Turbulent heat transfer in channels with irregular roughness

It is well known that rough surfaces affect turbulent ﬂows signiﬁcantly. How such surfaces affect turbu- lent heat transfer is less well understood. To gain more insight, we have performed a series of direct numerical simulations of turbulent heat transfer in a channel ﬂow with grit-blasted surfaces. An immersed boundary method is used to account for the rough surface. A source term in the thermal energy balance is used to maximise the analogy between the transport of heat and the transport of streamwise momentum. The wall roughness size is varied from k þ ¼ 15 to k þ =120. Turbulence statistics like mean temperature proﬁle, mean temperature ﬂuctuations and heat ﬂuxes are presented. The structure of the turbulent temperature ﬁeld is analysed in detail. Recirculation zones, which are the result of an adverse pressure gradient, have a profound effect on heat transfer. This is important as it leads to the wall-scaled mean temperature proﬁles being of larger magnitude than the mean velocity proﬁles both inside and outside the roughness layer. This means that the temperature wall roughness function D H þ k þ s ; Pr (cid:1) (cid:3) is different from the momentum wall roughness function D U þ k þ s (cid:1) (cid:3) . Since the bulk temperature and velocity depend on D H þ k þ s ; Pr (cid:1) (cid:3) and D U þ k þ s (cid:1) (cid:3) , it was shown that the Stanton number and the skin friction factor directly depend on D H þ k þ s ; Pr (cid:1) (cid:3) and D U þ k þ s (cid:1) (cid:3) , respectively. Therefore, the failure of the Reynolds analogy in fully rough conditions can be directly related to the difference between D H þ k þ s ; Pr (cid:1) (cid:3) and D U þ k þ s (cid:1) (cid:3) . (cid:1) 2020 The Authors. Published by Elsevier Ltd. ThisisanopenaccessarticleundertheCCBYlicense(http:// creativecommons.org/licenses/by/4.0/).


a b s t r a c t
It is well known that rough surfaces affect turbulent flows significantly. How such surfaces affect turbulent heat transfer is less well understood. To gain more insight, we have performed a series of direct numerical simulations of turbulent heat transfer in a channel flow with grit-blasted surfaces. An immersed boundary method is used to account for the rough surface. A source term in the thermal energy balance is used to maximise the analogy between the transport of heat and the transport of streamwise momentum. The wall roughness size is varied from k þ ¼ 15 to k þ =120. Turbulence statistics like mean temperature profile, mean temperature fluctuations and heat fluxes are presented. The structure of the turbulent temperature field is analysed in detail. Recirculation zones, which are the result of an adverse pressure gradient, have a profound effect on heat transfer. This is important as it leads to the wall-scaled mean temperature profiles being of larger magnitude than the mean velocity profiles both inside and outside the roughness layer. This means that the temperature wall roughness function DH þ k þ s ; Pr À Á is different from the momentum wall roughness function DU þ k þ s À Á . Since the bulk temperature and velocity depend on DH þ k

Introduction
Hydrodynamically rough surfaces can be found in many engineering systems. Typical causes of roughness include machining, fouling or corrosion. It is well known that wall roughness affects both the mean flow as well as the turbulent motion of a fluid, which in turn affects skin friction. A well-known result of the effects of roughness is summarised in the form of the Moodydiagram. However, as Flack [11] points out, it is impossible to predict exactly how a rough surface affects the flow in the transitionally rough regime. The effect of roughness on turbulent flows is typically summarised by a single function, which is called the wall roughness function. Recent work from Busse et al. [6] and Thakkar et al. [35] shows that direct numerical simulations of turbulent flows over realistic surfaces can be used to help determine which topological characteristics should be used to predict the wall roughness function. Thakkar et al. [36] have shown that direct numerical simulations of turbulent flow over a grit-blasted surface yields results that are very close to the results that were originally reported by Nikuradse [27].
While the effect of wall roughness on momentum transfer is being extensively investigated, less attention has been given to the effect of wall roughness on turbulent heat transfer. Early research by Martinelli [24], Nunner [28], Dipprey and Sabersky [9] and Yaglom and Kader [39] focussed on finding a relation for the Stanton number in pipes as a function of the skin friction factor, the Reynolds number and the molecular Prandtl number. Their methods combine analytical considerations with experimental results. A theoretical study by Katoh et al. [17] showed that the heat transfer enhancement is typically less than the drag enhancement if the molecular Prandtl number is less than the turbulent Prandtl number. DNS results of turbulent heat transfer in channels with transverse ribs by Nagano et al. [26] showed for Pr ¼ 0:71 that heat transfer increases less than drag does and thus, that the Reynolds analogy does not hold. Furthermore, they surmise that the pressure temperature gradient correlation is important in maintaining the turbulent heat flux. Later, Bons [5] evaluated the application of the Reynolds analogy for turbine flows and found that the accuracy of this analogy decreases dramatically as surface roughness height increases. Both Dipprey and Sabersky [9] and Bons [5] note that increases in the Stanton number are accompanied by even larger increases in the skin friction coefficient. Moreover, Bons [5]  introduces a pressure drag component to the skin friction coefficient and that there is no corresponding mechanism with respect to the Stanton number. This is logical, since it is known that the existence of mean pressure gradients is the cause of dissimilarity between velocity and temperature fields in perturbed thermal boundary layers, see for instance Kong et al. [21].
Recently, Leonardi et al. [23] and Orlandi et al. [29] used direct numerical simulations to study how regular roughness (such as bars and cubes mounted on the surfaces of a channel) can affect heat transfer. These studies focused on heat transfer modification through passive means. However, they did not show how irregular rough surfaces, such as surfaces affected by machining or corrosion, affect heat transfer. Forooghi et al. [13] showed DNS results of momentum and heat transfer in a channel with combustion deposits. They too found that the Reynolds analogy is less accurate with increasing wall roughness height. Forooghi et al. [12] subsequently investigated systematically the effect of effective surface slope, roughness density and roughness height on heat transfer. They note that an effect known as sheltering leads to a decrease in the Stanton number for high roughness densities and that different roughness shapes affect heat and momentum transfer differently. Furthermore, they presented a reasonable correlation between the Reynolds analogy factor and sand grain equivalent roughness height, k þ s . Another model for the Reynolds analogy was recently derived by Aupoix [4], which is dependent on both the momentum and scalar wall roughness functions and a novel turbulent Prandtl number relation. It was shown that this new model yields enhanced heat transfer predictions.
The goal of the present study is to gain insight into the modification of turbulent heat transfer by an irregular rough surface, which has been shown to act as a surrogate for Nikuradse-type roughness. In particular, we will present a detailed analysis of the effect of wall roughness on turbulent heat transfer statistics, structural effects and the parametrisation of these effects, such as the temperature wall roughness function DH þ and various Stanton number relations.

Numerical methods
To understand how wall roughness affects heat transfer, a numerical approach is used to solve the dimensionless incompressible Navier-Stokes equations and the transport equation for the temperature h with uniform heat generation Q: A simple channel geometry with roughness on both isothermal walls is considered. The domain size L x Â L y Â L z À Á is 5:63d Â 2d Â 2:815d in the streamwise, wall normal and spanwise directions. Periodic boundary conditions are used in the streamwise and spanwise directions. To account for the wall roughness, the immersed boundary method of Fadlun et al. [10] is used. At the first grid point outside the immersed boundary, a velocity is enforced that is based on the interpolation by a quadratic polynomial. Fig. 1 shows height maps of three different rough surfaces that are considered in this study. The first (Fig. 1a) is a filtered surface scan of a grit-blasted surface (see also Busse et al. [6] and Busse et al. [7]). The second (Fig. 1b) and third surface (Fig. 1c) are tiled versions of the first and are shrunk by a factor of two and four, respectively. The method of tiling was previously described by Thakkar et al. [36]. These surfaces have been shown to act as a surrogate for Nikuradse-type roughness. Fig. 2 shows a schematic of a rough surface in a channel. The surface height s is measured from the lowest point of the computational domain to the surface. The mean surface height is located at y=d ¼ 0.

Direct numerical simulations
Multiple new cases with irregular roughness are considered. The details of these cases are presented in Table 1. The roughness height is varied from þ , with k þ being the mean peak-to-valley-height. To calculate this quantity, the surface is first partitioned into 5 Â 5 tiles. The mean peak to valley height is then calculated as the average of the maximum peak-to-valley-heights of the different tiles, see Thakkar et al. [35]. Different Reynolds numbers (Re s u s d=m) are considered, varying from Re s ¼ 180 to Re s ¼ 720. The molecular Prandtl number Pr is assumed to be equal to unity. Furthermore, the rough walls are kept cool, h w ¼ 0, while the bulk of the fluid is volumetrically heated.

Domain size and validation
It is vital that the domain size is large enough to capture all scales of motion. Results by Coceal et al. [8] demonstrated that turbulence statistics of flows over rough walls can be captured accurately while using relatively small domain sizes. The focus of this study is to investigate how wall roughness affects heat transfer. Therefore, stream-and spanwise autocorrelations of the temperature fluctuations R h 0 h 0 were calculated first; these are shown in Fig. 3. These results inidicate that the domain is large enough to capture the largest thermal structures.
The aforementioned numerical methods were validated by simulating different cases that were previously described by Busse et al. [7], Thakkar et al. [36] and Kim et al. [20]. The Reynolds number for these cases is equal to 180. The details of the validation cases are listed in Table 1. Fig. 4a shows that there is excellent agreement between the previous and present results with regards to the mean velocity profiles. As a result, there is also very good agreement in terms of the wall roughness function DU þ . Fig. 4b shows the comparison between mean scalar profiles that were obtained by Kim and Moin [19] and the present code. There is again excellent agreement between the current code and the literature results.

Mean statistics
From the DNS various temperature statistics are obtained. We will first discuss the mean temperature profiles of the different cases. Subsequently, we will investigate how turbulent thermal fluctuations as well as the turbulent heat flux are affected.

Averaging procedure
Before discussing the mean statistics, we will first detail the averaging procedure that is employed. The mean temperature is obtained by plane averaging the time averaged temperature h over the area that is occupied by the fluid: where w is an indicator function that is equal to unity inside and zero outside the fluid, and where . . . ð Þ denotes the time averaged temperature and S denotes a plane parallel to the wall. Similarly, U is obtained by plane averaging the time averaged streamwise velocity u. Instantaneous fluctuations of the temperature are streamwise and wall normal velocities, respectively. These fluctuations give rise to the Reynolds stresses and heat fluxes. Unless noted otherwise, the mean statistics will be presented as plane-and timeaveraged quantities. . The mean temperature profiles show an increasing downward shift with increasing wall roughness height for the region y þ '10, while showing an upward shift for y þ /1, which means that there is increased heat transfer towards y=d ¼ 0. This is qualitatively similar to the velocity profiles that were shown by Busse et al. [7]. However, the magnitude of the downward shift is larger for the velocity profiles than it is for the temperature profiles; this difference will be discussed later in Section 5.1.

Mean temperature profile
From Fig. 6(a) it can be seen that the temperature defect profile, shows a good collapse for y=d'0:06; this value means that the collapse of the temperature defect profile starts much closer to the centre of the roughness layer (y=d ¼ 0) than the collapse of the velocity defect profile, which shows a collapse for y=d'0:20 (which was shown in Thakkar et al. [36]). By applying similarity arguments between momentum transfer and heat transfer, a log-law can be found for the mean temperature profile; see Landau and Lifshitz [22] or Kader and Yaglom [16]. According to Kader [15] the log-region of the mean temperature profile H (in the absence of wall roughness) can be described as, Analogous to the diagnostic function for momentum, c H can be used to determine the factor Pr t =j. Fig.6(b) shows a      reasonably flat profile of the temperature diagnostic function in the region w < y=d < 0:6, where w max 50=Re s ; k=d ð Þ .From Eq. (5), it follows that c H ¼ Pr t =j. After assuming that j ¼ 0:4; Pr t can be determined, see Table 2.The values of Pr t reported here fall within the range Pr t ¼ 0:85 À 1 that is typically assumed in literature.
The computed values are slightly higher than the values used by Abe and Antonia [1] (who use Pr t =j ¼ 1=0:46, and j ¼ 0:39 which means that Pr t % 0:85), or Kader and Yaglom [16] (Pr t ¼ 0:85) or Landau and Lifshitz [22] (Pr t ¼ 0:9). The slightly higher results could be due to the diagnostic approach, the low Reynolds number of the simulations, or due to roughness. Spalart and Strelets [34] proposed a method to determine an effective Prandtl number inside a recirculation bubble as Pr eff .
Far away from the wall, Pr eff % Pr t . Profiles of Pr eff are shown in Fig. 7. Pr eff is very similar in the k þ ¼ 0 and the k þ ¼ 15 cases. However, for larger k þ ; Pr eff shows a peak value just below the plane y=d ¼ 0. The peak value is more obvious in the k=d ¼ 1=6 cases. This peak indicates that the effective thermal diffusivity is much smaller here. Therefore, the region y=d < 0 can be said to act as a thermal resistance. Meinders et al. [25] showed that a recirculation zone behind a single wall mounted cube acts as a thermal resistance. This is further investigated in Section 4.3. Further away from the wall, Pr eff decreases with wall distance; a feature that is intrinsically linked to the unmixedness of the scalar, see Guezennec et al. [14], Abe and Antonia [1,2]. Furthermore, far away from the wall, Pr eff is larger in the simulations with roughness, than Pr t in the smooth channel case. These results indicate that roughness does indeed affect Pr t .

Turbulent thermal fluctuations
As was mentioned before, the thermal fluctuations due to the turbulent motion of the fluid are defined as h 0 ¼ h À h.

Reynolds stresses and heat fluxes
The temperature fluctuations h 0 and velocity fluctuations u 0 and v 0 give rise to significant mean Reynolds stresses and heat fluxes. In Fig. 9, we compare v 0 h 0 with v 0 u 0 and u 0 h 0 with u 0 u 0 for the range k þ ¼ 0 À 120. In all cases, v 0 h 0 is practically identical to v 0 u 0 . This is not the case for u 0 h 0 and u 0 u 0 . At small k þ ; u 0 h 0 and u 0 u 0 are very similar up to y þ % 30. However, at large wall roughness sizes, u 0 h 0 is smaller than u 0 u 0 . This difference between the streamwise heat flux and u 0 u 0 grows as k þ increases. At k þ ¼ 30, the difference is clearly visible for y þ > 15, but for larger k þ , this difference extends to well below y þ ¼ 10. The magnitude of both u 0 u 0 and u 0 h 0 increase near y þ ¼ 1 for increasing k þ . The increasing difference between u 0 h 0 and u 0 u 0 indicates that the correlation between u 0 and h 0 decreases as k þ increases.  Table 2 The turbulent Prandtl number Prt for cases with different roughness sizes. It is assumed that j ¼ 0:4.

Effects of local roughness
While the previous section showed the effect of wall roughness on typical turbulence statistics, in this section we will focus more closely on near wall effects.

Structures and recirculation
In the roughness sub-layer, the flow is noticeably affected by local roughness features. Fig. 10 shows that as k þ is increased, the thermal streaks h 0 < 0 become much shorter and less coherent.  Furthermore, for larger values of k þ , regions of h 0 < 0 do not necessarily coincide with regions where u 0 < 0 and high speed regions contain regions of both h 0 < 0 and h 0 > 0. This suggests that a velocity streak u 0 < 0 under the influence of wall roughness is less likely to consist of relatively cold fluid. As the similarity between the thermal and momentum streaks disappears for increasing k þ ; the correlation between u 0 and h 0 decreases. A decreasing correlation between the velocity and temperature streaks is a physical explanation why the difference between the streamwise Reynolds stress and the streamwise Reynolds heat flux increases with wall roughness size.
The dissimilarity between the scalar and streamwise velocity fields is discussed further by investigating instantaneous fluctuations in the y À z-plane. It is well known that in smooth channel flows interfaces between neighbouring eddies are more visible in the scalar field than in the streamwise velocity field (see Antonia et al. [3], Pirozzoli et al. [33] and Abe and Antonia [1]). Despite the lower Reynolds numbers in the current study, similar observa-  tions can be made in Fig. 11. Particularly striking is the fact that close to the rough wall, the interfaces in the scalar field are much sharper than the interfaces in the velocity field. This indicates the scalar is less well mixed compared to the velocity close to the rough wall than further away. This is in line with the large effective Prandtl numbers (in the region y=d < 0) that we discussed in Section 3.2. It was argued by Guezennec et al. [14] that the difference between scalar and velocity fields can be attributed to the role of the pressure; it is likely that this role is enhanced in channels with rough walls. Busse et al. [7] already showed that significant flow reversal zones exist in turbulent flows over the grit-blasted surface. Flow reversal zones typically exist near locations where @s=@x < 0. An instantaneous visualisation of a recirculation zone is shown in Fig. 12. The recirculation noticeably affects the near wall temperature field. Close to the windward slope (@s=@x > 0) relatively hot fluid (indicated by a light blue/white colour) is drawn in from the bulk, thereby creating a large temperature gradient at the surface. However, close to the downward slope, cold fluid (indicated by a dark blue colour) accumulates, which leads to a visibly smaller temperature gradient. The recirculation zones, formed as a result of adverse pressure gradients have a profound effect on the mean temperature and velocity field, as shown in Fig. 13. The strength of the recirculation is likely to increase as k þ increases, which results in lower mean velocity values. As k þ increases, hotter fluid can flow from the bulk region (Fig. 12, region A) into the roughness layer, which leads to higher mean temperatures inside the roughness layer (which is evident from Fig. 13). The relatively hot fluid then typically impinges on a windward slope (Fig. 12, region B). A thin thermal boundary layer is then formed that extends both in the streamwise direction (under the influence of mean streamwise momentum) as well as the opposite direction (under the influence of the recirculation zone and thus, an adverse pressure gradient), which ends as soon as it comes under the influence of  mean streamwise momentum (Fig. 12, region C). Fig. 13 showcases the fact that, while an adverse pressure gradient affects momentum, there is no corresponding mechanism for heat transfer. The local maximum in the mean temperature profile that is seen in the k þ ¼30, 60, 90 and 120 cases is the result of the irregular surface height in combination with the averaging procedure that was described in Section 3.1.

Quadrant analysis
It has been shown that wall roughness has a profound effect on the velocity and thermal streaks close to the wall. It is well known that such regions give rise to the Reynolds shear stress v 0 u 0 and heat flux v 0 h 0 , which in turn largely determine the skin friction factor and the Stanton number in smooth channels. Therefore, we will examine in more detail the structure of v 0 u 0 and v 0 h 0 using the quadrant analysis that was originally developed by Willmarth and Lu [38]. This method has proven to be a simple, but effective tool in the analysis of Reynolds shear stress, see Wallace [37]. The definition of the quadrants of the Reynolds shear stress is as The definition of the heat flux quadrants is found by replacing u 0 with h 0 ., i.e. Q1: In a smooth channel with internal heating, Q2 and Q4 are the most frequent quadrants. These quadrants are known as ejections and sweeps (or outward motion of cold fluid and inward motion of hot fluid for heat transfer), respectively.
In Fig. 14, we compare the different quadrants of the Reynolds shear stress u 0 v 0 with those of the turbulent heat flux v 0 h 0 . By comparing the quadrants for different wall roughness heights, it becomes clear that two distinct regions can be found, in which the quadrants show different behaviour. For different wall roughness heights, these regions show very similar behaviour. In the region y þ > k þ =3, the probability distributions of events in each quadrant for both the heat flux and Reynolds shear stress are very similar to each other, which is also typical of the behaviour for a smooth channel, see also Perry and Hoffman [31]. For the Reynolds shear stress, the probability distribution of the different quadrants are equipartioned for y þ < Àk þ =3 and then rapidly diverge in the region Àk þ =3 < y þ < k þ =3.
In the region y þ < k þ =3, the heat flux quadrants show different behaviour compared to the Reynolds shear stress quadrants. In this region, the probability of Q3 events in v 0 h 0 is enhanced, while the probability of Q4 in v 0 h 0 events is diminished, when compared to the region y þ > k þ =3. These results mean that the similarity in structure of the Reynolds stress and the heat flux breaks down for y þ < k þ =3.
We continue the analysis of the Reynolds shear stress and heat flux quadrants by investigating the relative contributions , where j refers to the quadrant number, which are shown in Fig. 15. As before, the different contributions of the Reynolds shear stress quadrants are very similar to those of the heat flux in the region y þ > k þ =3, which is again reminiscent of smooth channel behaviour (for a comparison with literature, see Abe and Antonia [1]). The relative contributions of the quadrants are different for y þ < k þ =3. The absolute value of the Reynolds shear stress in each quadrant grows rapidly in this region. Since the total Reynolds shear stress changes sign twice in this region, discontinuous behaviour can be seen. In the same region, the heat flux quadrants behave differently. The most notable difference is that the relative magnitude of Q4 is larger than Q2 for y þ < k þ 3, meaning that inward motions of relatively hot fluid are promoted.

Probability density functions
The structural changes seen in the previous section will be likely to affect the local heat transfer. An often-used nondimensional measure of the local heat transfer coefficient is the In smooth channels, this number is often related to the skin friction coefficient through the Reynolds analogy: St ¼ C f =2. However, since we investigate channels with rough walls, we will distinguish between the local frictional drag T x , which was shown by Peet and Sagaut [30] to be and the local form drag F x : In these equations, x is the unit streamwise vector,n is the unit normal vector. The subscript s means that the quantity is evaluated at the surface. Similarly, q is the local heat flux, defined as:  A local skin friction coefficient may be calculated as: while a local Stanton number may be calculated as: Please note that T x ; F x and q s are instantaneous values. We compare probability density functions of both St l and C l f for the k þ ¼ 15; 30; 60 cases at Re s ¼ 360 in Fig. 16. In the same figure, we also investigate the probabilities of St l and C l f , conditioned on the presence of recirculation zones, P St l ju < 0 or on the absence of recirculation P St l ju > 0 . All probability density functions in   The general shape of P St l and P C f À Á l appears qualitatively similar, except for the fact that C l f can assume negative values. Inside the recirculation zones, the Stanton number is likely to be smaller than the global average; this corresponds well with Meinders et al. [25] observation that a recirculation zone acts as a thermal resistance. As k þ increases, the area that is covered by a recirculation zone increases. The largest Stanton numbers are most likely to be found outside the recirculation zones. As k þ increases, P St l ju < 0 approaches P St l ju > 0 for large values of the Stanton number. The effects of the recirculation zone on the Stanton number are similar for the friction coefficient; inside recirculation zones, C l f is likely to be smaller than zero and as k þ increases, P C l f ju < 0 approaches P C l f ju > 0 for large friction coefficients.
The fact that positive skin friction factors can be found inside the recirculation zone indicates that the zone can move in time.

Correlation between surface features and heat transfer
In flows with smooth walls, the near wall streaks are known to largely determine local skin friction and heat transfer. In Section 4.1, local roughness protrusions were found to break-up the streaks, which at least partially determine the local skin friction and heat transfer. Furthermore, it was shown in Section 4.3 that inside recirculation zones, the Stanton number is likely to be much smaller than the average, which could be related to the accumulation of relatively cold fluid near downward slopes. Based on these observations, a correlation between either the local surface height s or the local surface slope @s=@x and the local Stanton number may exist. The correlation coefficents q s; St Þincreases with k þ . This means that at small k þ , the influence of the local surface height is more important, which may be related to the break-up of the streaks. For fully rough conditions, the recirculation zones are much more important as local negative slopes typically yield low Stanton numbers, while windward slopes yield large Stanton numbers. This observation may be related to results that were presentated by Forooghi et al. [12]; with increasing mean slope, the heat transfer enhancement is less than the drag enhancement. This could be a result of the recirculation zones acting as a thermal resistance, while in the same region both friction and form drag contribute to the drag increase.

Parametrisation
In the previous sections, we have presented mean statistics and analysed the effect of local surface roughness on heat transfer. In this section, we will investigate whether these effects can be parametrised in relations that are simple to use.

Roughness function
Analogous to the momentum wall roughness function, s that parametrises the roughness  effects on the temperature field in the log-region of the turbulent flow can be found. Fig. 18 where b 0 Pr ð Þ ¼ b Pr ð Þ À 7:4. The form of Eq. (12) resembles the asymptotic behaviour of DU þ that was found by Nikuradse [27] (i.e., DU þ k þ s À Á ¼ 2:5 ln k þ s À Á À 2:98 for k þ s > 70), with the exception of the last term on the right hand side.
Another estimate for DH þ k þ s ; Pr À Á can be obtained by considering a modification to b Pr ð Þ that was proposed by Yaglom and Kader [39] in order to account for wall roughness effects: Pr 2=3 À 0:2 À Prt j ln k þ À Á þ 9:5 and b s 12:5Pr 2=3 À 6. Please note that the value of b s % b. This results in the following temperature wall roughness function: where b 00 Pr ð Þ ¼ b Pr ð ÞÀ 9:5. The form of Eq. (15) is very similar to that of (12). However, Eq. (15) yields decreasing values of DH for increasing values of k þ for k þ s > 100, which Eq. (12) does not. Eqs. (12) and (14) are shown together with the DNS results in figure 18(b). The DNS results lie between both estimates. The dis-crepancy between the DNS results and Eq. (4) results partially from the fact that a fully rough regime is assumed by Kays and Crawford [18]. Moreover, the equation by Kays and Crawford [18] depends on the results that were obtained by Dipprey and Sabersky [9], who used a different kind of wall roughness. Furthermore, the factor 5 4 was determined by Pimenta et al. [32] from data that were obtained from a rough surface consisting of closely packed spheres.

Stanton number relations
The estimation of DH þ k which in turn is closely related to the Stanton number (for a volumetrically heated channel), see Abe and Antonia [1]. Similar equations can be found for the bulk velocity and the skin friction factor: and  [18] and Yaglom and Kader [39].  Table 3 show that the assumption of a log-law for the complete temperature profile (not including the near-wall and wake regions) does not lead to large errors. In other words, Eqs. (16) & (17) yield very good estimations of the heat transfer coefficient, as long as the temperature wall roughness function is known. Therefore, Eq. (12) yields better results than (14).
Different St-friction factor analogies have been presented in the past, such as: by Martinelli [24], by Nunner [28], and by Dipprey and Sabersky [9]. These relations are compared with the DNS results in Table 4. The Reynolds analogy yields an increasing error for larger k þ ; this result is in line with Bons [5] and Forooghi et al. [13]. The increase in error can be directly related to the difference in the bulk temperature and the bulk velocity (Eqs. (16) and (18)), which in turn stems from the difference between DH þ and DU þ that was shown in Fig. 18 More recently, researchers have sought after a relation between the Reynolds analogy factor RA 2St=C f and the wall roughness parameter k. Aupoix [4] derived the following relation: in which the subscript '0' refers smooth wall conditions. Forooghi et al. [12] proposed the following relation: Predictions by Eqs. (23) and (24) are compared with the DNS results in Table 5

Conclusions and discussion
We have analysed the effect of wall roughness on turbulent heat transfer. Wall roughness affects heat transfer and momentum transfer differently. The temperature and momentum wall roughness functions differ significantly: DH þ is substantially smaller than DU þ at Pr ¼ 1. This difference stems from the fact that high temperature fluid from the bulk region penetrates into the roughness layer. This results in the wall-scaled mean temperature profile being much larger than the mean velocity profile (which is largely negative due to the effect of pressure) inside the roughness layer. This translates into temperature values being larger than mean velocity values in the bulk region as well. The difference between DH þ and DU þ accounts directly for the failure of the Reynolds analogy for fully rough conditions. This is in-line with literature since the result that DU þ > DH þ can be traced to the fact that the velocity field is affected by pressure, while there is no corresponding mechanism for the thermal field.
Wall roughness also affects the near wall cycle. The velocity and temperature streaks are broken up by local wall roughness protrusions. This effect causes the time-and spatially-averaged temperature fluctuations to decrease for increasing wall roughness size.
While the wall normal Reynolds stress v 0 u 0 and heat flux v 0 h 0 decrease for larger wall roughness size, their respective magnitudes remain very similar for different k þ . In contrast, the structure of the wall normal Reynolds shear stress and the heat flux is affected. In the roughness layer, for y þ < k þ =3, quadrants of the Reynolds shear stress are affected differently than the quadrants of the heat flux. However, it should be noted that the magnitude of the heat flux u 0 h 0 is very small here. Therefore, it can be argued that the wall roughness effect on the quadrants of v 0 u 0 and v 0 h 0 contribute to the failure of the Reynolds analogy only in a minor fashion. The similarity and therefore correlation between the velocity and the temperature streaks decreases for increasing wall roughness, which results in a decrease in the magnitude of the streamwise heat flux u 0 h 0 . In smooth channels, only the wall normal heat flux v 0 h 0 contributes to the mean Stanton number. However, it is conceivable that the streamwise heat flux contributes to local Stanton numbers in parts of the channel where the surface slope is steep. Furthermore, the difference between u 0 u 0 and u 0 h 0 increases for larger k þ . Thus, it is also conceivable that the difference between u 0 u 0 and u 0 h 0 for increasing k þ contributes to the failure of the Reynolds analogy, as u 0 u 0 and u 0 h 0 contribute differently to C f and St, respectively. Table 4 Stanton number -friction factor analogy errors. Errors are calculated as is indicated in the caption of Table 3.

Reynolds analogy
Dipprey/Sabersky analogy  By studying probability density functions of the instantaneous Stanton number, the recirculation zones were found to have a detrimental effect on heat transfer. This observation, and that of the break-up of streaks has led to the conclusions that (1) for small k þ , the instantaneous Stanton number and the local height of the rough surface are well correlated and that (2) for k þ > 60, the slope and the instantaneous local Stanton number are correlated. Finally, it should be noted that the results in this paper were obtained on a single rough surface. It is well known that different rough surfaces yield different results for DU þ and the same could be true for DH þ . It is also unknown how the thermal field will be affected by wall roughness at Prandtl numbers not equal to unity. The relations for DH þ that were derived from empirical relations found in the literature do depend on the Prandtl number and may serve as a guideline for a general relation for DH þ in the future.