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Control of instabilities in non-Newtonian free surface fluid flows

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Abstract

Free surface flows in inclined channels can develop periodic instabilities that are propagated downstream as shock waves with well-defined wavelengths and amplitudes. Such disturbances are called “roll waves” and are common in channels, torrential lava, landslides, and avalanches. The prediction and detection of such waves over certain types of structures and environments are useful for the prevention of natural risks. In this work, a mathematical model is established using a theoretical approach based on Cauchy’s equations with the Herschel–Bulkley rheological model inserted into the viscous part of the stress tensor. This arrangement can adequately represent the behavior of muddy fluids, such as water–clay mixture. Then, taking into account the shallow water and the Rankine–Hugoniot’s (shock wave) conditions, the equation of the roll wave and its properties, profile, and propagation velocity are determined. A linear stability analysis is performed with an emphasis on determining the condition that allows the generation of such instabilities, which depends on the minimum Froude number. A sensitivity analysis on the numerical parameters is performed, and numerical results including the influence of the Froude number, the index flow and dimensionless yield stress on the amplitude, the wavelength of roll waves and the propagation velocity of roll waves are shown. We show that our numerical results were in agreement with Coussot’s experimental results (1994).

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Abbreviations

C * :

Dimensionless yield stress

Fr :

Froude number

Fr min :

Minimum Froude number

g :

Acceleration due to gravity

h :

Total depth of flow

h 0 :

Flow depth under uniform system

h 1 :

Flow depth before shock

h 2 :

Flow depth after shock

h c :

Critical flow depth

h z :

Domain of possible values for h 1

H :

Infinitesimal height value when the flow is disturbed

\( \hat{H} \) :

Magnitude of the disturbance

k :

Wavenumber of infinitesimal disturbances

K n :

Consistency rate of the Herschel–Bulkley fluid

l :

Characteristic width (channel width)

L :

Characteristic length

n :

Power-law index

P :

Pressure

P * :

Dimensionless pressure

t :

Time scale unit

u :

Velocity component in the x direction

u * :

Dimensionless velocity component in the x direction

\( \bar{u} \) :

Mean flow velocity

\( \bar{u}_{c} \) :

Mean flow velocity in h c

u 0 :

Velocity component in the flow direction under uniform system

\( \bar{u}_{0} \) :

Mean flow velocity under uniform system

U :

Vector velocity

U :

Roll wave propagation velocity

\( \hat{U} \) :

Infinitesimal value for the velocity flow when disturbed

w :

Velocity component in the z direction

w * :

Dimensionless velocity component in the z direction

x :

Abscissa in the Cartesian coordinate system

x * :

Dimensionless abscissa in the Cartesian coordinate system

x′:

Abscissa in the mobile system of coordinates

z :

Quota in the Cartesian coordinate system

z * :

Dimensionless quotas in the Cartesian coordinate system

z 0 :

Flow depth in the sheared region

α :

Velocity distribution coefficient

θ :

Slope of the channel

λ :

Length of roll wave

ρ :

Mass density

τ :

Shear stress

τ xz :

Shear stress acting on the x axis in the z direction

τ * :

Dimensionless shear stress

τ c :

Yield stress

τ b :

Wall stress

ω :

Frequency of infinitesimal disturbances

∆h :

Wave amplitude

References

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Author information

Authors and Affiliations

  1. Civil Engineering Department, FEIS/UNESP, Ilha Solteira, SP, 15385-000, Brazil

    Geraldo de Freitas Maciel

  2. Electrical Engineering Department, FEIS/UNESP, Ilha Solteira, SP, 15385-000, Brazil

    Fabiana de Oliveira Ferreira

  3. Mechanical Engineering Department, FEIS/UNESP, Ilha Solteira, SP, 15385-000, Brazil

    Guilherme Henrique Fiorot

Authors
  1. Geraldo de Freitas Maciel
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  2. Fabiana de Oliveira Ferreira
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  3. Guilherme Henrique Fiorot
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Corresponding author

Correspondence to Geraldo de Freitas Maciel.

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Technical Editor: Monica Naccache.

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de Freitas Maciel, G., de Oliveira Ferreira, F. & Fiorot, G.H. Control of instabilities in non-Newtonian free surface fluid flows. J Braz. Soc. Mech. Sci. Eng. 35, 217–229 (2013). https://doi.org/10.1007/s40430-013-0025-y

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  • DOI: https://doi.org/10.1007/s40430-013-0025-y

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