2.3.1 Subsonic cases

Figure 1. Simulation domain for present subsonic PG simulations. The yellow coloured regions represent sponge zones, blue shaded regions between $x_{end}$ and $x_{max}$ regions where grid stretching is applied in the streamwise direction to make spatial filtering effective. The red-bordered zone represents the main region of the simulations. (a) Simplified simulation domain for the calculation of physical mean values for sponge-zone application. (b) Simulation domain for the final calculation. The yellow coloured sponge regions force the unsteady flow to the previously calculated mean-flow values, which are set as new base-flow values.

A sketch of the subsonic simulation domain is presented in figure 1. The main region of the computational box (red framed in figure 1 a,b) is designed with a height of at least three boundary-layer thicknesses $\unicode[STIX]{x1D6FF}_{99,end}$ in the wall-normal $y$ -direction, measured at the end of the main region $x_{end}$ for the most restricting $\unicode[STIX]{x1D6FD}_{K}=1.05$ case. The width is approximately $\unicode[STIX]{x03C0}\,\unicode[STIX]{x1D6FF}_{99,end}$ in the spanwise $z$ -direction. Measured in boundary-layer thicknesses at the inflow of the simulation domain ( $Re_{\unicode[STIX]{x1D703}}\approx 300$ ), the main region of the computational box has a dimension of $340\,\unicode[STIX]{x1D6FF}_{99,0}\times 34\,\unicode[STIX]{x1D6FF}_{99,0}\times 12\unicode[STIX]{x03C0}\,\unicode[STIX]{x1D6FF}_{99,0}$ , respectively. The length of the PG set-up in the streamwise $x$ -direction thus corresponds to 73 % of the ZPG case discussed in Wenzel et al. (Reference Wenzel, Selent, Kloker and Rist2018b ) whereas the width corresponds to 150 %.

At the solid wall, the flow field is treated as fully adiabatic at every time instance with $(\text{d}T/\text{d}y)_{w}=0$ , which suppresses any heat exchange between the wall and fluid; the pressure at the wall is calculated by $(\text{d}p/\text{d}y)_{w}=0$ . For the velocity components, the no-slip impermeable wall BC is applied. At the outflow, the time derivative, respectively the complete space operator, is extrapolated with $\unicode[STIX]{x2202}\boldsymbol{Q}/\unicode[STIX]{x2202}t|_{N}=\unicode[STIX]{x2202}\boldsymbol{Q}/\unicode[STIX]{x2202}t|_{N-1}$ corresponding to a first-order extrapolation. At the top of the domain, $p$ , $T$ and thus also $\unicode[STIX]{x1D70C}$ are kept constant. The velocity components $u$ and $w$ are specified by $\text{d}/\text{d}y=0$ ; the wall-normal velocity component $v$ is calculated from the continuity equation under the assumption of $\text{d}\unicode[STIX]{x1D70C}/\text{d}y=0$ such that $\text{d}v/\text{d}y=-1/\unicode[STIX]{x1D70C}(\text{d}(\unicode[STIX]{x1D70C}u)/\text{d}x+\text{d}(\unicode[STIX]{x1D70C}w)/\text{d}z)$ . Caused by the non-characteristic behaviour of this BC, the additional application of numerical grid stretching and filtering and/or additional sponge zones is necessary if sound-wave reflection should be suppressed. At the inflow, a digital-filtering synthetic-eddy-method (SEM) approach is used to generate an unsteady, turbulent inflow condition, see Wenzel et al. (Reference Wenzel, Selent, Kloker and Rist2018b ) or Wenzel (Reference Wenzel2019). The spanwise direction is treated as periodic.

In order to allow a smooth transition process with a defined state from the pseudo-physical to a fully developed turbulent state, a ZPG region with a spatial extent of $30\unicode[STIX]{x1D6FF}_{99,0}$ is applied at the inflow of the domain before the PG is forced, see § 2.4. The induction distance for reaching fully developed turbulence is therefore the same as for the ZPG case at $M_{\infty }=0.5$ ( $\unicode[STIX]{x0394}x_{ind}/\unicode[STIX]{x1D6FF}_{99,0}\approx 28$ , see table 2 and Wenzel et al. (Reference Wenzel, Selent, Kloker and Rist2018b )), and similar for all PG distributions. A sponge region is applied outside the boundary layer at the inflow to prevent the included far-field flow from being unduly distorted by the transient process, see Schmidt (Reference Schmidt2014), Kurz & Kloker (Reference Kurz and Kloker2014) and Wenzel (Reference Wenzel2019). The sponge zone fixes the constant pressure region in the far field through the inflow region, which otherwise is strongly influenced by the retroactive effect of the PG applied further downstream. Only in the streamwise direction, geometrical grid stretching is applied to the numerical grid in the outflow region, see Wenzel (Reference Wenzel2019).

Due to the limited height of the simulation domain, the realizable total grid cell increase and thus the effect of the numerical filter is limited in the wall-normal direction in the far field of the domain. Possible reflections occurring at the far-field boundary are therefore suppressed by the application of additional sponge zones. These are to damp the unsteady flow to a prescribed mean flow not known a priori. To generate this mean flow, a precursor simulation is computed on a preliminary set-up for all PG cases, see figure 1(a). Weak reflections occurring at the far-field boundary are not treated by this set-up since they have been found to average out in the temporal mean. The accuracy of the resulting flow field has been verified by a comparison with the ZPG case given in Wenzel et al. (Reference Wenzel, Selent, Kloker and Rist2018b ). At the outlet of the simulation domain, a small sponge zone is applied to avoid a temporal drift of the outlet pressure. This sponge damps the flow field to a spatially undeveloped reference solution defined a priori. Hence, to minimize the distortion of the flow field, its effectiveness is reduced to 30 % of the inflow sponge. The resulting mean flow is used as a new base flow for the final set-up in a second step, which allows the application of additional sponge zones at the far-field boundary and the outflow region, both with the same effectiveness as the inflow sponge. A sketch of the final set-up is given in figure 1(b).

2.3.2 Supersonic cases

A sketch of the supersonic simulation domain is depicted in figure 2. Caused by the spatially hyperbolic behaviour of the supersonic flow, the pressure information is only allowed to move along the angle of the flow’s local characteristics. Thus, prescribed by the far-field BC at the top of the domain, the information of the particular PG is diagonally transmitted as Mach waves (shock waves if the PG is too strong) down to the plate (black solid lines in figure 2) where they are reflected (black dashed lines). The final pressure field results from the superposition of the prescribed, incoming and the reflected, outgoing waves. Both yield a complex flow field with lines of constant pressure being almost perpendicular to the wall for regions of fully established wave systems (cyan coloured solid lines). The induction length (inflow region in figure 2) therefore depends on the Mach number, the length of the ZPG inflow region and the height of the simulation domain. Hence, it is fixed by the particular numerical set-up.

Figure 2. Simulation domain for present supersonic PG simulations. The yellow and blue shaded regions represent sponged and grid-stretched zones, respectively. The main region of the simulation is red bordered. Pressure information introduced at the far field moves diagonally to the wall (black solid lines) and is reflected (black dashed lines). The final pressure field is depicted by cyan coloured solid lines.

For the present simulations, the main region of the computational box (red framed in figure 2) is designed with a height of at least three boundary-layer thicknesses $\unicode[STIX]{x1D6FF}_{99,end}$ in the wall-normal $y$ -direction and a width of approximately $\unicode[STIX]{x03C0}\,\unicode[STIX]{x1D6FF}_{99,end}$ in the spanwise $z$ -direction for the most restricting $\unicode[STIX]{x1D6FD}_{K}=0.69$ case. Measured in boundary-layer thicknesses at the inflow of the simulation domain ( $Re_{\unicode[STIX]{x1D703}}\approx 300$ ), the main region of the computational box has a dimension of $500\,\unicode[STIX]{x1D6FF}_{99,0}\times 34\,\unicode[STIX]{x1D6FF}_{99,0}\times 12\unicode[STIX]{x03C0}\,\unicode[STIX]{x1D6FF}_{99,0}$ , respectively. The length of the PG set-up in the streamwise $x$ -direction thus corresponds to 107 % of the ZPG cases discussed in Wenzel et al. (Reference Wenzel, Selent, Kloker and Rist2018b ) and 147 % of the previously presented subsonic PG cases.

The outflow is parabolized by neglecting the second $x$ -derivatives; biased difference stencils are used for the first $x$ -derivatives. At the top of the simulation domain, a subsonic characteristic outflow BC is applied according to Poinsot & Lele (Reference Poinsot and Lele1991) (see also Babucke (Reference Babucke2009), Franko & Lele (Reference Franko and Lele2014)). Its implementation allows upward travelling (outgoing) disturbances to leave the integration domain, whereas downward travelling (incoming) disturbances are set to zero in order to avoid reflections. This BC, in fact, allows the mean pressure to adapt at the upper boundary, see below. A detailed justification and discussion of its use is given in Wenzel (Reference Wenzel2019).

Following the same ideas as for the subsonic set-ups, a ZPG region with a spatial extent of $80\,\unicode[STIX]{x1D6FF}_{99,0}$ is applied at the inflow of the domain before the PG is forced (see next section for details). The induction distance until turbulence is fully developed is therefore the same as for the ZPG case at $M_{\infty }=2.0$ ( $\unicode[STIX]{x0394}x_{ind}/\unicode[STIX]{x1D6FF}_{99,0}\approx 60$ , see table 2 and Wenzel et al. (Reference Wenzel, Selent, Kloker and Rist2018b )), and identical for all different PG distributions. As for the subsonic cases, an identically dimensioned sponge region is applied at the inflow of the domain to absorb the shock system which is caused by the transient process of the inflow BC. It is noted that the prescribed wall-normal velocity component $v$ , to which the flow field is damped down in the sponge region, has to closely match the one generated by the boundary layer itself. A mismatch of both velocities causes a discontinuity (exemplarily given in figure 2 as dotted lines at the inflow), which is reflected by the upper boundary and thus affects the flow field in the whole simulation domain. A detailed derivation of an analytical velocity adaption procedure can be found in Gibis (Reference Gibis2018). The outflow region of the simulation domain is only slightly stretched and filtered to avoid reflections from the outflow BC in the subsonic near-wall regions of the boundary layer. It was found that no additional outflow sponge region is required.

3.2.1 Evaluation of the Rotta–Clauser parameter

Both the kinematic and the compressible results of the Rotta–Clauser parameter $\unicode[STIX]{x1D6FD}_{(K)}=(\unicode[STIX]{x1D6FF}_{(K)}^{\ast }/\overline{\unicode[STIX]{x1D70F}}_{w})(\text{d}p_{e}/\text{d}x)$ are shown in figure 10, where the kinematic displacement thickness is $\unicode[STIX]{x1D6FF}_{K}^{\ast }=\int _{0}^{\unicode[STIX]{x1D6FF}_{99}}(1-\overline{u}/u_{e})\,\text{d}y$ and the compressible displacement thickness $\unicode[STIX]{x1D6FF}^{\ast }=\int _{0}^{\unicode[STIX]{x1D6FF}_{99}}(1-(\overline{\unicode[STIX]{x1D70C}}\,\overline{u})/(\unicode[STIX]{x1D70C}_{e}u_{e}))\,\text{d}y$ . It is shown in Part 2 of this study (Gibis et al. Reference Gibis, Wenzel, Kloker and Rist2019) and by the DNS results discussed below that the $\unicode[STIX]{x1D6FD}_{K}$ -parameter correctly characterizes the self-similar state of the compressible TBL and hence enables a comparison of PG influences between compressible and incompressible flows. The compressible definition $\unicode[STIX]{x1D6FD}$ turns out to be less relevant because the compressible displacement thickness is not a good characterizing length scale for the outer part of the boundary layer, as shown in Gibis et al. (Reference Gibis, Wenzel, Kloker and Rist2019). Since all simulations start with a ZPG region at the inflow (see § 2.4), an induction distance is needed before the individual $\unicode[STIX]{x1D6FD}_{(K)}$ -values become approximately constant (grey dashed lines). Expressed in various Reynolds numbers, the start and the end of the (almost) constant regions are summarized in table 3(a) for all cases.

Figure 10. Clauser parameters $\unicode[STIX]{x1D6FD}_{(K)}=(\unicode[STIX]{x1D6FF}_{(K)}^{\ast }/\overline{\unicode[STIX]{x1D70F}}_{w})(\text{d}p_{e}/\text{d}x)$ for the computed results. Red lines represent the four subsonic cases, blue and cyan lines the six supersonic cases. Coloured horizontal lines represent curve fits of their solid line counterparts. Grey dashed lines represent induction regions, where $\unicode[STIX]{x1D6FD}_{(K)}$ is not yet constant. Expressed in various Reynolds numbers, both the start and the end of the coloured regions are summarized in table 3(a) for all cases.

An evaluation of the resulting $\unicode[STIX]{x1D6FD}_{(K)}$ -values in figure 10 yields almost perfectly constant values for the subsonic APG cases, slightly rising values for the supersonic APG cases and slightly decreasing values for the supersonic FPG case. Due to the additional consideration of density variations in the wall-normal direction, $\unicode[STIX]{x1D6FD}$ -values are increased compared to the $\unicode[STIX]{x1D6FD}_{K}$ -values. As shown in the next section, $\unicode[STIX]{x1D6FF}_{K}^{\ast }$ and $\unicode[STIX]{x1D6FF}^{\ast }$ and thus also $\unicode[STIX]{x1D6FD}_{K}$ and $\unicode[STIX]{x1D6FD}$ are not proportional to each other in their streamwise evolution for the supersonic cases due to the varying influence of compressibility. The $\unicode[STIX]{x1D6FD}_{K}$ -distributions are therefore slightly steeper compared to $\unicode[STIX]{x1D6FD}$ . Whereas the subsonic cases are characterized by $\unicode[STIX]{x1D6FD}_{K}=0.19$ , 0.58 and 1.05, the supersonic cases yield $\unicode[STIX]{x1D6FD}_{K}=0.15$ , 0.42, 0.55 and 0.69 for the APG and $\unicode[STIX]{x1D6FD}_{K}=-0.18$ for the FPG case. The corresponding $\unicode[STIX]{x1D6FD}$ -parameters are summarized in table 1.

Constant $\unicode[STIX]{x1D6FD}_{K}$ -values mainly characterize the resulting mean flow but do not inevitably warrant perfect self-similarity of higher moments in the streamwise direction, since influences resulting from history effects occurring in the induction sections cannot be excluded. Nevertheless, the achieved constancy of the distributions suggests an approximated state of local self-similarity for all PG cases for at least 50 average boundary-layer thicknesses $\unicode[STIX]{x1D6FF}_{99,av}$ but not exactly for $x\rightarrow \infty$ . Detailed investigations are given in Gibis et al. (Reference Gibis, Wenzel, Kloker and Rist2019).

3.2.2 Evolution of the boundary layer

Figure 11. Boundary-layer values for regions of constant $\unicode[STIX]{x1D6FD}_{(K)}$ . Subsonic and supersonic cases are coloured red and blue, respectively, the supersonic FPG case cyan. Depicted are (a) the boundary-layer thickness $\unicode[STIX]{x1D6FF}_{99}/\unicode[STIX]{x1D6FF}_{99,0}$ and the sonic lines in green solely for the supersonic cases; (b,c) the momentum thickness $\unicode[STIX]{x1D703}/\unicode[STIX]{x1D6FF}_{99}$ (symbols, left $y$ -axis) and the displacement thickness $\unicode[STIX]{x1D6FF}^{\ast }/\unicode[STIX]{x1D6FF}_{99,0}$ (lines, right $y$ -axis) for the sub- and supersonic cases, respectively; $(d)$ the kinematic momentum thickness $\unicode[STIX]{x1D703}_{K}/\unicode[STIX]{x1D6FF}_{99}$ and the kinematic displacement thickness $\unicode[STIX]{x1D6FF}_{K}^{\ast }/\unicode[STIX]{x1D6FF}_{99}$ solely for the supersonic cases.

All supersonic APG cases follow the same qualitative trends; thus only the cases $cAPG_{\unicode[STIX]{x1D6FD}_{K}=0.42}$ and $cAPG_{\unicode[STIX]{x1D6FD}_{K}=0.69}$ are depicted.

Figure 11(a) illustrates the spatial evolution of the boundary-layer thickness $\unicode[STIX]{x1D6FF}_{99}/\unicode[STIX]{x1D6FF}_{99,0}$ and the wall-normal position of the sonic line $\unicode[STIX]{x1D6FF}_{SL}/\unicode[STIX]{x1D6FF}_{99,0}$ in green (line style in accordance with the boundary-layer thickness), the latter of which separates sub- and supersonic parts of the supersonic boundary-layer cases. The ratios of the boundary-layer thicknesses $\unicode[STIX]{x1D703}/\unicode[STIX]{x1D6FF}_{99}$ and $\unicode[STIX]{x1D6FF}^{\ast }/\unicode[STIX]{x1D6FF}_{99}$ are depicted in figures 11(b) and 11(c) for the sub- and supersonic cases, respectively. Solely for the supersonic cases, the ratios of the kinematic boundary-layer thicknesses are given in figure 11(d). For all three panels (b,c,d), results depicted by symbols belong to the $y$ -axis located on the left-hand side of the plot; results depicted by lines belong to the $y$ -axis located on the right-hand side of the plot.

For the quasi-incompressible cases, the boundary-layer thickens strongly with increasing APG strength (figure 11 a) resulting in a maximum of twice the thickness of the ZPG case for the $iAPG_{\unicode[STIX]{x1D6FD}_{K}=1.05}$ case. When the pressure rises in subsonic flows, the velocity decreases more rapidly than the density increases. This results in a decompression of the streamtubes and thus an increase in the boundary-layer thickness (Spina, Smits & Robinson Reference Spina, Smits and Robinson1994).

Given in figure 11(b), the ratio of the momentum thickness $\unicode[STIX]{x1D703}/\unicode[STIX]{x1D6FF}_{99}$ (red symbols, left $y$ -axis) as well as the ratio of the displacement thickness $\unicode[STIX]{x1D6FF}^{\ast }/\unicode[STIX]{x1D6FF}_{99}$ (red lines, right $y$ -axis) are almost constant and thus almost proportional to each other in the streamwise direction, $\unicode[STIX]{x1D6FF}_{99}\propto \unicode[STIX]{x1D6FF}^{\ast }\propto \unicode[STIX]{x1D703}$ . As a consequence, also the shape factor $H=\unicode[STIX]{x1D6FF}^{\ast }/\unicode[STIX]{x1D703}$ is almost constant as will be further discussed in § 3.3.1.

For the compressible, supersonic APG cases, the boundary-layer thickness $\unicode[STIX]{x1D6FF}_{99}/\unicode[STIX]{x1D6FF}_{99,0}$ is first reduced and then increased compared to the $cZPG$ case and thus leads to an intersection point, see figure 11(a). Following Spina et al.’s (Reference Spina, Smits and Robinson1994) argument, this counterintuitive behaviour is caused by the unique character of the supersonic boundary layer that consists of both sub- and supersonic regions. In contrast to the subsonic regions, the density increases more rapidly than the velocity decreases in supersonic regions for APGs, meaning that streamtubes are compressed in the supersonic part above the green sonic line and decompressed in the subsonic part below the sonic line. When the subsonic region of the boundary layer becomes large, which means that the sonic line is noticeably lifted off the wall (approximately for $M_{\infty }\lessapprox 1.8$ , cf. Spina et al. (Reference Spina, Smits and Robinson1994)), the APG causes the boundary-layer thickness to increase and the wall shear stress to decrease. As given in table 3(a), the usable region of the $cAPG_{\unicode[STIX]{x1D6FD}_{K}=0.42}$ and $cAPG_{\unicode[STIX]{x1D6FD}_{K}=0.69}$ cases covers Mach-number ranges of $1.82\geqslant M_{e}\geqslant 1.50$ and $1.74\geqslant M_{e}\geqslant 1.33$ , respectively, which approximately confirms the Spina et al.’s (Reference Spina, Smits and Robinson1994) switch-over value of $M_{\infty }\lessapprox 1.8$ . For the compressible, supersonic $cFPG_{\unicode[STIX]{x1D6FD}_{K}=-0.18}$ case, the supersonic part dominates the boundary layer, which thus is thickened up in its spatial development compared to the $cZPG$ case.

Discussing the momentum and displacement thickness distributions for all supersonic cases in figure 11(c) next, only the ratio of the momentum thickness $\unicode[STIX]{x1D703}/\unicode[STIX]{x1D6FF}_{99}$ (symbols, left $y$ -axis) is approximately constant in the streamwise direction. The ratio of the displacement thickness $\unicode[STIX]{x1D6FF}^{\ast }/\unicode[STIX]{x1D6FF}_{99}$ (lines, right $y$ -axis) is strongly decreased for the APG and increased for the FPG cases, approximately leading to $\unicode[STIX]{x1D6FF}_{99}\propto \unicode[STIX]{x1D703}\not \propto \unicode[STIX]{x1D6FF}^{\ast }$ . As a consequence also the shape factor $H$ is not constant anymore for the compressible results as will be further discussed in § 3.3.1. An evaluation of the kinematic boundary-layer values in figure 11(d) can reproduce almost the same trends as for the nearly incompressible cases in figure 11(b), yielding again the approximate dependency of $\unicode[STIX]{x1D6FF}_{99}\propto \unicode[STIX]{x1D703}_{K}\propto \unicode[STIX]{x1D6FF}_{K}^{\ast }$ and thus the approximate constancy of $H_{K}=\unicode[STIX]{x1D6FF}_{K}^{\ast }/\unicode[STIX]{x1D703}_{K}$ .

3.3.1 Spatial evolution of mean-flow statistics

Figures 12, 13 and 14 give the distribution of the skin-friction coefficient $c_{f}=2\overline{\unicode[STIX]{x1D70F}}_{w}/(\overline{\unicode[STIX]{x1D70C}}_{w}u_{e}^{2})$ with $\overline{\unicode[STIX]{x1D70F}}_{w}=\overline{\unicode[STIX]{x1D707}\unicode[STIX]{x2202}u/\unicode[STIX]{x2202}y}_{w}\approx \overline{\unicode[STIX]{x1D707}}_{w}(\unicode[STIX]{x2202}\overline{u}/\unicode[STIX]{x2202}y)_{w}$ , the skin-friction velocity Reynolds number $Re_{\unicode[STIX]{x1D70F}}=\overline{\unicode[STIX]{x1D70C}}_{w}u_{\unicode[STIX]{x1D70F}}\unicode[STIX]{x1D6FF}_{99}/\overline{\unicode[STIX]{x1D707}}_{w}$ and the compressible (kinematic) shape factors $H_{(K)}=\unicode[STIX]{x1D6FF}_{(K)}^{\ast }/\unicode[STIX]{x1D703}_{(K)}$ as functions of the momentum-thickness Reynolds number $Re_{\unicode[STIX]{x1D703}_{(K)}}=\unicode[STIX]{x1D70C}_{e}u_{e}\unicode[STIX]{x1D703}_{(K)}/\unicode[STIX]{x1D707}_{e}$ . Additionally, as a function of the streamwise position $x/\unicode[STIX]{x1D6FF}_{99,0}$ and computed from $\overline{T}_{aw}/T_{e}=1+r(\unicode[STIX]{x1D6FE}-1)/2\,M_{e}^{2}$ , the streamwise evolution of the recovery factor $r$ is given in figure 15.

Figure 12. Skin-friction coefficient $c_{f}$ as function of $Re_{\unicode[STIX]{x1D703}}$ . The incompressible, re-calibrated correlation (black solid line) is given by $c_{f}=0.0274\,Re_{\unicode[STIX]{x1D703}}^{-0.27}$ , the compressible one by the van Driest II transformation (grey dashed lines, see White (Reference White2006)).

The $c_{f}$ -distribution (figure 12) is supplemented by a black solid line representing a re-calibrated incompressible reference following $c_{f}=0.0274\,Re_{\unicode[STIX]{x1D703}}^{-0.27}$ (re-calibration is based on the results from Wenzel et al. (Reference Wenzel, Selent, Kloker and Rist2018b )). Note that the re-calibrated $c_{f}$ -correlation also better represents the Schlatter & Örlü’s (Reference Schlatter and Örlü2010) incompressible dataset. Compressible references for $M_{\infty }=0.5$ and $2.0$ are computed from the van Driest II transformation (White Reference White2006) and are depicted as grey dashed lines; see also Wenzel et al. (Reference Wenzel, Selent, Kloker and Rist2018b ). The $Re_{\unicode[STIX]{x1D70F}}$ - and $H_{K}$ -distributions are supplemented by the incompressible references according to Schlatter & Örlü (Reference Schlatter and Örlü2010) and Chauhan, Monkewitz & Nagib (Reference Chauhan, Monkewitz and Nagib2009). Represented as grey dashed lines in figures 13 and 14, the compressible references are gained by shifting the incompressible correlations to fit the compressible ZPG results.

Figure 13. Skin-friction velocity Reynolds number $Re_{\unicode[STIX]{x1D70F}}$ as function of $Re_{\unicode[STIX]{x1D703}}$ . The incompressible correlation is given by $Re_{\unicode[STIX]{x1D70F}}=1.13\,Re_{\unicode[STIX]{x1D703}}^{0.843}$ (black solid line, see Schlatter & Örlü (Reference Schlatter and Örlü2010)), the compressible ones by the simple shift of the incompressible correlation (grey dashed lines).

Figure 14. (a) Shape factor $H$ and (b) kinematic shape factor $H_{K}$ as a function of $Re_{\unicode[STIX]{x1D703}}$ and $Re_{\unicode[STIX]{x1D703},K}$ , respectively. The incompressible correlation is given as the shape of the integrated incompressible composite profiles (Chauhan et al. Reference Chauhan, Monkewitz and Nagib2009). The compressible references are gained by a simple shift of the incompressible correlation.

For the $c_{f}$ -distributions in figure 12, both the sub- and the supersonic ZPG cases show excellent agreement with the compressibility transformed re-calibrated $c_{f}$ -correlation. In accordance to the literature database, the $c_{f}$ -distributions of the subsonic APG cases are shifted towards lower values for increasing APG strength. However, they almost follow the same slope as the subsonic ZPG case due to their (approximate) state of self-similarity. A comparison between the $iAPG_{\unicode[STIX]{x1D6FD}_{K}=1.05}$ case and the incompressible $\unicode[STIX]{x1D6FD}_{K}=1.00$ case of Kitsios et al. (Reference Kitsios, Sekimoto, Atkinson, Sillero, Borrell, Gungor, Jiménez and Soria2017) (red circles) shows good agreement. The supersonic APG cases are difficult to interpret due to the strong coupling of PG and compressibility effects. According to Spina et al. (Reference Spina, Smits and Robinson1994), the wall shear stress increases monotonically through the region of applied APGs for supersonic flows where the sonic line is close to the wall, meaning that $M_{e}\gtrsim 1.8$ . The $c_{f}$ -distribution, however, remains nearly constant due to the increase in $\unicode[STIX]{x1D70C}_{e}u_{e}^{2}$ . All supersonic APG cases are therefore in a comparable $c_{f}$ -range for $Re_{\unicode[STIX]{x1D703}}\lesssim 1700$ where local Mach numbers are about $M_{e}\approx 1.8$ , see table 3(a). For higher Reynolds numbers, compressibility effects become weaker due to a decrease of the local Mach number and the $c_{f}$ -distributions are increasingly dominated by PG effects. Since the PG is stronger for the $cAPG_{\unicode[STIX]{x1D6FD}_{K}=0.69}$ case compared to the $cAPG_{\unicode[STIX]{x1D6FD}=0.42}$ case, for instance, the $c_{f}$ -distribution of the $cAPG_{\unicode[STIX]{x1D6FD}_{K}=0.69}$ case is lower compared to the $cAPG_{\unicode[STIX]{x1D6FD}_{K}=0.42}$ case. The interpretation of the supersonic FPG is opposite: compressibility effects (decreasing $c_{f}$ ) dominate the PG effect (increasing $c_{f}$ ), which thus yields a steeper downwards running $c_{f}$ -slope compared to the ZPG case.

The interpretation of the $Re_{\unicode[STIX]{x1D70F}}$ -distributions in figure 13 follows the same arguments as for the $c_{f}$ -distribution due to their mutual dependency on the non-dimensional wall shear stress. Subsonic APG cases can be gained by shifting the ZPG results to higher $Re_{\unicode[STIX]{x1D703}}$ - and slightly increased $Re_{\unicode[STIX]{x1D70F}}$ -values. The supersonic APG and FPG cases show a consistent behaviour as the $c_{f}$ -distributions discussed before. Note that the $Re_{\unicode[STIX]{x1D70F}}$ -distributions for the supersonic APG cases are slightly curved in the double-logarithmic representation since they have to converge towards the incompressible distributions when decelerated to almost incompressible Mach numbers.

Calculated in its compressible formulation, the shape-factor distribution $H$ (figure 14 a) is more influenced by compressibility effects than by PG effects, which makes the interpretation of the compressible results more unambiguous. Whereas the subsonic APG cases again seem to be obtainable by shifting the ZPG distribution towards higher values, the shape factors of the supersonic APG cases become significantly smaller in the streamwise direction caused by the substantial decrease in the local Mach number. For instance, the Mach number of the $cAPG_{\unicode[STIX]{x1D6FD}_{K}=0.69}$ case is lower at the end of the simulation domain than the Mach number of the $cAPG_{\unicode[STIX]{x1D6FD}_{K}=0.55}$ case, and its shape factor is lower. For the supersonic FPG case, the reversed argument holds. When calculated without taking density variations in the wall-normal direction into account, compressibility influences are almost eliminated for the kinematic shape-factor distributions $H_{K}$ in figure 14(b). However, a quantitative comparison between sub- and supersonic cases seems not to be reasonable, see also figure 11(d). But especially for the self-similarity analysis in Gibis et al. (Reference Gibis, Wenzel, Kloker and Rist2019), the coalescence of sub- and supersonic kinematic shape factors will be utilized in the definition of appropriate length scales defining the state of the compressible PG TBL.

Figure 15. Distribution of the recovery factor $r$ as function of streamwise position $x/\unicode[STIX]{x1D6FF}_{99,0}$ . Red coloured lines represent the subsonic cases, blue and cyan lines the supersonic cases. Grey dotted lines denote the induction regions where $\unicode[STIX]{x1D6FD}_{(K)}$ is not yet constant, see figure 10.

The recovery factor $r$ is expected to be constant in the streamwise direction for self-similar flows, see Gibis et al. (Reference Gibis, Wenzel, Kloker and Rist2019), but not necessarily with the same value for different $\unicode[STIX]{x1D6FD}_{K}$ -cases. Figure 15 shows the various curves compared to the often used ZPG reference value $r=Pr^{1/3}\approx 0.892$ ; constancy of $r$ might be interpreted as an indicator for the state of self-similarity reached by the thermal boundary layer.

For the subsonic cases, an increasing APG means lower recovery factor values, with slightly rising trends in the streamwise direction. Whereas its value is approximately equal to $r\approx 0.88$ for the $iZPG$ case, its value is about $r\approx 0.82$ for the strongest APG case and thus decreased by 7 %. Note that the calculation of $r$ for low Mach numbers gets very sensitive and does not make sense for zero Mach number.

Although being much less influenced by PGs compared to the subsonic cases, similar behaviour is observed for the supersonic cases. Whereas $r$ is decreased by approximately 2 % for the strongest supersonic APG case compared to the ZPG case, the FPG case value is increased by approximately 1 %. Compared to the subsonic cases, $r$ is much more constant in the streamwise direction for $x/\unicode[STIX]{x1D6FF}_{99,0}\gtrapprox 300$ , but needs an induction distance of approximately $\unicode[STIX]{x0394}x/\unicode[STIX]{x1D6FF}_{99,0}\approx 100$ to reach the constant state. It is therefore expected that the thermal boundary layer requires more time/space to fully adapt to the self-similar state. The effects of the elongated induction distance on self-similarity of local mean-flow profiles are further discussed in Gibis et al. (Reference Gibis, Wenzel, Kloker and Rist2019).

3.3.2 Spatial evolution of averaged fluctuation statistics

Figure 16. Near-wall behaviour of the streamwise velocity fluctuation $u^{\prime +}=\sqrt{\overline{u^{\prime 2}}}/u_{\unicode[STIX]{x1D70F}}$ in inner scaling versus $Re_{\unicode[STIX]{x1D70F}}$ (cf. Schlatter & Örlü Reference Schlatter and Örlü2010). (a) Peak value of the wall-near maximum $u_{max}^{\prime +}$ , (b) the wall-normal position of the maximum $y^{+}\left(u_{max}^{\prime +}\right)$ and (c) the wall slope $(\unicode[STIX]{x2202}u^{\prime +}/\unicode[STIX]{x2202}y^{+})|_{w}$ . Black dashed lines are references for the ZPG cases taken from Wenzel et al. (Reference Wenzel, Selent, Kloker and Rist2018b ).

The peak value $u_{max}^{\prime +}$ of the Reynolds fluctuations in the streamwise direction $u^{\prime +}=\sqrt{\overline{u^{\prime 2}}}/u_{\unicode[STIX]{x1D70F}}$ and their wall-normal position $y^{+}(u_{max}^{\prime +})$ are given in figure 16(a,b), respectively. The wall slope $(\unicode[STIX]{x2202}u^{\prime +}/\unicode[STIX]{x2202}y^{+})|_{w}$ is given in figure 16(c). All distributions are plotted versus $Re_{\unicode[STIX]{x1D70F}}$ . In order to emphasize their spatial trends, all data (grey solid lines in the background) are approximated by a coloured linear regression. (The linear slopes are only rough approximations which are not expected to hold for higher Reynolds numbers. The supersonic cases, for instance, are decelerated to the incompressible regime and thus have to converge to the incompressible references.) Taken from Wenzel et al. (Reference Wenzel, Selent, Kloker and Rist2018b ), black dashed lines represent references for both ZPG cases.

For the subsonic cases in figure 16(a) first, the peak value of the streamwise velocity fluctuation $u_{max}^{\prime +}$ is increased by up to 10 % in the investigated $\unicode[STIX]{x1D6FD}_{K}$ - and $Re_{\unicode[STIX]{x1D70F}}$ -range for the strongest APG case. The slope of the linear regression is slightly increased. For the supersonic APG cases this shift is larger, since both the effect of a decreasing Mach number in the streamwise direction and APGs supplement each other. With opposing argumentation, a contrary behaviour is found for the supersonic FPG case.

For the subsonic cases, the wall-normal position of the fluctuations’ maximum value $y^{+}(u_{max}^{\prime +})$ in figure 16(b) is shifted towards the wall for lower and away from the wall for higher Reynolds numbers. For increasing APG strength, the slope of the linear regressions is increased. For the supersonic PG cases, both the effects of varying Mach number in the streamwise direction and PGs are opposed to each other, which leads to comparable distributions for the $cAPG_{\unicode[STIX]{x1D6FD}_{K}=0.55}$ and $cAPG_{\unicode[STIX]{x1D6FD}_{K}=0.69}$ cases for instance. Globally, the maxima are shifted away from the wall for $Re_{\unicode[STIX]{x1D70F}}$ larger than 350.

The wall slope in figure 16(c) shows a comparable behaviour for both sub- and supersonic cases, although the slopes of the supersonic cases are somewhat lower compared to the subsonic ones. More details will be discussed in the next section.

4.1.1 Shear-stress distributions

According to Guarini et al. (Reference Guarini, Moser, Shariff and Wray2000), the total shear stress $\overline{\unicode[STIX]{x1D70F}}_{t}$ writes

(4.4) $$\begin{eqnarray}\underbrace{\overline{\unicode[STIX]{x1D70F}}_{t}}_{total\,stress}=\underbrace{\overline{\unicode[STIX]{x1D707}}\frac{\unicode[STIX]{x2202}\overline{u}}{\unicode[STIX]{x2202}y}}_{mean\,stress}-\underbrace{\overline{\unicode[STIX]{x1D70C}u^{\prime \prime }v^{\prime \prime }}}_{turbulent\,stress}+\underbrace{\overline{\unicode[STIX]{x1D707}^{\prime }\left(\frac{\unicode[STIX]{x2202}u^{\prime }}{\unicode[STIX]{x2202}y}+\frac{\unicode[STIX]{x2202}v^{\prime }}{\unicode[STIX]{x2202}x}\right)}}_{stress\,correlation},\end{eqnarray}$$

where $f^{\prime \prime }$ denotes fluctuations about the Favre average $\tilde{f}=\overline{\unicode[STIX]{x1D70C}f}/\overline{\unicode[STIX]{x1D70C}}$ for an unsteady quantity $f$ . Normalized by the wall shear stress $\overline{\unicode[STIX]{x1D70F}}_{w}=\overline{\unicode[STIX]{x1D70C}}_{w}u_{\unicode[STIX]{x1D70F}}^{2}$ , the mean shear stress, the turbulent stress and the total stress are given in figure 17(a–c) for all cases, respectively. The distribution of the shear-stress correlation is almost zero and therefore not considered in the following. Based on the first-order expansion of the total shear-stress distribution $\overline{\unicode[STIX]{x1D70F}}_{t}\approx \overline{\unicode[STIX]{x1D70F}}_{w}+(\text{d}p_{e}/\text{d}x)y$ , the pressure-corrected total shear-stress distribution $\overline{\unicode[STIX]{x1D70F}}_{t,pc}=\overline{\unicode[STIX]{x1D70F}}_{t}-(\text{d}p_{e}/\text{d}x)y$ is given in figure 17(d). All data are extracted at $Re_{\unicode[STIX]{x1D70F}}=490$ except for the supersonic FPG case where data are extracted at $Re_{\unicode[STIX]{x1D70F}}=360$ .

Figure 17. Comparison of (a) the mean shear stress $\overline{\unicode[STIX]{x1D707}}(\unicode[STIX]{x2202}\overline{u}/\unicode[STIX]{x2202}y)/\overline{\unicode[STIX]{x1D70F}}_{w}$ , (b) the turbulent shear stress $-\overline{\unicode[STIX]{x1D70C}u^{\prime \prime }v^{\prime \prime }}/\overline{\unicode[STIX]{x1D70F}}_{w}$ , (c) the total shear stress $\overline{\unicode[STIX]{x1D70F}}_{t}$ and (d) the pressure-corrected total shear stress $\overline{\unicode[STIX]{x1D70F}}_{t,pc}=\overline{\unicode[STIX]{x1D70F}}_{t}-(\text{d}p_{e}/\text{d}x)y$ . Supersonic FPG data are extracted at $Re_{\unicode[STIX]{x1D70F}}=390$ , other data at $Re_{\unicode[STIX]{x1D70F}}=490$ . Red: ———  $iZPG$ , —— -  $\cdot$   $iAPG_{\unicode[STIX]{x1D6FD}_{K}=0.19}$ ,  — - —  $iAPG_{\unicode[STIX]{x1D6FD}_{K}=0.58}$ , –  $\cdot$  –  $\cdot$  –  $iAPG_{\unicode[STIX]{x1D6FD}_{K}=1.05}$ . Blue: ———  $cZPG$ , ——– –  $cAPG_{\unicode[STIX]{x1D6FD}_{K}=0.15}$ , — — -  $cAPG_{\unicode[STIX]{x1D6FD}_{K}=0.42}$ , – – – – –  $cAPG_{\unicode[STIX]{x1D6FD}_{K}=0.55}$ , - - - - - -  $cAPG_{\unicode[STIX]{x1D6FD}_{K}=0.69}$ . Cyan: $-\!-$  - - –  $cFPG_{\unicode[STIX]{x1D6FD}_{K}=-0.18}$ .

Given in figure 17(a), the mean shear-stress distribution is almost constant at the wall for all cases and only slightly influenced by both compressibility and PG effects in the buffer layer between the viscous sublayer and the log layer at $3\lessapprox y^{+}\lessapprox 50$ . For the subsonic cases, the mean shear-stress distributions are slightly decreased with increasing APG strength. The supersonic cases are almost identical since effects of decreasing Mach number (increasing) and increasing effect of APG (decreasing) cancel each other.

The non-dimensional turbulent stresses $-\overline{\unicode[STIX]{x1D70C}u^{\prime \prime }v^{\prime \prime }}/\overline{\unicode[STIX]{x1D70F}}_{w}$ in figure 17(b) are strongly influenced by PG effects over the entire boundary layer for both sub- and supersonic Mach numbers. For increasing APG strength a peak rises in the outer region whereas the distributions are reduced for FPGs. In contrast to the mean-flow distributions, the turbulent stresses are also noticeably influenced in the near-wall region.

Computed as the sum of (a) and (b), the total shear-stress distributions in figure 17(c) are constant for the sub- and supersonic ZPG cases up to approximately 30–40 wall units above the wall and virtually identical. For both the sub- and supersonic PG cases the distributions are increased with distance to the wall for APG cases ( $\text{d}p/\text{d}x>0$ ) and decreased for the FPG case ( $\text{d}p_{e}/\text{d}x<0$ ) by following $\text{d}\overline{p}/\text{d}x=\unicode[STIX]{x2202}\overline{\unicode[STIX]{x1D70F}}_{t}/\unicode[STIX]{x2202}y$ (Smits & Dussauge Reference Smits and Dussauge2006). If the PG term $(\text{d}p_{e}/\text{d}x)y$ is subtracted from the total shear-stress distributions as given in figure 17(d), the constant stress region near the wall is regained for all PG cases. Since the mean shear stresses in (a) are almost unaffected by PGs and almost constant at the wall, PG effects included in the total shear-stress distributions by $(\text{d}p_{e}/\text{d}x)y$ are mainly balanced by the turbulent stresses at the wall (Smits & Dussauge Reference Smits and Dussauge2006).

A comparison between the total shear-stress distributions for the sub- and supersonic PG cases in (c) shows an almost complete agreement between the $iAPG_{\unicode[STIX]{x1D6FD}_{K}=0.19}$ and $cAPG_{\unicode[STIX]{x1D6FD}_{K}=0.15}$ cases and the $iAPG_{\unicode[STIX]{x1D6FD}_{K}=0.58}$ and $cAPG_{\unicode[STIX]{x1D6FD}_{K}=0.55}$ cases. If incompressible and compressible PG cases are chosen to be characterized by the same kinematic Rotta–Clauser parameter $\unicode[STIX]{x1D6FD}_{K}$ , cases are thus constructed with the same total shear-stress distributions. This in turn implies the Mach-number invariance of $\overline{\unicode[STIX]{x1D70F}}_{t}/\overline{\unicode[STIX]{x1D70F}}_{w}$ for compressible PG cases,

(4.5) $$\begin{eqnarray}\left.\frac{\overline{\unicode[STIX]{x1D70F}}_{t}}{\overline{\unicode[STIX]{x1D70F}}_{w}}\right|_{PG}=f(PG)\neq f(M).\end{eqnarray}$$

As long as differences in the mean shear-stress distributions are not too large, also the turbulent stresses $-\overline{\unicode[STIX]{x1D70C}u^{\prime \prime }v^{\prime \prime }}/\overline{\unicode[STIX]{x1D70F}}_{w}$ are invariant to the Mach number, which further verifies the applicability of Morkovin’s scaling for the present compressible PG cases

(4.6) $$\begin{eqnarray}\left.\frac{-\overline{\unicode[STIX]{x1D70C}u^{\prime \prime }v^{\prime \prime }}}{\overline{\unicode[STIX]{x1D70F}}_{w}}\right|_{PG}\approx \left.\frac{\overline{\unicode[STIX]{x1D70C}}}{\overline{\unicode[STIX]{x1D70C}}_{w}}\frac{-\overline{u^{\prime }v^{\prime }}}{u_{\unicode[STIX]{x1D70F}}^{2}}\right|_{PG}=f(PG)\neq f(M).\end{eqnarray}$$

4.1.2 Pressure distributions

From the shear-stress distributions discussed in the previous section, compressibility and PG effects have been found to be almost uncoupled for the $-\overline{u^{\prime }v^{\prime }}$ fluctuations. If it is further assumed that Morkovin’s scaling also holds for the $\overline{v^{\prime 2}}$ -fluctuations, meaning that $(\overline{\unicode[STIX]{x1D70C}}/\overline{\unicode[STIX]{x1D70C}}_{w})(\overline{v^{\prime 2}}/u_{\unicode[STIX]{x1D70F}}^{2})\neq f(M)$ , the non-dimensionalized $y$ -momentum equation of the two-dimensional turbulent boundary-layer equations

(4.7) $$\begin{eqnarray}\frac{p_{e\approx w}-\overline{p}}{\overline{\unicode[STIX]{x1D70C}}_{w}u_{\unicode[STIX]{x1D70F}}^{2}}=\frac{\overline{\unicode[STIX]{x1D70C}}}{\overline{\unicode[STIX]{x1D70C}}_{w}}\frac{\widetilde{v^{\prime \prime 2}}}{u_{\unicode[STIX]{x1D70F}}^{2}}\neq f\left(M\right)\end{eqnarray}$$

implies that the pressure distributions $(p_{e\approx w}-\overline{p})/(\overline{\unicode[STIX]{x1D70C}}_{w}u_{\unicode[STIX]{x1D70F}}^{2})=(p_{e\approx w}-\overline{p})^{+}$ of corresponding PG cases are comparable for different Mach numbers. (The index ‘ $e\approx w$ ’ should emphasize that the pressure $p$ can be evaluated both at the edge of the boundary layer or at the wall.) If it is also assumed that comparable mean pressure distributions yield comparable pressure fluctuations $p^{\prime +}=\sqrt{\overline{p^{\prime 2}}}/\overline{\unicode[STIX]{x1D70F}}_{w}$ , also their wall-normal distributions should be comparable for corresponding $\unicode[STIX]{x1D6FD}_{K}$ -cases at different Mach numbers.

Figure 18. (a) Mean-flow pressure distribution $(\overline{p}-\overline{p}_{w})^{+}=(\overline{p}-\overline{p}_{w})/\overline{\unicode[STIX]{x1D70F}}_{w}$ at $Re_{\unicode[STIX]{x1D70F}}=490$ and (b) pressure fluctuations $p^{\prime +}=\sqrt{\overline{p^{\prime 2}}}/\overline{\unicode[STIX]{x1D70F}}_{w}$ at $Re_{\unicode[STIX]{x1D70F}}=730$ . ZPG references: Schlatter & Örlü (Reference Schlatter and Örlü2010) at $Re_{\unicode[STIX]{x1D70F}}=830$ (incompressible), Pirozzoli & Bernardini (Reference Pirozzoli and Bernardini2011) at $Re_{\unicode[STIX]{x1D70F}}=840$ ( $M_{\infty }=2.0$ ). Red: ———  $iZPG$ , —— -  $\cdot$   $iAPG_{\unicode[STIX]{x1D6FD}_{K}=0.19}$ , — - —  $iAPG_{\unicode[STIX]{x1D6FD}_{K}=0.58}$ ,  –  $\cdot$  –  $\cdot$  –  $iAPG_{\unicode[STIX]{x1D6FD}_{K}=1.05}$ . Blue: ———  $cZPG$ , ——– –  $cAPG_{\unicode[STIX]{x1D6FD}_{K}=0.15}$ , — — -  $cAPG_{\unicode[STIX]{x1D6FD}_{K}=0.42}$ ,  – – – – –  $cAPG_{\unicode[STIX]{x1D6FD}_{K}=0.55}$ , - - - - - -  $cAPG_{\unicode[STIX]{x1D6FD}_{K}=0.69}$ . Cyan: $-\!-$  - - –  $cFPG_{\unicode[STIX]{x1D6FD}_{K}=-0.18}$ .

The mean-pressure and the pressure-fluctuation distributions are depicted in figure 18(a,b), respectively. The mean-pressure fields are extracted at $Re_{\unicode[STIX]{x1D70F}}=490$ , where data of all cases except for the FPG case are available. Due to the SEM inflow boundary condition, the wall pressure fluctuations are increased by about ten per cent in the inflow region of the simulation domain for the subsonic cases. These initial disturbances have decayed after about half of the simulation domain, such that reliable pressure-fluctuation distributions are only available for the second half of the domain for the subsonic cases. The development of other flow quantities like density or temperature fluctuations are unaffected by the inflow boundary condition, see Wenzel et al. (Reference Wenzel, Selent, Kloker and Rist2018b ). For the supersonic cases, pressure fluctuations are almost immediately comparable to reference data near the inflow of the domain, as the general pressure-fluctuation level is higher inside the supersonic boundary layer. Hence, the pressure-fluctuation profiles given in figure 18(b) are extracted at $Re_{\unicode[STIX]{x1D70F}}=730$ to allow a meaningful comparison. Panel (b) is supplemented by incompressible and compressible ZPG reference data taken from Schlatter & Örlü (Reference Schlatter and Örlü2010) at $Re_{\unicode[STIX]{x1D70F}}=830$ , and Pirozzoli & Bernardini (Reference Pirozzoli and Bernardini2011) at $Re_{\unicode[STIX]{x1D70F}}=840$ and $M_{\infty }=2.0$ , respectively.

For increasing APG strength, the peak of the mean pressure distributions is lifted away from the wall and the profiles become fuller, see (a). The distribution of the subsonic ZPG case is slightly lower compared to the supersonic ZPG case. If evaluated at comparable $\unicode[STIX]{x1D6FD}_{K}$ -values, the distributions are almost identical for the sub- and supersonic APG cases as emphasized by the grey shaded regions. The pressure fluctuations in (b) are virtually unaffected by compressibility effects for the ZPG cases as already shown in Wenzel et al. (Reference Wenzel, Selent, Kloker and Rist2018b ) for instance, yielding an excellent agreement of the subsonic ZPG case and the incompressible and compressible ZPG references. With increasing APG strength, the fluctuation intensity increases for both sub- and supersonic cases but still remains almost identical for cases of comparable $\unicode[STIX]{x1D6FD}_{K}$ , yielding

(4.8) $$\begin{eqnarray}\left.\frac{p^{\prime 2}}{\overline{\unicode[STIX]{x1D70C}}_{w}u_{\unicode[STIX]{x1D70F}}^{2}}\right|_{PG}=f(PG)\neq f(M).\end{eqnarray}$$

4.1.3 Wall-normal velocity fluctuations $\overline{v^{\prime 2}}$

Based on the previous argumentation, the close agreement between the mean-pressure distributions of sub- and supersonic counterparts with similar $\unicode[STIX]{x1D6FD}_{K}$ has been implied by the assumed validity of Morkovin’s scaling for the wall-normal velocity fluctuations $\overline{v^{\prime 2}}$ (or vice versa). To verify the accuracy of the simplified $y$ -momentum equation of the turbulent boundary-layer equations for compressible PG cases (4.7) and thus the success of Morkovin’s scaling for the wall-normal velocity fluctuations $\overline{v^{\prime 2}}$ , these are given both in unscaled $\sqrt{\overline{v^{\prime 2}}}/u_{\unicode[STIX]{x1D70F}}$ and Morkovin-scaled $\sqrt{\overline{\unicode[STIX]{x1D70C}}/\overline{\unicode[STIX]{x1D70C}}_{w}}\sqrt{\overline{v^{\prime 2}}}/u_{\unicode[STIX]{x1D70F}}$ representation in figure 19(a,b), respectively. For better comparability, only sub- and supersonic cases with similar $\unicode[STIX]{x1D6FD}_{K}$ -values are considered. All results are extracted at the three Reynolds-number values $Re_{\unicode[STIX]{x1D70F}}=490$ , 610 and 730, diagonally staggered in ascending order. Represented as circled symbols in (b), some pressure distributions $\sqrt{(\overline{p}_{w}-\overline{p})/(\overline{\unicode[STIX]{x1D70C}}_{w}u_{\unicode[STIX]{x1D70F}}^{2})}$ are additionally given for selected cases.

Figure 19. Comparison of wall-normal Reynolds fluctuations at $Re_{\unicode[STIX]{x1D70F}}=490$ , 610 and 730 in (a) unscaled formulation $v^{\prime +}=\sqrt{\overline{v^{\prime 2}}}/u_{\unicode[STIX]{x1D70F}}$ and (b) density-scaled formulation $v_{M}^{\prime +}=\sqrt{\overline{\unicode[STIX]{x1D70C}}/\overline{\unicode[STIX]{x1D70C}}_{w}}\sqrt{v^{\prime 2}}/u_{\unicode[STIX]{x1D70F}}$ as lines. Filled square symbols in (b) represent the pressure distribution $\sqrt{\overline{p}_{w}-\overline{p}}/(\sqrt{\overline{\unicode[STIX]{x1D70C}}_{w}}u_{\unicode[STIX]{x1D70F}})$ as already given in figure 18(a). Only corresponding cases of comparable $\unicode[STIX]{x1D6FD}_{K}$ -values are given. All red: ———  $iZPG$ , —— -  $\cdot$ $iAPG_{\unicode[STIX]{x1D6FD}_{K}=0.19}$ , — - —  $iAPG_{\unicode[STIX]{x1D6FD}_{K}=0.58}$ . All blue: ———  $cZPG$ , ——– –  $cAPG_{\unicode[STIX]{x1D6FD}_{K}=0.15}$ , – – – – –  $cAPG_{\unicode[STIX]{x1D6FD}_{K}=0.55}$ .

If the pressure distributions and the corresponding Morkovin-scaled velocity fluctuations are compared in (b) first, an almost complete agreement can be established up to the boundary-layer edge $\unicode[STIX]{x1D6FF}_{99}^{+}=Re_{\unicode[STIX]{x1D70F}}$ . Thus, both the validity of (4.7) for compressible PG cases with similar $\unicode[STIX]{x1D6FD}_{K}$ -values and the approximate validity of Morkovin’s scaling for the wall-normal velocity fluctuations $\overline{v^{\prime 2}}$ are implied. If assessed against the untransformed distributions in (a), no complete scaling success can be found for the ZPG cases. While the incompressible distribution is noticeably above the compressible one in the unscaled representation in (a), the incompressible one is noticeably below the compressible one in the density-scaled representation in (b), see also Wenzel et al. (Reference Wenzel, Selent, Kloker and Rist2018b ). In contrast, the scaling success is better for the APG cases, mainly caused by the smaller Mach numbers of the APG cases at the given Reynolds numbers.

4.2.1 Mean-flow velocity profiles $\overline{u}$

The streamwise mean-flow velocity profiles $\overline{u}^{+}$ are given in figures 20 and 21 for all sub- and supersonic cases, respectively, the inner layer on the left, the outer layer on the right. All results are extracted at the same three Reynolds numbers $Re_{\unicode[STIX]{x1D70F}}=490$ , $610$ and $730$ if available, except for the supersonic FPG case extracted at $Re_{\unicode[STIX]{x1D70F}}=360$ . All velocity profiles are located in regions of well-developed $\unicode[STIX]{x1D6FD}_{K}$ -parameter distributions, see figure 10.

In the subsonic cases (figure 20) often reported results can be seen. Whereas the velocity profiles are shifted down to lower $\overline{u}^{+}$ -values in the logarithmic region with increasing $\unicode[STIX]{x1D6FD}_{K}$ , they are lifted in the wake region. Evaluated at $y^{+}\approx 50$ and $y^{+}=\unicode[STIX]{x1D6FF}_{99}^{+}=Re_{\unicode[STIX]{x1D70F}}$ , this shift is almost constant for all given Reynolds numbers.

Mainly due to the varying effect of compressibility in the streamwise direction, the supersonic APG results as given in figure 21 are more difficult to interpret. In the logarithmic region, both the opposing effects of compressibility (lifting for decreasing streamwise Mach numbers) and APGs (shifting down for increasing APG strength) balance each other, yielding the almost same distributions in the logarithmic layer for all APG cases in figure 21(a) (cf. also figure 12 or 17 a). In the wake region where both effects constructively overlap (lifting up for increasing APG strength and decreasing streamwise Mach number), $\overline{u}^{+}$ -values are increased with increasing $\unicode[STIX]{x1D6FD}_{K}$ , see $y^{+}=Re_{\unicode[STIX]{x1D70F}}=\unicode[STIX]{x1D6FF}_{99}^{+}$ in figure 21(b). However, due to the decreasing influence of compressibility in the streamwise direction, this shift is not expected to be constant for increasing Reynolds numbers. An assessment of the supersonic FPG case shows inverted trends.

4.2.2 The van Driest transformation

Figures 22 and 23 illustrate the untransformed and van Driest transformed mean-flow distributions $\overline{u}^{+}$ and $\overline{u}_{VD}^{+}$ (4.2), respectively, both for the subsonic $iAPG_{\unicode[STIX]{x1D6FD}_{K}=0.19}$ and $iAPG_{\unicode[STIX]{x1D6FD}_{K}=0.58}$ cases and the compressible $cAPG_{\unicode[STIX]{x1D6FD}_{K}=0.15}$ and $cAPG_{\unicode[STIX]{x1D6FD}_{K}=0.55}$ cases. Based on the total shear-stress distributions $\overline{\unicode[STIX]{x1D70F}}_{t}$ discussed in the previous section, the compressible cases should coincide with the incompressible ones if the van Driest transformation is successful.

Figure 20. Comparison of subsonic mean-flow velocity profiles $\overline{u}^{+}$ at $Re_{\unicode[STIX]{x1D70F}}=490$ , $610$ and $730$ . Black dotted lines denote the viscous sublayer $\overline{u}^{+}=y^{+}$ , dashed black lines the logarithmic region $\overline{u}^{+}=1/k\,\ln y^{+}+C$ with $k=0.41$ and $C=5.2$ . All red:  ———  $iZPG$ , —— -  $\cdot$   $iAPG_{\unicode[STIX]{x1D6FD}_{K}=0.19}$ , — - —  $iAPG_{\unicode[STIX]{x1D6FD}_{K}=0.58}$ , —— -  $\cdot$   $iAPG_{\unicode[STIX]{x1D6FD}_{K}=1.05}$ .

Figure 21. Comparison of supersonic mean-flow velocity profiles $\overline{u}^{+}$ at $Re_{\unicode[STIX]{x1D70F}}=490$ , $610$ and $730$ for ZPG/APG and $Re_{\unicode[STIX]{x1D70F}}=360$ for FPG cases. Further information are given in figure 20. All blue: ———  $cZPG$ , ——– –  $cAPG_{\unicode[STIX]{x1D6FD}_{K}=0.15}$ , — — -  $cAPG_{\unicode[STIX]{x1D6FD}_{K}=0.42}$ , – – – – –  $cAPG_{\unicode[STIX]{x1D6FD}_{K}=0.55}$ , - - - - - -  $cAPG_{\unicode[STIX]{x1D6FD}_{K}=0.69}$ . Cyan: $-\!-$  - - –  $cFPG_{\unicode[STIX]{x1D6FD}=-0.35}$ .

Figure 22. Comparison of sub- and supersonic mean-flow velocity profiles $\overline{u}^{+}$ at $Re_{\unicode[STIX]{x1D70F}}=490$ , $610$ and $730$ . Black dotted lines denote the viscous sublayer $\overline{u}^{+}=y^{+}$ , dashed black lines the logarithmic region $\overline{u}^{+}=1/k\,\ln y^{+}+C$ with $k=0.41$ and $C=5.2$ . Only corresponding cases of comparable $\unicode[STIX]{x1D6FD}_{K}$ -values are given. All red: ———  $iZPG$ , —— -  $\cdot$   $iAPG_{\unicode[STIX]{x1D6FD}_{K}=0.19}$ , — - —  $iAPG_{\unicode[STIX]{x1D6FD}_{K}=0.58}$ . All blue: ———  $cZPG$ , ——– –  $cAPG_{\unicode[STIX]{x1D6FD}_{K}=0.15}$ , - - - - - -  $cAPG_{\unicode[STIX]{x1D6FD}_{K}=0.55}$ .

Figure 23. Same as figure 22, but for van Driest transformed profiles  $\overline{u}_{VD}^{+}$ .

For the inner layer, figure 23(a), the supersonic velocity profiles $\overline{u}_{VD}^{+}$ are almost perfectly shifted into their subsonic counterparts, compare figures 22(a) and 23(a). In the wake region for $y^{+}\gtrapprox 100$ (figure 23 b), the subsonic $iAPG_{\unicode[STIX]{x1D6FD}_{K}=0.19}$ and supersonic $cAPG_{\unicode[STIX]{x1D6FD}_{K}=0.15}$ cases are almost identical, whereas the supersonic $cAPG_{\unicode[STIX]{x1D6FD}_{K}=0.55}$ case lies noticeably below the subsonic $iAPG_{\unicode[STIX]{x1D6FD}_{K}=0.58}$ case. In the ZPG case, however, the van Driest transformed distributions lie above the incompressible references if compared at the same $Re_{\unicode[STIX]{x1D70F}}$ , see Wenzel et al. (Reference Wenzel, Selent, Kloker and Rist2018b ). Consequently, the wake distribution of the van Driest transformed velocity profiles is increasingly underestimated with increasing APG strength. For the weak $cAPG_{\unicode[STIX]{x1D6FD}_{K}=0.15}$ case, the underestimating effect by the APG balances the overestimating effect by compressibility, whereas it is overcompensated for the stronger $cAPG_{\unicode[STIX]{x1D6FD}_{K}=0.55}$ case. Thus, with increasing $Re_{\unicode[STIX]{x1D70F}}$ and thus decreased Mach number in APG flows, also the success of van Driest’s transformation varies. In some contrast to the ZPG study however, the van Driest transformation can be denoted to be still approximately valid for the wake region of the present PG cases, since compressibility effects are significantly reduced, although not being completely excluded.

Figure 24. Comparison of subsonic streamwise Reynolds fluctuations $u^{\prime +}=\sqrt{\overline{u^{\prime 2}}}/u_{\unicode[STIX]{x1D70F}}$ at $Re_{\unicode[STIX]{x1D70F}}=490$ , $610$ and $730$ . Solid lines represent the ZPG case, non-solid lines APG cases. Arrows denote the direction of increasing APG strength $\unicode[STIX]{x1D6FD}_{K}$ . Red: ——— $iZPG$ , —— -  $\cdot$   $iAPG_{\unicode[STIX]{x1D6FD}_{K}=0.19}$ , — - —  $iAPG_{\unicode[STIX]{x1D6FD}_{K}=0.58}$ , –  $\cdot$  –  $\cdot$  –  $iAPG_{\unicode[STIX]{x1D6FD}_{K}=1.05}$ .

Figure 25. Comparison of supersonic streamwise Reynolds fluctuations $u^{\prime +}=\sqrt{\overline{u^{\prime 2}}}/u_{\unicode[STIX]{x1D70F}}$ at $Re_{\unicode[STIX]{x1D70F}}=490$ , $610$ and $730$ for ZPG/APG and $Re_{\unicode[STIX]{x1D70F}}=360$ for FPG cases. Blue: ———  $cZPG$ , ——– –  $cAPG_{\unicode[STIX]{x1D6FD}_{K}=0.15}$ , — — -  $cAPG_{\unicode[STIX]{x1D6FD}_{K}=0.42}$ , – – – – –  $cAPG_{\unicode[STIX]{x1D6FD}_{K}=0.55}$ , - - - - - - $cAPG_{\unicode[STIX]{x1D6FD}_{K}=0.69}$ . Cyan: $-\!-$  - - –  $cFPG_{\unicode[STIX]{x1D6FD}=-0.35}$ .

Figure 26. Comparison of sub- and supersonic streamwise Reynolds fluctuations $u^{\prime +}=\sqrt{\overline{u^{\prime 2}}}/u_{\unicode[STIX]{x1D70F}}$ at $Re_{\unicode[STIX]{x1D70F}}=490$ , $610$ and $730$ . Only corresponding cases of comparable $\unicode[STIX]{x1D6FD}_{K}$ -values are given. All red: —— -  $\cdot$   $iAPG_{\unicode[STIX]{x1D6FD}_{K}=0.19}$ , — - —  $iAPG_{\unicode[STIX]{x1D6FD}_{K}=0.58}$ . All blue: ——– –  $cAPG_{\unicode[STIX]{x1D6FD}_{K}=0.15}$ , - - - - - -  $cAPG_{\unicode[STIX]{x1D6FD}_{K}=0.55}$ .

Figure 27. Same as figure 26, but for density-scaled streamwise Reynolds fluctuations, $u_{M}^{\prime +}=\sqrt{\overline{\unicode[STIX]{x1D70C}}/\overline{\unicode[STIX]{x1D70C}}_{w}}\sqrt{\overline{u^{\prime 2}}}/u_{\unicode[STIX]{x1D70F}}$ .

4.2.3 Reynolds fluctuations $\overline{u^{\prime 2}}$

In the same representation as for the mean velocity profiles, the streamwise Reynolds-averaged velocity fluctuations $u^{\prime +}=\sqrt{\overline{u^{\prime 2}}}/u_{\unicode[STIX]{x1D70F}}$ are given in figures 24 and 25 separately for the sub- and supersonic cases, respectively.

At first and in extension of figure 16, the inner layer of the Reynolds fluctuations is revisited in figures 24(a) and 25(a). In comparison to the compressible ZPG cases (Wenzel et al. Reference Wenzel, Selent, Kloker and Rist2018b ), the Reynolds fluctuation profiles do not match for different PG cases near the wall for $y^{+}\lessapprox 10$ . The distributions are increased for APGs and decreased for FPGs causing higher and lower values for $(\unicode[STIX]{x2202}u^{\prime +}/\unicode[STIX]{x2202}y^{+})|_{w}$ , respectively, see figure 16(c). The position of the near-wall fluctuation maximum is increased for APGs and decreased for FPGs as already has been seen in figure 16(a,b). However, in combination with the effect of varying Mach number in streamwise direction, the interpretation of the peak’s absolute value is again complex for the supersonic cases. Consistently given for both the sub- and supersonic cases in figures 24(b) and 25(b), the well-known second maximum appears in the outer region of the boundary layer, see Skote et al. (Reference Skote, Henningson and Henkes1998) and Kitsios et al. (Reference Kitsios, Sekimoto, Atkinson, Sillero, Borrell, Gungor, Jiménez and Soria2017), for instance.

4.2.4 Morkovin scaling

Both in its unscaled $u^{\prime +}=\sqrt{\overline{u^{\prime 2}}}/u_{\unicode[STIX]{x1D70F}}$ and Morkovin-scaled representation $u_{M}^{\prime +}=\sqrt{\overline{\unicode[STIX]{x1D70C}}/\overline{\unicode[STIX]{x1D70C}}_{w}}\sqrt{\overline{u^{\prime 2}}}/u_{\unicode[STIX]{x1D70F}}$ , the Reynolds fluctuations are depicted in figures 26 and 27, respectively. As for the mean-flow distributions, only the subsonic $iAPG_{\unicode[STIX]{x1D6FD}_{K}=0.19}$ and $iAPG_{\unicode[STIX]{x1D6FD}_{K}=0.58}$ cases and the compressible $cAPG_{\unicode[STIX]{x1D6FD}_{K}=0.15}$ and $cAPG_{\unicode[STIX]{x1D6FD}_{K}=0.55}$ cases are depicted, for which the PG influence is comparable.

If compared in the unscaled representation in figure 26, both the inner and the outer peaks of the supersonic cases are reduced compared to the subsonic results. In the Morkovin-scaled representation in figure 27, both the inner and the outer peaks of the supersonic cases are slightly lifted above the subsonic results. It is noted, however, that this behaviour has been also observed in the ZPG study to a similar extent, see Wenzel et al. (Reference Wenzel, Selent, Kloker and Rist2018b ), and is therefore much more attributed to compressibility than to PG effects. Morkovin’s scaling thus also can be confirmed for the present PG cases to a comparable extent as for the ZPG cases, which further implies the Mach-number invariance of the non-dimensionalized streamwise $\overline{\unicode[STIX]{x1D70C}u^{\prime \prime 2}}/\overline{\unicode[STIX]{x1D70F}}_{w}$ stress

(4.9) $$\begin{eqnarray}\left.\frac{\overline{\unicode[STIX]{x1D70C}u^{\prime \prime 2}}}{\overline{\unicode[STIX]{x1D70F}}_{w}}\right|_{PG}\approx \left.\frac{\overline{\unicode[STIX]{x1D70C}}}{\overline{\unicode[STIX]{x1D70C}}_{w}}\frac{\overline{u^{\prime 2}}}{u_{\unicode[STIX]{x1D70F}}^{2}}\right|_{PG}=f(PG)\neq f(M).\end{eqnarray}$$