Analysis of the coherent and turbulent stresses of a numerically simulated rough wall pipe

A turbulent rough wall flow in a pipe is simulated using direct numerical simulation (DNS) where the roughness elements consist of explicitly gridded three-dimensional sinusoids. Two groups of simulations were conducted where the roughness semi-amplitude h+ and the roughness wavelength λ+ are systematically varied. The triple decomposition is applied to the velocity to separate the coherent and turbulent components. The coherent or dispersive component arises due to the roughness and depends on the topological features of the surface. The turbulent stress on the other hand, scales with the friction Reynolds number. For the case with the largest roughness wavelength, large secondary flows are observed which are similar to that of duct flows. The occurrence of these large secondary flows is due to the spanwise heterogeneity of the roughness which has a spacing approximately equal to the boundary layer thickness δ.


Introduction
Typically, a turbulent quantity can be decomposed into its mean and fluctuating component where u i is the time-averaged mean and u i is the fluctuation about the time-averaged mean (also known as the turbulent fluctuation) for component i. This decomposition is known as the Reynolds decomposition and is commonly used to analyse the turbulent fluctuations in a flow. However, in a rough wall flow, variations due to the unevenness of the spatial geometry must be taken into account when analysing the turbulent flow data. Therefore, a triple decomposition is applied to the turbulent quantity where it is decomposed into three components [1,2], Here, U i = u i is the spatial and temporal averaged mean, which is also known as the global mean andũ i = u i − U i is the spatial variation of the time-averaged flow around individual roughness elements (also known as the coherent or dispersive component). The sum ofũ i + u i = u i is the total fluctuation and contains both the turbulent and coherent fluctuations. For a smooth wall,ũ i = 0 and therefore U i = u i . The triple decomposition is a result from double averaging, where the governing equations are averaged over time and then in space. This technique has been used to analyse plant canopy flows (urban-canopy model) [3,4,5] and also in open-channel flow over rough beds [6,7,8] where the rough surfaces are spatially inhomogeneous. When conducting laboratory experiments, point measurement techniques such as hot wire anemometry and laser Doppler velocimetry gather data locally in space and only measures the turbulent fluctuations of the flow. This can result in misinterpretation of the data especially when conducting rough wall experiments where large spatial variation of velocity might exist in the flow. In this paper, we will investigate the importance of the coherent and turbulent fluctuations in a turbulent rough wall flow.

Computational setup
DNS is conducted in a pipe using CDP which is a second-order, energy conserving finite volume code. The rough wall pipe has a Cartesian 'O-grid' mesh and a body conforming grid is used to explicitly represent the roughness elements. The pipe has a length of L x = 4πR 0 where R 0 is the reference radius. Further details of the numerical method used can be found in [9].
Throughout the paper, the roughness cases are identified by the following identifying code where the first two digits represent the roughness height and the last three digits represent the streamwise or spanwise wavelength of the roughness elements (λ x = λ s ). Two groups of cases were simulated where in group S1, the roughness elements are geometrically scaled while maintaining a fixed h/λ ratio and in group S2, the wavelength of the roughness λ is varied while the roughness semi-amplitude h is held constant. The computational details and mean flow properties of the rough cases are tabulated in table 1. Table 1. Details of the computational mesh and mean flow properties of the cases simulated. N r,θ is the number of elements in an (r, θ) plane, N x the number of elements in the streamwise direction and N λx the number of elements per roughness wavelength. Δr + and Δx + are the radial and streamwise mean viscous grid spacings at the wall calculated using the local friction velocity U τ . The largest cells are located at the centre of the pipe where Δr + ≈ Δrθ + ≈ Δx + . k + a is the average roughness height and ES is the effective slope of the roughness (see [9]). Re τ and Re D are the friction and bulk Reynolds numbers and ΔU + is the Hama roughness function. S1: Geometrically increasing roughness (fixed h/λ)

Mean velocity profile
The mean velocity profile of the rough cases are plotted in figure 1. The downward shift in the mean velocity profile of the rough wall is quantified by the roughness function ΔU + which is measured 200 viscous wall units above the crest of the roughness elements. The roughness function increases with increasing roughness height and also with decreasing roughness wavelength. However decreasing the roughness wavelength from λ + = 424 to 212 only results in a marginal increase in the ΔU + as the surface is in the dense regime (ES = 2Λ = 0.72). From [9], it is found that case 60 424 is in the fully rough regime. Therefore, it is deduced that case 60 212 which has a higher ES (or solidity) is also in the fully rough regime.
Group S1 G r o u p S2

E E
Increasing h + Decreasing λ +

Stress profiles
The various fluctuating terms will be compared in this section using the triple decomposition method introduced in § 1. The stress profiles for the azimuthal turbulent u + θ,rms , total u + θ,rms and coherentũ + θ,rms fluctuations are plotted in figure 2. In figures 2 (a, d), the occurrence of a secondary peak is observed within the roughness canopy (y ≤ h). This peak increases with increasing h + (group S1) and also with increasing roughness wavelength λ + (group S2). The secondary peak is contributed by the coherent (or dispersive) fluctuating component of the flow which is time independent (see figures 2 (c, f )).
The peak of the azimuthal coherent stress profile is located at a wall-normal location which ranges from y + ≈ 10 − 20 and has a value which is comparable to the azimuthal turbulent stress. The peak in the azimuthal coherent stress profile is due to the channelling of the flow around the three-dimensional sinusoidal roughness elements. However, the coherent stress profiles quickly reduces to zero above the crest of the roughness elements (expect for case 60 848). This indicates that the effects of the azimuthal coherent stresses to the flow are small. However, for case of 60 848, the azimuthal coherent stress profile extends more than 200 viscous wall units above the crest of the roughness elements. This is because there are large stationary features occurring in the flow and this will be investigated further in § 5. The stress profiles for the total (figures 2 (a, d)) and turbulent (figures 2 (b, e)) fluctuation in the azimuthal direction for the rough cases collapses with the smooth wall in the outer region of the flow. Good collapse in the outer region of the flow is also obtained for the stress profiles of the total and turbulent fluctuations in the streamwise and radial direction (not shown here). Good collapse is observed as the total fluctuation and turbulent stresses scale with the friction Reynolds numbers. Therefore, our findings support Townsend's outer-layer similarity hypothesis when considering the second-order statistics. However, when analysing the contours of the streamwise premultiplied energy spectra, it is observed that the collapse occurs at a much larger wall-normal location as energy is redistributed to the wavelength corresponding to the roughness elements and its harmonics [10].
In general, the maximum coherent stresses in all three dimensions (streamwise and radial components, not shown here) increases with increasing geometrical scale (S1) and increasing wavelength (S2). In addition, with increasing h + and λ + , the influence of the coherent component is felt further away from the roughness and for case 60 848, extends even into the edge of the outer layer. As for the turbulent stresses, the profiles for the rough cases above the roughness elements are not too dissimilar with the smooth case profiles as the turbulent stresses scales predominantly with the friction Reynolds number.

Streamwise vorticity
The azimuthal coherent fluctuation stress profile for case 60 848 extends far above the crest of the roughness elements due to large secondary flows. Isosurfaces of the time-averaged streamwise vorticity ω + x of the unwrapped surface for case 60 848 are plotted in figures 3 (a, b). There is strong streamwise vorticity near the surface of the roughness which alternates between positive and negative values in the streamwise direction due to the topological changes of the three-dimensional sinusoidal roughness ( figure 3 (a)). Plotting isosurfaces of the timeaveraged streamwise vorticity where |ω + x | = 0.01 in figure 3 (b), we observe large secondary flows which extend up to 200 viscous units from the wall. Figures 3 (c, d) shows the contours of the time-averaged streamwise vorticity overlaid with the time-averaged azimuthal and radial velocity vectors along cross sectional slice I where the surface is locally rough and II where the surface is locally smooth. The zoom in view of these contours clearly shows the occurrence of these secondary flows which are similar to that observed in duct flows [11]. The sketch in figure 3 (e) illustrates the orientation of the secondary flow which alternates in the spanwise direction. Fluid is pushed upwards from the wall along the locally rough streamwise plane and is pushed downwards in the locally smooth streamwise plane. Secondary flows are also observed in the other rough cases (not shown here) but is confined to regions near the crest of the roughness. Large secondary flows occur in case 60 848 due to the spanwise heterogeneity of the roughness elements which are spaced approximately equal to the boundary layer thickness δ + = R + 0 = 540 ≈ L + s = 424 (for a pipe, the δ is fixed and is equal to the reference radius of the pipe R 0 ). This is consistent with the findings of [12] who found that δ-spaced roughness elements induces secondary flows on the scale of δ.

Conclusion
DNS in a rough wall pipe with explicitly gridded three-dimensional sinusoidal roughnesses is simulated at moderate Reynolds numbers where the height h + and the wavelength λ of the roughness elements are systematically altered. Increasing the roughness height and/or roughness wavelength results in an increase in the coherent fluctuation stresses. For case 60 848, the coherent fluctuation stress profile extends much further above the crest of the roughness elements compared to the other rough cases due to the occurrence of large time invariant secondary flows. These large secondary flows develop in the pipe due to the spanwise heterogeneity of the roughness which has a spanwise spacing approximately equal to the boundary layer thickness.