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Non-Hydrostatic Free Surface Flows

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Abstract

The computation of non-hydrostatic inviscid flows is considered here. For steady irrotational flows, the energy head is a constant in all the points of the domain of the flow, making use of the energy principle simple. However, both the energy and momentum principles for non-hydrostatic flows are presented and discussed. Extended Boussinesq-type energy and momentum equations are derived using Picard iteration method (Matthew in Proceedings of the ICE 91(3):187–201, 1991), based on an iterative solution of the Cauchy–Riemann equations. The first and second iterative cycles are detailed, and the ensuing analytical predictors for the velocity components are carefully tested using a complete two-dimensional potential flow model based on the x-ψ method. Another set of Boussinesq-type equations is derived based on an approximate treatment of the flow net formulating the Euler’s equations in intrinsic coordinates (Hager and Hutter in Acta Mechanica 51(3–4):31–48, 1984a; 53(3–4), 183–200, 1984b). Picard iteration method is also presented in topography following curvilinear coordinates. The development is used to derive a generalization of Dressler’s (Journal of Hydraulic Research 16(3):205–222, 1978) theory, in which cnoidal and solitary wavelike solutions are embedded. Applications for steady potential flows are detailed, including critical flows over spillways, flows in free overfalls, transitions from mild to steep slopes, and flows in vertical sluice gates. The inclusion of vorticity in non-hydrostatic models is presented with a specific approximation for flows in free overfalls. An introduction to the mathematical theory of irrotational water waves is given based on the Serre–Green–Naghdi equations. The frequency dispersion of non-hydrostatic water waves is explained using a small-amplitude wave. The cnoidal wave theory for finite-amplitude long waves is also discussed. The solitary wave theory is used to investigate the effect of linear and parabolic approximations for the vertical pressure distribution. Non-hydrostatic dam break waves over a rigid and horizontal bed are used to introduce the need of numerical models; a simple finite-difference scheme is employed to illustrate the behavior of these waves.

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Notes

  1. 1.

    In most books on fluid mechanics, ϕ and \( \psi \) are defined with the opposite signs.

  2. 2.

    In the processes of the above computations, expressions of the form (1 + a)n arise, which are approximated as (1 + na) on the basis that |a| ≪ 1. Another approximation typically used is exp(a) = 1 + a. This is used frequently without explicitly mentioning it.

  3. 3.

    The expansion ln(1 − a) = −a − a 2/2 − a 3/3 − … ≈ − a(1 + a/2) is used on the basis that |a| ≪ 1.

  4. 4.

    Nappe is a French word used by hydraulic engineers to define the free surface of a jet. Therefore, the free jet is defined by the upper and lower nappe profiles [z s  = z s (x) and z b  = z b (x), respectively].

  5. 5.

    From Appendix E and for flat channels, the approximate integration of the Euler equations along the equipotential lines (Hager and Hutter 1984a) agrees with the first Picard iteration cycle (Matthew 1991).

  6. 6.

    In weir flow, R b is negative (convex bottom profile), resulting in m < 0. In the original developments presented by Jaeger (1939) and Montes (1970), R b is to be used in absolute value, given that a negative sign was introduced into the governing equations. Therefore, m = +2 is the value used in these works.

  7. 7.

    If the phase speed c depends on the wave number k, the corresponding wave is called dispersive.

  8. 8.

    Note, on comparing Eq. (3.664)2 with (3.670)2, the negative sign on the right-hand side of Eq. (3.668).

Abbreviations

a :

Gate opening (m); also wave amplitude (m/s)

A :

Angular momentum function (m3); wave amplitude (m)

A, B, C, D :

Auxiliary variables

B :

Boussinesq’s mixed term (m4/s2)

c :

Wave celerity (m/s); also constant of integration (m)

c o :

Celerity of small gravity wave (m/s)

c w :

Celerity of shock front (m/s)

C d :

Weir discharge coefficient (–)

C D :

Spillway discharge coefficient at design head (–)

C c :

Contraction coefficient (–)

CFL :

Courant–Friedrichs–Lewy number (–)

d :

Still water depth (m)

D :

Constant of integration (–)

E :

Specific energy head (m)

f :

Function (m)

F :

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

F :

Froude number (–)

F o :

Froude number of hydrostatic flow (–); also Froude number of undisturbed flow (–)

F p :

Froude number of translating wave over still water (–)

F :

Force (N)

g :

Gravity acceleration (m/s2); also function (m/s)

G :

Recursion index (–)

h :

Flow depth measured vertically (m)

h c :

Critical depth for parallel-streamlined flow (m) = (q 2/g)1/3

h b :

Brink depth (m)

h o :

Uniform flow depth (m); also still water depth (m); also terminal jet thickness (m)

h p :

Effective pressure head (m)

h 1 :

Maximum flow depth of cnoidal wave (m)

h 2 :

Minimum flow depth of cnoidal wave (m)

h 3 :

q 2/(gh 1 h 2) (m)

h max :

Maximum flow depth of solitary wave (m)

H :

Total energy head (m)

H D :

Design energy head of spillway profile (m)

i :

Relative inclination (–); also node index in the x-direction (–)

j :

Node index in the ψ-direction (–)

k :

Recursion index (–); also wave number (m−1)

k 2 :

Modulus of incomplete elliptical integral of first kind (–)

K, K b , K G :

Curvature distribution parameters in Fawer’s theory (–)

m :

Inclination distribution parameter (–); pressure parameter (–); also parameter of the original Jaeger’s theory (–)

m o :

Radius of curvature distribution parameter in Jaeger’s theory (–)

m 1 :

Power-law exponent at inflow section (–)

M :

Vertical momentum (m3); also maximum value of j-index (–), also momentum function (m4/s2)

n :

Curvilinear coordinate along equipotential (m)

N :

Flow depth measured normal to bottom profile (m); also maximum value of j-index (–); also power-law exponent (–)

N o :

Length of equipotential curve (m)

p :

Pressure (N/m2); also auxiliary variable (–)

p b :

Bottom pressure (N/m2)

p s :

Interface pressure (N/m2)

p 1 :

Pressure excess over hydrostatic pressure at channel bottom (N/m2)

p be :

Bottom pressure at brink section of free overfall (N/m2)

q :

Unit discharge (m2/s)

q o :

Normalized unit discharge in slope break with rounded transition (–)

q p :

Progressive unit discharge (m2/s)

r :

Relative curvature (–); ratio of down- to upstream water depths in dam break problem (–); also roller thickness (m)

R :

Radius of streamline curvature (m)

R * :

Radius of circular-shaped equipotential line (m)

R s :

Radius of free surface (m)

R b :

Radius of channel bottom (m); also radius of circular arc transition (m)

s :

Curvilinear coordinate along streamline (m); also main stream profile of submerged jet (m)

S :

Specific momentum (m2)

S o :

Bottom slope (–)

t :

Vertical flow depth (m); also time (s)

u :

Velocity in the x-direction (m/s); also normalized variable (–); also incomplete elliptical integral of first kind (–)

u ξ :

Velocity in the ξ-direction (m/s)

u α :

Velocity at elevation z α (m/s)

U :

Mean flow velocity (m/s) = q/h

U o :

Amplitude of perturbation of flow velocity (m/s)

U :

Vector of conserved variables (m, m/s)

U c :

Mean critical flow velocity (m/s) = q/h c

V :

Local velocity (m/s)

w :

Velocity in the z-direction (m/s)

w e :

Velocity in the z-direction at gate edge (m/s)

w ζ :

Velocity in the ζ-direction (m/s)

x :

Horizontal coordinate (m)

X :

X/h c (–); also longitudinal coordinate in moving system of reference (m); also x/H D (–)

\( \overline{X} \) :

Modified X-coordinate of sharp-crested weir flow (–)

y :

H/h max (–); also coordinate in the horizontal plane, normal to x (m)

Y :

H/h c (–); also (y − 1)/(F 2 p  − 1) (–)

z :

Vertical elevation (m)

z s :

Free surface elevation (m)

z b :

Elevation of channel bottom (m)

z α :

Reference elevation (m)

Z :

Recursion index (–); also normalized variable (–); also z b /H D (–)

\( \overline{Z} \) :

Modified Z-coordinate of sharp-crested weir flow (–)

α :

Dispersion coefficient (–)

α N :

Nwogu-type dispersion coefficient (–)

α 1, α 2, α 3, α 4 :

Coefficients

β 1, β 2, β 3, β 4 :

Coefficients

Γ:

Incomplete gamma function (–)

γ :

Specific weight of water (N/m3)

Δ:

Step in the x-direction (m); also factor in cnoidal wave theory (–)

ε :

Lower nappe maximum elevation (m)

ε 1, ε 2 :

Curvilinear coefficients (–)

ζ :

Coordinate normal to channel bottom profile, normal to ξ (m); also water depth variation around static level (m); also normalized x-coordinate in solitary wave profile (–)

η :

Vertical coordinate above channel bottom (m)

θ :

Angle of streamline inclination with horizontal (rad)

κ :

Curvature of streamline (m−1); also denoted as κ s

κ b :

Curvature of bottom profile (m−1)

κ n :

Curvature of equipotential curve (m−1)

Λ :

Vorticity factor (–)

λ :

Non-hydrostatic correction coefficient in critical flow condition (–); also wavelength (m)

μ :

Dimensionless vertical coordinate (–)

ν :

Dimensionless curvilinear coordinate along equipotential line (–); also kinematic viscosity (m2/s)

ξ :

Curvilinear coordinate measured along bottom profile (m); also normalized x-coordinate in solitary wave profile (–)

П:

Function (m)

ρ :

Density of water (kg/m3)

τ :

R/R s (–)

ϕ :

Potential function (m2/s)

Φ1, Φ2 :

Functions of water surface profile equation (–)

φ :

Non-hydrostatic correction coefficient (–)

χ :

E/H D (–); also curvature parameter (–); also normalized x-coordinate in free jet profile (–); also parameter of cnoidal wave (–)

ψ :

Stream function (m2/s)

Ω:

Effective angular momentum function (m3); also curvilinear function (–); also vorticity (s−1); also normalized variable (–)

ω :

Constant in solitary wave profile (–); also recursion index (–); also normalized variable (–); also frequency (s−1)

b :

Relative to channel bottom

c :

Critical flow

crest:

Relative to crest section

d :

Relative to downstream boundary condition

o :

Relative to approach flow conditions

s :

Relative to free surface

u :

Relative to upstream boundary condition

*:

Relative to dimensionless value; also relative to star region in Riemann problem

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  1. University of Cordoba, Cordoba, Spain

    Oscar Castro-Orgaz

  2. ETH Zurich, Zürich, Switzerland

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Castro-Orgaz, O., Hager, W.H. (2017). Inviscid Channel Flows. 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_3

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