^{1}and M. Asai

^{2}

### Abstract

The question of whether a system of roughness elements has to be viewed either as a distributed roughness or a set of individual, hydrodynamically independent roughness elements has been considered. The answer has been given in the context of definition of hydraulic smoothness proposed by Floryan [Eur. J. Mech. B/Fluids **26**, 305 (2007)] where a roughness system that cannot destabilize the flow is viewed as hydraulically inactive. Linear stability characteristics have been traced from the distributed to the isolated roughness limits. It has been shown that an increase of distance between roughness elements very quickly stabilizes disturbances in the form of streamwise vortices; however, roughness elements placed quite far apart are able to affect evolution of disturbances in the form of traveling waves. Transition from the distributed to the isolated roughness limit is achieved much faster in the case of roughness elements in the form of “trenches” forming depressions below the reference surface than in the case of roughness elements in the form of “ridges” protruding above the reference surface.

This work has been carried out with partials support of NSERC of Canada and the Grant-in-Aid for Scientific Research from the Japan Society for Promotion of Science (No. 21560820). The authors would like to thank Dr. S. Krol for assistance in carrying out the required computations. Sharcnet of Canada provided the required computing resources.

I. INTRODUCTION

II. MODELING OF SURFACE ROUGHNESS

III. STATIONARY FLOW IN A CORRUGATED CHANNEL

IV. STABILITY ANALYSIS

V. DISCUSSION OF RESULTS

VI. CONCLUSIONS

### Key Topics

- Flow instabilities
- 21.0
- Reynolds stress modeling
- 19.0
- Rotating flows
- 18.0
- Boundary value problems
- 14.0
- Channel flows
- 13.0

## Figures

Geometry model. Roughness elements in the form of triangular “ridges.” (A) Distributed roughness; and (B) transition towards the isolated roughness limit.

Geometry model. Roughness elements in the form of triangular “ridges.” (A) Distributed roughness; and (B) transition towards the isolated roughness limit.

Geometry model. Roughness elements in the form of triangular “trenches.” (A) Distributed roughness and (B) transition towards the isolated roughness limit.

Geometry model. Roughness elements in the form of triangular “trenches.” (A) Distributed roughness and (B) transition towards the isolated roughness limit.

Geometry of the roughness elements. Elements 1-4 correspond to the “ridges” while 5-8 are opposite to 1-4 and correspond to the “trenches.” The sketch is not in scale. All elements have the same height (H = 0.032) and bases with the length B = π/2, π/4, π/8, and π/16; the slopes of the side walls are 2.3°, 4.6°, 9.25°, and 18.05°, respectively.

Geometry of the roughness elements. Elements 1-4 correspond to the “ridges” while 5-8 are opposite to 1-4 and correspond to the “trenches.” The sketch is not in scale. All elements have the same height (H = 0.032) and bases with the length B = π/2, π/4, π/8, and π/16; the slopes of the side walls are 2.3°, 4.6°, 9.25°, and 18.05°, respectively.

Sketch of the flow domain. The computational domain is contained between y = min(y_{L}) and y = 1.

Sketch of the flow domain. The computational domain is contained between y = min(y_{L}) and y = 1.

The neutral curves in the (Re, β) plane for disturbances in the form of streamwise vortices for the roughness system made of “ridges” with the base B = π/2 and spacing characterized using parameter κ = λ/B (larger values of κ corresponds to larger distances between the roughness elements).

The neutral curves in the (Re, β) plane for disturbances in the form of streamwise vortices for the roughness system made of “ridges” with the base B = π/2 and spacing characterized using parameter κ = λ/B (larger values of κ corresponds to larger distances between the roughness elements).

The neutral curves in the (Re, κ) plane for disturbances in the form of streamwise vortices for different values of the flow Reynolds number Re for the roughness system made of “ridges” with the base B = π/2 and spacing characterized using parameter κ = λ/B.

The neutral curves in the (Re, κ) plane for disturbances in the form of streamwise vortices for different values of the flow Reynolds number Re for the roughness system made of “ridges” with the base B = π/2 and spacing characterized using parameter κ = λ/B.

Curvature of streamlines of the base flow over roughness system made of “ridges” with the base B = π/2 and located at distances apart from each other corresponding to κ = 1, 1.5, 2, 3, 4 (see (A)–(E), respectively).

Curvature of streamlines of the base flow over roughness system made of “ridges” with the base B = π/2 and located at distances apart from each other corresponding to κ = 1, 1.5, 2, 3, 4 (see (A)–(E), respectively).

Variations of the critical Reynolds number for disturbances in the form of streamwise vortices as a function of the roughness spacing parameter κ = λ/B for the roughness system made of triangular elements with the base B = π/2, π/4, π/8, and π/16. Continuous and dashed lines correspond to the “ridges” and the “trenches,” respectively.

Variations of the critical Reynolds number for disturbances in the form of streamwise vortices as a function of the roughness spacing parameter κ = λ/B for the roughness system made of triangular elements with the base B = π/2, π/4, π/8, and π/16. Continuous and dashed lines correspond to the “ridges” and the “trenches,” respectively.

Variations of the critical Reynolds number for disturbances in the form of streamwise vortices as a function of the wavelength λ of the roughness system. Other conditions as in Fig. 8.

Variations of the critical Reynolds number for disturbances in the form of streamwise vortices as a function of the wavelength λ of the roughness system. Other conditions as in Fig. 8.

Curvature of streamlines of the base flow over roughness system made of “trenches” with the base B = π/2 and located at a distance away from each other corresponding to κ = 3.

Curvature of streamlines of the base flow over roughness system made of “trenches” with the base B = π/2 and located at a distance away from each other corresponding to κ = 3.

The neutral curves in the (Re, δ) plane for disturbances in the form of traveling waves for the roughness system made of “ridges” with the base B = π/2 and spacing characterized in terms of parameter κ = λ/B (larger values of κ corresponds to larger distances between the roughness elements). κ = ∞ denotes smooth surface.

The neutral curves in the (Re, δ) plane for disturbances in the form of traveling waves for the roughness system made of “ridges” with the base B = π/2 and spacing characterized in terms of parameter κ = λ/B (larger values of κ corresponds to larger distances between the roughness elements). κ = ∞ denotes smooth surface.

Variations of the critical Reynolds number for disturbances in the form of traveling waves as a function of the spacing parameter κ = λ/B for the roughness system made of triangular elements with the base B = π/2, π/4, π/8, and π/16. Continuous and dashed lines correspond to the “ridges” and to the “trenches”, respectively.

Variations of the critical Reynolds number for disturbances in the form of traveling waves as a function of the spacing parameter κ = λ/B for the roughness system made of triangular elements with the base B = π/2, π/4, π/8, and π/16. Continuous and dashed lines correspond to the “ridges” and to the “trenches”, respectively.

Variations of the critical Reynolds number for disturbances in the form of traveling waves as a function of the wavelength λ of the roughness system. Other conditions are as in Fig. 11.

Variations of the critical Reynolds number for disturbances in the form of traveling waves as a function of the wavelength λ of the roughness system. Other conditions are as in Fig. 11.

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