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Vertically Integrated Non-hydrostatic Free Surface Flow Equations

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Non-Hydrostatic Free Surface Flows

Abstract

Vertically integrated mass and momentum equations for unsteady non-hydrostatic flows over 3D terrain of a continuum mixture of fluids and solids are presented, thereby assuming material surfaces on the bottom and the top of the flowing layer. These are presented as evolution equations of the velocity field and stress tensor. Subsequently, the general method to mathematically determine the distributions of the vertical velocity and vertical non-hydrostatic stress is presented. Under the shallow flow approximation, use of depth-independent predictors for the velocity components in the horizontal plane, following Serre (La Houille Blanche 8(6–7):374–388; 8(12):830–887, 1953), provides closure to the computation of the velocity field. Using the stress tensor for turbulent clear-water flows, the particular one-dimensional problems of the undular (potential) bore propagation over horizontal topography and turbulent-uniform open-channel flows on a steep slope are presented. Steady potential flows over curved beds, and turbulent flows in undular jumps, are also discussed. The effect of movable beds on non-hydrostatic depth-averaged modelling is presented using Exner’s equation, assuming that the bed load is the dominant sediment transport mode. The conservation equations are presented in a compact conservative form, suitable for constructing numerical schemes using modern shock-capturing methods. Examples of suitable numerical schemes to solve the equations are discussed. Fawer’s (Etude de quelques écoulements permanents à filets courbes (Study of some steady flows with curved streamlines), 1937) theory and the weighted residual method (Steffler and Jin in Journal of Hydraulic Research 31(1):5–17, 1993) are used to highlight techniques to produce higher-order models not limited by the assumption of depth-independent velocity components.

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Notes

  1. 1.

    Fawer (1937), Iwasa (1955, 1956), Iwasa and Kennedy (1968), Mandrup-Andersen (1975, 1978), Marchi (1963, 1992, 1993), Matthew (1963, 1991), Engelund and Hansen (1966), Basco (1983), Hager (1983), Hager and Hutter (1984a, b), Montes (1986), Berger and Carey (1998a, b), Soares-Frazão and Zech (2002), Mohapatra and Chaudhry (2004), Dewals et al. (2006), Bose and Dey (2007, 2009), Chaudhry (2008), Castro-Orgaz and Hager (2009), Denlinger and O’Connel (2008).

  2. 2.

    Mei (1983), Antunes do Carmo et al. (1993), Nwogu (1993), Chen and Liu (1995), Wei et al. (1995), Wei and Kirby (1995), Madsen et al. (1997), Madsen and Schäffer (1998), Stansby and Zhou (1998), Chen et al. (1999, 2003), Kennedy et al. (2000), Lynett et al. (2002), Stansby (2003), Erduran et al. (2005), Musumeci et al. (2005), Lynett (2006), Chen (2006), Soares-Frazão and Guinot (2008), Mignot and Cienfuegos (2008), Kim et al. (2009), Kim and Lynett (2011), Brocchini (2013).

  3. 3.

    Examples in hydraulic engineering were studied by Boussinesq (1877), Yen (1973), Steffler and Jin (1993), Liggett (1994), Vreugdenhil (1994), Khan and Steffler (1996a, b), Jain (2001), whereas in rapid gravity-driven mass flows the scene has been set by Hutter and Savage (1988), Savage and Hutter (1989, 1991), Iverson (1997, 2005), Denlinger and Iverson (2004). For a review, see e.g., Pudasaini and Hutter (2007).

Abbreviations

a :

Shallow water wave celerity based on enhanced gravity (m/s)

A :

Finite volume area (m2)

CFL :

Courant–Friedrichs–Lewy number (–)

D :

Representative particle diameter (m)

f :

Weighting function (m)

F :

Vector of fluxes in x-direction (m2/s, m3/s2)

F i+1/2 :

Numerical flux in x-direction at cell interface (m2/s, m3/s2)

g :

Gravity acceleration (m/s2)

g′:

Enhanced gravity acceleration (m/s2)

G :

Vector of fluxes in y-direction (m2/s, m3/s2)

h :

Vertical flow depth (m)

h*:

Flow depth at star region in HLL Riemann solver (m)

H :

Vertical length scale (m)

i :

x-index for finite volume cell (–)

I :

Auxiliary variable (m2/s)

k :

Time index (–)

K :

Fawer exponent (–)

L :

Horizontal length scale (m)

M :

Momentum function (m2)

n :

Curvilinear coordinate normal to channel bottom (m) also bed porosity (–)

N :

Flow depth normal to channel bottom (m)

p :

Fluid pressure (N/m2)

p 1 :

Bottom pressure in excess of hydrostatic pressure (N/m2)

p 2 :

Midpressure in excess of pressure average at bottom and surface elevation (N/m2)

q :

Unit discharge (m2/s)

q b :

Unit bed load (m2/s)

q x :

Unit discharge in x-direction (m2/s)

q y :

Unit discharge in y-direction (m2/s)

q K :

Signal speed factor in HLL solver (–)

R :

Submerged specific gravity (–)

R ep :

Particle Reynolds number (–)

s :

Curvilinear coordinate along channel bed (m)

S o :

Bottom slope (–)

S f :

Friction slope (–)

S L :

Speed of left signal in HLL Riemann solver (m/s)

S R :

Speed of right signal in HLL Riemann solver (m/s)

S :

Vector of source terms (m/s, m2/s2)

t :

Time (s)

T ij :

Depth-averaged Reynolds stress (N/m2) with (i, j) = (x, y)

T :

Stress tensor (N/m2)

u :

Velocity in x-direction (m/s)

u :

Depth-averaged velocity vector (m/s, m/s)

u 1 :

Velocity at surface in excess of mean (m/s)

U :

Depth-averaged flow velocity in x-direction (m/s)

U :

Vector of conserved variables (m, m2/s)

v :

Velocity in y-direction (m/s)

V :

Depth-averaged flow velocity in y-direction also modulus of velocity (m/s)

w :

Velocity in z-direction (m/s)

\( \bar{w} \) :

Depth-averaged flow velocity in z-direction (m/s)

w 2 :

Middepth vertical velocity in excess of average (m/s)

x :

Horizontal coordinate (m)

y :

Horizontal coordinate normal to x (m)

z :

Vertical coordinate (m)

z b :

Bed elevation (m)

γ :

Specific weight of water (N/m3)

γ s :

Specific weight of solids (N/m3)

ε :

Shallowness parameter (–)

η :

Vertical distance above channel bottom (m)

θ :

Angle of bottom with horizontal (rad)

κ :

Streamline curvature (m−1)

\( \bar{\kappa } \) :

Depth-averaged streamline curvature (m−1)

λ :

Dispersive factor (–)

υ :

Kinematic viscosity (m2/s)

ρ :

Density (kg/m3)

σ ij :

Turbulent Reynolds stress, with (i, j) = (x, y, z) (N/m2)

τ b :

Boundary shear stress along bed in s-direction (N/m2)

τ ij :

Stress in continuum medium, with (i, j) = (x, y, z) (N/m2)

ω :

Weighting parameter (–)

Ω :

Control volume (m3)

s :

Relative to free surface

b :

Relative to bottom

L :

Relative to left state in Riemann problem

R :

Relative to right state in Riemann problem

*:

Relative to dimensionless quantity

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  1. University of Cordoba, Cordoba, Spain

    Oscar Castro-Orgaz

  2. ETH Zurich, Zürich, Switzerland

    Willi H. Hager

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Castro-Orgaz, O., Hager, W.H. (2017). Vertically Integrated Non-hydrostatic Free Surface Flow Equations. In: Non-Hydrostatic Free Surface Flows. Advances in Geophysical and Environmental Mechanics and Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-47971-2_2

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