# Transition delay using biomimetic fish scale arrays

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Sep 3, 2020

The surface, mimicking the array of overlapping scales along the flat plate in the low-turbulence facility, again show the generation of streamwise velocity streaks, similar as detected in our previous study on real fish bodies17. Contours of constant streamwise velocity in a cross-section of the boundary layer in Fig. 3A illustrate the velocity variation at 1.2 m from CFD simulation and experiment for one wavelength of the streak ((lambda _z = 50,hbox {mm})) for Setting-1. Note, that the result is periodic in spanwise direction with each row of scales. The Blasius contour at the right depicts the 2-D mean flow over the same wavelength28. A detailed comparison of velocity profiles in the high- and low-velocity regions behind the scales are shown in Fig. 3B,C, respectively. In the region of high velocity and low velocity regions the deviation from the reference Blasius profile depicts velocity deficit or increase behind the scale array. Spanwise averaged streamwise velocity in the streaky flow is compared with Blasius profile in Fig. 3D. The streaky flow produces a fuller velocity profile when compared with the reference flow which results in a shape factor of 2.47 instead of 2.59 which is comparable with the Large Eddy Simulation results reported in literature for the streaky base flow29. CFD and experimental results are comparable with only minor variations that may be attributed to the boundary conditions and the experimental uncertainties. Measurements at different locations further downstream prove that the streak persists in streamwise direction (not shown here).

This modulation of the velocity is fundamentally different from the streaky structure generated by the lift-up effect caused by a vortex generator or cylinder array30. The streaky structure generated by the overlapping scales is formed by a spanwise periodic flow very close to the surface, and the amplitude of the streak increases with the number of scale rows in the direction of flow. Setting-1 with about the same number of scales as Setting-2 but twice as thick produces a streak amplitude approximately twice as large compared to Setting-2 ((A_{st} sim 20%) for Setting-1 and (A_{st} sim 10%) for Setting-2) as shown in Fig. 4A. The streak amplitude is defined as in Eq. (1) and increases with the number of scale rows from the leading edge of the model, it is seen from Fig. 4A within X = 0.3 m to X = 0.6 m (It is the extent where the scales are placed on the tunnel). Afterwards it drops as a result of the decelerating trailing ramp. Once again the flow reorganises up to some extent to increase the streak amplitude and then the viscosity causes it to decay continuously downstream. Both models did not induce a bypass transition (instantly tripping laminar flow into turbulent), nor did they induce secondary streak instability as seen from hot-film signals and with flow visualisations.

begin{aligned} A_{st} = bigg [max _{y}{U(X,y,z)}-min _{y}{U(X,y,z)}bigg ]bigg /({2U_infty }) end{aligned}

(1)

To investigate the response of the boundary layer to the scaled surface with regard to the laminar-to-turbulent transition process, a controlled transition experiment with a representative Tollmien–Schlichting (TS) wave at a given frequency were performed, following the method invented in27. In Fig. 4A the neutral stability curve is shown as a black line for the present free-stream velocity of (U_infty = 0.086, hbox {m/s}). The neutral stability curve is given along the X-axis and also for comparison with non-dimensional parameters in similar studies based on the Reynolds number ((Re_x)). The area within the stability curve is the region in which, according to linear stability theory, infinitesimal disturbances will grow exponentially7. The velocity signals were measured for 60 s at a data acquisition frequency of 100 Hz at (Y = 10,hbox {mm}) from the wall from (X = 1.98,hbox {m}) to 5.92 m for the reference flat plate and fish scale array (Setting-1) cases. In the reference case, the induced small disturbances from the vibrating wire grow in the streamwise direction inside the instability region which can be inferred from the black (u_{rms}) curve from 2 to 2.6 m in Fig. 4B. Initially, the so-called primary instability mechanism increases the velocity fluctuations until secondary instability mechanism set in, afterwards the fluctuations increase rapidly until they reach a peak around 3.8 m in Fig. 4B. From there on the flow is turbulent as observed from the constant (u_{rms}) plateau after 4.5 m31. However, for the flow with fish scale array (Setting-1), as seen from the red line in Fig. 4B the fluctuation level (u_{rms}) remains almost constant until 4 m and it increases monotonically but with a lower rate when compared with the reference case. The local flow state can be defined generally by the intermittency parameter which classifies the flow into laminar, turbulent, and transitional32. The value in percentage indicates the nature of the flow, e.g.: zero represents laminar flow and 100% represents fully turbulent flow, and any value in between indicates how long the flow is turbulent in a given period of time. For the reference flat plate case, the flow is laminar until 3 m and becomes turbulent just after 4 m as shown in the black curve in Fig. 4C. For the case with fish scale array as shown in the red line in Fig. 4C, the flow remains laminar for a larger extent until 4.65 m and then becomes turbulent around 6 m. This results in a streamwise extension of laminar flow by about 1.65 m which corresponds to a 55% delay in transition location.

Next, we explore the flow by means of temporal velocity signals for the two cases previously discussed. Figure 5A displays the velocity signals subtracted from their mean values at 2, 2.5, 3.0 and 3.5 m for a period of 20 s for the reference case without fish scales. The amplitude of the velocity signals in Fig. 5A increases with streamwise coordinate X. The respective frequency spectra of these velocity signals is given in Fig. 5C, where the abscissa is normalised with respect to the vibration frequency ((F_0 = 0.2,hbox {Hz})) of the vibrating ribbon. At (X=2,hbox {m}) the spectrum displays a peak at (F/F_0=1) which indicates that the fluctuation energy is only from the wire’s fundamental vibrating frequency. Higher harmonics and subharmonics of the fundamental frequency (F_0) appear further downstream (see the peaks at 3 and 3.5 m) and the disturbance energy also increases compared to the spectrum at 2 m. At 3.5 m the energy is increased over all frequencies given in the plot indicating that the flow is becoming increasingly disturbed resulting in turbulence. On the contrary, the velocity magnitude for the flow with fish scale array remains within 2% at all locations, as shown in Fig. 5B. At the same time the fluctuation energy is very small compared to the flat plate case as shown in Fig. 5D. Additionally, the higher harmonic components in the flow are completely absent in the case of the fish scale array. This reflects the very low level of velocity fluctuations (u_{rms}) depicted by a red line in in Fig. 4B. The increase of (u_{rms}) beyond 4 m is due to uncontrolled background oscillations of the water tunnel and not necessarily due to a re-amplification of the TS wave.

Complementing information about the flow states in different cases is obtained from flow visualisation using the method of surface streakline generation. Individual potassium permanganate crystals were placed on the flat plate and, while dissolving as dye in the water, they visualize the flow close to the surface. These streak lines will be visible as compact dye lines if the flow is laminar, while, in contrast, they develop kinks and diffuse very quickly when the flow is turbulent. The locations where the visualisations has been done is shown in Fig. 6A for brevity. Figure 6B(Reference) shows the visualisation picture for the reference flat plate case from (X= 3.15) to 3.85 m. Certainly, the streaklines are visible for more than 70% of the picture indicating laminar flow with some instabilities. In Figure 6B(Reference), where the frame is from 3.55 to 4.25 m the streaklines have broken down because of the turbulent flow which is comparable with the hot-film measurements explained above. Figure 6B(Setting-1) portray the visualization picture for the same locations described previously, but for the case with fish scale array (Setting-1). The flow is completely laminar in the given locations and even for an additional location from 4.35 to 5.05 m as shown in Fig. 6B(Setting-1). For the case with the second set of fish scale array (Setting-2), the flow is still laminar in the above-mentioned locations until the end of the picture as shown in Fig. 6B(Setting-2). Hence, this visually proves that the fish scale array increases the laminar flow extent by delaying the transition and this result is in perfect agreement with the transition delay visualisations performed with cylindrical roughness elements19.

### Drag estimation

Flow over any body will experience drag that has two components, skin friction and pressure drag. For a flat plate, the drag is only from skin friction with the friction coefficient for laminar and turbulent flow given by Eqs. (2) and (3), respectively28. These equations were compared with Direct Numerical Simulation results and found to be comparable with similar kind of TS waves33.

begin{aligned} C_{fx_L}= & {} 0.664/sqrt{Re_x} end{aligned}

(2)

begin{aligned} C_{fx_T}= & {} 0.059/{Re_x}^{frac{1}{5}} end{aligned}

(3)

Hence, the total drag along a flat plate with laminar and turbulent flow regimes can be approximated by the summation of the drag components by the Eq. (4), where, (x_L) is the location from the leading edge of the flat plate where the flow is assumed to change from laminar to turbulent state.

begin{aligned} D_{Net} = D_L + D_T + D_P = int _{0}^{x_L} frac{1}{2}C_{fx_L} rho U_infty ^2 dx + int _{x_L}^{L} frac{1}{2}C_{fx_T} rho U_infty ^2 dx + D_P end{aligned}

(4)

For the sake of comparing the drag for both cases, the location (x_L) is assumed where the intermittency factor reaches a value of 50%. For the reference flat plate case the location (x_L) is estimated from the hot-film measurements to be at 3.3 m and for the case with the fish scale array (Setting-1) the location (x_L) is placed at 5.3 m. For Setting-2 the location (x_L) is chosen from the flow visualisation of about 4.3 m since we do not observe any turbulence even at Frame-2. Additionally, when a body like fish scale array is mounted on the flat plate it experiences added pressure drag ((D_P)). The value of this drag component is calculated from the CFD simulation for a length of (L_X = 1.2,hbox {m}) (as shown in Fig. 2C). The components of drag for the two cases are given in Table 1. The net drag (DNet) is reduced by 0.02 N with the fish scale array (Setting-1) and 0.008 N for Setting-2. This results in a net drag reduction of about 27% for Setting-1 and 10.7% for Setting-2 when compared with the reference flat plate case.

This result can be understood by using the skin friction plots as depicted in Fig. 7A. The laminar skin friction curve is shown as a dashed pink line and the turbulent skin friction curve as a dashed green line. Typical transition curves appear in all the cases considered here28. Generally, if a flow becomes turbulent the skin friction coefficient rises to almost twice its value for laminar flow at a particular location. The total drag of the surface is the area under the skin friction curve, therefore, the area under the curve reduces for fish scale array when compared with the reference flat plate case. Furthermore, the reduction in integral amplitude is proportional to the streak amplitude in the experiments considered here.

We have demonstrated that the fish scale array could delay transition to reduce the net drag. The underlying mechanism is the attenuation of the modulated TS waves due to the streamwise velocity streaks in the base flow. The latter produce a spanwise averaged flow with a steeper velocity gradient than the Blasius solution (reference flat plate case). This leads to a smaller shape factor which is known to stabilize the boundary layer29,34. This is also seen by the streamwise decay of the observed lambda-vortices for the scales. In comparison, in the classical transition scenario (regular Blasius flow) the two dimensional TS waves grow inside the boundary layer within the linear instability region and grow further until the amplitude of fluctuation increases above a critical amplitude, which is when three-dimensional undulations lead to the formation of strong (Lambda)-vortices35( non-linear flow regime based on H-type or K-type transition29) and ultimately to turbulence. The spanwise wavelength of these (Lambda)-vortices (spacing between the legs of the (Lambda)-vortex) is generally larger than half of the TS wavelength36. Herein, for the streaky base flow, the TS waves reorganise already early in the linear phase into weak (Lambda)-vortices as depicted in Fig. 7B due to the streamwise modulation of the flow. The spanwise wavelength of these weak (Lambda)-vortices is equal to the wavelength of the fish scale array, different from the wavelength on natural transition. In addition, because of the stabilizing effect of the smaller shape-factor of the boundary layer34 the weak (Lambda)-vortices decay in the downstream direction. This proposed mechanism follows similar arguments given in29 for the simulation of transition delay due to finite amplitude streaks.