4.2.1. Instantaneous flow

The figure 9 shows iso-contours of streamwise-velocity fluctuations normalised by the local external velocity $U_e(x,t)$ in a plane parallel to the wall located at $y^+_{0} = 5$, in an area close to $70\,\%$ chord.

Figure 9.

Streamwise fluctuating velocity iso-surfaces normalised by local external velocity $U_e(x,t)$. Here, $x^* = 0$ corresponds to $70\,\%$ of the chord and $t^*$ is defined as $t^* = {t}/({\delta _{(x^*,t)=0}/U_{b,1}})$.

The situation in figure 9$(a)$ corresponds to the initial flow for which many turbulent structures are visible, showing that the boundary layer is fully turbulent. In figure 9$(b)$, the shock is in the analysis region. Immediately behind the shock, the turbulent structures have almost extinguished. Compared with the accelerated velocity of the main flow, the fluctuations are now negligible. Only a few isolated disturbances persist: the flow seems quasi-laminar. After a delay, streamwise elongated streaks appear in figure 9$(c)$. Between figure 9$(b)$ and figure 9$(c)$, pre-existing turbulent structures are modulated to give rise to streaks. The time needed to observe streamwise streaks after the shock propagation can be interpreted as a receptivity process of the boundary layer to pre-existing turbulence. These streaks degenerate to turbulent spots.

The phase between the shock propagation and the first turbulent spots can be named pre-transition in relation with the work of He & Seddighi (Reference He and Seddighi2013) for an incompressible flow. Figure 9(d) reveals the coexistence of turbulent spots with streamwise streaks on the same abscissa: this is the transition stage. Turbulent spots become increasingly prominent during the transition stage, as illustrated on view $(e)$. This stage ends when the turbulence has totally contaminated the boundary layer: it is then fully turbulent with the properties of the new turbulence, which emerged in figure 9(d). The transition from the old turbulence to the new is therefore not monotonous but follows a temporal transition that has similarities with the spatial bypass boundary layer process. The temporal transition is completed in three stages: pre-transition, transition and fully turbulent boundary layer. These steps exhibit large similarities with the three phases of a bypass spatial process, namely: buffeted laminar boundary layer, transition and fully turbulent boundary layer. Moreover, these steps are very similar to those described by He & Seddighi (Reference He and Seddighi2013) observed in incompressible flow. Compressibility does not seem to significantly affect the turbulence dynamic provoked by an acceleration. The analysis of the mean and fluctuating motion which is now proposed helps to confirm this, as well as to reinforce the similarities with the bypass process.

4.2.2. Mean motion

Figure 10 shows the temporal evolution of the friction coefficient at $70\,\%$ of the chord (a delay effect due to flow advection is observed when the friction coefficient is extracted at a further abscissa; see, for instance, figure 8(c) to observe this delay). As observed in figure 7, a peak of friction follows the shock, which is here located at $(t^*=0^+)$. This peak is provoked by the sudden acceleration of the entire flow, which also generates a thin layer of high strain close to the wall, as we show in the following while analysing the velocity profile. This thin wall shear layer then diffuses into the boundary layer, causing a drop in the friction coefficient. This phase is the pre-transition step $(t^*<7.0)$. When the first turbulent spots appear, the friction coefficient begins to increase: this is the beginning of the transition phase $(t^*=7.0)$. When the transition ends, the friction is constant at the value corresponding to the $Re_{\theta }$ of the final regime $(t^*=16.0)$. The temporal evolution of the friction coefficient reveals strong similarities with the spatial development of the friction coefficient of a boundary layer undergoing a bypass transition.

Figure 10.

Temporal evolution of the friction coefficient at $70\,\%$ chord.

To go further in comparison with a spatial case, He & Seddighi (Reference He and Seddighi2013) proposed to investigate the temporal evolution of the boundary layer resulting only from the acceleration (i.e. shock propagation). Consequently, the temporal development of the boundary layer is investigated by examining the differential boundary layer,

(4.1) \begin{equation} \bar{u}\hat ( y/\delta, t^*) = \frac{\bar{u}(y/\delta, t^*) - \bar{u}(y/\delta, 0) }{ U_{b,1} - U_{b,0}}, \end{equation}

where $U_{b,0}$ ($U_{b,1}$) represents the external flow velocity for the initial (final) flow.

The graph on the left-hand side of figure 11 represents the differential velocity profile in the boundary layer as a function of $y/\delta (t)$. Immediately following the wave passage, $\bar {u}$ˆ is close to $1$ in almost the entire boundary layer except very close to the wall. This means that the acceleration imposed by the shock wave is constant throughout the boundary layer thickness which reacts as a whole, except the very near-wall region. The velocity profile recorded at $t^*=0.4$ shows how intense the shear layer is. The diffusion of this layer in the core of the boundary layer is the key phenomenon, initiating many mechanisms during the transient. This diffusion is very clear during the pre-transition phase (between $t^*=0.4$ and $t^*=7.0$), whereas the differential velocity profile in the outer region of the boundary layer remains essentially unchanged. Then, for $t^*>7.0$, the outer regions of the profile are affected, and the wall shear stress starts increasing: the transition phase has begun.

Figure 11.

Normalised differential velocity during the transient flow at $70\,\%$ of chord: blue, pre-transition; red, transition; black, fully turbulent.

The present pre-transition stage looks quite similar to a laminar boundary layer, according to the evolution of the friction coefficient presented previously. This is detailed further and the usual self-similarity of laminar flows are sought, such as that observed for the normalised velocity profile as a function of the normalised wall distance $\eta = y/\delta (x)$, see Schlichting & Gersten (Reference Schlichting and Gersten2017).

However, the differential velocity profiles ($\bar {u}\hat {}\,$), plotted against $y/\delta (t)$, do not show any self-similarity during this phase. The difficulty comes from the offset introduced by the initial value of the boundary layer thickness. This problem does not exist for conventional spatial analysis of a flat plate boundary layer because the thickness is null at the leading edge. For the present investigation, compensation is required to reveal the self-similarity: subtraction of the initial thickness of the boundary layer is performed so that our fictitious thickness is zero before the shock propagation. The velocity profiles are then displayed as a function of $\eta = {y}/({\delta (t) - \delta _0}) = {y}/{\sqrt {\nu t}}$. Figure 11 finally demonstrates that a self-similarity actually exists during the pre-transition stage. The temporal development of the differential boundary layer thus follows a laminar-like growth process during the pre-transition. This is consistent with the instantaneous flow visualisations. Indeed, downstream of the shock wave, despite some pre-existing residual turbulence, no turbulent spots can be observed until the beginning of the transition stage. Only streaks appear during this stage, which is typical of a buffeted laminar boundary layer.

A more direct comparison of the two transitional processes, spatial and temporal, is now proposed in figure 12. It is based on the mean flow features trough the friction coefficient. A reference velocity is required to synchronise those two plots, and have a good correspondence between $x$ (or here $Re_x$) and $t^*$. This reference velocity is obtained by minimising the difference between the synchronised temporal friction and the Blasius solution in the quasi-laminar region, which gives a value of $0.76U_{b,1}$. The final correspondence between the two mechanisms is striking. The deviation compared with the Blasius laminar prediction is observed at the same location. This suggests that the receptivity of those two boundary layers (stage $(0)$ on the figure 12) is quite similar. In the spatial case, it is a receptivity to external disturbances. In the temporal case, a receptivity to the residual old turbulence. The characteristic time of such a receptivity looks comparable in the two cases. Interestingly, the maximum turbulence rate in the initial boundary layer and that of the external flow in the initial spatial case are also comparable. Indeed, the maximum turbulence rate in the initial boundary layer is approximately equal to $5\,\%$ (when normalised by the external post-shock velocity) to $y^+_0 = 12$, which corresponds to the turbulence rate of the external flow during the initial regime. However, this turbulence is strongly anisotropic in one case, whereas it has isotropic properties in the other. However, the start of the transition phase (stage $(2)$ in figure 12) does not happen at the same time ($t^*=5.0$ for the spatial case against $t^*=7.0$ for the temporal case). This is certainly caused by a change in the mechanism at the origin of turbulent spots. Unlike spatial evolution, when the differential boundary layer becomes fully turbulent, friction becomes constant (stage $(3)$ on the figure 12). The transient regime then ends in the near-wall region and the flow there reaches an equilibrium corresponding to the characteristics of the new turbulence.

Figure 12.

Temporal evolution of friction coefficient based on differential velocity. Here, $C_f = {\tau _{w,\hat {u} }}/{1/2\rho U_{b,1}^2}$ with $\tau _{w,\hat {u}} = \mu ({\partial [ u(y/\delta,t^*) - u(y/\delta,0) ] }/{\partial y}$.

4.2.3. Fluctuating motion

The response of the fluctuating field, which initiate the different changes in the mean flow, is now detailed. It reinforces the understanding of the different phases of the process which were presented previously.

A representation of Reynolds stresses at $70\,\%$ of the chord during the establishment transient of the new equilibrium of the boundary layer is given in figure 13. The boundary layer characteristic thicknesses are also reported and the steps of the transitional process recalled. Reynolds stresses are normalised here by the external velocity of the boundary layer and thus reflect the turbulence influence on the mean flow. When the shock crosses the abscissa at $t^*=0$, the normalised Reynolds stresses experience a sudden drop and reach a very low level which persists all along the phase $(0)$. The turbulence influences on flow dynamics is weak compared with their contribution to the pre-shock flow $(t^*<0)$. However, a different normalisation (figure 14) shows that the absolute level of fluctuations is comparable to that of the pre-shock situation. The shock wave accelerates the flow but freezes the intensity of fluctuations at its initial level, all along this first phase (there is actually a very slight amplification of $u'_{rms}$ during this phase). This explains why the flow can be considered as quasi-laminar immediately after the wave passage and explains why the differential boundary layer is laminar during this stage. In addition, the receptivity process of the boundary layer to those pre-existing perturbations is underway, and the phase $(1)$ begins when the streamwise Reynolds stress $(u'_{rms})$ growth is significant.

Figure 13.

Reynolds stresses normalised by the external velocity $(U_e(x,t))$ during the transient flow: $(0)$ receptivity process; $(1)$ streak growth; $(2)$ transition; and $(3)$ fully turbulent. Here, $(0)$ and $(1)$ correspond to the pre-transition stage according to He & Seddighi (Reference He and Seddighi2013).

Figure 14.

Production and pressure strain terms for $u'_{rms}$ integrated over the boundary layer thickness and normalised by $\rho u_{\tau,0}^4/(\nu \delta _0)$.

Only the streamwise component takes benefits from a production term (see Cebeci & Cousteix Reference Cebeci and Cousteix2005) which explains why it is the first to respond. However, this response is not instantaneous, despite the very intense shear layer created by the wave near the wall. The production term $(- \rho \overline {u'v'} ({\partial \bar {U}}/{\partial y}))$ is the product of the normal mean velocity gradient and the turbulent shear stress. The latter is substantially zero near the wall, up to approximately $y^+_0=5$. However, pre-existing turbulent structure persists elsewhere in the boundary layer and provides the disturbances on which the new turbulence will emerge. Thus, the delay in the response of $u'_{rms}$ is the characteristic time required by the diffusion of the shear layer to reach heights where the turbulent shear stress is no longer negligible. Then the production term is activated and the streamwise elongated streaks form. Stages $(0)$ and $(1)$, cf. figure 13, correspond to the pre-transition stages according to He & Seddighi (Reference He and Seddighi2013). The increase in the growth rate of the production term as the pre-transition step is shown in figure 15, where the temporal evolution of the production integrated over the boundary layer thickness is given. During pre-transition, the turbulent kinetic energy is therefore essentially concentrated in the streamwise component of the fluctuating motion through streaks. At the end of this step, the turbulence's anisotropy level is, therefore, very high compared with that of an equilibrium turbulent boundary layer.

Figure 15.

Reynolds stresses normalised by the post-shock external velocity $(U_{b,1})$ during the transient flow: $(0)$ receptivity process; $(1)$ streak growth; $(2)$ transition; and $(3)$ fully turbulent. Here, $(0)$ and $(1)$ correspond to the pre-transition stage according to  He & Seddighi (Reference He and Seddighi2013).

The pre-transition stage ends as soon as the pressure strain term activates the turbulent kinetic energy redistribution through the responses of $v'_{rms}$ and $w'_{rms}$. These two stresses are thus activated together and indicate the appearance of turbulent spots and the beginning of the transition phase. In the present configuration, the transition starts for ${u'_{rms, max}/U_{b,1} \approx 10\,\%}$. This value is in agreement with the critical amplitude predicted by Vaughan & Zaki (Reference Vaughan and Zaki2011) for the streaks, at the beginning of the transition stage for a spatial bypass process. It is also interesting to note that this phase starts exactly at the time predicted by the criterion of He & Seddighi (Reference He and Seddighi2015), so-called $t^*_{cr} = 1.34\times 10^3({\nu _1}/{U_{b,1}\delta _0})({u'_{rms,max}}/{U_{b,1}})^{1.71}$. This seems to suggest that, for the study conditions, the temporal Reynolds number of the transition stage beginning is only dependent of initial turbulence intensity and not of the turbulence length scale, in the same way as those described by He & Seddighi (Reference He and Seddighi2015) for the acceleration of an incompressible turbulent boundary layer. This result is not in line with Brandt et al. (Reference Brandt, Schlatter and Henningson2004) findings, which revealed that the transition Reynolds number was dependent on the turbulence length scale for a spatial transition of a boundary layer subject to free-stream turbulence. It should be underlined that the result presented here should be treated with caution as no sensitivity of the results to the turbulence length scale was performed in this study.

In addition, the growth of the maximum of $u'_{rms}$ is observed inside the transition stage even though the redistribution term is already active. This is another example of similarities with the usual bypass process: it has been observed in figure 6 while analysing the spatial transition of the initial boundary layer. The nature of the structures containing the kinetic energy of the fluctuating motion changes continuously during the transition stage from elongated streaks to small turbulent spots, resulting in changes in the kinetic energy spectrum during this phase.

At stage $(3)$, the turbulence has contaminated the entire boundary layer thickness. The Reynolds stresses distribution become constant in time: the transition phase is over and a steady equilibrium state is found. The boundary layer is thus fully turbulent. Some boundary layer regions are fully turbulent before others because the response time differs according to the wall distance. This is why the transition appears completed at $t^*=16$ according to the friction coefficient, whereas at this stage, the outer region of the boundary layer is still only slightly affected by the new turbulence. In the fully turbulent region, the turbulence in the boundary layer thus recovers typical levels of anisotropy.

Finally, let us closely look at the turbulence dynamic behind the wave, which is also identical to that of a bypass spatial transition of the boundary layer. The temporal evolution of maximum turbulent kinetic energy and Reynolds stresses are presented in figure 16. During a bypass transition, the disturbances amplification is linearly related to the longitudinal distance or $Re_x$. This linearity is also observed in the temporal domain during the pre-transition stage, which corresponds to the developments of the streaks $(t^* \in [2.5, 7.0])$. The different delays in the response of Reynolds stresses are particularly evident.

Figure 16.

(a) Disturbances maximum along the plate. (b) Profile of $u'_{rms}/u'_{rms,max}$ for several $Re_x$. Here, $U_{b,0}$ corresponds to the boundary layer external velocity of the initial flow.

On the same figure 16, the velocity profiles $u'_{rms}/u'_{rms,max}$ are also proposed. During a bypass transition, these velocity profiles present a maximum whose distance to the wall is constant throughout the streak amplification phase, see Luchini (Reference Luchini2000). Here, the streak amplification happens when $t^* \in [2.5, 7.0]$. In that time interval, the location of this maximum is recorded constant, approximately at $y/\delta _u = 0.4$. This is closer to the wall than what is generally observed for a spatial bypass process $(y/\delta _u = 1.33)$. This difference probably comes from the location at which the perturbations initiating the streaks are found. For the spatial process, it comes from the free-stream turbulence: the outer part of the boundary layer is thus concerned. For the temporal process, it is the residual old turbulence which triggers the instability. It is thus present at far more important depths inside the boundary layer. However, the fundamental mechanism remains the same.