Exact mathematical formulas for wall-heat flux in compressible turbulent channel flows

The impact of bulk viscosity is unclear with considering the increased dilatational dissipation and compressibility effects in hypersonic turbulence flows. In this study, we employ direct numerical simulations to conduct comprehensive analysis of the effect of bulk viscosity on hypersonic turbulent boundary layer flow over a flat plate. The results demonstrate that the scaling relations remain valid even when accounting for large bulk viscosity. However, the wall-normal velocity fluctuations $v_{rms}^{\prime \prime }$ decrease significantly in the viscous sublayer due to the enhanced bulk dilatational dissipation. The intensity of travelling-wave-like alternating positive and negative structures of instantaneous pressure fluctuations $p_{rms}^{\prime }$ in the near-wall region decreases distinctly after considering the bulk viscosity, which is attributed mainly to the reduction of compressible pressure fluctuations $p_{c,rms}^{+}$ . Furthermore, the velocity divergence $\partial u_{i} / \partial x_{i}$ undergoes a significant decrease by bulk viscosity. In short, our results indicate that bulk viscosity can weaken the compressibility of the hypersonic turbulent boundary layer and becomes more significant as the Mach number increases and the wall temperature decreases. Notably, when the bulk-to-shear viscosity ratio of the gas reaches a few hundred levels ( $\mu _b/\mu =O(10^2)$ ), and mechanical behaviour of the near-wall region ( $\kern 0.06em y^+\le 30$ ) is of greater interest, the impact of bulk viscosity on the hypersonic cold-wall turbulent boundary layer may not be negligible.

Section: Velocity-related Statisticsmentioning

confidence: 99%

Effect of bulk viscosity on the hypersonic compressible turbulent boundary layer

Zheng,

Feng,

Zheng

2024

“…Now we investigate the wall heat flux by using integral analysis (Zhang & Xia 2020; Sun et al. 2021; Zhang, Song & Xia 2021; Wenzel, Gibis & Kloker 2022; Xu, Wang & Chen 2022). In the present study, we start with the internal energy equation (Mittal & Girimaji 2019), and its Reynolds-averaged form can be expressed as (see also (4.10)) Here, and , where the four terms are the energy transfer/exchange terms between and and between and in § 4.…”

Section: Analysis Of Wall Heat Transfermentioning

confidence: 99%

Direct numerical simulations of compressible turbulent channel flows with asymmetric thermal walls

Zhang,

Song,

Xia

2024

Self Cite

This paper extends the work of Tamano & Morinishi (J. Fluid Mech., vol. 548, 2006, pp. 361–373) by simulating supersonic turbulent channel flow with asymmetric thermal walls using a larger computational domain and a finer mesh. Direct numerical simulation is carried out for four cases with different thermal wall boundaries at the top wall at fixed $Ma=1.5$ , $Re=6000$ and $Pr=0.72$ , while the bottom wall is maintained at a constant temperature of $T_L$ equal to the reference temperature. These cases are referred to as the adiabatic case TAd, where the top wall is adiabatic; the pseudo-adiabatic case T32, where the top wall is isothermal with temperature $T_{w,t}=T_A$ ; the sub-adiabatic case T25, with $T_{w,t}=0.77T_A$ ; and the super-adiabatic case T40, with $T_{w,t}=1.24T_A$ . Here, $T_A=3.234$ is the mean temperature at the adiabatic wall in the TAd case. The objective of this study is to compare and contrast the TAd case with its corresponding T32 case, and to investigate the effect of the wall temperature difference between the two isothermal walls. Comparisons of the basic turbulent statistics, the heat transfer between the Favre-averaged mean-flow kinetic energy, the Favre-averaged turbulent kinetic energy and the Favre-averaged mean internal energy, as well as the wall heat transfer properties, indicate that the TAd case and its corresponding T32 case are generally equivalent. The only discernible difference is in the region very close to the top wall for the temperature-fluctuation-related quantities. The analysis reveals that the asymmetry of the thermal walls causes asymmetry in the flow and thermal fields. In addition, the transfer of the heat generated by the pressure dilatation and the viscous stress is facilitated by the turbulent heat flux term and the mean molecular heat flux term.

“…2015; Griffin, Fu & Moin 2021; Huang, Duan & Choudhari 2022; Cheng et al. 2024). The magnitude of should be high enough to obtain a discernible logarithmic region, which is a primary condition for investigating the multiscale eddies and their interactions in depth.…”

Section: Introductionmentioning

confidence: 99%

“…However, the semilocal friction Reynolds numbers Re * τ of their cases are not high (Re * τ = Re τ √ ( ρc / ρw )/( μc / μw ); ρ is the density, μ the dynamic viscosity; the subscript c refers to the quantities evaluated at the channel centre; • denotes the average value). We take special care of the Re * τ of the simulations because previous studies indicate that Re * τ can reasonably clarify the Reynolds-number effects on the statistics involving the thermodynamic and the velocity variables in compressible channel flows (Gerolymos & Vallet 2014;Patel et al 2015;Griffin, Fu & Moin 2021;Huang, Duan & Choudhari 2022;Cheng et al 2024). The magnitude of Re * τ should be high enough to obtain a discernible logarithmic region, which is a primary condition for investigating the multiscale eddies and their interactions in depth.…”

Section: Introductionmentioning

confidence: 99%

On the streamwise velocity, temperature and passive scalar fields in compressible turbulent channel flows: a viewpoint from multiphysics couplings

Cheng,

2024

It is generally believed that the velocity and passive scalar fields share many similarities and differences in wall-bounded turbulence. In the present study, we conduct a series of direct numerical simulations of compressible channel flows with passive scalars and employ the two-dimensional spectral linear stochastic estimation and the correlation function as diagnostic tools to shed light on these aspects. Particular attention is paid to the relevant multiphysics couplings in the spectral domain, i.e. the velocity–temperature ( $u-T$ ), scalar–temperature ( $g-T$ ) and velocity–scalar ( $u-g$ ) couplings. These couplings are found to be utterly different at a given wall-normal position in the logarithmic and outer regions. Specifically, in the logarithmic region, the $u-T$ and $u-g$ couplings are tight at the scales that correspond to the attached eddies and the very large-scale motions (VLSMs), whereas the $g-T$ coupling is robust in the whole spectral domain. In the outer region, $u-T$ and $u-g$ couplings are only active at the scales corresponding to the VLSMs, whereas the $g-T$ coupling is diminished but still strong at all scales. Further analysis indicates that although the temperature field in the vast majority of zones in a channel can be roughly treated as a passive scalar, its physical properties gradually deviate from those of a pure passive scalar as the wall-normal height increases due to the enhancement of the acoustic mode. Furthermore, the deep involvement of the pressure field in the self-sustaining process of energy-containing motions also drives the streamwise velocity fluctuation away from a passive scalar. The current work is an extension of our previous study (Cheng & Fu, J. Fluid Mech., vol. 964, 2023, A15), and further uncovers the details of the multiphysics couplings in compressible wall turbulence.