### Abstract

Double-averaging methodology is very convenient for investigating spatially heterogeneous flows such as boundary-layer flows over permeable walls. However, spatial averaging volumes suitable for boundary-layer flows over rough walls are very thin in the wall-normal direction whereas those for porous-media flows usually have similar length in all three directions. This scale mismatch can be addressed by allowing the averaging volume to vary in space so that its size can be adjusted to the physical characteristics of particular flow regions. This paper presents a new spatial averaging theorem derived for a spatially variable averaging volume that may contain a stationary solid phase and that may also extend beyond the boundary of the problem domain. The theorem provides the expression for the difference between the average of a spatial derivative and the derivative of the spatial average (of a general flow quantity), here named the commutation correction (CC) term. The CC term contains three parts accounting for (1) the presence of the solid phase in the averaging volume, (2) the averaging volume extending beyond the flow-domain boundary, and (3) the spatial variation of the averaging volume. The first two parts have been acknowledged in the literature; the third part is introduced in this paper and named the volume variation (VV) term. The averaging theorem is used to derive large-scale continuity and momentum equations, which contain the new VV term. The equations are applied to steady, uniform, microscopically two-dimensional flow over a permeable wall. The averaging volume for the boundary-layer flow above the wall is a thin wall-parallel layer; on crossing the wall surface, the volume grows until its height reaches the size required for the homogeneous porous layer. The averaging procedure and the magnitude of the VV term are illustrated by an example adopted from the literature that involves numerical simulation of two-dimensional open-channel flow over a bundle of circular cylinders.

Original language | English |
---|---|

Article number | 04014087 |

Number of pages | 11 |

Journal | Journal of Hydraulic Engineering |

Volume | 141 |

Issue number | 4 |

Early online date | 4 Dec 2014 |

DOIs | |

Publication status | Published - 1 Apr 2015 |

### Fingerprint

### Keywords

- Double-averaging
- Extended spatial averaging theorem
- Fluid-porous interface
- Variable averaging volume

### ASJC Scopus subject areas

- Water Science and Technology
- Civil and Structural Engineering
- Mechanical Engineering

### Cite this

*Journal of Hydraulic Engineering*,

*141*(4), [04014087]. https://doi.org/10.1061/(ASCE)HY.1943-7900.0000972

**Spatial averaging over a variable volume and its application to boundary-layer flows over permeable walls.** / Pokrajac, D.; de Lemos, M. J S.

Research output: Contribution to journal › Article

*Journal of Hydraulic Engineering*, vol. 141, no. 4, 04014087. https://doi.org/10.1061/(ASCE)HY.1943-7900.0000972

}

TY - JOUR

T1 - Spatial averaging over a variable volume and its application to boundary-layer flows over permeable walls

AU - Pokrajac, D.

AU - de Lemos, M. J S

N1 - The authors are grateful to the two anonymous reviewers for their thorough examination of the paper and their constructive comments.

PY - 2015/4/1

Y1 - 2015/4/1

N2 - Double-averaging methodology is very convenient for investigating spatially heterogeneous flows such as boundary-layer flows over permeable walls. However, spatial averaging volumes suitable for boundary-layer flows over rough walls are very thin in the wall-normal direction whereas those for porous-media flows usually have similar length in all three directions. This scale mismatch can be addressed by allowing the averaging volume to vary in space so that its size can be adjusted to the physical characteristics of particular flow regions. This paper presents a new spatial averaging theorem derived for a spatially variable averaging volume that may contain a stationary solid phase and that may also extend beyond the boundary of the problem domain. The theorem provides the expression for the difference between the average of a spatial derivative and the derivative of the spatial average (of a general flow quantity), here named the commutation correction (CC) term. The CC term contains three parts accounting for (1) the presence of the solid phase in the averaging volume, (2) the averaging volume extending beyond the flow-domain boundary, and (3) the spatial variation of the averaging volume. The first two parts have been acknowledged in the literature; the third part is introduced in this paper and named the volume variation (VV) term. The averaging theorem is used to derive large-scale continuity and momentum equations, which contain the new VV term. The equations are applied to steady, uniform, microscopically two-dimensional flow over a permeable wall. The averaging volume for the boundary-layer flow above the wall is a thin wall-parallel layer; on crossing the wall surface, the volume grows until its height reaches the size required for the homogeneous porous layer. The averaging procedure and the magnitude of the VV term are illustrated by an example adopted from the literature that involves numerical simulation of two-dimensional open-channel flow over a bundle of circular cylinders.

AB - Double-averaging methodology is very convenient for investigating spatially heterogeneous flows such as boundary-layer flows over permeable walls. However, spatial averaging volumes suitable for boundary-layer flows over rough walls are very thin in the wall-normal direction whereas those for porous-media flows usually have similar length in all three directions. This scale mismatch can be addressed by allowing the averaging volume to vary in space so that its size can be adjusted to the physical characteristics of particular flow regions. This paper presents a new spatial averaging theorem derived for a spatially variable averaging volume that may contain a stationary solid phase and that may also extend beyond the boundary of the problem domain. The theorem provides the expression for the difference between the average of a spatial derivative and the derivative of the spatial average (of a general flow quantity), here named the commutation correction (CC) term. The CC term contains three parts accounting for (1) the presence of the solid phase in the averaging volume, (2) the averaging volume extending beyond the flow-domain boundary, and (3) the spatial variation of the averaging volume. The first two parts have been acknowledged in the literature; the third part is introduced in this paper and named the volume variation (VV) term. The averaging theorem is used to derive large-scale continuity and momentum equations, which contain the new VV term. The equations are applied to steady, uniform, microscopically two-dimensional flow over a permeable wall. The averaging volume for the boundary-layer flow above the wall is a thin wall-parallel layer; on crossing the wall surface, the volume grows until its height reaches the size required for the homogeneous porous layer. The averaging procedure and the magnitude of the VV term are illustrated by an example adopted from the literature that involves numerical simulation of two-dimensional open-channel flow over a bundle of circular cylinders.

KW - Double-averaging

KW - Extended spatial averaging theorem

KW - Fluid-porous interface

KW - Variable averaging volume

UR - http://www.scopus.com/inward/record.url?scp=84924995973&partnerID=8YFLogxK

U2 - 10.1061/(ASCE)HY.1943-7900.0000972

DO - 10.1061/(ASCE)HY.1943-7900.0000972

M3 - Article

AN - SCOPUS:84924995973

VL - 141

JO - Journal of Hydraulic Engineering

JF - Journal of Hydraulic Engineering

SN - 0733-9429

IS - 4

M1 - 04014087

ER -