2 Simulations

Our work deals with a non-planar incompressible turbulent channel flow, studied via DNS. One of the channel walls is flat, and the other has a relatively small two-dimensional bump. Two bump profiles with the same height are considered, to produce an attached and a separated flow. Streamwise-travelling waves for the reduction of frictional drag are imposed at the lower non-planar wall, and their effect on the total drag is measured.

The DNS code, introduced by Luchini (Reference Luchini2016), solves the incompressible Navier–Stokes equations written in primitive variables on a staggered Cartesian grid. Second-order finite differences are used in every direction. The momentum equations are advanced in time by a fractional time stepping method using a third-order Runge–Kutta scheme. The Poisson equation for the pressure is solved by an iterative successive over-relaxation (known as SOR) algorithm. The non-planar wall is dealt with via an implicit immersed-boundary method, implemented in staggered variables to be continuous with respect to boundary crossing and numerically stable at all distances from the boundary (Luchini Reference Luchini2013, Reference Luchini2016).

Figure 1.

Sketch of the computational domain and the reference system. The bump is on the lower wall. The streamwise-periodic upstream domain (black) provides an inflow condition for the downstream one (blue).

The computational domain (a sketch is shown in figure 1, with the bump on the lower wall) is made by two streamwise-adjacent portions of similar length: the upstream volume with planar walls is streamwise-periodic, and feeds the downstream one where inflow and outflow conditions are used. Periodic conditions are used everywhere for the spanwise direction, and no-slip and no-penetration are enforced on the walls. The outflow condition extrapolates the velocity components according to

(2.1)$$\begin{eqnarray}{\displaystyle \frac{\unicode[STIX]{x2202}u_{i}}{\unicode[STIX]{x2202}t}}+U_{c}{\displaystyle \frac{\unicode[STIX]{x2202}u_{i}}{\unicode[STIX]{x2202}x}}=0,\quad i=1,2,3,\end{eqnarray}$$

where $U_{c}(z)$ is the profile of the mean convection velocity of turbulent fluctuations defined as in Quadrio & Luchini (Reference Quadrio and Luchini2003), and implemented as the mean velocity profile of the plane channel flow, modified in the near-wall region to have $U_{c}^{+}(z)\geqslant 10$. Alternative outflow conditions have been tested, finding negligible differences in the results.

The simulations are carried out at a bulk Reynolds number $Re_{b}=U_{b}h/\unicode[STIX]{x1D708}=3173$ which in the reference case corresponds to a friction Reynolds number of $Re_{\unicode[STIX]{x1D70F}}=u_{\unicode[STIX]{x1D70F}}h/\unicode[STIX]{x1D708}=200$ in the plane channel. In their definition, the length scale is $h$, half the distance between the plane walls, whereas the velocity scale is the bulk velocity $U_{b}$ in the former case and the friction velocity $u_{\unicode[STIX]{x1D70F}}$ in the latter. Unless otherwise noted (e.g. with the plus notation indicating viscous units), in the following, quantities are made dimensionless with $h$ and $U_{b}$.

The size of the computational domain is $(L_{x}^{p}+L_{x}^{np},L_{y},L_{z})=(24.56,\unicode[STIX]{x03C0},2)$ in the streamwise, spanwise and wall-normal directions, respectively. The flat walls are placed at $z=0$ and $z=2$. The upstream periodic portion, with streamwise length of $L_{x}^{p}=4\unicode[STIX]{x03C0}$, runs a standard channel flow DNS where the constant flow rate (known as CFR) condition is imposed (Quadrio, Frohnapfel & Hasegawa Reference Quadrio, Frohnapfel and Hasegawa2016), and has a spatial resolution of $(n_{x},n_{y},n_{z})=(360,312,241)$ discretization points. The downstream portion of the computational domain starts at $x=0$ with a length of $L_{x}^{np}=12$, over which 800 discretization points are non-uniformly distributed; grid and domain sizes in the spanwise and wall-normal directions are the same of the upstream domain, to avoid interpolation.

The grid is tuned for optimal use of computational resources while providing the necessary spatial resolution and smooth description of the bump geometry via the immersed-boundary method. The spanwise grid spacing is uniform at $\unicode[STIX]{x0394}y=0.01$; it corresponds to $\unicode[STIX]{x0394}y^{+}=2$ based on the inlet $u_{\unicode[STIX]{x1D70F}}$, and to $\unicode[STIX]{x0394}y^{+}=4$ close to the bump tip, where friction velocity is maximum. Streamwise resolution is uniform at $\unicode[STIX]{x0394}x=0.04$ or $\unicode[STIX]{x0394}x^{+}=8$ in the periodic part, but increases as the bump is approached, reaching up to $\unicode[STIX]{x0394}x^{+}=2$ (based on local $u_{\unicode[STIX]{x1D70F}}$). The wall-normal spacing is neither constant in $z$ nor symmetrical with respect to the centreline, since the bump is present on one wall only. A constant $\unicode[STIX]{x0394}z=0.001$ is adopted from the lower wall to $z=h_{b}$, where $h_{b}$ is the maximum bump height, and corresponds to $\unicode[STIX]{x0394}z^{+}=0.2$, based on the inlet $u_{\unicode[STIX]{x1D70F}}$. Then $\unicode[STIX]{x0394}z$ gradually increases until, at the centreline, the maximum value of $\unicode[STIX]{x0394}z=0.02$ is reached. The spacing then decreases again in the upper half of the channel, to reach $\unicode[STIX]{x0394}z=0.004$ at the upper wall. Overall, the largest streamwise spacing is six times the local Kolmogorov length $\unicode[STIX]{x1D702}$, and the wall-normal and spanwise spacings are everywhere less than $2\unicode[STIX]{x1D702}$. Near the bump the resolution is even higher; in the recirculation zone, the smallest dissipative scales are well resolved, with spacing in every direction equal to or lower than $\unicode[STIX]{x1D702}$.

Figure 2.

Bump geometries $G_{1}$ (blue line) and $G_{2}$ (red dashed line); they are identical up to the bump tip. Both have height of $h_{b}=0.0837$ (only a portion of the streamwise extent is shown; note the enlarged vertical axis). Here, $G_{1}$ leads to a mildly separated flow, while $G_{2}$ produces an attached flow.

The geometry of the bump, which is located on the lower wall, is two-dimensional and similar to the one considered by Marquillie et al. (Reference Marquillie, Laval and Dolganov2008), but with significantly smaller size, to reduce blockage and produce nearly undisturbed flow at the inlet and outlet sections. To enable reproducibility, the bump is analytically specified as the sum of two overlapping Gaussian curves, resulting in a smooth profile described by six parameters

(2.2)$$\begin{eqnarray}G_{1}(x)=a\exp \left[-\left({\displaystyle \frac{x-b}{c}}\right)^{2}\right]+a^{\prime }\exp \left[-\left({\displaystyle \frac{x-b^{\prime }}{c^{\prime }}}\right)^{2}\right].\end{eqnarray}$$

The parameters values chosen for the geometry $G_{1}$ are $a=0.0505,b=4,c=0.2922$ and $a^{\prime }=0.060425,b^{\prime }=4.36,c^{\prime }=0.3847$; they produce a bump with height $h_{b}=0.0837$. A second geometry $G_{2}$ is identical to $G_{1}$ in the fore part up to the tip, but a streamwise expansion factor of 2.5 is applied to the rear part. Both $G_{1}$ and $G_{2}$ are shown in figure 2, with the former geometry producing a mildly separated flow, and the latter a fully attached flow.

In terms of computational procedures, after reaching statistical equilibrium flow statistics are accumulated over a simulation time of $T=1000$. The time step is set at $\unicode[STIX]{x0394}t=1.5\times 10^{-3}$, corresponding to an average Courant–Friedrichs–Lewy (known as CFL) number of approximately 0.5. Simulations are carried out with and without StTW for the reduction of skin friction. This specific drag reduction technique, introduced by Quadrio et al. (Reference Quadrio, Ricco and Viotti2009), has been selected because of its interesting energetic properties, its large effect on the turbulent friction and the availability of successful experimental implementations starting from Auteri et al. (Reference Auteri, Baron, Belan, Campanardi and Quadrio2010). It should also be mentioned that a preliminary version of this work employed another spanwise forcing technique made by stationary waves, and the main findings were the same. The StTW are applied at the lower wall only, but including the periodic upstream part. The forcing translates into a non-homogeneous boundary condition for the spanwise velocity component at the wall as follows:

(2.3)$$\begin{eqnarray}V_{w}(x,t)=A\sin (\unicode[STIX]{x1D705}_{x}x-\unicode[STIX]{x1D714}t),\end{eqnarray}$$

where $V_{w}$ is the spanwise velocity at the wall, $A$ is its maximum amplitude, and $\unicode[STIX]{x1D705}_{x}$ and $\unicode[STIX]{x1D714}$ represent the spatial and temporal frequencies of the wave. The wall forcing produces a sinusoidal distribution of spanwise velocity which travels in the streamwise direction. The numerical values of the forcing parameters are chosen, based on existing information (e.g. Gatti & Quadrio Reference Gatti and Quadrio2016), to guarantee large amounts of skin-friction drag reduction in the plane channel. The selected values $A=0.75$, $\unicode[STIX]{x1D714}=\unicode[STIX]{x03C0}/10$ and $\unicode[STIX]{x1D705}_{x}=2$ yield 46 % of drag reduction in a plane channel at $Re_{\unicode[STIX]{x1D70F}}=200$. Owing to the rather large actuator intensity, the total power budget is only mildly positive, with 11 % of net power savings.

3 Instantaneous and mean flow fields

Figure 3.

Isosurfaces of $\unicode[STIX]{x1D706}_{2}^{+}=-0.04$ for an instantaneous flow field in the reference case. Isosurfaces are colour-coded with the coordinate $z$.

To begin with a qualitative picture of the flow, figure 3 portrays the appearance of turbulent vortical structures over the shorter bump $G_{1}$, that will be the focus of this section. The figure is for the reference simulation without StTW, and plots isosurfaces of the intermediate eigenvalue $\unicode[STIX]{x1D706}_{2}$ of the velocity gradient tensor (Jeong & Hussain Reference Jeong and Hussain1995), colour-coded with the coordinate $z$. Even though the height of the bump is quite limited, the localized increase of turbulent activity immediately downstream of the bump is readily appreciated.

Figure 4.

Colour plot of an instantaneous streamwise velocity field, in the plane $z=0.08$ over the bump $G_{1}$, for the reference case (a) and with StTW (b). Flow is from left to right, and the upstream periodic section ends at $x=0$.

Figures 4 and 5 show instantaneous colour plots of the streamwise and spanwise velocity components in the plane $z=0.08$, which lies just below the bump tip. Every figure compares the flow with (b) and without (a) StTW; the white vertical band shows the intersection of the cut plane with the bump. In figure 4, the elongated streaks of high/low streamwise momentum are clearly visible for the reference simulation upstream of the bump and immediately downstream. In the StTW case, by comparison, the overall velocity level is lower, and the range of fluctuations more limited overall, with a less evident streaky pattern. On the other hand, the streamwise modulation induced by StTW is noticed, particularly just upstream and downstream of the bump tip, where the distance between the wall and the cut plane is small.

Figure 5.

Colour plot of an instantaneous spanwise velocity field, in the plane $z=0.08$ over the bump $G_{1}$. Panels as in figure 4.

In figure 5, the spanwise velocity over the same $z$ plane is plotted. In the reference simulation the flow organization in turbulent structures can be appreciated, whereas a different picture emerges in the wall-forced case, where the spanwise forcing creates alternating bands of positive and negative spanwise velocity.

Figure 6.

Colour plot of the mean vertical velocity $\langle w\rangle$ for the bump $G_{1}$: (a) reference case; (b) StTW. Positive contours (continuous lines) are drawn for $\langle w\rangle =(0.05,0.065,0.08)$, and negative contours (dashed lines) are drawn for $\langle w\rangle =(-0.02,-0.015,-0.01)$. The thick black line indicates $\langle u\rangle =0$ and marks the boundary of the separated region.

Moving on to the analysis of the mean flow field, for which the operator $\langle \cdot \rangle$ implies averaging over time and the homogeneous spanwise direction, figure 6 plots a vertical plane with a colour map of the vertical velocity component $\langle w\rangle$, for a localized portion of the domain which includes the bump, namely $2.5\leqslant x\leqslant 7$. The plot shows that the peak of $w$ just ahead of the bump is decreased because of StTW. The thick contour line corresponds to $\langle u\rangle =0$ and visualizes the separated region with a recirculation bubble after the bump; the separated region is very small for the reference case with $G_{1}$, but the wall forcing somewhat increases its extension.

Figure 7.

Colour plot of the mean pressure $\langle p\rangle$ for the bump $G_{1}$: (a) reference case; (b) StTW. Positive contours (continuous lines) are drawn for $\langle p\rangle =(0.05,0.0525,0.055)$, and negative contours (dashed lines) are drawn for $\langle p\rangle =(-0.05,-0.04,-0.03)$.

The mean pressure distribution $\langle p\rangle$ is shown in figure 7; to ease comparison, the pressure levels of the two cases are offset such that they coincide at $x=0$. Pressure locally increases over the anterior part of the bump, while a local minimum appears shortly after the tip because of the negative wall curvature. In the StTW case, the local maximum before the bump is decreased, and similarly the local minimum at the bump tip shows lesser intensity.

4 Skin friction and pressure at the wall

The aerodynamic force includes contributions from friction and pressure. Friction and pressure at the wall are often expressed by local dimensionless coefficients, although their integral contribution is not straightforwardly related to the drag force. The friction coefficient is

(4.1)$$\begin{eqnarray}c_{f}(x)={\displaystyle \frac{2\langle \unicode[STIX]{x1D70F}_{w}\rangle (x)}{\unicode[STIX]{x1D70C}U_{b}^{2}}},\end{eqnarray}$$

and the pressure coefficient is

(4.2)$$\begin{eqnarray}c_{p}(x)={\displaystyle \frac{2\langle p\rangle (x)}{\unicode[STIX]{x1D70C}U_{b}^{2}}}.\end{eqnarray}$$

In (4.1), $\unicode[STIX]{x1D70C}$ is the fluid density and $\unicode[STIX]{x1D70F}_{w}=\unicode[STIX]{x1D707}~\hat{\boldsymbol{t}}\boldsymbol{\cdot }\unicode[STIX]{x2202}\boldsymbol{u}/\unicode[STIX]{x2202}n$, with $\hat{\boldsymbol{t}}$ the tangential unit vector and $\unicode[STIX]{x2202}/\unicode[STIX]{x2202}n$ the derivative in wall-normal direction. In (4.2), the pressure value $p(x)$, which can be arbitrarily shifted in an incompressible flow, is set to have $\langle p\rangle =0$ at the outlet section.

Figure 8.

Skin-friction distribution $c_{f}(x)$ over the wall with the bump. (a) Comparison between the reference case (grey) and the controlled case (blue) for bump $G_{1}$. (b) Local skin-friction reduction rate $r(x)$ for $G_{1}$ (blue) and $G_{2}$ (red dashed). The thin profiles at the bottom of the plots draw the two bumps, in arbitrary vertical units.

The quantity $c_{f}(x)$ is considered in figure 8 for the reference and controlled cases. Only the lower wall of the non-periodic portion $0\leqslant x\leqslant 12$ of the computational domain is shown. Indeed, the presence of the bump is felt on the $c_{f}$ distribution at the opposite wall too; however, owing to the small blockage this effect is minimal and therefore not shown here. Figure 8(a) plots the distribution of $c_{f}(x)$ itself, comparing the reference and the actuated flows for the bump $G_{1}$. In the reference simulation the friction coefficient decreases just before the bump, and then quickly grows to reach its maximum close to the bump tip. The maximum value is approximately three times that of the flat wall. Downstream of the tip, $c_{f}$ quickly drops towards zero. The flow separation (already discussed in figure 6), produces a locally negative $c_{f}$. After reattachment the friction distribution presents a mild overshooting, followed by a slow recovery towards the undisturbed planar-wall value. When StTW are applied, the behaviour of $c_{f}(x)$ is qualitatively similar, but quantitative changes are introduced, as friction is reduced everywhere by StTW. To quantify such changes, a local skin-friction reduction rate $r(x)$ is plotted in figure 8(b) for both bumps. Here, $r$ is defined as the relative change of $c_{f}(x)$ between the controlled and the reference flow

(4.3)$$\begin{eqnarray}r(x)=1-{\displaystyle \frac{c_{f}(x)}{c_{f,0}(x)}},\end{eqnarray}$$

where $c_{f,0}(x)$ is for the reference flow. Way upstream of the bump $G_{1}$, where the wall is flat, $r$ equals the value typical of StTW in the indefinite plane channel flow, namely $r=46\,\%$ at the present $Re$ and for the employed parameter values (see for example Gatti & Quadrio (Reference Gatti and Quadrio2016)). When the bump is approached, $r$ at first increases slightly above 50 % immediately upstream of the bump, and then decreases to 25 % over the anterior part of the bump. After the tip, when flow separation takes place, the quantity $r$ becomes meaningless; for example, the extrema of the separation bubble in the unforced case, identified by the zero points for $c_{f,0}$, correspond to points where $r$ diverges to infinity. There StTW are observed to cause an increase of both intensity and length of the separation bubble. After the reattachment point, differently from the reference case, no overshooting occurs for friction over StTW.

The longer bump $G_{2}$, though very similar, does not lead to flow separation. Here $r$ (red dashed line) is nearly identical to that for $G_{1}$ up to the bump tip, and then increases towards a local maximum of 70 % near $x=6$. Once again, the recovery towards the planar-wall value is quite slow, and the local drag reduction remains higher than the planar value in most of the computational domain after the bump tip.

Figure 9.

Pressure distribution $c_{p}(x)$ over the wall with the bump. (a) Comparison between the reference case (grey) and the controlled case (blue) for bump $G_{1}$. (b) Local difference between pressure coefficients $\unicode[STIX]{x0394}c_{p}(x)=c_{p}(x)-c_{p,0}(x)$ for $G_{1}$ (blue) and $G_{2}$ (red dashed). The thin profiles at the bottom of the plots draw the two bumps, in arbitrary vertical units.

Figure 9 plots the streamwise distribution of $c_{p}(x)$; the pressure levels of the two cases, which by definition coincide at the outlet section where the mean pressure is set at zero, have been adjusted such that they coincide at $x=0$ instead, as done already in figure 7. In the reference case, the local pressure increases before the bump, and so does the pressure coefficient. An evident minimum of $c_{p}$ is reached at the bump tip, followed by a relatively fast recompression. In the inlet and outlet portions of the computational domain, i.e. far enough from the bump, $c_{p}(x)$ presents the linear decrease (i.e. uniform mean pressure gradient) that is expected for a plane channel flow. As already commented upon for $c_{f}$, the bump affects the pressure coefficient on the opposite planar wall too, but this is not shown here as the changes are minimal.

The action of StTW at the inlet, where the local drag is friction-dominated, simply translates into a milder negative slope of the $c_{p}(x)$ curve, owing to the lower friction drag. More downstream, the positive pressure peak before the bump is noticeably reduced by StTW, thus anticipating that the pressure drag contribution associated with the anterior part of the bump will be reduced (see later § 4.2.2). The minimum of $c_{p}$ near the bump tip is also decreased by StTW, so that the pressure jump between the two local extrema is reduced by around 20 %. After the bump, say for $x>7$, the flow is not affected by the presence of the bump, and again only a reduced pressure gradient is visible.

In order to quantify changes in pressure distribution due to StTW, a local pressure reduction rate, in analogy with friction, cannot be used; in incompressible flows, pressure values can be shifted by a constant, making a ratio meaningless. For this reason we simply study the difference introduced by StTW as

(4.4)$$\begin{eqnarray}\unicode[STIX]{x0394}c_{p}(x)=c_{p}(x)-c_{p,0}(x).\end{eqnarray}$$

Figure 9(b) plots $\unicode[STIX]{x0394}c_{p}(x)$ for both geometries showing a very similar behaviour; the two curves almost coincide up to the bump tip, then for $G_{2}$ the milder slope in the aft part creates a slower recompression. The agreement of $\unicode[STIX]{x0394}c_{p}(x)$ up to the bump tip for the two geometries indicates that the effect of StTW on the pressure distribution is not dictated by the presence of a separation bubble.

4.1 Drag coefficients

To assess how StTW interact with the curved wall, simply comparing the drag force per unit area between the plane geometry and the one with the bump is not the best choice, because the bump introduces concentrated losses. In fact, in the limit of very large streamwise extent $L_{x}^{np}$ of the computational domain that includes the bump, concentrated losses become negligible and the same drag of the planar case is obtained.

Drag is usually quantified in two ways. The first one, meant to evaluate distributed losses $C_{d}^{d}$, uses the streamwise length of the domain under consideration as the reference length, and leads to the definitions below for friction and pressure contributions

(4.5a,b)$$\begin{eqnarray}C_{d,f}^{d}={\displaystyle \frac{2}{\unicode[STIX]{x1D70C}U_{b}^{2}~L_{x}}}~\hat{\boldsymbol{x}}\boldsymbol{\cdot }\int _{0}^{L_{x}}\unicode[STIX]{x1D707}(\unicode[STIX]{x1D735}\boldsymbol{u}+\unicode[STIX]{x1D735}\boldsymbol{u}^{\text{T}})\boldsymbol{\cdot }\boldsymbol{n}\,\text{d}\ell ,\quad C_{d,p}^{d}={\displaystyle \frac{2}{\unicode[STIX]{x1D70C}U_{b}^{2}L_{x}}}\hat{\boldsymbol{x}}\boldsymbol{\cdot }\int _{0}^{L_{x}}\langle p\rangle \boldsymbol{n}\,\text{d}\ell ,\end{eqnarray}$$

where $\hat{\boldsymbol{x}}$ is the unit vector in the $x$ direction. Obviously, in the flat channel only the friction contribution is present. (In the above definitions, note the use of capital letters to indicate global force coefficients, whereas lowercase was used above for local coefficients.)

To evaluate concentrated losses $C_{d}^{c}$, on the other hand, the obstacle contribution to drag is singled out computing the drag variation with respect to the planar case. Using the frontal area of the obstacle as a reference surface, or in this case the bump height $h_{b}$ as the reference length, $C_{d}^{c}$ becomes independent from the domain length

(4.6a,b)$$\begin{eqnarray}C_{d,f}^{c}={\displaystyle \frac{L_{x}^{np}}{h_{b}}}(\widetilde{C}_{d,f}^{d}-\overline{C}_{d,f}^{d}),\quad C_{d,p}^{c}={\displaystyle \frac{L_{x}^{np}}{h_{b}}}(\widetilde{C}_{d,p}^{d}-\overline{C}_{d,p}^{d}),\end{eqnarray}$$

where $\widetilde{C}_{d}^{d}$ and $\overline{C}_{d}^{d}$ indicate drag coefficients computed for the non-planar and planar wall, respectively. Obviously, when concentrated losses $C_{d}^{c}$ are evaluated for the controlled case, both $\widetilde{C}_{d}^{d}$ and $\overline{C}_{d}^{d}$ are computed in presence of StTW.

Table 1.

Drag coefficients for the bump $G_{1}$. Here, $C_{d,f}$ and $C_{d,p}$ are the friction and pressure components, respectively, with $C_{d}=C_{d,f}+C_{d,p}$. Distributed losses are computed according to (4.5) in the planar geometry while concentrated losses introduced by the bump are evaluated via (4.6). Figures are for the lower wall only.

In table 1, drag coefficients are reported for the bump $G_{1}$, computed over the lower wall only. Distributed losses are evaluated in the flat periodic domain while concentrated losses are computed in the non-periodic domain that includes the bump. In the planar case StTW reduce friction drag by 46 %, as expected, and no pressure drag is present. The friction component of the concentrated losses due to the bump is globally nearly zero in the reference case, implying that the friction coefficient computed over the entire wall with the bump almost coincides with that over the flat wall. This is non-obvious, because the bump has been observed (cf. figure 8) to introduce significant local variations. The pressure component, on the other hand, generates a considerable additional contribution to drag.

When StTW are applied, concentrated friction losses become negative, implying that the mean friction over the wall with the bump is lower than the friction over a controlled plane wall. This benefit, absent without flow control, is due to the slower downstream recovery of friction to its planar value and the lack of overshooting after the bump, as shown in figure 8. However, quantifying this benefit in terms of the percentage change of $C_{d,f}^{c}$ (not shown in table 1) would be meaningless, since the reference value is close to zero. The bump is responsible for a considerable pressure drag, and it is interesting to observe that StTW reduce this component too, by an amount of approximately 10 %. Overall, control by StTW leads to reduction of total concentrated losses by around 57 %.

Table 2.

Drag coefficients for the bump $G_{2}$.

Table 2 provides the same quantities for the milder bump $G_{2}$. The distributed losses in the planar case are obviously the same as $G_{1}$ (the small difference is attributed to the finite averaging time). Since $G_{2}$ is less steep after the tip, the concentrated losses decrease. The concentrated friction losses are clearly negative; pressure recovery is more effective and thus pressure losses are lower in both reference and controlled case. The presence of StTW also induces a pressure drag reduction, which is, however, less pronounced than in $G_{1}$. The overall outcome is a reduction of approximately 43 % in the concentrated losses.

It is noted explicitly that the reduction of concentrated losses reported in the rightmost column of tables 1 and 2 must be added to the distributed drag reduction to assess the global saving. This is discussed in the following paragraph.

4.2 Changes in friction and pressure drag over the curved wall

Drag changes induced by StTW are now assessed against the scenario in which StTW are assumed to simply reduce the friction component by the amount they would in a plane channel. We name this the ‘extrapolated amount’, and indicate it with an $(e)$ superscript.

The simulation for $G_{1}$ shows that the friction drag over the entire non-periodic portion is 92 % of the overall drag. Combining changes in distributed and concentrated losses, StTW reduce friction drag by 49.6 %, i.e. approximately 4 % more than the planar case. The pressure drag, representing 8 % of the total drag, is reduced by a relative 10.3 %. Bump $G_{2}$, albeit separation free, shows a similar behaviour. With friction drag accounting for 94 % of the total, StTW reduce friction component by 3 % more than the planar value, and the pressure component too is reduced by a further 3 %.

A physical explanation of the interrelation between friction and pressure drag reduction requires quantifying the influence of local stresses on the global drag budget. A local drag reduction coefficient $\unicode[STIX]{x0394}c_{d}(x)$ is defined as the difference of local contributions to drag coefficients, i.e. the integrands in (4.5), between the reference and the controlled case,

(4.7)$$\begin{eqnarray}\unicode[STIX]{x0394}c_{d}(x)=c_{d,0}(x)-c_{d}(x),\end{eqnarray}$$

where the subscript 0 indicates the reference case. It should be noted that a sign inversion is adopted here in comparison to (4.4), for drag benefits to yield positive $\unicode[STIX]{x0394}c_{d}(x)$. To weigh the local contribution on the integral budget, a global drag reduction rate $R(x)$ is further introduced as in Stroh et al. (Reference Stroh, Hasegawa, Schlatter and Frohnapfel2016): $R(x)$ is defined as the integral of $\unicode[STIX]{x0394}c_{d}(x)$ up to location $x$, expressed as a percentage of the drag calculated over the entire domain in the reference case,

(4.8)$$\begin{eqnarray}R(x)={\displaystyle \frac{\displaystyle \int _{0}^{x}\unicode[STIX]{x0394}c_{d}(x^{\prime })\,\text{d}x^{\prime }}{\displaystyle \int _{0}^{L_{x}}c_{d,0}(x^{\prime })\,\text{d}x^{\prime }}}.\end{eqnarray}$$

4.2.1 Friction drag reduction

Figure 10.

Changes in the skin-friction component of the total drag. (a) The computed $\unicode[STIX]{x0394}c_{d,f}$ (thick line) compared with the extrapolated $\unicode[STIX]{x0394}c_{d,f}^{(e)}$ (thin line with labels) for bump $G_{1}$. (b) Difference between computed and extrapolated friction drag reduction, for geometries $G_{1}$ (blue line) and $G_{2}$ (red dashed line). (c) Difference between actual $R_{f}$ and extrapolated integral budget $R_{f}^{(e)}$ for both geometries. The thin profiles at the bottom of the plot draw the two bumps, in arbitrary vertical units.

The friction component of $\unicode[STIX]{x0394}c_{d}$, namely $\unicode[STIX]{x0394}c_{d,f}$, is plotted in figure 10(a), together with the extrapolated value $\unicode[STIX]{x0394}c_{d,f}^{(e)}$ obtained by assuming that the planar friction reduction rate carries over to the non-planar geometry $G_{1}$. Obviously, the two curves tend to coincide far from the bump, while immediately upstream and over a large downstream extent the true friction drag reduction is larger than the extrapolated one. On the other hand, over the anterior part of the bump, where $r$ in figure 8 shows a local minimum, the true friction drag reduction is smaller than $\unicode[STIX]{x0394}c_{d,f}^{(e)}$.

To understand which area is specifically responsible for the extra friction drag reduction, the local difference between actual and extrapolated values is shown in figure 10(b), for both bump geometries, to demonstrate that the qualitative behaviour is the same, regardless of the presence of flow separation. The two curves coincide, as expected, at the inlet and in the fore part of the bump, whereas the lack of separation in $G_{2}$ makes the two curves quantitatively differ in the decelerating region after the bump tip. The integral budget $R_{f}$, i.e. the friction component of the integrated drag reduction introduced by (4.8), is plotted in figure 10(c) as a difference with respect to the extrapolated value $R_{f}^{(e)}$. For both bumps, the larger friction drag reduction after the bump tip, and its slower recovery of the planar value, already discussed in the context of figure 8, translate into a global friction drag reduction approximately 3 % larger than the value extrapolated from the planar case.

Figure 11.

Colour plot of the turbulent kinetic energy, in outer units, for (a) the bump $G_{1}$ and (b) $G_{2}$, with and without StTW.

Figure 11 links the differences in friction drag reduction observed after the bump tip, including where the wall is flat again, to the distribution of the turbulent kinetic energy $k=1/2\langle u_{i}^{\prime }u_{i}^{\prime }\rangle$. Its production $P$ is shown later in figure 12. It is worth mentioning that the fluctuating velocity field $u_{i}^{\prime }$ is defined by subtracting the local mean field and, for the StTW case, by employing a phase average to additionally remove the contribution of the spanwise Stokes layer.

In agreement with the literature, for the reference cases figure 11 shows two areas of high $k$: one just ahead of the bump, and the other, more intense, immediately after the bump tip, extending approximately one bump length and related to the strong adverse pressure gradient (Wu & Squires Reference Wu and Squires1998). Marquillie et al. (Reference Marquillie, Ehrenstein and Laval2011) discuss a similar picture based on the streamwise distribution of the maximum of turbulent kinetic energy. For the milder bump $G_{2}$ the strong peak after the tip is weakened, whereas the local maximum before the tip is unchanged. The controlled cases show both quantitative and qualitative differences. At the inlet, in agreement with the observations by Quadrio & Ricco (Reference Quadrio and Ricco2011) for the flat wall, the maximum value of $k$ (measured in outer units) is reduced by StTW and displaced at larger wall distances, and small values of $k$ are observed within the Stokes layer. The interaction with the bump appears to be minimal, with the peak value of $k$ remaining nearly constant along the streamwise coordinate, suggesting that the Stokes layer effectively hinders the propagation of the geometrical perturbation made by the bump into the buffer layer and above. There appears to be no substantial difference between $G_{1}$ and $G_{2}$.

Figure 12.

Colour plot of the production $P$ of turbulent kinetic energy. The level $P=0$ is indicated by the contour line. Panels as in figure 11.

In figure 12 the production $P$ of turbulent kinetic energy,

(4.9)$$\begin{eqnarray}P=-\langle u_{i}^{\prime }u_{j}^{\prime }\rangle {\displaystyle \frac{\unicode[STIX]{x2202}\langle u_{i}\rangle }{\unicode[STIX]{x2202}x_{j}}},\end{eqnarray}$$

is plotted to show that the spatial distribution of $k$ is consistent with that of its production: even for $P$, the strong streamwise variation of the reference case is significantly altered by the StTW. Over the bump without actuation, as already observed by Mollicone et al. (Reference Mollicone, Battista, Gualtieri and Casciola2017), the turbulent production increases slightly before the bump and then drops to slightly negative values in the accelerated region just before the bump tip. A large positive peak of turbulent production follows, beginning at the bump tip. The intensity of the local maxima is lower for $G_{2}$. When StTW are applied, the two cases show a similar behaviour: streamwise changes of $P$ are strongly inhibited, and only a local slightly negative minimum in the accelerating region can be detected, though both the extension and the absolute value of the minimum are considerably reduced. In addition, careful scrutiny of the various contributions to $P$ (not shown) reveals that the major cause for the difference between the uncontrolled and controlled flows rests with the field of the Reynolds stresses, while the gradients of the mean flow are much less affected.

The increase of turbulent activity is related to the friction increase producing the overshoot in the $c_{f}$ curves of figure 8 downstream of the bump. Figures 11 and 12 confirm that, with StTW, turbulent activity is inhibited and no overshooting is found in the friction distribution. It is the area immediately downstream the bump tip (see figure 10) that is associated with the extra friction reduction.

4.2.2 Pressure drag reduction

Figure 13.

Comparison of the contribution to pressure drag changes between $G_{1}$ (blue) and $G_{2}$ (red dashed). (a) $\unicode[STIX]{x0394}c_{d,p}(x)$; (b) integral budget $R_{p}$ for both geometries. The thin profiles at the bottom of the plot draw the two bump geometries.

A similar analysis is now carried out for pressure drag, for which simple extrapolation from the planar case would indicate no reduction at all. Figure 13 examines first the streamwise distribution of $\unicode[STIX]{x0394}c_{d,p}(x)$, the pressure component of (4.7). Panel (a) plots $\unicode[STIX]{x0394}c_{d,p}(x)$ for both bumps, and panel (b) shows $R_{p}(x)$, the pressure component of (4.8).

As already shown in figure 9, StTW reduce the positive pressure peak upstream of the bump, as well as the negative one near the bump tip. In terms of drag, the reduction of the first peak is beneficial, and translates into a local drag reduction and a positive $\unicode[STIX]{x0394}c_{d,p}(x)$ with a similar (albeit not identical) local maximum for both geometries, located at the same streamwise coordinate. Such agreement between the two geometries implies that the reduction of pressure drag in the anterior part of the bump is not related to the changes in the separation bubble. The attenuation of the first pressure peak alone produces 6 % of pressure drag reduction for the geometry $G_{1}$ and 10 % for $G_{2}$. The reduction in the intensity of the second, negative pressure peak starts upstream of the bump tip, and extends downstream. Because of the orientation of the surface normal, before the tip such changes are detrimental to drag reduction, and become beneficial after the tip. The negative peak of $\unicode[STIX]{x0394}c_{d,p}(x)$ is essentially identical for both geometries up to the bump tip; however, in the region downstream of the tip, the different local slope of the wall implies a different projection of wall-normal force in the horizontal direction. For this reason, the milder bump $G_{2}$ only partially benefits from the increased pressure recovery in the aft part of the bump. The global effect is therefore a 10 % pressure drag reduction for $G_{1}$, and only 3 % for $G_{2}$, as already shown in tables 1 and 2.

Pressure drag reduction has been shown to be unrelated to changes in the separation bubble produced by $G_{1}$. However, it is interesting to explore such changes, although they produce no significant effect in the global drag budget. The streamwise extent of the separated region is determined by looking at the zeros of the skin-friction distribution shown in figure 8. The detached region starts at $x_{d,0}=4.67$ for the reference case and at $x_{d}=4.6$ for the one with StTW: the two detachment points are very close to each other, with the controlled one moved slightly upstream by the control. The two reattachment points are at $x_{r,0}=5.03$ and $x_{r}=5.32$, respectively. Overall, StTW produce a longer separated region, with length $L_{s}=0.72$, twice the length of the reference case $L_{s,0}=0.36$. Equivalent information was already available from figure 6, where the spatial extent of the recirculating region was determined from the zero of the streamwise component of the velocity. The intensity of the recirculating flow, in terms of largest negative wall shear stress, increases by 60 % with StTW.

Figure 14.

Contour of the probability $\unicode[STIX]{x1D6FE}_{u}$ of non-reverse flow, without (a) and with (b) StTW over the bump $G_{1}$. Black lines identify contours of $\unicode[STIX]{x1D6FE}_{u}=(0.50,0.80,0.99)$ involved in the definitions by Simpson, Chew & Shivaprasad (Reference Simpson, Chew and Shivaprasad1981).

The inner structure of the separation bubble is further investigated, following Simpson et al. (Reference Simpson, Chew and Shivaprasad1981), by locally computing the probability $\unicode[STIX]{x1D6FE}_{u}$ that $u>0$. Incipient detachment is conventionally associated with $\unicode[STIX]{x1D6FE}_{u}=0.99$, i.e. backflow is observed only for 1 % of the time; intermittent transitory detachment takes place when $\unicode[STIX]{x1D6FE}_{u}=0.80$, transitory detachment when $\unicode[STIX]{x1D6FE}_{u}=0.50$, and full detachment when $\langle \unicode[STIX]{x1D70F}_{w}\rangle =0$. Figure 14 is a colour plot of $\unicode[STIX]{x1D6FE}_{u}$, along with the contour lines corresponding to the three values of $\unicode[STIX]{x1D6FE}_{u}$ mentioned above. Moreover, table 3 contains quantitative information regarding the detachment and reattachment points, as well as the spatial extent of the separated region.

Table 3.

Detachment and reattachment points for the reference and controlled cases, along with longitudinal extent deduced for specified values of the probability function $\unicode[STIX]{x1D6FE}_{u}$.

The plot confirms that a longer separation bubble is created by StTW, but adds the information that the recirculating region also undergoes qualitative changes. The reference flow presents a diffused interface between the core of the recirculation where $\unicode[STIX]{x1D6FE}_{u}$ is close to zero and the attached flow where $\unicode[STIX]{x1D6FE}_{u}=1$, whereas the case with control shows a sharper interface, hinting at a separation bubble that is almost steady and does not undergo temporal oscillations.

5 Global power budget

Streamwise-travelling waves are an active flow control technique, which requires actuation power and is capable of favourably altering the power budget in a turbulent plane channel flow (Quadrio et al. Reference Quadrio, Ricco and Viotti2009). The power budget related to the lower wall is now computed separately for the plane wall (i.e. the periodic simulation that feeds the portion of the channel with the bump) and the bumped wall, and reported in table 4 for the bump $G_{1}$. Figures are normalized with the power $P_{tot}$ due to the total drag of the non-actuated case.

Table 4.

Power budget for the bump $G_{1}$. Here, $P_{tot}$ is the power required to overcome the total drag produced by the lower wall, $P_{req}$ is the power required for actuation and $P_{net}=P_{tot}-P_{req}$ represents the net balance. Figures are for the lower wall only.

In the plane geometry, where no pressure drag is present, results agree with those by Gatti & Quadrio (Reference Gatti and Quadrio2016). The percentage reduction of pumping power is by definition identical to the change $\unicode[STIX]{x0394}C_{d,f}$ of the distributed losses already reported in table 1. The net power saving is positive but amounts to only 11.5 % of the reference total power, since in the present study StTW are made to work to maximize their drag-reducing effect, hence the required actuation power $P_{req}$ is quite large, namely 34 % of the total power.

Over the curved wall, StTW reduce friction drag by an amount that exceeds that of the plane wall, and decrease pressure drag too. To quantify their extra benefit, an additional column in table 4 reports the ‘extrapolated’ power budget, obtained via the assumption (already discussed in § 4.1) that the planar friction reduction carries over to the frictional drag component, with no effect on the pressure component. In table 4, the total power is reduced by 46.5 % instead of 41.8 %, with a 10 % relative improvement. Since actuation power is almost unchanged, the net savings become 15.3 % instead of 10.5 %, with a relative increment of almost a half. These specific figures obviously depend on the ratio between friction and pressure drag, i.e. on the bump geometry and the relative extent of the planar surface; a larger difference between actual and extrapolated net power saving can be expected for a larger bump.

Table 5.

Power budget for $G_{2}$, as in table 4.

The power budget for the bump $G_{2}$ is reported in table 5. The plane wall obviously shows the same figures as the previous case, the minor differences being due to finite averaging time. The non-planar wall is qualitatively similar too, with a total drag reduction of 46.3 % instead of 43.7 %. The net power saving is 15.1 % of the total power, i.e. larger than the extrapolated value of 12 %. Hence, for both attached and separated flow, StTW produce extra benefits when applied over curved walls.

Figure 15.

Streamwise mean velocity profile in the periodic plane channel. Comparison between reference flow (grey) and actuated flow on the lower wall (blue).

A final comment is about the power budget discussed above, which concerns only the lower wall with the bump. Indeed, additional benefits appear once the upper, flat wall is included. Even though the upper wall has no actuation, StTW applied on the lower wall induce an asymmetry in the mean streamwise velocity profile, so that friction is reduced by 4 % on the upper wall too. This asymmetry, which is also present in the planar case, is explained by the displacement of momentum towards the wall with lower friction. Figure 15 plots the two mean streamwise velocity profiles, with and without StTW, for the planar case; the velocity maximum is displaced towards the lower actuated wall. This extra benefit on the upper wall, obtained with no additional power, brings the global drag reduction for the whole channel containing the bump $G_{1}$ to 26 %, 3 % more than the extrapolated value, and a net power savings of 10 %, i.e. nearly doubling the extrapolated value.