2.1 Supersonic and jet inflow conditions

All simulations in this paper solve the unsteady 3D compressible Navier–Stokes equations directly without any modelling, using an in-house DNS code. The code has been developed and applied to studies of instability, transition and turbulence in high-speed flows (Sandham et al. Reference Sandham, Schlein, Wagner, Willems and Steelant2014; Sandham Reference Sandham2016). Here we provide a brief description of the main features of the code and explain the simulation conditions for the present study. The detailed governing equations of mass, momentum and total energy conservation for a 3D flow field can be found in Touber (Reference Touber2010) and thus are not repeated here.

The air inflow parameters (shown in table 1) are set in accordance with the $Ma=2.7$ experiments of Sun et al. (Reference Sun, Zhang, Zhao, Zhao and Liang2013) with stagnation pressure $P_{0}=101\,325~\text{Pa}$ and stagnation temperature $T_{0}=300~\text{K}$ . The supersonic inflow 99 % boundary-layer thickness, which is the same for all simulations, is estimated to be $\unicode[STIX]{x1D6FF}_{i}=5.12~\text{mm}$ , with the compressible (including density variations) boundary-layer displacement thickness ${\unicode[STIX]{x1D6FF}_{i}}^{\ast }=1.75~\text{mm}$ and the momentum thickness $\unicode[STIX]{x1D703}=0.38~\text{mm}$ and corresponding Reynolds numbers $Re_{\unicode[STIX]{x1D6FF}\ast }=15\,367$ and $Re_{\unicode[STIX]{x1D703}}=3337$ , respectively. A sketch of the computational domain can be found in the previous papers (Sun & Hu Reference Sun and Hu2018a ,Reference Sun and Hu b ) and for brevity is omitted here.

Two momentum flux ratios $J=2.3$ and $J=5.5$ of air injection are modelled. Both cases lead to a sonic jet with stagnation temperature $T_{0i}=300~\text{K}$ . The jet properties are shown in table 2. Since the turbulent levels originating from the jet orifice in the experiments are not known, a uniform profile across the jet orifice is implemented without any disturbance. It might be argued that turbulence would not be thoroughly resolved in the jet; however, it is believed that the overall behaviour of coherent structures in the jet and boundary layer will not be affected by this assumption (Sun & Hu Reference Sun and Hu2018a ,Reference Sun and Hu b ).

Table 1.

Air flow conditions for the simulations, including the Mach number $(Ma)$ , stagnation temperature ( $T_{0}$ ), stagnation pressure ( $P_{0}$ ), dimensional boundary-layer 99 % thickness ( $\unicode[STIX]{x1D6FF}_{i}$ ), momentum thickness ( $\unicode[STIX]{x1D703}$ ) and Reynolds number ( $Re_{\unicode[STIX]{x1D703}}$ ) at the inflow.

Table 2.

Jet conditions for the simulations, including jet-to-cross-flow momentum ratio $(J)$ , orifice diameter $(D)$ , Mach number ( $Ma_{j}$ ), stagnation temperature ( $T_{0j}$ ) and stagnation pressure ( $P_{0j}$ ).

2.2 Computational domain and boundary conditions

The use of a digital filter to generate the inflow boundary condition significantly reduces the length required for boundary-layer development (Xie & Castro Reference Xie and Castro2008), but nevertheless there should be enough distance provided in the simulation to allow the boundary layers to adjust to an equilibrium state upstream of the jet. Wang et al. (Reference Wang, Sandham, Hu and Liu2015) found that a distance of 12 times the inflow boundary-layer thickness is enough to obtain realistic turbulent mean and r.m.s. profiles from the digital filter inflow generator used. Accordingly, the initial length is set to be $L_{i}=12.7\unicode[STIX]{x1D6FF}_{i}\approx 65~\text{mm}$ for a fully developed turbulence generation. The jet mixing length is set to $L_{m}=90~\text{mm}$ including the near field (denoted as JISC section 1) with 10 mm upstream and 40 mm downstream of the jet and the far field (denoted as JISC section 2) with 40 mm downstream of JISC section 1. Therefore, the total length of the physical domain is $L_{x}=155~\text{mm}$ . The wall-normal extent of the domain is 18 mm, and a span of 14 mm is used. The jet is centred 75 mm downstream of the inlet with an orifice diameter of $D=2~\text{mm}$ . The grid used in the current simulation is kept the same as in the set-up in our previous work (Sun & Hu Reference Sun and Hu2018a ,Reference Sun and Hu b ). As analysed, the grid set-up ensures a fine DNS resolution in the near-wall region and a quasi-DNS (QDNS) resolution in the jet and the mainstream due to the unknown information about the smallest scales in the jet plume, as denoted by Sandham, Johnstone & Jacobs (Reference Sandham, Johnstone and Jacobs2017).

Large buffer layers with non-reflective outside boundaries are introduced with lengths of 15 mm, 11 mm and 12 mm at the outlet, on both the sides and the upper boundaries, respectively. The grid is smoothly stretched in the buffer layers towards the boundaries. The buffer layers, with coarsened mesh, help to damp fluctuations and minimize any possible reflections from the boundaries. Non-slip and adiabatic boundary conditions are enforced on the wall. An integrated characteristic boundary condition (Thompson Reference Thompson1987) is applied at the inflow, and an outflow condition with an integrated characteristic scheme is applied to the spanwise, top and outflow boundaries, which are located on the outer edge of the buffer layers. The jet orifice is defined directly in the surface wall uniform grid and set to a sonic condition with $J=2.3$ and $J=5.5$ , respectively.

2.3 Numerical methods and validation

This in-house code (Sandham, Li & Yee Reference Sandham, Li and Yee2002) uses a fourth-order central difference scheme for the spatial derivatives and a third-order explicit Runge–Kutta scheme for time advancement. The code makes use of an entropy splitting of the Euler terms and the Laplacian formulation of the viscous terms to improve the stability of the non-dissipative central scheme. To better resolve the steep variations in flow field around the injected jet, we developed a hybrid scheme which combines the entropy splitting central scheme with a fourth-order central-upwind weighted essentially non-oscillatory (WENO-CU4) scheme and is applied in the areas with high gradients (Sun & Hu Reference Sun and Hu2018b ).

The digital filter inflow generation method of Xie & Castro (Reference Xie and Castro2008) is used to provide inflow boundary conditions to generate a more realistic turbulent boundary layer, from predefined mean and r.m.s. profiles. The mean inflow profiles are generated from a similarity solution of the compressible boundary layer using the same approach as in Wang et al. (Reference Wang, Sandham, Hu and Liu2015). The inflow r.m.s. values are taken from the DNS results of Schlatter & Orlu (Reference Schlatter and Orlu2010) for a similar Reynolds number.

Turbulent boundary-layer mean profiles and distributions of r.m.s. values at $x/D=-7.5$ (corresponding to the position at 60 mm downstream of the inlet) have been evaluated, giving a good agreement with the existing data (Sun & Hu Reference Sun and Hu2018a ,Reference Sun and Hu b ). The approximate penetration boundary based on a concentration identification was obtained and compared with the empirical correlation. It was found that the current penetration result agrees well with previous experimental correlations for both $J=5.5$ and $J=2.3$ (Sun & Hu Reference Sun and Hu2018a ). The jet trajectory of $J=5.5$ and $J=2.3$ was extracted from the streamline originating from the centre of the jet exit on the time-averaged symmetry plane. It was compared with Schetz & Billig’s (Reference Schetz and Billig1966) prediction and showed reasonably good agreement (Sun & Hu Reference Sun and Hu2018a ). The above comparison validated the approach and the current simulation.

3.1 Instantaneous flow structures

Contours of typical instantaneous density fields for the supersonic boundary layer without and with the jet injection are compared in figure 2. Figure 2(a) reveals detailed structures of the boundary layer and unsteady jet penetration. The bow shock, barrel shock and separation bubbles introduced by the jet are well identified in the simulations. The upstream separation shock is not as strong as the other shocks. Upstream of the jet, a separation zone is observed and marked by the isolines of $u/U_{\infty }=0.0$ . The vortical structures, displayed by local density variations, around the jet break down to form the well-developed small turbulent eddies downstream. Analysis of the jet structure can be found in the previous literature, and therefore is not detailed here. The present study focuses on the boundary-layer developments downstream of the jet. It is found that the jet interacts with the upcoming boundary layer intensively and the interaction is still strong in the downstream region in the centreline plane $(z/D=0)$ . Figure 2(b) gives the flow field on the plane $z/D=3.0$ for all cases. From this side view, the vortices in the boundary layer of both $J=5.5$ and $J=2.3$ cases are observed to have been suppressed, especially for $J=5.5$ . The vortical structures are attenuated, and start to reorganize (denoted as recovery in this paper) after a long distance in the region downstream of the jet. For $J=5.5$ , this recovery distance is longer, and part of the boundary layer even approaches the laminar state, which means that higher $J$ leads to a more intensive vortex decay and even a relaminarization downstream of the jet.

Figure 2.

Density contours at $z/D=0.0$ (a) and $z/D=3.0$ (b) planes of the jet interacting with the incoming boundary layer. The red solid lines are the isolines of $u/U_{\infty }=0.0$ .

Figure 3.

Density contours on $y/D=0.08$ (a) and $y/D=1.0$ (b) planes for different cases ( $J=0$ , 2.3, 5.5 from top to bottom). The red solid lines are the isolines of $u/U_{\infty }=0.0$ .

To clearly identify the development of the turbulence in the boundary layer downstream of the jet, the near-wall streaks in the turbulent flow field are shown and analysed. The density contours at the same wall-normal distances ( $y/D=0.08$ and $y/D=1.0$ ) for $J=0$ , 2.3 and 5.5 are shown in figure 3 from a top view. Alternate low-magnitude (dark colour) and high-magnitude (light colour) density regions are observed in figure 3(a), which are caused by the near-wall streaks in the boundary layer. The flow field of the jet interaction cases demonstrates different flow characteristics compared to the flat plate. We can see that the classic streaks are developed along the streamwise direction in the undisturbed boundary layer of the flat plate upstream of the jet orifices. For $J=2.3$ and $J=5.5$ , the quasi-streamwise streaky structures crush into the separation region upstream of the jet. On the lateral of the jet, the streaks are interrupted by the bow shock induced by the jet. For both $J=2.3$ and $J=5.5$ cases, downstream of the jet lateral separation region, the vortical structures displayed by local density variations in the boundary layer are greatly weakened and almost disappear, accompanied by increased density (shown in grey-white colour). In the far field of $x/D>10.0$ , it can be seen from figure 3(a) that the boundary-layer turbulence in the lateral downstream of the jet at $y/D=0.08$ of $J=2.3$ has a quick recovery and small streaks intermittently appear in the wall-bounded flow. For $J=5.5$ , it is observed at $y/D=0.08$ that the turbulence decreases more noticeably than for $J=2.3$ .

Figure 3(b) shows density contours on the $y/D=1.0$ plane for all cases from a top view. Downstream of the jet lee, the high-speed flow impacts the shock waves generated by the collision of the flow running around the jet barrel. The collision shock waves are clearly shown in density contours on $y/D=1.0$ slices and they are twisted by the unsteady turbulent structures in the jet lee. The boundary layer interacts with the shock generated by the collision of the flow around the jet barrel. It is clear that these collision shocks induce the herringbone separation shown in figure 3(a). It is seen that the herringbone separation region is confined laterally and also in the wall-normal direction since its size decreases significantly at $y/D=1.0$ (figure 3 a) compared to at $y/D=0.08$ (figure 3 a). Flow over the separation reattaches downstream of the cross point of the separation branches. From figure 3(b), it is seen that the herringbone separation zone exists in a limited region and the high-speed flow in the lateral ( $z/D>3.0$ or $z/D<-3.0$ ) of the jet seems not to be affected by the herringbone zone. The region is affected slightly by the collision shock waves. The flow behind the collision shock falls into the reattachment zone and after a certain distance the streaky structures reorganize near the symmetry plane, as shown in figure 3(a). A comparison of figure 3(a) and figure 3(b), especially for $J=2.3$ , shows that at $y/D=0.08$ the streaks are regenerated earlier than $y/D=1.0$ , which suggests that the turbulence in the inner layer of the lateral boundary layer recovers more quickly than in the outer layer. A comparison between $J=5.5$ and $J=2.3$ shows that higher $J$ leads to a more intensified suppression phenomenon on the wall-bounded flow. For $J=5.5$ the intensified suppression causes the lateral flow to stay in a weakened turbulence state that has not been fully recovered even near the exit, which is $40D$ downstream of the jet. For both $J=2.3$ and $J=5.5$ cases, the central region downstream of the collision shock-affected region remains in a turbulence state with augmented intensity characterized by the twisted and corrugated streaks. In the jet lateral the streaks are intermittent and smooth, especially for the $J=5.5$ case, which is the character of low-intensity turbulence. This phenomenon demonstrates that the collision flow in the jet lee central zone leads to a local turbulence with larger intensity in the boundary layer than the jet lateral. A preliminary experimental observation on the slices of the flow field in appendix B confirms the jet lateral suppression phenomenon and the central augmentation phenomenon observed in figure 3.

Figure 4.

Isosurface of the streamwise velocity $(u/U_{\infty }=0.4)$ fields of the jet interaction with the incoming boundary layer: (a) $J=0.0$ , (b) $J=2.3$ , (c) $J=5.5$ .

Figure 5.

Isosurface of the streamwise velocity $(u/U_{\infty }=0.55)$ fields of the jet interaction with the incoming boundary layer: (a) $J=0.0$ , (b) $J=2.3$ , (c) $J=5.5$ .

Isosurfaces of the streamwise velocity in the 3D flow field coloured by the wall-normal distance are illustrated and compared in figures 4 and 5. For the flat plate $(J=0.0)$ , we can clearly see the classic elongated streamwise vortices in the isosurface of $u/U_{\infty }=0.4$ , which maintains similar characteristics within the whole domain. A mean velocity of $u/U_{\infty }=0.4$ usually corresponds to an inner layer location in the boundary layer, as was shown in Sun, Hu & Sandham (Reference Sun, Hu and Sandham2017). The streaky structures concentrate in the inner part of the flat-plate boundary layer and have no large liftup. For the $J=2.3$ and $J=5.5$ cases, the structures are similar to the flat-plate case upstream of the jet. Further downstream the near-wall streaks develop into a completely new pattern which exhibits a suppression of the streaky structures. The phenomenon is especially prominent for the $J=5.5$ case. The isosurfaces of $u/U_{\infty }=0.55$ given in figure 5 show the large-scale structures in the outer part of the boundary layer. Structures of the $J=0.0$ case (figure 5 a) show that many packets grow from the near-wall region to a higher wall-normal location. These structures occur with local mass and energy exchange between the inner layer and the outer layer. In figure 5(c) for the $J=5.5$ case, we can hardly see any packet structures in the lateral downstream of the separation bubble ahead of the jet orifice, which reflects a suppression of turbulence when the flow goes across the jet windward separation region. Usually when the packet structures occur less, the exchange between the inner layer and the outer layer would suffer (Sun et al. Reference Sun, Hu and Sandham2017). Compared with the $J=0.0$ case, jet interaction modifies the boundary layer and weakens the exchange process in the inner layer with the outer layer. Downstream of the jet leeward separation at $x/D>10.0$ , coherent structures in the near-wall region are regenerated during the recovery process and packet structures in the outer part of the boundary layer seem to grow with the recovery.

From the results shown in figures 4 and 5, we can find that the turbulence in the far field is affected by the history effect of the turbulence decay in the vicinity of the jet. Higher $J$ leads to a more intensive interaction and a more apparent turbulence decay, which needs a longer distance to recover. In the jet leeward region near the centreline, turbulence is not suppressed since the jet interaction with the main stream enhances the local turbulence intensity, especially when the collision shock has an intensified interaction with the boundary layer.

3.2 Mean flow properties and quantitative analysis on turbulent kinetic energy and Reynolds shear stress

This section further analyses the turbulence formation mechanism in the boundary layer around the jet with turbulent statistics. Since averaging over 100 characteristic times is still not enough to obtain smooth profiles of high-order statistics, data are further averaged over the region $z/D\in [3.0,3.5]$ . It is assumed that the boundary layer is approximately homogeneous in this spanwise zone and is not affected by the leeward separation zone; therefore averaging in the spanwise direction across $z/D\in [3.0,3.5]$ away from the jet is considered to be reasonable.

Profiles normal to the wall are extracted at several locations upstream and downstream of the jet to compare with the results at the same locations for the flat-plate case, as is illustrated in figure 6 for the streamwise velocity. In the vicinity downstream of the jet, where strong interaction is expected underneath the jet with the boundary layer, the velocity downstream of the jet is larger over most of the boundary layer than in the flat-plate case, and the downstream velocity profiles become fuller as $J$ increases. It is interesting that at $x/D=5$ the inner-layer velocity ( $y/D<0.15$ , shown in figure 7 a) of $J=5.5$ is larger than $J=2.3$ and $J=0.0$ , while at $x/D=35$ the inner-layer velocity ( $y/D<0.15$ shown in figure 7 b) is virtually the same for the three cases. This implies that in the far field downstream of the jet, the inner-layer velocity ( $y/D<0.15$ ) for both $J=2.3$ and $J=5.5$ almost recovers to the shape of the fully developed turbulent boundary layer of the $J=0.0$ case (flat plate), while the outer layer is not fully recovered.

Figure 8 gives the development of the boundary-layer 99 % thickness $\unicode[STIX]{x1D6FF}_{99}$ , the 90 % thickness $\unicode[STIX]{x1D6FF}_{90}$ and the displacement thickness $\unicode[STIX]{x1D6FF}^{\ast }$ along the streamwise direction for all cases. There $U_{e}$ is taken as the free-stream velocity at $y/D=3.0$ . This is not an accurate estimation but it provides a useful comparison to understand the evolution of the boundary-layer thickness. It is not surprising to see that the boundary-layer thickness of the jet cases has an obvious drop downstream of the jet. Since there is a strong interaction region at $4.0<x/D<20.0$ , an inaccurate prediction for the boundary-layer thickness $\unicode[STIX]{x1D6FF}_{99}$ is shown in figure 8. After $x/D=20.0$ , it is seen that the boundary-layer thickness of the jet cases is smaller than the flat-plate $(J=0.0)$ case, which reflects the decrease of the boundary-layer thickness in the jet lateral downstream of the windward separation. The boundary-layer thickness even drops below $0.8D$ after the jet for the $J=5.5$ case. It is found that $\unicode[STIX]{x1D6FF}_{90}$ is not sensitive to the jet interaction and could reflect the decrease of the boundary layer in the jet lateral side. The displacement thickness, $\unicode[STIX]{x1D6FF}^{\ast }$ can also be used to characterize the growth of the boundary layer,

(3.1) $$\begin{eqnarray}\unicode[STIX]{x1D6FF}^{\ast }=\int _{0}^{\infty }\left(1-\frac{\unicode[STIX]{x1D70C}}{\unicode[STIX]{x1D70C}_{e}}\frac{u}{U_{e}}\right)\text{d}y.\end{eqnarray}$$

Here the displacement thickness of different cases is compared in figure 8(c), and it is seen that $\unicode[STIX]{x1D6FF}^{\ast }$ of the $J=5.5$ case is the smallest downstream of the jet orifice. This phenomenon corresponds to the experimental nanoparticle laser scattering (NPLS) images given in appendix B where the boundary-layer thickness decreases in the jet lateral.

Figure 6.

Mean velocity profiles at different streamwise locations based on averaging over $z/D\in [3.0,3.5]$ , normalized by inflow free-stream velocity: (a)  $x/D=-10.0$ , (b)  $x/D=5.0$ , (c)  $x/D=20.0$ , (d)  $x/D=35.0$ .

Figure 7.

Mean velocity profiles at different streamwise locations in logarithmic coordinates, based on averaging over $z/D\in [3.0,3.5]$ , normalized by the inflow free-stream velocity.

Figure 8.

Boundary-layer thicknesses $\unicode[STIX]{x1D6FF}_{99}$ , $\unicode[STIX]{x1D6FF}_{90}$ and $\unicode[STIX]{x1D6FF}^{\ast }$ along the streamwise direction based on averaging over $z/D\in [3.0,3.5]$ , where $U_{e}$ is taken as the free-stream velocity at $y/D=3.0$ .

Time-averaged skin friction along the wall in the domain $z/D\in [3.0,3.5]$ is shown in figure 9. Usually the skin friction considering compressibility effects can be calculated, following Young’s (Reference Young1989) assumption, as $c_{f}/c_{fi}\approx (1+0.1165M_{\infty }^{2})^{-0.622}$ for an adiabatic wall. Here $c_{fi}$ denotes the skin friction at the same Reynolds number in the incompressible boundary layer by assuming a $1/7$ th power law. If we calculate $Re_{x}$ from $\unicode[STIX]{x1D6FF}^{vd}=0.37xRe_{x}^{-0.2}$ (the superscript $vd$ means the value under the van Driest transform (van Driest Reference van Driest1956)) and $c_{fi}$ is obtained as $c_{ft}=0.0592Re_{x}^{-0.2}$ , we get $c_{f}\approx 0.00209$ at $x/D=-7.5$ for the flat plate. The prediction is comparable to the numerical calculation $c_{f}\approx 0.00188$ .

The skin friction for different cases follows the theoretical result for a flat-plate turbulent boundary layer up to $x/D=-10.0$ , which indicates that the artificial inflow condition develops to a fully turbulent boundary layer before the jet. The skin friction coefficient decreases suddenly over $-10.0<x/D<-2.0$ in the separation region ahead of the jet. In the region $-2.0<x/D<-0$ , the skin friction coefficient recovers across the jet lateral separation. The wall pressure (shown in figure 9 b) rises in this region due to the bow shock effect. Downstream of the location $x/D=-1.0$ for $J=2.3$ and $x/D=-2.0$ for $J=5.5$ , wall pressure falls corresponding to an expansion in the supersonic flow. It is worth mentioning that the wall pressure starts to decease ahead of the jet, which means that the expansion starts upstream of the jet, not only in the jet lee, as seen in figure 9(b). For $x/D>7.5$ , the wall-pressure gradient is positive, which corresponds to the recovery process. In the region $3.0<x/D<8.0$ for $J=2.3$ and $3.0<x/D<10.0$ for $J=5.5$ , the skin friction coefficient decreases and reaches a local minimum. Downstream of this region, the skin friction quickly recovers towards an equilibrium state for $J=2.3$ , while the recovery for $J=5.5$ is much slower. At the location $x/D=40.0$ near the physical domain exit, the skin friction coefficient for both jet cases approaches the flat-plate value, which indicates that the inner-layer flow almost recovers to the fully developed turbulent boundary layer within the domain. These results are consistent with the analysis for streaks shown in figures 3–5 and velocity profiles in figures 6 and 7. It is identified that the expansion process exists across the jet separation bubble. As is well known, a sudden expansion suppresses the turbulence in the supersonic flows (Sun et al. Reference Sun, Hu and Sandham2017), which could be the reason that the streaks are weakened and the skin friction coefficient falls.

Figure 9.

Skin friction coefficient (a) and wall pressure (b) distributions along the wall averaged for the zone $z/D\in [3.0,3.5]$ .

Figure 10.

Contours of the Mach number (a,c) and static pressure (b,d) on different wall-parallel planes for the $J=5.5$ case. Streamlines are superimposed on the Mach number contours. The white solid line superimposed on the contour plots represents $u/U_{\infty }=0.0$ .

Further analysis of the mean flow is presented in figures 10 and 11, which show the contours of static pressure and Mach number at various locations. It is seen that the pressure downstream of the separation region in front of the jet decreases until the flow runs to the collision shock waves. Streamlines on the Mach number contours shown in figure 10 reflect a divergence in the lateral around the jet which corresponds to an expansion. A low-pressure zone exists in the jet lateral due to the blockage and expansion effects from the jet. The flow in the jet lateral is supersonic and the collision shock is induced. Cross-flow runs around the jet and impinges together in the jet lee, which leads to a local pressure increment due to the collision effects. Studies of the supersonic boundary-layer flow over an expansion corner (Arnette, Samimy & Elliott Reference Arnette, Samimy and Elliott1998) showed that sudden expansion would lead to local turbulence decay. The separation region behind the collision shocks in the jet lee on the $y/D=0.08$ plane has a large size, but almost disappears on $y/D=0.5$ , which means that the separation is restricted to $y/D\leqslant 0.5$ . The recirculating flow within the separation zone on the $y/D=0.08$ slice reattaches in a V-shaped reattachment region. The supersonic flow at the $y/D=0.5$ slice around the jet impinges into a concentrated flow behind the collision shocks. The reattached supersonic flow and the impinging flow near the wall lead to a pressure increment in the downstream region of the leeward separation bubble, as shown in figure 10.

The TKE contours on different wall-parallel planes for $J=5.5$ are plotted in figure 11, which shows the whole flow field including the jet wake and the far field. It is seen that the TKE in the separation region ahead of the jet is increased compared to the TKE upstream on both $y/D=0.08$ and $y/D=0.5$ planes. Downstream of the windward separation bubble, the TKE reduces significantly, which corresponds to the process of lateral supersonic flow running around the jet orifice. Downstream of the windward separation bubble, the TKE reduces significantly, which corresponds to the expansion process of the lateral supersonic flow running around the jet orifice. In the far field of the jet lateral, the TKE shows a slow recovery. The herringbone separation zone behind the collision shocks has a lower TKE level due to the recirculating flow. Outside of the herringbone zone, the TKE increases noticeably due to the shock interaction with the boundary layer and the reattachment of the jet leeward flow, which is the typical feature of boundary layer–shock wave interactions (Andreopoulos, Agui & Briassulis Reference Andreopoulos, Agui and Briassulis2000; Touber & Sandham Reference Touber and Sandham2011). Above the herringbone zone $(y/D>0.5)$ , the TKE is intensified due to the interaction of the collision shocks and the flow impingement. In the region downstream in the jet far field, the TKE is higher in the centreline than that on the lateral along the span.

Figure 11.

TKE contours on different wall-parallel planes of the $J=5.5$ case. The white solid line superimposed on the contour plots represents $u/U_{\infty }=0.0$ .

Representative TKE profiles averaged over $z/D\in [3.0,3.5]$ along the streamwise direction are given in figures 12 and 13. Results for the flat-plate case are also plotted for comparison. At $x/D=5.0$ , the TKE is much higher for both jet cases due to the intensive interaction of the jet with the boundary-layer flow. As shown in figure 11, the TKE at $x/D=5.0$ is smaller than that within the zone of $-2.0<x/D<3.0$ since the flow over the windward separation region has a higher TKE, which indicates that TKE reduction occurs after the windward separation. It is also seen that the jet lateral supersonic streamlines diverge in figure 10 at $x/D=5.0$ , which corresponds to the expansion process and leads to the turbulence decay. At $x/D=20.0$ , the TKE reduction continues and a ravine with low TKE magnitude forms between the inner layer and the outer layer. This phenomenon is more obvious for the $J=5.5$ case (see the blue arrow points in figure 12 d) at $x/D=35.0$ although the boundary layer is recovering. In the inner layer, the TKE is increased for both jet cases compared to the flat-plate case, which indicates that the near-wall turbulence remains at a higher level after the flow goes across the jet windward separation region. The ravine in the TKE profile is very clear compared to the flat-plate case, which reveals a significant TKE decay near the inner layer. In the outer layer, the interaction between the jet and the boundary-layer flow exists and the local TKE magnitude deviates significantly from the flat-plate case. As the boundary layer grows, the TKE ravine extends to the outer layer due to the history effect of the turbulence suppression near the jet orifice. Recalling the analysis for figures 2–5, the disappearance of the streaky structure reflects the local turbulence laminarization, which originates from the significant turbulence decay near the jet orifice. Figure 13 shows the TKE averaged over the zone $z/D\in [-0.5,0.5]$ in the boundary layer at $x/D=35.0$ . It is seen that the TKE is kept at a high level due to the interaction of the collision shock and the impingement of the lateral supersonic flow running around the jet, as analysed for figures 10 and 11. Corresponding to figures 2–5, along the streamwise domain within $z/D\in [-0.5,0.5]$ , an abundance of turbulence is generated and the turbulent vortical structures create the high-level TKE downstream of the jet.

Figure 12.

TKE profile predictions at different locations along the belt $z/D\in [3.0,3.5]$ , normalized by the square of the inflow free-stream velocity: (a)  $x/D=-10.0$ , (b)  $x/D=5.0$ , (c)  $x/D=20.0$ , (d)  $x/D=35.0$ .

Figure 13.

TKE profiles at $x/D=35.0$ averaged over $z/D\in [-0.5,0.5]$ , normalized by the square of the inflow free-stream velocity.

Figure 14 shows the evolution of r.m.s. velocity components at different streamwise locations for the $J=5.5$ case compared with the profiles on the flat plate. At $x/D=5.0$ , velocity fluctuations of $\langle u^{\prime }v^{\prime }\rangle$ are suppressed in the inner layer of $y^{+}<20$ . In the outer layer, turbulence is enhanced significantly due to the interaction of the jet with the boundary-layer flow. Comparing the r.m.s. velocity components at $x/D=20.0$ and $x/D=35.0$ , it is clearly seen that the turbulence in the inner layer recovers to the same level as for the flat-plate flow; however, it is still not recovered thoroughly in other parts of the boundary layer due to the history effect of the turbulence decay in the vicinity of the jet. It is obvious that a ravine (indicated by the arrow in figure 14 d) exists in the profile of the shear stress even at $x/D=35.0$ compared to the flat-plate case. Since the shear stress is mainly associated with large-scale structures, it is consistent with figures 4 and 5, which shows that the large structures are less coherent in the lateral downstream of the jet. The reduction of the turbulence level between the inner and outer layers highlights a weakening of the large-scale structures in the jet lateral flow. Owing to the turbulence suppression in this region, the exchange of turbulence energy between the inner layer and the outer layer is reduced, which results in a turbulent energy accumulation in the inner layer since the turbulence in the inner layer stays in local equilibrium, as seen in the profiles of figure 12. The exchange intensity of turbulence energy between the inner layer and the outer layer is reduced. The r.m.s. velocity components at $x/D=35.0$ averaged in $z/D\in [-0.5,0.5]$ are kept at a high level and no indication of decay exists in the profiles of the $J=5.5$ case compared to the flat-plate case, as shown in figure 15, which reveals that the turbulence stress retains a strong intensity along the streamwise domain within $z/D\in [-0.5,0.5]$ downstream of the jet.

Figure 14.

R.m.s. velocity profiles at different locations averaged over $z/D\in [3.0,3.5]$ , plotted in wall unit coordinates, normalized by the inflow free-stream velocity: (a)  $x/D=-10.0$ , (b)  $x/D=5.0$ , (c)  $x/D=20.0$ , (d)  $x/D=35.0$ . Solid lines represent the flat-plate case, and dash-dotted lines represent the $J=5.5$ case. R.m.s. velocities of the streamwise (▵), wall-normal (♦) and spanwise (▸) components, together with the Reynolds shear stress (●), are shown.

Figure 15.

R.m.s. velocity profile predictions at $x/D=35.0$ in the belt $z/D\in [-0.5,0.5]$ , normalized by the inflow free-stream velocity. Solid lines represent the $J=5.5$ case and dash-dotted lines represent the flat-plate case for r.m.s. velocities of the streamwise (▵), wall-normal (♦) and spanwise (▸) components, together with the Reynolds shear stress (●).

The evolutions of the TKE and the shear stress $\langle u^{\prime }v^{\prime }\rangle$ along $y/D=0.08$ and $y/D=0.6$ in the domain $z/D\in [3.0,3.5]$ are shown in figures 16 and 17, respectively, where a similar trend of the different evolution processes for jet cases can be seen. The amplification of the TKE and $\langle u^{\prime }v^{\prime }\rangle$ near the jet orifice is due to the interaction of the jet bow shock with the boundary layer (Andreopoulos et al. Reference Andreopoulos, Agui and Briassulis2000), which results in the windward separation region in front of the jet. It is interesting to see that the absolute magnitudes of both the TKE and $\langle u^{\prime }v^{\prime }\rangle$ start to decease ahead of the jet, which corresponds to the analysis of the pressure distribution in figure 9(b). From figures 16 and 17, we can see that the evolution of the TKE and $\langle u^{\prime }v^{\prime }\rangle$ depends on the momentum flux ratio $J$ . As $J$ increases, the peak values of TKE and $\langle u^{\prime }v^{\prime }\rangle$ decrease in the lateral downstream of the jet. The $\langle u^{\prime }v^{\prime }\rangle$ value is reduced to almost zero along $y/D=0.6$ for $J=5.5$ , indicating that turbulence is greatly suppressed and even laminarization happens. The magnitude of the TKE and $\langle u^{\prime }v^{\prime }\rangle$ decreases to local minima soon after the separation at $y/D=0.08$ , then gradually redevelops outwards. Along $y/D=0.08$ downstream of the jet orifice, the peak of the TKE and $\langle u^{\prime }v^{\prime }\rangle$ first occurs around $x/D=18.0$ for $J=2.3$ , where the inner layer is basically recovered; while at $y/D=0.6$ for $J=2.3$ , the peak of the TKE and $\langle u^{\prime }v^{\prime }\rangle$ first occurs around $x/D=25.0$ , suggesting that the outer boundary layer takes a longer distance to recover than the inner layer. Compared to $J=2.3$ , the peak of the TKE and $\langle u^{\prime }v^{\prime }\rangle$ of $J=5.5$ appears further downstream, which means that wall turbulence regeneration in the near-wall region is slower for higher $J$ . For $J=2.3$ , the process of regeneration of wall turbulence takes place within a shorter distance than $J=5.5$ , which again can be attributed to the interaction induced by the jet plume with the boundary layer. Compared to $J=2.3$ , the peak of the TKE and $\langle u^{\prime }v^{\prime }\rangle$ of $J=5.5$ appears further downstream, which means that wall turbulence regeneration in the near-wall region is slower for higher  $J$ .

Figure 16.

TKE along the streamwise direction at (a)  $y/D=0.08$ and (b)  $y/D=0.6$ in the domain $z/D\in [3.0,3.5]$ for different cases, normalized by the square of the inflow free-stream velocity.

Figure 17.

The velocity $\langle u^{\prime }v^{\prime }\rangle$ along the streamwise direction at (a)  $y/D=0.08$ and (b)  $y/D=0.6$ in the domain $z/D\in [3.0,3.5]$ of different cases, normalized by the square of the inflow free-stream velocity.