3.1. Mode classification

Four distinct modes of the flag are identified according to the free-stream velocity $U$ and the distance $d$ between the wall and the flagpole. The classification of the modes depends on the location of the wall. When a wall is installed above the flag, the wall distance $d$ does not affect the initial configuration of the flag at $U$ = 0 (figure 1ai), and the initial configuration is determined by gravity and the bending rigidity of the flag. Under a fluid flow, the fluid force exerted on the flag is balanced by the weight and internal bending force of the flag, yielding quasistatic deformation of the sheet with negligible vibrations: quasistatic equilibrium mode in figure 2(ai) and supplementary movie 1 available at https://doi.org/10.1017/jfm.2023.931. As $U$ increases beyond a critical value $U_{c1}$, the flag becomes unstable and flutters asymmetrically due to the effect of gravity (figure 2aii). With a further increase in $U$, the amplitude of the flag increases accordingly, and the flag begins to contact the wall periodically (figures 2aiii and 2aiv).

Figure 2. (a) Superimposed images of the flag for the upper wall: (i) quasistatic equilibrium mode;(ii) flutter mode; (iii) partial contact mode; (iv) saturated contact mode. (b) Superimposed images of the flag for the lower wall: (i) quasistatic equilibrium mode; (ii) partial contact mode; (iii) saturated contact mode. See supplementary movies 1 and 2 for (a) and (b), respectively.

By contrast, when a wall is installed below the flag, the wall distance $d$ affects the initial configuration of the flag at $U$ = 0 (figure 1aii), and the flag is in contact with the wall in the entire range of $d$ examined in this study. As $U$ increases, the quasistatic equilibrium state directly transitions to the periodic contact state at a critical velocity $U_{c1}$ (figure 2b and supplementary movie 2). In the postequilibrium state, the flutter mode without contact, which corresponds to figure 2(aii) for the upper wall, does not occur, and the flag contacts the wall periodically once it begins to flutter. The chaotic contact mode reported in the previous studies with a vertical flag and a nearby vertical wall (Bae et al. Reference Bae, Lee, Kim, Ha, Lee, Park, Choong, Kim, Wang and Kim2014; Lee et al. Reference Lee, Kim and Kim2021) is not observed for the flow velocity range available in the present study.

The general sequence of contact between the flag and the wall is illustrated in figure 3. In the case of an upper wall, the contact process is determined by how close the flagpole is to the wall, because the downward gravitational force interrupts the contact. For small $d$ ($d\leq 2.0$ cm), the contact commences at a specific point in the middle of the flag and propagates downstream, increasing the area of contact (figures 3ai–3aiii). The contact area then decreases until contact occurs at the trailing edge of the flag (figures 3aiv and 3av). Finally, the flag is fully separated from the wall (figure 3avi). This contact process is similar to when a flag makes contact under the condition that the influence of gravity is negligible, e.g. a vertical flag near a vertical wall (Bae et al. Reference Bae, Lee, Kim, Ha, Lee, Park, Choong, Kim, Wang and Kim2014). However, when $d$ is large ($d\geq$2.5 cm), contact begins near the trailing edge of the flag and propagates only slightly upstream (figures 3bi–3biii). The subsequent process is similar to the small-$d$ case until the flag is fully separated from the wall (figures 3biv–3bvi).

Figure 3. Sequences of contact process for the upper wall: (a) small wall distance $d$ and (b) large $d$.

For the lower wall, the contact process is similar to that of the upper wall with small $d$, as depicted in figure 3(a). Regardless of $d$, the contact starts at a specific point in the middle of the flag and propagates downstream. Gravity acting downwards makes the contact begin in the middle of the flag and propagate along the streamwise direction, even for the large-$d$ case. This is the distinction in the contact process with a large $d$ between the upper wall and lower wall. In summary, as our flag model is horizontal and sufficiently soft for large deflection, the influence of gravity is sufficiently strong that the distance $d$ and the location of the wall significantly affect the contact process.

The contact state is subdivided into partial and saturated contact modes for both the upper and lower walls (figure 2) according to the dramatic change in the contact process across a threshold of $U$. To analyse the contact process in depth, two quantities are considered. The location at which contact begins is identifiable from the images, and the streamwise distance between the leading edge and the first contact point is defined as $X_c$ (figure 4a). In addition, the duration from when the flag begins to contact the wall at a certain point to when it becomes fully separated from the wall is defined as $T_c$; $T_c = 0$ in the quasistatic equilibrium and flutter modes. Because the dynamics of our model during contact are strongly affected by gravity and wall distance and is generally different from that of a vertical flag with a nearby vertical wall, the new terms for contact modes are introduced to describe the contact process more properly instead of employing the terms used by Lee et al. (Reference Lee, Kim and Kim2021).

Figure 4. (a) Definition of streamwise distance $X_c$ between the leading edge and the first contact point for (i) small $d$ and (ii) large $d$. Here (b) $X_c$ and (c) contact duration $T_c$ with respect to free-stream velocity $U$ for the upper wall $([d, h] = [1.5\ {\rm cm}, 1.0\ {\rm mm}])$. The green and yellow areas denote the partial contact and saturated contact modes, respectively.

Irrespective of the wall location (either upper or lower) and distance $d$, $X_c$ gradually decreases with increasing $U$, implying that a greater area of the flag contacts the wall during the contact process (figure 4b): partial contact mode. However, above a certain $U$, $X_c$ resides in a narrow range. This saturation of $X_c$ indicates that the contact is fully developed: saturated contact mode. Furthermore, the contact duration $T_c$ increases rapidly with $U$ in the contact state, before reaching a plateau (figure 4c). For the same model configuration (upper or lower wall), wall distance $d$ and flag thickness $h$, the threshold value of $U$ beyond which $T_c$ remains almost unchanged coincides with the threshold corresponding to the saturation of $X_c$ (figure 4b,c). In this study, the critical velocity for the transition between the partial and saturated contact modes are termed $U_{c2}$ for both the upper and lower walls; the critical conditions for mode transitions are discussed in § 3.2.

3.2. Mode distribution and critical velocity

To quantitatively analyse the interaction between a flag and a surrounding fluid, many previous studies have adopted a dimensionless flow velocity $U^*=U(\rho _f l^3 /B)^{1/2}$, which represents the ratio of the fluid force exerted on the flag to its internal bending force (e.g. Connell & Yue Reference Connell and Yue2007; Eloy et al. Reference Eloy, Lagrange, Souilliez and Schouveiler2008; Kim et al. Reference Kim, Cossé, Cerdeira and Gharib2013, Reference Kim, Kang and Kim2017). However, the gravitational force is also important in our flag model. The bending force per unit width, $F_b$, scales as $B /l^2$ for large deflections, where the bending stiffness per unit width $B =Eh^3/12(1-\nu ^2)$ and the gravitational force per unit width $F_g$ scales as $\rho _s g h l$. The ratio of the two forces, $F_b/F_g \sim B/\rho _s g h l^3$, is 0.24 for a flag thickness $h$ of 1 mm and 0.06 when $h$ is 0.5 mm. The gravitational force is dominant over the bending force in our model, making it acceptable to define the dimensionless flow velocity based on the ratio of fluid force to gravitational force.

As mentioned in § 3.1, for the upper-wall case, the distance between the wall and the flagpole, $d$, does not affect the initial configuration of the flag at $U=0$. The fluid force exerted on the flag scales with the vertical downward deflection $\delta$ of the flag trailing edge, from the horizontal line on the flagpole, in the absence of flow (figure 1ai). Even for the fluttering flag, the fluid force is more relevant to $\delta$ than to $d$ because $\delta$ is greater than $d$ in our model. The fluid force per unit width then scales as $F_f \sim \rho _f U^2 \delta$. In the previous studies regarding the interaction of an elastic sheet with a fluid flow, the large deflection of the sheet was scaled as the length of the elastic sheet (Jung, Song & Kim Reference Jung, Song and Kim2021; Kim et al. Reference Kim, Lahooti, Kim and Kim2021; Lee, Joung & Kim Reference Lee, Joung and Kim2022). Because of the use of soft material in this study, the deflection of the flag is comparable to the flag length in the absence of flow and sufficiently large in the most range of the quasistatic equilibrium state. Therefore, $\delta$ scales as the flag length $l$, leading to $F_f \sim \rho _f U^2 l$.

This scaling of the fluid force is the same as that found in previous studies on fluttering flags in the absence of gravity and contact. With the gravitational force per unit width $F_g \sim \rho _s g h l$, a new dimensionless flow velocity for the upper wall is suggested as

(3.1)\begin{equation} U^*=U\left(\frac{\rho_f}{\rho_s g h}\right)^{1/2} =Fr (m^*)^{{-}1/2}, \end{equation}

which indicates the relative magnitude of the fluid force with respect to the gravitational force. Here $U^*$ can also be defined using the Froude number $Fr(=U/(g l)^{1/2}$) and the mass ratio $m^*(=\rho _s h / \rho _f l)$.

For the lower-wall case, it is clear that the fluid force $F_f$ is influenced by $d$ because the flag is initially in contact with the wall and $d$ determines the initial configuration of the flag at $U=0$; note that we limit the range of $d$ so that the flag is in contact with the lower wall at $U=0$. The fluid force scales reasonably as $F_f \sim \rho _f U^2 d$, while the scaling of the gravitational force $F_g$ is the same as in the upper-wall case. Considering the ratio of the two forces, a new dimensionless flow velocity for the lower wall is defined as

(3.2)\begin{equation} \hat{U}=U\left(\frac{\rho_f d}{\rho_s g h l}\right)^{1/2}=Fr (m^*)^{{-}1/2}(d^*)^{1/2}, \end{equation}

where $\hat {U}$ can also be expressed in terms of $Fr$, $m^*$ and the wall distance ratio $d^*(=d/l)$.

Using the dimensionless flow velocity ($U^*$ for the upper wall and $\hat {U}$ for the lower wall) and the distance ratio $d^*(=d/l)$, the flag modes described in § 3.1 are illustrated in figure 5. For the upper wall (figure 5a), the dimensionless first critical velocity $U^*_{c1}$ between the quasistatic equilibrium mode and flutter mode is independent of $d^*$. Because the flag is deflected downwards and the initial configuration of the flag is independent of the wall distance, $d^*$ is not a factor in the transition between these two modes. However, the dimensionless second critical velocity $U_{c2}^*$ between the partial contact mode and saturated contact mode increases monotonically with $d^*$. As $d^*$ becomes greater, the transition to the saturated contact mode occurs at a higher dimensionless flow velocity $U^*$. Another critical velocity for the transition between the flutter mode and partial contact mode, which appears only for the upper wall, is not examined because the present study aims to compare critical conditions for the upper-wall and lower-wall cases.

Figure 5. (a) Distribution of four modes in terms of $U^*$ in (3.1) and $d^*$ for the upper wall ($m^*=19.16$). (b) Distribution of three modes in terms of $\hat {U}$ in (3.2) and $d^*$ for the lower wall ($m^*=19.16$).

The trends in the dimensionless critical velocities for the lower wall are similar to those for the upper wall, although we have different definitions of the dimensionless flow velocity, (3.1) for the upper wall and (3.2) for the lower wall. By adopting a dimensionless flow velocity appropriate for each configuration, it is possible to characterize the mode transition trends consistently. In figure 5(b), the dimensionless first critical velocity $\hat {U}_{c1}$ exhibits little variation with $d^*$, whereas the dimensionless second critical velocity $\hat {U}_{c2}$ increases monotonically with $d^*$. Because the wall distance $d$ is included in the definition of $\hat {U}$ in (3.2), the relationship between the dimensional wall distance $d$ and the dimensional critical velocities $U_{c1}$ and $U_{c2}$, with the other dimensional parameters unchanged, cannot be directly captured from figure 5(b).

In contrast to our upper-wall configuration and the vertical flag/wall configuration of Lee et al. (Reference Lee, Kim and Kim2021) in which the first critical velocity $U_{c1}$ is insensitive to $d$ for $d = 1.5\unicode{x2013}5.5$ cm and $l= 5\unicode{x2013}16$ cm, $U_{c1}$ decreases monotonically with increasing $d$ for our lower-wall configuration, over the entire range of $d$ considered in this study (figure 6a). As $d$ increases, a greater fluid force is imposed on the flag, making the transition to the postequilibrium state easier. For the lower wall with the smallest $d$ = 1.0 cm, the gap between the flagpole and the wall is too narrow, and the fluid force is not sufficient to induce the transition to the postequilibrium state. For this reason, the value of $U_{c1}$ is omitted for $d$ = 1.0 cm in figure 6(a). In this study, the range of $d$ for the lower wall is smaller than the initial downward deflection $\delta$ of the trailing edge for the upper wall (figure 1ai). The magnitude of the fluid force exerted on the sheet is smaller for the lower wall than for the upper wall at a given $U$ in the quasistatic equilibrium state. Consequently, $U_{c1}$ for the lower wall exceeds that for the upper wall under the same wall distance $d$ and flag thickness $h$ (figure 6a).

Figure 6. (a) First critical velocity $U_{c1}$ and (b) second critical velocity $U_{c2}$ with respect to wall distance $d$ for both the upper-wall and lower-wall configurations: $h = 1.0$ mm ($m^* = 19.16$) and $h = 0.5$ mm ($m^* = 9.58$).

As the second critical velocity $U_{c2}$ is a threshold that subdivides the postequilibrium state into partial and saturated contact modes, its analysis should be conducted from a different point of view to that of $U_{c1}$ related to the equilibrium state. When the flag is in the equilibrium state, the fluid force scales with the flag length $l$ for the upper wall and the wall distance $d$ for the lower wall. However, when the flag reaches the periodic contact modes, the effect of the incoming flow is mainly determined by the oscillation amplitude of the flag, which is generally comparable to or greater than the wall distance $d$. Therefore, the relationship between $U_{c2}$ and $d$ is expected to have a similar tendency, regardless of the wall location (upper wall or lower wall). Indeed, $U_{c2}$ increases monotonically with respect to $d$ for both wall locations (figure 6b). When the fluttering flag contacts the wall periodically, the $y$-coordinate of the flag's centre of mass, $y_G$, changes periodically as well, and the vertical displacement of $y_G$ increases with $d$. The flag requires more potential energy to make saturated contact. As the potential energy of the flag comes from the kinetic energy of the incoming flow, $U_{c2}$ increases with $d$.

Although the tendency of $U_{c2}$ with respect to $d$ is similar between the two wall locations, the magnitude of $U_{c2}$ at the same $d$ is clearly affected by the wall location, and $U_{c2}$ for the upper wall exceeds that for the lower wall in most conditions (figure 6b). This is because, for the upper wall, the presence of gravity acting in the opposite direction interrupts the fully developed contact of the flag with the wall. Moreover, for both wall locations, $U_{c2}$ becomes greater with increasing flag thickness $h$ for a given $d$ because a greater fluid force is necessary for the heavier flag (figure 6b); this is also true for $U_{c1}$ (figure 6a). This trend with respect to $h$ can be attributed to the increase in internal bending force induced by the enhanced bending stiffness, in addition to the increase in the gravitational force.

Thus far, we have reported the general trends of the critical velocities. Next, using simple scaling analysis, we derive quantitative relationships between the wall distance and two critical velocities ($U_{c1}$ and $U_{c2}$) for the upper wall and lower wall, respectively. For the upper wall, the fluid force acting on the flag is balanced by the gravitational and bending forces in the quasistatic equilibrium. As mentioned above, the gravitational force is generally more significant than the bending force for our flag model. Therefore, it is assumed that the transition from quasistatic equilibrium mode to flutter mode is initiated when the fluid force becomes greater than the gravitational force. The fluid force per unit width $F_f \sim \rho _f U^2 l$ and the gravitational force per unit width $F_g \sim \rho _s g h l$ obey the following scaling relations at the first critical condition $U = U_{c1}$:

(3.3a)$$\begin{gather} \rho_f U_{c1}^2 l \sim \rho_s g h l, \end{gather}$$

(3.3b)$$\begin{gather}Fr_{c1} (m^*)^{{-}1/2} \sim \mathrm{const}. \end{gather}$$

Corresponding to $U = U_{c1}$, the first critical Froude number is defined as $Fr_{c1}$ = $U_{c1} /(g l)^{1/2}$. The term $Fr_{c1} (m^*)^{-1/2}$ on the left-hand side of (3.3b) is actually the same as $U_{c1}^*$ in (3.1). As shown in figure 5(a), $Fr_{c1} (m^*)^{-1/2}$ ($=U_{c1}^*$) is independent of $d^*$.

Regarding the lower wall, the wall distance $d$, rather than $l$ (or $\delta$), is the key length parameter in determining the fluid force in the equilibrium state. The fluid force $F_f \sim \rho _f U^2 d$ instead of $\rho _f U^2 l$. The gravitational force $F_g \sim \rho _s g h l$ is the same as for the upper wall. Accordingly, at $U = U_{c1}$,

(3.4a)$$\begin{gather} \rho_f U_{c1}^2 d \sim \rho_s g h l, \end{gather}$$

(3.4b)$$\begin{gather}Fr_{c1} (m^*)^{{-}1/2} (d^*)^{1/2} \sim \mathrm{const}. \end{gather}$$

The contact force from the lower wall is neglected because only the trailing edge of the flag is in contact during the transition between the quasistatic equilibrium and partial contact modes. The term $Fr_{c1} (m^*)^{-1/2} (d^*)^{1/2}$ on the left-hand side of (3.4b) is identical to $\hat {U}_{c1}$ in (3.2). Relation (3.4b) is consistent with the experimental results in figure 5(b), in which $Fr_{c1} (m^*)^{-1/2} (d^*)^{1/2}$ ($=\hat {U}_{c1}$) exhibits little variation with respect to $d^*$. To validate (3.4b), figure 6(a) is reconstructed in figure 7 using dimensionless variables ($Fr_{c1}$, $m^*$, and $d^*$). The fitting curve $Fr_{c1} (m^*)^{-1/2}= 0.75 (d^*)^{-1/2}$ from scaling relation (3.4b) is in good agreement with the experimental results.

Figure 7. First critical condition $Fr_{c1}(m^*)^{-1/2}$ with respect to distance ratio $d^*$ for the lower wall. The symbols denote experimental data, and the black dashed line denotes the fitting curve from scaling relation (3.4b).

As for the scaling analysis of the second critical velocity $U_{c2}$, it is more appropriate to consider energy rather than force because $U_{c2}$ is defined in the midst of the postequilibrium state. In our model, four energy quantities of the flag need to be accounted for: strain energy ($E_s$); kinetic energy ($E_{k, flag}$); potential energy ($E_p$); and the energy dissipated during the contact process ($E_c$). As scaling analysis is based on the relation between two dominant physical quantities, it is necessary to identify the dominant energy component of the flag, which is balanced with the fluid kinetic energy input imposed on the flag ($E_{k, fluid}$).

We first calculate each energy component from the experimental data to compare the magnitudes of the energy components. The strain energy $E_s$ of the flag per unit width is defined as $E_s =\frac {1}{2}B\int _{0}^{l} \kappa ^2\,\mathrm {d}s$, where $\kappa$ ($=(\mathrm {d}^2y/\mathrm {d}\kern 0.06em x^2)/[1+(\mathrm {d}\kern 0.05em y/\mathrm {d}\kern 0.06em x)^2]^{3/2}$) is the curvature of the flag. The curvature is computed from the position data obtained from the images. The sheet is discretized equally into $200$ segments along its length. For each segment, the central difference scheme is used to acquire $\mathrm {d}^2y/\mathrm {d}\kern 0.06em x^2$ and $\mathrm {d}\kern 0.05em y/\mathrm {d}\kern 0.06em x$. In addition to the bending energy, the stretching energy may be considered for the strain energy. However, the tension acting on the flag and the resultant change in the flag length are negligible. For this reason, the stretching energy is assumed to be zero. The kinetic energy $E_{k, flag}$ of the flag per unit width is defined as $E_{k, flag}=\frac {1}{2} \rho _s h \int _{0}^{l} v^2 \,\mathrm {d}s$, where $v$ ($=[(\mathrm {d}\kern 0.06em x/\mathrm {d}t)^2 + (\mathrm {d}\kern 0.05em y/\mathrm {d}t)^2]^{1/2}$) is the velocity magnitude of the flag. For each segment, the central difference scheme is used to obtain $\mathrm {d}\kern 0.06em x/\mathrm {d}t$ and $\mathrm {d}\kern 0.05em y/\mathrm {d}t$. The potential energy $E_p$ of the flag per unit width is defined as $\rho _s g h \int _{0}^{l} y \,\mathrm {d}s$, where $y$ is the vertical position of the flag. As $y=0$ is set at the leading edge (flagpole), as shown in figure 1, the potential energy can be negative when the flag is located below the flagpole.

The three energy components ($E_s$, $E_{k,flag}$ and $E_p$) of the flag at the second critical velocity are compared in figure 8. Although the energy is dissipated as acoustic energy during the contact process, the overall magnitude of the dissipated energy $E_c$ appears to be negligible because the material of the flag does not absorb energy, and the duration of the contact is very small compared with the period of flutter. The peak-to-peak value of $E_p$ in one cycle is dominant over those of the other energy components for both the upper and lower walls. Although figure 8 presents the data at the second critical condition for two specific cases of $[d, h]=[1.5\ {\rm cm}, 1.0\ {\rm mm}]$, the potential energy of the flag is found to be dominant in the postequilibrium state regardless of the wall location, $d$, and $h$. Based on this result, it is reasonable to consider the fluid kinetic energy $E_{k,fluid}$ and the potential energy of the flag $E_{p}$ for scaling analysis.

Figure 8. Time histories of strain energy $E_s$ (black), kinetic energy $E_{k, flag}$ (yellow) and potential energy $E_p$ (red) of the flag per unit width at the second critical velocity $U_{c2}$ for the (a) upper wall and (b) lower wall. $[U, d, h] = [9.79\ {\rm m}\ {\rm s}^{-1}, 1.5\ {\rm cm},\ 1.0\ {\rm mm}]$ for the upper wall and [10.18 m s$^{-1}$, 1.5 cm, 1.0 mm] for the lower wall.

The two energy quantities $E_{k, fluid}$ and $E_p$ should be balanced in a cycle-averaged sense in terms of scaling: $\bar {E}_{k,fluid} \sim \bar {E}_p$. The kinetic energy $E_{k, fluid}$ of the fluid per unit width is defined as the total power imposed on the sheet by the uniform flow during a cycle. Here $E_{k, fluid} = \int _{0}^{T}\int _{0}^{l} \boldsymbol {F}_{\boldsymbol {f}} \boldsymbol {{\cdot }} \boldsymbol {v} \,\mathrm {d}s \,\mathrm {d}t$, where $\boldsymbol {F}_{\boldsymbol {f}}$ is the fluid force vector acting on the flag per unit area, $\boldsymbol {v}$ is the velocity vector of the flag and $T$ is the period of flutter. From this integral expression, the fluid kinetic energy can be scaled without using the flutter period $T$. The fluid force $\boldsymbol {F}_{\boldsymbol {f}}$ scales as $F_f \sim \rho _f U^2$, the dynamic pressure induced by the free-stream velocity. Here $\int _{0}^{T} \boldsymbol {v} \,\mathrm {d}t$ scales as the peak-to-peak oscillation amplitude $A$ of the flag, and $\int _{0}^{l} \mathrm {d}s=l$. Consequently, $\bar {E}_{k, fluid} \sim \rho _f U^2 l A$. At postequilibrium, the vertical displacement $\Delta y_G$ for the centre of the flag determines the potential energy. As $d$ increases, the magnitude of $\Delta y_G$ required for saturated contact increases, which indicates that the potential energy may scale with $d$. Indeed, $\bar {E}_p$ computed from our experimental results tends to increase linearly with $d$ for both the upper wall and lower wall; see Appendix A. Thus, the cycle-averaged potential energy of the flag per unit width is regarded as $\bar {E}_p \sim \rho _s g h l d$.

As the peak-to-peak amplitude $A$ defined in figure 1(b) is a dependent variable that should be obtained from experiments, it is necessary to employ a different length scale based on the input parameters. For the upper wall, the oscillation amplitude of the flag is almost independent of $d$, and $A^* (=A/l)$ converges to a specific value as the dimensionless flow velocity increases (as reported in § 3.3). Hence, it is reasonable to argue that $A$ scales as the flag length $l$, leading to $\bar {E}_{k, fluid} \sim \rho _f U^2 l^2$. At the second critical condition $U = U_{c2}$ of the upper wall, the following scaling relations are established from the energy balance $\bar {E}_{k, fluid} \sim \bar {E}_p$:

(3.5a)$$\begin{gather} \rho_f U_{c2}^2 l^2 \sim \rho_s g h l d, \end{gather}$$

(3.5b)$$\begin{gather}Fr_{c2} (m^*)^{{-}1/2} \sim (d^*)^{1/2}, \end{gather}$$

where the second critical Froude number $Fr_{c2} = U_{c2} /(g l)^{1/2}$ and $Fr_{c2} (m^*)^{-1/2} = U_{c2}^*$. As depicted in figure 5(a), $U_{c2}^*$ increases monotonically with $d^*$.

In contrast to the upper wall, the peak-to-peak amplitude $A$ of the flag for the lower wall is distinctly affected by $d$; a detailed explanation will be given in § 3.3. Thus, the effect of $d$ must be included when scaling the oscillation amplitude. The amplitude of the flag can be divided into two parts based on $y=0$ (figure 1bii). In the lower-wall case, the amplitude in the region $y<0$ is equal to $d$, and we assume that the amplitude in the region $y>0$ is proportional to the flag length $l$, having the form $\alpha l$ with the proportional constant $\alpha$ to be determined later. As for the amplitude scale, $A \sim d + \alpha l$ is adopted instead of $A \sim l$. From the energy balance between $\bar {E}_{k, fluid}\sim \rho _f U^2 l(d+\alpha l)$ and $\bar {E}_p$, at the second critical condition $U = U_{c2}$ of the lower wall,

(3.6a)$$\begin{gather} \rho_f U_{c2}^2 l (d + \alpha l) \sim \rho_s g h l d, \end{gather}$$

(3.6b)$$\begin{gather}Fr_{c2} (m^*)^{{-}1/2} (d^*)^{1/2} \sim d^* (d^* + \alpha)^{{-}1/2}. \end{gather}$$

Here, $Fr_{c2} (m^*)^{-1/2}(d^*)^{1/2} = \hat {U}_{c2}$. Relation (3.6b) captures the trend of the experimental results, with $\hat {U}_{c2}$ increasing monotonically with $d^*$ in figure 5(b).

The second critical velocity $U_{c2}$ in figure 6(b) is reconstructed in dimensionless form, using $Fr_{c2}$, $m^*$ and $d^*$, in figure 9. For a given $m^*$, the fitting curves from scaling relations (3.5b) and (3.6b) are also included. For the upper wall in figure 9(a), $Fr_{c2} (m^*)^{-1/2} = C (d^*)^{1/2}$ from (3.5b), where the fitting constant $C$ is 5.42 for $m^*=19.16$ and 3.76 for $m^*=9.58$. The values predicted from the scaling analysis are in good agreement with the experimental results. Note that the constant $C$ depends on $m^*$, although $m^*$ is explicitly included in the scaling relation. Although the kinetic energy of the flag is not as dominant as the potential energy of the flag and is not included on the right-hand side of (3.5a), it is not actually negligible according to figure 8. The kinetic energy of the flag is proportional to the mass ratio (flag thickness $h$) of the flag. Therefore, the dimensionless velocity ${U_{c2}^*(=Fr_{c2} (m^*)^{-1/2})}$ is expected to increase as the kinetic energy of the flag becomes stronger (i.e. when $m^*$ increases), which is demonstrated experimentally in figure 9(a).

Figure 9. Second critical condition $Fr_{c2} (m^*)^{-1/2}$ with respect to wall distance ratio $d^*$ for the (a) upper wall and (b) lower wall. The symbols denote experimental data, and the dashed lines denote the fitting curves from scaling relation (3.5b) for (a) and from scaling relation (3.6b) for (b).

For the lower wall (figure 9b), $\alpha$ in (3.6b) is obtained empirically from the experimental data in the postequilibrium state; $\alpha = (A-d)/l$. The value of $\alpha$ averaged over seven different $d^*$ cases is 0.55 for both $m^*=19.16$ and 9.58 (see Appendix B). Thus, $Fr_{c2} (m^*)^{-1/2} = C (d^*)^{1/2} (d^*+0.55)^{-1/2}$ from (3.6b), where the fitting constant $C$ is 4.19 for $m^*=19.16$ and 2.52 for $m^*=9.58$. The constant $C$ varies with respect to $m^*$, as for the upper wall. When $d^*$ is relatively low ($d^*=0.15$), the curve of the scaling relation deviates notably from the experimental values. The value of $\alpha$ is fixed in the fitting process, although it varies with respect to $d$ and $U$ (Appendix B). If $\alpha$ is chosen using only the small-$d^*$ cases, the fitting curve in figure 9(b) will be good in the small-$d^*$ region but give poor predictions in the large-$d^*$ region. Despite such limitations, we can successfully predict the trend of the second critical velocity by applying simple scaling relations based on energy balance.

For the given $d$ in the range considered in this study, $l > d+\alpha l$. From the above scaling relations for $\bar {E}_{k, fluid}$ and $\bar {E}_p$, $\bar {E}_p$ of the upper wall is larger than that of the lower wall, and accordingly the peak-to-peak value of $E_p$ in one cycle for the upper wall exceeds that of the lower wall, as exemplified in figure 8. The kinetic energy of the flag, $E_k$, also exhibits similar trends when compared between the upper wall and lower wall. That is, the presence of gravity and the location of the wall greatly affect the temporal distributions of the energy components of the flag as well as its critical conditions.

3.3. Flag dynamics in postequilibrium state

This section examines some representative quantities, such as the contact force, oscillation amplitude and oscillation frequency, of the postequilibrium flag dynamics. The contact force exerted on the wall by the fluttering flag, which is measured with a load cell, exhibits periodic peaks following the oscillation period of the flag (figure 10). The temporal profile of the contact force during the contact and separation process differs between the partial and saturated contact modes and can thus be used to quantitatively distinguish the two modes, in addition to the initial contact point $X_c$ and contact duration $T_c$ presented in figure 4.

Figure 10. (a) Contact force $F_c$ of partial contact mode, $[U, d, h] = [9.39\ {\rm m}\ {\rm s}^{-1},\ 3.0\ {\rm cm},\ 1.0\ {\rm mm}]$, and (b) $F_c$ of saturated contact mode, $[U, d, h] = [11.38\ {\rm m}\ {\rm s}^{-1},\ 3.0\ {\rm cm},\ 1.0\ {\rm mm}]$, for the lower-wall configuration.

In the partial contact mode, the first peak appears as the flag makes contact with the wall (figure 10a). After the first peak, the contact force does not decrease to zero but increases again, producing a second peak with a slightly smaller magnitude than the first peak. After the second peak, the contact force returns to zero when the flag separates from the wall. The contacting parts do not separate simultaneously from the wall in the partial contact mode, but instead, some parts remain in contact while the others separate. This phenomenon leads to a relatively weak second peak. As the partial contact mode transitions to the saturated contact mode with increasing free-stream velocity $U$, only one stronger peak arises per cycle (figure 10b). In the saturated contact mode, the contacting parts separate from the wall rapidly and almost simultaneously in contrast to the partial contact mode, thereby yielding a single strong peak. Although $U$ increases by only 1.99 m s$^{-1}$ from the partial contact mode in figure 10(a) to the saturated contact mode in figure 10(b), the peak contact force increases dramatically, from 1.1 to 3.7 N. This feature of the contact force is common regardless of the wall location and other geometric parameters; during the transition of the contact mode, two peaks become a single peak, and the magnitude at the peak increases remarkably.

With the same wall distance, the magnitude of the peak contact force $F_{c,p}$ strongly depends on whether the wall is located above or below the flag. Here $F_{c,p}$ is acquired by averaging 10 successive force peaks for contact modes. For flutter mode, the flag rarely contacts the upper wall, and thus $F_{c,p}$ is computed by dividing the sum of force peaks during 10 oscillation cycles by 10. For the lower wall, when the wall distance ratio is large (i.e. $d^*= 0.40$), sufficient fluid kinetic energy $E_{k,fluid}$ is supplied because it scales with $d + \alpha l$ according to (3.6a). Thus, along with the positive contribution of the downward gravitational force, the large input of the fluid kinetic energy causes $F_{c,p}$ in the lower-wall case to exceed that of the upper-wall case significantly, and the difference between them becomes greater as $U$ increases (figure 11b). With a large $d^*$, the fluttering flag must move upward to a higher location to contact the upper wall. Because the energy supplied to the flag should be used to enhance the potential energy at the moment of contact, the peak contact force $F_{c,p}$ of the upper wall decreases notably from that of the small-$d^*$ case for the same $U$; compare figure 11(b) with figure 11(a).

Figure 11. Peak of contact force $F_{c,p}$ versus free-stream velocity $U$ for (a) $d^*=0.20$ and (b) $d^*=0.40$ ($m^*=19.16$): upper wall (circles) and lower wall (squares). The red and blue arrows indicate critical velocities for the upper and lower walls, respectively.

The trend in $F_{c,p}$ differs notably for a small distance ratio $d^*= 0.20$ (figure 11a). $F_{c,p}$ in the lower-wall configuration is less than that of the upper-wall configuration in the intermediate range of $U$ ($=$ 6–11 m s$^{-1}$), although gravity acts to enhance the contact force for the lower wall. As mentioned above, the fluid kinetic energy $E_{k,fluid}$ scales with $d + \alpha l$ for the lower wall. In the small-$d^*$ condition, $E_{k,fluid}$ for the lower wall is generally smaller than that of the upper wall, eventually leading to a smaller $F_{c,p}$. However, in the large-$U$ regime ($U > 11$ m s$^{-1}$), the oscillation amplitude of the lower-wall case becomes sufficiently large, resulting in greater fluid kinetic energy. Thus, $F_{c,p}$ of the lower-wall case is comparable to that of the upper wall.

The contact force for the lower wall increases rapidly at a specific free-stream velocity $U$ (e.g. 9.39 m s$^{-1}$ in figure 11a and 6.30 m s$^{-1}$ in figure 11b) as the quasistatic equilibrium mode switches directly to the partial contact mode. However, for the upper wall, the contact force increases with a relatively gentle slope from a specific free-stream velocity $U$ (e.g. 6.30 m s$^{-1}$ in figure 11a and 8.22 m s$^{-1}$ in figure 11b), because the quasistatic equilibrium mode does not switch directly to the partial contact mode, but goes through the flutter-mode regime. In the flutter mode, the oscillations of the flag are not perfectly periodic, and the flag can rarely contact the upper wall with small strength, which results in non-zero $F_{c,p}$, albeit minor.

From the perspective of triboelectric energy harvesting applications, it is necessary to address which wall configuration is desirable for system design. Under the same free-stream velocity and flag parameters, the lower-wall configuration appears more advantageous because gravity acts positively. However, as shown in figure 11, the desirable wall position for energy harvesting based on the peak contact force depends on the wall distance ratio $d^*$. According to figure 12, which presents the difference in peak contact force between the upper wall and lower wall ($\Delta F_{c,p}=F_{c,p}^U - F_{c,p}^L)$, the lower wall produces the larger peak contact force over the most range of $U$ at large distance ratios ($d^*\geq 0.30$). Conversely, the upper wall yields the larger peak contact force at a small distance ratio ($d^* = 0.20$). Furthermore, under the small distance ratio, triboelectric energy can be generated in a broader range of free-stream velocity with the upper-wall configuration because the flow velocity threshold for initiating contact is much lower than with the lower-wall configuration (figure 11a).

Figure 12. Difference in peak contact force $\Delta F_{c,p}(=F_{c,p}^U - F_{c,p}^L)$ between the upper and lower walls with respect to free-stream velocity $U$ ($d^* = 0.20\unicode{x2013}0.40$ and $m^*=19.16$).

Although the dimensional $F_{c,p}$ has been used to explain the trends in the contact force, the contact force coefficient $F_{c,p}^*(=F_{c,p}/(\rho _f U^2 S))$, which is normalized by the surface area $S(=lw)$ of the flag and the free-stream velocity, may be a more suitable parameter than $F_{c,p}$ for evaluating how effectively the contact force is generated. Here $F_{c,p}^*$ is presented with respect to the dimensionless flow velocity $U^*$ for the upper wall in figure 13(a) and with respect to $\hat {U}$ for the lower wall in figure 13(b).

Figure 13. Contact force efficient $F_{c,p}^*$ with respect to (a) $U^*$ for the upper wall and (b) $\hat {U}$ for the lower wall ($d^* = 0.20\unicode{x2013}0.40$ and $m^*=19.16$).

The dimensional $F_{c,p}$ tends to increase monotonically with $U$, regardless of the wall location and wall distance (figure 11). By contrast, $F_{c,p}^*$ tends to increase and then decrease with the dimensionless flow velocity ($U^*$ or $\hat {U}$) for most wall distance ratios (figure 13). The optimal maximum value of $F_{c,p}^*$ is clearly identified at $d^*\leq 0.30$ for the upper wall and at $d^*\geq 0.30$ for the lower wall. In the low-dimensionless-velocity regime, $F_{c,p}^*$ generally increases with the dimensionless velocity. However, as the flow velocity increases further, the slope of the dimensional $F_{c,p}$ versus $U$ declines because the flag transitions from the partial contact mode to saturated contact mode and undergoes fully developed contact. Accordingly, $F_{c,p}^*$ decreases after reaching its maximum. Furthermore, figure 13 shows opposite trends of $F_{c,p}^*$ with respect to $d^*$ between the two wall locations; $F_{c,p}^*$ tends to decrease with increasing $d^*$ for the upper wall, while it tends to increase with increasing $d^*$ for the lower wall. For the lower wall, the wall distance is a more important parameter in determining the magnitude of the fluid kinetic energy and contact force than for the upper wall, as mentioned above. Hence, $F_{c,p}^*$ exhibits more distinct variations among the five $d^*$ cases at a given flow velocity.

The peak-to-peak amplitude $A$ of the flag (figure 1b) is normalized by the flag length $l$: $A^*=A/l$. The normalized amplitude $A^*$ displays different tendencies depending on the wall location. For the upper wall, when contact begins to occur in the partial contact mode, the amplitude above the flagpole is limited to the wall distance $d$ (figure 1bi). Therefore, $A^*$ cannot continue to increase with $U^*$, but converges to a certain value for each distance ratio $d^*$ (figure 14ai). The converged $A^*$ in the high-$U^*$ regime tends to be greater for larger values of $d^*$, although its variations are minor between $d^*=0.30\unicode{x2013}0.40$. By contrast, for the lower wall, the amplitude above the flagpole is not limited, and the amplitude below the flagpole is equal to $d$ because the fluttering flag contacts the wall in all cases (figure 1bii). The amplitude above the flagpole increases with the free-stream velocity $U$, and thus, at a given $U$, the peak-to-peak amplitude $A$ becomes greater for larger values of $d$. Interestingly, if the dimensionless velocity $\hat {U}$, which includes $d$ in its definition, is employed, the curves of normalized amplitude $A^*$ collapse, and the values of $A^*$ are similar for a given $\hat {U}$ regardless of $d^*$ (figure 14aii).

Figure 14. (a) Normalized peak-to-peak amplitude $A^*$ and (b) Strouhal number $St(=fA/U)$ with respect to dimensionless flow velocity: (i) upper wall and (ii) lower wall ($m^*=19.16$).

In both the upper-wall and lower-wall configurations, the dimensional frequency $f$ tends to increase monotonically with the free-stream velocity $U$. To make the frequency dimensionless, the flag length $l$ or the peak-to-peak amplitude $A$ can be regarded as the characteristic length. In our model, $A$, which exhibits different trends depending on the wall location, better represents the model configuration; moreover, $l$ is independent of the wall location. That is, the Strouhal number $St(=fA/U)$, reflecting the relative magnitude of the flutter speed of the flag tip to the free-stream velocity, is used to represent the frequency of the flag. The overall trends of $St$ versus the dimensionless flow velocity are quite similar to those of $A^*$ (figure 14b). Here $St$ suddenly increases and thereafter becomes saturated for the upper-wall case, while it continues to increase and has a similar value irrespective of $d^*$ for the lower-wall configuration.

For both the upper and lower walls, $St$ is restricted below $0.10$, which is markedly smaller than $St (=0.20\unicode{x2013}0.40)$ for a vertical flag with negligible gravity effects near a vertical wall (Lee et al. Reference Lee, Kim and Kim2021) and a typical flag model without an adjacent wall and gravity effects (Shelley, Vandenberghe & Zhang Reference Shelley, Vandenberghe and Zhang2005; Connell & Yue Reference Connell and Yue2007). Compared with the previous flag models without gravity, a significant portion of the fluid kinetic energy is converted into the potential energy of the flag, thereby reducing the kinetic energy of the flag. Hence, the flutter speed of the flag becomes relatively low compared with the previous flag models without gravity for a given free-stream velocity, which eventually results in a decrease in $St$. In contrast, the range of the normalized amplitude $A^*$ in our model is comparable to that of previous flag models, $A^* = 0.4\unicode{x2013}1.0$ (Connell & Yue Reference Connell and Yue2007; Eloy et al. Reference Eloy, Lagrange, Souilliez and Schouveiler2008; Virot, Amandolese & Hémon Reference Virot, Amandolese and Hémon2013; Lee et al. Reference Lee, Kim and Kim2021), because the downward deflection of the flag by gravity offsets the negative effect of the reduced kinetic energy of the flag on the oscillation amplitude.

For flag models, hysteresis generally occurs in a transition phase between equilibrium and postequilibrium states, and a bistable state exists within a certain range of the free-stream velocity (Eloy et al. Reference Eloy, Lagrange, Souilliez and Schouveiler2008; Eloy, Kofman & Schouveiler Reference Eloy, Kofman and Schouveiler2012; Kim et al. Reference Kim, Cossé, Cerdeira and Gharib2013). Whether gravity and contact affect the emergence of hysteresis in our model is now examined. The width $e$ of hysteresis is defined as $e=(U_{c1,i}-U_{c1,d})/U_{c1,i}$ (Eloy et al. Reference Eloy, Kofman and Schouveiler2012). Here $U_{c1,i}$ is the first critical velocity at the transition from the quasistatic equilibrium mode, which is measured while the free-stream velocity increases; and $U_{c1,d}$ is the first critical velocity at the transition to the quasistatic equilibrium mode, and is measured while the free-stream velocity decreases. Hysteresis with a subcritical bifurcation is identified from the plots of normalized amplitude $A^*$ for both profiles of increasing and decreasing free-stream velocities (figure 15a); in the figure, $U_{c1,i}^* = 1.27$ and $U_{c1,d}^* =0.99$. For both the upper and lower walls, the value of $e$ is mostly between 2.6 %–7.8 % (figure 15b). Furthermore, to exclude the effect of contact and solely consider the effect of gravity, the case in which contact does not occur is also considered by removing the wall. The value of $e$ averaged over 10 trials is 5.1 % for the non-contact case in the absence of a wall, which is similar to the $e$ values of contact cases. That is, external factors such as gravity and contact do not alter the existence of hysteresis, which is regarded as an inherent characteristic of a fluttering flag.

Figure 15. (a) Hysteresis of normalized peak-to-peak amplitude $A^*$ for the upper wall: increasing free-stream velocity (solid) and decreasing free-stream velocity (dashed); $[d^*, m^*] = [0.15, 19.16]$. (b) Width of hysteresis $e$ for the upper wall (circle) and the lower wall (square); $m^*$ = 19.16. The red dashed line denotes the value of $e$ for the case without a wall.