structure modifications were explored by numerical experiments, carried out using Large Eddy Simulation techniques. The wall geometry was made using an archetypal artificial method, thus to achieve irregular two- and three-dimensional shapes. The proposed roughness shapes are highly irregular and are characterised by high and small peaks, thus it can be considered a practical realistic roughness. Their effects are analysed comparing the turbulence quantities over smooth, 2D and 3D rough walls of fully developed channel flow at relatively low friction Reynolds number Res ¼ 395. Both transitional and fully rough regimes have been investigated. The two rough surfaces were built in such a way that the same mean roughness height and averaged mean deviation is obtained. Despite of this, very different quantitative and qualitative results are generated. The analysis of the mean quantitative statistics and turbulence fluctuations shows that deviations are mainly concentrated in the inner layer. These results support the Townsend’s similarity hypothesis. Among the geometrical parameters, which characterise the wall geometries, roughness slope correlates well with the roughness function DU þ . Specifically, a logarithmic law is proposed to predict the downward shift of the velocity profile for the transitional regime. Instantaneous view of turbulent organised structures display differences in small-scale structures. The flow field over rough surfaces is populated with coherent structures shorter than those observed over flat planes. The comparative analysis of both streaks and wall-normal vortical structures shows that 2D and 3D irregularities have quite different effects. The results highlight that 3D rough wall are representative of a more realistic surface compared to idealised 2D roughness. Ó 2015 Elsevier Inc. All rights reserved.

1. Introduction Rough surfaces are encountered in a wide range of engineering and environmental applications. In the literature, several studies have been carried out in the last decades, starting from the pioneeristic Nikuradse’s experiments, aimed to clarify and to understand the effects of turbulent flows over non-smooth surfaces. The natural degradation or modification of flat surfaces in time is one of the main causes of roughness appearance and increase. The natural wall roughness is highly irregular, with peaks and caves of different heights, oscillating around a mean value (see, e.g. the turbine surface analysed by Barros and Christensen (2014)). It is clear that the analysis of turbulent flows over rough surface is of primary interest for physical and engineering ⇑ Corresponding author. E-mail addresses: [email protected] (M. De Marchis), [email protected] (B. Milici), [email protected] (E. Napoli). http://dx.doi.org/10.1016/j.ijheatfluidflow.2015.07.002 0142-727X/Ó 2015 Elsevier Inc. All rights reserved.

purposes. Several studies were, thus, carried out by means of experiments or numerical simulations aimed to reproduce roughness through two- or three-dimensional regular elements arranged over a flat plate (see Djenidi et al. (1999), Leonardi et al. (2003), Ikeda and Durbin (2007), Volino et al. (2011), Dritselis (2014) for 2D elements and Hong et al. (2011), Boppana et al. (2010) for 3D elements). Despite the above contributions gave fundamental elements in order to understand the physical processes arising in turbulent flows over rough surface, this kind of idealised roughness is affected by the length scales of the roughness elements and by their distribution, so that they could not reflect the behaviour of a real roughness, as already observed by Barros and Christensen (2014) or Yuan and Piomelli (2014). In order to overcome this lack, in the last years researches have been carried out focusing the attention on the effects of irregular surfaces. The analysis were mainly focused on laboratory experiments. One of the first experimental analysis of turbulent flow over irregular rough surface was performed by Van Rij et al. (2002a,b), who investigated a kind of

M. De Marchis et al. / International Journal of Heat and Fluid Flow 56 (2015) 108–123

three-dimensional roughness spreaded over a flat surface. Their analysis was focused to the determination of the equivalent sand grain roughness ks . Zhang et al. (2003) proposed a method for the determination of the skin friction number over rough turbine airfoils. Later, Subramanian et al. (2004) pointed out that the equivalent sand grain roughness gives inconsistent results in terms of flow characteristics and analysed the effect of a realistic strong irregular roughness on the mean velocity profile and turbulent intensities. Following several researches, the analysis was carried out also in term of outer layer similarity. The effect of real irregular rough walls on turbulence quantities was extensively investigated measuring the flow field though the particle-image velocimetry (PIV). Wu and Christensen (2006) reproduced the surface topology of the replicated turbine-blade roughness analysing the Reynolds stress enhancement induced by roughness. They found that surface roughness significantly increases the ejections and sweeps, compared to the smooth-wall case, whereas the inward and outward interaction contributions seem to be unchanged. Later the authors, applying the same PIV methodology, focused the attention on the validity of Townsend’s wall similarity hypothesis Townsend (1976). In contrast with some researches, but coherently with others, the authors found that, when scaling with inner variables (i.e. the friction velocity us ), a collapse of the statistical quantities is achieved in the region located beyond the roughness sublayer 5k, with k the roughness parameter. A similar analysis was performed by Mejia-Alvarez and Christensen (2010), who focused the attention on the relative impact of various topographical scales within irregular surface roughness on a turbulent boundary layer under both developing- and developed-flow conditions. Looking at the developed flow, the authors found that all the irregular surfaces analysed reproduce the outer-layer characteristics for flow over smooth surface, in accordance with Townsend’s wall similarity hypothesis. Differences between smooth walls, irregular and regular roughness was recently investigated by Birch and Morrison (2011), looking at the similarity hypothesis, according to Bakken et al. (2005). They found the interesting issue that regularly woven meshes lead to a lack of similarity in the outer region, whereas for irregular rough surface the Townsend’s hypothesis holds. This result is coherent with the findings of Krogstad and Antonia (1999), who observed that regular woven mesh affects the outer layers. At the same time, the result of Birch and Morrison (2011) is consistent with the researches conducted over irregular rough surfaces (see Bons et al. (2001) and Wu and Christensen (2007) and literature therein cited). The above cited experimental researches investigated the effects of wall roughness in terms of mean statistical quantities and less attention was given to turbulence structure modification induced by the irregularities. On the other hand, the effect of rough wall on turbulence structures was deeply investigated over 2D or 3D regular rough surfaces (see among others, Coceal et al., 2006; Ashrafian and Anderson, 2006; Ikeda and Durbin, 2007; Volino et al., 2011; Djenidi et al., 2008; Hong et al., 2011; Lee et al., 2011). It is only recently that experimental analysis have been carried out in the light of turbulence structures. Mejia-Alvarez and Christensen (2013) and Mejia-Alvarez et al. (2014) observed that near-wall turbulence production cycles, typical of smooth-wall, are modified by the roughness. The analysis of turbulent flow over irregular rough surfaces, carried out by Barros and Christensen (2014), revealed that the spanwise irregularities produce secondary flows that could be hide by classical 2D roughness. Specifically, Barros and Christensen (2014) found that low and high momentum regions are alternated along the spanwise direction. The authors attributed this specific behaviour to the heterogeneity of the roughness in the spanwise direction. These results clearly show the importance of further investigation on the subject.

109

One of the main problems connected with experiments is related to the possibility of carrying out velocity measurements in the near-wall regions, especially when irregular rough surfaces are considered. This problem can be overcome by the use of numerical simulation techniques, even at low-Reynolds number. Conversely, due to computational cost and difficulty in reproducing real irregularities by means of non-uniform meshes, few numerical researches have been conducted till now. Starting from the analysis of Bhaganagar et al. (2004), where a 3D rough surface (egg carton like) was reproduced using a sinusoidal function applied along the streamwise and spanwise directions; later, Singh et al. (2007) performed DNS of the fully rough flow over three-dimensional roughness elements. In both studies 3D wall roughness was built with 3D peaks and troughs of the same height, thus hiding the possible effects of the randomness of peak and valley heights. Looking at numerical analysis of turbulent flow over real rough surfaces, Cardillo et al. (2013) performed DNS of turbulent boundary layer over an irregular rough wall, reproducing the primitive topographical shape already studied by Brzek et al. (2007), throughout laboratory experiments. The authors confirmed the ability of the numerical methods to reproduce the experimental data. Recently, Yuan and Piomelli (2014) compared the effect of four types of realistic roughness, having different topographical characteristics, with the sand-grain roughness, through Large Eddy Simulation (LES) technique. First and second order statistic were analysed, showing that, both in transitional or fully rough regimes, the effects are mainly confined in the roughness sublayer. The authors focused also the attention on roughness parametrisation, finding new insight in the critical effective slope ES, defined by Napoli et al. (2008) and later analysed by Schultz and Flack (2009). The effects of irregular rough surfaces, derived by topographical shape of real turbine blade damaged by spallation, was studied by Nagabhushana Rao et al. (2014) applying Large Eddy Simulation technique. Single and coupled effects of incoming free stream turbulence and surface roughness were systematically studied. The analysis showed that streaks induced by free-stream turbulence are intermittent in nature and streaks due to roughness are steady for a given topology of the rough surface. The literature cited above shows that numerical models are powerful tools for investigation and comprehension of turbulent physical processes which arise over complicated surfaces. Despite of this, several efforts are still required. Starting from the analysis of De Marchis and Napoli (2012), where the effect of irregular rough surfaces are highlighted through statistical quantities, the current study is aimed at extending previous results looking at the effect of irregular rough walls on turbulence structures, in transitional and fully rough regimes. In order to do this, the flow over a realistic irregular 3D roughness has been analysed in the light of the results achieved over an idealised 2D rough surface. The proposed realistic rough walls can be considered representative of naturally degraded surfaces. Resolved LES, also named quasi-DNS after Spalart et al. (1997), of turbulent channel flows with rough walls has been performed imposing a constant pressure gradient. Flow conditions and details of numerical code and simulations are described in Section 2. The results are highlighted in Section 3 followed by the concluding remarks in Section 4. 2. Computational framework The analysis has been performed numerically by solving the mass and momentum conservation equations, for incompressible newtonian fluid:

@ui @ui uj 1 @ 2 ui @p @ sij þ  þ þ þ P di1 ¼ 0 Res @xj @xj @xi @xj @t @xj

ð1Þ

110

@ui ¼0 @xi

M. De Marchis et al. / International Journal of Heat and Fluid Flow 56 (2015) 108–123

ð2Þ

where the variables are made non-dimensional with the friction velocity u and the channel half-width d; xi is the i-th coordinate (with x1 ; x2 and x3 the streamwise, spanwise and wall-normal directions, respectively), t is the time, ui is the i-th component of the fil is the filtered pressure field in kinematic units tered velocity field, p (i.e., divided by the fluid density), sij ¼ ui uj  ui uj is the sub-grid scale turbulent stress tensor, Res ¼ u d=m is the frictional Reynolds number, P is the imposed non-dimensional pressure gradient to drive the flow and dij is the Kronecker function (dij ¼ 1 for i ¼ j; dij ¼ 0 for i – j). In the Eqs. (1) and (2), the symbol ðÞ denotes filtered quantities. The subgrid-scale (SGS) stress tensor sij is modelled using the Dynamic Mixed Model (DMM) of Zang et al. (1993):

sij ¼ ðui uj  ui uj Þ  2C D2 jSjSij

ð3Þ

where ui uj  ui uj is the modified Leonard term and C is the dynamic Smagorinsky closure coefficient. The characteristic length of the filter width is D ¼ ðDx1  Dx2  Dx3 Þ1=3 with Dx1 ; Dx2 and Dx3 the grid spacing in the streamwise, spanwise and wall normal directions, qffiffiffiffiffiffiffiffiffiffiffiffi respectively. Sij is the resolved strain-rate tensor and jSj ¼ 2Sij Sij . The momentum and continuity Eqs. (1) and (2) are resolved using the LES/DNS finite-volume numerical code PANORMUS, an open source model available at www.panormus3d.org. The code is second-order accurate both in time and space. The AdamsBashfort method is used as the time advancement scheme, whereas a fractional-step technique is applied to overcome the pressure– velocity decoupling, typical of incompressible flows. The pressure Poisson equation is solved through a line-SOR technique in conjunction with a multigrid V-cycle accelerator, thus ensuring a fast

convergence (further details on the numerical procedure can be found in De Marchis and Napoli, 2008; De Marchis et al., 2011; Napoli et al., 2008; Milici et al., 2014). The numerical simulations are performed reproducing the rough wall using the same methodologies already proposed by De Marchis and Napoli (2012). Fig. 1 shows the computational domain. Periodic boundary conditions are imposed in both streamwise and spanwise directions and no-slip conditions are enforced at the rough walls. Different methods can be found in literature to resolve the rough wall both in regular or irregular geometries. Recently, Anderson (2013), van Nimwegen et al. (2015) and Busse et al. (2015) enforced the no-slip boundary condition on the immersed boundary, by introducing an extra momentum force in the predictor step and a mass source in the corrector step, to analyse the effect of irregular roughness on the turbulent flow. In the present analysis a body conforming grid is used and the no slip condition is applied at each point of the roughness elements. As pointed out by Leonardi et al. (2006), the boundary fitted method can cause a computational overload, nevertheless the method is clearly efficient for irregular three-dimensional roughness. Following the previous study of De Marchis and Napoli (2012) and Bhaganagar and Hsu (2009), 3D rough walls have been generated superimposing, over a flat plate, the oscillations obtained by means of the equation:

rðx1 ; x2 Þ ¼

  n2   n1 X 2ipx1 X 2jpx2  Ai sin Bj sin Lx1 =2 Lx2 =2 i¼1 j¼1

ð4Þ

where r is the wall distance from the flat reference surface, Lxi is the channel length in the i-th direction, ni is the number of sinusoidal functions, Ai and Bj are the amplitude of the sinusoidal function and Lx1 =2i and Lx2 =2j are wave-length of the i-th function in the

Fig. 1. 3D plot of the irregular surfaces. Top: 3D; bottom: 2D. Colouring is base on x3 , red colour indicates the highest roughness peaks x3 > 0, whereas blue is used for the cavities x3 < 0. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

111

M. De Marchis et al. / International Journal of Heat and Fluid Flow 56 (2015) 108–123 Table 1 Domain size, number of cells, mesh resolution. Case

Res

Lx1 =d

Lx2 =d

Lx3 =d

ðN x1 ; N x2 ; N x3 Þ

Dxþ 1

Dxþ 2

Dxþ 3;min

Dxþ 3;max

Smooth 2D 3D

395 395 395

2p 4p 4p

p p p

2 2 2

128, 64, 64 256, 64, 64 256, 128, 64

19 19 19

19 19 9.5

1.6 1.6 1.6

25 26 26

streamwise and spanwise directions, respectively. The two-dimensional roughness has been obtained applying the same algorithm just in the streamwise direction. For details on the mesh generation see De Marchis et al. (2010). Looking at the inset of Fig. 1, it can be observed that the proposed mesh is fine enough to capture even the small scales of the irregularities. Even though it is reproduced through the superimposition of sinusoidal functions, the 3D shape is clearly similar to those investigated by Yuan and Piomelli (2014) and Nagabhushana Rao et al. (2014), where degraded turbine blades are reproduced. Nevertheless the proposed shape is slightly smoothed with respect to the real roughness. The numerical simulations for flat, as well as 2D and 3D irregular rough surfaces, have been carried out at Res ¼ u d=m ¼ 395. The results of the numerical simulations, in term of roughness effect, have been analysed in the light of the classical simulation over a flat channel flow, driven by the same pressure gradient. Table 1 summarises data of domain size, cell number and grid resolution. The domain length in the streamwise direction was duplicated for the rough cases. This was necessary to increase the number of random sinusoidal functions and generate a more realistic synthetic roughness. For all the three dimensions the mesh resolution is quite similar to the values of other analyses carried out using LES method (Yuan and Piomelli, 2014). Furthermore, in De Marchis and Napoli (2012) two-point correlations in both stream- and spanwise directions showed that the domain size and resolution are adequate to resolve the turbulence structures. It is well known that one of the macroscopic effects of the roughness is the mean velocity reduction. In literature this loss is called RoughnessFunction DU þ , defined as the downward shift of the streamwise mean velocity profile in the log region. This shift has been deeply investigated and related to several geometrical parameters, both for regular and irregular roughness geometries. Furthermore, these geometrical quantities have been analysed in the light of the equivalent sand grain roughness ks , which is representative of the resistance induced by uniform sand as proposed in the archetypal research of Nikuradse. ks is a single parameter able to reproduce the effect of wall roughness on the mean flow and it is used to estimate the velocity through logarithmic wall law. For a review on the experimental analysis focused on the equivalent sand grain see Jimenez (2004). Regular rough surfaces (i.e. 2D transverse bar) can be parameterized using the elements height (h or k). Conversely, in irregular rough surfaces it is difficult to identify a single value of the roughness elements, due to random nature of peaks and valleys. The mean value of the oscillations is usually adopted as the representative geometrical parameter, but it has been deeply demonstrated that the mean roughness height cannot be considered representative of the roughness effect, whereas the shape (or texture) of the roughness must be taken into account. Following recent researches Schultz and Flack (2009), Wu and Christensen (2007), Flack and Schultz (2010), Wu and Christensen (2010), Yuan and Piomelli (2014), Mejia-Alvarez and Christensen (2013), and Nagabhushana Rao et al. (2014), the rough wall has been analysed in the light of statistical moments up to fourth order. In the present research, due to the nature of the sinusoidal functions the centerline of the roughness vanishes and cannot be considered a real metric for the roughness characterisation. The first parameter is the averaged absolute deviation rðx1 ; x2 Þ,

which coincides with the average value of magnitude of the roughness, already defined by Bons et al. (2001); then a second order moment, the root mean square r rms of wall oscillations, has been calculated. Furthermore, in order to give a quantitative view of the asymmetry of the roughness, we considered also the third order moment Sk (Skewness) which highlights the tendency of the oscillations toward peaks or valleys. To check the randomness of the rough wall, the fourth order statistics K u (Kurtosis) has been calculated. Following Napoli et al. (2008), to take into account the irregular roughness shape, the Effective Slope ES has been evaluated. The above parameters have been calculated according to the equations:

Z 1 1 j rðx1 ; x2 Þ j dx1 dx2 d Lx1 Lx2 Lx1 Lx2 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 XNx1 XNx2 r rms ¼ rðx1 ; x2 Þ2 i¼1 j¼1 dN r¼

Sk ¼

N x1 N x2 X X 1 rðx1 ; x2 Þ3  NÞ i¼1 j¼1

ðr 3rms

ð5Þ

N x1 N x2 X X 1 rðx1 ; x2 Þ4 4 ðr rms  NÞ i¼1 j¼1  Z  @rðx1 ; x2 Þ 1  dx1 ES ¼  @x  L

Ku ¼

x1

Lx1

1

with N ¼ ðNx1 þ 1Þ  ðNx2 þ 1Þ. Table 2 summarises the values of the above mentioned parameters. The maximum value of the roughness peak is also reported, as well as the value of the equivalent sand grain roughness ks . Based on accepted historical classification, the equivalent sand grain values, reported in Table 2, confirm that the rough walls are generated in both the transitional regime (r ¼ 0:024) and fully rough regimes (r ¼ 0:050). Unfortunately, for þ surfaces other than Nikuradse sand grain, the fact that ks P 70 is not sufficient to conclude that the flow is in the fully rough regime, in fact, due to the different kind of roughness existing in nature, the þ critical value of ks can be affected by the specific rough surface. To ensure that the fully rough regime is achieved it is necessary to find evidence that the viscous effects are negligible. According to Schultz and Flack (2005, 2007) and Flack et al. (2005) in fully rough regime the near-wall peak of the u01;rms profiles disappears over rough walls. This can be justified considering that the peak in the streamwise Reynolds normal stress on a smooth wall is primarily due to viscous effects. As will be investigated in the following, the profiles of streamwise component of turbulent intensities confirm that the fully rough regime is achieved, when r ¼ 0:050. The negative skewness values achieved in both 2D rough walls show that deeper cavities are more frequent than taller peaks. The values are consistent with those obtained by Yuan and Piomelli (2014) over real irregular roughness shapes. Looking at the kurtosis parameter, it can be observed that in all cases the roughness disagrees from the normal distribution. The achieved values are consistent with those observed over irregular rough walls derived by degraded turbine Flack and Schultz (2010), Yuan and Piomelli (2014), Mejia-Alvarez and Christensen (2013), and Nagabhushana Rao et al. (2014). 2D roughness are characterised by K u < 3,

112

M. De Marchis et al. / International Journal of Heat and Fluid Flow 56 (2015) 108–123

Table 2 Statistics of the rough walls. r and r þ : average absolute deviations of the heights of the rough walls made non-dimensional with the channel half-width d and the wall units, respectively. Similarly, r rms and rþ rms are the root mean squares of the wall oscillations, made non-dimensional with the channel half-width d and the wall units, respectively. ks and þ ks are the equivalent sand-grain roughness (obtained through the matching with the logarithmic wall-law) made non-dimensional with the channel half-width d and the wall units, respectively. r max and r þ max are the highest roughness peaks. The values of the effective roughness ES, as defined in Napoli et al., 2008 is also reported. Sk and K u are skewness and kurtosis, respectively. Case

r

rþ

r rms

rþ rms

Sk

Ku

ES

ks

ks

r max

rþ max

2D 2D 3D 3D

0.024 0.050 0.024 0.050

9.50 19.75 9.50 19.75

0.029 0.060 0.033 0.070

11.45 13.05 23.70 27.65

0.19 0.19 0.00 0.00

2.37 2.37 4.45 5.10

0.10 0.20 0.10 0.24

0.058 0.20 0.057 0.36

22.91 79.00 22.91 142.2

0.07 0.14 0.17 0.37

27.65 55.30 67.15 146.15

indicating high values of peaks and cavities with respect to the mean values. Conversely, 3D distributions have K u > 3, revealing a distribution with a peak around a specific value. The above results can be well understood through the analysis of the probability density function (p.d.f.) of roughness amplitude about the mean elevation, plotted in Fig. 2a and b. Fig. 2(a) clearly show that roughness p.d.f. distribution is not Gaussian and is characterised by two main peaks, one for the cavities and the second for the peaks. On the other hand, coherently with the achieved value of K u ¼ 5:10, the p.d.f. of 3D rough wall is characterised by a peak around the null value, higher than the gaussian distribution. Besides, Fig. 2c and d plot the p.d.f. of the streamwise surface gradient for 2D and 3D roughness. A similar analysis was carried out by Mejia-Alvarez and Christensen (2010) who compared the p.d.f. of the streamwise and spanwise surface gradient achieved for three different surface roughness. The first surface was a real

turbine-blade roughness, whereas the other rough walls were two low-order representation of the real surface. Looking at the p.d.f. of a real rough wall reported by Mejia-Alvarez and Christensen (2010) the reader can argue that the p.d.f. of a real rough surface has a clear skewness in the streamwise direction, and this is an effect of the flow-created rough surfaces (e.g. spallation, deposition of foreign materials, scouring) that tend to be skewed in the streamwise direction. On the other hand, this behaviour is less evident for the low-order representation of the roughness, where the p.d.f. of the slope of the roughness is symmetric. In the present analysis the p.d.f of the roughness slope, shown in Fig. 2c and d have a symmetric pattern, clearly similar to the p.d.f of the synthetic rough surface shown by Mejia-Alvarez and Christensen (2010). In this framework, even though the proposed surface roughness can be considered a realistic geometry, the tendency of real roughness to be skewed toward the mean flow

8

10

(a)

7

8 7

5

6

p.d.f

p.d.f

(b)

9

6

4

5 4

3

3

2

2

1 0 -0.30

þ

1 -0.20

-0.10

0.00

0.10

0.20

0 -0.50

0.30

-0.25

0.00

0.25

0.50

x3/

x3/ 2.5

2.5

(c)

(d)

1.5

1.5

p.d.f

2

p.d.f

2

1

1

0.5

0.5

0 -2.00 -1.50 -1.00 -0.50

0.00

x 3/ x 1

0.50

1.00

1.50

2.00

0 -2.00 -1.50 -1.00 -0.50

0.00

0.50

1.00

1.50

2.00

x3/ x1

Fig. 2. Probability density function (p.d.f.) of the roughness oscillation around the mean value (continuos line) vs gaussian distribution with equivalent root mean square (dashed line). (a) 2D rough geometry (r ¼ 0:05d); (b) 3D rough geometries (r ¼ 0:05d). Probability density function (p.d.f.) of the streamwise surface gradient (continuos line) vs gaussian distribution with equivalent root mean square (dashed line). (c) 2D rough geometry (r ¼ 0:05d); (d) 3D rough geometries (r ¼ 0:05d). P.d.f. are relative to the bottom wall. Similar results are obtained for the upper wall and for r ¼ 0:024d, not shown here.

M. De Marchis et al. / International Journal of Heat and Fluid Flow 56 (2015) 108–123

direction is not well captured. Some deviations between the proposed roughness and real rough surface can be thus attributed to this peculiarity. The equivalent sand-grain height has been calculated equating, in the log region, the mean streamwise velocity profile with the wall law for the mean velocity in a rough-wall turbulent boundary layer. According to Ligrani and Moffat (1986), in the transitional regime the equivalent sand grain roughness has been estimated through the equation:

Uþ ¼

      1 PG ln xþ3 þ C s þ 8:5  Cs  lnðReks Þ  sen 2 j j 1

ð6Þ

ln G¼



ln

Reks 5



70 ; Reks ¼

1

ð10Þ

The formula

ks  u

5



m

ð7Þ

When the fully rough regime is achieved, the following equation is applied:

j

directions) and time ðhUisst ¼ hu1 ðx1 ; x2 ; x3 ; tÞisst Þ obtaining statistical quantities along the vertical direction (i.e. velocity profiles), averaging over time and in spanwise direction ðhUist ¼ hu1 ðx1 ; x2 ; x3 ; tÞist Þ, obtaining contour mean values along x1  x3 planes, used for 2D rough surfaces, and averaging over time only ðhUit ¼ hu1 ðx1 ; x2 ; x3 ; tÞit Þ. The last average has been used for 3D rough surfaces, due to the heterogeneity of the roughness along the stream- and spanwise directions. Once the average has been defined, the corresponding turbulent quantities have been calculated. Eq. (10) is used to calculate the resolved turbulent fluctuations in the smooth channel flow.

u00i ðx1 ; x2 ; x3 ; tÞ ¼ ui ðx1 ; x2 ; x3 ; tÞ  hu1 ðx1 ; x2 ; x3 ; tÞisst

where

113

lnðxþ3 Þ þ C s  DU þ ¼

1

j

 þ ln xþ3 =ks þ 8:5

ð8Þ

In Eqs. (6)–(8) the mean velocity, as well as the roughness function, is given by the numerical mean velocity profile, C s is the classical additive constant for smooth case. The values of ks , reported in Table 2, have been also compared with the values calculated using the equation proposed by Flack and Schultz (2014), here reported:

ks ¼ f ðkrms ; Sk Þ  4:43rrms ð1 þ Sk Þ1:37

ð9Þ

The authors correctly observed that ks is not a physical measure of the surface roughness thus, in order to provide an engineering predictive tool, it must be related to physical roughness scales as the root-mean-square roughness height r rms and the skewness ðSk Þ, at least in the fully rough regime. Using the data reported in Table 2, for the cases having r ¼ 0:050, the ks given by Eq. (9) are equal to 0.19 and 0.31, that are quite similar to those achieved using the log law theory, thus partially supporting the equation proposed by Flack and Schultz (2014). The equation fails for the other cases where the transitional regime holds. The numerical simulations have been carried out until the steady state was achieved. Then the simulations were continued for a period of 40 non-dimensional time units tu =d for the 2D roughness and 80 tu =d for the 3D roughness, to collect an adequate number of field in order to ensure that also the statistical steady-state is achieved. The statistical convergence has been already confirmed in De Marchis and Napoli (2012) throughout the linear pattern of the total shear stress. 3. Results In the following, the effects of irregular rough surfaces are analysed in the light of averaged quantities, turbulent fluctuating and instantaneous fields. Due to the different nature of the domains here considered (smooth wall, 2D or 3D rough walls), different kinds of average have to be defined. For the sake of clarity, the notation adopted is specified here. Indicating with ui ðx1 ; x2 ; x3 ; tÞ the instantaneous velocity, the mean value can be achieved averaging both in time and along the homogeneous directions. The symbol hisst indicates quantities averaged in time and along the streamwise and spanwise direction; hist denotes average in time and spanwise direction, whereas hit is applied for statistical quantities obtained averaging over time. Following the above notations, the mean streamwise velocity profile can be calculated in three different ways: averaging over space (both streamwise and spanwise

u00i ðx1 ; x2 ; x3 ; tÞ ¼ ui ðx1 ; x2 ; x3 ; tÞ  hu1 ðx1 ; x2 ; x3 ; tÞist

ð11Þ

is used to calculate the resolved part of the turbulent fluctuations in the 2D rough wall case. Finally, for 3D roughness the formula reported in Eq. (12) is applied:

u00i ðx1 ; x2 ; x3 ; tÞ ¼ ui ðx1 ; x2 ; x3 ; tÞ  hu1 ðx1 ; x2 ; x3 ; tÞit

ð12Þ

3.1. Mean velocity In Fig. 3 the mean velocity profiles are plotted in wall units. In all cases, the mean velocity profile is shifted down due to the energy loss induced by the roughness. The plot shows that fully rough regime is verified for the highest value of r, both for 2D and 3D walls, whereas in cases characterised by r ¼ 0:024d the transitional regime is obtained. Fig. 3 shows the non-dimensional mean velocity profiles as a function of xþ 3  d for both smooth and rough cases. Here d is known as the zero-plane displacement and, as specified in De Marchis and Napoli (2012), is calculated fitting the mean velocity profile in order to collapse the smooth and rough profiles in the log region (details can be found in Flack and Schultz, 2010). Coherently with the observations of De Marchis and Napoli (2012), the fitting procedure shows that the displacement d is very close to the value of the minimum wall height and the mean roughness-height plane, both in 2D and 3D rough surfaces. Hereafter xþ 3 is used to denote the distances, taking into account the virtual origin. Deep modifications arise in the roughness sublayer, defined as the region extending from the wall to about 5 times the roughness height or the mean roughness height (among others Tachie et al., 2003; Ashrafian et al., 2004; Flack et al., 2005), whereas in log region the profiles have the same slope, achieved also in flat case. This result suggests the validity of the Townsend wall similarity hypothesis. The similarity is also confirmed through the velocity defect, plotted in Fig. 3(b). The profiles, as pointed out by De Marchis and Napoli (2012), show that the effects of both 2D and 3D wall roughness are mainly confined in the inner layer, whereas in the most of the outer layer all profiles are quite similar. To further investigate on the inner-outer interaction, Fig. 4 plots the mean velocity profiles, obtained averaging just in time, achieved in specific points (caves, peaks or flat regions). Fig. 4(a) shows the profiles obtained for 2D case for rþ ¼ 9:5, whereas Fig. 4(b) plots those achieved over two-dimensional roughness, having r þ ¼ 19:75. It is clear that, mean velocities are locally affected by the roughness. Negative velocities are verified close to the wall in correspondence of grooves (profiles P1 and P6), where typical recirculation regions are captured. Conversely, positive mean velocities are obtained over peaks (profiles P3 and P5) and almost null values are verified in zero plane regions (profiles P2 and P4). To ensure the clarity, Fig. 4(c) plots the shape of the bottom wall for the two-dimensional rough geometries. In the

114

M. De Marchis et al. / International Journal of Heat and Fluid Flow 56 (2015) 108–123

(a)

(b)

Fig. 3. (a) Mean velocity profiles. (b) Inner layer scaling of the mean velocity defect. -r-, 2D rough wall r þ ¼ 9:50; --, 3D rough wall r þ ¼ 9:50; -N-, 2D rough wall r þ ¼ 19:75; xþ þ 1 3 -j-, 3D rough wall r þ ¼ 19:75; Dot dashed line, smooth wall case. Dashed line, limit of the fully rough wall hU þ i ¼ log þ 8:5 with ks ¼ 70. þ 1 sst j k s

(a)

(b)

(c)

Fig. 4. (a) Mean velocity profiles sampled along the streamwise direction for 2D wall roughness having r þ ¼ 9:5. (b) Mean velocity profiles sampled along the streamwise direction for 2D wall roughness having r þ ¼ 19:75. (c) Profile of the wall roughness.

figure, dashed vertical lines indicate the position P1–P6 where the vertical profiles have been calculated. All the profiles collapse each other in the region beyond the roughness sublayer, whereas in the region lower than 5rþ the roughness strongly modifies fluid velocities, with alternating positive or negative values, induced by the alternating peaks or valleys along the streamwise direction. Similar consideration can be done for 3D rough cases, not reported. This result is in agreement with the above observation that the effects of the roughness are confined in a region close to the wall and does not affect the outer layer.

Looking at Fig. 3(a), it can be observed that overall the main effects of the wall roughness on the turbulent flows is the reduction of the streamwise velocity with respect to smooth-wall conditions. The shift of the velocity profile, in the vertical direction, is the so called roughness function DU þ . In rough walls, due to increase of resistance induced by the wall irregularities, the mean velocity is reduced despite the same pressure gradient drives the current. In fully rough regime, even though 2D and 3D rough walls are characterised by the same value rþ ( 20), the roughness function obtained with the 3D roughness is significantly higher than

M. De Marchis et al. / International Journal of Heat and Fluid Flow 56 (2015) 108–123

that obtained in the 2D case (DU þ ¼ 9:1 for 3D case and DU þ ¼ 7:6 for 2D one). This result was already pointed out by De Marchis and Napoli (2012), who concluded that the effects of 3D roughness on the mean velocities are higher than those of 2D roughness. Conversely, in transitional rough regime the velocity profiles achieved over 2D and 3D rough walls collapse each other, thus reaching the same value of roughness function DU þ ¼ 4:0. It seems that in the transitional regime the shape of the roughness, 2D or 3D, does not affect the mean velocity profile, whereas in fully rough regime the shape has a fundamental role. Already Antonia and Krogstad (2001) observed that two surfaces with very different geometries (2D circular rods and 3D mesh screen surfaces), having similar roughness height, produce the same value of the downward shift. A possible explanation for the two different behaviours can be found in the surface slope of the roughness. Starting from the original analysis of Napoli et al. (2008), recently, several authors (Yuan and Piomelli, 2014; Flack and Schultz, 2014; Anderson, 2013; Mejia-Alvarez and Christensen, 2013) observed that when the roughness is not extremely steep, the roughness function DU þ is related to the ‘‘effective slope’’ ES. As reported in Table 2, the rough surfaces in the transitional regime have the same values of ES = 0.10, whereas the 2D and 3D rough walls, having r þ ¼ 19:75, are characterised by ES = 0.20 and ES = 0.24, respectively. The same value of ES = 0.10 is in agreement with the overlap of the mean velocity profiles in the log region. Conversely, the higher value of ES for the fully rough 3D shape is coherent with the higher value of DU þ . Recently, Volino et al. (2011) compared 2D and 3D rough surfaces. The authors found that the 3D woven mesh (having k=d ¼ 0:014, see Volino et al. (2007)) causes less intense effects than 2D bars, when large 2D bars ðk=d ¼ 0:031Þ are considered (Volino et al., 2009); whereas 3D rough surface causes higher values of the roughness function than 2D bars, when small 2D bars are considered ðk=d ¼ 0:0062Þ. This can be attributed to the values of k=d. The irregular 2D roughness analysed in the proposed research, shares some features with small 2D bars rather than large 2D shapes. Furthermore, in the present research, even though 2D and 3D rough surfaces are build thus to have the same r, 3D rough surface is characterised by higher values of ES, as reported in Table 2. The highest value of ES over the 3D surface (at least for r ¼ 0:05) is responsible of the highest values of DU þ and ks , showing that when regular rough elements are considered, the roughness height k can be well representative of the roughness effects, whereas in irregular rough conditions (both 2D and 3D) the texture of the roughness must be considered. Fig. 5 presents the dependence of the roughness function on ES. The numerical results of Napoli et al. (2008); Yuan and Piomelli,

12 10

ΔU +

8 6 4 2 0 0.1

1.0

ES Fig. 5. Dependence of the roughness function on the effective slope on various surfaces. Open symbols, Napoli et al. (2008); --, Schultz and Flack (2009); -J-, Yuan and Piomelli (2014); -.-, Mejia-Alvarez and Christensen (2013); -j-, 3D rough wall r þ ¼ 19:75; -N; 2D rough wall r þ ¼ 19:75; -r-; 3D rough wall r þ ¼ 9:50; - - - ES = 0.35; bold line: Eq. (13).

115

2014 and the experimental results of Mejia-Alvarez and Christensen (2013) and Schultz and Flack (2009) are reported together with the present results. The vertical dashed line corresponds to the value of ES = 0.35, suggested by Schultz and Flack (2009) to distinguish the waviness regime to the roughness one. Later, Mejia-Alvarez and Christensen (2013) observed that ES = 0.35 represents a limit between slope-dependent and height-dependent flow, whereas the waviness regime can be attributed to the region where ES < 0.15, region dominated by the viscous drag, as shown by Napoli et al. (2008). Furthermore, Mejia-Alvarez and Christensen (2013) defined the region 0.15 < ES < 0.35 ‘‘transition’’ regime, where the dependence of U þ on ES weakens and its dependence on roughness height strengthens as Re increases. In Fig. 5 a thin vertical line correspond to the value of ES = 0.15. All the results obtained by the authors fall in the region characterised by ES < 0.35, where the ES dependence on DU þ assumes a logarithmic distribution. Following the log-law proposed by Hama (1954), who observed that the roughness function is related to the equivalent sand grain through a logarithmic law, Fig. 5 plots the following equation:

DU þ ¼

1

j

lnðES2 Þ  B  C

ð13Þ

where j is the von Kármán constant, B and C are the typical log law constant for smooth and rough surfaces, equal to 5.0 and 8.5, respectively. The Eq. (13) was obtained by fitting all pairs of data (ES  DU þ ) reported in Fig. 5. It can be observed that in the waviness regime (following the definition of Mejia-Alvarez and Christensen (2013)) all the data are aligned with the bold line representative, in the semi-logarithmic plane, of Eq. (13). Some deviations can be observed in the transition regime for few surfaces (see some of the results of Yuan and Piomelli, 2014), nevertheless the proposed log-law can be used to predict the roughness function once the effective slope is known. The above analysis shows that rough walls strongly modify the mean statistical quantities especially in the roughness sublayer. Furthermore, the velocity profiles averaged in time, plotted in Fig. 4 pointed out that roughness locally modifies the current, respect to the classical homogeneous characteristic of the current observed over flat plates. In the picture, in fact, can be observed the presence of alternating accelerating or recirculation regions, coherently with the presence of peaks or cavities. The local effect of different wall roughness can be better highlighted through contour plots of velocity distribution. Fig. 6 presents the contour values of the mean streamwise velocity, averaged in time < U 1 >t . The upper panels, Fig. 6a–c, show the streamwise velocity distribution in three subsequent x2  x3 planes for flat, 2D and 3D rough walls having r þ ¼ 9:50, respectively. For clarity, only half channel is plotted in the streamwise and wall-normal directions. Accordingly, the lower panels, Fig. 6d-e-f, show the streamwise velocity distribution for flat, 2D rough wall having rþ ¼ 19:75 and 3D rough wall characterised by r þ ¼ 19:75, respectively. To improve the clarity, in all cases half channel is plotted in the vertical direction, furthermore, for the rough surfaces only, half channel in the streamwise direction is plotted. In Fig. 6b–e, the velocity distribution over 2D rough walls reveals that the mean velocity is homogeneous in the outer layer. Conversely, in the near wall region the distribution is affected by the position of the vertical plane. Specifically, in the cavity region high negative velocity are observed, whereas they assume positive values in correspondence of the peaks, coherently with the findings of Fig. 4. Due to high recirculation regions, this result is enhanced for r þ ¼ 19:75. Interestingly, even though differences are clearly visible along the streamwise direction, the velocity distribution is homogeneous along the spanwise direction. This is a feature of the roughness

116

M. De Marchis et al. / International Journal of Heat and Fluid Flow 56 (2015) 108–123

þ Fig. 6. Contour of the streamwise component of the velocity, averaged in time hU þ 1 it , along spanwise planes. (a) Smooth channel. (b) 2D rough wall r ¼ 9:50. (c) 3D rough wall r þ ¼ 9:50. (d) Smooth channel. (e) 2D rough wall r þ ¼ 19:75. (f) 3D rough wall r þ ¼ 19:75. For clarity, only half channel is plotted in the streamwise and wall-normal directions.

shape. The artificial 2D behaviour of the roughness can hide important differences along the spanwise direction. Focusing the attention on 3D rough surfaces (Fig. 6c and f), a completely different velocity distribution appears. Threedimensional roughness elements are responsible for a marked heterogeneity both in the streamwise and spanwise directions. Similarly to the recent findings of Barros and Christensen (2014), along the x2 axis alternating high momentum and low momentum flow pathways occur. Due to the difference in the roughness heights, here the intermittent behaviour is clearly related to the highest roughness peaks and cavities. 3D rough walls are more realistic than 2D shapes, avoiding artificial systematic behaviour of the current. In Fig. 7 the wall normal component of the velocity over x2  x3 planes is plotted. Again, for clarity only half channel is plotted in the streamwise and wall-normal directions. The overall effect of the roughness is the huge increase of the wall normal velocity, close to the wall, respect to the flat case. Comparing Fig. 7(a) with Fig. 7(b) and (c) (similarly Fig. 7(d) with Fig. 7(e) and (f)), it can be argued that the wall normal velocity in smooth case is one order lower in magnitude than those obtained in the rough cases. Moreover, the comparison between 2D and 3D rough surface shows that, even for the same value of r þ , 3D roughness enhances the wall normal velocity even more than 2D case. As expected, the more the mean roughness height increases the more the wall normal velocity in the near wall region grows up. The comparison between 2D and 3D rough wall corrugations reveals that 2D corrugation produces almost the same effect along the spanwise direction, whereas 3D hills causes heterogeneity in streamwise as well as in spanwise direction. This effect can strongly modify the turbulence structures that populate the near wall region.

3.2. Reynolds stresses Figs. 8–10 show the normal components of the total Reynolds pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi stress tensor ui;rms ¼ hu0i  u0i isst , where u0i is normalised by us . The total Reynolds stresses are calculated considering that hu0i  u0i isst ¼ hu00i  u00i isst þ hu00i  u00i isgs , where u00i are the resolved turbulent fluctuations, whereas hu00i  u00i isgs is calculated according to the

Dynamic Mixed Model of Zang et al. (1993). The wall-normal distances are plotted in logarithmic scale in order to better highlight the differences in the inner-region. Overall, the roughness damps the peak of the streamwise turbulent intensities and increases both spanwise and wall-normal components. Looking at the streamwise component, it can be observed (Fig. 8) that, when r ¼ 0:050, the peak of the streamwise component of the turbulent intensities disappears and the profile is nearly flat. This is indicative that the viscous effects are negligible and that fully rough regime is obtained. Away from the wall, profiles collapse. Specifically, profiles overlap the smooth profile in the þ outer-region located for xþ 3 > 5r . This result supports the wall-similarity hypothesis, already discussed through the mean velocity profiles. Similar results were found by Mejia-Alvarez and Christensen (2010), in their experiments of turbulent flow over irregular rough surfaces. Specifically, the authors observed that outside the roughness sublayer the Townsend’s similarity holds and found a damping of the peak of the streamwise turbulent intensity. Looking at the streamwise component of the turbulent intensities, Fig. 8 shows that 3D rough surfaces cause higher peak reduction than 2D shape. Furthermore, increasing the roughness mean height the peak of turbulent intensities is less evident and for 3D roughness characterised by r þ ¼ 19:75 a nearly flat profile is achieved. The present results partially confirm the findings of Antonia and Krogstad (2001), different surfaces (2D roughness and 3D irregularities) with the same DU þ ¼ 4:0 involve differences in the Reynolds stresses, even though in the present research the effect are confined in the roughness sublayer. Similar considerations can be made for the spanwise and wall-normal components. Interestingly, the spanwise component shows a behaviour slightly different from the other components. Specifically, profiles obtained for 2D rough surfaces, both for r þ ¼ 9:50 and rþ ¼ 19:75, overlap each other. The same behaviour occurs for the 3D configurations. Furthermore, 2D rough surfaces, in the near wall region, have higher deviations than 3D rough surfaces. This is a topographical effect of the roughness, which cause the simultaneous impingement of the flow along all the cross section, as already observed through the streamwise and wall-normal components of the mean velocity, shown in Figs. 6 and 7. Conversely, the wall-normal component of the turbulent intensities, shown in Fig. 10, is not hugely

117

M. De Marchis et al. / International Journal of Heat and Fluid Flow 56 (2015) 108–123

þ Fig. 7. Contour of the wall normal component of the velocity, averaged in time hU þ 3 it , along spanwise planes. (a) Smooth channel. (b) 2D rough wall r ¼ 9:50. (c) 3D rough wall r þ ¼ 9:50. (d) Smooth channel. (e) 2D rough wall r þ ¼ 19:75. (f) 3D rough wall r þ ¼ 19:75. For clarity, only half channel is plotted in the streamwise and wall-normal directions.

3.5

1

3 0.75

2

u3,rms

u1,rms

2.5

1.5

0.5

1 0.25 0.5 0 +

100

200300

0 +

x3

x3

Fig. 8. Profiles of the streamwise component of the turbulent intensities pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u01;rms ¼ hu01 u01 isst , where u01 is normalised by us . r-, 2D rough wall r þ ¼ 9:50; --, 3D rough wall r þ ¼ 9:50; -N-, 2D rough wall r þ ¼ 19:75; -j-, 3D rough wall r þ ¼ 19:75; and Dot dashed line, smooth wall case. Thin vertical line and dashed þ þ þ vertical line xþ 3 ¼ 5r for r ¼ 9:50 and r ¼ 19:75, respectively.

100

200300

Fig. 10. Profiles of the wall-normal component of the turbulent intensities pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u03;rms ¼ hu03 u03 isst , where u03 is normalised by us . r-, 2D rough wall r þ ¼ 9:50; --, 3D rough wall r þ ¼ 9:50; -N-, 2D rough wall r þ ¼ 19:75; -j-, 3D rough wall r þ ¼ 19:75; Dot dashed line, smooth wall case. Thin vertical line and dashed vertical þ line xþ for r þ ¼ 9:50 and r þ ¼ 19:75, respectively. For clarity, only half 3 ¼ 5r channel is plotted in the streamwise direction.

1.5

when r þ ¼ 9:50 and slight deviation are observed when rþ ¼ 19:75. This specific behaviour can be explained considering that the u03;rms is clearly related to the roughness function, as discussed by Orlandi et al. (2006) and Orlandi and Leonardi (2008). The authors, in fact, observed, throughout experimental and numerical data, a strong correlation between the roughness function and the turbulent intensities in the wall normal direction u03;rms at the crest plane. Furthermore, the roughness function

u2,rms

1

0.5

0 +

x3

100

200300

Fig. 9. Profiles of the spanwise component of the turbulent intensities pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u02;rms ¼ hu02 u02 isst , where u02 is normalised by us . r-, 2D rough wall r þ ¼ 9:50; --, 3D rough wall r þ ¼ 9:50; -N-, 2D rough wall r þ ¼ 19:75; -j-, 3D rough wall r þ ¼ 19:75; Dot dashed line, smooth wall case. Thin vertical line and dashed vertical þ þ þ line xþ 3 ¼ 5r for r ¼ 9:50 and r ¼ 19:75, respectively.

affected by the roughness shape rather by the mean roughness height. Fig. 10 reveals that, in the roughness sublayer, the augmentation of the intensities for 2D and 3D rough walls is the same

DU þ is strongly connected with the roughness height, as pointed out by several authors (among others Jimenez, 2004; Leonardi et al., 2003), thus confirming that the roughness height is the main parameter affecting the turbulent intensities in the wall normal direction u03;rms . The reduction of the streamwise turbulent intensities and the increase of the spanwise and wall-normal components confirms the tendency toward turbulence isotropization, even to Figs. 6 and 7 clearly showed the heterogeneity of the flow in the roughness sublayer, as already observed by Barros and Christensen (2014) and Mejia-Alvarez et al. (2014). This heterogeneity is more and more evident for 3D rough surfaces.

118

M. De Marchis et al. / International Journal of Heat and Fluid Flow 56 (2015) 108–123

3.3. Turbulence structures The effects of irregular rough walls can be further highlighted looking at turbulence (or quasi-coherent) structures. In this section streaks and vortical structures are analysed. Streaks, as already pointed out by Hutchins et al. (2012), are coherent structures that can have different meaning with respect to the case of study. Here, coherently with the observations of Hutchins and Marusic (2007,) we refer to structures of slow fluid motion, mostly observed in near-wall regions, by means of streamwise velocity fluctuations and defined as Low- and High-Momentum Regions (LMRs– HMRs). Usually low-speed regions are flanked by high speed regions on either side. In smooth-wall layers, streaks assume a streamwise elongated shape (De Marchis et al., 2010). Wall roughness can selectively modify this coherent pattern. In this scenario, roughness shape and density can alter the effects of turbulent structures distribution. The coherent structures will be analysed in the light of contour plots or 3D iso-surfaces. For the sake of clarity, hereafter for the rough cases only half channel will be shown. The considerations work well for the whole domain. Fig. 11 shows resolved instantaneous streamwise velocity fluctuations, normalised by the r.m.s. of the streamline velocity, plotting the x1 x2 -plane at distance from the wall xþ 3 ¼ 15, corresponding to x3 =d ¼ 0:038. The upper panels, Fig. 11a–c, show the streaky structures occurring over flat plate, 2D rough wall for r þ ¼ 9:50 and 3D rough wall for r þ ¼ 9:50, respectively. Accordingly, the lower panels, Fig. 11d–f, show streaks for rþ ¼ 19:75. The plot shows that, overall, roughness causes an abrupt reduction of both low- and high-momentum regions length in the streamwise direction. Fig. 11(c) and (f) reveals that 3D roughness produces an enlargement of streaks space in the spanwise direction, coherently with the observed tendency towards turbulence isotropization. Nevertheless, comparing the effects of 2D and 3D rough walls, it can be argue that the former rough abruptly segregates the structures in the whole spanwise direction, whereas 3D irregularities preserve a more elongated shape in the streamwise direction with a meandering shape through the 3D hills. A similar meander behaviour was observed by Lee et al. (2011), over 3D cube roughness,

staggered in the streamwise direction. As expected, increasing the roughness height a streamwise reduction and spanwise enlargement of the coherent structures is verified. De Marchis et al. (2010) observed that one the effects of 2D rough walls is the displacement of elongated streaks away from the wall. Fig. 12 shows streaks at a distance from the mean wall height of about 35 wall units, corresponding to x3 =d ¼ 0:09. The structures assume a more elongated shape, in the streamwise direction, mainly for 2D rough wall having the highest value of rþ . The comparison between Figs. 11 and 12 shows that when r þ is equal to 9.50 the streaky structures over 2D rough walls at xþ 3 ¼ 15 (x3 =d ¼ 0:038) preserve an elongated shape, with respect þ to those observed at xþ 3 ¼ 35 (x3 =d ¼ 0:09). When r ¼ 19:75 the opposite holds. Focusing the attention on coherent structures over 3D rough walls and comparing Fig. 11 with 12 the reader can argue that streaks are longer for rþ ¼ 19:75, confirming that the effects of 3D rough walls can be different from those observed over 2D rough walls. The qualitative results, shown above, have been confirmed by the quantitative analysis of the streaks dimension. Following Dennis and Nickels (2011a), the length is found by locating the start of a structure (the most upstream location of the iso-surface) and then searching downstream, within the iso-surface, until the most downstream point within the surface is found. The analysis has been carried out looking at the approximate distribution of lengths rather than their precise values. Considering that streaks dimension in the spanwise direction is clearly shorter, it is difficult to correctly estimate the streak spacing following the criterion proposed by Dennis and Nickels (2011b), thus the spanwise length has been determined according to Chernyshenko and Baig (2005), who observed that the spanwise autocorrelation function of the streamwise two-point correlation reaches a minimum at a distance in the spanwise direction and that the streak spacing can be defined as twice Dmin . As known Dmin is in function of the wall normal distance. In Table 3 the streak þ þ dimensions in the streamwise lx1 and spanwise direction lx2 are reported. According to literature findings, the streaks in the þ streamwise direction have a mean length of about lx1  1000,

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ hu01  u01 it at x1  x2 plane at xþ Fig. 11. Contour plot of u01 3 ¼ 15, corresponding to x3 =d ¼ 0:038. (a) Smooth channel; (b) 2D rough wall r ¼ 9:50; (c) 3D rough wall þ r ¼ 9:50; (d) Smooth channel; (e) 2D rough wall r þ ¼ 19:75; and (f) 3D rough wall r þ ¼ 19:75. For clarity, only half channel is plotted in the streamwise direction.

119

M. De Marchis et al. / International Journal of Heat and Fluid Flow 56 (2015) 108–123

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ þ hu01  u01 it at x1  x2 plane at xþ Fig. 12. Contour plot of u01 3 ¼ 35, corresponding to x3 =d ¼ 0:09. (a) Smooth channel; (b) 2D rough wall r ¼ 9:50; (c) 3D rough wall r ¼ 9:50; (d) Smooth channel; (e) 2D rough wall r þ ¼ 19:75; and (f) 3D rough wall r þ ¼ 19:75. For clarity, only half channel is plotted in the streamwise direction.

Table 3 þ þ Streak length in the streamwise lx1 and spanwise lx2 directions. The lengths are reported in wall units. l

þ

smooth

2D r þ ¼ 9:50

2D r þ ¼ 19:75

3D r þ ¼ 9:50

3D r þ ¼ 19:75

þ lx1 þ lx2

1100

590

200

980

390

120

140

200

150

220

þ

whereas lx2  120. Overall, roughness reduces the streamwise length and increases the spanwise dimension. Coherently with the qualitative findings observed throughout Figs. 11 and 12, streaks over 3D rough surfaces are greater than 2D shapes. Specifically, the dimension of the low speed region achieved for 3D rþ ¼ 9:50, equal to 980, is clearly close to the value achieved over smooth surfaces. The main difference is the meandering shapes observed through the hills. Increasing the roughness height ðrþ ¼ 19:75Þ the length dramatically reduces, whereas the spanwise dimension is twice the value observed over a smooth surface. Fig. 13 shows 3D iso-surfaces of streak structures, where segregation of the coherent zone can be well observed in the streamwise direction with respect to smooth wall. 3D visualisation confirms that three-dimensional peaks and cavities have less influence than 2D roughness. The inner region shown in Fig. 13(c) and (f), in fact, are populated by coherent zones clearly more elongated than turbulent structures plotted in Fig. 13(b) and (e). Nevertheless, 3D rough walls cause the augmentation of the streaks in the streamwise and spanwise directions, higher than 2D geometries. This result is in agreement with the highest values of the roughness function, plotted in Fig. 3(a). Recently, similar Mejia-Alvarez and Christensen (2013) observed low-momentum regions (LMRs) and high-momentum regions (HMRs), typical of fluctuating velocity fields, in the ensemble-averaged field of the streamwise velocity and defined these coherent regions as low-momentum pathway (LMP) and by high-momentum pathways (HMPs). The authors observed, through the ensemble-averaged streamwise velocity, that roughness induces a channelling exhibiting significant spatial heterogeneity in the form of a spanwise-localised low-momentum pathway (LMP) bounded by high-momentum pathways (HMPs).

In order to confirm the qualitative results observed above, the coherent regions of vorticity are analysed. Specifically, Fig. 14 shows the contour plots of the wall-normal vorticity in the x1 x2 -plane at xþ 3 ¼ 15, corresponding to x3 =d ¼ 0:038. Fig. 14(a) and (d) show the typical behaviour of the vortical structures achieved over smooth walls: they are elongated in the streamwise direction and assume alternatively positive and negative values. Wall-normal vorticity over 2D rough walls is clearly shorter than that observed over flat plates but seems to be aligned with the mean flow direction. Similarly to streak structures, also vorticity is abruptly segregated by the 2D roughness (Fig. 14(b)–(e)). Furthermore, increasing the roughness height the coherence along the streamwise direction totally disappears. Conversely, vortical structures over 3D geometries (Fig. 14(c)) preserve a behaviour similar to that observed in smooth channels (Fig. 14(a). Nevertheless, vorticity over 3D irregularities meander significantly in the spanwise direction. Even though, increasing the mean roughness height, roughness segregates the vortical structures, a channelling effects among 3D hills and cavities is observed also in Fig. 14(f), where some elongated structures hold up. The effect of roughness features on the wall-normal vorticity can be better addressed through 3D visualisation of iso-surfaces. Fig. 15(c) shows that, 3D rough surface having r þ ¼ 9:50 preserves the elongated structures in the mean flow direction, whereas for rþ ¼ 19:75, the segregation is more evident (Fig. 15(f)). Overall, accordingly with the above observations, 2D irregularities cause a dramatic reduction of the streamwise coherence. Despite of this, structures over 3D rough walls seem to be larger in the spanwise direction than those over 2D irregularities. For high roughness

120

M. De Marchis et al. / International Journal of Heat and Fluid Flow 56 (2015) 108–123

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ hu01  u01 it ¼ 1:50 at x1  x2 plane at xþ Fig. 13. 3D iso-surface of streak structures for u01 3 ¼ 35. (a) Smooth channel; (b) 2D rough wall r ¼ 9:50; (c) 3D rough wall þ r ¼ 9:50; (d) Smooth channel; (e) 2D rough wall r þ ¼ 19:75; and (f) 3D rough wall r þ ¼ 19:75. For clarity, only half channel is plotted in the streamwise direction.

þ Fig. 14. Contour of the wall-normal vorticity at x1  x2 plane at xþ 3 ¼ 15, corresponding to x3 =d ¼ 0:038. (a) Smooth channel; (b) 2D rough wall r ¼ 9:50; (c) 3D rough wall r þ ¼ 9:50; (d) Smooth channel; (e) 2D rough wall r þ ¼ 19:75; and (f) 3D rough wall r þ ¼ 19:75.

elements the coherent structures are segregated into small structures. On the other hand, when the current flows over small 3D rough peaks the vortical structures are only partially segregated and they turn around the obstacle. Moreover, the zoomed-in view confirms that 2D roughness alters the vortical structures in the whole spanwise direction. The reduction of the streamwise length and the enlargement in spanwise direction is followed by an augmentation of their inclination in the spanwise and wall normal direction. Fig. 16 clearly points out the increase of the inclination of the structures with roughness height. To identify a vortex other

methods are available in literature, Chakraborty et al. (2005) have reviewed popular vortex identification criteria and their limitations. According to these studies, well established methods are the Q-criterion, D-criterion and k2 -criterion, not analysed here. A detailed discussion about the effect of the roughness on single vortex will be reported in a forthcoming research. The proposed detailed view of the quasi-coherent structures reveals that, even though 2D and 3D roughness may share some statistical quantities (i.e. mean velocity values), 3D roughness effects are quite different from those observed over 2D rough walls.

M. De Marchis et al. / International Journal of Heat and Fluid Flow 56 (2015) 108–123

121

Fig. 15. 3D iso-surface of the wall-normal vorticity. (a) Smooth channel; (b) 2D rough wall r þ ¼ 9:50; (c) 3D rough wall r þ ¼ 9:50; (d) Smooth channel; (e) 2D rough wall r þ ¼ 19:75; and (f) 3D rough wall r þ ¼ 19:75. xx3 ¼ 0:20xx3;max . For clarity, only half channel is plotted in the streamwise direction.

Fig. 16. Zoomed-in view of 3D iso-surface of the wall-normal vorticity. (a) Smooth channel; (b) 2D rough wall r þ ¼ 9:50; (c) 3D rough wall r þ ¼ 9:50; (d) Smooth channel; (e) 2D rough wall r þ ¼ 19:75; and (f) 3D rough wall r þ ¼ 19:75. xx3 ¼ 0:20xx3;max .

4. Summary and concluding remarks Two- and three-dimensional irregular rough surfaces are analysed in light of their effect on the turbulence structures. The study is carried out through the numerical technique of Large Eddy Simulations, applied to a fully developed turbulent channel flows, having a relatively low friction Reynolds number Res ¼ 395. Both the transitional and fully rough regimes are covered for 2D and 3D surfaces. The proposed surfaces can be considered representative of a realistic (i.e. natural irregular surfaces) roughness shape.

The first- and second-order statistics calculated in the smooth-wall and over the four rough walls showed that the roughness effect are mainly confined in the roughness sublayer thus supporting the wall similarity hypothesis. The analysis of the downward shift of the mean velocity profile reveals that in fully rough regime, 2D and 3D rough surfaces, having the same mean roughness heights, give rise to different values of the roughness function. Specifically, 3D roughness is significantly higher than that obtained in the 2D case (DU þ ¼ 9:1 for 3D case and DU þ ¼ 7:6 for 2D one). On the other hand, in transitional rough

122

M. De Marchis et al. / International Journal of Heat and Fluid Flow 56 (2015) 108–123

regime the velocity profiles achieved over 2D and 3D rough walls collapse each other, thus reaching the same value of roughness function DU þ . The result is attributed to the dominant effect of the geometrical parameter called effective slope. The two configurations in transitional regime are, in fact, characterised by the same value of ES. Interestingly, a logarithmic law is proposed to find a relation between ES and DU þ . The proposed equation well reproduced the effect of real rough surfaces, idealised 2D roughness or 3D regular or irregular geometries. The log-law is valid in the waviness regime only. The profiles of the rms’s of the velocity fluctuations show that the deviations from the smooth wall conditions, both for the 2D and the 3D roughness, are confined in a region close to the wall and that in the external region an overlap of all the profiles is obtained. The turbulence intensity analysis confirms that the 3D roughness induces higher effects than the 2D one, increasing the tendency toward the turbulence isotropization. pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Nevertheless, the spanwise component u02;rms ¼ hu02 u02 isst reveals that 2D rough surfaces enhance the difference from the smooth wall cases, with respect to the 3D rough shapes. This particular effect shows that 2D rough surfaces may not be fully representative of real roughness. The analysis, throughout contour plots of the wall-normal and streamwise mean velocity showed, coherently with recent finding over real rough surfaces, that 2D rough surfaces hide a spanwise heterogeneity of the flow, clearly visible over 3D irregularities. This heterogeneity is reflected on the modification of the turbulence structure. The results, herein presented, revealed, in fact, that streaks and wall-normal vortical structures are dramatically destroyed in the spanwise direction when 2D rough walls are encountered, whereas a somewhat of elongated shapes are preserved over 3D rough geometries, with a typical meandering behaviour. The results observed over 3D rough surfaces share several features with the results observed, through experiments of numerical analysis, over real rough surfaces, showing that the achieved findings can be considered representative of the effect of real roughness surfaces on the turbulence.

References Anderson, W., 2013. An immersed boundary method wall model for high-Reynoldsnumber channel flow over complex topography. Int. J. Num. Meth. Fluids 71 (12), 1588–1608. Antonia, R.A., Krogstad, P.A., 2001. Turbulent structure in boundary layers over different types of surface roughness. Fluid Dyn. Res. 28, 139–157. Ashrafian, A., Anderson, H.I., Manhart, M., 2004. DNS of turbulent flow in a rodroughness channel. Int. J. Heat Fluid Flow 25, 373–383. Ashrafian, A., Anderson, H.I., 2006. The structure of turbulence in a rod-roughened channel. Int. J. Heat Fluid Flow 27, 65–79. Bakken, O.M., Krogstad, P.A., Ashrafian, A., Andersonn, H.I., 2005. Reynolds number effects in the outer layer of the turbulent flow in channel with rough walls. Phys. Fluids 17 (065101). Barros, J.M., Christensen, K.T., 2014. Observations of turbulent secondary flows in a rough-wall boundary layer. J. Fluid Mech. 748, 1–13. Bhaganagar, K., Kim, J., Coleman, G., 2004. Effect of roughness on wall-bounded turbulence. Flow Turbul. Combust. 72, 463–492. Bhaganagar, K., Hsu, T.J., 2009. Direct numerical simulations of flow over twodimensional and three-dimensional ripples and implication to sediment transport: steady flow. Coast. Eng. 38, 320–331. Birch, D.M., Morrison, J.F., 2011. Similarity of the streamwise velocity component in very-rough-wall channel flow. J. Fluid Mech. 668, 174–201. Boppana, V.B.L., Xie, Z.T., Castro, I.P., 2010. Large-eddy simulation of dispersion from surface sources in arrays of obstacles. Boundary-Layer Meteorol. 135, 433–454. Bons, J.P., Taylor, R.P., McClain, S.T., Rivir, R., 2001. The many faces of turbine surface roughness. J. Turbomach. 123 (4), 739–748. Brzek, B., Cal, R.B., Johansson, G., Castillo, L., 2007. Inner and outer scalings in rough surface zero pressure gradient turbulent boundary layers. Phys. Fluids 19, 065101. Busse, A., Lutzner, M., Sandham, D., 2015. Direct numerical simulation of turbulent flow over a rough surface based on a surface scan. Comput. Fluids. http:// dx.doi.org/10.1016/j.compfluid.2015.04.008. Cardillo, J., Chen, Y., Araya, G., Newman, J., Jansen, K., Castillo, L., 2013. DNS of a turbulent boundary layer with surface roughness. J. Fluid Mech. 729, 603–637. Chakraborty, P., Balachandar, S., Adrian, R.J., 2005. On the relationships between local vortex identification schemes. J. Fluid Mech. 535, 189–214.

Chernyshenko, S.I., Baig, M.F., 2005. The mechanism of streak formation in nearwall turbulence. J. Fluid Mech. 544, 99–131. Coceal, O., Thomas, T.G., Castro, I.P., Belcher, S.E., 2006. Mean flow and turbulence statistics over groups of urban-like cubical obstacles. Boundary-Layer Meteorol. 121, 491–519. De Marchis, M., Napoli, E., 2008. The effect of geometrical parameters on the discharge capacity of meandering compound channels. Adv. Water Res. 31, 1662–1673. De Marchis, M., Napoli, E., Armenio, V., 2010. Turbulence structures over irregular rough surfaces. J. Turb. 11 (3), 1–32. De Marchis, M., Ciraolo, G., Nasello, C., Napoli, E., 2011. Wind- and tide-induced currents in the Stagnone lagoon (Sicily). Environ. Fluid Mech. http://dx.doi.org/ 10.1007/s10652-011-9225-0. De Marchis, M., Napoli, E., 2012. Effects of irregular two-dimensional and threedimensional surface roughness in turbulent channel flows. Int. J. Heat Fluid Flow 36, 7–17. Dennis, D., Nickels, T.B., 2011a. Experimental measurement of large-scale threedimensional structures in a turbulent boundary layer. Part 1. Vortex packets. J. Fluid Mech. 673, 180–217. Dennis, D., Nickels, T.B., 2011b. Experimental measurement of large-scale threedimensional structures in a turbulent boundary layer. Part 2. Long structures. J. Fluid Mech. 673, 218–244. Djenidi, L., Elevarasan, R., Antonia, R.A., 1999. The turbulent boundary layer over transverse square cavities. J. Fluid Mech. 395, 271–294. Djenidi, L., Antonia, R.A., Antonia, M.A., Anselmet, F., 2008. A turbulent boundary layer over a two-dimensional rough wall. Exp. Fluids. 44, 37–47. Dritselis, C.D., 2014. Large eddy simulation of turbulent channel flow with transverse roughness elements on one wall Int. J. Heat Fluid Flow. http:// dx.doi.org/10.1016/j.ijheatfluidflow.2014.08.008. Flack, K.A., Schultz, M.P., Shapiro, T.A., 2005. Experimental support for Townsend’s Reynolds number similarity hypothesis on rough walls. Phys. Fluids 17 (035102-035102-9), 1–9. Flack, K.A., Schultz, M.P., 2010. Review of hydraulic roughness scales in the fully rough regime. J. Fluids Eng. 132, 1–10. Flack, K.A., Schultz, M.P., 2014. Roughness effects on wall-bounded turbulent flows. Phys. Fluids 26 (103105), 1–17. Hama, F.R., 1954. Boundary layer characteristics for smooth and rough surfaces. Trans. Soc. Nav. Arch. Marine Engrs. 62, 333–358. Hong, J., Katz, J., Schultz, M.P., 2011. Near-wall turbulence statistics and flow structures over three-dimensional roughness in a turbulent channel flow. J. Fluid Mech. 667, 1–37. Hutchins, N., Marusic, I., 2007. Evidence of very long meandering features in the logarithmic region of turbulent boundary layers. J. Fluid Mech. 579, 1–28. Hutchins, N., Marusic, I., 2007. Large-scale influence in near wall turbulence. Phil. Trans. R. Soc. A 365, 647–664. Hutchins, N., Chauhan, K., Marusic, I., Monty, J., Klewicki, J., 2012. Towards reconciling the large-scale structure of turbulent boundary layers in the atmosphere and laboratory. Boundary-Layer Meteorol. 145, 273–306. Ikeda, T., Durbin, P., 2007. Direct simulations of a rough-wall channel flow. J. Fluid Mech. 561, 235–263. Jimenez, J., 2004. Turbulent flows over rough walls. Annu. Rev. Fluid Mech. 36, 173– 196. Krogstad, P.A., Antonia, R.A., 1999. Surface roughness effects in turbulent boundary layers. Exp. Fluids 27, 450–460. Lee, J.H., Sung, H.J., Krogstad, P.A., 2011. Direct numerical simulation of the turbulent boundary layer over a cube-roughened wall. J. Fluid Mech. 669, 397– 431. Leonardi, S., Orlandi, P., Smalley, R.J., Djenidi, L., Antonia, R.A., 2003. Direct numerical simulations of turbulent channel flow with transverse square bars on one wall. J. Fluid Mech. 491, 229–238. Leonardi, S., Orlandi, P., Djenidi, L., Antonia, R.A., 2006. Guidelines for modeling a 2D rough wall channel flow. Flow Turbul. Combust. 77, 41–57. Ligrani, P.M., Moffat, R.J., 1986. Structure of transitionally rough and fully rough turbulent boundary layers. J. Fluid Mech. 162, 69–98. Mejia-Alvarez, R., Christensen, K.T., 2010. Low-order representations of irregular surface roughness and their impact on a turbulent boundary layer. Phys. Fluids 22, 015106. Mejia-Alvarez, R., Christensen, K.T., 2013. Wall-parallel stereo particle-image velocimetry measurements in the roughness sublayer of turbulent flow overlying highly irregular roughness. Phys. Fluids 25, 115109. Mejia-Alvarez, R., Wu, Y., Christensen, K.T., 2014. Observations of meandering superstructures in the roughness sublayer of a turbulent boundary layer. Int. J. Heat Fluid Flow 48, 43–51. Milici, B., De Marchis, M., Sardina, G., Napoli, E., 2014. Effects of roughness on particle dynamics in turbulent channel flows: a DNS analysis. J. Fluid Mech. 739, 465–478. Napoli, E., Armenio, V., De Marchis, M., 2008. The effect of the slope of irregularly distributed roughness elements on turbulent wall-bounded flows. J. Fluid Mech. 613, 385–394. Nagabhushana Rao, V., Jefferson-Loveday, R., Tucker, P.G., Lardeau, S., 2014. Large eddy simulations in turbines: influence of roughness and free-stream turbulence. Flow Turbul. Combust. 92 (1-2), 543–561. Orlandi, P., Leonardi, S., Antonia, R.A., 2006. Turbulent channel flow with either transverse or longitudinal roughness elements on one wall. J. Fluid Mech. 561, 279–305.

M. De Marchis et al. / International Journal of Heat and Fluid Flow 56 (2015) 108–123 Orlandi, P., Leonardi, S., 2008. Direct numerical simulation of three-dimensional turbulent rough channels: parameterization and flow physics. J. Fluid Mech. 606, 399–415. Schultz, M.P., Flack, K.A., 2005. Outer layer similarity in fully rough turbulent boundary layers. Exp. Fluids 38, 328–340. Schultz, M.P., Flack, K.A., 2007. The rough-wall turbulent boundary layer from the hydraulically smooth to the fully rough regime. J. Fluid Mech. 580, 381–405. Schultz, M.P., Flack, K.A., 2009. Turbulent boundary layers on a systematically varied rough wall. Phys. Fluids 21 (015104), 1–9. Singh, K.M., Sandham, N.D., Williams, J.J., 2007. Numerical simulation of flow over rough bed. J. Hydraul. Eng. 133 (4), 386–398. Spalart, P.R., Jou, W.-H., Strelets, M., Allmaras, S.R., 1997. Comments on the feasibility of LES for wings, and on a hybrid RANS/LES approach. In: First AFOSR Int. Conference on DNS/LES, in Advance in DNS/LES, pp. 137–147. Subramanian, C.S., King, P.I., Reeder, M.F., Ou, S., Rivir, R.B., 2004. Effects of strong irregular roughness on the turbulent boundary layer. Flow Turbul. Combust. 72, 349–368. Tachie, M.F., Bergstrom, D.J., Balachandar, R., 2003. Roughness effects in the lowReH open-channel turbulent boundary layers. Exp. Fluids 33, 338–346. Townsend, A.A., 1976. The Structure of Turbulent Shear Flow, second ed. Cambridge University Press. van Nimwegen, A.T., Schutte, K.C.J., Portela, L.M., 2015. Direct numerical simulation of turbulent flow in pipes with an arbitrary roughness topography using a combined momentum–mass source immersed boundary method. Comput. Fluids 108 (15), 92–106.

123

Van Rij, J.A., Belnap, B.J., Ligrani, P.M., 2002a. Analysis and experiments on threedimensional irregular surface roughness. J. Fluids Eng., Trans. ASME 124 (3), 671–677. Van Rij, J.A., Belnap, B.J., Ligrani, P.M., 2002b. Analysis and experiments on threedimensional irregular surface roughness. J. Fluid Mech. 580, 381–405. Volino, R.J., Schultz, M.P., Flack, K.A., 2007. Turbulence structure in rough- and smooth-wall boundary layers. J. Fluid Mech. 592, 263–293. Volino, R.J., Schultz, M.P., Flack, K.A., 2009. Turbulence structure in a boundary layer with two-dimensional roughness. J. Fluid Mech. 635, 75–101. Volino, R.J., Schultz, M.P., Flack, K.A., 2011. Turbulence structure in boundary layers over periodic two- and three-dimensional roughness. J. Fluid Mech. 676, 172– 190. Wu, Y., Christensen, K.T., 2006. Reynolds-stress enhancement associated with a short fetch of roughness in wall turbulence. AIAA J. 44 (12), 3098–3106. Wu, Y., Christensen, K.T., 2007. Outer-layer similarity in the presence of a practical rough-wall topography. Phys. Fluids 19, 1–15. Wu, Y., Christensen, K.T., 2010. Spatial structure of a turbulent boundary layer with irregular surface roughness. J. Fluid Mech. 655, 380–418. Yuan, J., Piomelli, U., 2014. Estimation and prediction of the roughness function on realistic surfaces. J. Turb. 15 (6), 350–365. Zang, Y., Street, R.L., Koseff, J.R., 1993. A dynamic mixed subgride-scale model and its application to turbulent recirculating flows. Phys. Fluids 12, 3186–3196. Zhang, Q., Ligrani, M.P., Lee, S.W., 2003. Determination of rough-surface skin friction coefficients from wake profile measurements. Exp. Fluids 35, 627–635.