Derive a generalized storage-capacity law

Derive a robust generalized mathematical formulation for the storage capacity \(M\) of the profile-concatenation algorithm across arbitrary Reynolds numbers and wall-normal degrees of freedom.

Background

The storage capacity MM controls the candidate pool used to sort and concatenate one-dimensional step-like velocity profiles and therefore governs their effective streamwise spacing and the coherence of the synthesized two-dimensional field. Earlier work proposed an empirical dependence on friction Reynolds number, while the present paper finds that extending the wall-normal domain introduces an additional dependence on wall-normal degrees of freedom.

The paper reports that optimal values of MM were mapped empirically but that no mathematically closed relationship valid across arbitrary Reynolds numbers and wall-normal extents has been obtained. Establishing such a formulation would provide a systematic basis for selecting MM and avoid both excessive smoothing and insufficient spatial coherence.

References

A robust, generalized formulation for the model storage capacity M, which fundamentally governs the corresponding streamwise spacing between the concatenated 1-D velocity profiles, has not yet been mathematically closed across arbitrary Reynolds numbers and wall-normal degrees of freedom. Resolving these two limitations requires a systematic refinement of the profile concatenation methodology.

— A stochastic modeling framework to generate 2-D rough-wall high-Reynolds-number turbulent boundary layers  (2609.10236 - Ehsani et al., 9 Sep 2026) in Section 6, “Limitations,” item 6