Mapping-based exact-integral formulation of skin-friction transformations for zero-pressure-gradient compressible turbulent boundary layers
Abstract
A long-standing route to efficient surface-drag prediction in zero-pressure-gradient compressible turbulent boundary layers is to map the skin-friction coefficient Cf and momentum-thickness Reynolds number Reθ onto their `incompressible' counterparts. Reassessment against an extensive DNS database shows that existing formulations do not consistently recover the reference incompressible skin-friction behaviour, even when transformed data exhibit improved collapse. We define the mapped `incompressible' state as a constant-property counterpart of the physical compressible boundary layer and derive the transformation factors from prescribed mean-velocity and wall-normal-coordinate mappings. This definition-first approach links skin-friction scaling to the full-layer accuracy of the underlying velocity transformation and exposes inherited outer-layer errors. Van Driest's theory is recast in a finite-Re exact-integral form, with the classical vD I and II transformations recovered as leading-order asymptotic reductions. Their limitations at finite Reynolds numbers are quantified, and the historical success of vD II is traced to a fortuitous cancellation of truncation errors. The exact-integral formulation then yields modified transformations assessed through a priori scaling and standalone a posteriori prediction of Cf from prescribed macroscopic and wall-thermal inputs. The VIPL-based modified transformation gives the best overall performance. Across 0.30 ≤ M∞ ≤ 13.64 and −0.55 ≤ varTheta ≤ 2.85, its prediction errors remain below 11%, with a mean error of 3.07%. Overall, the analysis places skin-friction transformations on a mapping-based exact-integral footing, relating them directly to prescribed mean-flow mappings while avoiding the leading-order asymptotic truncations that limit classical van Driest theory at finite Reynolds numbers.
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BibTeX
@misc{zhu2026mappingbased,
title = {{Mapping-based exact-integral formulation of skin-friction transformations for zero-pressure-gradient compressible turbulent boundary layers}},
author = {Xuke Zhu and Xiaoshuo Yang and Yongchao Ji and Shiyi Chen and Zhenhua Xia},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.02650},
eprint = {2609.02650},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.02650},
}
RIS
TY - UNPB
TI - Mapping-based exact-integral formulation of skin-friction transformations for zero-pressure-gradient compressible turbulent boundary layers
AU - Xuke Zhu
AU - Xiaoshuo Yang
AU - Yongchao Ji
AU - Shiyi Chen
AU - Zhenhua Xia
PY - 2026
DA - 2026/09/02
PB - arXiv
DO - 10.48550/arxiv.2609.02650
UR - https://arxiv.org/abs/2609.02650
AB - A long-standing route to efficient surface-drag prediction in zero-pressure-gradient compressible turbulent boundary layers is to map the skin-friction coefficient C_f and momentum-thickness Reynolds number Re_θ onto their `incompressible' counterparts. Reassessment against an extensive DNS database shows that existing formulations do not consistently recover the reference incompressible skin-friction behaviour, even when transformed data exhibit improved collapse. We define the mapped `incompressible' state as a constant-property counterpart of the physical compressible boundary layer and derive the transformation factors from prescribed mean-velocity and wall-normal-coordinate mappings. This definition-first approach links skin-friction scaling to the full-layer accuracy of the underlying velocity transformation and exposes inherited outer-layer errors. Van Driest's theory is recast in a finite-Re exact-integral form, with the classical vD I and II transformations recovered as leading-order asymptotic reductions. Their limitations at finite Reynolds numbers are quantified, and the historical success of vD II is traced to a fortuitous cancellation of truncation errors. The exact-integral formulation then yields modified transformations assessed through a priori scaling and standalone a posteriori prediction of C_f from prescribed macroscopic and wall-thermal inputs. The VIPL-based modified transformation gives the best overall performance. Across 0.30 ≤ M_∞ ≤ 13.64 and −0.55 ≤ varTheta ≤ 2.85, its prediction errors remain below 11%, with a mean error of 3.07%. Overall, the analysis places skin-friction transformations on a mapping-based exact-integral footing, relating them directly to prescribed mean-flow mappings while avoiding the leading-order asymptotic truncations that limit classical van Driest theory at finite Reynolds numbers.
ER -