4.1. The POD modes

The POD eigenvalues were utilized to estimate the cumulative energy content using (B7). Figure 6 shows the cumulative energy plotted versus the number of POD modes used to estimate it. The range of angles of attack is $\alpha =9.25^{\circ }$–$9.8^{\circ }$ for ${\textit {Re}}_c = 5\times 10^4$ and $\alpha =10.25^{\circ }$–$10.8^{\circ }$ for ${\textit {Re}}_c = 9\times 10^4$ on the top part of the figure, and $\alpha =9.8^{\circ }$–$10.5^{\circ }$ and $\alpha =10.8^{\circ }$–$11.2^{\circ }$ on the bottom part of the figure. The angles of attack of $9.8^{\circ }$ at ${\textit {Re}}_c=5\times 10^4$ and $10.8^{\circ }$ at ${\textit {Re}}_c=9\times 10^4$ are duplicated in both panels because they are the angles at which the POD modes have the fastest convergence to the total energy; that is, the minimum number of POD modes used to attain more than $99\,\%$ of the total energy. Thus, they are displayed in both panels to compare the convergence of POD modes at other angles of attack. These are the angles of attack at which the LFO has the maximum amplitude of oscillations (Eljack et al. Reference Eljack, Soria, Elawad and Ohtake2021). At the stall angle of attack, the cumulative energy converges slowly towards $1$ as the number of POD modes approaches $250$, as seen in the figure. The cumulative energy converges faster as the angle of attack increases. The fastest cumulative energy convergence was achieved at the angles of attack of maximum LFO. The convergence process of the POD energy slows down as the angle of attack increases above the angle of attack of maximum LFO.

Figure 6.

Cumulative energy of the POD modes ($\kappa _t(m)$) estimated by summing the POD eigenvalues over $m$ POD modes. The arrows indicate the direction in which the angle of attack $\alpha$ increases.

4.2. The DMD modes

The DMD spectra were estimated for the lift coefficient, the drag coefficient, the pressure, the streamwise velocity and the wall-normal velocity. The DMD spectra of all flow variables were used to identify the most dominant DMD flow modes. There is one low-frequency peak in the spectrum of each of the analysed flow variables featuring the low-frequency oscillating mode one (LFO-Mode-1) for all of the investigated angles of attack. However, the spectrum of the drag coefficient exhibits a low-frequency peak at a different Strouhal number from that of the LFO-Mode-1. Thus, the DMD flow mode that dominates the drag spectrum represents a second low-frequency oscillating mode (LFO-Mode-2). The LFO-Mode-1 and the LFO-Mode-2 exchange the dominance of the spectra of the lift coefficient, the drag coefficient, the pressure and the streamwise velocity. There is no significant high-frequency peak in any of the spectra of these variables. However, there is a pronounced high-frequency peak in the spectrum of the wall-normal velocity featuring the HFO mode. Hence, the DMD spectra of the lift coefficient, the drag coefficient and the wall-normal velocity were used to identify the LFO-Mode-1, the LFO-Mode-2 and the HFO mode, respectively. Once each of the three modes was identified, its frequency was used to locate its corresponding dynamic mode in the spectrum of the other flow variables. Thus, the DMD spectra of the flow variables show three dominant modes. However, each of the three dominant modes is only dominant in the DMD spectrum of the variable used to identify it.

Figure 7 shows the DMD spectra for the five variables. The left-hand side of the figure shows the spectra for angles of attack of $9.25^{\circ }$–$10.5^{\circ }$ at Reynolds number of $5\times 10^4$ and the right-hand side shows the spectra for angles of attack of $10.25^{\circ }$–$11.2^{\circ }$ at Reynolds number of $9\times 10^4$. The points in the spectra representing the LFO-Mode-1, the LFO-Mode-2 and the HFO are indicated by filled black circles, filled red circles and filled blue circles, respectively. The LFO-Mode-1 and the LFO-Mode-2 dominate the spectra of the lift coefficient and the drag coefficient, respectively, for all of the investigated angles of attack and Reynolds numbers of $5\times 10^4$ and $9\times 10^4$. At Reynolds number of $5\times 10^4$, the LFO-Mode-1 dominates the spectra of the pressure, the streamwise velocity and the wall-normal velocity at angles of attack higher than the angle of attack of $9.5^{\circ }$ and lower than the angle of attack of $10.1^{\circ }$. It is worth mentioning that the LFO phenomenon initiates at angles of attack higher than the angle of attack of $9.25^{\circ }$, becomes fully developed at the angle of attack of $9.8^{\circ }$ and loses momentum at angles of attack higher than the angle of attack of $10.0^{\circ }$. The LFO-Mode-1 dominates the DMD spectra of the pressure and streamwise velocity at all of the investigated angles of attack at Reynolds number of $9\times 10^4$.

Figure 7.

DMD spectra of (a) the lift coefficient ($\textit{C}_{\!_\textit{L}}$), (b) the drag coefficient ($\textit{C}_{\!_\textit{D}}$), (c) the pressure ($\kern 1.5pt p$), (d) the streamwise velocity ($u_1$) and (e) the wall-normal velocity ($u_2$). The filled black circles denote the LFO-Mode-1, the filled red circles display the LFO-Mode-2 and the filled blue circles indicate the HFO mode.

The spectra of the wall-normal velocity are dominated by one low-frequency mode and a broad band of high-frequency modes for each of the investigated angles of attack. At Reynolds number of $5\times 10^4$ and angle of attack of $9.25^{\circ }$ the spectrum is dominated by the HFO mode as seen in the left-hand side of the figure. As the angle of attack increases above $9.25^{\circ }$, a sudden and drastic change occurs, and the dominant mode shifts to the LFO-Mode-1 and the LFO-Mode-2. The two low-frequency modes dominate the spectra of the wall-normal velocity component until the angle of attack is raised above $10.0^{\circ }$; then, the HFO mode dominates the spectra again, as seen in the figure. At Reynolds number of $9\times 10^4$, the HFO mode dominates the spectra at angles of attack lower than or equal to the angle of attack of $10.6^{\circ }$. Whereas, the LFO-Mode-1 dominates the spectra at angles of attack higher than or equal to the angle of attack of $10.8^{\circ }$. It is worth mentioning that the LFO phenomenon is pronounced in the amplitude of oscillation and uniform in its lifecycle at angles of attack higher than or equal to $\alpha = 10.8^{\circ }$. In summary, the LFO-Mode-1 and the LFO-Mode-2 (absolute instability) dominate the spectrum when the LFO phenomenon exists in the flow, and the HFO mode (convective instability) dominates the spectrum when the LFO does not exist in the flow. Thus, the LFO phenomenon occurs in the flow when there is absolute instability. The HFO mode has an insignificant magnitude in the DMD spectrum of the lift coefficient at all of the investigated angles of attack. This is indicative that the HFO mode does not contribute directly to the oscillations in the lift coefficient. It is worth noting that the DMD analysis decomposes the fluctuating flow into many low-frequency modes as seen in figure 7. Whereas, the POD method recovers only two low-frequency modes that reconstruct the fluctuating flow favourably as will be discussed later.

Figure 8 shows plots of the Strouhal number of the most dominant flow modes versus the angle of attack. The profile of the Strouhal number of the LFO-Mode-1 for the lift coefficient, the pressure and the streamwise velocity is similar to that obtained using the fast Fourier transform algorithm by Eljack et al. (Reference Eljack, Soria, Elawad and Ohtake2021). The wall-normal velocity component is dominated by a high-frequency flow mode at angles of attack of $\alpha \leqslant 9.25^{\circ }$ and $\alpha \geqslant 10.1^{\circ }$, and dominated by a low-frequency mode at angles of attack of $9.4^{\circ } \leqslant \alpha \leqslant 10.0^{\circ }$ for Reynolds number of $5\times 10^4$. At Reynolds number of $9\times 10^4$, the HFO dominates at angles of attack lower than or equal to $10.6^{\circ }$. The vertical axis is broken into two ranges in the plot for the wall-normal velocity component to capture both the low-frequency and the high-frequency ranges that dominate it.

Figure 8.

Strouhal number (St) of the most dominant flow modes versus the angle of attack and Reynolds number for the LFO-Mode-1 (left) and the HFO mode (right).

4.3. The SMD modes

Time history of the lift coefficient ($\textit{C}_{\!_\textit{L}}(t)$) was utilized to obtain the orthonormal spatial SMD mode ($\boldsymbol {\varPsi }_{\textit{C}_{\!_\textit{L}}}$) using (2.2) and (2.3). The flow field was then projected onto $\boldsymbol {\varPsi }_{\textit{C}_{\!_\textit{L}}}$ to obtain the SMD coefficient corresponding to the lift coefficient (${a}_{\textit{C}_{\!_\textit{L}}}(t)$) using (2.4). After that, the orthonormal spatial SMD mode ($\boldsymbol {\varPsi }_{\textit{C}_{\!_\textit{D}}}$) and the SMD coefficient corresponding to the drag coefficient (${a}_{\textit{C}_{\!_\textit{D}}}(t)$) were duly estimated. The fluctuating flow along the wake of the airfoil mimics the oscillating behaviour of the wall-normal velocity. Thus, the wall-normal velocity component was probed at four locations downstream of the airfoil trailing edge, and the most dominant frequency was determined. The value of the frequency was used to estimate the rate at which the HFO mode repeats periodically; consequently, phase-averaging of the fluctuating flow at this frequency was performed to recover the spatial HFO mode. Equation (2.3) was then used to obtain the orthonormal spatial SMD mode ($\boldsymbol {\varPsi }_{v}$), where the subscript $v$ denotes the wall-normal velocity used to extract the mode. After that, the fluctuating flow field was projected onto $\boldsymbol {\varPsi }_{v}$ to obtain the SMD coefficient corresponding to the HFO mode (${a}_{v}(t)$). The SMD modes corresponding to the lift and the drag coefficients represent the LFO-Mode-1 and LFO-Mode-2, respectively. Figure 9 shows that the SMD modes corresponding to the lift coefficient, the drag coefficient and the wall-normal velocity compare very well with their corresponding POD modes. The LFO phenomenon is fully developed at an angle of attack of $9.8^{\circ }$ and Reynolds number of $5\times 10^4$, and an angle of attack of $10.8^{\circ }$ and Reynolds number of $9\times 10^4$. Thus, the LFO repeats periodically with some disturbed cycles. However, at lower angles of attack, the LFO becomes stochastic, and the POD and SMD coefficients do not compare with each other.

Figure 9.

Comparison of the POD coefficients (blue line) and the SMD coefficients (red line) for (a,b) the LFO-Mode-1, (c,d) the LFO-Mode-2 and (e,f) the HFO mode.

4.4. The spatial LFO-Mode-1

Figure 10 shows the DMD spectrum of the lift coefficient and the DMD spatial mode corresponding to the LFO-Mode-1. The DMD spectrum shows the growth rates of the LFO-Mode-1 and the LFO-Mode-2 indicated by the filled black and red circles, respectively. The growth rates of the LFO-Mode-1 and the LFO-Mode-2 are both positive at this angle of attack, indicating that the two flow modes are growing. At other angles of attack, the growth rates of the LFO-Mode-1 and the LFO-Mode-2 have negative or positive signs, indicating that the LFO-Mode-1 and LFO-Mode-2 are decaying or growing, respectively. The LFO is quasi-periodic; thus, the growth rates for the leading DMD modes should be zero. However, the data from which the DMD spectrum was extracted spans four low-frequency cycles. Should a longer data record be used, a zero growth rate would have been obtained. The size of the circles denotes the relative amplitude of each dynamic mode. The relative magnitude of the LFO-Mode-1 and the LFO-Mode-2 is proportional to the angle of attack and reaches its maximum amplitude at the angle of attack of maximum LFO. At angle of attack of $9.8^{\circ }$ at Reynolds number of $5\times 10^4$ and angle of attack of $11.0^{\circ }$ at Reynolds number of $9\times 10^4$, the LFO-Mode-1 and LFO-Mode-2 contain more than $50\,\%$ of the turbulent kinetic energy and the remaining kinetic energy is almost equally distributed among higher modes as seen in figure 6. Therefore, only a few low-frequency modes are shown in figure 10, and the higher modes become insignificant. This is indicative that the LFO process is fully developed at these angles of attack, and the oscillating flow is more pronounced in magnitude and more coherent in shape. The relative amplitude of the LFO-Mode-1 and the LFO-Mode-2 becomes inversely proportional to the angle of attack when the angle increases above the angle of attack of maximum LFO. The figure shows only the DMD spectrum of the lift coefficient; however, the DMD spectra of the drag, the pressure, the streamwise velocity and the wall-normal velocity are similar to that of the lift coefficient. The LFO-Mode-1 features the triad of vortices as seen in figure 10(b). The triad of vortices is driven by the oscillating streamwise velocity component across the laminar portion of the separated shear layer. The magnitude of the oscillating velocity increases as the angle of attack increases (Eljack & Soria Reference Eljack and Soria2020).

Figure 10.

The LFO-Mode-1 for angle of attack of $9.8^{\circ }$ and Reynolds number of $5\times 10^4$. (a) The DMD spectrum of the lift coefficient, where $\omega _r$ is the growth rate and $St$ is the Strouhal number. (b) Streamline patterns superimposed on colour maps of the spanwise vorticity $\omega _z$ for the DMD spatial mode.

Figure 11 shows the spatial LFO-Mode-1 constructed using the POD, the DMD and the SMD methods. The POD spatial mode for the LFO-Mode-1 was constructed by multiplying its orthonormal spatial POD mode by the average amplitude of its corresponding coefficient $\overline {|b^{(1)}(t)|}$. The POD method recovers the flow modes based on their energy content. Thus, the spatial LFO-Mode-1 constructed using the POD method is contaminated by other high-frequency flow modes. Hence, the shape and evolution of the triad of vortices are obscured by high-frequency flow modes. However, POD captured the general features of this flow mode. In the DMD method, the LFO-Mode-1 is the dynamic mode corresponding to the low-frequency peak in the DMD spectrum of the lift coefficient. Thus, the LFO-Mode-1 represents the instability that features globally oscillating flow around the airfoil, the oscillating pressure along the airfoil chord and the process that creates and sustains the triad of vortices. Hence, the triad of vortices and the general features of this flow mode are accurately captured by the DMD method. However, the LFO-Mode-1 is presented by several dynamic modes; whereas, it was objectively recovered by a single POD mode. Equation (2.2) was used to calculate the pressure, the streamwise velocity and the wall-normal velocity corresponding to the spatial SMD mode of the LFO-Mode-1 based on the time history of the lift coefficient. The zoom-in box shows the triad of vortices constructed using the SMD method. The lift coefficient is dominated by a low frequency corresponding to the LFO-Mode-1. Consequently, the spatial LFO-Mode-1 constructed using the SMD method is similar to that constructed using the DMD method and constitutes a coherent triad of vortices. Furthermore, the SMD method effectively recovered the LFO-Mode-1 in a single flow mode. Thus, the SMD method combines the strengths of the POD and the DMD methods by objectively recovering the flow mode and capturing the instability. As seen in the figure, the LFO-Mode-1 is a global flow mode that drives and sustains the triad of vortices. The spatial LFO-Mode-1 creates an adverse oscillating pressure gradient when the sign of its amplitude is positive, and a favourable oscillating pressure gradient when the sign of its amplitude is negative (see figure 9). The LFO-Mode-1 preserves shape over all of the investigated angles of attack; however, the magnitude of oscillation increases as the angle of attack increases.

Figure 11.

Streamline patterns superimposed on colour maps of the oscillating pressure field for the spatial LFO-Mode-1 constructed using the POD, the DMD and the SMD methods: (a,c,e) $\alpha =9.8^{\circ }$ and ${\textit {Re}}_c = 5\times 10^4$; (b,d,f) $\alpha =11.0^{\circ }$ and ${\textit {Re}}_c = 9\times 10^4$.

4.5. The spatial LFO-Mode-2

The spatial LFO-Mode-2 constructed using the POD, the DMD and the SMD methods is shown in figure 12. The POD mode corresponding to the LFO-Mode-2 was constructed by multiplying its orthonormal spatial POD mode by the average amplitude of its corresponding POD coefficient $\overline {|b^{(2)}(t)|}$. The time history of the drag coefficient was used to estimate the pressure, streamwise velocity and wall-normal velocity corresponding to the SMD construction of the LFO-Mode-2. The LFO-Mode-2 originates and evolves on the suction surface of the airfoil and features the expansion and advection of the upstream vortex of the TCV (Eljack & Soria Reference Eljack and Soria2020). The LFO-Mode-2 constructed using the POD and the SMD methods are similar in shape and magnitude as seen in the figure. The POD and the SMD analyses extracted only two low-frequency modes. Whereas, the DMD analysis decomposed the low-frequency oscillating flow into many low-frequency modes. The low-frequency modes extracted by the POD and the SMD methods are more coherent and energetic compared with those extracted by the DMD method. Furthermore, the DMD and the SMD methods accurately captured the instability. Thus, combining the three methods provided all the information that governs and controls the dynamics of the flow.

Figure 12.

Streamline patterns superimposed on colour maps of the oscillating pressure field for the spatial LFO-Mode-2 constructed using the POD, the DMD and the SMD methods: (a,c,e) $\alpha =9.8^{\circ }$ and ${\textit {Re}}_c = 5\times 10^4$; (b,d,f) $\alpha =11.0^{\circ }$ and ${\textit {Re}}_c = 9\times 10^4$.

4.6. The spatial HFO mode

Figure 13 shows the spatial HFO mode constructed using the POD, the DMD and the SMD methods. As seen in the figure, the HFO mode features large oscillations that originate at the leading edge and advect towards the trailing edge. At the trailing edge, the HFO mode interacts with the local flow instability, stretched by the strong shear, and energizes the vortex shedding. The magnitude of the oscillation is proportional to the angle of attack. It is interesting to note that there is no high-frequency peak in the spectra of the lift and the drag coefficients. The aerodynamic coefficients are direct consequences of variations in the flow field adjacent to the airfoil surface. Furthermore, the pressure field communicates the dynamics of the flow away from the wall to the flow field adjacent to the wall. However, when low-frequency modes of significant amplitude are present in the flow, the high-frequency modes do not much affect the flow variables near the wall and will not be reflected in the aerodynamic forces. Consequently, such an important flow feature will not show up as a peak in the spectra of the aerodynamic coefficients unless enough blocks of data are used and the spectra are smoothed out.

Figure 13.

Streamline patterns superimposed on colour maps of the oscillating wall-normal velocity field for the spatial HFO mode constructed using the POD, the DMD and the SMD methods: (a,c,e) $\alpha =9.8^{\circ }$ and ${\textit {Re}}_c = 5\times 10^4$; (b,d,f) $\alpha =11.0^{\circ }$ and ${\textit {Re}}_c = 9\times 10^4$.

Figure 7 shows that the spectra of the wall-normal velocity peak significantly at high frequency. Whereas, the spectra of the streamwise velocity and the pressure show no significant peak at high frequency. This is indicative that the HFO is primarily driven by the wall-normal velocity component. However, the streamwise velocity and the pressure indirectly affect the behaviour of the HFO mode. Furthermore, the spectra of the wall-normal velocity are interchangeably dominated by the HFO mode and the two low-frequency modes. Thus, the two low-frequency modes and the HFO mode are interlinked and play a profound role in the dynamics of the flow. Copious amounts of research work have concentrated on the characteristics of the HFO mode downstream of the airfoil trailing edge and the evolution of the Kelvin–Helmholtz instability in the vicinity of the leading edge (Jones, Sandberg & Sandham Reference Jones, Sandberg and Sandham2008, Reference Jones, Sandberg and Sandham2010; Almutairi, Eljack & Alqadi Reference Almutairi, Eljack and Alqadi2017). Most of those studies found that there are many aspects of similarities and concluded that the HFO mode generates acoustic waves that travel upstream, energizes the Kelvin–Helmholtz instability in the vicinity of the leading edge and forces early transition. However, the HFO mode originates in the vicinity of the airfoil leading edge, then travels downstream towards the airfoil trailing edge and evolves along the wake of the airfoil as seen in figure 13. Hence the similarity between the characteristics of the HFO mode and the evolution of the Kelvin–Helmholtz instability. An experiment in a water-tunnel or solving the incompressible Navier–Stokes equations numerically would answer the question of whether or not the acoustic waves generated by the HFO mode affect the transition process and the LFO phenomenon.

It is worth noting that the HFO flow along the wake of the airfoil is sinusoidal. Thus, the summation of all Fourier modes would approximate this flow accurately. However, each one of the Fourier modes does not necessarily exist in the flow. On the contrary, the POD algorithm looks for the most energetic structures in the flow convected at the same frequency and collects them in a single flow mode. The POD method continues this optimization process for all frequencies until there is not much energy left in the remaining flow. The SMD method selectively phase-averages at the chosen frequency. Consequently, the SMD method recovers the time-averaged shape of flow structures that physically exist in the flow field. The flow modes recovered by the POD and the SMD methods converge towards similar flow modes when the fluctuating flow field is closely approximated by Fourier modes.

5.1. Temporal evolution of the flow modes

The POD eigenfunctions ($\boldsymbol {\phi }^{(n)}$) are used to reconstruct the flow modes and as basis functions for low-dimensional modelling without much attention to their amplitude or how they evolve in time. In the present study, the POD eigenfunctions were scaled with their corresponding eigenvalues ($\boldsymbol {\lambda }^{(n)}$), and the percentage of the scaled amplitude of each of the temporal POD modes is estimated at each instant in time using (B9). Figure 14 shows time histories of the POD eigenfunctions, the percentage of energy content in each POD mode and the fluctuating lift and the fluctuating drag coefficients at $\alpha =9.8^{\circ }$ and ${\textit {Re}}_c = 5\times 10^4$, and angles of attack of $10.8^{\circ }$ and $11.0^{\circ }$ at ${\textit {Re}}_c = 9\times 10^4$. The angles of attack of $9.8^{\circ }$ and $11.0^{\circ }$ are the angles of attack at which the LFO has the maximum amplitude of oscillations. The LFO at the angle of attack of $10.8^{\circ }$ exhibits three uniform cycles followed by a disturbed cycle, thus representing a special case that is worth presenting. As seen in the figure, the scaled POD eigenfunctions make more sense since they display the temporal evolution of the flow modes with their actual amplitude of energy content compared with other POD modes, and most importantly, they mimic time histories of the fluctuating lift and the fluctuating drag coefficients. At angles of attack where the flow field exhibits an LFO, the LFO-Mode-1 peaks at more than 75 %, and the LFO-Mode-2 peaks at more than 25 %. However, additional data at various Reynolds numbers and angles of attack are required to verify these thresholds.

Figure 14.

Time histories of (a–c) the POD eigenfunctions ($\boldsymbol {\phi }^{(n)}(t)$), (d–f) percentage of the energy content in POD mode $n$ ($\boldsymbol {\beta }^{(n)}(t)$) and (g–i) the fluctuating lift and the fluctuating drag coefficients. The black, red and blue lines indicate the LFO-Mode-1, the LFO-Mode-2 and the HFO mode, respectively. The black and red lines display the fluctuating lift and the fluctuating drag coefficients, respectively.

The LFO-Mode-1 and the LFO-Mode-2 signals are correlated, and their magnitudes are interlinked. That is, if the LFO-Mode-1 peaks at a relatively high amplitude, so will the LFO-Mode-2, and vice versa. When the periodic cycle of the LFO-Mode-1 is disturbed, so is that of the LFO-Mode-2 as seen in the figure. The two low-frequency modes drive the dynamics of the LFO and play a contradicting role. That is, the LFO-Mode-1 decays to a minimum, during which the LFO-Mode-2 grows to a maximum. Thus, when the LFO-Mode-1 vanishes and the flow reaches a temporary equilibrium, the LFO-Mode-2 amplifies and starts a new disequilibrium. The process continues periodically as discussed by Eljack & Soria (Reference Eljack and Soria2020). Therefore, these two low-frequency modes are interlinked and govern the dynamics of the LFO. The LFO-Mode-1 generates the triad of vortices and feeds energy into the upstream vortex of the TCV. When the upstream vortex of the triad of vortices is saturated with energy, it expands, and the LFO-Mode-2 dominates the flow until the oscillating flow switches its direction, then the LFO-Mode-1 takes over again. The HFO mode fluctuates more energetically when the flow is separated as seen in the figure. When the LFO-Mode-2 peaks during an attached flow phase, the upstream vortex of the TCV expands and separates the flow. Vortex shedding from the separated flow energizes the shedding along the wake. Consequently, the HFO mode is energized. That is, the interaction takes place, and the HFO mode becomes more energetic whenever the LFO-Mode-2 peaks to a maximum value. When the upstream vortex expands and advects downstream, it gets attracted by the low-pressure region at the trailing edge and consequently energizes the trailing-edge shedding and the HFO mode as seen in the spatial HFO mode in figure 13.

The time history of the percentage of the energy content in LFO-Mode-1 is not Gaussian as seen in figure 14. The signal exhibits sharp peaks at higher amplitudes and troughs at smaller amplitudes. When the percentage is positive, the oscillating pressure of the LFO-Mode-1 has a negative value in the vicinity of the LSB and a positive value in the vicinity of the trailing edge as seen in figure 11. Consequently, an adverse oscillating pressure works against the LFO-Mode-2 and delays the ejection of the upstream vortex. Thus, the percentage of LFO-Mode-1 peaks sharply at higher amplitudes. On the contrary, the oscillating pressure is favourable when the percentage of the LFO-Mode-1 is negative. Thus, the oscillating pressure helps the LFO-Mode-2 in ejecting the upstream vortex, and the LFO-Mode-1 exhibits troughs at smaller amplitudes.

Another interesting observation is the phase shift in the time histories of the LFO-Mode-1, the LFO-Mode-2, the fluctuating lift and the fluctuating drag. The LFO-Mode-1 and the fluctuating lift are leading the LFO-Mode-2 and the fluctuating drag by a phase shift of ${\rm \pi} /2$. The LFO-Mode-1 has a peak when the flow is attached and a trough when the flow is fully separated. Consequently, the fluctuating lift coefficient mimics the LFO-Mode-1. The upstream vortex ejects and separates (attaches) the flow whenever the LFO-Mode-2 has a peak (trough). Consequently, the oscillating pressure on the suction surface of the airfoil, the pressure difference across the airfoil in the streamwise direction and the fluctuating drag coefficient oscillate accordingly. Thus, the fluctuating drag coefficient mimics the LFO-Mode-2. The pressure field makes the flow feel the wall, and the dynamics of the flow creates oscillations in the aerodynamic forces. However, there is a time lag between the aerodynamic forces and the POD coefficients. Thus, the LFO-Mode-1 and the LFO-Mode-2 are systematically leading the fluctuating lift and the fluctuating drag with a few degrees in phase shift as seen in the figure.

5.2. Reconstruction of the flow field

The orthonormal POD spatial modes and the POD coefficients can be used to reconstruct the original flow field using any subset of the POD modes. The flow field was reconstructed and probed at selected locations using the LFO-Mode-1; the LFO-Mode-1 and the LFO-Mode-2; and the LFO-Mode-1, the LFO-Mode-2 and the HFO mode. A comparison of the reconstructed signals probed at different locations shows that the streamwise velocity, the wall-normal velocity and the pressure have interesting behaviour in the vicinity of the leading edge, mid-chord and downstream of the trailing edge on the suction surface of the airfoil. Figure 15 shows comparisons of the original LES data and the reconstructed POD data for the streamwise velocity, the wall-normal velocity and the pressure at angle of attack of $9.8^{\circ }$ and Reynolds number of $5\times 10^4$. The leading-edge, the mid-chord and the trailing-edge probe were chosen to be in a location inside the upstream vortex of the TCV, at the mid-chord point close to the wall and at about $0.4$ chords downstream of the trailing edge, respectively. The grey solid line indicates the original LES data. The black, red and blue solid lines display the reconstructed data using the LFO-Mode-1; the LFO-Mode-1 and the LFO-Mode-2; and the LFO-Mode-1, the LFO-Mode-2 and the HFO mode, respectively. Figures 15(a,d,g), 15(b,e,h) and 15(c,f,i) show signals of the flow variables in the vicinity of the airfoil leading edge, mid-chord and downstream of the trailing edge, respectively. The three most dominant POD modes (the LFO-Mode-1, the LFO-Mode-2 and the HFO mode) reconstructed the oscillating flow favourably. However, these three dominant flow modes contain a maximum of $65\,\%$ of the energy of the flow in all of the investigated angles of attack. Thus, at least $35\,\%$ of the energy content is not accounted for by these three dominant modes. It is noted that these three dominant flow modes represent the periodic motion of the flow ‘total fluctuations’ and not the random motion of the flow ‘turbulent fluctuations’. Thus, the remaining $35\,\%$ of energy content is ‘turbulent fluctuation’; therefore, it is not reflected by these three modes.

Figure 15.

Time histories of the LES data of (a–c) the streamwise velocity, (d–f) the wall-normal velocity and (g–i) the pressure probed at the vicinity of the leading edge, mid-chord and downstream of the trailing edge on the suction surface of the airfoil compared with their corresponding POD reconstruction of the data at angle of attack of $9.8^{\circ }$ and Reynolds number of $5\times 10^4$. Grey solid line, LES data; black solid line, reconstructed data using the LFO-Mode-1; red solid line, reconstructed data using the LFO-Mode-1 and the LFO-Mode-2; blue solid line, reconstructed data using the LFO-Mode-1, the LFO-Mode-2 and the HFO mode.

The leading-edge probe shows that the velocity components and the pressure are reconstructed mostly by the LFO-Mode-1 and the LFO-Mode-2; thus, the HFO mode does not contribute much to any of the flow variables. This is indicative that the HFO mode does not directly influence the flow at this location. Furthermore, the original data of the velocity components are mostly recovered by the LFO-Mode-2. While the LFO-Mode-1 has a small and uniform effect, it is the LFO-Mode-2 that shaped the signal to its original LES form. This is indicative that the velocity components in this location are mostly influenced by the expansion and advection of the upstream vortex of the TCV rather than being influenced by the LFO-Mode-1. The pressure field is mostly recovered by the LFO-Mode-1 in this location with a small and limited effect of the LFO-Mode-2 and the HFO mode. The trailing-edge probe shows that the flow variables are overwhelmed by the HFO mode and its subharmonics. However, the low-frequency pattern exists in all of the flow variables at this location. Thus, the velocity components are influenced exclusively by the LFO-Mode-1 and the HFO mode. Whereas, the pressure at this location is influenced by the LFO-Mode-1 and the HFO mode in addition to a limited effect of the LFO-Mode-2.

5.3. Spatial evolution of the flow field

Figure 16 shows the reconstructed fluctuating flow using the LFO-Mode-1, the LFO-Mode-2 and the HFO mode at angle of attack of $10.8^{\circ }$ and Reynolds number of $9\times 10^4$. The main panel displays streamlines of the fluctuating flow superimposed on colour maps of the spanwise vorticity, and the top-left inset shows streamline patterns of the reconstructed instantaneous flow superimposed on colour maps of the instantaneous streamwise velocity. Time histories of the lift and drag coefficients are also shown in the bottom-right inset. The LSB and the triad of vortices are visualized by the streamline patterns of the instantaneous and fluctuating flow, respectively. The LFO-Mode-1 extracts energy from the mean flow and stores it in the triad of vortices. When the sign of the LFO-Mode-1 is positive, the oscillating flow rotates around the airfoil in the clockwise direction and flows in the streamwise direction on the suction surface of the airfoil. Consequently, the oscillating flow adds momentum to the boundary layer and helps it to remain attached against the APG. At a specific energy threshold of the LFO-Mode-2, the upstream vortex expands and separates the flow. However, the process reverses its direction as the LFO-Mode-1 switches its sign, and the process continues in a periodic manner (Eljack & Soria Reference Eljack and Soria2020). The HFO mode oscillates at a relatively small amplitude, but it tends to be more energetic when the LFO-Mode-2 peaks. As discussed before, the upstream vortex expands and advects downstream whenever the LFO-Mode-2 peaks. Consequently, the HFO mode fluctuates at a relatively higher amplitude during the separated phase of the flow field.

Figure 16.

Streamline patterns of the POD reconstruction of the fluctuating flow superimposed on colour maps of the spanwise vorticity (main panel), streamline patterns of the POD reconstruction of the instantaneous flow superimposed on the instantaneous streamwise velocity (top-left inset) and plot of time histories of the fluctuating lift and the drag coefficients (bottom-right inset). (a) Attached phase of the flow. (b) Separated phase of the flow (multimedia view).

The evolution of the flow shows how the triad of vortices switches the direction of the fluctuating flow and separates/attaches the instantaneous flow field as seen in supplementary movie 1 available at https://doi.org/10.1017/jfm.2024.855. The reconstructed oscillating flow using the LFO-Mode-1, the LFO-Mode-2 and the HFO mode is added to the mean flow to obtain the reconstructed instantaneous flow field. The flow is attached, and the LFO-Mode-1 is at its maximum amplitude at the flow time of $258.25$. The flow is attached, the oscillating flow direction is clockwise and the triad of vortices is present and in their most coherent state. At the flow time of $258.5$, the upstream vortex of the triad of vortices pops up above the downstream vortex of the triad of vortices. At the flow time of $259.0$, the upstream vortex pops a little more above the upstream vortex of the TCV. At the flow time of $259.5$, the upstream vortex starts to slide above the downstream vortex of the TCV. The downstream vortex starts to merge with the upstream vortex and energizes it at the flow time of $260.0$. While the downstream vortex merges with the upstream vortex, the latter continue to expand until the two vortices form one vortex at the flow time of $260.5$. At the flow time of $261.0$, the recently formed vortex expands abruptly. It is interesting to note that the LFO-Mode-2 peaks after this instant in time, as shown in figure 14 at angle of attack of $10.8^{\circ }$. After that, the recently formed vortex continues to expand and starts to change the direction of the oscillating flow. The secondary vortex is present and intact during this process. The oscillating flow then changes its direction of rotation from the clockwise to the counter-clockwise direction. At the flow time of $264.0$, the secondary vortex vanishes, and a new downstream vortex forms. As the oscillating flow continues to change its direction of rotation, a new upstream vortex starts to form at the flow time of $266.0$. It is interesting to note how the vorticity changes sign after this instant in time and how it creates the new upstream vortex of the TCV that is rotating in the counter-clockwise direction. The newly formed upstream vortex grows in size and strength at flow times of $267.5$ and $268.5$. The flow is fully separated, and the LFO-Mode-1 is at its minimum amplitude at the flow time of $269.0$. The oscillating flow direction is counter-clockwise, and the triad of vortices is present and in their most coherent state. Consequently, the lift coefficient drops to its minimum value as seen in the figure. Reconstruction of the flow field provided a detailed description of how the flow separates. When the cycle of the LFO proceeds to a flow time larger than $270.0$, the process reverses its direction, and the fully separated flow starts to attach. Thus, a similar sequence of events takes place, as shown in supplementary movie 1. The evolution of the oscillating flow is animated to reveal how the flow separates and attaches. Thus, a detailed description of the LFO phenomenon is given, and the underlying mechanism is discussed in detail. It is important to mention that the POD spatial mode corresponding to LFO-Mode-1 does not exhibit the triad of vortices. However, the reconstruction of the flow field using the three dominant flow modes clearly shows the evolution and dynamics of the triad of vortices. None of the POD modes recover the triad of vortices as discussed in § 4.4, but the summation of two or more of the most dominant POD modes reconstructs a flow field that exhibits the triad of vortices and their evolution.

Figure 17 shows the reconstructed fluctuating flow using the SMD flow modes corresponding to the LFO-Mode-1, the LFO-Mode-2 and the HFO mode. The flow is attached at the starting point in time at $t=206.0$ as seen in figure 14. The evolution of the oscillating flow is visualized in step-by-step snapshots that show how the triad of vortices separates the flow. When the cycle of the LFO proceeds to flow time ${>}217.5$, the process reverses its direction, and the fully separated instantaneous flow starts to attach as seen in figure 14. Thus, a similar sequence of events takes place, and the SMD method accurately reconstructs the flow field using the most dominant flow modes.

Figure 17.

(a–r) Streamline patterns superimposed on colour maps of the spanwise vorticity ($\omega _z$) for the SMD reconstruction of the fluctuating flow at $\alpha =9.8^{\circ }$ and Reynolds number of $5\times 10^4$.

5.4. The effect of the spanwise extent on the LSB and the LFO phenomenon

Figure 5 shows that the LES data of the present work compare very well with the LES data of Alferez et al. (Reference Alferez, Mary and Lamballais2013) and Alferez (Reference Alferez2014) representing the flow field about a NACA0012 airfoil of aspect ratio of one. However, we would like to examine the effect that the narrow span of $0.5$ chords used in the present work can have on the obtained results. For instance, stall cells would appear at relatively high angles of attack; however, they might be suppressed by the narrowness of the span, consequently questioning the accuracy of the data and the generality of the conclusions based on them.

Previous studies on two-dimensional airfoil stalling characteristics have shown a low-frequency and highly unsteady flow or a steady large-scale three-dimensional structure. The latter is termed stall cells, and there has been a considerable amount of research focused on their structure (Winkelman & Barlow Reference Winkelman and Barlow1980; Dallmann Reference Dallmann1983; Winkelman Reference Winkelman1990; Yon & Katz Reference Yon and Katz1998; Rodríguez & Theofilis Reference Rodríguez and Theofilis2010, Reference Rodríguez and Theofilis2011; He et al. Reference He, Gioria, Pérez and Theofilis2017). Winkelman & Barlow (Reference Winkelman and Barlow1980) carried out oil-flow visualizations of the flow field on the suction surface of stalled wings of a Clark Y airfoil section at chord Reynolds numbers of $24.5\times 10^4$, $26\times 10^4$ and $38.5\times 10^4$. The ends of the wing are flush with the tunnel side walls or splitter plates and on plane rectangular wings of infinite aspect ratio. They observed that mushroom-shaped cells started to form at the onset of stall on the two-dimensional model. The three-dimensional cells coexisted with trailing-edge stall cells on the wing's surface several degrees above the stalling angle of attack. They tentatively sketched a flow field model showing the general features of a leading-edge separation bubble and trailing-edge separation. Winkelman (Reference Winkelman1990) took up the work of Winkelman & Barlow (Reference Winkelman and Barlow1980) and measured the spectra of the velocity in the wake of a rectangular wing having the same airfoil section. The spectra of the wake measurements did not show any indications of a dominant low-frequency mode. Yon & Katz (Reference Yon and Katz1998) used fine-thread tuft-flow visualization and high-frequency response pressure transducer measurements to investigate the unsteady features of the stall cells. They studied the flow field around a wing of aspect ratio ranging from $2$ to $6$ with a NACA0015 airfoil section. They imposed the two-dimensionality using end plates that effectively eliminated the tip flow. The authors reported the existence of a low-frequency mode in their pressure spectrum but with a considerably small amplitude of oscillation.

Broeren & Bragg (Reference Broeren and Bragg2001) studied five different airfoil configurations (NACA2414, NACA64A010, LRN-1007, E374 and Ultra-Sport) by measuring the wake velocity across the spanwise direction and using mini-tufts for flow visualization. They found that all the stall types were dependent on the type of airfoil. They concluded that the LFO phenomenon always occurs in airfoils that exhibit a thin-airfoil stall or a combination of thin-airfoil and trailing-edge stall. Most importantly, they found that the low-frequency unsteadiness is essentially two-dimensional. Their conclusions were in good agreement with that of Zaman et al. (Reference Zaman, McKinzie and Rumsey1989), who observed that the LFO takes place with airfoils exhibiting either trailing-edge- or thin-airfoil-type stalls but does not take place with the leading-edge-type stall. To this end, it is evident that the LFO phenomenon is inherently two-dimensional by nature, and neither the imposed periodic boundary condition nor the spanwise width of the computational domain affects the accuracy of the data and the conclusions based on them.

To further examine the effect of spanwise width on the shape of the LSB and the LFO phenomenon, an LES was carried out at an angle of attack of $9.4^{\circ }$, Reynolds number of $5\times 10^4$ and a spanwise width of two chords. The time history of the lift coefficient at each location in the spanwise direction $\textit{C}_{\!_\textit{L}}(x_3, t)$ was used to estimate the time-averaged surplus of the oscillating flow on the fly to obtain the three-dimensional LFO-Mode-1. Figure 18 shows the time-averaged shape of the LSB and the triad of vortices. A uniform seeding of the streamlines was done by drawing a line parallel to the spanwise direction and specifying a certain number of points along this line. Each of the streamlines that pass by any of these points is automatically completed and allowed to extend in both directions; hence, the visualization of the streamlines is user-independent. Despite the irregularities encountered in the LSB and the triad of vortices, the flow field is essentially two-dimensional, and stall cells are not present in any form as seen in the figure.

Figure 18.

Streamline patterns of the mean flow and the SMD mode corresponding to the lift coefficient plotted around a NACA0012 airfoil of spanwise width of two chords at an angle of attack of $9.4^{\circ }$ and Reynolds number of $5\times 10^4$ showing the time-averaged shape of the LSB (a) and the triad of vortices (b).

5.5. The effect of compressibility on the LFO phenomenon

Classically, the bursting of the LSB and its associated LFO used to be investigated in compressible flow settings. Experiments are carried out in wind-tunnels rather than in water-tunnels and compressible Navier–Stokes equations are solved in numerical simulations rather than solving the incompressible set of equations. The vast majority of investigations being compressible is mainly due to the widely accepted hypothesis that there is an acoustic wave feedback mechanism involved. It is assumed that the acoustic waves generated at the trailing edge of the airfoil travel upstream and interact with some receptivity mechanism in the vicinity of the leading edge (Jones et al. Reference Jones, Sandberg and Sandham2008, Reference Jones, Sandberg and Sandham2010; Almutairi et al. Reference Almutairi, Eljack and Alqadi2017; Kurelek, Kotsonis & Yarusevych Reference Kurelek, Kotsonis and Yarusevych2018; Pröbsting & Yarusevych Reference Pröbsting and Yarusevych2021). Such feedback would force the shear layer to undergo early transition and reattach the flow. The movement of the transition location along the shear layer, causing late transition/early transition, is synchronized with the flow separation and reattachment, respectively. Thus, the acoustic wave feedback mechanism generates and sustains the LFO phenomenon. On the contrary, flow forcing using a selected set of frequencies could remove the LFO as reported by Zaman et al. (Reference Zaman, Bar-Sever and Mangalam1987) and Eljack, Alqadi & Almutairi (Reference Eljack, Alqadi and Almutairi2018). Such a conflicting role of the acoustic excitation was also noted by Zaman et al. (Reference Zaman, McKinzie and Rumsey1989).

Almutairi, Alqadi & Eljack (Reference Almutairi, Alqadi and Eljack2015) and Almutairi et al. (Reference Almutairi, Eljack and Alqadi2017) applied the DMD method to the pressure field of the flow field around a NACA0012 airfoil at ${\textit {Re}}_c = 1.3\times 10^5$, $M = 0.4$ and angle of attack of $11.5^{\circ }$. The authors sampled the instantaneous pressure field on the $x$–$y$ plane at a non-dimensional frequency of $658$, and the data span about one low-frequency cycle ($25$ non-dimensional time units). The DMD identified two dominant flow modes: a low-frequency mode at Strouhal number of $0.008$ featuring the bursting and reformation cycle of the LSB and a high-frequency mode featuring the trailing-edge shedding frequency. They concluded that the trailing-edge shedding induces acoustic waves that travel upstream and excite the separated shear layer via some receptivity mechanism and forces it to undergo early transition and reattach the separated flow. Once the flow is attached, the strong vortex shedding downstream of the trailing edge dies down, and the acoustic feedback is cut off; consequently, the flow separates again, and the separation/reattachment process repeats periodically. However, the authors applied the DMD method to the pressure field only; therefore, the authors did not show streamlines corresponding to the low-frequency mode. Consequently, the triad of vortices discovered and explained by Eljack & Soria (Reference Eljack and Soria2020) was not presented in their work.

The model of the LFO phenomenon presented by Eljack & Soria (Reference Eljack and Soria2020) and the discussion in § 5.3 above show that the circumstances that govern the stability of the LSB and its associated LFO phenomenon can be met in both compressible and incompressible flows. Figure 19 shows the variation of the mean velocity above and below that of the mean flow due to attachment and separation of the flow, respectively. The production of kinetic energy increases and an early transition occurs when the flow is attached due to the increase in the velocity gradient across the separated shear layer and vice versa. Thus, the movement of the transition location is interlinked with the oscillating flow induced by the LFO-Mode-1 and the LFO-Mode-2. Consequently, the oscillating flow affects the velocity gradient across the separated shear layer and the development of the Kelvin–Helmholtz instability. In conclusion, previous work on the LFO suggests that the acoustic wave feedback mechanism is crucial for the LFO phenomenon. Whereas, the present work shows that an interplay of three vortices induces the LFO phenomenon without excluding the acoustic waves scenario.

Figure 19.

(a,b) The scaled mean streamwise velocity (${U}_1/{U}_{\infty }$) at the location of maximum production of turbulent kinetic energy along the separated shear layer at the stall angle of attack and the angle of attack of maximum LFO. Black line, time-averaged flow; red line, time-average attached flow; blue line, time-average separated flow.

Rodríguez et al. (Reference Rodríguez, Lehmkuhl, Borrell and Oliva2013, Reference Rodríguez, Lehmkuhl, Borrell and Oliva2015) carried out several direct numerical simulations in which they solved the incompressible Navier–Stokes equations. They investigated the flow field around a NACA0012 airfoil at Reynolds number of $5\times 10^4$ and angles of attack of $5^{\circ }$, $8^{\circ }$, $9.25^{\circ }$ and $12^{\circ }$. The authors reported that they started the simulations from an initially homogeneous flow field that introduced some numerical disturbances. However, the authors did not clearly state whether or not they captured a global LFO. They have not shown time histories of the aerodynamic coefficients; thus, the reader cannot determine if the LFO phenomenon was captured. The only low-frequency flow phenomenon reported by the authors is the shear-layer flapping on the suction surface of the airfoil at Strouhal number of $0.021$. To the best of the author's knowledge, these are the only incompressible studies that investigated the LFO in the flow field around an airfoil at near-stall conditions. However, further investigations are required before a conclusion on whether or not an LFO phenomenon can be captured in an incompressible flow. Therefore, experimental measurements in a water-tunnel or solving the incompressible Navier–Stokes equations in a direct numerical simulation are merited. In the case of the numerical simulation, starting the simulation using clean flow might help capture the phenomenon. Whereas, in the water-tunnel experiment, adjusting the free-stream turbulence to the right level is crucial for capturing the phenomenon.