Study of drag reduction using periodic spanwise grooves on incompressible viscous laminar flows

Introducing surface corrugations to alter the boundary layer flow is a proven way to reduce the fluid drag on a surface. In this study, we examine the effect of periodic, infinitely long spanwise grooves on the laminar boundary layer over a plate for global Reynolds numbers $\left(\mathrm{R}{\mathrm{e}}_L\right)$ between 1000 and 25 000. By employing numerical simulations in two-dimensional domains, we investigate the flow and pressure evolution over surfaces containing rectangular grooves that are perpendicular to the flow and infinitely long in the spanwise direction, and compare them to a flat plate. We characterize the flow interactions near the grooves based on their width-to-depth aspect ratio (AR). Below a certain aspect ratio, a primary vortex fills the space inside each groove. These vortices allow the free stream to “slip over” the grooved regions and result in less skin friction on the wall. However, the interaction between the flow and the grooves’ vertical walls leads to a pressure drag. We study the behavior of the individual drag components over a wide range of aspect ratios (0.2 < AR < 200) and compare the total drag reduction in each case. Based on the simulation results, the transverse grooves in the laminar regime can reduce the total drag up to 10% in comparison to a flat plate, despite increasing the wetted surface area of the plate. Conversely, at some aspect ratios the grooves cause a total drag increase of more than 200%. We observe that by increasing the aspect ratio, the free stream bends more toward the grooves; ultimately at a certain aspect ratio, denoted by $\mathrm{A}{\mathrm{R}}^{*}$, the flow breaks apart the underlying circulation and reaches the bottom of the grooves. When this happens, the flow shear on the grooves’ bottom walls combined with the high-pressure drag exerted on the vertical walls can lead to a net increase in the drag. Therefore, the aspect ratio of the grooves is a critical parameter in optimizing drag reduction in transverse geometries.

Figure 1

(Left) Schematic of the plate with embedded grooves. The grooves are oriented perpendicularly to the flow direction which is shown by the arrow. (Right) Schematic of the simulation domain with the boundary conditions. We consider transverse grooves that are embedded in the plate with no protrusion into the main flow, and extend infinitely into the plane. As a result, a two-dimensional model is used to investigate the flow dynamics and drag properties.

Figure 2

The influence of the number of elements in the computational grid on the drag coefficient of a flat plate $\left(C_D\right)$ at $\mathrm{R}{\mathrm{e}}_L\approx {10}^4$. The dashed line represents the value obtained from the Blasius solution $\left(1.328/\sqrt{\mathrm{R}{\mathrm{e}}_L}\approx 0.0136\right)$.

Figure 3

Isopressure contours at the leading edge of a flat plate and nondimensional velocity profiles at different locations along the plate for (a) a domain without the symmetry region and (b) a domain with the symmetry region. Without such region, the leading-edge effects are propagated in the plate direction, causing a pressure gradient along the plate. By extending the domain prior to the plate, the pressure effects of the leading edge are impeded in the plate direction and instead, they propagate backward. $\eta =y/x\sqrt{\mathrm{R}{\mathrm{e}}_x}$ represents the similarity variable in the Blasius solution. For a computational domain without the symmetry region the profiles deviate from the Blasius solution, especially as they get closer to the leading edge of the plate. The velocity profiles over a plate that is preceded by a symmetry region precisely follow the Blasius solution.

Figure 4

The effect of the extended symmetry region length over the plate length $\left(L_s/L_p\right)$ on the drag coefficient $\left(C_D\right)$ at $\mathrm{R}{\mathrm{e}}_L\approx {10}^4$. To get accurate drag coefficients that are within 1% of the Blasius solution, an entry length of almost the same size as the plate is required for the simulations.

Figure 5

Steady streamline patterns in the first groove of the periodic textures for different aspect ratios (AR) at $\mathrm{R}{\mathrm{e}}_L\approx {10}^3$ and $\mathrm{R}{\mathrm{e}}_L\approx {10}^4$. The behavior of streamlines can be classified into two major types. At smaller ARs, a primary circulation fills the groove and the free stream slips over the grooves. At larger ARs, the flow starts penetrating the grooves and eventually reaches the bottom of the grooves. In this state, the underlying vortex is split into two smaller vortices. We use $\mathrm{A}{\mathrm{R}}^{*}$ to show the critical aspect ratio beyond which the flow fully penetrates the groove. As the $\mathrm{R}{\mathrm{e}}_L$ increases, this transition occurs at higher ARs.

Figure 6

Unsteady streamline patterns at $\mathrm{R}{\mathrm{e}}_L\approx 2.5\times {10}^4$. (a) At smaller AR, the streamlines show a periodic behavior where the patterns are repeated over time. (b) At larger AR, the streamline patterns become more unpredictable as the interaction of the flow and the grooves’ trailing walls causes vortex shedding inside and above the grooves.

Figure 7

Effect of the vortical flows in the grooves on their average lifting pressure for different aspect ratios (AR) and global Reynolds numbers $\left(\mathrm{R}{\mathrm{e}}_L\right)$. The location of the maximum for each curve corresponds to the $\mathrm{A}{\mathrm{R}}^{*}$ at that $\mathrm{R}{\mathrm{e}}_L$. After this point, the inertia of the main flow breaks apart the main vortex, causing a reduction in the net force per width of the vortices. At higher $\mathrm{R}{\mathrm{e}}_L$, the flow induces vortices with stronger circulations which improves the ability of the grooves for holding the main flow up to larger AR.

Figure 8

Nondimensional velocity profiles at different locations along a single groove. $\eta =y/x\sqrt{\mathrm{R}{\mathrm{e}}_x}$ is the similarity variable in the Blasius solution, which is extended for $y<0$ to represent part of the profile inside the groove. Each location along the groove is distinguished by a colored line. The dashed line shows the Blasius profile of a flat plate and the solid black line corresponds to the velocity profile on the land region of a grooved plate. The negative value of $u$/$U$ shows the backward motion of the fluid within the grooves due to the flow circulation.

Figure 9

Average effective slip length $\overline{l_s}$ over textured plates as a function of the grooves’ aspect ratios (AR) at different global Reynolds numbers $\left(\mathrm{R}{\mathrm{e}}_L\right)$. The slip lengths are calculated for the grooves where the free stream shows minimal bending $\left(\mathrm{AR}\ll \mathrm{A}{\mathrm{R}}^{*}\right)$.

Figure 10

Momentum thickness (θ) along the whole plate with different groove aspect ratios (AR) at global Reynolds number of $\mathrm{R}{\mathrm{e}}_L\approx {10}^4$. The grooves’ layout associated with each AR is shown at the bottom of each plot.

Figure 11

The momentum thickness different (Δθ) along textured plates with different ARs at a global Reynolds number of $\mathrm{R}{\mathrm{e}}_L\approx {10}^4$. The grooves’ layout associated with each AR is shown at the bottom of each plot. $\mathrm{\Delta }\theta =\theta \left(y:0\to H\right)-\theta \left(y:-C\to H\right)$ shows the momentum gain or loss as a result of the flow in the grooves. Based on this definition, a negative value indicates a loss of momentum because of the presence of the grooves in the flow.

Figure 12

The evolution of local skin friction $\left(C_f\right)$ and pressure coefficient $\left(C_p\right)$ along the first few trenches with aspect ratio (AR) of 2.5 at $\mathrm{R}{\mathrm{e}}_L={10}^4$. The dashed line shows the local coefficients for a flat plate $(C_f=0.664/\sqrt{\mathrm{R}{\mathrm{e}}_x}$ and $C_p=0)$.

Figure 13

Comparison of local skin friction $\left(C_f\right)$ and pressure coefficient $\left(C_p\right)$ in a single trench for different aspect ratios (AR). χ represents a normalized length scale for the grooves, projecting the different grooves’ widths between zero and 1.

Figure 14

Transient behavior of the streamlines over the whole grooved plate and their corresponding fluctuation in the drag coefficients. (a) At smaller AR, the transient vortices are confined within the grooves and both skin friction and pressure coefficients show a periodic trend. In this situation, the time-averaged total drag coefficient is still lower than a flat plate. (b) At larger AR, interaction of the flow with the trailing wall of the groove that act as a blunt body, causes oscillations in the flow similar to a vortex shedding phenomenon. During this process the flow loses a significant portion of its momentum and as a consequence, the plate experiences a much higher drag in comparison to a flat plate.

Figure 15

(a) Average skin friction coefficient $\left({\overline{C}}_f\right)$ and (b) average pressure coefficient $\left({\overline{C}}_p\right)$ over the whole textured plate with different groove aspect ratios (AR) and roughness values $\left(\varepsilon =C/L_p\right)$ at $\mathrm{R}{\mathrm{e}}_L\approx {10}^4$.

Figure 16

Percentage reduction of total drag coefficient for different aspect ratios (AR) and roughness values (ε) at $\mathrm{R}{\mathrm{e}}_L\approx {10}^4$. $\mathrm{\Delta }{C}_D$ is the signed difference of the drag coefficient of a grooved plate and a flat surface, and $C_{D0}$ is the total drag coefficient of a flat plate $\left(C_{D0}={\overline{C}}_f=1.328/\sqrt{\mathrm{R}{\mathrm{e}}_L}\right)$. The negative values represent drag reduction, while the positive values show drag increase in comparison to a flat plate of the same nominal length.

Figure 17

Comparison of (a) average skin friction coefficient $\left({\overline{C}}_f\right)$ and (b) average pressure coefficient $\left({\overline{C}}_p\right)$ for different aspect ratios (AR) and roughness values (ε) at three different global Reynolds numbers $\left(\mathrm{R}{\mathrm{e}}_L\right)$. We can generally notice valley-shaped plots for ${\overline{C}}_f$ and bell-shaped behavior for ${\overline{C}}_p$, indicating their opposite effects in the total drag. Presence of the grooves allows for creation of vortices which help with drag reduction by reducing skin friction. However, the vertical walls of each groove act like blunt bodies in the flow, causing a net pressure force, thus increasing the drag.

Figure 18

Comparison of the total drag reduction $\left(DR\right)$ over the grooved surfaces with different groove aspect ratios (AR) and global Reynolds numbers $\left(\mathrm{R}{\mathrm{e}}_L\right)$. Each plot represents a different roughness value $\left(\varepsilon =C/L_p\right)$.