3.1 Instantaneous flow fields

Figure 6. Instantaneous streamwise velocity contour plots in $x$ – $y$ plane for all five simulations.

Figure 7. Instantaneous streamwise velocity contour plots in $x$ – $z$ plane at $y^{+}=15$ in (a,b), $y=0.5$ in (c,d) and $y^{+}=15$ (from the top wall) in (e,f). Simulation A1 in (a,c,e) and simulation A3 in (b,d,f).

Figure 6 shows instantaneous contour plots of streamwise velocity in the $x$ – $y$ plane for all the simulations. As expected, investing the bump, the flow velocity increases at the channel restriction and separates behind the bump with the formation of an intense shear layer between the bulk flow and the separation bubble close to the bottom wall. With increasing Reynolds number, cases A1, A2 and A3 progressively, structures at smaller scales appear. The separated region behind the bump becomes smaller, more attached to the bump and less protruded in the streamwise direction. It is interesting to compare, at least qualitatively, the flow snapshots for cases A1 and A3 with the smoke patterns used to visualise the boundary layer on a convex wall as shown in p. 91 of the classical album by Van Dyke (Reference Van Dyke1982). Indeed, in case A3 the flow impinging the bump is clearly characterised by small-scale structures while case A1, even though nominally turbulent, appears smoother. Under this respect, case A3 can be regarded as producing a turbulent boundary layer between the external turbulent stream and the wall able to better withstand the adverse pressure gradients before separation. The separated region in front of the bump is also smaller at high Reynolds number. On the other hand, the separated region behind the bump becomes larger as the bump becomes bluffer. At the top wall, the boundary layer thickens after the constriction due to the adverse pressure gradient but separation is not observed. This effect is more evident for the lower Reynolds number cases, probably due to the higher extension of the recirculation bubble.

Figure 7 shows instantaneous contour plots of streamwise velocity in $x$ – $z$ planes at three wall-normal distances for simulations A1 and A3 to qualitatively compare Reynolds number effects. At the higher Reynolds number, smaller structures are clearly present and the spacing between region of high and low speed is greatly reduced. In the far downstream region and close to the walls, this is consistent with the expected scaling of streak spacing in internal units (Robinson Reference Robinson1991). In particular, figure 7(a,b) shows $x$ – $z$ -planes close to the bottom wall at $y^{+}=15$ . This wall distance corresponds to the classical buffer layer of the channel flow. The empty region in the plot represents the intersection of the plane with the bump. The recirculation region behind the bump is characterised, at this distance from the wall, by negative velocity and small-scale structures. The size of the region where reverse flow occurs decreases when increasing the Reynolds number. Downstream, the small-scale structures elongate in the flow direction and resemble the streaky structures found in turbulent planar channel flow. The $x{-}z$ -planes for simulations A1 and A3, in (c) and (d), respectively, just touch the top of the bump which is indicated by the continuous zero velocity line at $x=4$ in the contour plot. The acceleration of the flow just before the bump and its deceleration just behind can be observed. The small-scale structures far away from the bump increase their length downstream. For simulation A3, the flow structures generated in the shear layer appear confined in a smaller region since the flow is more attached to the bump’s surface and the separated region protrudes less in the streamwise direction. Figure 7(e,f) shows the $x{-}z$ -planes close to the top wall, at $y^{+}=15$ . The effect of the bump on the velocity is still present and a clear velocity increase is seen at the bump’s location, due to the cross-section restriction. This is followed by a low velocity region corresponding to the end of the separation bubble at the opposite wall. In this region, the turbulent structures maintain a streamlined, streaky shape and no separated region is present.

3.2 Mean velocity and turbulent fluctuations

In the statistical analysis to follow, the average of a generic quantity $q$ is indicated with the angular brackets, $\langle q\rangle$ , or with the capital letter, $Q$ , according to convenience, while the fluctuation is indicated with the apex, $q^{\prime }$ . Details of the recirculation bubble for all the simulations are shown in figure 8, providing the contour plot of the mean streamwise velocity $\langle u_{x}\rangle$ normalised with the bulk velocity. The black solid line highlights the zero isolines to better appreciate the mean flow reversal.

Figure 8. Average streamwise velocity $\langle u_{x}\rangle$ contour plots with isoline at $\langle u_{x}\rangle =0$ for all five simulations.

By progressively restricting the section, the first part of the bump makes the flow velocity increase consistently with the pressure decrease which is mechanically responsible for the acceleration. After the top of the bump the flow decelerates and a strong adverse pressure gradient occurs. This produces a backward flow near the bottom wall, giving rise to flow detachment. The bubble becomes smaller and more attached to the bump starting from the lower Reynolds number (A1) to the higher Reynolds number (A3). The profiles recuperate positive average velocity at the bottom wall behind the bump at an earlier $x$ -position for simulation A3 compared to simulation A2 or A1. Concerning the effect of the geometry, the bubble becomes larger starting from the streamlined bump in simulation A1 to the more bluff geometry in simulation C1. However, the mean position of the reattachment point in the streamwise direction is basically independent of the bump width, at $x=7$ for all three geometries at $Re=2500$ , while it clearly depends on the Reynolds number.

Figure 9. Mean axial velocity, $\langle u_{x}\rangle$ , at six positions (a–f) corresponding to the stations in figure 1. Simulations A1, B1 and C1 are represented by solid, dashed and dashed-dotted blue lines respectively. Simulations A2 and A3 are represented by solid green and red lines respectively.

The above discussion is confirmed in detail by considering the mean velocity profiles extracted from figure 8 and reported in figures 9 and 10 at the downstream positions shown in figure 1. Simulations A1, B1 and C1 are represented by solid, dashed and dashed-dotted blue lines respectively whilst simulations A2 and A3 are represented by solid green and red lines respectively. Figure 9 shows mean streamwise velocity profiles for all five simulations. For the station shown in (a), the profiles are almost independent of the Reynolds number (solid lines) while they depend strongly on the geometry (broken lines), suggesting that the mean flow has already achieved a sort of asymptotic state. The profile for case C1 extends down to the bottom wall with a slightly negative velocity, indicating a small recirculating region ahead of the bump. At the tip of the bump, (b), all the profiles essentially exhibit the same behaviour. The recirculation is already well developed at the station of (c) for the bluffest bulge, case C1. Further downstream, (d), the wall-normal extension of the backward flow are well evident for all cases except for the highest Reynolds number (case A3), where the recirculation is more attached to the wall and extends less streamwise. Since the extension of the bubble is larger for the lower Reynolds number cases, it is still present at the station corresponding to figure 9(e). In contrast, reattachment already occurred for the higher Reynolds number (cases A2–A3). Further downstream, (f), the flow is attached for all conditions.

Figure 10. Mean wall-normal velocity, $\langle u_{y}\rangle$ , at six positions (a–f) corresponding to the stations in figure 1. Simulations A1, B1 and C1 are represented by solid, dashed and dashed-dotted blue lines respectively. Simulations A2 and A3 are represented by solid green and red lines respectively.

Figure 10 shows the mean wall-normal velocity profiles. Wall-normal velocities are particularly intense as the flow invests the bump, (a). At the tip of the bump, (b), the positive (away from the bottom wall) wall-normal velocity peak is higher for the less streamlined bump (C1), indicating that the flow is strongly converging towards the opposite wall, leading to a contraction of the effective section (vena contracta). The wall-normal velocity is progressively reduced downstream, to eventually become negative. At intermediate stations, see e.g. (c), the wall-normal velocity is negative in the outer flow, indicating the trend toward reattachment to the lower wall. Approaching the wall, $\langle u_{y}\rangle$ becomes zero at the edge of the recirculation bubble. Inside the bubble $\langle u_{y}\rangle$ is positive, indicating that the profile is traversing the fore part of the bubble, recirculating clockwise. Moving further downstream, the external flow still moves toward the lower wall, but now the aft part of the bubble is reached, implying a negative wall-normal velocity also inside the bubble. Finally, the reattachment point is reached and the wall-normal average velocity tends to vanish in the entire channel section, starting to recover nearly parallel-flow conditions expected far away from the bump. The shorter bubble length leads to a more abrupt reattachment, as seen by the large negative wall-normal average velocity in the red plot of (c). The bluffest configuration induces an evident secondary recirculation bubble just below the primary one where the bump ends, see also figure 8. Note that in this figure a small bubble is also apparent just ahead of the bump.

Figure 11. Mean streamwise velocity fluctuations, $\langle u_{x}^{\prime 2}\rangle$ , at six positions (a–f) corresponding to the stations in figure 1. Simulations A1, B1 and C1 are represented by solid, dashed and dashed-dotted blue lines respectively. Simulations A2 and A3 are represented by solid green and red lines respectively.

The next figures present the mean profiles for the fluctuating quantities, i.e. $\langle u_{x}^{\prime 2}\rangle$ , $\langle u_{y}^{\prime 2}\rangle$ and $\langle u_{x}^{\prime }u_{y}^{\prime }\rangle$ , at the same stations addressed above for the mean velocity profiles. Figure 11 shows mean streamwise velocity fluctuation profiles $\langle u_{x}^{\prime 2}\rangle$ for all five simulations. These are particularly strong close to the walls or inside the shear layer above the recirculating region, for all cases. The fluctuations are maximum in the high Reynolds number case (A3) and in the most bluff case (C1). In the former, the high velocity fluctuations are due to the higher Reynolds number, whilst in the latter, the fluctuations are fed by the strong flow separation induced by the bump shape. In the lower Reynolds number simulations, the regions with higher velocity fluctuations are larger, corresponding to a larger shear layer. The presence of the bump also results in an increase in fluctuations at the top wall, particularly evident for the lower Reynolds number simulations. The change in geometry also affects the maxima reached by the fluctuations. In particular, due to the increased size of the recirculating region, simulation C1 exhibits peak values which are comparable to simulation A3 in the shear layer, see (c). In correspondence of the recirculating region, the peaks are higher for case C1 with respect to the cases at the same Reynolds number, B1 and A1. The profiles for the wall-normal velocity fluctuations $\langle u_{y}^{\prime 2}\rangle$ and the Reynolds stress, $\langle u_{x}^{\prime }u_{y}^{\prime }\rangle$ , follow a similar trend. The profiles are shown in figures 12 and 13 for all five simulations. The positive values of Reynolds stress confirm the presence of the small recirculation bubble ahead of the bump, figure 13(a).

Figure 12. Mean wall-normal velocity fluctuations, $\langle u_{y}^{\prime 2}\rangle$ , at six positions (a–f) corresponding to the stations in figure 1. Simulations A1, B1 and C1 are represented by solid, dashed and dashed-dotted blue lines respectively. Simulations A2 and A3 are represented by solid green and red lines respectively.

Figure 13. Reynolds stress, $\langle u_{x}^{\prime }u_{y}^{\prime }\rangle$ , at six positions (a–f) corresponding to the stations in figure 1. Simulations A1, B1 and C1 are represented by solid, dashed and dashed-dotted blue lines respectively. Simulations A2 and A3 are represented by solid green and red lines respectively.

Figure 14. Dependence of the reattachment point normalised with the height of the obstacle $x_{r}/h_{b}$ , on the Reynolds number based on $h_{b}$ , $Re_{h_{b}}=Re\,h_{b}$ . $h_{b}$ is the bump height normalised with the nominal channel half-height, $h_{0}$ . Closed black symbols refer to experimental measurements, open black symbols to numerical simulations (mainly LES). Red dashed line is the exponential fit of the experimental data (Kähler et al. Reference Kähler, Scharnowski and Cierpka2016). Open red symbols are extracted from the present DNS (Mellen et al. Reference Mellen, Frölich, Rodi, Deville and Owens2000; Temmerman et al. Reference Temmerman, Leschziner, Mellen and Fröhlich2003; Fröhlich et al. Reference Fröhlich, Mellen, Rodi, Temmerman and Leschziner2005; Peller & Manhart Reference Peller and Manhart2006; Šarić et al. Reference Šarić, Jakirlić, Breuer, Jaffrézic, Deng, Chikhaoui, Fröhlich, Von Terzi, Manhart and Peller2007; Hickel et al. Reference Hickel, Kempe and Adams2008; Breuer et al. Reference Breuer, Peller, Rapp and Manhart2009; Rapp & Manhart Reference Rapp and Manhart2011; Diosady & Murman Reference Diosady and Murman2014; Kähler et al. Reference Kähler, Scharnowski and Cierpka2016).

The position of the reattachment point clearly does not depend on the dimension of the obstacle in the streamwise direction, but on the Reynolds number. These observations are in agreement with both numerical and experimental data already available in the literature. Figure 14 shows the reattachment point normalised with the height of the bump, $x_{r}/h_{b}$ as a function of the Reynolds number based on the bulk velocity and on the height of the bump, $Re_{h_{b}}=Re\,h_{b}$ (note that $h_{b}$ is the dimensionless bump height). Data have been collected in Kähler et al. (Reference Kähler, Scharnowski and Cierpka2016) from several experiments (closed symbols) and numerical simulations, mainly LES, (open symbols), whilst the open red circles are the present simulations.

Note that the geometries in these experiments and simulations only qualitatively corresponds to ours. We therefore focus the comparison on the re-attachment point whose position is mainly controlled by bump height and Reynolds number, quite independently of the detailed geometry. The red dashed line in the figure is a power-law fit, based on all experimental results, which asymptotically reaches $3.71\,x/h_{b}$ for large Reynolds numbers. The reattachment location scales with $Re_{h_{b}}$ to the power $-1.4$ . The reattachment position moves further upstream with increasing Reynolds number due to the stronger turbulent mixing, i.e. due to a higher turbulent momentum transfer towards the wall. The present DNS data agree well with the experimental results of Kähler et al. (Reference Kähler, Scharnowski and Cierpka2016) and of Rapp & Manhart (Reference Rapp and Manhart2011). Overall, this compilation of data shows a significant scatter of data obtained by turbulence modelling and a certain inability of the models to capture the bubble reattachment position, see also the discussion in Kähler et al. (Reference Kähler, Scharnowski and Cierpka2016) for more details. Concerning the present DNS, the slight differences with the experiments can be attributed to details of the turbulence investing the bump and the confinement effect of the upper wall.

3.3 Pressure, drag and friction coefficients

The instantaneous pressure is decomposed as the sum of a contribution linearly decreasing in the streamwise direction, which is associated with the instantaneous pressure drop $\unicode[STIX]{x0394}p(t)$ across the channel, plus a departure $\tilde{p}$ from the linear law,

(3.1) $$\begin{eqnarray}\displaystyle p(\boldsymbol{x},t)=-\frac{\unicode[STIX]{x0394}p}{L_{x}}x+\tilde{p}(\boldsymbol{x},t). & & \displaystyle\end{eqnarray}$$

In the present simulations $\unicode[STIX]{x0394}p(t)$ does not significantly fluctuate in time, oscillating within 1 % at most, although its value is in principle continuously adjusted to keep the flow rate rigorously constant. The drag coefficient in terms of the present dimensionless variables is

(3.2) $$\begin{eqnarray}\displaystyle C_{d}=4\frac{\unicode[STIX]{x0394}P}{L_{x}}=-\frac{2}{L_{x}}\int _{walls}\langle \boldsymbol{t}\rangle \boldsymbol{\cdot }\boldsymbol{e}_{x}\,\text{d}l, & & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x0394}P=\langle \unicode[STIX]{x0394}p\rangle$ is the average pressure drop, $\boldsymbol{t}$ is the (dimensionless) traction at the wall (pressure plus shear force) and $\boldsymbol{e}_{x}$ is the unit vector in the streamwise direction. Wherever needed (see e.g. § 3.4.2) the fluctuation of the pressure drop will be denoted by $\unicode[STIX]{x0394}p^{\prime }(t)$ .

The drag increases moving from the most streamlined (A1) to the most bluff profile (C1), see table 2. On the other hand, the drag coefficient decreases with the increase in Reynolds number, from simulation A1, A2 to A3. For purpose of comparison, the drag in a planar channel at the same flow rate is $C_{d}^{channel}=4(Re_{\unicode[STIX]{x1D70F}}^{0}/Re_{b})^{2}$ . The drag coefficient can be decomposed in form and friction components, namely

(3.3a,b ) $$\begin{eqnarray}\displaystyle C_{d}^{form}=\frac{2}{L_{x}}\boldsymbol{e}_{x}\boldsymbol{\cdot }\int _{walls}P\boldsymbol{n}\,\text{d}l,\quad C_{d}^{friction}=-\frac{2}{L_{x}}\boldsymbol{e}_{x}\boldsymbol{\cdot }\int _{walls}\unicode[STIX]{x1D707}\,\frac{\unicode[STIX]{x2202}\boldsymbol{U}}{\unicode[STIX]{x2202}n}\,\text{d}l, & & \displaystyle\end{eqnarray}$$

where $\boldsymbol{n}$ is the unit normal exiting the fluid domain. The form drag coefficient increases by 50 % going from the most streamlined to most bluff shape. The observed decrease of the form drag coefficient with increasing Reynolds number is more than compensated by the larger velocity for flows in the same geometry implying, as obvious, the increase of the corresponding contribution to the resistance, $D^{form}\propto U_{b}^{2}C_{d}^{form}$ . In the present geometry, the friction component is dominated by the straight part of the channel and its value is not significantly different from the one expected in a planar channel, see the small difference $C_{d}^{friction}-C_{d}^{channel}$ in table 2. This confirms that most of the bump-induced drag should be interpreted as form drag.

Table 2. Drag coefficient decomposed into form and friction contributions and comparison against an equivalent planar channel. See text for definitions.

Figure 15. Average pressure $\langle p\rangle$ contour plots for all five simulations.

Given its relevance in determining the drag, the mean pressure field is shown in figure 15 for all simulations. Approximately, the qualitative behaviour is similar for all cases. A high pressure region occurs just before the bump where a stagnation point occurs. Further downstream the pressure decreases reaching its minimum in the high velocity region, at the top of the bump. The pressure field behind the bump is strongly influenced by the shape and dimension of the recirculation bubble. When the separation region is significant, cases A1, B1 and C1, a second pressure minimum develops inside the recirculation bubble. The high Reynolds number simulations where performed on the most streamlined geometry, which produces a smaller separation compared to the other geometries. With increasing Reynolds number, cases A2 and A3, the flow more easily faces the adverse pressure gradient due to enhanced turbulent mixing, resulting in delayed separation. As a consequence, pressure recovery behind the bump is more effective.

Figure 16. Modified pressure coefficient at the top and bottom walls of the domain, (a) and (b) respectively. Simulations A1, B1 and C1 are represented by solid, dashed and dashed-dotted blue lines respectively. Simulations A2 and A3 are represented by solid green and red lines respectively.

Given the average pressure drop along the channel, the effect of the bump on the wall pressure is better addressed in terms of a departure-pressure coefficient

(3.4) $$\begin{eqnarray}\displaystyle C_{p}^{t/b}(x)=2\langle \tilde{p}\rangle |_{y=2/y=0}, & & \displaystyle\end{eqnarray}$$

where the superscripts ‘ $t$ ’ and ‘ $b$ ’ refer to the top and bottom wall respectively. Adding the linear term $-2x\unicode[STIX]{x0394}P/L_{x}$ recovers the standard definition of $C_{p}$ . Figure 16 shows the departure pressure at both walls. The effect of the bump extends to the opposite wall, with $C_{p}^{t}$ presenting a trough just after the bump tip (at $x=4$ ) and recovering downstream. The trough is smaller and shifted closer to the bump tip with increasing Reynolds number. At constant Reynolds number, the trough is more pronounced and farther away from the tip when the geometry is bluffer, i.e. simulations B1 and C1. At the bottom wall, as the flow reaches the bump, $C_{p}^{b}$ initially increases producing a small recirculation just ahead of the bump. The pressure then decreases to its minimum slightly ahead of the tip. These trends become stronger for the bluffer geometries. Immediately downstream of the tip, the pressure abruptly rises reaching the separation point. In the recirculation bubble the pressure remains almost constant, with the extension of the plateaux becoming smaller at increasing Reynolds number (red line). The extension of the plateaux is almost independent of the geometry, where $C_{p}^{b}$ decreases for bluffer bumps.

Figure 17. Mean skin friction coefficient along $x$ at the top and bottom walls of the domain in (a) and (b) respectively. Simulations A1, B1 and C1 are represented by solid, dashed and dashed-dotted blue lines respectively. Simulations A2 and A3 are represented by solid green and red lines respectively.

Figure 17 shows the mean skin friction coefficient $C_{f}=2\unicode[STIX]{x1D70F}_{w}$ . At the top wall, $C_{f}$ is always positive, showing that the adverse pressure gradient (see figure 16) is too mild to induce average flow separation on the wall opposite to the bump. The maximum and minimum $C_{f}$ occur just before the bump tip and at the end of the bubble, respectively. At the bottom wall, the skin friction coefficient ahead of the bump becomes slightly negative due to the small recirculation bubble and reaches a positive peak as the flow approaches the tip of the bump. For the different Reynolds numbers, the position of the peak coincides but the maximum reduces with increasing Reynolds number. The peak is shifted towards the tip with increasing bluffness of geometry. A skin friction plateaux is observed behind the tip, along the bump where the cross-stream section of the channel increases. At the bubble, consistently with the backward flow at the wall, the skin friction is negative, with increasing absolute value for bluffer geometries and lower Reynolds numbers.

3.4 Mean kinetic energy and turbulent kinetic energy budgets

The Reynolds decomposition entails the splitting of the total kinetic energy into two parts, $K=K_{M}+k_{T}$ , where $K_{M}=1/2\,\langle u_{i}\rangle \langle u_{i}\rangle$ is the kinetic energy of the mean flow and $k_{T}=1/2\langle u_{i}^{\prime }u_{i}^{\prime }\rangle$ is the turbulent kinetic energy. In the literature, little attention is typically paid to the kinetic energy of the mean field and most interest is focused on the turbulent contribution. This is motivated by the usually simple flow configuration where the mean balance equation for $K_{M}$ is trivial. In the present case both mean and turbulent kinetic energy need to be dealt with explicitly. The reason is that the bump breaks the streamwise homogeneity and induces strong mean wall-normal velocities. This gives rise to non-trivial mean and turbulent spatial energy fluxes, dissipation and turbulent kinetic energy production.

3.4.1 Mean kinetic energy

The (stationary) mean flow kinetic energy equation reads

(3.5) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}{\unicode[STIX]{x1D6F7}_{M}}_{j}}{\unicode[STIX]{x2202}x_{j}}=-\unicode[STIX]{x1D700}_{M}-\unicode[STIX]{x1D6F1}+\frac{\unicode[STIX]{x0394}P}{L_{x}}U_{x},\end{eqnarray}$$

where $\unicode[STIX]{x1D700}_{M}=1/Re(\unicode[STIX]{x2202}U_{i}/\unicode[STIX]{x2202}x_{j})(\unicode[STIX]{x2202}U_{i}/\unicode[STIX]{x2202}x_{j})$ is the mean flow energy dissipation rate per unit volume and $\unicode[STIX]{x1D6F1}=-\langle u_{i}^{\prime }u_{j}^{\prime }\rangle \unicode[STIX]{x2202}U_{i}/\unicode[STIX]{x2202}x_{j}$ is the turbulent kinetic energy production. $U_{x}\unicode[STIX]{x0394}P/L_{x}$ is the external average power input. The spatial flux,

(3.6) $$\begin{eqnarray}{\unicode[STIX]{x1D6F7}_{M}}_{j}=U_{j}K_{M}+U_{j}\langle \tilde{p}\rangle -\frac{1}{Re}\frac{\unicode[STIX]{x2202}K_{M}}{\unicode[STIX]{x2202}x_{j}}+U_{i}\langle u_{i}^{\prime }u_{j}^{\prime }\rangle ,\end{eqnarray}$$

redistributes energy across the flow, overall providing zero net contribution to the power.

Figure 18 shows the terms in (3.5) normalised with the total average power injection per unit volume, i.e. the total dissipation rate, $\int (\unicode[STIX]{x1D700}_{M}+\unicode[STIX]{x1D700}_{T})\,\text{d}V$ where $\unicode[STIX]{x1D700}_{T}=\langle 1/Re(\unicode[STIX]{x2202}u_{i}^{\prime }/\unicode[STIX]{x2202}x_{j})(\unicode[STIX]{x2202}u_{i}^{\prime }/\unicode[STIX]{x2202}x_{j})\rangle$ is the turbulent dissipation rate density. In the figure, the turbulent kinetic energy production rate, $-\unicode[STIX]{x1D6F1}$ , is shown in the background colour plot whilst solid isolines (mostly concentrated near the bump wall) represent the mean field dissipation, $\unicode[STIX]{x1D700}_{M}$ . Vectors correspond to the spatial flux $\unicode[STIX]{x1D6F7}_{Mj}$ .

Given the behaviour of the mean streamwise velocity, see figure 8, the mean energy input, $U_{x}\unicode[STIX]{x0394}P/L_{x}$ , is largest at the bump tip. On the other hand, the production $-\unicode[STIX]{x1D6F1}$ , is concentrated in the detaching shear layer well behind the bump, where the largest fluctuation intensities are attained, as discussed in the previous section. This region acts as a sink of mean energy and is fed by the mean energy flux, ${\unicode[STIX]{x1D6F7}_{M}}_{j}$ , that is crucial in redistributing energy from the external input to the turbulent production. With respect to case A1, taken as a basis for comparison, the maximum turbulent production increases by almost 50 % for the bluffer geometry and by 400 % at the maximum Reynolds number. By definition, turbulent production is the product of mean flow gradients and Reynolds stresses. For the given geometry, the mean gradients in the shear layer slightly depend on Reynolds number, as shown in figure 9(c) which corresponds to the section of maximum production. This suggests that the mean field already attained an almost Reynolds independent state. On the other hand, turbulent stresses increase significantly at this section, see figure 13(c), resulting in the increased peak production apparent in figure 18. In general, the position of the energy production region depends, through the shear layer, on the dimensions and the position of the separation bubbles. Changing geometry at fixed, lower Reynolds number, the strength of the mean gradients in the same section, now in figure 9(d), are only marginally affected by the change in geometry. On the other hand, the Reynolds stresses are greatly enhanced passing from a streamlined to a bluff configuration, figure 13(d), consistent with the increasing peak energy production from case A1 to case C1 in figure 18.

Figure 18. Mean kinetic energy balance equation: turbulent kinetic energy production $-\unicode[STIX]{x1D6F1}$ (background colour), mean energy dissipation $\unicode[STIX]{x1D700}_{M}$ (solid isolines) and mean energy spatial flux $\unicode[STIX]{x1D6F7}_{M}$ (vectors).

Although hardly apparent in figure 18, for the considered cases the mean flow dissipation rate is not irrelevant, and contributes of order 40 % of the total dissipation in the system, consistently with significant mean velocity gradients, observed at the bump wall where the flow is abruptly accelerated.

Figure 19. Turbulent kinetic energy balance equation: turbulent kinetic energy production $\unicode[STIX]{x1D6F1}$ (background colour), turbulent energy dissipation $\unicode[STIX]{x1D700}_{M}$ (solid isolines) and turbulent energy spatial flux $\unicode[STIX]{x1D6F7}_{M}$ (vectors).

Figure 20. Norm of the deviatoric component of the Reynolds stress tensor, $\Vert \unicode[STIX]{x1D657}\Vert =\sqrt{\unicode[STIX]{x1D623}_{ij}\unicode[STIX]{x1D623}_{ij}}$ with $\unicode[STIX]{x1D623}_{ij}=\langle u_{i}^{\prime }u_{j}^{\prime }\rangle /\langle u_{k}^{\prime }u_{k}^{\prime }\rangle -1/3\,\unicode[STIX]{x1D6FF}_{ij}$ , and $\unicode[STIX]{x1D6FF}_{ij}$ the components of the identity tensor.

3.4.2 Turbulent kinetic energy

The balance equation for the turbulent kinetic energy reads

(3.7) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6F7}_{Tj}}{\unicode[STIX]{x2202}x_{j}}=-\unicode[STIX]{x1D700}_{T}+\unicode[STIX]{x1D6F1}+\left\langle \frac{\unicode[STIX]{x0394}p^{\prime }(t)}{L_{x}}u_{x}^{\prime }\right\rangle ,\end{eqnarray}$$

where, as anticipated, $\unicode[STIX]{x1D700}_{T}$ is the turbulent kinetic energy dissipation rate, $\unicode[STIX]{x1D6F1}$ , here with the opposite sign with respect to (3.5), is the production and $\langle \unicode[STIX]{x0394}p^{\prime }\,u_{x}^{\prime }/L_{x}\rangle$ is the external source of fluctuating energy. The spatial flux,

(3.8) $$\begin{eqnarray}\unicode[STIX]{x1D6F7}_{Tj}=U_{j}k_{T}+\frac{1}{2}\langle u_{i}^{\prime }u_{i}^{\prime }u_{j}^{\prime }\rangle +\langle \tilde{p}^{\prime }u_{j}^{\prime }\rangle -\frac{1}{Re}\frac{\unicode[STIX]{x2202}k_{T}}{\unicode[STIX]{x2202}x_{j}},\end{eqnarray}$$

contributes zero net power when integrated over the whole domain. The energy locally provided by the fluctuations of pressure difference between inlet and outlet $\langle \unicode[STIX]{x0394}p^{\prime }\,u_{x}^{\prime }/L_{x}\rangle$ is negligible, $\max _{x,y}\langle \unicode[STIX]{x0394}p^{\prime }\,u_{x}^{\prime }/L_{x}\rangle \simeq 10^{-5}\,\max _{x,y}\unicode[STIX]{x1D6F1}$ .

Figure 19 shows the turbulent kinetic energy production, turbulent energy dissipation and spatial fluxes for all the simulations. The terms are normalised with the overall power injected in the system which, in the statistically steady state, is balanced by the total dissipation rate, $\int (\unicode[STIX]{x1D700}_{M}+\unicode[STIX]{x1D700}_{T})\,\text{d}V$ . The production term injects most energy in the shear layer behind the bump. From the shear layer the energy follows different paths, see the vector field in the figure where local energy release is associated with the (positive) divergence of the energy flux. The turbulent energy is transferred towards the centre of the channel, into the separation bubble or towards the wall, in particular behind the bump under the separation bubble. From the analysis of the dissipation field, $\unicode[STIX]{x1D700}_{T}$ , part of the energy is found to be locally dissipated in the shear layer and in the separation bubble. Most of the energy is dissipated at the bottom wall after the bump (note the isolines of dissipation concentrated in that region).

Figure 21. Norm of the deviatoric component of the pseudo dissipation tensor, $\Vert \unicode[STIX]{x1D659}\Vert =\sqrt{\unicode[STIX]{x1D625}_{ij}\unicode[STIX]{x1D625}_{ij}}$ , where $\unicode[STIX]{x1D625}_{ij}=\unicode[STIX]{x1D716}_{ij}/\unicode[STIX]{x1D716}_{kk}-1/3\,\unicode[STIX]{x1D6FF}_{ij}$ and $\unicode[STIX]{x1D716}_{ij}=2/Re\langle (\unicode[STIX]{x2202}u_{i}^{\prime }/\unicode[STIX]{x2202}x_{k})(\unicode[STIX]{x2202}u_{j}^{\prime }/\unicode[STIX]{x2202}x_{k})\rangle$ is the pseudo dissipation tensor.

3.4.3 Large- and small-scale anisotropy

The anisotropy of the large turbulent scales is described by the deviatoric component of the Reynolds stress, $\unicode[STIX]{x1D623}_{ij}=\langle u_{i}^{\prime }u_{j}^{\prime }\rangle /\langle u_{k}^{\prime }u_{k}^{\prime }\rangle -1/3\unicode[STIX]{x1D6FF}_{ij}$ , where $\unicode[STIX]{x1D6FF}_{ij}$ denotes the Kronecker symbol. Note that in isotropic conditions, $\unicode[STIX]{x1D623}_{ij}$ is identically zero. An overall measure of anisotropy is given by the norm $\Vert \unicode[STIX]{x1D657}\Vert =\sqrt{\unicode[STIX]{x1D623}_{ij}\unicode[STIX]{x1D623}_{ij}}$ , figure 20. The anisotropy is particularly significant in the near-wall region and in the shear layer, while the bulk flow and the recirculation bubble are almost isotropic, consistent with the Reynolds stress profiles of figures 11–13. Increasing the Reynolds number, the anisotropic regions become progressively smaller, squeezed closer to the wall, on one side, and more concentrated in the shear layer, on the other. The extension of the isotropic region in the bulk widens, whilst it shrinks with the recirculation bubble in the separated region. As a result of the change in geometry, the bluffest bump produces the highest anisotropic content.

Figure 21 reports the norm, $\Vert \unicode[STIX]{x1D659}\Vert =\sqrt{\unicode[STIX]{x1D625}_{ij}\unicode[STIX]{x1D625}_{ij}}$ , of the deviatoric component, $\unicode[STIX]{x1D625}_{ij}=\unicode[STIX]{x1D716}_{ij}/\unicode[STIX]{x1D716}_{kk}-1/3\,\unicode[STIX]{x1D6FF}_{ij}$ , of the pseudo-dissipation tensor, $\unicode[STIX]{x1D716}_{ij}=2/Re\langle (\unicode[STIX]{x2202}u_{i}^{\prime }/\unicode[STIX]{x2202}x_{k})(\unicode[STIX]{x2202}u_{j}^{\prime }/\unicode[STIX]{x2202}x_{k})\rangle$ . $\Vert \unicode[STIX]{x1D659}\Vert$ provides a measure of the small-scale anisotropy content (Antonia, Djenidi & Spalart Reference Antonia, Djenidi and Spalart1994; Pumir, Xu & Siggia Reference Pumir, Xu and Siggia2016) and therefore, as $\Vert \unicode[STIX]{x1D659}\Vert$ approaches zero, isotropic behaviour of the smallest scales is achieved. The small scales in the recirculation bubble and in the bulk of the flow are isotropic, consistently with the isotropy of the large scales in the same regions. Strong anisotropy persists in the near-wall regions and in the shear layer. The behaviour of $\Vert \unicode[STIX]{x1D659}\Vert$ is strongly dependent on the Reynolds number which ultimately sets the separation between the largest and the smallest scales. The regions of small-scale isotropy progressively increase with the Reynolds number, basically due to the shrinking of the large-scale anisotropy regions. However, anisotropy still persists at small scales, irrespective of the Reynolds number, near the walls and in the shear layer. This behaviour can be explained and understood by addressing the dynamics of a turbulent flow in presence of strong shear. In isotropic conditions the turbulence is forced at the largest scales comparable with the integral scale $L_{0}=(2k_{T})^{3/2}/\unicode[STIX]{x1D716}_{T}$ and is dissipated by viscosity at the Kolmogorov scale $\unicode[STIX]{x1D702}=(\unicode[STIX]{x1D708}^{3}/\unicode[STIX]{x1D716}_{T})^{1/4}$ . In the inertial range ( $\unicode[STIX]{x1D702}\ll r\ll L_{0}$ ) the energy is simply transferred from the large to the small scales. In turbulent shear flows the shear scale $L_{S}=\sqrt{\unicode[STIX]{x1D716}_{T}/S^{3}}$ , extensively discussed in Casciola et al. (Reference Casciola, Gualtieri, Benzi and Piva2003), where $S$ is the shear rate, plays a crucial role in explaining the dynamics. Basically the shear scale identifies the range of scales $L_{S}<r<L_{0}$ where the turbulence is driven by the (anisotropic) production of turbulent kinetic energy due to the Reynolds stresses. In the range of scales below $L_{S}$ , $\unicode[STIX]{x1D702}<r<\unicode[STIX]{x0141}_{S}$ , the dynamics of the turbulent fluctuations is driven by the process of energy cascade typical of isotropic flows, see e.g. Marati, Casciola & Piva (Reference Marati, Casciola and Piva2004), Cimarelli, De Angelis & Casciola (Reference Cimarelli, De Angelis and Casciola2013) for a detailed analysis of the energy paths in a planar channel. It follows that the dynamics of a shear flow is described by two dimensionless parameters. The first one is the shear intensity $S^{\ast }=S(2k_{T})/\unicode[STIX]{x1D716}_{T}$ , see e.g. Lee, Kim & Moin (Reference Lee, Kim and Moin1990), that can be recast in terms of the shear scale as $S^{\ast }=(L_{0}/L_{S})^{2/3}$ , (Casciola et al. Reference Casciola, Gualtieri, Jacob and Piva2007). The shear intensity measures the separation between the shear scale and the integral scale thus providing the extension of the range of scales directly affected by the production mechanisms. The second parameter is the Corrsin parameter $S_{c}=S\sqrt{\unicode[STIX]{x1D708}/\unicode[STIX]{x1D716}_{T}}$ (Corrsin Reference Corrsin1958) that can be recast in terms of the shear scale as $S_{c}=(\unicode[STIX]{x1D702}/L_{S})^{2/3}$ . The Corrsin parameter measures the extension of the range between the shear scale and the Kolmogorov scale where the flow is driven by the inertial cascade. Clearly, only in the range of scales below the shear-scale isotropisation of turbulent fluctuations can take place. The shear scale can be evaluated in spatially non-homogeneous flows by considering the norm of the local mean velocity gradient $L_{S}=\sqrt{\unicode[STIX]{x1D716}_{T}/(\unicode[STIX]{x2202}_{j}U_{i}\unicode[STIX]{x2202}_{j}U_{i})^{3/2}}$ . In our case the shear scale is a field $L_{S}(x,y)$ . In a similar way the local integral scale $L_{0}=(2k_{T})^{3/2}/\unicode[STIX]{x1D716}_{T}$ and the local Kolmogorov scale $\unicode[STIX]{x1D702}=(\unicode[STIX]{x1D708}^{3}/\unicode[STIX]{x1D716}_{T})^{1/4}$ can be considered. The shear strength $S^{\ast }$ and the Corrsin parameter $S_{c}$ are position dependent. When $S^{\ast }$ is large the whole range of scales is dominated by production and there is no room left for isotropy recovery at small scales. On the contrary, an isotropy recovery range is available where the Corrsin parameter is small. Figures 22 and 23 provide the fields $S^{\ast }$ and $S_{c}$ respectively, for the different Reynolds numbers and geometries considered in this paper. A joint analysis of $S^{\ast }$ and $S_{c}$ provides the physical interpretation of the observed anisotropy (figures 20 and 21). In the bulk region, $S^{\ast }$ decreases, denoting weak production of turbulent kinetic energy since the shear scale approaches the integral scale. Concurrently, $S_{c}$ is small, indicating a large separation between shear and Kolmogorov scale. This behaviour is generic and the only relevant changes are observed when the Reynolds number is increased, A1–A3. At large Reynolds number, the spatial region in the bulk where isotropisation occurs is broadened. The relative position of integral, shear and Kolmogorov scales in the bulk explains why the small scales are isotropic (figure 21). The conditions are different near the wall and in the shear layer. In these regions, see figure 22, $S^{\ast }$ is large and the whole range of scales is now dominated by turbulent kinetic energy production. Concurrently $S_{c}$ is of order one, i.e. the shear scale is forced on the Kolmogorov scale (figure 23). This behaviour is again generic for the cases we address. In a nutshell, near the wall and in the shear layer there is no room for the formation of the inertial range where isotropisation can take place. The flow is driven by the anisotropic mechanisms of turbulent kinetic energy production, see figure 20 and the anisotropy persists down to the smallest scales (figure 21).

Figure 22. Shear intensity $S^{\ast }=S(2k_{T})/\unicode[STIX]{x1D716}_{T}=(L_{0}/L_{S})^{2/3}$ in the flow domain for all simulations.

Figure 23. Corrsin parameter $S_{c}=S\sqrt{\unicode[STIX]{x1D708}/\unicode[STIX]{x1D716}_{T}}=(\unicode[STIX]{x1D702}/L_{S})^{2/3}$ in the flow domain for all simulations.

3.4.4 Invariant maps

The anisotropy invariant map (AIM) originally introduced by Lumley & Newman (Reference Lumley and Newman1977), Lumley (Reference Lumley1979) provides a description of the different anisotropic states of the large turbulent scales. They are quantified in terms of the invariants of the anisotropy tensor, i.e. the deviatoric component of the Reynolds stress, $b_{ij}$ , namely $\text{I}=b_{ii}=0$ , $\text{II}=b_{ij}\,b_{ji}$ and $\text{III}=b_{ij}\,b_{jk}\,b_{ki}$ . The admissible states of the flow must lie within a (curvilinear) triangle of the II–III plane. This constraint comes from the requirement that the eigenvalues of $b_{ij}$ should be real and the squared velocity fluctuation in the principal direction must be positive. The admissible region is delimited above by the line $\text{II}=2/9+2\text{III}$ corresponding to statistically two-dimensional turbulence, i.e. the fluctuation intensity in one of the eigen-directions vanishes. The other two limiting lines, $\text{II}=3/2(16/9\,\text{III}^{2})^{1/3}$ , represent axisymmetric turbulence, i.e. the fluctuation intensity in two eigen-directions are identical. In the left branch ( $\text{III}<0$ ), the fluctuation intensity in the third eigen-direction is smaller than the other two (pancake turbulence). In the right branch ( $\text{III}>0$ ), the third component is larger than the other two (cigar-like turbulence). The corners correspond to: the isotropic state ( $\text{II}=0,\text{III}=0$ ), the two-component isotropic state ( $\text{II}=-1/36,\text{III}=1/6$ ), and the one-component state ( $\text{II}=2/9,\text{III}=2/3$ ). In many applications, turbulence modelling exploits the idea of eddy viscosity which assumes that the Reynolds stress tensor is proportional (via the eddy-viscosity) to the mean strain rate tensor, see Speziale (Reference Speziale1991), Gatski & Speziale (Reference Gatski and Speziale1993). Modelling is particularly challenging for separated flows. The AIM is helpful to directly take into account the anisotropy of the flow, see Jovičić, Breuer & Jovanović (Reference Jovičić, Breuer and Jovanović2006), Kumar et al. (Reference Kumar, Frohnapfel, Jovanović, Breuer, Zuo, Hadzić and Lechner2009) and the general discussion in Jovanovic (Reference Jovanovic2013) where II and III are used to compute the length scale appearing in the eddy viscosity thus including anisotropic effects in the model.

Figure 24. Anisotropy invariant map for case C1 at: (a) the tip of the bump; (b) end of the bump; (c) inside the recirculation region. Note that (a–c) correspond to stations (b–d) in figure 1. The colour legend represents the $y$ coordinate.

Figure 24 shows the AIM for simulation C1 at three stations. At the tip of the bump, (a), close to the bump wall at $y=0.5$ , turbulence is essentially two-dimensional. As $y$ increases, the points in the plot approach the lower left branch indicating axisymmetric turbulence. As expected, close to the centreline of the channel, the flow becomes isotropic. As the top wall is approached, the flow follows a trend similar to what is found in a planar channel (Gilbert & Kleiser Reference Gilbert, Kleiser, Braza and Nogues1991) becoming two-dimensional again (red points) at the top wall. However, in our case due to the adverse pressure gradient at the top wall the trajectory followed in the map by the orange/red points is shifted away from the right branch, departing from the axisymmetric state typical of the planar channel flow. Figure 24(b) corresponds to the end of the bump which exhibits a more complex AIM. As the $y$ coordinate increases, the initial axisymmetric state is hardly reached and the trajectory in the II–III plane follows an inner path towards the opposite axisymmetric state ( $\text{III}>0$ ) as the shear layer is crossed (light blue). At the centreline, the flow is again isotropic and follows an inner path towards the top wall. These results are typical of separated flows as found, e.g. in the backward facing step configuration (Le et al. Reference Le, Moin and Kim1997). Figure 24(c) shows the trajectory at a station closer to the reattachment point. Flow close to the lower wall is completely axisymmetric ( $\text{III}<0$ ) until the shear layer is reached (light blue). The turbulence shifts from axisymmetric contraction to axisymmetric expansion, similar to (b). The flow is isotropic at the centreline and follows the same trend discussed for (a) and (b) as the top wall is reached. The analysis of the AIM suggests that the flow we are addressing is rather complex to model, due to the presence of the recirculating region behind the bump and the adverse pressure gradient along the top wall.