4.1. Sinusoidal grooves

We begin discussion by considering travelling wave disturbances, which connect to the classical TS waves in the limit of $S\rightarrow 0$ . The full disturbance spectrum for $\mathit{Re}=6000$ , $S=0.035$ and ${\it\beta}=0.7$ for the ‘two-dimensional’ waves with ${\it\delta}=1.02$ and ${\it\mu}=0$ is displayed in figure 4. The term ‘two-dimensional’ is given in quote marks, as disturbances are always three-dimensional owing to the groove-imposed modulations, but reduce to the purely two-dimensional form for $S\rightarrow 0$ . The dominant eigenvalue is located at the tip of branch A (wall modes; Mack Reference Mack1976) and its location is weakly affected by the grooves. Comparison of the disturbance velocity eigenfunctions for $S=0$ and $S\neq 0$ shows small differences in the $x$ - and $y$ -velocity components and the formation of wall layers with highly intense spanwise motions (figure 5). The results displayed in figure 6(a) demonstrate that this mode connects to the TS waves for $S\rightarrow 0$ . The flow topology illustrated in figure 7 is very similar to that found in the smooth channel, i.e. it consists of a set of spanwise rolls propagating in the streamwise direction. The rolls are modified by the spanwise movements concentrated in the wall layers, with the fluid periodically flowing away from the corrugation peaks near the corrugated wall and towards the peaks at the opposite wall, with these directions reversed every quarter wavelength (see figure 8). Very strong spanwise movement in the wall layers occurs at the beginning of the disturbance wavelength ( $x=0$ , see figure 8 a), followed by a very strong vertical movement in the channel centre a quarter wavelength further downstream ( $x={\it\lambda}_{x}/4$ ; see figure 8 b). The presence of grooves adds structures in the central part of the channel, which can be described as very weak vertical rolls (see figure 9 a); there is no trace of these structures in the wall layers (see figure 9 b).

Figure 5.

Disturbance velocity eigenfunctions (a)  $g_{u}^{(n)}$ ( $n=0$ , 1, 2), (b)  $g_{v}^{(n)}$ ( $n=0,1,2$ ) and (c)  $g_{w}^{(n)}$ ( $n=1,2,3$ ) associated with the most unstable eigenvalue for $\mathit{Re}=6000$ , $S=0.035$ , ${\it\beta}=0.7$ , ${\it\delta}=1.02$ and ${\it\mu}=0$ , normalized by the condition $\max _{0\leqslant y\leqslant 1}|g_{u}^{(0)}(y)|=1$ . The solid and dashed lines correspond to the real and imaginary parts, respectively. Eigenfunctions for the TS waves in a smooth channel are shown in (a) and (b) using grey lines.

Figure 6.

Variations of the growth rate ${\it\sigma}_{i}$ and the frequency ${\it\sigma}_{r}$ of the ‘two-dimensional’ disturbances ( ${\it\mu}=0$ ) as a function of the groove amplitude $S$ for the groove geometry described by (2.8) with ${\it\beta}=0.7$ for the flow Reynolds number $\mathit{Re}=6000$ and disturbances with streamwise wavenumber ${\it\delta}=1.02$  (a) and ${\it\delta}=0.3$  (b). Dashed lines identify the imaginary and real parts of the most unstable TS waves with ${\it\delta}=1.02$ in the smooth channel (a) and the least stable Squire mode with ${\it\delta}=0.3$ in the smooth channel (b).

Figure 7.

Disturbance velocity vectors in the ( $x,y$ )-plane in the smooth channel (a) and in the corrugated channel at $z={\it\lambda}_{z}/4$  (b) for the TS waves for the same conditions as in figure 5 and normalized by the condition $\max _{y_{L}\leqslant y\leqslant 1,\,0\leqslant z\leqslant {\it\lambda}_{z}}|u_{D}(0,y,z,0)|=1$ .

Figure 8.

Disturbance velocity vectors in the ( $y,z$ )-plane at $x=0$ (a) and $x={\it\lambda}_{x}/4$  (b) for the TS waves for the same conditions as in figure 5 and normalized in the same manner as in figure 7.

Figure 9.

Disturbance velocity vectors in the ( $x,z$ )-plane at $y=0$ (a) and $y=-0.9$  (b) for the TS waves for the same conditions as in figure 5 and normalized in the same manner as in figure 7.

Figure 10 displays the spectrum for the same flow conditions and channel geometry but for a different class of disturbances, e.g. for ${\it\delta}=0.3$ and ${\it\mu}=0$ . The unstable eigenvalue is located at the tip of branch P (centre modes) and its location is strongly affected by the groove amplitude. This mode connects to the Squire mode in the limit of $S\rightarrow 0$ as demonstrated in figure 6(b). The velocity eigenfunction for the limiting Squire mode is displayed in figure 11 and demonstrates that the fluid movement occurs only in the spanwise direction, resulting in the three-dimensionalization of the velocity field in a nominally two-dimensional flow even when the spanwise disturbance wavenumber ${\it\mu}=0$ . Figure 11 also displays velocity eigenfunctions for the leading mode from the spectrum shown in figure 10 and demonstrates a significant change in the disturbance flow field. The $x$ - and $z$ -velocity components are $O(1)$ while the $y$ -component is significantly smaller, suggesting that the disturbance motion retains a nearly planar character, i.e. it is confined mostly to the ( $x,z$ )-plane. The topology of the disturbance velocity field illustrated in figure 12 suggests that this field is made up of rows of counter-rotating rolls oriented across the channel and propagating in the streamwise direction. There are two rows of rolls per one groove wavelength, one centred at the widest channel opening and the other one centred at the narrowest opening, with the former one being stronger.

Figure 10.

Disturbance spectrum for flow in a channel with geometry defined by (2.8) with ${\it\beta}=0.7$ and $S=0.035$ for $\mathit{Re}=6000$ and disturbances with ${\it\delta}=0.3$ and ${\it\mu}=0$ computed using $N_{S}=10$ Fourier modes and $N_{T}=250$ Chebyshev polynomials. Black crosses identify elements of the spectrum. The Squire spectrum for the smooth channel is given for comparison purposes and is identified using grey circles. The width of branch S is dictated by the number of Fourier modes used in the solution.

Figure 11.

Disturbance velocity eigenfunctions (a)  $g_{u}^{(n)}$ ( $n=1$ , 2, 3), (b)  $g_{v}^{(n)}$ ( $n=1$ , 2, 3) and (c)  $g_{w}^{(n)}$ ( $n=0$ , 1, 2) associated with the most unstable eigenvalue for $\mathit{Re}=6000$ , $S=0.035$ , ${\it\beta}=0.7$ , ${\it\delta}=0.3$ and ${\it\mu}=0$ , normalized by the condition $\max _{0\leqslant y\leqslant 1}|g_{w}^{(0)}(y)|=1$ . The solid and dashed lines correspond to the real and imaginary parts, respectively. Eigenfunctions for the Squire mode in a smooth channel are shown in (c) using grey lines.

Figure 12.

Disturbance velocity vectors in the ( $y,z$ )-plane at $x=0$ (a) and at $x={\it\lambda}_{x}/4$ (b), in the ( $x,z$ )-plane at $y=0$  (c), and in the ( $x,y$ )-plane at $z={\it\lambda}_{z}/4$  (d) for the new mode for the same conditions as in figure 11 and normalized by the condition $\max _{y_{L}\leqslant y\leqslant 1,\,0\leqslant z\leqslant {\it\lambda}_{z}}|u_{D}(0,y,z,0)|=1$ .

We shall now address the question of the origin of the new instability. It is known that the TS instability is driven by shear and a sufficient increase of $\mathit{Re}$ will lead to its eventual stabilization. This process is illustrated in figure 13(a), displaying variations of the amplification rate ${\it\sigma}_{i}$ as a function of ${\it\delta}$ for increasing $\mathit{Re}$ . The initial destabilization is followed by stabilization when $\mathit{Re}$ becomes sufficiently large. Figure 13(b) displays the results of a similar study for the new mode and demonstrates that the amplification rate becomes $\mathit{Re}$ -independent once $\mathit{Re}$ becomes large enough. These results demonstrate that the instability is driven by an inviscid mechanism. It is simple to show that the mean flow vorticity has local extrema and thus the vorticity dynamics drives the instability through the so-called inflection point instability (Fjørtoft Reference Fjørtoft1950). Figure 14 displays the distribution of the mean flow $y$ -vorticity component, which can be viewed as a measure of the ‘driving force’, as well as the distribution of the dissipation function ${\it\Phi}_{B}$ , which can be viewed as a measure of the ‘opposing force’. Quote marks are used here to underline the qualitative character of these terms. The ‘driving force’ increases monotonically in the downward direction while the dissipation function is smallest around the channel mid-line, with the local minima in the widest and narrowest channel openings, and increases with distance away from the channel centre. The optimal conditions for the initiation of the instability are on both sides of the widest channel opening slightly below the channel mid-line, as this is where the ratio of the ‘driving’ and ‘opposing’ forces is the highest. Another argument explaining the location of the most intense instability motion can be made by looking at the spanwise variations of the local Reynolds number $\mathit{Re}_{loc}$ , which is based on the maximum velocity and the channel half-height at a given spanwise $z$ -location, i.e. 

The variations of $\mathit{Re}_{loc}$ illustrated in figure 15 demonstrate that $\mathit{Re}_{loc}$ is largest at the widest channel opening and suggests that this is where the balance between the inertial and viscous forces is most favourable for the instability.

Figure 13.

Variations of the growth rate ${\it\sigma}_{i}$ of the ‘two-dimensional’ disturbances ( ${\it\mu}=0$ ) as a function of the streamwise wavenumber ${\it\delta}$ for flow in a channel with geometry described by (2.8) with $S=0.035$ and ${\it\beta}=0.7$ for the TS waves (a) and the new mode (b).

Figure 14.

Distributions of the mean flow dissipation function ${\it\Phi}_{B}$ (black lines) and the mean flow normal vorticity component ${\it\eta}_{B}=\partial u_{B}/\partial z$ (grey lines) in the middle of the channel with geometry described by (2.8) with $S=0.035$ and ${\it\beta}=0.7$ .

Figure 15.

Variations of the local Reynolds number $\mathit{Re}_{loc}$ defined by (4.1) for flow in a channel with geometry defined by (2.8) with $S=0.035$ and ${\it\beta}=0.7$ for $\mathit{Re}=6000$ .

The dominant eigenvalues identified through the analysis of the spectra shown in figures 4 and 10 determine the critical stability conditions. Their tracing through the parameter space provides the means for characterization of the instability. Figure 16(a,b) displays the neutral curves in the ( $\mathit{Re},{\it\delta}$ )-plane for ${\it\beta}=1$ and typical groove amplitudes, and for $S=0.05$ and typical groove wavenumbers. These results demonstrate the significantly different wavelengths of the most unstable TS waves and the most unstable new mode, as the critical wavenumber for the new mode is less than half of that for the TS waves. They also demonstrate a large sensitivity of the new mode, and the relative insensitivity of the TS waves, to the groove amplitude, as well as a rapid decrease of the critical Reynolds number $\mathit{Re}_{c}$ for the new mode as $S$ increases. The last observation is consistent with the results displayed in figure 6(b), which show a rapid increase of the amplification rate as $S$ increases.

Figure 16.

Neutral curves in the ( $\mathit{Re},{\it\delta}$ )-plane for the ‘two-dimensional’ disturbances ( ${\it\mu}=0$ ) in a channel with geometry defined by (2.8). (a) Results for ${\it\beta}=1$ and different  $S$ . (b) Results for $S=0.05$ and different ${\it\beta}$ . Solid and dotted lines correspond to the new mode and the TS waves, respectively.

The effects of the groove wavenumber ${\it\beta}$ are illustrated in figure 17. It can be seen that the new mode can be induced by grooves with a finite range of ${\it\beta}$ and that the range of ‘active’ ${\it\beta}$ increases rapidly with an increase of both $S$  (a) and $\mathit{Re}$  (b). The term ‘active’ refers to ${\it\beta}$ values that are able to induce this instability. An increase of $\mathit{Re}$ corresponds to the reduction of the ‘opposing force’, while an increase of $S$ corresponds to the increase of the ‘driving force’. The latter effect is illustrated in figure 18, which demonstrates a rapid increase of ${\it\eta}_{B}=\partial u_{B}/\partial z$ with an increase of $S$ while, at the same time, the dissipation function remains nearly unchanged (not shown). The same figure also demonstrates a rapid decrease of ${\it\eta}_{B}=\partial u_{B}/\partial z$ for small and large ${\it\beta}$ values, which explains why such grooves are unable to support the new instability.

Figure 17.

Neutral curves in the ( ${\it\beta},{\it\delta}$ )-plane for the ‘two-dimensional’ disturbances ( ${\it\mu}=0$ ) in a channel with geometry defined by (2.8). (a) Results for $\mathit{Re}=7000$ and different $S$ . (b) Results for $S=0.05$ and different $\mathit{Re}$ . Solid and dotted lines correspond to the new mode and the TS waves, respectively.

Figure 18.

Variations of the maximum of the mean flow vertical vorticity component ${\it\eta}_{B}=\partial u_{B}/\partial z$ at $y=0$ as a function of the groove wavenumber ${\it\beta}$ in a channel with geometry defined by (2.8) with $S=0.035$ .

So far, the discussion has only considered ‘two-dimensional’ waves, i.e. ${\it\mu}=0$ . The results displayed in figure 19 demonstrate that such waves play the critical role for the new mode, as the oblique waves are more attenuated. A similar conclusion for the TS waves has been reported by Moradi & Floryan (Reference Moradi and Floryan2014).

Figure 19.

Variations of the critical Reynolds number $\mathit{Re}_{c}$ for disturbances with the wavevector $\boldsymbol{q}=({\it\delta},{\it\mu})$ as a function of its inclination angle ${\it\theta}=\pm \text{tan}^{-1}({\it\mu}/{\it\delta})$ for the groove geometry described by (2.8) with $S=0.05$ . Results correspond to the new mode with $|\boldsymbol{q}|=({\it\delta}^{2}+{\it\mu}^{2})^{1/2}=0.3$ (black lines) and $|\boldsymbol{q}|=0.32$ (grey lines).

Figure 20 displays variations of the critical Reynolds number $\mathit{Re}_{c}$ as a function of the groove geometry, i.e.  ${\it\beta}$ and $S$ , for the TS waves and for the new mode. The range of the unstable ${\it\beta}$ values for the new mode is clearly delineated together with the remarkable decrease of $\mathit{Re}_{c}$ when $S$ increases. Variations of $\mathit{Re}_{c}$ for the TS waves can be viewed as marginal when compared with variations of $\mathit{Re}_{c}$ for the new mode. Figure 21 provides a zoom-in on the range of parameters where the new mode plays the critical role and, at the same time, expands the range of $S$ covered by the analysis; it provides detailed information about the critical Reynolds number $\mathit{Re}_{c}$ , the critical wavenumber ${\it\delta}_{c}$ , the critical frequency ${\it\sigma}_{r,c}$ and the critical phase speed $c_{c}={\it\sigma}_{r,c}/{\it\delta}_{c}$ . It can be seen that $\mathit{Re}_{c}$ can be reduced down to $\mathit{Re}_{c}<750$ for $S=0.1$ . The range of unstable ${\it\beta}$ values has a rather steep, weakly $S$ -dependent bound on the side of large ${\it\beta}$ , while it keeps expanding on the side of small ${\it\beta}$ as $S$ increases. The critical wavenumber ${\it\delta}_{c}$ changes from ${\sim}1$ for large ${\it\beta}$ to ${\sim}0.025$ for small ${\it\beta}$ for the new mode while ${\it\delta}_{c}$ for the TS waves changes marginally. The critical frequency for the new mode changes from ${\sim}1$ for large ${\it\beta}$ to ${\sim}0.025$ for small ${\it\beta}$ , while it remains nearly constant for the TS waves. The critical phase speed remains nearly constant at $c_{c}\approx 1$ for the new mode and at $c_{c}\approx 0.26$ for the TS waves. In summary, at the onset one can observe the new mode with wavelength varying widely depending on the groove wavenumber but with a phase speed that is nearly always the same and approximately four times greater than the TS wave phase speed.

Figure 20.

Variations of the critical Reynolds number $\mathit{Re}_{c}$ as a function of the groove wavenumber ${\it\beta}$ and the groove amplitude $S$ in a channel with geometry defined by (2.8). Solid and dotted lines correspond to the ‘two-dimensional’ new mode ( ${\it\mu}=0$ ) and the two-dimensional TS waves ( ${\it\mu}=0$ ), respectively. The thick line separates zones where each mode plays the critical role. The zoom-in on the range of ${\it\beta}$ where the new mode determines the critical conditions together with an extension to larger values of $S$ is given in figure 21.

Figure 21.

Variations of the critical stability parameters as a function of the groove wavenumber ${\it\beta}$ and the groove amplitude $S$ . Other conditions are as in figure 20. Variations of the critical Reynolds number $\mathit{Re}_{c}$ , the critical wavenumber ${\it\delta}_{c}$ , the critical frequency ${\it\sigma}_{r,c}$ and the critical phase speed $c_{c}={\it\sigma}_{r,c}/{\it\delta}_{c}$ are displayed in panels (a)–(d), respectively.

The reader may note that ${\it\beta}$ and ${\it\delta}_{c}$ for the new mode satisfy the relation ${\it\delta}_{c}\approx {\it\beta}/2$ (see figure 21 b), which suggests that a parametric resonance is contributing to the instability. One can look at the grooves as one-dimensional spatially distributed forcing imposed on the flow and characterized by the wavevector ( $0,{\it\beta}$ ). Studies of simple model problems involving such forcing demonstrate that the dynamical system may exhibit wavenumber locking and responses that extend into two spatial dimensions, i.e. the system response may have a component in the direction orthogonal to the forcing wavevector (Manor, Hagberg & Meron Reference Manor, Hagberg and Meron2008, Reference Manor, Hagberg and Meron2009). Analysis of natural convection in a horizontal layer subject to one-dimensional spatial temperature modulations demonstrates that the system response is driven by a combination of the buoyancy-driven instability and the spatial parametric resonance, with each mechanism dominating over a different range of forcing wavelengths and the wavenumber locking between the forcing and the system response observed in the latter case (Hossain & Floryan Reference Hossain and Floryan2013). The forcing reduces the system symmetry to translational symmetry. Addition of a pressure-gradient-driven flow further affects the symmetry, with the system response showing preference for the flow direction (Hossain & Floryan Reference Hossain and Floryan2015). Such a system still exhibits characteristics associated with the spatial parametric resonance if the flow Reynolds number is small enough and the forcing is strong enough. Shear layers always have a preferred direction, e.g. the flow direction, and thus the effect of parametric resonance may be difficult to identify. The analysis is further complicated by the simultaneous appearance of shear, vorticity (inflection point mechanism) and centrifugal instability mechanisms. Floryan (Reference Floryan1997) studied pressure-gradient-driven channel flow modified by forcing in the form of streamwise suction or blowing and demonstrated that one possible response involves the formation of streamwise vortices, i.e. structures characterized by a wavevector with a component orthogonal to the forcing wavevector. Forcing generated by the transverse surface corrugations has led to similar conclusions for the pressure-gradient-driven flows as well as for the kinematically driven flows (Floryan Reference Floryan2002, Reference Floryan2007). The system response to forcing in the form of longitudinal grooves furthermore underscores the importance of the spatial parametric resonance.

The results presented in figures 20 and 21 provide the basis for the identification of conditions that guarantee that the flow remains laminar in the presence of grooves, i.e. when the grooved surface behaves as a hydraulically smooth surface. Here we shall use the definition proposed by Floryan (Reference Floryan2007), which states that the corrugated wall can be viewed as hydraulically smooth as long as it is unable to create system bifurcation. It is convenient to introduce the global critical Reynolds number defined as

i.e. it is defined as the minimum of $\mathit{Re}_{c}$ over all groove wavenumbers for a given groove amplitude for a particular instability mode. Figure 22 displays variations of $\mathit{Re}_{c,g}$ as a function of $S$ for the TS waves and for the new instability mode. The zone attached to the bottom left corner identifies conditions when the flow remains stable regardless of the spatial distributions of the grooves. When $S<0.02$ , increase of $\mathit{Re}$ will eventually lead to the onset of the TS wave instability and the corresponding conditions define the travelling wave limitation. When $S>0.02$ , increase of $\mathit{Re}$ will eventually activate the new mode and the corresponding conditions define the new mode limitation. The latter curve can be approximated as $\mathit{Re}_{c,g}=4.505S^{-1.925}$ and it defines the upper limit of $S$ that guarantees flow stability. Conditions below the limiting curves guarantee flow stability for any groove distributions and thus walls with such grooves can be viewed as hydraulically smooth. The reader may note that not every groove distribution corresponding to the conditions above these curves must lead to flow instability.

Figure 22.

Variations of the global critical Reynolds number $\mathit{Re}_{c,g}$ as a function of the groove amplitude $S$ for flow through a channel fitted with grooves of different shapes. Solid grey lines correspond to sinusoidal grooves defined by (2.8). Black dashed, dotted, solid and dash-dotted lines correspond to triangular grooves with $a_{1}=b_{1}={\it\lambda}/2$ and $d_{1}=0$ , universal trapezoidal grooves with $a_{2}=b_{2}={\it\lambda}/8$ and $c_{2}=d_{2}=3{\it\lambda}/8$ , rectangular grooves with $c_{3}=d_{3}={\it\lambda}/2$ and circular-segment grooves with $c_{4}=d_{4}={\it\lambda}/2$ , respectively (see figure 23 for definitions of geometric parameters). Correlation for the new instability mode for the most dangerous shape, i.e. the rectangular grooves, has the form $\mathit{Re}_{c,g}=4.505S^{-1.925}$ .

4.2. Grooves with arbitrary shapes

The above discussion was focused on grooves with shapes described by a single Fourier mode, i.e. (2.8). We shall now enquire how the new instability mode is affected by changes in the groove shape. It has been shown by Floryan (Reference Floryan2007) in the analysis of the stability of channel flow modified by transverse grooves, and by Moradi & Floryan (Reference Moradi and Floryan2014) in the analysis of the stability of the TS mode in a channel with longitudinal grooves, that groove shapes can be replaced with an acceptable accuracy by the leading mode from the Fourier expansion describing their geometry. These conclusions form the basis of the ROG model. It remains to be shown if the ROG model remains valid for the new mode. Figure 23 describes four general shapes used in the analysis, including rectangular grooves. The Fourier representation of the latter shape suffers from the Gibbs phenomenon, which has been controlled using a filtering technique based on the ‘raised cosine’ method (Canuto et al. Reference Canuto, Hussaini, Quarteroni and Zang2006). Figure 24 displays the neutral curves for the test grooves with their shapes replaced by a gradually increasing number of Fourier modes from Fourier expansions representing their shapes. It can be seen that the computed critical Reynolds numbers rapidly converge as the number of Fourier modes used for the geometry description increases. These results demonstrate that the ROG model can indeed be used in the analysis of the new mode and thus the results described in § 4.1 can be used to estimate the stability characteristics of arbitrary grooves with an acceptable accuracy.

Figure 23.

Groove shapes used in this study: (a) triangular grooves, (b) trapezoidal grooves, (c) rectangular grooves, and (d) circular-segment grooves with $c_{4}\geqslant 4S$ and $R=c_{4}^{2}/(16S)+S$ .

Figure 24.

The neutral curves in the ( $\mathit{Re},{\it\delta}$ )-plane for grooves with amplitude $S=0.05$ and wavenumber ${\it\beta}=1$ . Dashed, dotted, solid and dash-dotted lines correspond to triangular grooves with $a_{1}=b_{1}={\it\lambda}/3$ , $d_{1}={\it\lambda}/3$ , trapezoidal grooves with $a_{2}=b_{2}={\it\lambda}/6$ , $c_{2}=d_{2}={\it\lambda}/3$ , rectangular grooves with $c_{3}=2{\it\lambda}/3$ , $d_{3}={\it\lambda}/3$ , and circular-segment grooves with $c_{4}=2{\it\lambda}/3$ , $d_{4}={\it\lambda}/3$ , respectively (see figure 23 for notation). (b) An enlargement of the box shown in (a). All grooves were represented in the analysis using one, three and five leading Fourier modes taken from the Fourier expansions describing their geometry.

Figure 22 illustrates the variations of $\mathit{Re}_{c,g}$ for all these grooves. It can be seen that the TS wave limitation is little affected by changes of the groove shape. The curve defining the new mode limitations shifts as a function of the groove shape but the difference between the critical Reynolds number for the actual groove and for its reduced geometry representation remains below 10 %. Rectangular grooves appear to be the most effective in initiating the new instability mode, and thus one can define the parameter range where the grooved surface behaves as a hydraulically smooth surface using results for this particular shape.

Figure 25.

The neutral curves in the ( $\mathit{Re},{\it\delta}$ )-plane for channels fitted with the optimal equal-depth grooves. Groove geometry is represented by the universal trapezoid with $a_{2}=b_{2}={\it\lambda}/8$ , $c_{2}=d_{2}=3{\it\lambda}/8$ and ${\it\beta}=0.7$ (see figure 23 b for notation). Results for sinusoidal grooves are given for reference (dotted lines). (b) An enlargement of the box shown in (a).

4.3. Optimal grooves

The stability characteristics of the optimal grooves are of interest, as such grooves provide the largest possible drag reduction (Mohammadi & Floryan Reference Mohammadi and Floryan2013b ). The shapes of such grooves depend on the type of constraints. The channel geometry used for analysis is expressed as

and the groove height ( $S_{L,max}$ ) and depth ( $S_{L,min}$ ) are subject to constraints of the form

In the case of equal-depth grooves, $S_{L,max}=S_{L,min}=S$ . In the case of unequal-depth grooves, the height is set by (4.4a,b ) while the depth is determined by the optimization process (Mohammadi & Floryan Reference Mohammadi and Floryan2013b ). The shape of the optimal, equal-depth grooves can be well approximated by a universal trapezoid with $a_{2}=b_{2}={\it\lambda}/8$ and $c_{2}=d_{2}=3{\it\lambda}/8$ (figure 23 b). The neutral stability curves for such grooves displayed in figure 25 demonstrate a significant loss of stability when compared with the simple sinusoidal grooves. The shapes of the unequal-depth grooves are illustrated in figure 26(a). If the $y$ -coordinate is rescaled using the peak-to-bottom distance as the length scale $\bar{y}_{L}=(y_{L}+1-S_{L,max})/(S_{L,min}+S_{L,max})$ , and the $z$ -coordinate is rescaled using the groove width at half-height $W_{half}$ as the length scale $\bar{z}=(z-z_{0})/W_{half}$ , the groove shape can be approximated using a Gaussian function of the form $\bar{y}=-\text{e}^{-4\bar{z}^{2}}$ (Mohammadi & Floryan Reference Mohammadi and Floryan2013b ). Figure 26(b) displays the neutral curves for these grooves. While one cannot compare them directly with the simple sinusoidal grooves, the critical Reynolds number is similar to that found in the case of the optimal equal-depth grooves (compare figure 26 b with figure 25 a) and suggests significant flow destabilization.

Figure 26.

The optimal unequal-depth grooves. (a) Geometry of the optimal grooves with ${\it\beta}=0.7$ and $S_{L,max}=0.04$ , 0.05 and 0.06. (b) The neutral curves in the ( $\mathit{Re},{\it\delta}$ )-plane for flow in a channel fitted with these grooves. Groove geometry is represented in the stability analysis by the universal Gaussian function (see text for details).