Aerodynamics and Percolation: Unfolding Laminar Separation Bubble on Airfoils

As a fundamental phenomenon of fluid mechanics, recent studies suggested laminar-turbulent transition belonging to the universality class of directed percolation. Here, the onset of a laminar separation bubble on an airfoil is analyzed in terms of the directed percolation model using particle image velocimetry data. Our findings indicate a clear significance of percolation models in a general flow situation beyond fundamental ones. We show that our results are robust against fluctuations of the parameter, namely, the threshold of turbulence intensity, that maps velocimetry data into binary cells (turbulent or laminar). In particular, this percolation approach enables the precise determination of the transition point of the laminar separation bubble, an important problem in aerodynamics.

Popular Summary

Fluid turbulence underlies a range of phenomena, from the design of airplane wings to the movement of gas in stars. And yet it remains one of the big unsolved challenges in physics. In particular, it is still not clear how smooth (or laminar) fluid flow transitions to a flow that is turbulent. Typically, this transition occurs via the disordered emergence of turbulent spots. Some recent theoretical work has taken some important steps toward understanding this transition using a set of statistical models known as “directed percolation,” which mimic the filtering of a fluid through a porous material. It remains to be seen, however, if this approach has any practical relevance to real-life turbulence. Employing a sophisticated technique for visualizing fluid flow, we show how to use percolation models to properly characterize how turbulence originates on an airfoil.

Using particle image velocimetry in a wind tunnel, we record the formation of a laminar separation bubble (LSB), a gap that opens up between an airfoil and a laminar fluid flow that triggers turbulence in its wake. We then analyze these data using a directed percolation model. This model allows us to determine physical parameters that are critical to characterizing the LSB. Our results show that a directed percolation model can determine the location of turbulence transition on an airfoil with a precision much higher than other methods in fluid dynamics.

This breakthrough—the first experimental evidence that directed percolation models can characterize flow over an airfoil—can be extended to other aerodynamic phenomena, efficient models in computational fluid dynamics, and better estimates of fatigue loads on wind turbines.

Figure 1

(a) Photograph of a LSB on airfoil CK220. The laser light sheet illuminates the particle-seeded inflow from the right-hand side and perpendicular to the airfoil’s surface, resulting in a shadow at the leading edge. The inflow is coming from the left-hand side. The LSB can be identified as a region without smoke between the airfoil’s surface and the ambient flow. False colors are used for better visualization. (b) Labeled zoom, emphasizing the flow topology at the LSB. The LSB occurs just behind the airfoil’s thickest cross section, where streamlines (in blue) separate from the surface. Further downstream, when the LSB is at its maximum expansion, the laminar shear layer destabilizes, resulting in transition to turbulence. High momentum perpendicular to the airfoil’s surface contained in the shear layer enables reattachment and the formation of a turbulent boundary layer.

Figure 2

Experimental setup consisting of (a) a HSPIV system and (b) a CK220 airfoil. A laminar inflow of $u_{\infty }=11\mathrm{m}/\mathrm{s}$ approaches the airfoil from the left-hand side, forming a LSB on the suction side. The light sheet (in green) is adjusted (c) tangentially to the surface approximately at the location where the LSB is observed. The coordinate system has its origin at the midspan leading edge and defines the main flow direction as the $x$ direction, the spanwise direction as the $y$ direction, and the normal to wind tunnel wall as the $z$ direction. The chord length and span of the airfoil are denoted $c$ and $s$, respectively.

Figure 3

Illustration of measured and computed quantities. The plane of space and the control parameter (${\mathrm{Re}}_x$) is formed by the airfoil’s spanwise and chordwise directions, where ${\mathrm{Re}}_x=\mathrm{Re}(x,u_{\infty })$ is based on the distance $x$ from the leading edge and the inflow velocity $u_{\infty }$. Here, ${\mathrm{Re}}_x$ provides a measure for the flow’s likelihood to undergo transition from laminar flow into a LSB and, therefore, serves as a control parameter. The presence of a LSB is indicated by means of turbulence intensity (TI) exceeding a certain threshold denoted as turbulent phase. The instantaneous turbulence intensity is derived from 10 temporally consecutive velocity fields at one respective position ${\mathrm{Re}}_x$ in a plane crossing the laminar separation bubble. Transition from laminar flow into the LSB happens after turbulent clusters merge significantly. At this critical point ${\mathrm{Re}}_{\mathrm{c}}$, characteristic turbulent clusters occur, depicted as an evolution in time (vertical plane). Averaging over time at each position ${\mathrm{Re}}_x$, the fraction of turbulent cells, $\rho$, is shown next to the plane of TI. Following $\rho$ in the streamwise direction, transition from the laminar boundary layer flow into the LSB and subsequent reattachment are obvious. The first phase transition represents the LSB’s onset, which is important for our work.

Figure 4

(a) Three illustrations of cluster distributions in $t$ (time) and $y$ (space) derived for ${\mathrm{TI}}_{\mathrm{th}}=0.8$: (Left panel) Reynolds number below the critical value ${\mathrm{Re}}_c$, showing a dominance of laminar clusters. (Right panel) Reynolds number above ${\mathrm{Re}}_c$, where turbulent cells dominate. (Middle panel) Reynolds number at the computed ${\mathrm{Re}}_c$ for which laminar and turbulent clusters span the entire system. (b) Turbulent fraction $\rho$ (circles), as a function of ${\mathrm{Re}}_x$, where $\rho$ is the fraction of cells with a turbulence intensity larger than a threshold ${\mathrm{TI}}_{\mathrm{th}}$. Results are shown for three different threshold values. The corresponding best fits (solid lines) above the critical point cross the control parameter axis at the critical Reynolds number ${\mathrm{Re}}_c$. Experimental uncertainty of $\rho$ is estimated by the standard error of 10 subsets constituting the total data set.

Figure 5

(a) Illustration of how the critical Reynolds number ${\mathrm{Re}}_c$ is determined: (i) One first adjusts a parabola (red line) around the three highest values (stars) of the derivative of the turbulent fraction $[d\rho /(d{\mathrm{Re}}_{\mathrm{x}})]$ (circles); (ii) one then performs a best fit, expecting the critical Reynolds number ${\mathrm{Re}}_c$ lying close to the mathematical maximum (red dashed line) of the parabola respecting the spatial resolution of the experiment. Based on this determined interval, ${\mathrm{Re}}_c$ is estimated after Eq. (2). The obtained ${\mathrm{Re}}_c$ is marked in the inset (gray dashed line) along with weights $a_1$ and $a_2$ used for computation of distributions of laminar cluster sizes at ${\mathrm{Re}}_c$ (see Fig. 6). Here, ${\mathrm{TI}}_{\mathrm{th}}=0.8$. (b) Compensated plots [after Eq. (3)] for each value of the turbulence intensity threshold: fraction of turbulent cells rescaled to the one for $1+1\mathrm{D}$ directed percolation predictions as a function of the reduced Reynolds number $ϵ$.

Figure 6

(a) Size distribution of laminar clusters in space at the estimated critical Reynolds number, i.e., consecutive laminar states when sweeping in the $y$ direction at a given time $t$, for three illustrative values of turbulence intensity thresholds and (b) their corresponding size distribution of laminar clusters in time, i.e., consecutive laminar states when sweeping in the time $t$ while keeping $y$ constant. Solid lines show the theoretical distribution predicted by the directed percolation model at the transition. For comparative purposes, the turbulence intensity threshold values are the same as the ones in Fig. 4, where the critical value for each $\mathrm{TI}$ is given. Cluster lengths are shown in multiples of the spatial and temporal resolution, respectively.

Figure 7

Systematic analysis of the critical exponents sensitivity to the threshold of turbulence intensity imposed in the directed percolation model: the exponent $\beta$ and the transverse fractal dimension ${\mu }_{\perp }$. The dashed lines indicate the theoretical value of the exponents in directed percolation. For a turbulence intensity ${\mathrm{TI}}_{\mathrm{th}}>0.64$, the critical exponents of directed percolation are obtained within numerical errors (see text). Gray symbols show the dependence of the critical Reynolds number on the threshold. For the sake of clarity, only three illustrative error bars of ${\mathrm{Re}}_c$ are shown.

Figure 8

(a) Rescaled turbulent fraction $\widehat{\rho }$ as a function of $ϵ$ for a variety of possible critical Reynolds numbers, ${\mathrm{Re}}_{c^{\prime }}\in [{\mathrm{Re}}_{\mathrm{c}},{\mathrm{Re}}_{\mathrm{c}}\pm 75,{\mathrm{Re}}_{\mathrm{c}}\pm 150]$. Where $\widehat{\rho }$ becomes constant, ${\mathrm{Re}}_c$ can be estimated best. (b) Derivative of $\widehat{\rho }$ as a function of ${\mathrm{Re}}_{c^{\prime }}$ including standard errors. The critical Reynolds number is determined by the zero crossing of ${\widehat{\rho }}^{\prime }$. The uncertainty of ${\mathrm{Re}}_c$ is estimated by propagation of errors. Here, ${\mathrm{TI}}_{\mathrm{th}}=0.93$.

Figure 9

Illustrations of cluster distributions in time and space at criticality. In the right diagram, a typical snapshot obtained from the experimental HSPIV data is depicted. In the left diagram, the (detailed) result from a DP simulation is compared with the corresponding coarse-grained visualization.

Figure 10

Turbulent fraction for the baseline DP model (in blue) according to Ref. [5], compared with a modified model (in red) that allows a noisy parameter $r$, which changes in time according to a standard normal distribution with a standard deviation of $\sigma \approx 0.02$ and a fixed average $⟨r⟩$. This average value plays the role of the “new” control parameter. In the inset, the respective compensated representation indicates a first-order phase transition for both cases. Please note that the critical point is adjusted to each model.

Figure 11

(a) Size distribution of laminar clusters in space at the estimated critical value (i.e., turbulent spreading) for one illustrative binary flow field generated by applying the baseline DP model (in blue) as presented in Ref. [5]. The baseline model is then modified according to experimental constraints, allowing a fluctuating control parameter (in red) and considering metrological coarse graining (in gray). (b) The corresponding size distribution of laminar clusters in time, depicted analogously. Dashed lines show the theoretical distribution predicted by the directed percolation model at the transition. Cluster lengths are shown in multiples of the discretization used in the simulation.