Self-similar compressible turbulent boundary layers with pressure gradients. Part 2. Self-similarity analysis of the outer layer

Abstract

A thorough self-similarity analysis is presented to investigate the properties of self-similarity for the outer layer of compressible turbulent boundary layers. The results are validated using the compressible and quasi-incompressible direct numerical simulation (DNS) data shown and discussed in the first part of this study; see Wenzel et al. (J. Fluid Mech., vol. 880, 2019, pp. 239–283). The analysis is carried out for a general set of characteristic scales, and conditions are derived which have to be fulfilled by these sets in case of self-similarity. To evaluate the main findings derived, four sets of characteristic scales are proposed and tested. These represent compressible extensions of the incompressible edge scaling, friction scaling, Zagarola–Smits scaling and a newly defined Rotta–Clauser scaling. Their scaling success is assessed by checking the collapse of flow-field profiles extracted at various streamwise positions, being normalized by the respective scales. For a good set of scales, most conditions derived in the analysis are fulfilled. As suggested by the data investigated, approximate self-similarity can be achieved for the mean-flow distributions of the velocity, mass flux and total enthalpy and the turbulent terms. Self-similarity thus can be stated to be achievable to a very high degree in the compressible regime. Revealed by the analysis and confirmed by the DNS data, this state is predicted by the compressible pressure-gradient boundary-layer growth parameter $\unicode[STIX]{x1D6EC}_{c}$, which is similar to the incompressible one found by related incompressible studies. Using appropriate adaption, $\unicode[STIX]{x1D6EC}_{c}$ values become comparable for compressible and incompressible pressure-gradient cases with similar wall-normal shear-stress distributions. The Rotta–Clauser parameter in its traditional form $\unicode[STIX]{x1D6FD}_{K}=(\unicode[STIX]{x1D6FF}_{K}^{\ast }/\bar{\unicode[STIX]{x1D70F}}_{w})(\text{d}p_{e}/\text{d}x)$ with the kinematic (incompressible) displacement thickness $\unicode[STIX]{x1D6FF}_{K}^{\ast }$ is shown to be a valid parameter of the form $\unicode[STIX]{x1D6EC}_{c}$ and hence still is a good indicator for equilibrium flow in the compressible regime at the finite Reynolds numbers considered. Furthermore, the analysis reveals that the often neglected derivative of the length scale, $\text{d}L_{0}/\text{d}x$, can be incorporated, which was found to have an important influence on the scaling success of common ‘low-Reynolds-number’ DNS data; this holds for both incompressible and compressible flow. Especially for the scaling of the $\bar{\unicode[STIX]{x1D70C}}\widetilde{u^{\prime \prime }v^{\prime \prime }}$ stress and thus also the wall shear stress $\bar{\unicode[STIX]{x1D70F}}_{w}$, the inclusion of $\text{d}L_{0}/\text{d}x$ leads to palpable improvements.

Self-similar compressible turbulent boundary layers with pressure gradients. Part 1. Direct numerical simulation and assessment of Morkovin’s hypothesis

“…Both the kinematic and the compressible results of the Rotta-Clauser parameter β (K) = (δ * (K) /τ w )(dp e /dx) are shown in figure 10, where the kinematic displacement thickness is δ * K = δ 99 0 (1 − u/u e ) dy and the compressible displacement thickness δ * = δ 99 0 (1 − (ρ u)/(ρ e u e )) dy. It is shown in Part 2 of this study (Gibis et al 2019) and by the DNS results discussed below that the β 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 β 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 FIGURE 10.…”

“…The equilibrium character of the investigated flows allows for a direct comparison of sub-and supersonic cases. It turned out that the kinematic Rotta-Clauser parameter β K built by the incompressible boundary-layer displacement thickness as the length scale is the right similarity parameter, see also Gibis et al (2019). Thus, by regarding both incompressible and compressibility transformed compressible counterparts, the isolated effects of continuous PGs can be evaluated, and the validity of the most common compressibility transformations scrutinized.…”

“…Therefore, all cases should be characterized by long distances of a constant kinematic Rotta-Clauser parameter β K = (δ * K /τ w )(dp e /dx), where δ * K is the kinematic displacement thickness (see § 3.2.1), τ w the mean wall shear stress and dp e /dx the pressure gradient in streamwise direction evaluated at the edge of the boundary layer. It was found that those regions still yield the state of approximate streamwise self-similarity in the compressible regime, see § 4.1.1 and Gibis et al (2019) for a detailed discussion. The use of both sub-and supersonic inflow Mach numbers should further allow for a meaningful comparison between the incompressible/subsonic and the compressible/supersonic regimes.…”

“…The present investigations are therefore subdivided into two parts, the present first part which is mainly focused on the DNS results, and a second part which is mainly focused on the derivation/validation of compressible self-similarity, see Gibis et al. (2019).…”

“…https://doi.org/10.1017/jfm.2019.670 DNS of self-similar compressible TBLs with pressure gradients 255 online) Snapshots of vortices by iso-surfaces of the λ 2 -criterion with λ 2 = −0.05u e (x)/u ∞,0 for (a) the supersonic cFPG βK =−0.18 case, (b) the supersonic cZPG case and (c) the strongest supersonic cAPG βK =0.69 case at Re τ = 490 (FPG at Re τ = 360). Colour depicts the fluctuation amplitude of the streamwise velocity component u /u ∞,0 in the same representation as in figure 8, see also figures 29 and 30.layer, as shown inGibis et al (2019). Since all simulations start with a ZPG region at the inflow (see § 2.4), an induction distance is needed before the individual β (K) -values become approximately constant (grey dashed lines…”