## Abstract

A semi-empirical model is presented that describes the development of a fully developed turbulent boundary layer in the presence of surface roughness with length scale k_{s} that varies with streamwise distance x. Interest is centred on flows for which all terms of the von Kármán integral relation, including the ratio of outer velocity to friction velocity U^{+}_{∞} ≡ U_{∞}/u_{ζ}, are streamwise constant. For Rex assumed large, use is made of a simple log-wake model of the local turbulent mean-velocity profile that contains a standard mean-velocity correction for the asymptotic fully rough regime and with assumed constant parameter values. It is then shown that, for a general power-law external velocity variation U_{∞} ∼ x^{m}, all measures of the boundary-layer thickness must be proportional to x and that the surface sand-grain roughness scale variation must be the linear form k_{s}(x) = αx, where x is the distance from the boundary layer of zero thickness and is a dimensionless constant. This is shown to give a two-parameter (m,α) family of solutions, for which U^{+}_{∞} (or equivalently C_{f}) and boundary-layer thicknesses can be simply calculated. These correspond to perfectly self-similar boundary-layer growth in the streamwise direction with similarity variable z/(αx), where z is the wall-normal coordinate. Results from this model over a range of α are discussed for several cases, including the zero-pressure-gradient (m = 0) and sink-flow (m =-1) boundary layers. Trends observed in the model are supported by wall-modelled large-eddy simulation of the zero-pressure-gradient case for Re_{x} in the range 10^{8}-10^{10} and for four values of α. Linear streamwise growth of the displacement, momentum and nominal boundary-layer thicknesses is confirmed, while, for each α, the mean-velocity profiles and streamwise turbulent variances are found to collapse reasonably well onto z/(αx). For given α, calculations of U^{+}_{∞} obtained from large-eddy simulations are streamwise constant and independent of Re_{x} when this is large. The present results suggest that, in the sense that U^{+}_{∞} (α, m) is constant, these flows can be interpreted as the fully rough limit for boundary layers in the presence of small-scale linear roughness.

Original language | English (US) |
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Pages (from-to) | 26-45 |

Number of pages | 20 |

Journal | Journal of Fluid Mechanics |

Volume | 818 |

DOIs | |

State | Published - May 10 2017 |

### Bibliographical note

Publisher Copyright:© 2017 Cambridge University Press.

## Keywords

- turbulence modelling
- turbulence simulation
- turbulent boundary layers

## ASJC Scopus subject areas

- Condensed Matter Physics
- Mechanics of Materials
- Mechanical Engineering
- Applied Mathematics