2.1. Streamwise domain-size study

In order to accurately predict statistical moments of the velocity field, the streamwise computational domain size $L_x$ must be held sufficiently large such that all dominant coherent flow structures may fully evolve in the streamwise direction and their wavelengths can be captured in a numerical simulation. To this purpose, the spatial two-point correlation function of streamwise velocity fluctuation ($R_{uu}$) is calculated for both ribbed duct cases at two streamwise locations. For a ribbed duct flow, the streamwise two-point auto-correlation coefficient is defined as (Townsend Reference Townsend1980; Volino et al. Reference Volino, Schultz and Flack2009)

(2.3)\begin{equation} R_{uu}(x_{ref},{\rm \Delta} x)=\frac{\langle u'(x_{ref})u'(x_{ref}+{\rm \Delta} x)\rangle}{\sqrt{\langle u^{\prime 2}(x_{ref})\rangle\langle u^{\prime 2}(x_{ref}+{\rm \Delta} x)}\rangle}, \end{equation}

where $u'$ represents the streamwise velocity fluctuations and ${\rm \Delta} x$ denotes the relative displacement from the reference point located at $x_{ref}$, such that $x=x_{ref}+{\rm \Delta} x$. The streamwise coordinate of the reference points are $x_{ref}/H= 29.0$ and $34.0$, while the vertical coordinate of the reference points is fixed at $y_{ref}/H=-3.8$. In terms of the relative streamwise coordinate, the reference points are located near the leeward and windward sides of a rib at $x_{ref}'/H=2.0$ and 7.0, respectively.

Figure 2 shows that for both rib cases, the profile of $R_{uu}$ drops drastically to almost zero at the two ends. This indicates that the velocity statistics are not affected by the streamwise domain size, such that a domain size of $L_x=64H$ is sufficiently large for capturing the characteristic length scales of dominant turbulence structures in the streamwise direction. From figure 2, it is also evident that the decaying rates of $R_{uu}$ in the inclined rib case are slower than those of the V-shaped rib case at both reference points ($x_{ref}/H=29.0$ and 34.0), indicating that the streamwise length scales of turbulent eddies of the inclined rib case are larger than those of the V-shaped rib case. By comparing figure 2(a) with 2(b), it is apparent that the decaying rate of $R_{uu}$ is slightly slower at the reference point near the leeward face of the rib ($x_{ref}/H=29.0$) than at the reference point near the windward face of the rib ($x_{ref}/H=34.0$). More specifically, the value of the streamwise integral length scale (i.e. $L_{xx}=\int _0^{\infty }R_{uu}(x)\, \textrm {d} x$) decreases by $16.5\,\%$ and $4.7\,\%$ as $x_{ref}/H$ increases from $29.0$ to $34.0$ in the inclined and V-shaped rib cases, respectively.

Figure 2.

Streamwise profiles of two-point auto-correlations for ribbed duct flows of two rib cases at different streamwise locations ($x/H$). The vertical coordinate of the reference points is fixed at $y_{ref}/H=-3.8$. The streamwise coordinates of the reference points are (a) $x_{ref}/H= 29.0$ and (b) $x_{ref}/H=34.0$ (in terms of the relative streamwise coordinate, the reference points are located near the leeward and windward of a rib at $x_{ref}'/H=2.0$ and $x_{ref}'/H=7.0$, respectively). The characteristic streamwise length scale ($L_{xx}$) determined based on the streamwise two-point auto-correlation coefficient.

Figure 3 shows the temporal energy spectra of the streamwise velocity fluctuations. The spatial reference points considered here are the same as in figure 2 for the calculation of the streamwise two-point auto-correlation coefficient. Figure 3 shows that for both inclined and V-shaped rib cases, the difference between the highest and lowest frequencies is of the order of four. This implies that the statistical sampling range is deemed to be sufficient to resolve all significant temporal scales of the studied turbulent flows. From figure 3 it is also observed that the energy spectrum is strongly influenced by the rib geometry, as the $-5/3$ slope (which is a characteristic of equilibrium turbulence regime, Pope Reference Pope2000) is apparently absent near the leeward and windward faces of the rib (at $x_{ref}/H=29.0$ and 34.0, respectively). This indicates the presence of small-scale structures with a high degree of anisotropy due to the disturbances from the ribs.

Figure 3.

Non-dimensionalized temporal energy spectra of streamwise velocity fluctuations for both inclined and V-shaped rib cases. The streamwise coordinates of the reference points are (a) $x_{ref}/H= 29.0$ and (b) $x_{ref}/H=34.0$. The comparison of the two ribbed flow cases is conducted at the elevation that is slightly above the rib crest (with the spatial reference point being identical to that used in figures 2a and 2b).

2.2. Grid resolution

The number of body-fitted grid points used in the current simulations are $N_x \times N_z \times N_y = 1424 \times 168 \times 162$ in the streamwise, vertical and spanwise directions, respectively. The mesh is non-uniform in all three directions, and is refined near all solid surfaces. To ensure that the local resolution of this grid is sufficient for conducting DNS of turbulent flow over different rib geometries, the contours of the ratio between the maximal dimension of a grid cell in all three directions (i.e. ${{\varDelta }}= \max ({{\varDelta }_{x}},{{\varDelta }_{y}},{{\varDelta }_{z}})$) and Kolmogorov length scale, $\eta =({\nu }^{3}/\varepsilon )^{0.25}$, in the central ($x$–$y$) plane (located at $z/H=0.0$) are plotted in figure 4. Here, $\varepsilon =\nu \langle {a}'_{ij} {a}'_{ij} \rangle$ is the TKE dissipation rate, with ${a'_{ij}}={\partial {{u}_{i}'}}/{\partial x_{j}}$ being the fluctuating velocity gradient tensor. In their DNS study of turbulent channel flows, Moser & Moin (Reference Moser and Moin1987) indicated that the grid size requires to be of the same order as the Kolmogorov length scale (i.e. $O(\varDelta /\eta )\sim O(1)$) in order to accurately capture the TKE dissipation pertaining to the smallest scales of turbulence.

Figure 4.

Contours of the ratio between the grid resolution and Kolmogorov length scale (${{\varDelta }}/\eta$) in the central plane at ${z}/{H}=0.0$ for two rib cases. (a) Inclined rib case and (b) V-shaped rib case.

Figure 4 shows that the maximum magnitude of ${{\varDelta }_{\max }}/\eta$ occurs in the inter-rib region under the rib height, which is approximately 5.3 and 6.3 for inclined and V-shaped rib cases, respectively. It is therefore expected that the mesh resolutions employed suffice to capture small turbulence motions at the Kolmogorov scale level in the two rib cases. From figure 4 it is evident that the appearance of a local maximum in the contours of ${{\varDelta }_{\max }}/\eta$ is associated with a destruction mechanism of the upstream flow structures near the rib elements. A detailed explanation of the rib geometry effects on turbulence structures will be provided in § 4. The spatial resolutions used in this research are comparable to those used by Ismail et al. (Reference Ismail, Zaki and Durbin2018), who conducted a DNS study of turbulent flows in a channel with rough-to-smooth step changes based on a second-order accurate finite-difference computer code.

3.1. Mean velocity field

Figure 5 shows the time-averaged streamlines superimposed onto the mean streamwise velocity field in the central vertical ($x$–$y$) plane located at $z/H=0.0$ for two rib cases. From figure 5 it is observed that both the contours of $\langle u\rangle /U_b$ and the mean streamlines are influenced significantly by the rib geometries. This inevitably leads to changes in the spatial distribution of skin friction and pressure coefficients, which will be discussed separately in this section. From figure 5(a) it is seen that for the inclined rib case, no apparent separation bubbles are present at the square duct midspan in the near-wall region below the rib height. This phenomenon is mainly attributed to the occurrence of large vortices unique for the inclined rib flow case, which reduce the tendency of generating separation bubbles in the leeward side of the rib. These physical features are consistent with the observations of Bonhoff et al. (Reference Bonhoff, Parneix, Leusch, Johnson, Schabacker and Bölcs1999) who studied the effects of inclined ribs on the turbulence statistics in a rectangular duct flow using a planar PIV. By contrast, for the V-shaped rib case as shown in figure 5(b), a large single separation bubble (marked with ‘A’) exists in the inter-rib region below the rib height. From figure 5(b) it is observed that due to the downdraft of the mean flow, the separation bubble behind the rib is squeezed towards the leeward face of the V-shaped rib, which further leads to the formation of a region of high pressure values. This well explains the pressure drag difference between the inclined and V-shaped rib cases shown previously in table 1. By comparing figures 5(a) with 5(b), it is seen that for the inclined rib case, the highest streamwise momentum level as indicated by the magnitude of $\langle u\rangle /U_b$ appears in regions well above the rib crest (for $y/H>-3.7$), whereas for the V-shaped rib cases, the maximum value of $\langle u\rangle /U_b$ occurs near the rib crest due to an enhanced pressure difference. As is evident in figure 5(b), the reattachment length in the V-shaped rib case is observed to be $4.0H$ which is comparable to $4.1H$ reported in Fang et al. (Reference Fang, Yang, Wang, Tachie and Bergstrom2017) based on their LES study of turbulent flows over transverse and V-shaped ribs at a Reynolds number of $Re_b=10\ 000$.

Figure 5.

Contours of the mean streamwise velocity $\langle u\rangle /U_b$ superimposed with in-plane mean streamlines in the central ($x$–$y$) plane (located at $z/H=0.0$) for two rib cases. (a) Inclined rib case and (b) V-shaped rib case.

Figure 6 compares the vertical profiles of the non-dimensionalized mean streamwise and vertical velocity profiles (i.e. $\langle u\rangle /U_b$ and $\langle v\rangle /U_b$, respectively) of the two rib cases along the central vertical lines located at $(x'/H,z/H)=(2.0, 0.0)$, (4.5, 0.0) and (7.0, 0.0). From figure 6 it is evident that the maximum value of $\langle u\rangle /U_b$ in both rib cases occurs above the rib height and then decreases monotonically as the relative downstream distance from the rib ($x'/H$) increases from 2.0 to 7.0, a pattern that is remarkably different from those in a canonical 2-D turbulent boundary layer over a flat plate (Miyake et al. Reference Miyake, Tsujimoto and Nakaji2001; Leonardi et al. Reference Leonardi, Orlandi, Djenidi and Antonia2004; Ikeda & Durbin Reference Ikeda and Durbin2007; Burattini et al. Reference Burattini, Leonardi, Orlandi and Antonia2008; Volino et al. Reference Volino, Schultz and Flack2009) or in a transverse rib-roughened duct flow (Coletti et al. Reference Coletti, Maurer, Arts and Di Sante2012; Labbé Reference Labbé2013; Mahmoodi-Jezeh & Wang Reference Mahmoodi-Jezeh and Wang2020). As an example, for the 2-D riblet flow over a flat plate, the magnitude of $\langle u\rangle /U_b$ reaches its maximum near the central region of the channel (Miyake et al. Reference Miyake, Tsujimoto and Nakaji2001; Volino et al. Reference Volino, Schultz and Flack2009). The existence of this mean streamwise velocity peak further causes the appearance of the strong shear layers (as indicated by the magnitude of mean streamwise velocity gradient $\textrm {d}\langle u\rangle /{\textrm {d} y}$ near the rib crest), which is often accompanied by an increase in the value of the TKE production rate (i.e. $-\langle u'_iu'_j \rangle {\partial \langle u_i \rangle }/{\partial x_j}$). From figure 6 it is also clear that for both ribbed duct cases, the vertical gradient of $\langle u\rangle$ is positive in the region near the ribbed bottom wall (for $-5.0< y/H<-4.0$), which indicates an acceleration of the mean flow in the downstream direction. However, as the duct centre is approached, the vertical gradient of $\langle u\rangle$ becomes negligible and negative in the inclined and V-shaped rib cases, respectively. In contrast to the monotonic variation of the mean streamwise velocity $\langle u\rangle /U_b$ with an increasing relative streamwise distance $x'/H$, the mean vertical velocity $\langle v\rangle /U_b$ exhibits a complex behaviour near the ribbed bottom wall. As demonstrated in figure 6, in the region near the rib crest at $x'/H=2.0$, the value of $\langle v\rangle /U_b$ is positive and negative in the inclined and V-shaped rib cases, respectively. This further confirms that the mean flow field is sensitive to the rib geometry. As seen in figure 6(a), the magnitude of $\langle v\rangle /U_b$ in the inclined rib case remains unchanged as the vertical distance from the rib crest increases. However, as is clear in figure 6(b), the magnitude of $\langle v\rangle /U_b$ in the V-shaped rib case decreases as the duct centre approaches, a feature that is consistent with the qualitative results shown previously in figure 5(b). From figures 5 and 6, it is understood that the magnitudes of the mean streamwise velocity gradient $|\textrm {d}\langle u\rangle /{\textrm {d} y}|$ reaches its maximum near the rib crest (in the vertical direction) and at the relative streamwise location $x'/H=2.0$ in the two rib cases. In view of this, in the remainder of our analysis, we need to pay close attention to the flow physics occurring at this special relative streamwise location $x'/H=2.0$.

Figure 6.

Comparison of the non-dimensionalized mean streamwise and vertical velocity profiles (indicated by the black and blue colours, respectively) at three relative downstream locations from the rib (for $x'/H=2.0$ (dash-dot-dot), 4.5 (dashed) and 7.0 (solid)) in the central plane (located at $z/H=0$) for two ribbed duct flow. The horizontal red dashed line demarcates the rib crest. (a) Inclined rib case and (b) V-shaped rib case.

Figure 7 shows the contours of non-dimensionalized mean streamwise vorticity (defined as $\langle \omega _x\rangle =\partial \langle w\rangle /\partial y-\partial \langle v\rangle /\partial z$) superimposed with the mean spanwise-vertical velocity streamlines in the ($y$–$z$) plane at the relative streamwise location $x'/H=2.0$ for two ribbed duct cases. From figure 7 it is seen that the pattern of the secondary flow is strongly influenced by the rib geometry, causing substantial variations in momentum transfer in the cross-stream plane. As is evident in figure 7(a), for the inclined rib case, the secondary flow appears as a large streamwise-elongated vortex, which occupies almost the entire cross-section of the square duct. Owing to the interaction between this large-scale circulation of the fluid with the four boundary layers developed over the duct walls, high-level negatively valued mean streamwise vorticity $\langle \omega _x\rangle$ appears near all four sidewalls. From figure 7(b) it is clear that the vortex pattern of the mean flow in the V-shaped rib case is drastically different from that in the inclined rib case since the secondary flow develops into a pair of large symmetrical counter-rotating vortices in the cross-stream directions. This figure also shows that due to the angled ribs in the V-shaped rib case, the mean streamwise vorticity generated in the near-rib region is convected sideways and upwards by the secondary flow, interacting with the boundary layers over the two vertical sidewalls, and creating a region with high values of $\langle \omega _x\rangle$ near the sidewalls above the rib height ($y/H=-2.5$).

Figure 7.

Contours of non-dimensionalized mean streamwise vorticity ($\langle \omega _x\rangle /(U_b/\delta ))$ superimposed with the mean spanwise-vertical velocity streamlines in the ($y$–$z$) plane at the relative streamwise location $x'/H=2.0$ for the two rib cases. For clarity, only large dominant vortices are shown in the cross-stream plane. (a) Inclined rib case and (b) V-shaped rib case.

Figure 8 compares the spanwise profiles of non-dimensionalized mean streamwise and vertical velocity profiles (i.e. $\langle u\rangle /U_b$ and $\langle v\rangle /U_b$, respectively) along three elevated lines positioned at $(x'/H, y/H)=(2.0, -3.8), (2.0, -2.5)$ and $(2.0, 0.0)$ of the two ribbed duct cases. From figure 8 it is clear that the profiles of $\langle u\rangle /U_b$ and $\langle v\rangle /U_b$ for the inclined rib case are asymmetrical in the cross-stream directions, but symmetrical for the V-shaped rib case at all three vertical locations. Figure 8 also shows that the presence of the cross-stream secondary flow motion influences significantly the spanwise distributions of both $\langle u\rangle /U_b$ and $\langle v\rangle /U_b$. From figure 8(a) it is seen that for the inclined rib case, the spanwise profile of $\langle u\rangle /U_b$ at all three vertical locations is skewed toward one side of the duct, with its magnitude peaks approximately at $z/H=-1.7$. However, for the V-shaped rib case as shown in figure 8(b), the profile of $\langle u\rangle /U_b$ peaks not only at the duct centre but also near the two vertical sidewalls (located at $z/H=\pm 5.0$). By comparing figure 8(a) with 8(b), it is apparent that the highest value of $\langle u\rangle /U_b$ occurs at $y/H=-2.5$ in the inclined rib case, and at $y/H=-3.8$ in the V-shaped rib case, indicating that the highest level of streamwise momentum occurs in the lower-half duct (for $y/H<0.0$) in both rib cases. This observation is consistent with the previous analysis of figure 5 and with the observations of Gao & Sundén (Reference Gao and Sundén2004b) and Fang et al. (Reference Fang, Yang, Wang, Tachie and Bergstrom2015), who conducted PIV experiments of ribbed duct flows with a similar geometrical set-up.

Figure 8.

Spanwise profiles of the non-dimensionalized mean streamwise and vertical velocity profiles (indicated by the black and blue colours, respectively) along three elevated lines positioned at $y/H=-3.8$ (dash-dot-dot), $-2.5$ (dashed) and $0.0$ (solid) for the two rib cases. The relative streamwise coordinate of the point is fixed at $x'/H=2.0$. (a) Inclined rib case and (b) V-shaped rib case.

For the mean vertical flow motion, figure 8(a) shows that the profile of $\langle v\rangle /U_b$ in the inclined rib case manifests two distinct peaks, one positively valued and one negatively valued, located at $z/H=-5.0$ and 5.0, respectively. This clearly indicates upward- and downward-moving of the flow near the sidewalls of the duct, which is a direct consequence of the appearance of the mean secondary flows in the cross-stream plane (see figure 7a). However, as shown in figure 8(b), the magnitude of $\langle v\rangle /U_b$ in the V-shaped rib case peaks close to the two vertical sidewalls of the duct. Furthermore, the peak magnitude $\langle v\rangle /U_b$ decreases as the distance from the sidewalls increases. This trend reflects the fact that the mean flow near the sidewalls is pushed upwards (corresponding to the positive sign of $\langle v\rangle /U_b$) and then convected downwards near the central region (corresponding to the negative sign of $\langle v\rangle /U_b$). From figure 8, it is observed that for both rib cases, the secondary flow effect on the mean flow field is the largest at $y/H=-2.5$, hence, our study needs to be refined to investigate the influence of secondary flow on the major characteristics of the flow field at this specific vertical position later in § 3.3.

3.2. Viscous and pressure drags

The effects of rib geometry on the mean flow field can be further investigated through an analysis of the skin friction and pressure coefficients, defined as ${{C}_{f}}={{\tau }_{w}}/(\rho U_{b}^{2}/2)$ and ${{C}_{p}}={\langle p \rangle }/{(\rho {{U}_{b}^{2}} /2)}$, respectively, where ${\tau }_{w}$ represents the local total wall friction stress calculated as $\tau _w = \mu [ (\partial \langle u \rangle /\partial y)^2+(\partial \langle w \rangle /\partial y )^2]_{w}^{1/2}$. Figure 9 shows the distributions of the skin friction ${{C}_{f}}$ on the bottom wall located at $y/H= -0.5$ for two ribbed duct cases. To explain the influence of the mean flow structures on the local $C_f$ value, the mean velocity streamlines are also superimposed. From figure 9 it is evident that owing to the presence of ribs and strong secondary flows, the magnitude of $C_f$ varies considerably along the cross-stream directions, indicating that this 3-D rib-roughened duct flow is remarkably inhomogeneous in the spanwise direction. As seen in figure 9(a), the highest value of $C_f$ in the inclined rib case appears at the corner on the leeward side of the upstream ribs near the vertical sidewall of the duct (at $z/H=5.0$). This is due to the fact that the secondary flow induces a strong downwash of mean flows towards the ribbed bottom wall (see figures 7a and 8a), further resulting in a large magnitude of the streamwise velocity gradient in the vertical direction. By contrast, in the V-shaped rib case as shown in figure 9(b), the highest friction coefficient values appear near the leeward face of the rib elements in the central region (for $-2.5< z/H<2.5$). This peak of $C_f$ coincides exactly with the core of vortex A exhibited previously in figure 5(b). Furthermore, it is apparent that both streamlines and vortex A (featuring high values of $C_f$) evolve along the V-shaped ribs and then become aligned with the main stream in regions near the sidewalls.

Figure 9.

Contours of the skin friction coefficient $C_f$ displayed in the ($x$–$z$) plane on the bottom wall located at $y/H=-5.0$ for the two rib cases. (a) Inclined rib case and (b) V-shaped rib case. The presentation of the contour plots of $C_f$ is superimposed with the mean velocity streamlines. Given the spanwise symmetry of the mean flow in the V-shaped rib case, streamlines are plotted only in one-half of the duct in panel (b) for clarity.

Figure 10 compares the spatial distributions of pressure coefficient $C_p$ of the two ribbed duct cases over the ribbed bottom wall located at $y/H=-5.0$. From figures 10(a) and 10(b), it is evident that similar to the skin friction coefficient, both the pressure coefficient level and its spatial distribution are sensitive to the rib geometry. Figure 10(a) clearly demonstrates the spatial evolution of the streamlines of the mean flow and shows how the flow turns as it passes over the inclined ribs. First, flow intensively interacts with the windward face of the upstream rib near the vertical sidewall of the duct (at $z/H=5.0$), causing the flow to become stagnant and create a high pressure region (marked with ‘I’), then the flow in the inter-rib region is driven toward the other sidewall (at $z/H=-5.0$) and impinges onto the windward side of the downstream rib, which results in the generation of the second and third high pressure regions (marked with ‘II’ and ‘III’, respectively). As a result, the mean separation bubble occurs near region II at $z/H=-5.0$, as shown in figure 10(a). From figure 10(b) it is observed that the mean streamline topology in the V-shaped rib case is substantially different from that of the inclined rib case. Figure 10(b) shows that the magnitude of $C_p$ reaches its maximum near the windward face of the V-shaped ribs (marked with ‘I’) and near the reattachment point (marked with ‘II’). Furthermore, owing to the angled ribs in the V-shaped rib case, the mean streamlines diverge from the duct midspan towards the sidewalls below the rib height and impinge on the two vertical sidewalls of the duct, leading to the formation of a region with high values of $C_p$ (marked with ‘III’). By comparing figures 9 with 10, it is concluded that the pressure difference between the windward and leeward faces of the ribs greatly contribute to increasing the total drag (see table 1). Furthermore, if we compare the mean flow features of the full inclined ribbed duct shown in figure 10(a) with that of one-half the V-shaped duct (for $z/H=-5.0$) shown in figure 10(b), certain similarity can be observed in terms of the streamline and $C_p$ contour patterns. More specifically, the mean flow pattern exhibited in the V-shaped ribbed duct is almost a pair of mirror reflections of that of the inclined rib case. The same can be concluded by comparing figures 9(a) and 9(b) with respect to the spatial distribution of the $C_f$ values.

Figure 10.

Contours of the pressure coefficient $C_p$ displayed in the ($x$–$z$) plane on the bottom wall located at $y/H=-5.0$ for the two rib cases. (a) Inclined rib case and (b) V-shaped rib case. The presentation of the contour plots of $C_p$ is superimposed with the mean velocity streamlines. Given the spanwise symmetry of the mean flow in the V-shaped rib case, streamlines are plotted only in one-half of the duct in panel (b) for clarity.

3.3. Reynolds stress distributions

Figure 11 compares the contours of the non-dimensionalized mean Reynolds normal stress components ($\langle u'u'\rangle /U_b^2, \langle v'v'\rangle /U_b^2$ and $\langle w'w'\rangle /U_b^2$) in the central ($x$–$y$) plane (located at $z/H=0.0$) for two rib cases. From figure 11(a) it is seen that for the inclined rib case, the magnitude of the streamwise Reynolds normal stress $\langle u'u'\rangle$ peaks in the neighbourhood of the rib crest and then decreases as the downstream distance from the leading edge of the rib ($x'/H=0.0$) increases. This enhancement in the magnitude of $\langle u'u'\rangle$ immediately downstream of the rib crest is a result of the occurrence of the boundary-layer separation near the leading edge of the rib, which also produces strong spanwise vortex shedding. However, as shown in figure 11(b), the primary peak of $\langle u'u'\rangle$ in the V-shaped case occurs in the inter-rib region (below the rib height) in the lee of the ribs. This is due to the negative values of $\langle v\rangle /U_b$ induced by the secondary flows, which results in a downwash of high momentum flow from duct centre to the ribbed wall (see figures 5b, 6b and 7b). By comparing figure 11(c) with 11(d), it is observed that the highest levels of the vertical Reynolds normal stress $\langle v'v'\rangle$ in the inclined rib case are mainly concentrated around the rib height; however, in the V-shaped rib case, large values of $\langle v'v'\rangle$ are primarily confined within a small region below the rib height. As is seen clearly in figure 11(e), in the inclined rib case the contours of the spanwise Reynolds normal stress $\langle w'w'\rangle$ exhibit two distinct peaks, one in the near-wall region below the rib height and one near the rib height. The occurrence of these two peaks is a unique feature of the inclined rib-roughened duct flow. By contrast, in the V-shaped rib case the maximum value of $\langle w'w'\rangle$ is only observed within the cavity (for $2.0< x'/H<4.0$), which nearly coincides with those of $\langle u'u'\rangle$ and $\langle v'v'\rangle$ shown in figures 11(b) and 11(d), respectively.

Figure 11.

Contours of non-dimensionalized Reynolds normal stress components ($\langle u'u'\rangle /U_b^2, \langle v'v'\rangle /U_b^2$ and $\langle w'w'\rangle /U_b^2$) in the central ($x$–$y$) plane for the inclined rib case (a,c,e) and V-shaped rib case (b,d, f). To facilitate a clear visual comparison, contour values less than 20 % of the peak value of Reynolds normal stresses are clipped off. The cross symbol $\times$ marks the local peak positions of Reynolds normal stresses.

To further investigate the effects of rib geometry on the velocity field, figure 12 compares the vertical profiles of non-dimensionalized Reynolds normal stresses ($\langle u'u'\rangle /U_b^2, \langle v'v'\rangle /U_b^2$ and $\langle w'w'\rangle /U_b^2$) of two rib cases along the central vertical lines located at $(x'/H,z/H)=(2.0, 0.0)$, (4.5, 0.0) and (7.0, 0.0). From figure 12 it is observed that for both ribbed duct cases, the highest turbulent levels (as indicated by the magnitudes of Reynolds normal stresses) occur around the rib height at the relative streamwise location $x'/H=2.0$, where the shear strength is the largest (see figure 6). The shear layer generated over the rib crest separates immediately after the rib, shedding downstream energetically and causing an enhancement in the variance of velocity fluctuations. Furthermore, it is observed that for both rib cases, the peak value of all the three Reynolds stress components reduces monotonically near the rib crest as $x'/H$ increases from 2.0 to 7.0. From figure 12 it is clear that owing to the disturbances from the ribs, the magnitudes of Reynolds normal stresses in both inclined and V-shaped rib cases are significantly amplified on the ribbed bottom wall side than on the smooth top wall side. Also, it is apparent that although there exists a drastic difference in the patterns of the Reynolds normal stresses profiles below the rib height ($y/H<-4.0$) between these two ribbed duct cases, they all collapse to a single profile above the rib height ($y/H>-4.0$). From figures 12(a), 12(c) and 12(e), it is evident that for the inclined rib case, the magnitudes of normal components reduce considerably as the duct centre is approached, especially on the ribbed side. By contrast, for the V-shaped rib case as shown in figures 12(b), 12(d) and 12( f), the magnitudes of normal components increase and peak near the duct centre, contributing to a local enhancement of TKE.

Figure 12.

Comparison of the non-dimensionalized Reynolds normal stress components ($\langle u'u'\rangle /U_b^2, \langle v'v'\rangle /U_b^2$ and $\langle w'w'\rangle /U_b^2$) at three relative streamwise locations (for $x'/H=2.0$, 4.5 and 7.0) in the central vertical plane (located at $z/H=0$) for the inclined rib case (a,c,e) and V-shaped rib case (b,d, f). The horizontal red dashed line demarcates the rib crest. In order to show clearly the profiles of the Reynolds stresses below the rib crest, they are partially enlarged and replotted in inset graphs in panels (b), (d) and (e).

Figure 13 compares the vertical profiles of Reynolds shear stress ($\langle u'v'\rangle /U_b^2$) along the central vertical lines located at $(x'/H, z/H)=(2.0, 0.0)$, (4.5, 0.0) and (7.0, 0.0) for the two ribbed duct flow cases. From figure 13 it is seen that similar to the trend of $\langle u'u'\rangle$ shown previously in figures 12(a) and 12(b) (for inclined and V-shaped rib cases, respectively), the Reynolds shear stress component $\langle u'v'\rangle$ peaks around the rib height for both rib cases, with the magnitude decreasing progressively as the relative streamwise distance $x'/H$ increases. Since the Reynolds shear stress is a reflection of the sweeping or ejection events, it implies that the strength of both events is significant near the rib crest but becomes trivial as $x'/H$ increases from 2.0 to 7.0. Later in § 4 we will refine our study by examining the rib geometry effects on turbulent motions through a quadrant analysis of the ejection and sweep events. From figure 13(a) it is clear that as the duct centre is approached, the magnitude of $\langle u'v'\rangle$ gradually increases and a positive peak occurs in its profile at $y/H=2.3$ in the inclined rib case. By contrast, for the V-shaped rib case as seen from figure 13(b), the magnitude of $\langle u'v'\rangle$ in all three streamwise locations becomes trivial in the central region of the duct. In fact, the profile of $\langle u'v'\rangle$ in the V-shaped rib case becomes almost a vertical straight line with a small magnitude in the region $-4.0< y/H<4.0$.

Figure 13.

Profiles of the non-dimensionalized Reynolds shear stress ($\langle u'v'\rangle /U_b^2$) at three relative streamwise locations (for $x'/H=2.0$, 4.5 and 7.0) in the central vertical plane (located at $z/H=0$) for the two rib cases. The horizontal red dashed line demarcates the rib crest. (a) Inclined rib case and (b) V-shaped rib case.

Figure 14 compares the spanwise profiles of the Reynolds normal (${{{\langle {{u}'{u}'}\rangle }}}, {{{\langle {{v}'{v}'}\rangle }}}$ and ${{{\langle {{w}'{w}'}\rangle }}}$) and shear (${{{\langle {{{u}'}{{v}'}}\rangle }}}$) stresses along an elevated line positioned at $(x'/H, y/H)=(2.0, -2.5)$ for the two rib cases. From figure 14 it is evident that for both inclined and V-shaped rib cases, the highest Reynolds stress levels appear near the vertical sidewalls and decay in magnitude as the duct centre is approached. This physical feature is mainly attributed to the difference of the rib geometries and associated secondary flow patterns in the cross-stream plane shown previously in figure 7. Figure 14(a) shows that for the inclined rib case, the magnitudes of the three normal components are comparable in value near the sidewall (located at $z/H=-5.0$), indicating that turbulence tends to be locally isotropic in this region. However, the three Reynolds normal stress magnitudes are significantly different near the other sidewall (located at $z/H=5.0$), with the magnitude of ${{{\langle {{u}'{u}'}\rangle }}}$ being approximately 1.9 and 8.7 times larger than those of ${{{\langle {{v}'{v}'}\rangle }}}$ and ${{{\langle {{w}'{w}'}\rangle }}}$, respectively. This implies that ${{{\langle {{u}'{u}'}\rangle }}}$ makes the primary contribution to the TKE among the three Reynolds normal stress components and also indicates the anisotropic states of turbulence in this region (near the side wall located at $z/H=5.0$). Given spanwise symmetry of the mean flow field in the V-shaped ribbed duct case, only one-half of the Reynolds stress profiles are displayed in figure 14(b). From figure 14(b) it is seen that near the two vertical sidewalls, the magnitude of ${{{\langle {{w}'{w}'}\rangle }}}$ is larger than those of ${{{\langle {{u}'{u}'}\rangle }}}$ and ${{{\langle {{v}'{v}'}\rangle }}}$ in the V-shaped rib case. This phenomenon reflects the fact that TKE is mostly contributed by ${{{\langle {{w}'{w}'}\rangle }}}$ (instead of $\langle u'u'\rangle$ and $\langle v'v'\rangle$), a physical feature that is drastically different from that of an inclined ribbed duct flow.

Figure 14.

Spanwise profiles of Reynolds normal (${{{\langle {{u}'{u}'}\rangle }}}, {{{\langle {{v}'{v}'}\rangle }}}$, and ${{{\langle {{w}'{w}'}\rangle }}}$) and shear (${{{\langle {{{u}'}{{v}'}}\rangle }}}$) stresses along an elevated line positioned at $(x'/H, y/H)=(2.0, -2.5)$ for the two rib cases. (a) Inclined rib case and (b) V-shaped rib case. In panel (b), for the V-shaped rib-roughened duct, only one-half of the duct is plotted given the spanwise symmetry of the profiles (the horizontal dashed line on the top of the panel demarcates the symmetric centre for $z/H=0.0$). Note that the spanwise profiles of Reynolds normal and shear stresses cross through only one rib, as they are plotted at the relative streamwise location $x'/H=2.0$.

Figure 15 shows the contours of the instantaneous TKE (defined as $k_t=(u^{\prime 2}+v^{\prime 2}+w^{\prime 2})/2$) in the central ($x$–$y$) plane located at ${z}/{H}=0.0$ for two different rib cases. From figure 15 it is seen that owing to the mechanisms such as vortex shedding and secondary flows, the TKE generated around the rib elements are deflected and then spread to the smooth top wall in both inclined and V-shaped rib ducts. By comparing figures 15(a) and 15(b), it is apparent that in the region well above the rib height, the intensity of turbulence as indicated by the value of $k_t$ in the V-shaped rib case is greater than that in the inclined rib case.

Figure 15.

Contours of instantaneous TKE $k_t=(u^{\prime 2}+v^{\prime 2}+w^{\prime 2})/2$ (non-dimensionalized using ${U}_{b}^2$) in the central ($x$–$y$) plane located at ${z}/{H}=0.0$ for two different rib cases. (a) Inclined rib case and (b) V-shaped rib case.

3.4. Anisotropy invariant map

A detailed analysis of the effects of rib geometry on turbulence anisotropy can be conducted through the anisotropy invariant map (AIM) proposed by Lumley & Newman (Reference Lumley and Newman1977). This AIM provides a description of the local flow topologies associated with the Reynolds stress anisotropy tensor, defined as

(3.1)\begin{equation} {{b}_{ij}}=\frac{\langle {{u}_{i}'}{{u}_{j}'} \rangle }{\langle {{u}_{k}'}{{u}_{k}'} \rangle }-\frac{1}{3}{\delta}_{ij}.\end{equation}

The Reynolds stress anisotropy tensor has three invariants in the context of an incompressible flow, one zero (i.e. $\mbox {I}=b_{ii}=0.0$) and two non-trial invariants (i.e. $\mbox {II}=b_{ij}b_{ji}$ and $\mbox {III}=b_{ij}b_{jk}b_{ki}$). Invariant II quantifies the degree of turbulence anisotropy, while invariant III indicates the nature of the turbulence anisotropy (Lumley Reference Lumley1979; Antonia, Djenidi & Spalart Reference Antonia, Djenidi and Spalart1994; Oyewola, Djenidi & Antonia Reference Oyewola, Djenidi and Antonia2004). There are three distinct characteristic states of turbulence associated with the so-called Lumley triangle: the axisymmetric turbulence state where one eigenvalue of the Reynolds stress tensor is either larger ($\mbox {III}>0.0$) or smaller ($\mbox {III}<0.0$) than the other two which are precisely equal in magnitude, and the two-component turbulence state in which one eigenvalue of the Reynolds stress tensor vanishes more rapidly than the other two (i.e. statistically 2-D turbulence). All possible turbulent states must lie within the Lumley triangle, because states outside of the Lumley triangle relate to non-realizable Reynolds stresses. Limiting points at the ends of the three defining curves of the Lumley triangle correspond to the isotropic turbulence state ($\mbox {II}=0.0$ and $\mbox {III}=0.0$), two-component isotropic turbulence state ($\mbox {II}=1/6$ and $\mbox {III}=-1/36$) and one-component turbulence state ($\mbox {II}=2/3$ and $\mbox {III}=2/9$).

Figure 16 compares the AIMs of the Reynolds stress anisotropy tensor along the central vertical line positioned at $(x'/H, z/H)=(2.0, 0.0)$ for both rib cases. From figure 16(a) it is seen that in the region very close to the ribbed bottom wall (for $-5.0< y/H<-4.9$), the state of $b_{ij}$ in the inclined rib case lies on the two-component turbulence state. This is simply because the magnitude of $\langle v'v'\rangle$ is negligibly small in comparison with those of $\langle u'u'\rangle$ and $\langle w'w'\rangle$ in this region (see figures 12a, 12c and 12e). However, as the vertical distance from the ribbed bottom wall increases to the range of $-4.9< y/H<-3.8$, the vertical normal stress $\langle v'v'\rangle$ makes increasingly important contributions to the TKE causing the turbulence flow to move towards the axisymmetric state. In the region well above the rib crest (for $-3.8< y/H<4.0$), the state of $b_{ij}$ turns towards the isotropic state where TKE is evenly distributed among the three normal components (see figures 12a, 12c and 12e). As the smooth top wall is approached (for $4.0< y/H<5.0$), the state of $b_{ij}$ falls back to the two-component turbulence state due to the enhanced damping effect of the smooth top wall on the vertical normal stress $\langle v'v'\rangle$. The wall anisotropy near the smooth top wall is similar to those of a smooth plane-channel flow observed by Lee & Choi (Reference Lee and Choi2001). As is evident in figure 16(b), the invariants in the V-shaped rib case exhibit a similar trend to that of the inclined rib case for most of the vertical positions, with the exception of the region just below the rib crest (at $y/H=-4.25$). At this elevation $y/H=-4.25$, the value of TKE is dominated by $\langle u'u'\rangle$ (see figures 12b, 12d and 12f), resulting in an enhanced degree of turbulence anisotropy in the V-shaped rib case.

Figure 16.

Anisotropy invariant map based on $\mbox {II}=b_{ij}b_{ji}$ and $\mbox {III}=b_{ij}b_{jk}b_{ki}$ of the Reynolds stress anisotropy tensor along the central vertical line positioned at $(x'/H, z/H)=(2.0, 0.0)$ for both rib cases. (a) Inclined rib case and (b) V-shaped rib case. The solid black line represents a two-component turbulence state (defined as $\mbox {II}=2/9+2\mbox {III}$), while both dashed and dash-dotted black lines indicate axisymmetric turbulence states (defined as $\mbox {II}=-6(\mbox {III}/6)^{2/3}$ and $\mbox {II}=6(\mbox {III}/6)^{2/3}$, respectively). These lines form the boundaries of the so-called Lumley triangle (Lumley & Newman Reference Lumley and Newman1977).

3.5. TKE budget analysis near the rib-roughened wall

To further elucidate the effects of rib geometry on turbulence stresses discussed in § 3.3, the turbulent transport equation of TKE (defined as $k=\langle u'_iu'_i \rangle /2$) can be studied, which reads as

(3.2)\begin{equation} 0=\varPi_k+T_k+D_k+P_k+\varepsilon_k+C_k.\end{equation}

Here, $\varPi _k, T_k, D_k, P_k, \varepsilon _k$ and $C_k$ denote the pressure diffusion term, turbulence diffusion term, viscous diffusion term, turbulent production term, dissipation term and mean convection term, defined as

(3.3a–c)$$\begin{gather} \varPi_k={-}\frac{1}{\rho} \frac{\partial \langle p'u'_j \rangle}{\partial x_j},\quad T_k={-}\frac{1}{2} \frac{\partial \langle u'_iu'_iu'_j \rangle}{\partial x_j},\quad D_k=\nu \frac{\partial^2 k}{\partial x^2_j}, \end{gather}$$

(3.4a–c)$$\begin{gather}P_k={-} \langle u'_iu'_j \rangle \frac{\partial \langle u_i \rangle}{\partial x_j},\quad \varepsilon_k={-}\nu \langle \frac{\partial u'_i}{\partial x_j}\frac{\partial u'_i}{\partial x_j} \rangle,\quad C_k={-}\langle u_j \rangle \frac{\partial k}{\partial x_j}, \end{gather}$$

respectively.

Figure 17 shows the profiles of TKE budget terms along the central vertical line positioned at $(x'/H, z/H)=(2.0, 0.0)$ for the two rib cases. In order to clearly demonstrate the profiles of the budget terms around the rib crest, they are plotted partially in figure 17. From the figure, it is clear that the turbulence energy transfer in the vertical direction is sensitive to the rib geometry, as the TKE budgets of these two rib cases are not only different in values but also dissimilar in patterns. As shown in figure 17(a), for the inclined rib case, the budget balance of TKE in the inter-rib region (for $-5.0< y/H<-4.5$) is dominated by viscous diffusion $D_k$ and turbulence diffusion $T_k$ as the source terms; and by convection $C_k$, production $P_k$ and dissipation $\varepsilon _k$ as the sink terms. Furthermore, it is observed from figure 17(a) that the turbulence diffusion term $T_k$ changes its sign four times near the ribbed bottom wall (for $-5.0< y/H<-3.0$), indicating that this term can be either a source or a sink in the inclined ribbed duct flow case. The appearance of a large positively valued turbulent diffusion term $T_k$ below and above the rib height (at $y/H=-4.6$ and $-3.5$, respectively) clearly indicates that highly turbulent fluid is transported away from these regions. From figure 17(a) it is also apparent that owing to the disturbances from the ribs, a dominant TKE source (as indicated by a distinct peak of turbulent production term $P_k$) appears above the rib crest, which is primarily balanced by the four sink terms (i.e. by turbulence diffusion, dissipation, convection and viscous diffusion). This peak of $P_k$ coincides with that in the streamwise normal stress $\langle u'u'\rangle$ shown previously in figure 12(a). By contrast, as shown in figure 17(b), in the V-shaped rib case, all dominant TKE budget terms peak in a region below the rib height. This feature is fully consistent with the quantitative results shown in figures 12(b), 12(d) and 12( f), in the sense that high levels of Reynolds normal stress components are also mainly concentrated in the region slightly below the rib crest. In the region very close to the ribbed bottom wall (for $-5.0< y/H<-4.7$), the TKE budget in the V-shaped rib case is primarily balanced between the viscous diffusion and dissipation terms (i.e. $D_k$ and $\varepsilon _k$), as the source and sink terms, respectively. However, as the distance from the bottom wall increases in the vertical direction (for $-4.4< y/H<-4.0$), the magnitudes of $D_k$ and $P_k$ become increasingly dominant TKE sources and reach their maxima just below the rib height, balanced by the dissipation term $\varepsilon _k$, convection term $C_k$ and turbulent diffusion term $T_k$. From figure 17 it is apparent that among all six TKE budget terms of (3.2), the pressure diffusion term $\varPi _k$ makes the smallest contribution to the TKE balance in both rib cases; therefore, it can be concluded that near the rib elements, the other five budget terms contribute considerably to the TKE balance, and the production and dissipation are not the only significant terms.

Figure 17.

Vertical profiles of the budget terms of the TKE transport equation along the central vertical line positioned at $(x'/H, z/H)=(2.0, 0.0)$ for both rib cases. (a) Inclined rib case and (b) V-shaped rib case. The budget terms are non-dimensionalized using the duct half-height, $\delta$, and bulk velocity, ${U}_{b}$. In order to show clearly the profiles of the budget terms around the rib crest, they are plotted partially in panels ($a$) and ($b$). The vertical dash-dotted line demarcates the position of the rib crest.