2.1 Problem description

We consider a two-dimensional free shear layer as a model problem to examine the effectiveness of thermal actuation. Two incoming streams at velocities $U_{1}$ and $U_{2}$ , initially separated by a splitter plate, at Mach numbers of $M_{1}=U_{1}/a_{\infty }=0.4$ (top) and $M_{2}=U_{2}/a_{\infty }=0.1$ (bottom) with the same free-stream acoustic speed, $a_{\infty }$ , and pressure, $p_{\infty }$ , form a spatially developing shear layer from the trailing edge, as illustrated in figure 1. Following the definition by Papamoschou & Roshko (Reference Papamoschou and Roshko1988), the mean convective Mach number for the shear layer is $M_{c}=(U_{1}-U_{2})/(2a_{\infty })=0.15$ , with the density ratio $\unicode[STIX]{x1D70C}_{1}/\unicode[STIX]{x1D70C}_{2}$ being unity ( $\unicode[STIX]{x1D70C}_{1}=\unicode[STIX]{x1D70C}_{2}=\unicode[STIX]{x1D70C}_{\infty }$ ). The plate-thickness-based Reynolds number, $Re_{w}\equiv \unicode[STIX]{x1D70C}_{\infty }\bar{U}w/\unicode[STIX]{x1D707}_{\infty }$ , is set to $2500$ , where $\bar{U}=(U_{1}+U_{2})/2$ is the convection velocity, $w$ is the plate thickness and $\unicode[STIX]{x1D707}_{\infty }$ is the free-stream dynamic viscosity. The boundary layers on both sides of the plate are prescribed with the same momentum thickness, $\unicode[STIX]{x1D703}_{0}$ , which is chosen to be $5\,\%$ , $10\,\%$ and $25\,\%$ of the splitter plate thickness, $w$ , causing the momentum-thickness-based Reynolds number $Re_{\unicode[STIX]{x1D703}_{0}}\equiv \unicode[STIX]{x1D70C}_{\infty }\bar{U}\unicode[STIX]{x1D703}_{0}/\unicode[STIX]{x1D707}_{\infty }$ to range from $125$ to $625$ .

2.2 Computational approach

The governing equations for the free shear layer are the compressible Navier–Stokes equations,

with the equation of state for ideal gas,

where $\unicode[STIX]{x1D6FE}$ is the specific heat ratio. The non-dimensional variables in (2.1)–(2.4) are the spatial coordinate $\tilde{x}_{i}$ , time $\tilde{t}$ , density $\tilde{\unicode[STIX]{x1D70C}}$ , velocity $\tilde{u} _{i}$ , pressure $\tilde{p}$ , energy $\tilde{e}$ and temperature $\tilde{T}$ according to the non-dimensionalization of

We note that the shear layer roll-up wavelength, $\unicode[STIX]{x1D706}_{n}$ , may also be used for the non-dimensionalization of the spatial coordinate $x_{i}$ in order to provide fair comparisons between different values of $\unicode[STIX]{x1D703}_{0}$ . In this paper, it will be clearly stated which length scale is being used for non-dimensionalization. The dimensionless parameters appearing in the governing equations are the acoustic Reynolds number and Prandtl number, given by

where $\unicode[STIX]{x1D6FC}_{\infty }$ is the free-stream thermal diffusivity.

To simulate the flow, we consider the two-dimensional set of equations and use the compressible flow solver CharLES (Khalighi et al. Reference Khalighi, Ham, Moin, Lele, Schlinker, Reba and Simonich2011a ,Reference Khalighi, Nichols, Ham, Lele and Moin b ; Brès et al. Reference Brès, Ham, Nichols and Lele2017), which utilizes a second-order-accurate finite-volume method and a third-order Runge–Kutta scheme for time stepping. We use $\unicode[STIX]{x1D6FE}=1.4$ , $Re=10\,000$ and $Pr=0.7$ in the present study. These values are representative of standard air. The temperature-varying dynamic viscosity, $\unicode[STIX]{x1D707}(T)$ , is evaluated by a power law as $\unicode[STIX]{x1D707}=\unicode[STIX]{x1D707}_{\infty }(T/T_{\infty })^{0.76}$ (see Garnier, Adams & Sagaut Reference Garnier, Adams and Sagaut2009). The power law models the dynamic viscosity variation well for standard air in the range $T/T_{\infty }\in [0.5,1.7]$ . This range is suitable for the current study with local thermal inputs, where we observe that the temperature fluctuation is within $22\,\%$ about $T_{\infty }$ .

The computational domain, as illustrated in the left part of figure 1, has a streamwise extent of $x/w\in [-15,400]$ and a maximum vertical extent of $y/w\in [-200,200]$ , similar to the domain set-up used by Sharma, Bhaskaran & Lele (Reference Sharma, Bhaskaran and Lele2011). The splitter plate has an elliptic trailing edge with an eccentricity of $0.866$ , with its tip positioned at the origin, $x/w=y/w=0$ . The domain is discretized with a structured body-fitted mesh around the splitter plate. The near-wall resolution is $\unicode[STIX]{x0394}y^{+}\equiv \unicode[STIX]{x0394}yu^{\ast }/\unicode[STIX]{x1D708}_{\infty }=0.25$ (see Choi & Moin Reference Choi and Moin2012), where $u^{\ast }=\sqrt{\unicode[STIX]{x1D70F}_{0}/\unicode[STIX]{x1D70C}_{\infty }}$ is the wall shear velocity and $\unicode[STIX]{x1D70F}_{0}$ is the wall shear stress. To resolve the interaction between the shear layer and the localized periodic heat forcing, the wall-normal grid stretching rate is limited such that the $\unicode[STIX]{x0394}y^{+}$ only reaches a maximum of $1.25$ within $50w$ from the wall. The overall grid size is approximately $4.4\times 10^{5}$ . The direct numerical simulation (DNS) results from this mesh were verified by comparing the flow field with that from a finer mesh, where the grid was refined in the sheared region and had a size of $6.8\times 10^{5}$ grid points. No noticeable differences were found in the results from these two meshes.

Two streams are introduced to the computational domain from the inlet boundary (left) using a Blasius boundary layer velocity profile. Their density profiles are computed using the Crocco–Busemann relation, with the same far-field temperature and pressure. A sponge zone is applied along the top, bottom and outlet (right) boundaries to damp out existing acoustic waves and vortical structures (Freund Reference Freund1997). The target states of the sponge zone at the top and bottom boundaries are set to $[U_{1},\unicode[STIX]{x1D70C}_{\infty },p_{\infty }]$ and $[U_{2},\unicode[STIX]{x1D70C}_{\infty },p_{\infty }]$ respectively. At the outlet, the time-average flow is set to be the target state of the sponge zone.

In this study, the effects of momentum thickness, the forcing frequency and the amplitude are examined in the context of modifying the characteristics of the shear layer. The flow control set-up using local periodic heating and the associated input parameters are discussed below.

2.3 Local periodic heating

The thermal actuator model is implemented in the numerical simulations through a boundary condition. Over the elliptic trailing edge of the splitter plate, an oscillatory energy-flux (unsteady Neumann) boundary condition that introduces local periodic heating is prescribed to the energy equation (2.3), along with a no-slip boundary condition for the momentum equation (2.2). In the present study, we consider two forms of periodic heating. First, we consider a temporally oscillatory heat flux with zero mean as

where the net heating input to the local fluid is zero over the forcing period. We also consider another forcing input $\dot{q}^{p}$ which introduces a positive direct current (DC) offset to the expression of $\dot{q}$ in (2.7) such that the heat flux is always positive over the duty cycle,

In the expressions for $\dot{q}$ and $\dot{q}^{p}$ , $f^{+}$ and $\hat{q}$ denote the forcing frequency and amplitude respectively. For each choice of $\unicode[STIX]{x1D703}_{0}/w$ , the forcing frequency, $f^{+}$ , is normalized by the corresponding baseline shear layer roll-up frequency, $f_{n}$ , and is varied from $f^{+}/f_{n}=0.500$ to $1.25$ , over which we observe forcing effects. Since the heat flux amplitude, $\hat{q}$ , accounts for the forcing power that is introduced to the shear layer flow, we normalize the total heating power from the trailing edge by the characteristic kinetic energy flux across the momentum thickness as

where

is the cyclic average heating power introduced by the actuator. This normalization can account for the use of a higher forcing amplitude, $\hat{q}$ , for a greater initial momentum thickness, $\unicode[STIX]{x1D703}_{0}$ , for a fair comparison of forcing requirements across different values of $\unicode[STIX]{x1D703}_{0}$ , since a thicker shear layer is expected to necessitate a higher level of perturbation inputs to modify the shear layer. We parameterize the normalized forcing power in this study in the range of $E^{+}=0.296$ – $2.96$ . This value is of the same order of magnitude of those used in plasma-based flow control works (Samimy et al. Reference Samimy, Kim, Kastner, Adamovich and Utkin2007; Corke et al. Reference Corke, Enloe and Wilkinson2010; Sinha et al. Reference Sinha, Alkandry, Kearney-Fischer, Samimy and Colonius2012; Akins et al. Reference Akins, Singh and Little2015). We also note that the maximum temperature fluctuation observed over the surface of the actuator is within $22\,\%$ about the free-stream temperature with the use of the largest forcing amplitude, $E^{+}=2.96$ , in $\unicode[STIX]{x1D703}_{0}/w=0.25$ cases.

Figure 2.

Baseline flow characterization for $\unicode[STIX]{x1D703}_{0}/w=0.1$ . (a) An instantaneous snapshot of the normalized spanwise vorticity $\unicode[STIX]{x1D714}_{z}^{\ast }=\unicode[STIX]{x1D714}_{z}\unicode[STIX]{x1D703}_{0}/\bar{U}$ . (b) Time-average velocity fluctuation magnitude $\Vert \boldsymbol{u}^{\prime }\Vert ^{2}=(u_{x}^{\prime 2}+u_{y}^{\prime 2})/\bar{U}^{2}$ . (c) Spatial growth of momentum thickness ( $\unicode[STIX]{x1D703}/\unicode[STIX]{x1D703}_{0}$ ). All share the same normalized streamwise coordinate, $x/\unicode[STIX]{x1D706}_{n}$ , where $\unicode[STIX]{x1D706}_{n}=\bar{U}/f_{n}$ is the baseline roll-up wavelength.

3.1 Shear layer characteristics

Let us first characterize the baseline flows at three momentum thicknesses specified at the inlet, namely $\unicode[STIX]{x1D703}_{0}/w=0.05$ , $0.1$ and $0.25$ . We focus our interest in the region within the streamwise extent of $10\unicode[STIX]{x1D706}_{n}$ , where $\unicode[STIX]{x1D706}_{n}=\bar{U}/f_{n}$ is the fundamental shear layer roll-up wavelength. Figure 2 shows a snapshot of the normalized instantaneous spanwise vorticity (a), the velocity fluctuation magnitude (b) and the momentum thickness profile (c) for the baseline flow with $\unicode[STIX]{x1D703}_{0}/w=0.1$ . In this work, the momentum thickness is computed using the density and streamwise velocity profiles in the shear layer as

where the overbar denotes the time-averaged quantity based on the flow statistics collected over more than 200 roll-up periods. Comparing the spatial growth of momentum thickness with the instantaneous flow field, we classify the shear layer flow into three regions: (i) the shear layer roll-up region, where the momentum thickness exhibits the first rapid growth behind the trailing edge; (ii) the isolated vortex region, where each vortex formed remains compact as an individual and convects closely along the centreline; (iii) the vortex merging region, where the vortices start to deviate from the centreline while convecting downstream, as the initiation of the vortex merging process. This last region can be identified by the second rapid growth of the momentum thickness. These three regions can also be characterized by the streamwise change in velocity fluctuation magnitude. In regions (i) and (iii), the spatial extent and the magnitude of fluctuation are both increasing, but region (ii) shows no apparent changes in the growth of the momentum thickness except for a rippling variation in the momentum thickness profile. The cause of this ripple is that the orientation of the vortex evolves as the vortex convects downstream, which is discussed in appendix A. Later, in § 5, the forcing effect of periodic heating on the characteristics of these three regions is examined in further detail.

3.2 Influence of the splitter plate thickness

Due to the finite thickness of the splitter plate, a small recirculation region of length $x_{r}$ develops directly behind the blunt trailing edge of the splitter plate, as illustrated by the time-average streamlines in figure 1 (bottom right). This streamline pattern for two boundary layers merging behind the trailing edge of a finite-thickness splitter plate was also observed by Laizet, Lardeau & Lamballais (Reference Laizet, Lardeau and Lamballais2010) and Sharma et al. (Reference Sharma, Bhaskaran and Lele2011). The recirculation region can be viewed as a wake behind the finite-thickness splitter plate and results in a deficit in the profile streamwise velocity. In the theoretical study by Zhuang & Dimotakis (Reference Zhuang and Dimotakis1995), it was reported that a wake instability mode can be introduced to the shear layer which has a deficit in its streamwise velocity profile due to a wake component, and the growth rates of both the wake mode and the fundamental shear layer mode increase when the streamwise velocity profile has a greater wake deficit. Mehta (Reference Mehta1991) experimentally showed that the splitter plate wake plays a dominant role in the development of the shear layer. The wake increases the turbulence levels in the near-field and shortens the development distance to achieve self-similarity in shear layer profiles. Remarks by Mehta (Reference Mehta1991) on the lack of simple scaling for the shear layer with the presence of the wake also suggest that a nonlinear interaction is taking place between the wake and the shear layer. Laizet et al. (Reference Laizet, Lardeau and Lamballais2010) performed DNS of shear layers developing behind three different trailing edge geometries and reported that the presence of the wake can lead to different turbulent structures and shear layer spreading rates. We note, however, that no trace of von Kármán shedding is observed in the simulations considered here, in spite of the presence of the recirculation region.

Figure 3.

Time-average streamwise velocity profiles at $x/w=0.3$ over $y/\unicode[STIX]{x1D706}_{n}$ for three baseline cases, showing a stronger wake deficit for lower $\unicode[STIX]{x1D703}_{0}$ .

Figure 4.

(a) Momentum thickness growth for baseline flows with $\unicode[STIX]{x1D703}_{0}/w=0.05$ , $0.1$ and $0.25$ . (b) Transverse velocity spectra for three baseline cases at $x/\unicode[STIX]{x1D706}_{n}=2.5$ . The shear layer roll-up frequency $f_{n}$ for each corresponding case is determined by the peak of its spectrum.

Table 1.

Baseline characteristics of the shear layer for different values of the momentum thickness $\unicode[STIX]{x1D703}_{0}/w$ : length of recirculation region $x_{r}/w$ ; roll-up frequency $f_{n}w/\bar{U}$ ; roll-up Strouhal number $St_{\unicode[STIX]{x1D703}_{0}}=f_{n}\unicode[STIX]{x1D703}_{0}/\bar{U}$ ; roll-up wavelength $\unicode[STIX]{x1D706}_{n}/w$ .

The lengths of the recirculation regions for each baseline case with varied momentum thickness $\unicode[STIX]{x1D703}_{0}/w$ are summarized in table 1, where the change in the wake length, $x_{r}/w$ , is limited to $6\,\%$ while $\unicode[STIX]{x1D703}_{0}/w$ is increased five times. At the streamwise station $x/w=3.0$ , the time-average streamwise velocity profile over the normalized transverse coordinate $y/\unicode[STIX]{x1D706}_{n}$ is presented in figure 3. It should be noted that the normalization of $y/\unicode[STIX]{x1D706}_{n}$ causes the velocity minimum to appear at different location for different choices of $\unicode[STIX]{x1D703}_{0}/w$ , since the value of $\unicode[STIX]{x1D706}_{n}$ is influenced by the value of $\unicode[STIX]{x1D703}_{0}/w$ . In figure 3, we observe that the wake deficit increases in both its transverse extent and its maximum velocity deficit for a thinner incoming boundary layer. In figure 4(a), we show the momentum thickness profile over the streamwise coordinate normalized by the corresponding roll-up wavelength for each case. As discussed previously using $\unicode[STIX]{x1D703}_{0}/w=0.1$ for the baseline characterization, the three-region growth patterns are also observed in the other two baseline cases. The initiation of merging takes place at $x/\unicode[STIX]{x1D706}_{n}\approx 6$ for all cases. However, thicker $\unicode[STIX]{x1D703}$ for the isolated vortex region and delayed roll-up are observed for thinner $\unicode[STIX]{x1D703}_{0}/w$ . These observations can be attributed to the stronger effect of the wake when $\unicode[STIX]{x1D703}_{0}$ is thinner. The effect of the plate wake can also be seen in the shear layer roll-up Strouhal number, $St_{\unicode[STIX]{x1D703}_{0}}=f_{n}\unicode[STIX]{x1D703}_{0}/\bar{U}$ , in table 1. We also note that the values of $St_{\unicode[STIX]{x1D703}_{0}}$ are lower than the prediction from linear instability theory (Monkewitz & Huerre Reference Monkewitz and Huerre1982; Ho & Huerre Reference Ho and Huerre1984), $St_{\unicode[STIX]{x1D703}_{0}}=0.032$ , in all baseline cases. This also suggests that the effective initial momentum thickness is greater than that on the high-speed side, $\unicode[STIX]{x1D703}_{0}$ , because of the presence of the wake. Therefore, for thinner $\unicode[STIX]{x1D703}_{0}$ , where the wake effect is more pronounced, a greater deviation of $St_{\unicode[STIX]{x1D703}_{0}}$ from $0.032$ is observed. The stronger wake effect on thin $\unicode[STIX]{x1D703}_{0}$ can also be inferred from figure 4(b), where the transverse velocity spectra at $x/\unicode[STIX]{x1D706}_{n}=2.5$ for three baseline cases are shown. The clean peaks in the cases of $\unicode[STIX]{x1D703}_{0}/w=0.1$ and $0.25$ suggest a synchronized shear layer roll-up and shedding, whereas the cause of the less prominent peak and broader frequency content in the case of $\unicode[STIX]{x1D703}_{0}/w=0.05$ can be attributed to the nonlinear interaction between the shear layer roll-up and the wake which redistributes the energy of the roll-up frequency to other nearby frequencies.

3.3 Local stability analysis

To determine regions with linear growth of the perturbations and predict the corresponding growth rate for the Kelvin–Helmholtz instability, local linear spatial stability analysis on the baseline flow is also performed in the onset region and the isolated vortex region of the shear layer. We compute the corresponding spatial growth rate at a specified perturbation frequency using the time-average density and streamwise velocity profiles as the base flow for the compressible Rayleigh equation (Sabatini & Bailly Reference Sabatini and Bailly2014).

Figure 5 shows the stability analysis results for $\unicode[STIX]{x1D703}_{0}/w=0.1$ baseline flow. Time-average flow profiles at two streamwise stations (shown in a) are chosen about which to perform the instability analysis. The first station $x_{1}$ , corresponding to $(x_{1}-x_{r})/\unicode[STIX]{x1D706}_{n}=0.24$ or $x_{1}/\unicode[STIX]{x1D706}_{n}=1.22$ , is located slightly downstream of the recirculation region but upstream of the shear layer roll-up region. In figure 2, the station $x_{1}/\unicode[STIX]{x1D706}_{n}=1.22$ is located in the flat region before the first sudden growth of the momentum thickness. Similarly, we choose the second station $x_{2}$ , where $(x_{2}-x_{r})/\unicode[STIX]{x1D706}_{n}=3.1$ or $x_{2}/\unicode[STIX]{x1D706}_{n}=4.06$ , in the isolated vortex region indicated by the second flat region and before the second sudden growth. In figure 5, the most amplified frequencies and the corresponding spatial growth rates are $St_{\unicode[STIX]{x1D703}_{0}}=0.0198,\unicode[STIX]{x1D6FC}_{i}\unicode[STIX]{x1D706}_{n}=2.74$ and $St_{\unicode[STIX]{x1D703}_{0}}=0.0111,\unicode[STIX]{x1D6FC}_{i}\unicode[STIX]{x1D706}_{n}=0.97$ for stations $x=x_{1}$ and $x=x_{2}$ respectively. We compare these values with the results from DNS shown in figure 6. In figure 6(a), the spectra of transverse velocity are plotted over a series of streamwise locations. The symbol ○ (blue) (referred to as mode 1 hereafter) represents the first spatially amplified mode and ○ (red) (referred to as mode 2 hereafter) depicts the second mode. Their corresponding frequencies in terms of Strouhal numbers are $St_{\unicode[STIX]{x1D703}_{0}}=0.0204$ and $0.0108$ for modes 1 and 2 respectively, as captured in (b). Compared with the frequencies predicted by linear stability theory, the differences are only approximately $1.5\,\%$ and $3\,\%$ for modes 1 and 2 respectively. The spatial growth of the modal amplitude can be seen in figure 6(c). Additionally, in (c), the amplitude growth rates of modes 1 and 2 are fitted with exponential growth functions in the ranges $0\leqslant x/\unicode[STIX]{x1D706}_{n}\leqslant 1.6$ and $2.5\leqslant x/\unicode[STIX]{x1D706}_{n}\leqslant 5.5$ respectively. We also find that the growth rates $\unicode[STIX]{x1D6FC}_{i}\unicode[STIX]{x1D706}_{n}$ from the fitting differ by only $0.7\,\%$ and $2\,\%$ from the predictions of the linear theory for roll-up and merging modes respectively. The fitted function deviates from the simulation data ○ (blue) and ○ (red) past $x/\unicode[STIX]{x1D706}_{n}=1.5$ and $6.5$ respectively. These deviations are expected since, referring to figure 2, these two stations have already reached the vortex formation and merging regions where the momentum thickness exhibits a sudden streamwise growth and the parallel flow assumption is no longer valid.

Figure 5.

Spatial growth rates from linear stability analysis at two streamwise stations for a $\unicode[STIX]{x1D703}_{0}/w=0.1$ baseline. Both agree with the results from DNS shown in figure 6. Theoretical growth rates at forcing frequencies that appear in § 5 are given in table 2.

Table 2.

The theoretical spatial growth rates obtained at forcing frequencies appearing in § 5 at stations $x=x_{1}$ and $x_{2}$ . These values of the growth rates will be compared with the growth rates of the forcing mode later in figures 10 and 14.

Figure 6.

Transverse velocity spectra over the centreline: (a) the spatial development of the spectra; (b) view from the amplitude–frequency plane; (c) streamwise profiles of modal amplitudes for the roll-up and merging modes with fitted exponential growth functions, where $x_{1}^{\ast }=(x-x_{1})/\unicode[STIX]{x1D706}_{n}$ and $x_{2}^{\ast }=(x-x_{2})/\unicode[STIX]{x1D706}_{n}$ . The frequencies and growth rates of modes 1 and 2 agree with the linear stability theory prediction in figure 5 with less than $3\,\%$ difference.

The agreement between the results from DNS and the linear stability analysis on the baseline flow provides deep insight into the growth of perturbations in the aforementioned regions. As we discuss later, the frequencies in the vicinity of that of mode 2 are also considered as candidates for the actuation frequency of the local periodic forcing in § 5. The growth rate prediction at the forcing frequency will also be compared with the forced cases later in discussions related to figures 10 and 14.

5.1 Spreading enhancement ( $f^{+}\approx 0.5f_{n}$ )

We examine the forcing effect when the actuation frequency is close to the first subharmonic of the roll-up frequency with the use of $\dot{q}$ . Two forcing frequencies are examined, namely $f^{+}/f_{n}=0.575$ and $0.500$ . The forcing frequency $f^{+}/f_{n}=0.575$ is selected for being close to the frequency of mode 2 found in figure 6. The exact first subharmonic $f^{+}/f_{n}=0.500$ is also chosen. For fair comparison, all cases presented in this section are based on the same forcing power of $E^{+}=0.741$ . In the discussion below, we first highlight the common forcing effects for these two forcing frequencies. We then discuss the differences between the two forced cases.

We start our detailed discussion on cases with $\unicode[STIX]{x1D703}_{0}/w=0.1$ . The baseline and forced flows are compared in figures 9 and 10. It is found that the forcing is able to change the vortex dynamics by encouraging the merging downstream while keeping the momentum thickness unchanged in the region dominated only by fundamental roll-up. We first track the centre of each individual vortex after roll-up using the $Q$ -criterion. By tracking all formed vortices over a time span of $100$ roll-up periods, we visualize the vortex trajectories by black transparent dots and the contour lines for a representative instantaneous spanwise vorticity field in figure 9(a–c). On observing the trajectories, it is clear that, as the vortices convect downstream, they spread away from the centreline ( $y/\unicode[STIX]{x1D706}_{n}=0$ ) earlier upstream in the forced flows compared with the baseline case. This deviation from the centreline serves as an initiation for the vortex merging process (Winant & Browand Reference Winant and Browand1974; Ho & Huang Reference Ho and Huang1982). While the vortex merging process takes place, the flow passes through a stage where a pair of vortices become vertically aligned, which then results in steep spatial growth of the momentum thickness. For example, the controlled case with $f^{+}/f_{n}=0.500$ shows the vertical alignment to take place at $x/\unicode[STIX]{x1D706}_{n}\approx 7$ . By observing the momentum thickness in figure 10(a), we find that this steep growth indeed corresponds to where the trajectories demonstrate deviation from the centreline for each case. With forcing at these two frequencies, the momentum thickness does not exhibit any change from that of the baseline until this second growth takes place farther upstream compared with the baseline.

A merging criterion is set up such that two vortices are considered to be merged if their instantaneous mutual distance is less than $0.2\unicode[STIX]{x1D706}_{n}$ . By setting this threshold, we show the average streamwise location along with the variation (standard deviation) in where merging takes place by the green-shaded regions for each case in figure 9(a–c). Again, by comparing the shaded regions from the forced cases with the baseline, it is found that the merging process completes upstream with reduced spatial variation. Observations made from the vortex tracking and the mean flow momentum thickness both enable us to draw the conclusion that the forcing is able to change the vortex dynamics by encouraging the merging downstream while keeping the momentum thickness unchanged in the region dominated only by fundamental roll-up.

Since the vortex merging process is dominated by the subharmonics of the roll-up frequency, we now examine the flow over the frequency domain and focus on low-frequency components with $f/f_{n}\leqslant 1$ . The transverse velocity spectra along the centreline are shown in figure 9(d–f) for each corresponding case in (a–c). On comparing the forced cases with the baseline case, we find that the forcing mode in each forced case is amplified downstream, which suggests that both control inputs ( $f^{+}/f_{n}=0.500$ and $0.575$ ) are efficiently leveraging the shear layer instability. Moreover, the growth rates of these two forcing modes are both well predicted by the local stability analysis, as previously depicted in figure 5. The energy distribution among the spectra does not show noticeable change from the baseline case until these two forcing modes start to be amplified at $x/\unicode[STIX]{x1D706}_{n}\approx 4$ , as shown in figure 9(d–f). As discussed in § 3.3, this streamwise station is in the isolated vortex region with linear perturbation growth. This location also corresponds to where the momentum thickness starts to depart from that of the baseline with forcing, as seen in figure 10(a). The excited low-frequency instability wave induced by the periodic heating accelerates the emergence of the vortex merging process within the shear layer. This observation agrees with those from the experimental study by Ho & Huang (Reference Ho and Huang1982) and the numerical simulation by Kourta et al. (Reference Kourta, Braza, Chassaing and Haminh1987).

Next, we shift our attention to the difference between forcing effects with $f^{+}/f_{n}=0.575$ and $0.500$ . In figure 9(b–c), even though in both forced cases the vortex starts to deviate from the centreline earlier than it does in baseline flow, the deviation takes place earlier for the $f^{+}/f_{n}=0.575$ case than the $f^{+}/f_{n}=0.500$ case. Correspondingly, the merging location is also farther upstream with $f^{+}/f_{n}=0.575$ . Moreover, the second growth in momentum thickness is also taking place earlier with $f^{+}/f_{n}=0.575$ , as shown in figure 10(a). Such observations can be explained by the greater growth rate in the forcing mode that both predicted by stability analysis and exhibited in figure 10(b). The vortex merging process takes place in a more repetitive manner when forcing is introduced at the frequency with higher growth rate, leading to a lower variation in the vortex merging location, visualized by the narrower green-shaded region for $f^{+}/f_{n}=0.575$ . By observing all merging processes identified by our criterion and examining all collected flow field snapshots, we find that $17\,\%$ of vortices are not paired with any other vortices with $f^{+}/f_{n}=0.500$ , whereas only $13\,\%$ of vortices do not go through pairing processes with $f^{+}/f_{n}=0.575$ in the streamwise window of $x/\unicode[STIX]{x1D706}_{n}<14$ .

After the vortices undergo merging, on the other hand, the vortex trajectories exhibit less vertical variation in the subharmonic forced case than they do in the case with $f^{+}/f_{n}=0.575$ . These observations also agree with those in the development of momentum thickness. The $\unicode[STIX]{x1D703}/\unicode[STIX]{x1D703}_{0}$ profiles also appear to be thinner for $x/\unicode[STIX]{x1D706}_{n}>8$ for the case with $f^{+}/f_{n}=0.500$ . By examining the spatial evolution of the frequency spectra in figures 9(e–f) and 10(b), we find that, even though the amplitude decay in the roll-up mode takes place upstream compared with the baseline case for both forced cases, the $f^{+}/f_{n}=0.500$ case shows a faster decay rate than the $f^{+}/f_{n}=0.575$ case. The forcing mode for $f^{+}/f_{n}=0.500$ maintains its high level of modal amplitude while marching downstream, and suppresses other lower-frequency modes from growing, as highlighted by the cleaner spectra in the green circled region in figure 9(e) compared with those of both the baseline and $f^{+}/f_{n}=0.575$ cases. This reduction is due to the lower spillage from the modal energy at the forcing frequency. Since the growth of other low-frequency modes causes the shear layer to keep spreading downstream, the relatively flat region in momentum thickness in the subharmonic forced case can also be attributed to the absence of these lower-frequency modes, as shown in figure 10(a) for $x\geqslant 8$ .

Figure 11.

Normalized ensemble averaged vortex circulation ( $\bar{\unicode[STIX]{x1D6E4}}^{\ast }$ ) and its variation (vertical bars) for cases with $\unicode[STIX]{x1D703}_{0}/w=0.1$ . We normalized the average circulation, $\bar{\unicode[STIX]{x1D6E4}}$ , by the characteristic vorticity that is fed to each vortex in the roll-up period, $\unicode[STIX]{x1D6FA}_{z}/f_{n}$ , from two streams. Low-frequency forcing does not significantly change the mean strength but introduces greater variation in the strength of each vortex.

As pointed out in § 4, the thermal actuation can produce oscillatory vorticity flux over the trailing edge. This extra fluctuating vorticity generation is in turn fed to each formed vortex and leads to variation in its strength from one vortex to another. This is shown in figure 11, where the normalized circulation at each streamwise station is averaged over each formed vortex. We observed that, in both forced cases, the mean vortex strength does not significantly change from that of the baseline, but the variation in strength is larger. The variation in vortex strength can tip over the spatial balance from one vortex to another, resulting in its trajectory deviation from the centreline. The spread of the shear layer can be attributed to both the random trajectory deviation and the vortex merging process. As the strength variation in $f^{+}/f_{n}=0.575$ is greater than that in $f^{+}/f_{n}=0.500$ , the trajectory deviation and merging are indeed taking place earlier for $f^{+}/f_{n}=0.575$ than for $f^{+}/f_{n}=0.500$ .

Figure 12.

Transverse velocity spectra at $x/\unicode[STIX]{x1D706}_{n}=3.0$ ; the shedding is observed to lock onto the forcing frequency when $f^{+}/f_{n}\in [1.00,1.25]$ : (a) $\unicode[STIX]{x1D703}_{0}/w=0.05$ ; (b) $\unicode[STIX]{x1D703}_{0}/w=0.1$ . The spectra are shifted vertically for graphical clarity.

The forcing effects with $f^{+}/f_{n}=0.500$ and $0.575$ in the cases of initial momentum thicknesses of $\unicode[STIX]{x1D703}_{0}/w=0.05$ and $0.25$ are qualitatively similar to the case of $\unicode[STIX]{x1D703}_{0}/w=0.1$ discussed above. Additional discussions for these two $\unicode[STIX]{x1D703}_{0}/w$ cases are provided in appendix B.

5.2 Excitation of fundamental roll-up ( $f^{+}\approx f_{n}$ )

Let us discuss the forcing effects when the forcing frequency is close to $f_{n}$ with the use of $\dot{q}$ (2.7). We consider forcing frequencies of $f^{+}/f_{n}=1.0$ , $1.16$ and $1.25$ with $\unicode[STIX]{x1D703}_{0}/w=0.1$ and $\unicode[STIX]{x1D703}_{0}/w=0.05$ . All cases presented in this section are of the same forcing power of $E^{+}=0.741$ . Although we have also studied forced flow with forcing frequency up to $f^{+}/f_{n}=16$ , no significant change in the forced two-dimensional shear layer is observed for cases with $f^{+}/f_{n}\geqslant 2$ .

Figure 12 shows the transverse velocity spectra for the baseline and forced cases with $f^{+}/f_{n}=1.00$ , $1.16$ and $1.25$ at a streamwise station of $x/\unicode[STIX]{x1D706}_{n}=3$ on the centreline. As discussed in § 3, the stronger wake effect in $\unicode[STIX]{x1D703}_{0}/w=0.05$ baseline flow leads to a less synchronized shear layer roll-up and a spectrum with a broader distribution of energy across frequencies. With periodic heating at the frequencies considered here, we find that the forcing is able to lock the roll-up onto the forcing frequency, indicated by the prominent peak at the corresponding forcing frequency for each case for both $\unicode[STIX]{x1D703}_{0}/w=0.05$ and $0.1$ in figures 12(a) and 12(b) respectively.

Analogously to the discussion for low-frequency forced cases, in figures 13 and 14 we show the vortex trajectories and spatial revolution of transverse velocity spectra and momentum thicknesses for the baseline and three forced cases. The thermal forcing input serves as an excitation to the fundamental instability wave and encourages the shear layer roll-up at the forcing frequency. In the cases with $f^{+}/f_{n}=1.0$ and $1.16$ , the thermal forcing results in earlier shedding than that of the baseline. This can be seen by comparing the instantaneous vorticity contours in the initial shedding region: in both cases of $f^{+}/f_{n}=1.0$ and $1.16$ , the vorticity sheet formed from $x/\unicode[STIX]{x1D706}_{n}=0$ starts to carry a streamwise instability wave earlier than the baseline. Earlier roll-up can also be identified from the first rapid growth of $\unicode[STIX]{x1D703}/\unicode[STIX]{x1D703}_{0}$ in figure 14(a).

Figure 13.

Vortex trajectories (a–c) and modal amplitude growth (d–f) for $\unicode[STIX]{x1D703}_{0}/w=0.1$ cases. With $f^{+}$ close to $f_{n}$ , the forcing is able to excite the shear layer roll-up and accelerate the roll-up process.

Figure 14.

(a) Momentum thickness growth and (b) modal amplitude growth for cases with $\unicode[STIX]{x1D703}_{0}/w=0.1$ and $f^{+}/f_{n}=1.0$ , $1.16$ and $1.25$ .

Figure 15.

Vortex strength for cases of $\unicode[STIX]{x1D703}_{0}/w=0.1$ with $f^{+}/f_{n}=1.0$ , $1.16$ and $1.25$ . In the control cases where the roll-up frequency is locked onto the actuation frequency, the mean strength of the formed vortex is changed accordingly. With greater variation in the strength of each vortex, enhanced spreading of the shear layer is also observed.

With $f^{+}/f_{n}=1.0$ , synchronized shear layer roll-up due to forcing can also be observed from the uniform spacing between each vortex in the vortex train, as shown in figure 13(a). Moreover, the forcing is able to suppress the vortex deviating from the centreline. This leads to an extended isolated vortex region with forcing, suggested by the wider flat region in the range $1\leqslant x/\unicode[STIX]{x1D706}_{n}\leqslant 5$ in figure 14(a).

The spatial development of the frequency spectra and the modal amplitude growth are shown in figures 13(d–f) and 14(b) respectively. For the three forced cases, the amplitudes of the modes with their frequencies corresponding to the actuation frequencies start to grow immediately behind the trailing edge at $x/\unicode[STIX]{x1D706}_{n}=0$ , and their growth rates are well predicted by the local stability analysis, as shown in figure 14(b). When the forcing frequency departs farther from $f_{n}$ , the growth rate decreases, as observed for $x/\unicode[STIX]{x1D706}_{n}\leqslant 2$ . This can be attributed to the instability characteristics of the shear layer. When the forcing frequency is closer to $f_{n}$ , the frequency with the highest spatial growth rate for the perturbation, the forcing effort grows faster since the growth rate is approaching its maximum.

The frequency spectra of the three forced cases all have distinct peaks at the forcing frequency initially for $x/\unicode[STIX]{x1D706}_{n}\leqslant 3$ . As $x/\unicode[STIX]{x1D706}_{n}>3$ , lower-frequency modes start to grow and the momentum thickness increases accordingly, as shown in figure 14(a). With higher $f^{+}$ , the modal amplitude at the corresponding forcing frequency also decays spatially earlier in the range $5\leqslant x/\unicode[STIX]{x1D706}_{n}\leqslant 7$ . Earlier decay of the modal amplitude at the forcing frequency also coincides with the earlier growth of lower-frequency modes in the full spectra in figure 13(d–f). Especially for the case of $f^{+}/f_{n}=1.25$ , lower-frequency modes start to develop farther upstream compared with the baseline flow, resulting in a faster second growth of $\unicode[STIX]{x1D703}/\unicode[STIX]{x1D703}_{0}$ .

Figure 16.

(a) Momentum thickness growth and (b) modal amplitude growth with high-frequency forcing for the $\unicode[STIX]{x1D703}_{0}/w=0.05$ cases. The accelerated roll-up and amplitude growth in the forcing mode are observed to be in qualitative agreement with cases of $\unicode[STIX]{x1D703}_{0}/w=0.1$ .

In the isolated vortex region for the forced cases, thinner $\unicode[STIX]{x1D703}$ is observed when using higher $f^{+}$ . This can be explained by the weaker vortex strength considering that the total vorticity flux is introduced from the inlet boundary at the same rate. With these compact vortices convecting at the mean shear layer velocity, the higher vortex passage frequency leads to a lower amount of vorticity being possessed by the individual vortices, which can be confirmed from figure 15. For the cases with $f^{+}/f_{n}=1.16$ and $1.25$ , the mean vortex strengths are lower than those from the baseline and $f^{+}/f_{n}=1.0$ cases. With weaker vortices, the reduced mixing of momentum from two streams leads to a smaller momentum thickness in the isolated vortex region. Moreover, the strength variations in the cases of $f^{+}/f_{n}=1.16$ and $1.25$ are greater than that of the baseline. This again leads to faster shear layer spreading than the baseline, similarly to the low-frequency forced cases. With $f^{+}/f_{n}=1.0$ , on the other hand, the smaller variation in vortex strength leads to a more stable isolated vortex region, leading to a wider streamwise extent than that of the baseline, as shown in figure 14(a).

For cases with $\unicode[STIX]{x1D703}_{0}/w=0.05$ , the lock-on of the shear layer roll-up frequency and the extended isolated vortex region are still observed when forcing is invoked with $f^{+}/f_{n}=1.00$ , $1.16$ and $1.25$ , as shown in figures 12 and 16. However, we also note that for $\unicode[STIX]{x1D703}_{0}/w=0.25$ cases, no significant changes can be seen in the shear layer for $f^{+}/f_{n}=1.00$ and $1.16$ , even when using a forcing amplitude of $E^{+}=0.724$ , four times the value considered earlier in this section.

5.3 Roll-up delay (positive-mean heating)

In this section, the forcing effects in the shear layer from using $\dot{q}^{p}$ (2.8) are investigated. This form of forcing, $\dot{q}^{p}$ , is composed of a positive DC offset added to the oscillatory $\dot{q}$ . The positive DC offset introduces net positive heat transfer from the actuator and raises the mean temperature of the fluid adjacent to the actuator. With the temperature-varying viscosity model discussed in § 2.2, this locally increased temperature leads to a higher viscosity.

Figure 17.

Vortex trajectories (a–c) and modal amplitude growth (d–f) for $\unicode[STIX]{x1D703}_{0}/w=0.1$ cases. The merging location and its spatial variation are depicted by the green-shaded region (a–c) in each case.

Figure 18.

(a) Momentum thickness growth and (b) modal amplitude growth for low-frequency forcing ( $\unicode[STIX]{x1D703}_{0}/w=0.1$ ).

Figure 19.

Vortex strength for cases of $\unicode[STIX]{x1D703}_{0}/w=0.1$ . In the cases with roll-up delay, the vortex strength is observed to have greater variation and encourage spreading of the shear layer downstream. All cases are for $\unicode[STIX]{x1D703}_{0}/w=0.1$ .

We consider representative forced cases with $E^{+}=0.741$ . The effects of the locally increased viscosity are shown in figure 17(a–c). Comparing the cases with $f^{+}/f_{n}=0$ (i.e. constant heating) and $0.500$ with the baseline, the shear layer roll-up is delayed due to the locally increased viscosity. The delayed shear layer roll-up leads to an elongated recirculation region behind the trailing edge and introduces a stronger wake effect. As a consequence, the roll-up is observed to be less synchronized, as depicted in figure 17(d–f) by the less prominent peak in the roll-up mode in the spectra of the DC heating and $f^{+}/f_{n}=0.500$ cases. The growth in the roll-up mode is also observed to reach a lower maximum amplitude than it does in the baseline flow, as shown in figure 18(b). With the stronger wake effect and the less synchronized roll-up, the vortex strength exhibits greater variation, as shown in figure 19. By the same mechanism as discussed in the previous sections, the higher variation leads to early deviations of the trajectories from the centreline, as well as early merging of the vortices. Both of these effects cause the shear layer to spread faster downstream in the cases with DC heating and $f^{+}/f_{n}=0.500$ . The delayed roll-up and accelerated spreading can also be observed in the growth of $\unicode[STIX]{x1D703}/\unicode[STIX]{x1D703}_{0}$ from figure 18(a). With the subharmonic forcing added, the $f^{+}/f_{n}=0.500$ case exhibits a greater growth in $\unicode[STIX]{x1D703}$ compared with the DC heating case. In the case with $f^{+}/f_{n}=1.16$ , the oscillatory component of $\dot{q}^{p}$ dominates over the DC component and still accelerates the shear layer roll-up. The changes in the vortex trajectories (figure 17 a–c), vortex strength (figure 19), $\unicode[STIX]{x1D703}/\unicode[STIX]{x1D703}_{0}$ growth (figure 18 a) and forcing mode growth (figure 18 b) show similar traits to those observed in the case using $\dot{q}$ with $f^{+}/f_{n}=1.16$ .

Figure 20.

Effects of forcing amplitude ( $E^{+}$ ) in the DC heating and subharmonic forced cases with $\dot{q}^{p}$ . All cases are for $\unicode[STIX]{x1D703}_{0}/w=0.1$ .

In the subharmonic forced case with $\dot{q}^{p}$ , the accelerated merging built upon the delayed roll-up draws additional interest. Motivated by the observations, for cases of DC heating and $f^{+}/f_{n}=0.500$ , we further examine the effects of the forcing amplitude, $E^{+}$ , on the growth of $\unicode[STIX]{x1D703}$ , as shown in figure 20. We observe that the shear layer roll-up location, suggested by the first growth in $\unicode[STIX]{x1D703}$ , is delayed farther downstream with increased $E^{+}$ in both DC heating and subharmonic forced cases. Meanwhile, the acceleration in the spreading (second growth of $\unicode[STIX]{x1D703}$ ) is also enhanced by the subharmonic forcing with increasing $E^{+}$ .

Comparing the forcing effects of $\dot{q}$ and $\dot{q}^{p}$ , we observe that the oscillatory components from both types impose similar forcing effects on the shear layer. The positive DC component in $\dot{q}^{p}$ is able to delay the shear layer roll-up when it is not excited by the oscillatory component.