A simple model of axisymmetric turbulent boundary layers along long thin circular cylinders

Abstract

Useful empirical and semi-empirical models of the turbulent boundary layer (TBL) and skin friction evolution along planar geometries are not applicable for axisymmetric thin cylinder flows. Their dissimilarity is readily detectable once the TBL thickness exceeds the cylinder radius (a). Although several recent empirically based axisymmetric models recognize this fact, their acceptable fidelity is either restrictive or deficient for general applicability. Herein, we correct this deficit by building a simple model for the specific canonical class of axisymmetric turbulent flows along long thin cylinders with a zero streamwise pressure gradient. Streamwise growth of the TBL thickness (δ/a), integral scales [displacement (δ*/a) and momentum thicknesses (θ/a)] and skin friction coefficient (Cf) can be estimated along the cylinder length via the respective axial mean velocity profile in wall units. This profile is given by Spalding's formula with algebraic expressions for the two input parameters (κ, κβ) that cover all turbulent Reynolds numbers. The necessary database for empirically tuning Spalding's parameters entails both experimental measurements and new numerical computations. Our present-day understanding of the axisymmetric TBL is replicated by the simple model where δ/a, δ*/a, and θ/a grow slower than the planar-type flow with Cf comparatively elevating once δ/a > O(1). These differences manifest themselves in the radial impact imposed by the thin cylinder transverse curvature. Interestingly, the axial-based Reynolds numbers Rea ≈ 7500 and a+ ≈ 350 at δ/a ≈ 21 mark earliest signs of a homogeneous streamwise state (constant Cf) near the cylinder wall. Owning a simple model of axisymmetric turbulent flows along thin cylinders eliminates expensive and timely experiments and/or computations. Its practicality targets both the Naval and oceanographic communities.