- Email: [email protected]
Scale-by-scale energy budget in a turbulent boundary layer over a rough wall
International Journal of Heat and Fluid Flow xxx (2015) xxx–xxx
Contents lists available at ScienceDirect
International Journal of Heat and Fluid Flow journal homepage: www.elsevier.com/locate/ijhff
Scale-by-scale energy budget in a turbulent boundary layer over a rough wall Md. Kamruzzaman, L. Djenidi, R.A. Antonia, K.M. Talluru ⇑ School of Engineering, University Drive, The University of Newcastle, Callaghan, 2308, NSW, Australia
a r t i c l e
i n f o
Article history: Received 27 January 2015 Received in revised form 9 April 2015 Accepted 11 April 2015 Available online xxxx Keywords: Scale-by-scale budget Structure function Rough wall boundary layer
a b s t r a c t Hot-wire velocity measurements are carried out in a turbulent boundary layer over a rough wall consisting of transverse circular rods, with a ratio of 8 between the spacing (w) of two consecutive rods and the rod height (k). The pressure distribution around the roughness element is used to accurately measure the mean friction velocity (U s ) and the error in the origin. It is found that U s remained practically constant in the streamwise direction suggesting that the boundary layer over this surface is evolving in a self-similar manner. This is further corroborated by the similarity observed at all scales of motion, in the region 0:2 6 y=d 6 0:6, as reflected in the constancy of Reynolds number (Rk ) based on Taylor’s microscale and the collapse of Kolmogorov normalized velocity spectra at all wavenumbers. A scale-by-scale budget for the second-order structure function hðduÞ2 i (du ¼ uðx þ rÞ uðxÞ, where u is the fluctuating streamwise velocity component and r is the longitudinal separation) is carried out to investigate the energy distribution amongst different scales in the boundary layer. It is found that while the small scales are controlled by the viscosity, intermediate scales over which the transfer of energy (or hðduÞ3 i) is important are affected by mechanisms induced by the large-scale inhomogeneities in the flow, such as production, advection and turbulent diffusion. For example, there are non-negligible contributions from the large-scale inhomogeneity to the budget at scales of the order of k, the Taylor microscale, in the region of the boundary layer extending from y=d ¼ 0:2 to 0.6 (d is the boundary layer thickness). Ó 2015 Elsevier Inc. All rights reserved.
1. Introduction The investigation of a turbulent boundary layer over rough walls is of fundamental and practical importance. From a fundamental view point, the knowledge of how a boundary layer responds to roughness (size and geometry, for example) can help in terms of understanding the dynamics of the boundary layer. This should in turn help to devise better management control strategies in flow situations, e.g., heat exchangers, or flow over a river bed, which involve flow over rough surfaces. The motivation for studying rough wall flows is well summarized by Antonia and Djenidi (2010) as follows: ‘‘Apart from the wide engineering applications associated with rough walls (e.g., Nikuradse, 1933; Perry et al., 1969; Raupach et al., 1991 and Jiménez, 2004) there are several compelling scientific reasons for studying flows over rough walls. First, our understanding of the turbulence structure near the vicinity of rough walls has lagged significantly behind that for the canonical smooth ⇑ Corresponding author. E-mail address: [email protected] (K.M. Talluru).
wall, for which streaks are observed throughout the viscous region and are important in the context of bursting. Eliminating the viscous layer through the introduction of roughness elements and examining the effect this has on both the inner and outer regions should be sufficient incentive for studying rough wall flows with vigour. Secondly, it is almost intuitive that the turbulence close to a drag-augmenting surface should be more isotropic than that over a smooth wall, thus facilitating somewhat the modelling of the near-wall region. Thirdly, a turbulent boundary layer which develops over a rough wall is more likely to satisfy the requirements of self-preservation (self-similarity along the streamwise direction) than a smooth wall boundary layer.’’ (taken from Antonia and Djenidi (2010)). Of particular interest amongst rough surfaces, is the rough wall consisting of transverse circular rods or square bars, which has attracted a lot of attention, (see, for example, Raupach, 1981; Bandyopadhyay and Watson, 1988; Bakken et al., 2005; Keirsbulck et al., 2002; Krogstad and Antonia, 1999). The reason for this interest relies on the relatively simple geometry of the roughness elements which allows parametric studies (e.g., Leonardi et al., 2003; Krogstad et al., 2005). However, although a
http://dx.doi.org/10.1016/j.ijheatfluidflow.2015.04.004 0142-727X/Ó 2015 Elsevier Inc. All rights reserved.
Please cite this article in press as: Kamruzzaman, M., et al. Scale-by-scale energy budget in a turbulent boundary layer over a rough wall. Int. J. Heat Fluid Flow (2015), http://dx.doi.org/10.1016/j.ijheatfluidflow.2015.04.004
2
Md. Kamruzzaman et al. / International Journal of Heat and Fluid Flow xxx (2015) xxx–xxx
larger number of studies of a turbulent boundary layer over this particular 2D rough wall have already been undertaken (Raupach, 1981; Bandyopadhyay and Watson, 1988; Krogstad and Antonia, 1994; Djenidi et al., 1999; Keirsbulck et al., 2002; Leonardi et al., 2003; Lee and Sung, 2007), there are still unresolved issues. For example, there is no consensus yet on whether the outer region of the boundary layer on this rough wall is affected by the roughness when compared to that of a smooth wall boundary layer (e.g., Krogstad et al., 1992; Antonia and Krogstad, 2001; Jiménez, 2004; Schultz and Flack, 2005; Antonia and Djenidi, 2010), casting doubt on the validity of Townsend’s wall-similarity hypothesis (Townsend, 1976). Also, the accurate measurement of the skin friction velocity is problematic and challenging and can lead to erroneous conclusions if incorrectly obtained. It is further relevant to note that most of the studies of turbulent boundary layers focused mainly on one-point statistics (e.g., mean velocity, Reynolds stresses). While these statistics are of great importance and provide information on the dynamical response of the boundary layer to the surface changes, they do not shed light on how the energy transfer amongst various scales in the flow might be altered. Relatively few studies of turbulent boundary layers over rough walls used two-point statistics to gain some insight into the structure of the flow (Saikrishnan et al., 2007, 2012). These were mainly aimed at measuring the spatial correlations. We carry out a two-point analysis of a turbulent boundary layer over a 2D rough wall with the aim to assess the energy distribution at any particular scale r (ranging from the smallest to the largest in the flow) as well as the way the energy is transferred amongst the scales. The focus of the work reported in this paper is on the scaleby-scale analysis of the second-order velocity structure function, hðduÞ2 i (du ¼ uðx1 þ rÞ uðx1 Þ, where u is the streamwise velocity fluctuation and r is the streamwise separation; the angular brackets denote time averaging). Notice that when r is very large then hðduÞ2 i ¼ 2hu2 i. If local isotropy is assumed, then the transport equation of hðduÞ2 i is given generically by Danaila et al. (2001),
hðduÞ3 i þ 6m
@ 4 hðduÞ2 i þ Iu ¼ hir; @r 5
ð1Þ
where m is the kinematic viscosity of the fluid, and hi is the mean turbulent kinetic energy dissipation rate. The first and second terms of the left side of Eq. (1) represent the energy transfer and the viscous diffusion of energy, respectively. The third term, Iu , accounts for the inhomogeneity or non-stationarity associated with the large scales; it can also account for the mean shear that exists in wallbounded flows. Eq. (1) corresponds to the two-point velocity correlation function, first written by Von Karman and Howarth (1938) and effectively represents the scale-by-scale (SBS) budget of the turbulent energy at a location in the flow. At large r, Eq. (1) reduces to the (one-point) energy budget equation; in the limit r ! 0, it reduces to the (one-point) transport equation of hi (e.g., Danaila
et al., 1999). Different versions of Eq. (1) have been proposed for a turbulent channel flow (Danaila et al., 2001) and self-preserving turbulent round jet (Burattini et al., 2005). The term Iu is different between these two cases and reflects the difference in the one-point energy budget. For example, in the case of a channel flow, where the flow is stationary, Iu near the wall includes the effects of shearing and the spatial inhomogenity while on the centre-line of a round jet, Iu accounts for the non-stationarity and spatial inhomogeneity. Clearly, Iu not only varies from flow to flow, but it also varies from location to location within the same flow. The objective of the present work is to extend the SBS approach to a turbulent boundary layer over a rough wall. Saikrishnan et al. (2007, 2012) performed a SBS analysis in a turbulent boundary layer and turbulent channel flow over a smooth wall. The motivation for considering a rough wall stems from the fact that the flow should be more isotropic in comparison to a smooth wall (see, for example, Shafi and Antonia, 1995, 1997; Antonia and Shafi, 1999) and thus one may expect that local isotropy is more adequately satisfied, thus making Eq. (1) more suitable for studying rough walls than smooth walls.
2. Experimental facility and measurements The wind tunnel measurements are conducted in a turbulent boundary layer developing over a rough wall made up of circular rods mounted transversely on the tunnel floor and spanning the full width of the test section with w=k ¼ 8, where w is the streamwise pitch and k is the roughness height (see Fig. 1). The roof of the wind tunnel is adjusted to obtain a nominally zero-pressure gradient (Dp=ð1=2qU 2 Þ ¼ 0:5%) in the streamwise direction. The velocity fluctuations are measured using a single hot-wire probe. The hot-wire (diameter, d ¼ 2:5 lm, and length, l=d ¼ 200) is etched from a coil of Pt-10% Rh and operated using an in-house built constant temperature anemometer (CTA) with an over-heat ratio of 1.5. The hot-wire is calibrated in situ against the Pitot-static tube positioned in the undisturbed free stream flow before and after every experiment at 15 different speeds ranging between 0 m/s and 16 m/s. A linear interpolation in time (see Talluru et al., 2014) is employed between the two calibrations to correct for any drift in the hot-wire voltage that occurs during the course of an experiment. Only a minimal drift is noticed in the hot-wire due to short duration of our experiments. For the results reported here, measurements are made at a streamwise distance x ¼ 2:54 m and 40 logarithmically spaced points are taken between 0:001 6 y=d 6 1:5 in the wall-normal direction. At each measurement point, samples are collected for a duration of 120 s at a sampling frequency of 20 kHz. Experiments were taken at different free stream speeds giving Reh (Reynolds number based on the momentum thickness) in the range of 7000 and 14,000 while the ratio d=k (d is the boundary layer thickness) remained nominally about 60. In addition, measurements are made at five different streamwise
y
x
(Traversing hot-wire)
Flow
k
w
Fig. 1. Schematic representation of the rough wall and hot-wire probe used in the experiment.
Please cite this article in press as: Kamruzzaman, M., et al. Scale-by-scale energy budget in a turbulent boundary layer over a rough wall. Int. J. Heat Fluid Flow (2015), http://dx.doi.org/10.1016/j.ijheatfluidflow.2015.04.004
3
Md. Kamruzzaman et al. / International Journal of Heat and Fluid Flow xxx (2015) xxx–xxx
locations between 1.4 m 6 x 6 2:8 m at a free stream velocity of 14 m/s. All the boundary layer characteristics in these experiments are summarized in Table 1. An attractive feature of this particular roughness is that, the pffiffiffiffiffiffiffiffiffiffiffi friction velocity, U s (U s ¼ sw =q, where sw is the wall shear stress and q is the fluid density), can be determined reliably from the form drag estimated by pressure tapping the individual roughness elements, see Kamruzzaman et al., 2014 for a full description of the method. This is also illustrated in Fig. 2 which shows the pressure distributions around the roughness element for two Reynolds numbers (Reh ¼ 8500 and 12,632). Interestingly, we found that there is a complete collapse of the pressure distribution profiles obtained between 8500 6 Reh 6 12; 632 (all results are not shown here for brevity) indicating that C D has become independent of Re. In addition, for the present configuration of roughness elements
Table 1 Boundary layer properties for different measurements carried out in this study. Case I: Different streamwise locations at a free stream velocity, U 1 14 m/s; Case II: At two different free stream velocities, U 1 10 m/s and 14 m/s at x = 2.54 m. x (m)
U 1 (m/s)
d99 (m)
d (m)
h (m)
U s (m/s)
Res
Reh
Case I 1.48 1.94 2.24 2.54 2.84
13.89 13.82 13.76 14.14 14.26
0.059 0.076 0.077 0.091 0.10
0.0168 0.0214 0.0222 0.0244 0.0268
0.0091 0.0119 0.0126 0.0140 0.0154
0.93 0.92 0.91 0.91 0.90
3460 4410 4480 5300 5830
8120 10,500 11,100 12,600 14,100
Case II 2.54 2.54
9.8 14.04
0.092 0.093
0.0245 0.0240
0.0136 0.0140
0.65 0.90
3830 5360
8550 12,630
0.06
0.04 θ
θ
U
θ
3. Results 3.1. Mean velocity and turbulence intensity profiles The mean velocity and the velocity variance profiles at free stream velocities, 10 m/s and 14 m/s are reported in Fig. 3(a) and (b) respectively. The data are presented in a log-linear plot and normalized by U s and d. Note that the origin y ¼ 0 is taken as the location that corresponds to the centroid of the moments of the pressure forces acting on the roughness elements (see Jackson, 1981; Leonardi et al., 2003; Kamruzzaman et al., 2014, for more details) and does not therefore rely on the existence of a log law. Fig. 3(a) shows that the mean velocity distributions over the rough wall display a convincing log law region, which is nearly identical for the two values of Reh ; the slope of this region is smaller than that over the smooth wall, thus making the determination of DU=U s , or roughness function, somewhat ambiguous. In the turbulence intensity profiles, shown in Fig. 3(b), we note that the intensity drops as we approach the wall instead of presenting a peak at y=d 0:002 or yþ 15 contrary to what we observe in a smooth wall boundary layer. The absence of the near-wall peak in the turbulence intensity profile indicates that the near-wall turbulence structure on the present rough wall is very different to that on a smooth wall. Accordingly, one expects that the turbulence production mechanism on this wall to be different to that on a smooth wall. This may have important consequences in terms of turbulence modelling on this particular rough surface.
Δ
1/2ρ
2
0.02
= 8500 = 12632
(w=k ¼ 8), the skin friction contribution to the total drag on the rough wall is practically zero (C v 0; C v is the coefficient of skinfriction drag) (see Leonardi et al., 2003, for more details). This facilitated us to estimate U s from C D alone. Further, it was verified that U s is practically constant with respect to x, confirming the claim made by Smalley et al. (2001) that this boundary layer is likely to be closer to self-preservation than a smooth wall boundary layer. The fact that the pressure distributions are equal for both Reynolds numbers further supports the claim.
0
3.2. 2nd and 3rd order structure functions −0.02
Before proceeding with the 2nd and 3rd order structure functions, it is worthwhile looking at how the Taylor microscale
−0.04
−0.06
0
50
100
150
200
250
300
350
400
θ Fig. 2. Distribution of the pressure around a rod element at two Reynolds numbers, Reh ¼ 8500 and 12,632 using symbols ( ) and ( ) respectively.
Reynolds number (Rk ¼ u0 k=m, with k ¼ u0 =hð@u=@xÞ2 i1=2 and u0 is the rms value of velocity fluctuations) varies across the boundary layer. It is to be noted that we employed Taylor’s hypothesis of frozen turbulence to project the temporal velocity fluctuations in the streamwise direction using a convection velocity equal to that of the local mean velocity. Fig. 4 shows such a variation of Rk for
25
5
20
4
15
3
=12.5
Δ
10
2
5
1
0
−2
10
−1
10
0
10
−2
10
−1
10
0
10
0
Fig. 3. Profiles of (a) mean velocity and (b) turbulence intensity for x = 2.54 m at Reh ¼ 8500 () and Reh ¼ 12; 632 () normalized using U s and d. The dashed line in (a) represents the smooth wall log region and is shown only as a reference.
Please cite this article in press as: Kamruzzaman, M., et al. Scale-by-scale energy budget in a turbulent boundary layer over a rough wall. Int. J. Heat Fluid Flow (2015), http://dx.doi.org/10.1016/j.ijheatfluidflow.2015.04.004
4
Md. Kamruzzaman et al. / International Journal of Heat and Fluid Flow xxx (2015) xxx–xxx 500
at y=d ¼ 0:03 collapse fairly well with the other two. Looking at the
300
results of hðduÞ3 i in Fig. 5(b), we note that all the three profiles collapse well at small values of r but at larger r, the differences show up. Particularly, the profile at y=d ¼ 0:03 deviates from the other two profiles. All the above discussion shows the effect of Rk on
200
hðduÞ3 i when the Rk at those locations is nominally the same.
400
the structure functions; there is a better collapse of hðduÞ2 i) and Looking further, as we expect, hðduÞ2 i=u02 approaches a value of
100
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
Fig. 4. Variation of Rk as a function of wall-normal distance (y=d) for x = 2.54 m at Reh ¼ 8500 and 12,600 using symbols ( ) and ( ) respectively.
x = 2.54 m for two Reynolds numbers, Reh ¼ 8500 and 12,600. Rk increases rapidly with the distance from the wall, then reaches a plateau with a value of about 300 and 370 for Reh ¼ 8500 and 12,600, respectively. The plateau extends from y=d 0:2 to about 0.6 and beyond this latter value, Rk decreases. The constancy of Rk suggests that there exists a relatively non-negligible part (40%) of the boundary layer in which the small scale motions may exist in a ‘‘quasi-equilibrium’’ state. It is interesting to note that such a plateau region has not been widely reported for Rk in a smooth wall turbulent boundary layer. Oyewola et al. (2008) is one of the few studies who presented the results of Rk over a smooth wall. However, their results do not show a clear plateau of Rk . Rather, it appears that there is a local minima for Rk . Further, the results of Oyewola et al. (2008) show a sharp rise of Rk in the near wall region, which is physically incorrect. This is because the measurements of Oyewola et al. (2008) were made using an X-wire, which is known to show erroneous results, particularly in the near wall region (Antonia and Luxton, 1971). In the present rough wall study, the constant value of Rk in the region 0:2 6 y=d 6 0:6 suggests that k may be a relevant scaling length in that region of the flow. This is indeed confirmed in Fig. 5(a) and (b) which show the profiles of the second2
3
(hðduÞ i) and third-order (hðduÞ i) velocity structure functions at three different locations in the wall-normal direction, y ¼ 0:03d (yþ ¼ 87), y ¼ 0:23d (yþ ¼ 690) and y ¼ 0:4d (yþ ¼ 1200). Here, the second- and the third-order structure functions are normalized by u02 and u03 respectively while the separation distance (r) is normalized by k. The data are plotted in a log–log scale in order to highlight the salient features of the second- and the third-order structure functions. The profiles of hðduÞ2 i for y=d ¼ 0:23 and 0:4 collapse very well. This is expected because both these points lie within the plateau region of Rk (see Fig. 4). Surprisingly, the results
2 at a large separation while hðduÞ3 i=u03 becomes zero. Notice that despite the reasonable value of the Reynolds number based on Taylor microscale (Rk ¼ 187, 287 and 310) the inertial range, represented by the law r2=3 , is not yet established. The data show, though, a tendency to follow the 2/3 law, suggesting that a further increase in Rk will lead to its emergence. On the other hand, the ratio hðduÞ2 i=u02 approaches the expected r 2 behaviour as r ! 0. Similarly, hðduÞ3 i=u03 does not present a clear inertial range (r 1 ) although it appears to approach the r 3 limit as r ! 0. It is of interest to determine whether self-preservation (denoted as SP. Here, self-preservation refers to a similarity solution based on one length scale and one velocity scale as the flow develops in the streamwise direction (Townsend, 1976) reflected in the constancy of the friction velocity (U s ), leads to SP of the velocity structure functions. The difficulty to pursue a proper SP analysis on hðduÞ2 i and hðduÞ3 i is due to the fact that in a turbulent boundary layer, the term Iu in Eq. (1) is made up of different terms representing various large-scale mechanisms contributing to the SBS energy budget, such as production, advection and diffusion. One can perform an approximate SP analysis based on a simplification, where the boundary layer is considered as a simple shear flow with a mean velocity gradient in the direction normal to the wall. Such an approach was carried out on the transport equation of hðdqÞ2 i ¼ hðduÞ2 i þ hðdv Þ2 i þ hðdwÞ2 i by Danaila et al. (2004) in a turbulent channel flow and Saikrishnan et al. (2007, 2012) in a turbulent boundary layer. The equation could be expressed as,
hðdqÞ2 dui þ 2m
@ 4 hðdqÞ2 i þ Iq ¼ hir: @r 3
ð2Þ
where the term Iq is mainly made up of the turbulent diffusion (TD) and the mean shear (S) (Danaila et al., 2004; Saikrishnan et al., 2007) viz.
S ¼ 2
1 r2
TD ¼
1 r2
Z
r
s2
@U hdudv ids @y
ð3Þ
s2
@hv þ ðdqÞ2 i ds; @y
ð4Þ
0
Z 0
r
where s is a dummy variable,
v þ ¼ ðv ðx; y; tÞ þ v ðx þ r; y; tÞ
and
ðdqÞ2 ¼ ðduÞ2 þ ðdv Þ2 þ ðdwÞ2 . The expressions (3) and (4) represent
Fig. 5. Profiles of normalized structure functions: (a) second order hðduÞ2 i=u02 and (b) third-order hðduÞ3 i=u02 at x = 2.54 m and Reh ¼ 8500.
Please cite this article in press as: Kamruzzaman, M., et al. Scale-by-scale energy budget in a turbulent boundary layer over a rough wall. Int. J. Heat Fluid Flow (2015), http://dx.doi.org/10.1016/j.ijheatfluidflow.2015.04.004
Md. Kamruzzaman et al. / International Journal of Heat and Fluid Flow xxx (2015) xxx–xxx
5
turbulent production and turbulent diffusion, respectively, at a scale r. Note though that the effects of large-scale inhomogeneity in the x direction are not accounted for in the term Iq . Accordingly, Eq. (2) is not correct for a turbulent boundary layer, whose thickness constantly increases with x, and thus is not consistent with the one-point energy budget. For example, when r is large, Eq. (2) does not yield the correct single point energy budget because the advection term is missing in the equation. We nevertheless use it as a first approximation and, if one assumes SP then we may seek solutions of the form,
hðdqÞ2 i ¼ u2 f ðr=lÞ
ð5Þ
hðdqÞ2 dui ¼ u3 gðr=lÞ
ð6Þ
hdudv i ¼ u2 hðr=lÞ
ð7Þ
hv þ ðdqÞ2 i ¼ U 1 u2 pðr=lÞ
ð8Þ
where u and l are the velocity and length scales that are functions of time; f ; g; h and p are dimensionless functions to be determined. After substituting these expressions in (2), the following SP conditions are obtained.
Rl ¼
u l
m
in the near wall region. This would justify the use of this equation by Saikrishnan et al. (2007) in the near wall region of a smooth wall turbulent boundary layer. 3.3. Scale by scale energy budgets
¼ C1;
ð9aÞ
2
SU l U 1 ¼ C2; m d 2 l U1 ¼ C3 ; m d 2 hil ¼ C4 ; mu2
ð9bÞ ð9cÞ ð9dÞ
where Rl is a scaling Reynolds number, SU @ðU=U 1 Þ=@ðy=dÞ is the non-dimensional mean velocity gradient and C 1 ; C 2 ; C 3 and C 4 are constants; d is a scaling length for the distance normal to the wall. In a first approximation, we assume that it is equal to the boundary layer thickness. Clearly, if one assumes u ¼ U s is constant then l must also be constant (independent of x and y). This is not possible for a smooth wall turbulent boundary layer, where U s decreases with increasing x, showing that U s cannot be a scaling velocity, at least, in the context of the SBS equation. Condition (9d) leads to,
u2 l ¼ C4m : hi 2
qffiffiffiffiffi Fig. 6. Profiles of pkffiffid Um1 (9c) as a function of wall-normal distance (y=d) for x = 1.48 m (); 1.94 m ( ); 2.24 m ( ); 2.54 m ( ); 2.84 m ( ).
ð10Þ
And if one assumes u ¼ u0 then Eq. (10) suggests that l k. qffiffiffiffi Accordingly, from (9c) we have pkffiffid Um1 = constant. To test this relaqffiffiffiffi tion, we report in Fig. 6, the profiles of pkffiffid Um1 as a function of y=d at several x positions. There is a good collapse of these profiles (the collapse extends well into the outer region of the boundary layer), indicating that the SP condition (9c) is well satisfied. A similar result was shown previously by Segalini et al. (2011) for the transverse Taylor microscale in a smooth wall turbulent boundary layer. Segalini et al. (2011) used mixed scaling based on the results of Alfredsson and Johansson (1990), who used an empirical approach to obtain mixed scaling. Here, we obtained the same result in a rigorous manner using an analytical approach that is based on the two-point energy budget equation. The collapse of qffiffiffiffi U1 pkffiffi m as a function of y=d further indicates that the boundary layer d thickness, d, can be used as scaling length, for this rough wall at least and over the present streamwise distance. Interestingly, notice the very good collapse of the profiles when y=d 6 0:1, where the advection term in the single point energy budget may be considered negligible, thus suggesting that Eq. (2) may be approximately valid
It was already stated that the term Iu in Eq. (1) (or Iq in Eq. (2)) represents large-scale effects on the energy and it varies from flow to flow and from location to location in a given flow. These largescale effects represent various contributions to the one-point energy budget equation. Thus, one should determine such an energy budget everywhere across the boundary layer in order to properly estimate the term Iu . This requires careful measurements of all the velocity components, a scope for future boundary layer studies in Turbulence community. In the meantime, Iu can be inferred indirectly from Eq. (1), once all the other terms are measured. While the first and second terms on the left side of the equation are relatively straightforward to measure, the one on the right side presents a major challenge, particularly in wall-bounded flows. Indeed, measurements of several velocity derivatives are required to obtain hi accurately. A crude estimate of hi is provided using local isotropy:
hiiso ¼ 15m
* + 2 @u ; @x
ð11Þ
We also used a new method, termed as ‘Spectral chart’, for estimating hi (Djenidi and Antonia, 2012), at least in a region of the boundary layer away from the near-wall region, where the method given in Eq. (11) is not reliable. The method is based on a spectral chart and provides reliable values of hi. We compared the values given by the spectral chart and Eq. (11) and we found that the error did not exceed about 14% for y=d P 0:2. It is to be noted that hi is used as a normalizing factor and any error in it would not affect the relative trend of different terms in Eq. (1). In the following, we used hiiso for displaying the terms of the scale-by-scale energy budgets. The term Iu is obtained by difference of the known terms in Eq. (1) and accounts for the large-scale contributions from advection, production and turbulent diffusion. Fig. 7 shows the individual terms of Eq. (1), normalized by 4=5hir, for y ¼ 0:23d at several streamwise positions (x = 1.94 m, 2.24 m, 2.54 m and 2.84 m); this y-position falls well in the region of constant Rk as seen in Fig. 4. It is quite interesting to observe the relatively good collapse of the individual terms, suggesting that the SBS budget may follow SP fairly well, corroborating the results of Fig. 6. As expected, the viscous term dominates the budget for small separations and drops to zero as the separation becomes
Please cite this article in press as: Kamruzzaman, M., et al. Scale-by-scale energy budget in a turbulent boundary layer over a rough wall. Int. J. Heat Fluid Flow (2015), http://dx.doi.org/10.1016/j.ijheatfluidflow.2015.04.004
6
Md. Kamruzzaman et al. / International Journal of Heat and Fluid Flow xxx (2015) xxx–xxx
Fig. 7. Terms of Eq. (1) for y=d ¼ 0:23 at x = 1.94 m (Squares), 2.24 m (Circles), 2.54 m (Triangles) and 2.84 m (Crosses). The asterisk means normalization by 4=5hir.
large. It is expected that this term should reach a value of 1 as r approaches zero. The third-order structure function becomes the dominant contributor to the budget when r=k P 0:4 and remains so until r=k ¼ 20. Beyond this separation, it becomes smaller than Iu . This latter quantity dominates the budget at large separations and decreases as r=k decreases. Notice though that it presents a local peak in the range 0:1 6 r=k 6 3 , reflecting contributions from the large-scale motions in this range of scales. These contributions are observed across the boundary layer as illustrated in Fig. 8, which shows Iu for different y-positions (y=d = 0.029, 0.23, 0.3 and 0.5) at x = 2.54 m. Note the collapse of the Iu profiles in Fig. 8 for y=d = 0.23, 0.3 and 0.5, as one may have expected since this range of y=d corresponds to constant Rk (see Fig. 4). When y=d ¼ 0:029, there is no local peak in the distribution of Iu , suggesting that the large-scale mechanisms responsible for the local peak at high values of y=d are either not present or have a small contribution in the near wall region. Fig. 9 shows the SBS budget at two y-positions, y=d ¼ 0:029 and 0.5 at a different streamwise location, x = 2.84 m to illustrate the effect of Rk on the energy budget across the boundary layer. The three terms of the budget at y=d ¼ 0:029 do not collapse with its counterpart, y=d ¼ 0:5, reflecting the variation of Rk between the two streamwise locations. Note that while the maximum of hðduÞ3 i is about 0.85 for y=d = 0.029, it is around 1 for y=d = 0.5 (the fact that it is slightly higher than 1 induces a negative value on Iu ; this overestimate may be related to an error in the estimate of hi and/or due to the effects of the outer region intermittency).
Fig. 9. Terms of Eq. (1) for x = 2.84 m, y=d ¼ 0:029 (blue symbols) and y=d ¼ 0:5 (red symbols). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Fig. 10. Kolmogorov normalized velocity spectra Eu for x = 2.84 m Reh ¼ 12; 600. The asterisk represents Kolmogorov normalization.
and
the existence of an inertial range or a 5/3 region in the velocity spectrum, or at least that it is relatively well approached. Fig. 10, which shows the Kolmogorov normalized velocity spectra for x = 2.84 m, seems to confirm these observations. The spectra for y=d P 0:029 follows relatively well the slope 5/3. The spectrum for y=d = 0.03 deviates from the others over the entire range of the wavenumber, reflecting certainly the inadequacy of estimating hi via Eq. (11) at a distance so close to the wall.
Note that hðduÞ3 i is also about 1 in Fig. 7. These results suggest 4. Conclusions Hot-wire velocity measurements have been made in a turbulent boundary layer over a rough wall consisting of transverse circular rods, with a ratio of 8 between the spacing of two consecutive rods and the height of the rod. This study looked at the scale-by-scale budget of the velocity variance. It is found that the Taylor microscale Reynolds number is constant in the region 0:2 6 y=d 6 0:6 of the boundary layer. The results also showed that the boundary layer has almost reached a self-preservation state at all scales of motion, in the region 0:2 6 y=d 6 0:6, reflected in the constancy of the Taylor microscale Reynolds number and the collapse of the Kolmogorov normalized velocity spectra at all wavenumbers. Self-preservation appears to be also observed in the SBS budget in the same region of the boundary layer. While the small scales are controlled by the viscosity, intermeFig. 8. Term Iu for y=d ¼ 0:029 (Squares), 0.23 (Triangles), 0.3 (Circles) and 0.5 (diamonds) and x = 2.54 m; The asterisk represents normalization by 4=5hir.
diate scales over which the transfer of energy (or hðduÞ3 i) is important are affected by mechanisms, represented in Iu of Eq. (1), which
Please cite this article in press as: Kamruzzaman, M., et al. Scale-by-scale energy budget in a turbulent boundary layer over a rough wall. Int. J. Heat Fluid Flow (2015), http://dx.doi.org/10.1016/j.ijheatfluidflow.2015.04.004
Md. Kamruzzaman et al. / International Journal of Heat and Fluid Flow xxx (2015) xxx–xxx
could be identified once an estimate of the one-point energy budget is made. For example, the data showed that the effects of the large-scale inhomogeneity are felt at scales as small as k in the region 0:2 6 y=d 6 0:6. It is likely that these mechanisms include production, advection and turbulent diffusion. Acknowledgment LD and RAA acknowledge the financial support of the Australian Research Council. References Alfredsson, P.H., Johansson, A.V., 1990. The art of detection of turbulence structures. Near-Wall Turb. 1, 888–896. Antonia, R.A., Djenidi, L., 2010. On the outer layer controversy for a turbulent boundary layer over a rough wall. In: IUTAM Symposium on the Physics of Wall-Bounded Turbulent Flows on Rough Walls. Springer, pp. 77–86. Antonia, R.A., Krogstad, P.A., 2001. Turbulence structure in boundary layers over different types of surface roughness. Fluid Dyn. Res. 28 (2), 139–157. Antonia, R.A., Luxton, R.E., 1971. The response of a turbulent boundary layer to a step change in surface roughness. Part 1–Smooth to rough. J. Fluid Mech. 48 (04), 721–761. Antonia, R.A., Shafi, H.S., 1999. Small scale intermittency in a rough wall turbulent boundary layer. Exp. Fluids 26 (1-2), 145–152. Bakken, O.M., Krogstad, P.A., Ashrafian, A., Andersson, H.I., 2005. Reynolds number effects in the outer layer of the turbulent flow in a channel with rough walls. Phys. Fluids 17 (6). Bandyopadhyay, P.R., Watson, R.D., 1988. Structure of rough-wall turbulent boundary layers. Phys. Fluids 31 (7), 1877–1883. Burattini, P., Antonia, R.A., Danaila, L., 2005. Scale-by-scale energy budget on the axis of a turbulent round jet. J. Turb. 6. Danaila, L., Anselmet, F., Zhou, T., Antonia, R.A., 1999. A generalization of Yaglom’s equation which accounts for the large-scale forcing in heated decaying turbulence. J. Fluid Mech. 391, 359–372. Danaila, L., Anselmet, F., Zhou, T., Antonia, R.A., 2001. Turbulent energy scale budget equations in a fully developed channel flow. J. Fluid Mech. 430, 87–109. Danaila, L., Anselmet, F., Zhou, T., 2004. Turbulent energy scale-budget equations for nearly homogeneous sheared turbulence. Flow Turb. Combust. 72 (2–4), 287– 310. Djenidi, L., Antonia, R.A., 2012. A spectral chart method for estimating the mean turbulent kinetic energy dissipation rate. Exp. Fluids 53 (4), 1005–1013. Djenidi, L., Elavarasan, R., Antonia, R.A., 1999. The turbulent boundary layer over transverse square cavities. J. Fluid Mech. 395, 271–294. Jackson, P.S., 1981. On the displacement height in the logarithmic velocity profile. J. Fluid Mech. 111, 15–25. Jiménez, J., 2004. Turbulent flows over rough walls. Annu. Rev. Fluid Mech. 36, 173– 196. Kamruzzaman, M., Talluru, K.M., Djenidi, L., Antonia, R.A., 2014. An experimental study of turbulent boundary layer over 2d transverse circular bars. In: 19th Australasian Fluid Mechanics Conference, Melbourne, Australia.
7
Keirsbulck, L., Labraga, L., Mazouz, A., Tournier, C., 2002. Surface roughness effects on turbulent boundary layer structures. J. Fluids Eng. 124 (1), 127–135. Krogstad, P.A., Antonia, R.A., 1994. Structure of turbulent boundary layers on smooth and rough walls. J. Fluid Mech. 277, 1–21. Krogstad, P.A., Antonia, R.A., 1999. Surface roughness effects in turbulent boundary layers. Exp. Fluids 27 (5), 450–460. Krogstad, P.A., Antonia, R.A., Browne, L.W.B., 1992. Comparison between rough- and smooth-wall turbulent boundary layers. J. Fluid Mech. 245, 599–617. Krogstad, P.A., Andersson, H.I., Bakken, O.M., Ashrafian, A., 2005. An experimental and numerical study of channel flow with rough walls. J. Fluid Mech. 530, 327– 352. Lee, S.H., Sung, H.J., 2007. Direct numerical simulation of the turbulent boundary layer over a rod-roughened wall. J. Fluid Mech. 584, 125–146. 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. Nikuradse, J., 1933. Stromungsgesetz in rauhren rohren, vdi-forsch. 361 (Engl. transl. 1950: Laws of flow in rough pipes.). Tech. rep., NACA TM 1292. Oyewola, O., Djenidi, L., Antonia, R.A., 2008. Response of mean turbulent energy dissipation rate and spectra to concentrated wall suction. Exp. Fluids 44 (1), 159–165. Perry, A.E., Schofield, W.H., Joubert, P.N., 1969. Rough wall turbulent boundary layers. J. Fluid Mech. 37 (02), 383–413. Raupach, M.R., 1981. Conditional statistics of Reynolds stress in rough-wall and smooth-wall turbulent boundary layers. J. Fluid Mech. 108, 363–382. Raupach, M.R., Antonia, R.A., Rajagopalan, S., 1991. Rough-wall turbulent boundary layers. App. Mech. Rev. 44 (1), 1–25. Saikrishnan, N., Longmire, E.K., Marusic, I., 2007. Analysis of scale energy budgets in wall turbulence using dual plane piv. In: 16th Australasian Fluid Mechanics Conference. Saikrishnan, N., De Angelis, E., Longmire, E.K., Marusic, I., Casciola, C.M., Piva, R., 2012. Reynolds number effects on scale energy balance in wall turbulence. Phys. Fluids 24 (1), 015101. Schultz, M.P., Flack, K.A., 2005. Outer layer similarity in fully rough turbulent boundary layers. Exp. Fluids 38 (3), 328–340. Segalini, A., Örlü, R., Schlatter, P., Alfredsson, P.H., Rüedi, J.D., Talamelli, A., 2011. A method to estimate turbulence intensity and transverse Taylor microscale in turbulent flows from spatially averaged hot-wire data. Exp. Fluids 51 (3), 693– 700. Shafi, H.S., Antonia, R.A., 1995. Anisotropy of the reynolds stresses in a turbulent boundary layer on a rough wall. Exp. Fluids 18 (3), 213–215. Shafi, H.S., Antonia, R.A., 1997. Small-scale characteristics of a turbulent boundary layer over a rough wall. J. Fluid Mech. 342, 263–293. Smalley, R.J., Antonia, R.A., Djenidi, L., 2001. Self-preservation of rough-wall turbulent boundary layers. Euro. J. Mech.–B/Fluids 20 (5), 591–602. Talluru, K.M., Kulandaivelu, V., Hutchins, N., Marusic, I., 2014. A calibration technique to correct sensor drift issues in hot-wire anemometry. Meas. Sci. Tech. 25 (10), 105304. Townsend, A.A., 1976. The Structure of Turbulent Shear Flow. Cambridge University Press. Von Karman, T., Howarth, L., 1938. On the statistical theory of isotropic turbulence. Proc. Roy. Soc. London A – Math. Phys. Sci. 164 (917), 192–215.
Please cite this article in press as: Kamruzzaman, M., et al. Scale-by-scale energy budget in a turbulent boundary layer over a rough wall. Int. J. Heat Fluid Flow (2015), http://dx.doi.org/10.1016/j.ijheatfluidflow.2015.04.004
Contents lists available at ScienceDirect
International Journal of Heat and Fluid Flow journal homepage: www.elsevier.com/locate/ijhff
Scale-by-scale energy budget in a turbulent boundary layer over a rough wall Md. Kamruzzaman, L. Djenidi, R.A. Antonia, K.M. Talluru ⇑ School of Engineering, University Drive, The University of Newcastle, Callaghan, 2308, NSW, Australia
a r t i c l e
i n f o
Article history: Received 27 January 2015 Received in revised form 9 April 2015 Accepted 11 April 2015 Available online xxxx Keywords: Scale-by-scale budget Structure function Rough wall boundary layer
a b s t r a c t Hot-wire velocity measurements are carried out in a turbulent boundary layer over a rough wall consisting of transverse circular rods, with a ratio of 8 between the spacing (w) of two consecutive rods and the rod height (k). The pressure distribution around the roughness element is used to accurately measure the mean friction velocity (U s ) and the error in the origin. It is found that U s remained practically constant in the streamwise direction suggesting that the boundary layer over this surface is evolving in a self-similar manner. This is further corroborated by the similarity observed at all scales of motion, in the region 0:2 6 y=d 6 0:6, as reflected in the constancy of Reynolds number (Rk ) based on Taylor’s microscale and the collapse of Kolmogorov normalized velocity spectra at all wavenumbers. A scale-by-scale budget for the second-order structure function hðduÞ2 i (du ¼ uðx þ rÞ uðxÞ, where u is the fluctuating streamwise velocity component and r is the longitudinal separation) is carried out to investigate the energy distribution amongst different scales in the boundary layer. It is found that while the small scales are controlled by the viscosity, intermediate scales over which the transfer of energy (or hðduÞ3 i) is important are affected by mechanisms induced by the large-scale inhomogeneities in the flow, such as production, advection and turbulent diffusion. For example, there are non-negligible contributions from the large-scale inhomogeneity to the budget at scales of the order of k, the Taylor microscale, in the region of the boundary layer extending from y=d ¼ 0:2 to 0.6 (d is the boundary layer thickness). Ó 2015 Elsevier Inc. All rights reserved.
1. Introduction The investigation of a turbulent boundary layer over rough walls is of fundamental and practical importance. From a fundamental view point, the knowledge of how a boundary layer responds to roughness (size and geometry, for example) can help in terms of understanding the dynamics of the boundary layer. This should in turn help to devise better management control strategies in flow situations, e.g., heat exchangers, or flow over a river bed, which involve flow over rough surfaces. The motivation for studying rough wall flows is well summarized by Antonia and Djenidi (2010) as follows: ‘‘Apart from the wide engineering applications associated with rough walls (e.g., Nikuradse, 1933; Perry et al., 1969; Raupach et al., 1991 and Jiménez, 2004) there are several compelling scientific reasons for studying flows over rough walls. First, our understanding of the turbulence structure near the vicinity of rough walls has lagged significantly behind that for the canonical smooth ⇑ Corresponding author. E-mail address: [email protected] (K.M. Talluru).
wall, for which streaks are observed throughout the viscous region and are important in the context of bursting. Eliminating the viscous layer through the introduction of roughness elements and examining the effect this has on both the inner and outer regions should be sufficient incentive for studying rough wall flows with vigour. Secondly, it is almost intuitive that the turbulence close to a drag-augmenting surface should be more isotropic than that over a smooth wall, thus facilitating somewhat the modelling of the near-wall region. Thirdly, a turbulent boundary layer which develops over a rough wall is more likely to satisfy the requirements of self-preservation (self-similarity along the streamwise direction) than a smooth wall boundary layer.’’ (taken from Antonia and Djenidi (2010)). Of particular interest amongst rough surfaces, is the rough wall consisting of transverse circular rods or square bars, which has attracted a lot of attention, (see, for example, Raupach, 1981; Bandyopadhyay and Watson, 1988; Bakken et al., 2005; Keirsbulck et al., 2002; Krogstad and Antonia, 1999). The reason for this interest relies on the relatively simple geometry of the roughness elements which allows parametric studies (e.g., Leonardi et al., 2003; Krogstad et al., 2005). However, although a
http://dx.doi.org/10.1016/j.ijheatfluidflow.2015.04.004 0142-727X/Ó 2015 Elsevier Inc. All rights reserved.
Please cite this article in press as: Kamruzzaman, M., et al. Scale-by-scale energy budget in a turbulent boundary layer over a rough wall. Int. J. Heat Fluid Flow (2015), http://dx.doi.org/10.1016/j.ijheatfluidflow.2015.04.004
2
Md. Kamruzzaman et al. / International Journal of Heat and Fluid Flow xxx (2015) xxx–xxx
larger number of studies of a turbulent boundary layer over this particular 2D rough wall have already been undertaken (Raupach, 1981; Bandyopadhyay and Watson, 1988; Krogstad and Antonia, 1994; Djenidi et al., 1999; Keirsbulck et al., 2002; Leonardi et al., 2003; Lee and Sung, 2007), there are still unresolved issues. For example, there is no consensus yet on whether the outer region of the boundary layer on this rough wall is affected by the roughness when compared to that of a smooth wall boundary layer (e.g., Krogstad et al., 1992; Antonia and Krogstad, 2001; Jiménez, 2004; Schultz and Flack, 2005; Antonia and Djenidi, 2010), casting doubt on the validity of Townsend’s wall-similarity hypothesis (Townsend, 1976). Also, the accurate measurement of the skin friction velocity is problematic and challenging and can lead to erroneous conclusions if incorrectly obtained. It is further relevant to note that most of the studies of turbulent boundary layers focused mainly on one-point statistics (e.g., mean velocity, Reynolds stresses). While these statistics are of great importance and provide information on the dynamical response of the boundary layer to the surface changes, they do not shed light on how the energy transfer amongst various scales in the flow might be altered. Relatively few studies of turbulent boundary layers over rough walls used two-point statistics to gain some insight into the structure of the flow (Saikrishnan et al., 2007, 2012). These were mainly aimed at measuring the spatial correlations. We carry out a two-point analysis of a turbulent boundary layer over a 2D rough wall with the aim to assess the energy distribution at any particular scale r (ranging from the smallest to the largest in the flow) as well as the way the energy is transferred amongst the scales. The focus of the work reported in this paper is on the scaleby-scale analysis of the second-order velocity structure function, hðduÞ2 i (du ¼ uðx1 þ rÞ uðx1 Þ, where u is the streamwise velocity fluctuation and r is the streamwise separation; the angular brackets denote time averaging). Notice that when r is very large then hðduÞ2 i ¼ 2hu2 i. If local isotropy is assumed, then the transport equation of hðduÞ2 i is given generically by Danaila et al. (2001),
hðduÞ3 i þ 6m
@ 4 hðduÞ2 i þ Iu ¼ hir; @r 5
ð1Þ
where m is the kinematic viscosity of the fluid, and hi is the mean turbulent kinetic energy dissipation rate. The first and second terms of the left side of Eq. (1) represent the energy transfer and the viscous diffusion of energy, respectively. The third term, Iu , accounts for the inhomogeneity or non-stationarity associated with the large scales; it can also account for the mean shear that exists in wallbounded flows. Eq. (1) corresponds to the two-point velocity correlation function, first written by Von Karman and Howarth (1938) and effectively represents the scale-by-scale (SBS) budget of the turbulent energy at a location in the flow. At large r, Eq. (1) reduces to the (one-point) energy budget equation; in the limit r ! 0, it reduces to the (one-point) transport equation of hi (e.g., Danaila
et al., 1999). Different versions of Eq. (1) have been proposed for a turbulent channel flow (Danaila et al., 2001) and self-preserving turbulent round jet (Burattini et al., 2005). The term Iu is different between these two cases and reflects the difference in the one-point energy budget. For example, in the case of a channel flow, where the flow is stationary, Iu near the wall includes the effects of shearing and the spatial inhomogenity while on the centre-line of a round jet, Iu accounts for the non-stationarity and spatial inhomogeneity. Clearly, Iu not only varies from flow to flow, but it also varies from location to location within the same flow. The objective of the present work is to extend the SBS approach to a turbulent boundary layer over a rough wall. Saikrishnan et al. (2007, 2012) performed a SBS analysis in a turbulent boundary layer and turbulent channel flow over a smooth wall. The motivation for considering a rough wall stems from the fact that the flow should be more isotropic in comparison to a smooth wall (see, for example, Shafi and Antonia, 1995, 1997; Antonia and Shafi, 1999) and thus one may expect that local isotropy is more adequately satisfied, thus making Eq. (1) more suitable for studying rough walls than smooth walls.
2. Experimental facility and measurements The wind tunnel measurements are conducted in a turbulent boundary layer developing over a rough wall made up of circular rods mounted transversely on the tunnel floor and spanning the full width of the test section with w=k ¼ 8, where w is the streamwise pitch and k is the roughness height (see Fig. 1). The roof of the wind tunnel is adjusted to obtain a nominally zero-pressure gradient (Dp=ð1=2qU 2 Þ ¼ 0:5%) in the streamwise direction. The velocity fluctuations are measured using a single hot-wire probe. The hot-wire (diameter, d ¼ 2:5 lm, and length, l=d ¼ 200) is etched from a coil of Pt-10% Rh and operated using an in-house built constant temperature anemometer (CTA) with an over-heat ratio of 1.5. The hot-wire is calibrated in situ against the Pitot-static tube positioned in the undisturbed free stream flow before and after every experiment at 15 different speeds ranging between 0 m/s and 16 m/s. A linear interpolation in time (see Talluru et al., 2014) is employed between the two calibrations to correct for any drift in the hot-wire voltage that occurs during the course of an experiment. Only a minimal drift is noticed in the hot-wire due to short duration of our experiments. For the results reported here, measurements are made at a streamwise distance x ¼ 2:54 m and 40 logarithmically spaced points are taken between 0:001 6 y=d 6 1:5 in the wall-normal direction. At each measurement point, samples are collected for a duration of 120 s at a sampling frequency of 20 kHz. Experiments were taken at different free stream speeds giving Reh (Reynolds number based on the momentum thickness) in the range of 7000 and 14,000 while the ratio d=k (d is the boundary layer thickness) remained nominally about 60. In addition, measurements are made at five different streamwise
y
x
(Traversing hot-wire)
Flow
k
w
Fig. 1. Schematic representation of the rough wall and hot-wire probe used in the experiment.
Please cite this article in press as: Kamruzzaman, M., et al. Scale-by-scale energy budget in a turbulent boundary layer over a rough wall. Int. J. Heat Fluid Flow (2015), http://dx.doi.org/10.1016/j.ijheatfluidflow.2015.04.004
3
Md. Kamruzzaman et al. / International Journal of Heat and Fluid Flow xxx (2015) xxx–xxx
locations between 1.4 m 6 x 6 2:8 m at a free stream velocity of 14 m/s. All the boundary layer characteristics in these experiments are summarized in Table 1. An attractive feature of this particular roughness is that, the pffiffiffiffiffiffiffiffiffiffiffi friction velocity, U s (U s ¼ sw =q, where sw is the wall shear stress and q is the fluid density), can be determined reliably from the form drag estimated by pressure tapping the individual roughness elements, see Kamruzzaman et al., 2014 for a full description of the method. This is also illustrated in Fig. 2 which shows the pressure distributions around the roughness element for two Reynolds numbers (Reh ¼ 8500 and 12,632). Interestingly, we found that there is a complete collapse of the pressure distribution profiles obtained between 8500 6 Reh 6 12; 632 (all results are not shown here for brevity) indicating that C D has become independent of Re. In addition, for the present configuration of roughness elements
Table 1 Boundary layer properties for different measurements carried out in this study. Case I: Different streamwise locations at a free stream velocity, U 1 14 m/s; Case II: At two different free stream velocities, U 1 10 m/s and 14 m/s at x = 2.54 m. x (m)
U 1 (m/s)
d99 (m)
d (m)
h (m)
U s (m/s)
Res
Reh
Case I 1.48 1.94 2.24 2.54 2.84
13.89 13.82 13.76 14.14 14.26
0.059 0.076 0.077 0.091 0.10
0.0168 0.0214 0.0222 0.0244 0.0268
0.0091 0.0119 0.0126 0.0140 0.0154
0.93 0.92 0.91 0.91 0.90
3460 4410 4480 5300 5830
8120 10,500 11,100 12,600 14,100
Case II 2.54 2.54
9.8 14.04
0.092 0.093
0.0245 0.0240
0.0136 0.0140
0.65 0.90
3830 5360
8550 12,630
0.06
0.04 θ
θ
U
θ
3. Results 3.1. Mean velocity and turbulence intensity profiles The mean velocity and the velocity variance profiles at free stream velocities, 10 m/s and 14 m/s are reported in Fig. 3(a) and (b) respectively. The data are presented in a log-linear plot and normalized by U s and d. Note that the origin y ¼ 0 is taken as the location that corresponds to the centroid of the moments of the pressure forces acting on the roughness elements (see Jackson, 1981; Leonardi et al., 2003; Kamruzzaman et al., 2014, for more details) and does not therefore rely on the existence of a log law. Fig. 3(a) shows that the mean velocity distributions over the rough wall display a convincing log law region, which is nearly identical for the two values of Reh ; the slope of this region is smaller than that over the smooth wall, thus making the determination of DU=U s , or roughness function, somewhat ambiguous. In the turbulence intensity profiles, shown in Fig. 3(b), we note that the intensity drops as we approach the wall instead of presenting a peak at y=d 0:002 or yþ 15 contrary to what we observe in a smooth wall boundary layer. The absence of the near-wall peak in the turbulence intensity profile indicates that the near-wall turbulence structure on the present rough wall is very different to that on a smooth wall. Accordingly, one expects that the turbulence production mechanism on this wall to be different to that on a smooth wall. This may have important consequences in terms of turbulence modelling on this particular rough surface.
Δ
1/2ρ
2
0.02
= 8500 = 12632
(w=k ¼ 8), the skin friction contribution to the total drag on the rough wall is practically zero (C v 0; C v is the coefficient of skinfriction drag) (see Leonardi et al., 2003, for more details). This facilitated us to estimate U s from C D alone. Further, it was verified that U s is practically constant with respect to x, confirming the claim made by Smalley et al. (2001) that this boundary layer is likely to be closer to self-preservation than a smooth wall boundary layer. The fact that the pressure distributions are equal for both Reynolds numbers further supports the claim.
0
3.2. 2nd and 3rd order structure functions −0.02
Before proceeding with the 2nd and 3rd order structure functions, it is worthwhile looking at how the Taylor microscale
−0.04
−0.06
0
50
100
150
200
250
300
350
400
θ Fig. 2. Distribution of the pressure around a rod element at two Reynolds numbers, Reh ¼ 8500 and 12,632 using symbols ( ) and ( ) respectively.
Reynolds number (Rk ¼ u0 k=m, with k ¼ u0 =hð@u=@xÞ2 i1=2 and u0 is the rms value of velocity fluctuations) varies across the boundary layer. It is to be noted that we employed Taylor’s hypothesis of frozen turbulence to project the temporal velocity fluctuations in the streamwise direction using a convection velocity equal to that of the local mean velocity. Fig. 4 shows such a variation of Rk for
25
5
20
4
15
3
=12.5
Δ
10
2
5
1
0
−2
10
−1
10
0
10
−2
10
−1
10
0
10
0
Fig. 3. Profiles of (a) mean velocity and (b) turbulence intensity for x = 2.54 m at Reh ¼ 8500 () and Reh ¼ 12; 632 () normalized using U s and d. The dashed line in (a) represents the smooth wall log region and is shown only as a reference.
Please cite this article in press as: Kamruzzaman, M., et al. Scale-by-scale energy budget in a turbulent boundary layer over a rough wall. Int. J. Heat Fluid Flow (2015), http://dx.doi.org/10.1016/j.ijheatfluidflow.2015.04.004
4
Md. Kamruzzaman et al. / International Journal of Heat and Fluid Flow xxx (2015) xxx–xxx 500
at y=d ¼ 0:03 collapse fairly well with the other two. Looking at the
300
results of hðduÞ3 i in Fig. 5(b), we note that all the three profiles collapse well at small values of r but at larger r, the differences show up. Particularly, the profile at y=d ¼ 0:03 deviates from the other two profiles. All the above discussion shows the effect of Rk on
200
hðduÞ3 i when the Rk at those locations is nominally the same.
400
the structure functions; there is a better collapse of hðduÞ2 i) and Looking further, as we expect, hðduÞ2 i=u02 approaches a value of
100
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
Fig. 4. Variation of Rk as a function of wall-normal distance (y=d) for x = 2.54 m at Reh ¼ 8500 and 12,600 using symbols ( ) and ( ) respectively.
x = 2.54 m for two Reynolds numbers, Reh ¼ 8500 and 12,600. Rk increases rapidly with the distance from the wall, then reaches a plateau with a value of about 300 and 370 for Reh ¼ 8500 and 12,600, respectively. The plateau extends from y=d 0:2 to about 0.6 and beyond this latter value, Rk decreases. The constancy of Rk suggests that there exists a relatively non-negligible part (40%) of the boundary layer in which the small scale motions may exist in a ‘‘quasi-equilibrium’’ state. It is interesting to note that such a plateau region has not been widely reported for Rk in a smooth wall turbulent boundary layer. Oyewola et al. (2008) is one of the few studies who presented the results of Rk over a smooth wall. However, their results do not show a clear plateau of Rk . Rather, it appears that there is a local minima for Rk . Further, the results of Oyewola et al. (2008) show a sharp rise of Rk in the near wall region, which is physically incorrect. This is because the measurements of Oyewola et al. (2008) were made using an X-wire, which is known to show erroneous results, particularly in the near wall region (Antonia and Luxton, 1971). In the present rough wall study, the constant value of Rk in the region 0:2 6 y=d 6 0:6 suggests that k may be a relevant scaling length in that region of the flow. This is indeed confirmed in Fig. 5(a) and (b) which show the profiles of the second2
3
(hðduÞ i) and third-order (hðduÞ i) velocity structure functions at three different locations in the wall-normal direction, y ¼ 0:03d (yþ ¼ 87), y ¼ 0:23d (yþ ¼ 690) and y ¼ 0:4d (yþ ¼ 1200). Here, the second- and the third-order structure functions are normalized by u02 and u03 respectively while the separation distance (r) is normalized by k. The data are plotted in a log–log scale in order to highlight the salient features of the second- and the third-order structure functions. The profiles of hðduÞ2 i for y=d ¼ 0:23 and 0:4 collapse very well. This is expected because both these points lie within the plateau region of Rk (see Fig. 4). Surprisingly, the results
2 at a large separation while hðduÞ3 i=u03 becomes zero. Notice that despite the reasonable value of the Reynolds number based on Taylor microscale (Rk ¼ 187, 287 and 310) the inertial range, represented by the law r2=3 , is not yet established. The data show, though, a tendency to follow the 2/3 law, suggesting that a further increase in Rk will lead to its emergence. On the other hand, the ratio hðduÞ2 i=u02 approaches the expected r 2 behaviour as r ! 0. Similarly, hðduÞ3 i=u03 does not present a clear inertial range (r 1 ) although it appears to approach the r 3 limit as r ! 0. It is of interest to determine whether self-preservation (denoted as SP. Here, self-preservation refers to a similarity solution based on one length scale and one velocity scale as the flow develops in the streamwise direction (Townsend, 1976) reflected in the constancy of the friction velocity (U s ), leads to SP of the velocity structure functions. The difficulty to pursue a proper SP analysis on hðduÞ2 i and hðduÞ3 i is due to the fact that in a turbulent boundary layer, the term Iu in Eq. (1) is made up of different terms representing various large-scale mechanisms contributing to the SBS energy budget, such as production, advection and diffusion. One can perform an approximate SP analysis based on a simplification, where the boundary layer is considered as a simple shear flow with a mean velocity gradient in the direction normal to the wall. Such an approach was carried out on the transport equation of hðdqÞ2 i ¼ hðduÞ2 i þ hðdv Þ2 i þ hðdwÞ2 i by Danaila et al. (2004) in a turbulent channel flow and Saikrishnan et al. (2007, 2012) in a turbulent boundary layer. The equation could be expressed as,
hðdqÞ2 dui þ 2m
@ 4 hðdqÞ2 i þ Iq ¼ hir: @r 3
ð2Þ
where the term Iq is mainly made up of the turbulent diffusion (TD) and the mean shear (S) (Danaila et al., 2004; Saikrishnan et al., 2007) viz.
S ¼ 2
1 r2
TD ¼
1 r2
Z
r
s2
@U hdudv ids @y
ð3Þ
s2
@hv þ ðdqÞ2 i ds; @y
ð4Þ
0
Z 0
r
where s is a dummy variable,
v þ ¼ ðv ðx; y; tÞ þ v ðx þ r; y; tÞ
and
ðdqÞ2 ¼ ðduÞ2 þ ðdv Þ2 þ ðdwÞ2 . The expressions (3) and (4) represent
Fig. 5. Profiles of normalized structure functions: (a) second order hðduÞ2 i=u02 and (b) third-order hðduÞ3 i=u02 at x = 2.54 m and Reh ¼ 8500.
Please cite this article in press as: Kamruzzaman, M., et al. Scale-by-scale energy budget in a turbulent boundary layer over a rough wall. Int. J. Heat Fluid Flow (2015), http://dx.doi.org/10.1016/j.ijheatfluidflow.2015.04.004
Md. Kamruzzaman et al. / International Journal of Heat and Fluid Flow xxx (2015) xxx–xxx
5
turbulent production and turbulent diffusion, respectively, at a scale r. Note though that the effects of large-scale inhomogeneity in the x direction are not accounted for in the term Iq . Accordingly, Eq. (2) is not correct for a turbulent boundary layer, whose thickness constantly increases with x, and thus is not consistent with the one-point energy budget. For example, when r is large, Eq. (2) does not yield the correct single point energy budget because the advection term is missing in the equation. We nevertheless use it as a first approximation and, if one assumes SP then we may seek solutions of the form,
hðdqÞ2 i ¼ u2 f ðr=lÞ
ð5Þ
hðdqÞ2 dui ¼ u3 gðr=lÞ
ð6Þ
hdudv i ¼ u2 hðr=lÞ
ð7Þ
hv þ ðdqÞ2 i ¼ U 1 u2 pðr=lÞ
ð8Þ
where u and l are the velocity and length scales that are functions of time; f ; g; h and p are dimensionless functions to be determined. After substituting these expressions in (2), the following SP conditions are obtained.
Rl ¼
u l
m
in the near wall region. This would justify the use of this equation by Saikrishnan et al. (2007) in the near wall region of a smooth wall turbulent boundary layer. 3.3. Scale by scale energy budgets
¼ C1;
ð9aÞ
2
SU l U 1 ¼ C2; m d 2 l U1 ¼ C3 ; m d 2 hil ¼ C4 ; mu2
ð9bÞ ð9cÞ ð9dÞ
where Rl is a scaling Reynolds number, SU @ðU=U 1 Þ=@ðy=dÞ is the non-dimensional mean velocity gradient and C 1 ; C 2 ; C 3 and C 4 are constants; d is a scaling length for the distance normal to the wall. In a first approximation, we assume that it is equal to the boundary layer thickness. Clearly, if one assumes u ¼ U s is constant then l must also be constant (independent of x and y). This is not possible for a smooth wall turbulent boundary layer, where U s decreases with increasing x, showing that U s cannot be a scaling velocity, at least, in the context of the SBS equation. Condition (9d) leads to,
u2 l ¼ C4m : hi 2
qffiffiffiffiffi Fig. 6. Profiles of pkffiffid Um1 (9c) as a function of wall-normal distance (y=d) for x = 1.48 m (); 1.94 m ( ); 2.24 m ( ); 2.54 m ( ); 2.84 m ( ).
ð10Þ
And if one assumes u ¼ u0 then Eq. (10) suggests that l k. qffiffiffiffi Accordingly, from (9c) we have pkffiffid Um1 = constant. To test this relaqffiffiffiffi tion, we report in Fig. 6, the profiles of pkffiffid Um1 as a function of y=d at several x positions. There is a good collapse of these profiles (the collapse extends well into the outer region of the boundary layer), indicating that the SP condition (9c) is well satisfied. A similar result was shown previously by Segalini et al. (2011) for the transverse Taylor microscale in a smooth wall turbulent boundary layer. Segalini et al. (2011) used mixed scaling based on the results of Alfredsson and Johansson (1990), who used an empirical approach to obtain mixed scaling. Here, we obtained the same result in a rigorous manner using an analytical approach that is based on the two-point energy budget equation. The collapse of qffiffiffiffi U1 pkffiffi m as a function of y=d further indicates that the boundary layer d thickness, d, can be used as scaling length, for this rough wall at least and over the present streamwise distance. Interestingly, notice the very good collapse of the profiles when y=d 6 0:1, where the advection term in the single point energy budget may be considered negligible, thus suggesting that Eq. (2) may be approximately valid
It was already stated that the term Iu in Eq. (1) (or Iq in Eq. (2)) represents large-scale effects on the energy and it varies from flow to flow and from location to location in a given flow. These largescale effects represent various contributions to the one-point energy budget equation. Thus, one should determine such an energy budget everywhere across the boundary layer in order to properly estimate the term Iu . This requires careful measurements of all the velocity components, a scope for future boundary layer studies in Turbulence community. In the meantime, Iu can be inferred indirectly from Eq. (1), once all the other terms are measured. While the first and second terms on the left side of the equation are relatively straightforward to measure, the one on the right side presents a major challenge, particularly in wall-bounded flows. Indeed, measurements of several velocity derivatives are required to obtain hi accurately. A crude estimate of hi is provided using local isotropy:
hiiso ¼ 15m
* + 2 @u ; @x
ð11Þ
We also used a new method, termed as ‘Spectral chart’, for estimating hi (Djenidi and Antonia, 2012), at least in a region of the boundary layer away from the near-wall region, where the method given in Eq. (11) is not reliable. The method is based on a spectral chart and provides reliable values of hi. We compared the values given by the spectral chart and Eq. (11) and we found that the error did not exceed about 14% for y=d P 0:2. It is to be noted that hi is used as a normalizing factor and any error in it would not affect the relative trend of different terms in Eq. (1). In the following, we used hiiso for displaying the terms of the scale-by-scale energy budgets. The term Iu is obtained by difference of the known terms in Eq. (1) and accounts for the large-scale contributions from advection, production and turbulent diffusion. Fig. 7 shows the individual terms of Eq. (1), normalized by 4=5hir, for y ¼ 0:23d at several streamwise positions (x = 1.94 m, 2.24 m, 2.54 m and 2.84 m); this y-position falls well in the region of constant Rk as seen in Fig. 4. It is quite interesting to observe the relatively good collapse of the individual terms, suggesting that the SBS budget may follow SP fairly well, corroborating the results of Fig. 6. As expected, the viscous term dominates the budget for small separations and drops to zero as the separation becomes
Please cite this article in press as: Kamruzzaman, M., et al. Scale-by-scale energy budget in a turbulent boundary layer over a rough wall. Int. J. Heat Fluid Flow (2015), http://dx.doi.org/10.1016/j.ijheatfluidflow.2015.04.004
6
Md. Kamruzzaman et al. / International Journal of Heat and Fluid Flow xxx (2015) xxx–xxx
Fig. 7. Terms of Eq. (1) for y=d ¼ 0:23 at x = 1.94 m (Squares), 2.24 m (Circles), 2.54 m (Triangles) and 2.84 m (Crosses). The asterisk means normalization by 4=5hir.
large. It is expected that this term should reach a value of 1 as r approaches zero. The third-order structure function becomes the dominant contributor to the budget when r=k P 0:4 and remains so until r=k ¼ 20. Beyond this separation, it becomes smaller than Iu . This latter quantity dominates the budget at large separations and decreases as r=k decreases. Notice though that it presents a local peak in the range 0:1 6 r=k 6 3 , reflecting contributions from the large-scale motions in this range of scales. These contributions are observed across the boundary layer as illustrated in Fig. 8, which shows Iu for different y-positions (y=d = 0.029, 0.23, 0.3 and 0.5) at x = 2.54 m. Note the collapse of the Iu profiles in Fig. 8 for y=d = 0.23, 0.3 and 0.5, as one may have expected since this range of y=d corresponds to constant Rk (see Fig. 4). When y=d ¼ 0:029, there is no local peak in the distribution of Iu , suggesting that the large-scale mechanisms responsible for the local peak at high values of y=d are either not present or have a small contribution in the near wall region. Fig. 9 shows the SBS budget at two y-positions, y=d ¼ 0:029 and 0.5 at a different streamwise location, x = 2.84 m to illustrate the effect of Rk on the energy budget across the boundary layer. The three terms of the budget at y=d ¼ 0:029 do not collapse with its counterpart, y=d ¼ 0:5, reflecting the variation of Rk between the two streamwise locations. Note that while the maximum of hðduÞ3 i is about 0.85 for y=d = 0.029, it is around 1 for y=d = 0.5 (the fact that it is slightly higher than 1 induces a negative value on Iu ; this overestimate may be related to an error in the estimate of hi and/or due to the effects of the outer region intermittency).
Fig. 9. Terms of Eq. (1) for x = 2.84 m, y=d ¼ 0:029 (blue symbols) and y=d ¼ 0:5 (red symbols). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Fig. 10. Kolmogorov normalized velocity spectra Eu for x = 2.84 m Reh ¼ 12; 600. The asterisk represents Kolmogorov normalization.
and
the existence of an inertial range or a 5/3 region in the velocity spectrum, or at least that it is relatively well approached. Fig. 10, which shows the Kolmogorov normalized velocity spectra for x = 2.84 m, seems to confirm these observations. The spectra for y=d P 0:029 follows relatively well the slope 5/3. The spectrum for y=d = 0.03 deviates from the others over the entire range of the wavenumber, reflecting certainly the inadequacy of estimating hi via Eq. (11) at a distance so close to the wall.
Note that hðduÞ3 i is also about 1 in Fig. 7. These results suggest 4. Conclusions Hot-wire velocity measurements have been made in a turbulent boundary layer over a rough wall consisting of transverse circular rods, with a ratio of 8 between the spacing of two consecutive rods and the height of the rod. This study looked at the scale-by-scale budget of the velocity variance. It is found that the Taylor microscale Reynolds number is constant in the region 0:2 6 y=d 6 0:6 of the boundary layer. The results also showed that the boundary layer has almost reached a self-preservation state at all scales of motion, in the region 0:2 6 y=d 6 0:6, reflected in the constancy of the Taylor microscale Reynolds number and the collapse of the Kolmogorov normalized velocity spectra at all wavenumbers. Self-preservation appears to be also observed in the SBS budget in the same region of the boundary layer. While the small scales are controlled by the viscosity, intermeFig. 8. Term Iu for y=d ¼ 0:029 (Squares), 0.23 (Triangles), 0.3 (Circles) and 0.5 (diamonds) and x = 2.54 m; The asterisk represents normalization by 4=5hir.
diate scales over which the transfer of energy (or hðduÞ3 i) is important are affected by mechanisms, represented in Iu of Eq. (1), which
Please cite this article in press as: Kamruzzaman, M., et al. Scale-by-scale energy budget in a turbulent boundary layer over a rough wall. Int. J. Heat Fluid Flow (2015), http://dx.doi.org/10.1016/j.ijheatfluidflow.2015.04.004
Md. Kamruzzaman et al. / International Journal of Heat and Fluid Flow xxx (2015) xxx–xxx
could be identified once an estimate of the one-point energy budget is made. For example, the data showed that the effects of the large-scale inhomogeneity are felt at scales as small as k in the region 0:2 6 y=d 6 0:6. It is likely that these mechanisms include production, advection and turbulent diffusion. Acknowledgment LD and RAA acknowledge the financial support of the Australian Research Council. References Alfredsson, P.H., Johansson, A.V., 1990. The art of detection of turbulence structures. Near-Wall Turb. 1, 888–896. Antonia, R.A., Djenidi, L., 2010. On the outer layer controversy for a turbulent boundary layer over a rough wall. In: IUTAM Symposium on the Physics of Wall-Bounded Turbulent Flows on Rough Walls. Springer, pp. 77–86. Antonia, R.A., Krogstad, P.A., 2001. Turbulence structure in boundary layers over different types of surface roughness. Fluid Dyn. Res. 28 (2), 139–157. Antonia, R.A., Luxton, R.E., 1971. The response of a turbulent boundary layer to a step change in surface roughness. Part 1–Smooth to rough. J. Fluid Mech. 48 (04), 721–761. Antonia, R.A., Shafi, H.S., 1999. Small scale intermittency in a rough wall turbulent boundary layer. Exp. Fluids 26 (1-2), 145–152. Bakken, O.M., Krogstad, P.A., Ashrafian, A., Andersson, H.I., 2005. Reynolds number effects in the outer layer of the turbulent flow in a channel with rough walls. Phys. Fluids 17 (6). Bandyopadhyay, P.R., Watson, R.D., 1988. Structure of rough-wall turbulent boundary layers. Phys. Fluids 31 (7), 1877–1883. Burattini, P., Antonia, R.A., Danaila, L., 2005. Scale-by-scale energy budget on the axis of a turbulent round jet. J. Turb. 6. Danaila, L., Anselmet, F., Zhou, T., Antonia, R.A., 1999. A generalization of Yaglom’s equation which accounts for the large-scale forcing in heated decaying turbulence. J. Fluid Mech. 391, 359–372. Danaila, L., Anselmet, F., Zhou, T., Antonia, R.A., 2001. Turbulent energy scale budget equations in a fully developed channel flow. J. Fluid Mech. 430, 87–109. Danaila, L., Anselmet, F., Zhou, T., 2004. Turbulent energy scale-budget equations for nearly homogeneous sheared turbulence. Flow Turb. Combust. 72 (2–4), 287– 310. Djenidi, L., Antonia, R.A., 2012. A spectral chart method for estimating the mean turbulent kinetic energy dissipation rate. Exp. Fluids 53 (4), 1005–1013. Djenidi, L., Elavarasan, R., Antonia, R.A., 1999. The turbulent boundary layer over transverse square cavities. J. Fluid Mech. 395, 271–294. Jackson, P.S., 1981. On the displacement height in the logarithmic velocity profile. J. Fluid Mech. 111, 15–25. Jiménez, J., 2004. Turbulent flows over rough walls. Annu. Rev. Fluid Mech. 36, 173– 196. Kamruzzaman, M., Talluru, K.M., Djenidi, L., Antonia, R.A., 2014. An experimental study of turbulent boundary layer over 2d transverse circular bars. In: 19th Australasian Fluid Mechanics Conference, Melbourne, Australia.
7
Keirsbulck, L., Labraga, L., Mazouz, A., Tournier, C., 2002. Surface roughness effects on turbulent boundary layer structures. J. Fluids Eng. 124 (1), 127–135. Krogstad, P.A., Antonia, R.A., 1994. Structure of turbulent boundary layers on smooth and rough walls. J. Fluid Mech. 277, 1–21. Krogstad, P.A., Antonia, R.A., 1999. Surface roughness effects in turbulent boundary layers. Exp. Fluids 27 (5), 450–460. Krogstad, P.A., Antonia, R.A., Browne, L.W.B., 1992. Comparison between rough- and smooth-wall turbulent boundary layers. J. Fluid Mech. 245, 599–617. Krogstad, P.A., Andersson, H.I., Bakken, O.M., Ashrafian, A., 2005. An experimental and numerical study of channel flow with rough walls. J. Fluid Mech. 530, 327– 352. Lee, S.H., Sung, H.J., 2007. Direct numerical simulation of the turbulent boundary layer over a rod-roughened wall. J. Fluid Mech. 584, 125–146. 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. Nikuradse, J., 1933. Stromungsgesetz in rauhren rohren, vdi-forsch. 361 (Engl. transl. 1950: Laws of flow in rough pipes.). Tech. rep., NACA TM 1292. Oyewola, O., Djenidi, L., Antonia, R.A., 2008. Response of mean turbulent energy dissipation rate and spectra to concentrated wall suction. Exp. Fluids 44 (1), 159–165. Perry, A.E., Schofield, W.H., Joubert, P.N., 1969. Rough wall turbulent boundary layers. J. Fluid Mech. 37 (02), 383–413. Raupach, M.R., 1981. Conditional statistics of Reynolds stress in rough-wall and smooth-wall turbulent boundary layers. J. Fluid Mech. 108, 363–382. Raupach, M.R., Antonia, R.A., Rajagopalan, S., 1991. Rough-wall turbulent boundary layers. App. Mech. Rev. 44 (1), 1–25. Saikrishnan, N., Longmire, E.K., Marusic, I., 2007. Analysis of scale energy budgets in wall turbulence using dual plane piv. In: 16th Australasian Fluid Mechanics Conference. Saikrishnan, N., De Angelis, E., Longmire, E.K., Marusic, I., Casciola, C.M., Piva, R., 2012. Reynolds number effects on scale energy balance in wall turbulence. Phys. Fluids 24 (1), 015101. Schultz, M.P., Flack, K.A., 2005. Outer layer similarity in fully rough turbulent boundary layers. Exp. Fluids 38 (3), 328–340. Segalini, A., Örlü, R., Schlatter, P., Alfredsson, P.H., Rüedi, J.D., Talamelli, A., 2011. A method to estimate turbulence intensity and transverse Taylor microscale in turbulent flows from spatially averaged hot-wire data. Exp. Fluids 51 (3), 693– 700. Shafi, H.S., Antonia, R.A., 1995. Anisotropy of the reynolds stresses in a turbulent boundary layer on a rough wall. Exp. Fluids 18 (3), 213–215. Shafi, H.S., Antonia, R.A., 1997. Small-scale characteristics of a turbulent boundary layer over a rough wall. J. Fluid Mech. 342, 263–293. Smalley, R.J., Antonia, R.A., Djenidi, L., 2001. Self-preservation of rough-wall turbulent boundary layers. Euro. J. Mech.–B/Fluids 20 (5), 591–602. Talluru, K.M., Kulandaivelu, V., Hutchins, N., Marusic, I., 2014. A calibration technique to correct sensor drift issues in hot-wire anemometry. Meas. Sci. Tech. 25 (10), 105304. Townsend, A.A., 1976. The Structure of Turbulent Shear Flow. Cambridge University Press. Von Karman, T., Howarth, L., 1938. On the statistical theory of isotropic turbulence. Proc. Roy. Soc. London A – Math. Phys. Sci. 164 (917), 192–215.
Please cite this article in press as: Kamruzzaman, M., et al. Scale-by-scale energy budget in a turbulent boundary layer over a rough wall. Int. J. Heat Fluid Flow (2015), http://dx.doi.org/10.1016/j.ijheatfluidflow.2015.04.004