3.1. The dissimilar scaling of the velocity

In this section, we consider the applicability of the scaling laws discussed in § 1.1 to the mean velocities. For convenience, we non-dimensionalise the time-averaged velocities by the free stream velocity according to the following expressions:

(3.1) \begin{equation} u^{\ast}=\frac {\overline {u_t}}{U_\infty }, \qquad v^{\ast}=\frac {\overline {v_n}}{U_\infty }. \end{equation}

We begin by examining the near-wall flow region at zero angle of attack, which is expected to resemble a boundary layer as the synthetic jets are inactive. In this scenario, the mean velocity should follow the logarithmic law in the overlap region for sufficiently high Reynolds numbers. We define the characteristic thickness as $\delta \equiv n_{0.99}$ , which aligns with the conventional boundary layer definition. Profiles of the dimensionless mean tangential velocity $u^{\ast}$ for the baseline case are shown in figure 7, where the point at $n_{0.99}$ is marked by a closed symbol. Evidently, the profiles of $u^{\ast}$ in this case closely resemble that of a turbulent boundary layer, with the semilogarithmic plots indicating the presence of a logarithmic region near the wall. Now, let us examine the profiles of $u^{\ast}$ at $\alpha ={15.0}^\circ$ and $C_\mu ={1.25\times 10^{-3}}{}$ , as shown in figure 8. In controlled cases, the tangential velocity profile may exhibit a peak near the wall, similar to turbulent wall jets. Nevertheless, we continue to use $\delta \equiv n_{0.99}$ instead of $n_1$ for our analyses as the precise determination of $n_1$ is uncertain in experiments (George et al. Reference George, Abrahamsson, Eriksson, Karlsson, Löfdahl and Wosnik2000).

Figure 7. Sectional profiles of the dimensionless mean tangential velocity in linear and semilogarithmic scales for $\alpha = 0$ and $C_\mu =0$ test case (baseline) at three locations: (a) $s/c = 0.45$ ( $\times$ ), (b) $s/c = 0.55$ ( ) and (c) $s/c = 0.65$ ( $+$ ).

Figure 8. Sectional profiles of the dimensionless mean tangential velocity in linear and semilogarithmic scales for $\alpha = {15.0}^\circ$ and $C_\mu ={1.25\times 10^{-3}}{}$ test case at three locations: (a) $s/c = 0.40$ ( $\times$ ), (b) $s/c = 0.50$ ( ) and (c) $s/c = 0.60$ ( $+$ ).

A first glance at figure 8 reveals a significant departure from classical behaviour. Although some logarithmic behaviour remains present near the wall, the remainder of the profile appears predominantly linear, a shape that is distinct from both turbulent boundary layers and wall jets. The shape of the velocity profiles suggest that the flow should be divided into three distinct shear layers, as proposed by Barenblatt, Chorin & Prostokishin (Reference Barenblatt, Chorin and Prostokishin2005) for a plane turbulent wall jet. In their study, Barenblatt et al. (Reference Barenblatt, Chorin and Prostokishin2005) scaled the inner and outer layers using power laws and noted that the data in a middle region beneath the maximum velocity location could not be collapsed onto a single curve. Following their approach, we scale the shear flow twice, first using $n_{1/2}$ and then $n_{0.99}$ as characteristic length scales, with $u_{0.99}$ as the characteristic velocity scale. The resulting scaled velocity profiles for $\alpha ={12.5}^\circ$ and $C_\mu ={1.25\times 10^{-3}}{}$ are presented in figure 9. While the overall pattern is similar to that observed by Barenblatt et al. (Reference Barenblatt, Chorin and Prostokishin2005), with data scattering in the middle region, the middle region in this case is considerably wider and exhibits a pronounced linear behaviour. Following Barenblatt et al. (Reference Barenblatt, Chorin and Prostokishin2005), we also refer to this middle layer as a mixing layer, a term that shall become more meaningful in § 3.2, § 3.3 and § 3.4.

Figure 9. Scatter of the mean tangential velocity due to an improper scaling approach, collected from $\alpha = {12.5}^\circ$ and $C_\mu ={1.25\times 10^{-3}}{}$ test case at locations $s/c = 0.40$ ( $\square$ ), $s/c = 0.50$ ( $\circ$ ) and $s/c = 0.60$ ( ).

Figure 10. Scatter of the mean tangential and normal velocity defects due to improper scaling parameters, collected from $\alpha = {15.0}^\circ$ and $C_\mu ={0.25\times 10^{-3}}{}$ test case at locations $s/c = 0.10$ ( $\square$ ), $s/c = 0.20$ ( $\circ$ ), $s/c = 0.30$ ( ) and $s/c = 0.40$ ( $\times$ ).

Figure 11. The dissimilar scaling of the mean velocity components at locations $s/c = 0.10$ ( $\square$ ), $s/c = 0.20$ ( $\circ$ ), $s/c = 0.30$ ( ), $s/c = 0.40$ ( $\times$ ) and $s/c = 0.50$ ( ) for four controlled cases: (a) $\alpha = {12.5}^\circ$ and $C_\mu ={1.25\times 10^{-3}}{}$ , (b) $\alpha = {12.5}^\circ$ and $C_\mu ={0.25\times 10^{-3}}{}$ , (c) $\alpha = {15.0}^\circ$ and $C_\mu ={1.25\times 10^{-3}}{}$ , (d) $\alpha = {15.0}^\circ$ and $C_\mu ={0.25\times 10^{-3}}{}$ .

In general, this scaling approach for turbulent wall jets, as also implied by the works of George et al. (Reference George, Abrahamsson, Eriksson, Karlsson, Löfdahl and Wosnik2000) and Barenblatt et al. (Reference Barenblatt, Chorin and Prostokishin2005), is effective only at high enough Reynolds numbers. The dispersion of the tangential velocity data in the middle layer is unsurprising since the Reynolds number ${Re}_\delta =U_\infty \delta / \nu$ was at most an order of ${O}(10^4)$ in these experiments. We now adopt an alternative scaling approach for the tangential velocity based on a modified form of von Kármán’s defect law,

(3.2) \begin{equation} \widehat {U}\left ({ \dfrac {n}{n_1} }\right ) = \dfrac {u^{\ast}-u^{\ast}_1}{u^{\ast}_s} ,\end{equation}

where $u^{\ast}_s$ is the tangential characteristic velocity. Overall, with a correct choice of $u^{\ast}_s$ , (3.2) effectively collapses the mean tangential velocity profiles onto a single curve across all test cases. The present experimental data suggests that $u^{\ast}_s$ may not necessarily be identical to the friction velocity. For clarity, consider the scaling applied to the test case with $\alpha = {15.0}^\circ$ and $C_\mu ={0.25\times 10^{-3}}{}$ . In this case, the tangential velocity profiles align well on a single curve when the velocity defect $u^{\ast}-u^{\ast}_{0.99}$ is scaled by $u^{\ast}_s=u^{\ast}_{0.99}$ . However, when the slope of the logarithmic layer, was used for scaling, a pronounced data dispersion was observed, as shown in figure 10(a). This behaviour was mostly observed for $\alpha = {12.5}^\circ$ and $\alpha = {15.0}^\circ$ cases, and not for the $\alpha = {7.5}^\circ$ test case.

Despite the success of the defect law in capturing the behaviour of the mean tangential velocity, the mean normal velocity could not be scaled using the same characteristic velocity $u^{\ast}_s$ . Attempts to scale the mean normal velocity or its defect using $u^{\ast}_s$ and length scale $\delta$ resulted in data scattering around one end of the profiles. An example is presented in figure 10(b), where the normal velocity defect $v^{\ast}-v^{\ast}_{0.99}$ is scaled by $u^{\ast}_s=u^{\ast}_{0.99}$ . The scatter in the mean normal velocity upon scaling by $u^{\ast}_s$ suggests a dissimilar scaling of the time-averaged velocity components. In other words, though the consecutive profiles could still be aligned, the appropriate scaling parameter for the normal velocity may not be the same as that for the tangential velocity. We now propose the following scaling law for the normal velocity:

(3.3) \begin{equation} \widehat {V}\left ({ \dfrac {n}{n_1} }\right ) = \dfrac {v^{\ast}-v^{\ast}_1}{v^{\ast}_1} .\end{equation}

For the $\alpha ={12.5}^\circ$ and $\alpha ={15.0}^\circ$ cases, only over a finite arclength along the aerofoil, scaling the data using (3.2) and (3.3) with $u^{\ast}_s = u^{\ast}_1$ led to a remarkable collapse of both velocity components onto a single curve. The results for the four test conditions at $\alpha ={12.5}^\circ$ and $\alpha ={15.0}^\circ$ are presented in figure 11. More discussion on the choice for $u^{\ast}_s$ and similarity of the velocity components shall be presented in §§ 3.3 and 3.4.

3.2. The formation of the mixing layer

So far, we have established that the two velocity components in the middle layer may not scale with the same characteristic velocity. However, a physical explanation for this behaviour is still needed. This dissimilar scaling may be linked to the normal free stream velocity. To illustrate this, consider the aerofoil at two angles of attack, as shown in figure 12. From the wall’s perspective, the free stream actually has two velocity components $u^{\ast}_\infty$ and $v^{\ast}_\infty$ . For all angles of attack within $0 \leqslant \alpha \leqslant {90}^\circ$ , the normal free stream velocity $v^{\ast}_\infty$ switches signs from positive to negative at the zero surface-slope location. For a NACA 0025 aerofoil, this reversal occurs within $0 \leqslant X/c \leqslant 0.3$ . Upstream of this point, the free stream pushes the synthetic jet towards the wall. Conversely, downstream of the reversal point, the free stream tends to pull the synthetic jet away from the wall. As the angle of attack increases, the reversal point shifts farther upstream, while the magnitude of the normal free stream velocity also increases. Hence, downstream of the reversal point, the normal velocity on the upper edge of the synthetic jet may eventually become comparable to the tangential velocity. The relative strength of the normal to tangential velocity decides the formation of the mixing layer, which depends on both the angle of attack $\alpha$ and the momentum coefficient $C_\mu$ .

Figure 12. A schematic showing the sign of the tangential and normal free stream velocity, as observed from the wall, for two angles of attack.

Figure 13. Contour plots of the coherent streamwise velocity at a phase angle of $\phi ={0}^\circ$ and an angle of attack of $\alpha ={12.5}^\circ$ for two momentum coefficients: (a) $C_\mu ={1.25\times 10^{-3}}{}$ and (b) $C_\mu ={0.25\times 10^{-3}}{}$ .

Now, let us examine figures 13 and 14 to understand how the previous discussion relates to the present experiments. The coherent structures for various angles of attack and momentum coefficients were identified using the triple decomposition method (Hussain & Reynolds Reference Hussain and Reynolds1970), expressed by the following relation:

(3.4) \begin{equation} \widetilde {\,{u_x}} = \left \langle {u_x} \right \rangle - \overline {{u_x}}. \end{equation}

The synthetic jet coherent structures initially appear as pairs of counter-rotating vortices, as illustrated by the vector plots superimposed on the streamwise coherent velocity contours $\widetilde {\,{u_x}}$ in figures 13(a) and 13(b). As the flow evolves, differences between the high- and low-momentum actuation cases become evident. In the high-momentum case, the upper portion of the coherent structures penetrates the cross-flow, whereas the lower portion adheres to the wall. For the low-momentum case, on the other hand, the upper and lower portions of the coherent structures remain closer together and dissipate earlier than in the high-momentum case.

Generally, there exists a strong shear between the wall and the core of these structures, which themselves act as a significant source of cross-stream perturbations. Still, the mixing layer does not appear unless these structures are sufficiently distant from the wall since the wall significantly dampens the wall-normal velocity fluctuations. Hence, at low angles of attack and high momentum coefficients, as the normal velocity component remains relatively weak, the structures are highly organised and their core adheres to the wall. As the angle of attack increases or the momentum coefficient decreases, the free stream eventually manages to pull the coherent structures away from the wall. The substantial normal velocity fluctuations caused by the synthetic jet structures combined with the high resident shear near the wall leads to the emergence of the mixing layer, extending between the boundary layer and the central core of the jet structures. Farther downstream, these structures gradually lose momentum, becoming disorganised and dispersing throughout the mixing layer and jet layer before ultimately dissipating. This behaviour is particularly evident in the cases for $\alpha ={12.5}^\circ$ and $\alpha ={15.0}^\circ$ , as shown in figures 14(c) and 14(d). Overall, the angle of attack and the momentum coefficient exhibit similar yet inverse effects on the development of the mixing layer. Increasing the angle of attack or decreasing the momentum coefficient enhances the normal velocity magnitude relative to the tangential velocity, pushing the onset of the mixing layer upstream and increasing its spreading rate.

Figure 14. Contour plots of the coherent streamwise velocity at a phase angle of $\phi ={0}^\circ$ and a momentum coefficient of $C_\mu ={0.25\times 10^{-3}}{}$ for four angles of attack: (a) $\alpha =0$ , (b) $\alpha ={7.5}^\circ$ , (c) $\alpha ={12.5}^\circ$ and (d) $\alpha ={15.0}^\circ$ .

Figure 15. A schematic showing the mean tangential velocity profile at finite and infinite Reynolds numbers.

3.3. The asymptotic solution

This section introduces an analytical solution for the mixing layer under asymptotic conditions. Let us consider a quasi-two-dimensional turbulent flow along a convex wall with a surface curvature of $\varrho$ , where the tangential and normal velocity components are both significant. The flow has evolved over an arclength of $\lambda$ from its onset at $s=s_0$ , reaching a characteristic thickness of $\delta$ and a Reynolds number of ${Re}_\delta =U_\infty \delta /\nu$ , as illustrated in figure 15. We seek a solution in the limit of $\delta /\lambda \ll 1$ at infinite Reynolds number, where $\delta /\lambda$ is the spreading rate. The curvature ratio $\varrho \delta$ , which was introduced by Dean (Reference Dean1927) for flow inside a toroidal pipe, quantifies the ratio of centrifugal to inertial forces.

To begin the analysis, we conduct an order of magnitude examination of the velocities. The time-averaged tangential velocity has a magnitude comparable to the free stream velocity $U_\infty$ . Since the contribution of the normal velocity gradient to flow continuity may become comparable to that of the tangential velocity gradient, the order of the normal velocity can be estimated using $\partial \overline {v_n}/\partial n \sim \partial \overline {u_t}/\partial s$ . The order of magnitude and the dimensionless form of the velocities and length scales are summarised as follows:

(3.5) \begin{align} s-s_0 &\sim {O}(\lambda ), & n &\sim {O}(\delta ), & \overline {u_t} &\sim {O}(U_\infty ), & \overline {v_n} &\sim {O}\left(\frac {U_\infty \delta }{\lambda }\right)\!,\\ \nonumber s^{\ast}&=\frac {s-s_0}{\lambda }, & n^{\ast} &=\frac {n}{\lambda } ,& u^{\ast} &=\frac {\overline {u_t}}{U_\infty }, & v^{\ast} &=\frac {\overline {v_n}}{U_\infty }. \end{align}

For an incompressible flow, the time-averaged continuity equation in the curvilinear coordinates $(s,n)$ is expressed by (3.6). The dimensionless form of this equation is derived by substituting the corresponding dimensionless variables. The result, normalised by the highest-order velocity gradient $U_\infty /\delta$ , is (3.7),

(3.6) \begin{align} \frac {\partial \overline {u_t}}{\partial s} + \frac {\partial (1 + \varrho n)\overline {v_n} }{\partial n} &= 0 ,\end{align}

(3.7) \begin{align} \underbrace { \frac {\delta }{\lambda } \frac {\partial u^{\ast}}{\partial s^{\ast}} }_{ {O}(\frac {\delta }{\lambda }) }+ \underbrace { \frac {\delta }{\lambda } \frac {\partial v^{\ast}}{\partial n^{\ast}} }_{ {O}(\frac {\delta }{\lambda }) } + \underbrace { \varrho \delta \frac {\partial n^{\ast} v^{\ast}}{\partial n^{\ast}} }_{ {O}\left(\varrho \delta \frac {\delta }{\lambda }\right) } &= 0.\end{align}

Our objective here is to determine the velocity profiles at infinite Reynolds number, where the viscous sublayer has vanished, and the inner, middle and outer flow regions are essentially decoupled. The boundary condition for the normal velocity at the wall remains $v^{\ast}(s^{\ast},n^{\ast}=0)=0$ as there is no suction or blowing. Expanding the normal velocity about the wall by Taylor series, and considering the limit $\delta /\lambda \ll 1$ , the series can be truncated at first order,

(3.8) \begin{equation} v^{\ast} = v^{\ast}(s^{\ast},0) + n^{\ast} \frac {\partial v^{\ast}}{\partial n^{\ast}}(s^{\ast},0) + {O}\big({n^{\ast}}^2\big) \xrightarrow [n^{\ast} \sim {O}\left ({ \frac {\delta }{\lambda } }\right )]{\frac {\partial v^{\ast}}{\partial n^{\ast}}(s^{\ast},0)=F(s^{\ast})} v^{\ast} = F(s^{\ast})n^{\ast} + {O} \left ({ \frac {\delta ^2}{\lambda ^2} }\right )\!. \end{equation}

The asymptotic form of the normal velocity can be determined through similarity, in contrast to plane shear flows, where the cross-stream velocity must be obtained from continuity. By defining the similarity variable $\xi = n^{\ast}/n^{\ast}_1$ and enforcing self-similarity on the normal velocity defect, we have

(3.9) \begin{align} \widetilde {V}(\xi ) = \frac {v^{\ast}-v^{\ast}_1}{v^{\ast}_1} = \frac {F(s^{\ast})\big(n^{\ast} - n^{\ast}_1\big)}{F(s^{\ast}) n^{\ast}_1} \longrightarrow \widetilde {V}(\xi ) &= \xi -1 .\end{align}

Hence, under the given conditions, the normal velocity must collapse onto a linear profile described by $\widetilde {V}(\xi )=\xi -1$ , a trend that is already evident in figure 11. Next, by applying flow continuity and imposing self-similarity on the tangential velocity defect, we derive the asymptotic form of the tangential component,

(3.10) \begin{align} &\frac {\partial u^{\ast}}{\partial s^{\ast}} + \frac {\partial (1+\varrho \lambda n^{\ast})v^{\ast}}{\partial n^{\ast}} = 0 \longrightarrow u^{\ast} = -\int _0^{s^{\ast}} F(\varsigma ){\rm d}\varsigma - 2\lambda n^{\ast} \int _0^{s^{\ast}} \varrho (\varsigma )F(\varsigma ) {\rm d}\varsigma + G(n^{\ast}), \nonumber \\ &u^{\ast}-u^{\ast}_1 = \left ({ -2\lambda \displaystyle \int _0^{s^{\ast}} \varrho (\varsigma )F(\varsigma ){\rm d}\varsigma }\right ) \big(n^{\ast} - n^{\ast}_1\big) + G(n^{\ast})-G\big(n^{\ast}_1\big) \xrightarrow {G(n^{\ast}) = A n^{\ast}+C} \nonumber \\ &\left .\begin{aligned} \widetilde {U}(\xi ) = \frac {u^{\ast}-u^{\ast}_1}{u^{\ast}_s} &= \frac {\left ({ A-2\lambda \displaystyle \int _0^{s^{\ast}} \varrho (\varsigma )F(\varsigma ){\rm d}\varsigma }\right )\big(n^{\ast} - n^{\ast}_1\big)}{u^{\ast}_s}\\ u^{\ast}_s &= \left ({ A-2\lambda \displaystyle \int _0^{s^{\ast}} \varrho (\varsigma )F(\varsigma ){\rm d}\varsigma } \right ) n^{\ast}_1 \end{aligned}\right \rbrace \longrightarrow \widetilde {U}(\xi ) = \xi -1. \end{align}

Equation (3.10) conveys two key messages. First, when the Reynolds number reaches infinity and the spreading rate remains sufficiently low, the tangential velocity also follows the linear profile $\widetilde {U}(\xi ) = \xi -1$ . Second, at infinite Reynolds number, the characteristic velocity required for scaling is given by $u^{\ast}_s = (A - 2\lambda \int _0^{s^{\ast}} \varrho (\varsigma )F(\varsigma ) {\rm d}\varsigma )n^{\ast}_1$ , which generally differs from the maximum velocity $u^{\ast}_1$ or the friction velocity. Nonetheless, in this limit, the viscous sublayer will perish, and the tangential velocity boundary condition at the wall becomes $u^{\ast}_0 = -\int _0^{s^{\ast}} F(\varsigma ) {\rm d}\varsigma +C$ . Thus, the characteristic velocity can be rewritten as $u^{\ast}_s = u^{\ast}_1 - u^{\ast}_0 = (A - 2\lambda \int _0^{s^{\ast}} \varrho (\varsigma )F(\varsigma ) {\rm d}\varsigma )n^{\ast}_1$ .

It can be proven that the velocities can be scaled relative to any arbitrary cross-stream location $n_\varepsilon$ , a property reminiscent of a plane mixing layer. This scaling property can be established by following the steps outlined here

(3.11) \begin{align} u^{\ast}_\varepsilon = \varepsilon u^{\ast}_1 + (1-\varepsilon ) u^{\ast}_0 \longrightarrow \frac {u^{\ast}_\varepsilon -u^{\ast}_1}{u^{\ast}_s} = \varepsilon -1 = \widetilde {U}\left(\frac {n^{\ast}_\varepsilon }{n^{\ast}_1}\right) = \frac {n^{\ast}_\varepsilon }{n^{\ast}_1} - 1 \longrightarrow \nonumber \frac {n_\varepsilon }{n_1} = \varepsilon &,\\ \left .\begin{aligned} \frac {u^{\ast}-u^{\ast}_\varepsilon }{u^{\ast}_s} &= \frac {u^{\ast}-u^{\ast}_1}{u^{\ast}_s} - \frac {u^{\ast}_\varepsilon -u^{\ast}_1}{u^{\ast}_s} = \xi - \varepsilon \\ \frac {v^{\ast}-v^{\ast}_\varepsilon }{v^{\ast}_s} &= \frac {v^{\ast}-v^{\ast}_1}{v^{\ast}_s} - \frac {v^{\ast}_\varepsilon -v^{\ast}_1}{v^{\ast}_s} = \xi - \varepsilon \end{aligned}\right \rbrace \xrightarrow {\zeta =\xi -\frac {n_\varepsilon }{n_1}=\xi -\varepsilon } \begin{aligned} \widetilde {U}_\varepsilon (\zeta ) &= \frac {u^{\ast}-u^{\ast}_\varepsilon }{u^{\ast}_s} = \zeta \\ \widetilde {V}_\varepsilon (\zeta ) &= \frac {v^{\ast}-v^{\ast}_\varepsilon }{v^{\ast}_s} = \zeta \end{aligned} &. \end{align}

In the first line, it is shown that the tangential velocity achieves a value of $u_\varepsilon$ at the cross-stream position $n_\varepsilon /n_1=\varepsilon$ . The subsequent steps reveal that the velocity deficits, when scaled about $\xi =\varepsilon$ by $u^{\ast}_s$ and $v^{\ast}_s$ , assume the forms $\widetilde {U}_\varepsilon (\zeta )=\zeta$ and $\widetilde {V}_\varepsilon (\zeta )=\zeta$ , where $-\varepsilon \leqslant \zeta \leqslant 1-\varepsilon$ . The asymptotic solution of the velocity is summarised as follows:

(3.12) \begin{align} v^{\ast} &= F(s^{\ast}) n^{\ast} ,\end{align}

(3.13) \begin{align} u^{\ast} &= -\int _0^{s^{\ast}} F(\varsigma ){\rm d}\varsigma + \left ({ A-2\lambda \displaystyle \int _0^{s^{\ast}} \varrho (\varsigma )F(\varsigma ){\rm d}\varsigma } \right ) n^{\ast} + C. \end{align}

3.4. The incomplete similarity

In § 3.3, we derived an asymptotic solution for the flow at infinite Reynolds number. In this limit, the flow behaves as an equilibrium mixing layer, both velocities scale with their respective velocity differences, and the shape of the velocity profile remains unchanged across all locations. At finite Reynolds numbers, however, two overlap regions appear at the edges of the middle layer. In this case, the flow is no longer purely a mixing layer or a boundary layer, but instead exhibits characteristics of both. Since the 1930s, two principal models have been proposed to describe the overlap regions: the logarithmic law (von Kármán Reference von Kármán1930) and the power law (Barenblatt Reference Barenblatt1993). Although both are derived with comparable rigour, the logarithmic law relies on the assumption that molecular viscosity is negligible. In this context, the logarithmic and power laws are expressed as follows:

(3.14) \begin{align} \widetilde {L}(\xi ) &= \ln (\xi ), \end{align}

(3.15) \begin{align} \widetilde {L}(\xi ) &= \xi ^\epsilon -1 .\end{align}

The extent to which the flow resembles either a boundary layer or a mixing layer strongly depends on the local Reynolds number ${Re}_\delta$ . To initiate our analysis, we examine a relatively low Reynolds number case with $\alpha ={7.5}^\circ$ and $C_\mu ={0.25\times 10^{-3}}{}$ . Our results show that at these lower Reynolds numbers, especially during the early stages of mixing layer development, both overlap regions conform to the logarithmic law. This behaviour is illustrated in the scaled velocity profiles shown at four representative locations in figure 16. Notably, at $s/c=0.1$ , the velocity profile can be entirely described by the logarithmic law. As the local Reynolds number increases, a transition occurs, and the mixing layer emerges at some streamwise location. However, since the characteristic velocity remains close to the friction velocity, the logarithmic law continues to provide a reasonable description of both overlap regions. The profile shape gradually evolves with increasing local Reynolds number, incorporating a linear segment associated with the mixing layer. Consequently, the tangential velocity profile remains Reynolds-number-dependent, and its similarity is incomplete.

Figure 16. Evolution of the mixing layer for $\alpha = {7.5}^\circ$ and $C_\mu ={0.25\times 10^{-3}}{}$ test case compared with the logarithmic law (solid line) at four locations: (a) $s/c = 0.10$ ( $\square$ ), (b) $s/c = 0.20$ ( $\circ$ ), (c) $s/c = 0.30$ ( ) and (d) $s/c = 0.40$ ( $\times$ ).

Figure 17. Evolution of the mixing layer for $\alpha = {15.0}^\circ$ and $C_\mu ={1.25\times 10^{-3}}{}$ test case compared with the power law (solid line) at four locations: (a) $s/c = 0.20$ ( $\circ$ ), (b) $s/c = 0.40$ ( $\times$ ), (c) $s/c = 0.50$ ( ) and (d) $s/c = 0.60$ ( $+$ ).

At high Reynolds numbers, such as in cases with $\alpha = {12.5}^\circ$ and ${15.0}^\circ$ , the scaling becomes largely insensitive to variations in the local Reynolds number as these cases are relatively close to the equilibrium mixing layer. In these cases, the characteristic velocity required to align successive profiles closely approximates the mixing layer velocity difference $u^{\ast}_1-u^{\ast}_0$ . However, the inner overlap region lies much closer to the wall, where the molecular viscosity can no longer be neglected. As a result, the power law is required to achieve consistent scaling across both overlap regions. Unlike the low Reynolds number regime, where the overlap region shape remains fixed, the shape in this high Reynolds number regime varies with the Reynolds number, due to the Reynolds number dependence of the power-law exponent. Figure 17 shows the velocity profiles at four selected locations for the case with $\alpha = {15.0}^\circ$ and $C_\mu ={1.25\times 10^{-3}}{}$ . The profiles are overlaid with both power law fits and the asymptotic mixing layer line, highlighting the increasing linearity of the profiles and the proximity of the flow to the equilibrium mixing layer.

The preceding discussions, particularly figures 16 and 17, demonstrate that, regardless of the specific law used to describe the overlap regions, the entire velocity profile may be interpreted as a boundary layer influenced by contributions from the mixing layer. Following Coles (Reference Coles1956), we propose an analogue of the law of the wake to represent the effect of the mixing layer embedded within a turbulent boundary layer,

(3.16) \begin{equation} \widehat {U}(\xi ) = \widetilde {L}(\xi ) - \varPi \widehat {M}(\xi ). \end{equation}

By analogy with Coles’ law of the wake, we refer to the parameter $\varPi$ in (3.16) as the mixing strength. Using the current experimental data, we aim to investigate the shape of the mixing function $\widehat {M}(\xi )$ for finite values of $\varPi$ . The most direct approach to determining the shape of the mixing function is to subtract the equilibrium mixing layer profile from a boundary layer profile that is infinitesimally close to it. Mathematically, this limiting state corresponds to the power-law exponent $\epsilon$ approaching unity. In this asymptotic state, the mixing function $\widehat {M}(\xi )$ takes the following form:

(3.17) \begin{align} \widetilde {\varPi } &= \max \left ({ \widetilde {L}(\xi )-\widetilde {U}(\xi ) }\right ) = \epsilon ^{\frac {\epsilon }{(1-\epsilon )}}-\epsilon ^{\frac {1}{(1-\epsilon )}}, \end{align}

(3.18) \begin{align} \widetilde {M}(\xi ) &= \lim _{\epsilon \rightarrow 1} \frac {\widetilde {L}(\xi )-\widetilde {U}(\xi )}{\widetilde {\varPi }} = \lim _{\epsilon \rightarrow 1} \frac {\xi ^\epsilon -\xi }{\widetilde {\varPi }} = -\xi \ln (\xi ). \end{align}

The asymptotic shape of the mixing function, along with the variation of mixing strength as $\epsilon \rightarrow 1$ , is illustrated in figure 18. Evidently, while the mixing strength approaches zero in this limit, the mixing function itself converges to a well-defined shape. Specifically, it tends towards $\widetilde {M}(\xi ) = -\xi \ln (\xi )$ , as shown in figure 18(a).

Figure 18. The asymptotic shape of the mixing function and the variations of the mixing strength with respect to the power-law exponent.

Figure 19. Distribution of the mixing function for $\alpha = {7.5}^\circ$ and $C_\mu ={0.25\times 10^{-3}}{}$ test case compared with the asymptotic shape (solid line) at two locations: (a) $s/c = 0.20$ ( $\circ$ ) and (b) $s/c = 0.40$ ( $\times$ ).

Figure 20. Distribution of the mixing function for $\alpha = {15.0}^\circ$ and $C_\mu ={0.25\times 10^{-3}}{}$ test case compared with the asymptotic shape (solid line) at two locations: (a) $s/c = 0.20$ ( $\circ$ ) and (b) $s/c = 0.60$ ( $+$ ).

If the mixing function remains self-similar, we would expect the shape observed in figure 18(a) to be preserved, regardless of the law used to describe the base boundary layer. To investigate this, we examine the contribution of the mixing layer for both cases presented earlier in figures 16 and 17. The results, obtained by subtracting either the logarithmic law or the power law from the velocity profiles, are shown in figures 19 and 20, respectively. These figures confirm our earlier hypothesis. Irrespective of the boundary layer description, the subtraction yields the same shape observed in figure 18(a). A notable feature of the mixing function is its asymmetry, which distinguishes it from the wake function observed by Coles (Reference Coles1956). Consequently, the sine-squared function, commonly employed to represent the wake function, is not well-suited for describing the mixing function. Overall, using any smooth empirical function $\widetilde {Q}(\xi )$ that accurately captures the shape shown in figure 18(a), we can express the tangential velocity profile as follows:

(3.19) \begin{equation} \widehat {U}(\xi ) = \widetilde {L}(\xi ) - \varPi \widetilde {Q}(\xi ) .\end{equation}

Figure 21. Distribution of the scaled normal velocity compared with the asymptotic shape (solid line) at locations $s/c = 0.20$ ( $\circ$ ), $s/c = 0.30$ ( ), $s/c = 0.40$ ( $\times$ ) and $s/c = 0.50$ ( ) for two test cases: (a) $\alpha = {7.5}^\circ$ and $C_\mu ={0.25\times 10^{-3}}{}$ and (b) $\alpha = {15.0}^\circ$ and $C_\mu ={1.25\times 10^{-3}}{}$ .

We conclude this section by comparing the distribution of the normal velocity with its asymptotic shape, as shown in figure 21. In contrast to the tangential velocity, the normal velocity can be consistently scaled using the velocity difference $v^{\ast}_1$ , with the data closely following the asymptotic line across the range of Reynolds numbers tested. A more detailed discussion of the normal velocity is deferred to § 3.5.

3.5. Curvature effects

In this section, we examine whether the surface curvature influences the velocity gradients $\partial v^{\ast}/\partial n^{\ast}$ and $\partial u^{\ast}/\partial n^{\ast}$ , focusing on regions with high local Reynolds numbers where the flow behaves more like a mixing layer. The effect of surface curvature on the normal velocity is inherently complex. As discussed in § 3.2, the normal free stream velocity, which drives the gradient $\partial v^{\ast}/\partial n^{\ast}$ , is non-zero and, according to (2.2), depends on both the angle of attack and the local slope of the aerofoil, with $v^{\ast}_\infty =\sin (\alpha -\beta )$ . As a result, it is not possible to decouple the curvature effects from those of the pressure gradient. Therefore, our interest here lies in understanding the combined influence of these effects on the normal velocity gradient $\partial v^{\ast}/\partial n^{\ast}$ . The experimental data indicate that, for cases with lower angles of attack and higher momentum coefficients, specifically up to $\alpha ={12.5}^\circ$ and $C_\mu ={1.25\times 10^{-3}}{}$ , the slope of the normal velocity gradient remains largely unchanged. In contrast, for the remaining cases where the combined effect of curvature and pressure gradient is more pronounced, the gradient slope gradually decreases along the aerofoil. For illustration, two representative cases are compared in figure 22.

Figure 22. The $s$ -contours of the dimensionless mean normal velocity at locations $s/c = 0.20$ ( $\circ$ ), $s/c = 0.30$ ( ), $s/c = 0.40$ ( $\times$ ), $s/c = 0.50$ ( ) and $s/c = 0.60$ ( $+$ ) for two test cases: (a) $\alpha = {12.5}^\circ$ and $C_\mu ={1.25\times 10^{-3}}{}$ and (b) $\alpha = {15.0}^\circ$ and $C_\mu ={1.25\times 10^{-3}}{}$ .

Figure 23. The $n$ -contours of the dimensionless mean normal velocity at locations $n/c = 0.02$ ( $\square$ ), $n/c = 0.04$ ( $\circ$ ), $n/c = 0.06$ ( ), $n/c = 0.08$ ( $\times$ ), $n/c = 0.10$ ( ) and $n/c = 0.12$ ( $+$ ) for two test cases: (a) $\alpha = {12.5}^\circ$ and $C_\mu ={1.25\times 10^{-3}}{}$ and (b) $\alpha = {15.0}^\circ$ and $C_\mu ={0.25\times 10^{-3}}{}$ .

To further investigate this behaviour, we expand the slope of the normal velocity gradient in the asymptotic solution $F(s^{\ast})$ about the onset of the mixing layer using the Taylor series,

(3.20) \begin{equation} F(s^{\ast}) = F(0) + \frac {{\rm d}F}{{\rm d}s^{\ast}}(0) s^{\ast} + \frac {1}{2}\frac {{\rm d}^2F}{{\rm d}s^{\ast 2}}(0) s^{\ast 2} + {O}\big ( s^{\ast 3} \big ). \end{equation}

In general, near the onset, $F(s^{\ast})$ can be approximated by its zeroth-order term. Experimental observations show that when the mixing layer is allowed to develop naturally, which is under minimal combined effects of pressure gradient and surface curvature, the function $F(s^{\ast})$ remains approximately constant over the full development length. This implies that the normal velocity gradient $\partial v^{\ast}/\partial n^{\ast}$ is preserved. Such behaviour was observed in three previously discussed cases, that is both $\alpha ={7.5}^\circ$ configurations, as well as the $\alpha = {12.5}^\circ$ and $C_\mu ={1.25\times 10^{-3}}{}$ case. In contrast, when the combined influence of surface curvature and pressure gradient increases, $F(s^{\ast})$ can no longer be truncated at the zeroth order and begins to exhibit higher-order behaviour. For example, in the $\alpha = {12.5}^\circ$ and $C_\mu ={0.25\times 10^{-3}}{}$ and $\alpha = {15.0}^\circ$ and $C_\mu ={1.25\times 10^{-3}}{}$ cases, the function $F(s^{\ast})$ exhibits a linear decrease. In the most extreme case, that is $\alpha = {15.0}^\circ$ and $C_\mu ={0.25\times 10^{-3}}{}$ with the highest angle of attack and the lowest momentum coefficient, even a first-order approximation proved insufficient, and $F(s^{\ast})$ exhibited a parabolic decay. The $n$ -contours of the dimensionless mean normal velocity for two representative cases are shown in figures 22 and 23, illustrating both the invariance of the normal velocity profile and the parabolic decay observed under strong curvature and pressure gradient conditions. Figure 23 highlights the need for higher-order Taylor series truncations at increased angles of attack and reduced momentum coefficients.

Figure 24. The spreading of the boundary layer for four test conditions: (a) $\alpha = {7.5}^\circ$ and $C_\mu ={0.25\times 10^{-3}}{}$ ( $\square$ ), (b) $\alpha = {12.5}^\circ$ and $C_\mu ={0.25\times 10^{-3}}{}$ ( $\circ$ ), (c) $\alpha = {15.0}^\circ$ and $C_\mu ={1.25\times 10^{-3}}{}$ ( ), (d) $\alpha = {15.0}^\circ$ and $C_\mu ={0.25\times 10^{-3}}{}$ ( $\times$ ).

A similar argument can be made for the spreading rate. It is well known that the spreading rate of turbulent wall jets is nonlinear and smaller than that of free jets. While the exact comparison between the spreading rate of this flow and a turbulent wall jet for a given momentum input remains unclear, the flow appears to spread at the very least linearly, when it is not disturbed. The spreading rate generally becomes nonlinear as the angle of attack increases or the momentum coefficient decreases, as shown in figure 24. The maximum spreading rate for the extreme case with $\alpha = {15.0}^\circ$ and $C_\mu ={0.25\times 10^{-3}}{}$ was around 25 %.

Unlike the normal velocity, surface curvature directly affects the tangential velocity gradient $\partial u^{\ast}/\partial n^{\ast}$ . The $s$ -contours of the dimensionless mean tangential velocity for two test cases are shown in figure 25. According to the asymptotic solution, the tangential velocity gradient is given by $\partial u^{\ast}/\partial n^{\ast} = A-2\lambda \int _0^{s^{\ast}} \varrho (\varsigma )F(\varsigma ){\rm d}\varsigma$ , indicating a clear dependence on curvature, even under conditions where the flow evolves naturally and $F(s^{\ast}) = F(0) = B$ . To isolate the effect of curvature, we consider two cases in which the normal velocity remains nearly constant along the streamwise direction, as shown in figure 25. In fact, looking at figure 26, we observe that the slope of the profiles changes more rapidly upstream, where the curvature magnitude is larger. Eventually, as the curvature diminishes closer to the trailing edge, the slope gradually stabilises and approaches a constant value.

Figure 25. The $s$ -contours of the dimensionless mean tangential velocity at locations $s/c = 0.10$ ( $\square$ ), $s/c = 0.20$ ( $\circ$ ), $s/c = 0.30$ ( ), $s/c = 0.40$ ( $\times$ ), $s/c = 0.50$ ( ) and $s/c = 0.60$ ( $+$ ) for two test cases: (a) $\alpha = {7.5}^\circ$ and $C_\mu ={0.25\times 10^{-3}}{}$ ( $\square$ ) and (b) $\alpha = {12.5}^\circ$ and $C_\mu ={1.25\times 10^{-3}}{}$ ( $\circ$ ).

Figure 26. Distribution of the curvature parameters with respect to the arclength for a NACA 0025 aerofoil.

Next, we examine the rotation of the fluid elements around the wall using the mean spanwise vorticity for the cases where $F(s^{\ast}) = F(0) = B$ . The time-averaged vorticity equation in curvilinear coordinates $(s,n)$ is stated by (3.21). The dimensionless form of this equation, normalised by the highest-order velocity gradient $U_\infty /\delta$ , is presented in (3.22) below,

(3.21) \begin{align} \frac {\partial \overline {v_n} }{\partial s} - \frac {\partial (1 + \varrho n)\overline {u_t} }{\partial n} &= (1 + \varrho n) \overline {\varOmega _z}, \end{align}

(3.22) \begin{align} \underbrace { \frac {\delta }{\lambda }\frac {\partial u^{\ast} }{\partial n^{\ast}} }_{ {O}(1) } + \underbrace { \varrho \delta \frac {\partial n^{\ast} u^{\ast} }{\partial n^{\ast}} }_{{O}(\varrho \delta )} - \underbrace { \frac {\delta }{\lambda } \frac {\partial v^{\ast} }{\partial s^{\ast}} }_{{O}\left( \frac {\delta ^2}{\lambda ^2}\right)} &= -(1 + \varrho \lambda n^{\ast}) \frac {\overline {\varOmega _z}\delta }{U_\infty } .\end{align}

Under the conditions $\delta /\lambda \ll 1$ , $\varrho \delta \ll 1$ and $F(s^{\ast}) = B$ , the time-averaged vorticity can be determined by neglecting terms of order $\delta ^2/\lambda ^2$ and $\varrho ^2\delta ^2$ and substituting the asymptotic velocity expressions into (3.22),

(3.23) \begin{align} &\left .\begin{aligned} u^{\ast} &= \left ({ A-2B\lambda \displaystyle \int _0^{s^{\ast}} \varrho (\varsigma ){\rm d}\varsigma } \right ) n^{\ast} - Bs^{\ast}+ C\\ v^{\ast} &= B n^{\ast} \end{aligned}\right \rbrace \xrightarrow {\dfrac {1}{1+\varrho \lambda n^{\ast}} = 1-\varrho \lambda n^{\ast} + {O}(\varrho ^2\delta ^2)} \nonumber \\ &\qquad \qquad \frac {\overline {\varOmega _z}\lambda }{U_\infty } = \varrho \lambda (Bs^{\ast}-C)(1-\varrho \lambda n^{\ast}) + \left ({ 2B\lambda \int _{0}^{s^{\ast}} \varrho (\varsigma ){\rm d}\varsigma -A }\right )(1+\varrho \lambda n^{\ast}) .\end{align}

In the limit of $\varrho \lambda \ll 1$ , another simplification can be applied to the vorticity distribution given in (3.23). Under this condition, the variations in the radius of curvature across the entire length of the mixing layer $\lambda$ become negligible. Consequently, we expect $\int _{0}^{s^{\ast}} \varrho (\varsigma ){\rm d}\varsigma \approx \varrho s^{\ast}$ .

The conditions $\varrho \delta \ll 1$ and $\varrho \lambda \ll 1$ are well satisfied over the second-half of the aerofoil ( $s/c \gt 0.15$ ), as shown in figure 26. Note that for the highest Reynolds number cases, $\delta /c \sim {O}(10^{-1})$ , and therefore $\varrho \delta \sim {O}(10^{-2})$ within $s/c \gt 0.15$ . With these considerations, (3.23) simplifies to

(3.24) \begin{equation} \varOmega ^{\ast} = \frac {\overline {\varOmega _z}}{\varrho U_\infty } = (Bs^{\ast}-C)(1-\varrho \lambda n^{\ast}) + \left ({ 2B s^{\ast}-\frac {A}{\varrho \lambda } }\right )(1+\varrho \lambda n^{\ast}). \end{equation}

Equation (3.24) carries significant implications regarding the effects of curvature on the mixing layer. First, as $\varrho \delta \rightarrow 0$ , the vorticity behaves as $\varOmega ^{\ast} = 3Bs-C-Ar_0/\lambda$ , suggesting that $\varOmega ^{\ast}$ is expected to increase linearly along the surface. Second, fluid elements rotate along the wall with a vorticity magnitude $\overline {\varOmega _z} = \varrho U_\infty \varOmega ^{\ast}$ , which is proportional to the local surface curvature $\varrho$ . As a result, the mixing layer tends to adhere to the wall, with this adherence gradually weakening as the mixing layer develops along the surface due to the positive normal velocity gradient $F(0) = B$ . The asymptotic behaviour of the vorticity is evident from the $n$ -contour plots of the dimensionless vorticity for the two test cases shown in figure 27. The tendency of the mixing layer to adhere to the wall is also apparent in the contour plots presented earlier in figures 13(a) and 14(b).

Figure 27. The $n$ -contours of the dimensionless mean spanwise vorticity at locations $n/c=0.02$ ( $\circ$ ), $n/c = 0.03$ ( ), $n/c = 0.04$ ( $\times$ ), $n/c = 0.05$ ( ), $n/c = 0.06$ ( $+$ ) and $n/c = 0.08$ ( ), demonstrating the clockwise rotation of fluid elements inside the mixing layer for two test cases: (a) $\alpha = {7.5}^\circ$ and $C_\mu ={0.25\times 10^{-3}}{}$ and (b) $\alpha = {12.5}^\circ$ and $C_\mu ={1.25\times 10^{-3}}{}$ .