2.0 CURRENT MODELLING APPROACH

The combination of flow features present in dimples makes for a challenging task, for any turbulence model, to accurately predict the dynamic and thermal fields in these flows. Thus, employing higher order turbulence closures could prove to be beneficial.

A systematic study of three modelling strategies within the RANS approach has been carried out. These include a linear Eddy Viscosity Model (EVM), a Non-Linear Model (NLEVM) and a Reynolds Stress transport Model (RSM).

As a linear eddy viscosity model, the Launder and Sharma^{(Reference Launder and Sharma19)} k-ε scheme, with the near-wall lengthscale correction term from Yap^{(Reference Yap20)}, has been used. The cubic non-linear EVM of Craft et al^{(Reference Craft, Launder and Suga21)} has also been tested, as has the stress transport model of Hanjalić and Jakirlić^{(Reference Hanjalić, Jakirlić, Launder and Sandham22)}. The low-Reynolds number form of each model was exclusively used. Table 2 shows a summary of the tested models and the different modelling classes they belong to.

Table 2

Tested models and their modelling classes

The simulations have been carried out using the finite-volume-based STREAM code^{(Reference Lien and Leschziner23)}, with the SIMPLE pressure correction algorithm. The second order accurate UMIST scheme of Lien and Leschziner^{(Reference Lien and Leschziner24)} has been used to approximate the convection terms for the velocity and pressure correction equations, while the convection of turbulent quantities has been discretised using the first order accurate UPWIND scheme. The meshing strategy has been taken from Wang et al^{(Reference Wang, Yeo and Khoo11)}, where a simple structured grid was first created to fit the channel section and then stretched in the wall-normal direction to fill up the dimple volume. Figure 1 shows the computational grids used for the tested cases. As can be seen from the figure, the meshes have been refined near walls to ensure a value of y ⁺ less than unity. Grids of up to around two million cells have been used with 2.4 m cells used for the single dimple case and 1.8 m cells for the dimple array case. Tests have been carried out to ensure the mesh density was sufficient to fully resolve the flow field and achieve grid independent solutions.

Figure 1.

The computational grid for both geometry cases.

3.0 FLOW OVER A SINGLE DIMPLE

The natural starting point for this investigation is to study the flow over a single dimple in a channel. This serves as a first step in exploring the best way to tackle more realistic and complex geometry configurations.

The case considered here is that studied experimentally by Kwon et al^{(Reference Kwon, Hwang and Cho2)}. The experimental data was reported at a Reynolds number, based on bulk velocity and channel height, of Re _H = 5,000 with a dimple print diameter to channel height ratio of D/H = 2.0 and a ratio of dimple depth to channel height of δ/H = 0.4. The Prandtl number was taken to be Pr = 0.71. A schematic of the tested domain is shown in Fig. 2, with W/H = 8 and L/H = 16.

Figure 2.

Schematic of the single dimple test channel.

At the inlet, fully-developed channel flow profiles have been provided for the dynamic field quantities, whilst a uniform temperature has been assigned. Zero gradient conditions have been applied to all variables except pressure at the outlet, with a bulk mass-correction algorithm employed to speed up convergence. The no-slip condition has been enforced on both walls, with two thermal boundary conditions tested. In the first, a uniform heat flux was imposed on the dimpled surface, while in the second, the dimpled wall was set at a constant temperature higher than that of the fluid. In both cases, the upper wall was treated as adiabatic. At the span-wise (z-direction) boundaries, periodic conditions have been applied. It is worth noting that the size of the domain is large enough for the boundary conditions not to have an effect on the dimple region.

The experiments reported that no dominant frequencies were found in the frequency spectrum which suggests that no large-scale unsteadiness was detected. Thus, the main simulations for this case have been conducted as steady state. To test the validity of this assumption, a time dependent simulation has been carried out using the Launder and Sharma^{(Reference Launder and Sharma19)} k-ε model without the Yap^{(Reference Yap20)} lengthscale correction term. The simulation resulted in a steady state solution, and thus the steady RANS approach has been deemed suitable for this case.

Figure 3 shows the near-wall streamlines predicted in the present study. It can be seen that all models successfully capture the formation of the dimple’s vortices, with the linear EVM and RSM predicting a pair of symmetric vortices. This vortex pair forms due to flow recirculation upstream along the dimple curve, following its impingement on the dimple downstream half. These structures tie in nicely with the flow patterns reported by Kwon et al^{(Reference Kwon, Hwang and Cho2)} and Isaev et al^{(Reference Isaev, Leontev and Baranov3)} in the laminar and turbulent flow regimes, respectively.

Figure 3.

Stream traces on a curved surface close to the dimpled wall.

The NLEVM, on the other hand, predicts a single asymmetric vortex cell that is inclined to the flow direction by nearly 45°. This is similar to the features reported by Isaev et al^{(Reference Isaev, Leontiev, Kudryavtsev and Pyshnyi6)}, however, for much deeper dimple depths (δ/H > 0.628). To investigate this further, another simulation has been conducted using the NLEVM for a shallower dimple depth of δ/H = 0.2 with the other geometric factors being the same. In this case, the model does predict a symmetric pair of vortices, in line with other model predictions, which could lead to the conclusion that the non-linear model may have amplified the effect of dimple depth.

One can clearly see from Fig. 4 that all models return more or less the same position for the reattachment point within the dimple, with a slightly thicker recirculation region predicted by the NLEVM.

Figure 4.

Stream traces across the centreline of the dimple at z = 0.0.

Figure 5 shows an iso-surface of the Q-criteria within the dimple. The figure emphasises the previous conclusion with both the EVM and RSM returning symmetric coherent structures within the dimple, while the NLEVM produces an asymmetric one.

Figure 5.

Q-criteria iso-surface (Q = 0.2) for the single dimple case with δ/H = 0.4 and D/H = 2.0.

The velocity profiles, in Figs 6 and 7, are in general good agreement with the experimental data reported by Kwon et al^{(Reference Kwon, Hwang and Cho2)}. However, in Fig. 6, the measured data does not extend far into the dimple region where the model predictions start to deviate from each other. The figure shows that all models predict similar velocity profiles over the dimple until just under the dividing surface (between the channel section and the dimple, at y/H = 0.0) with a steep velocity gradient above and across the upper part of the dimple. Further in the dimple, the NLEVM shows slightly different behaviour, at least partly as a result of the different flow field (asymmetric vortex) predicted by the model. The differences in prediction between the models become more clear in the wake of the dimple, Fig. 7, where the NLEVM asymmetric flow pattern shifts the wake flow away from the dimple centreline and off the sampling point.

Figure 6.

Profiles of the stream-wise velocity, U, at a point on the centreline in the upstream half of the dimple, (x, z) = (− 0.15D, 0.0), as indicated by the cross on the schematic, compared to the experimental data reported by Kwon et al^{(Reference Kwon, Hwang and Cho2)}.

Figure 7.

Profiles of the stream-wise velocity, U, at a point on the centreline in the wake of the dimple, (x, z) = (0.85D, 0.0), as indicated by the cross on the schematic, compared to the experimental data reported by Kwon et al^{(Reference Kwon, Hwang and Cho2)}.

The contours of the turbulent kinetic energy over the dimple, in Figs 8 and 9, show that the peak levels of k occur in the shear layer and close to the impingement region around the dimple rear rim. The asymmetric flow pattern of the NLEVM model is clearly visible in the cross-stream plane and the model generally returns lower levels of k.

Figure 8.

Contours of the turbulent kinetic energy across the dimple centreline at z = 0.0.

Figure 9.

Contours of the turbulent kinetic energy in the cross-stream plane at x = 0.0.

Furthermore, it can be seen from Fig. 10 that all models, apart from the NLEVM, predict a high peak k level to occur close to the dividing surface between the dimple and the channel, partly as a result of the steep velocity gradients occurring across the separated shear layer in this region. The lower levels of k predicted by the NLEVM are also clear in this figure.

Figure 10.

Profiles of the turbulent kinetic energy k at a point on the centreline in the upstream half of the dimple, (x, z) = (− 0.15D, 0.0), as indicated by the cross on the schematic.

After the dimple, as shown in Fig. 11, the same pattern persists. Tests indicated that this difference in behaviour is mainly due to the form of the c _μ coefficient used in the turbulent viscosity formulation of the NLEVM, rather than the non-linear terms appearing in its stress-strain relation. In this case, c _μ is taken as a function of local strain rate and reduces significantly in regions of high strain (such as the separated shear layer), resulting in lower turbulent viscosity and turbulent kinetic energy levels.

Figure 11.

Profiles of the turbulent kinetic energy k at a point on the centreline in the wake of the dimple, (x, z) = (0.85D, 0.0), as indicated by the cross on the schematic.

Reported u′ levels by Kwon et al^{(Reference Kwon, Hwang and Cho2)} suggest that the NLEVM slightly under-predicts, and other models over-predict, k levels in the dimple, while all models under-predict k in the dimple wake region.

Finally, the Reynolds stress anisotropy does not seem to be a major factor in the flow field prediction in this case. Hence, the use of the linear stress-strain relation (Boussinesq approximation) did not hinder the performance of the EVM.

The disparity in flow structures predicted by the models is also reflected in the contours of the Nusselt number shown in Figs 12 and 13. These figures show that, for both wall boundary conditions, the symmetric flow field predicted when using the EVM and RSM leads, expectedly, to symmetric distribution of Nusselt number contours, with the highest levels of heat transfer enhancement being located in the dimple downstream half, where the flow impinges on the dimple surface. The NLEVM model, however, shows asymmetric contours of Nu with a peak location that does not coincide with the flow impingement zone. A reduction in heat transfer is also evident in the upstream part of the dimple.

Figure 12.

Contours of the Nusselt number at the dimpled wall normalised by that of a plane channel, for the case of a constant temperature wall thermal boundary condition.

Figure 13.

Contours of the Nusselt number at the dimpled wall normalised by that of a plane channel, for the case of a constant heat flux wall thermal boundary condition.

It is worth noting that for a constant temperature wall boundary condition, as shown in Fig. 12, the contour lines of the Nusselt number extend further in the lateral direction when compared to those of the constant heat flux case, as shown in Fig. 13. This is consistent with experimental findings reported by Kwon et al^{(Reference Kwon, Hwang and Cho2)} (for the constant temperature BC case) and Mahmood et al^{(Reference Mahmood, Moon, Glezer, Hill, Nelson and Ligrani12)} (for the constant heat flux case).

Figure 14 shows profiles of the span-wise averaged Nusselt number for the constant temperature case, with the averaging carried out over the area shown in the contour plot, Fig. 12, and the resulting averaged Nu normalised by that of a plane channel. The figure shows that the model predictions are generally in good qualitative agreement with the experimental data in the sense that all models reproduce the reduction in the upstream half of the dimple as well as the significant heat transfer enhancement in the downstream part. It can also be seen from Fig. 14 that the RSM outperforms the other two models as it provides a quantitative prediction of Nu, in the dimple area, that is more inline with the levels reported in the experiment.

Figure 14.

The span-wise averaged Nusselt number, normalised by that of a plane channel, compared to the experimental data reported by Kwon et al^{(Reference Kwon, Hwang and Cho2)}.

4.0 FLOW OVER AN ARRAY OF DIMPLES

The case considered is based on the DNS performed by Wang et al^{(Reference Wang, Yeo and Khoo11)}. The geometry of the tested channel is shown in Fig. 15 where, in this case, W/H = π, L/H = 2π. The dimple surface geometry factors were taken as: P/H = π/2, D/H = 1.0 and δ/H = 0.2.

Figure 15.

Schematic of the dimple array.

The flow was driven by a fixed stream-wise mean pressure gradient (Π). The Reynolds number, ${\rm Re}_{{}_\Pi } = {0.5 u_{{}_\Pi } \, H}/{\nu }$ , is taken as 180, to match the DNS data, where $u_{{}_\Pi }$ is a velocity scale defined as $u_{{}_\Pi }=\sqrt{{0.5 \Pi \, H}/{\rho }}$ , corresponding to the usual friction velocity in a plane channel case. The Prandtl number is taken as 0.71. Periodic boundary conditions were applied for all transported quantities in both the stream- and span-wise directions. On the walls, the no-slip condition was enforced with a uniform heat flux boundary condition applied on the dimpled surface, while the upper wall was considered adiabatic.

Unlike in the single dimple case, no information about the flow field statistical steadiness, or lack of, can be easily inferred from the study of Wang et al^{(Reference Wang, Yeo and Khoo11)}. Moreover, Ligrani et al^{(Reference Ligrani, Harrison, Mahmmod and Hill13)} and Won et al^{(Reference Won, Zhang and Ligrani25)} have reported, through flow visualisation in a similar set-up to the one described here, that the main vortex pair formed within the dimple is shed periodically from the cavity. This suggests that the presence of a large scale unsteadiness may be likely for this case. Thus, time-dependent simulations were conducted to investigate the flow in this case, with the timestep taken small enough to ensure CFL numbers less than unity. The EVM and the RSM converged to steady state solutions from which the results below are presented. As for the NLEVM, the predicted flow field remained unsteady, and thus, the long-term temporal averages are presented here.

Figure 16 shows the near-wall streamlines, over a curved surface parallel to the dimpled wall. It can be seen (similarly to what was described in the single dimple case) that both the EVM and RSM still maintain the symmetric structures, similar to those of the DNS, in the dimples, while the NLEVM predicts an asymmetric, single vortex cell instead. However, it is worth noting that in the flow field predicted by the non-linear EVM, the inclination of these asymmetric vortices changes direction, from −45° to 45° and vice versa, from one row of dimples to the following. This alternating inclination of vortices is also clearly visible in the instantaneous flow field, forcing the flow out of the dimple to follow the side of the vortex inclination. This leads to the formation of neighbouring fast and slow flow lanes where the former appear on the sides of the dimples where shedding occurs and the latter lanes on the opposite sides.

Figure 16.

Stream traces on a curved surface close to the dimple array. DNS data reported by Wang et al^{(Reference Wang, Yeo and Khoo11)}.

In Fig. 17, the profiles of $U^+=U/u_{{}_\Pi }$ are shown at the symmetry point between the dimples. From the DNS data, one can clearly see that the viscous sub-layer and a log-law region (albeit below the usual log-law line) remain present near the dimpled wall. However, they are separated by a buffer region where the velocity gradient is sharply reduced. The RSM clearly reproduces all three regions in its profile, but significantly over-predicts the U ⁺ gradient in the buffer region. The NLEVM captures the general level of U ⁺ but fails to produce the buffer region. This could be due to the fact that, with the asymmetric shedding described previously, the sampling point plotted is no longer in the wake of the dimple shedding. The profiles predicted by the EVM show a slender buffer region, but it also significantly over-predicts the U ⁺ gradient in the buffer region and slightly under-predicts the U ⁺ level in the log-law region.

Figure 17.

Profiles of the stream-wise velocity, U ⁺, at the symmetry point between dimples, compared to DNS data reported by Wang et al^{(Reference Wang, Yeo and Khoo11)}.

Through the dimple’s centre, as shown in Fig. 18, all model predictions are generally in good agreement with the DNS data, particularly in the recirculation region. The NLEVM extends this good agreement further across the shear layer, while the other two models better match the DNS in the remaining part of the channel.

Figure 18.

Profiles of the mean stream-wise velocity, U ⁺, through dimple centre, compared to DNS data reported by Wang et al^{(Reference Wang, Yeo and Khoo11)}.

Figure 19 shows profiles of the turbulent shear stress, ${\overline{uv}}^+ = \overline{uv}/{u^2_{{}_\Pi }}$ , at the symmetry point between the dimples. Results from all the models show an increase in the value of ${\overline{uv}}^+$ near the dimpled wall compared to that of a plane channel. The EVM tends to over-predict the values of ${\overline{uv}}^+$ in this region, while the results from the NLEVM and RSM better match the levels reported by the DNS across most of the channel. However, near the wall, the NLEVM slightly under-predicts the peak ${\overline{uv}}^+$ magnitude, while the RSM overpredicts it.

Figure 19.

Profiles of the turbulent shear stress, ${\overline{\hbox\textit{uv}}}^+$ , at the symmetry point between dimples, compared to DNS data reported by Wang et al^{(Reference Wang, Yeo and Khoo11)}.

In Fig. 20, the profiles of the turbulent shear stress through a dimple centre are shown. It can be seen that the NLEVM returns significantly lower levels of turbulence within the dimple area and across the shear layer compared to the other two models. The EVM and RSM give similar peak levels of ${\overline{uv}}^+$ and, in general, predict increased turbulence levels within the dimple.

Figure 20.

Profiles of the turbulent shear stress ${\overline{\hbox\textit{uv}}}^+$ through dimple centre.

Figures 21 and 22 show the turbulent normal stresses at the symmetry point between dimples predicted by the NLEVM and the RSM, respectively. Both models return the correct stress distribution with the RSM, expectedly, outperforming the NLEVM when it comes to quantitative comparison with DNS. The EVM predictions were not included here since the model produces a nearly isotropic normal stress distribution in this case. This stems from a well-known deficiency in the Boussinesq approximation which underpins all linear eddy viscosity models.

Figure 21.

Comparison between the NLEVM predicted turbulent normal stresses (lines) and DNS data (symbols) at the symmetry point between dimples. DNS data was taken from Wang et al^{(Reference Wang, Yeo and Khoo11)}.

Figure 22.

Comparison between the RSM predicted turbulent normal stresses (lines) and DNS data (symbols) at the symmetry point between dimples. DNS data was taken from Wang et al^{(Reference Wang, Yeo and Khoo11)}.

Contours of the Nusselt number over the dimpled wall, normalised by that of a plane surface at the same Reynolds number, are shown in Fig. 23. Similarly to the discussion in the single dimple case, the models predict increased values of the Nusselt number in the downstream half of the dimples, where the flow impinges on the dimpled wall. Moreover, all models show reduced levels of Nu in the upstream part of the dimple due to flow separation. Following their predicted flow fields, the EVM and RSM report a symmetric distribution of the Nusselt number on the dimple surface, while the NLEVM returns an asymmetric Nu distribution, due to the inclined vortex structure predicted in the dimple. On average, all models report an enhancement in Nusselt number levels, up to twice or more than that of a plane channel in parts of the dimpled wall. However, reduction in heat transfer over the upstream part of the dimples is also evident.

Figure 23.

Contours of the Nusselt number across the dimple array normalised by that of a plane channel.