List of symbols

A cross-section area (m2);

Aab cross-section area (m2) of air bubble in a particular direction (App. C);

AW clear-water flow cross-section area (m2);

A1,A2 constants (see App. A);

A3,A4 constants (see App. A);

A90 air-water flow cross-section area (m2) (i.e. for C < 0.90);

a specific interface area (m− l);

a1, a2 constants;

B coefficient of proportionality (see App. G);

B′ integration constant (defined by WOOD 1984) (see App. G);

B1, B2, B3 constants (see App. A);

B90 free-surface width (m) of the air-water flow (i.e. at y = Y90);

C air concentration defined as the volume of air per unit volume of air and water; it is also called void fraction;

CDS downstream dissolved gas concentration (kg/m3);

CT characteristic air concentration defined as : CT= C(y=dT) (see App. G);

CUS upstream dissolved gas concentration (kg/m3);

Cb air concentration next to a solid boundary;

Cd drag coefficient;

Ce mean air concentration in uniform equilibrium self-aerated flow;

Cgas concentration of dissolved gas in water (kg/m3);

Cmax maximum air concentration in the air bubble diffusion layer;

(Cmax)o initial maximum air concentration;

(Cmax)I maximum air concentration at inflow;

Cmean mean air concentration defined in terms of 90%-air content :
(1 - Cmean) * Y90 = d in two-dimensional flow down a rectangular cross-section channel;
mean=1A90*∫0Y90∫C*dz*dy in partially-filled pipes and channels of irregular cross-section;

(Cmean)cl mean air concentration on the channel centreline;

(Cmean)max maximum mean air concentration along a chute;

(Cmean)1 mean air concentration at the inflow of pre-entrained hydraulic jump;

Co initial air content;

Cp specific heat at constant pressure (J/kg.K) (see App. A);

Csat gas saturation concentration in water (kg/m3) (see App. A);

Csed mean volumetric sediment concentration;

Csound sound celerity (m/s) (see App. A and D);

Cv specific heat at constant volume (J/kg.K) (see App. A);

C* mean air concentration at x = 0 in self-aerated flow (Chap. 10);

Chl chlorinity;

ch chord length (m);

chab air bubble chord length (m) (defined in App. K);

D 1- internal pipe diameter (m);
2- orifice diameter (m);

DH hydraulic diameter (m): DH = 4*AW/PW;

Dm molecular diffusivity (m2/s) of gas (see App. A);

Dp penetration depth (m) of air bubbles (measured vertically below the free-surface);

Dr turbulent diffusivity (m2/s) in the radial direction;

Ds sediment diffusion coefficient (m2/s);

Dt turbulent diffusivity (m2/s) of air bubbles in air-water flow;

Dy turbulent diffusivity (m2/s) in the y-direction;

Dl integration constant of the logarithmic velocity profile in turbulent boundary layers along a smooth plate (see App. F);

D2 integration constant of the logarithmic velocity profile in turbulent boundary layers along rough wall (see App. F);

Do ‘diffusivity’ term (m2/s) related to the longitudinal velocity gradient (see App. I);

D′ dimensionless turbulent diffusivity : D′ = Dt/((ur)Hyd*cosα*Y90);

D′′ dimensionless turbulent diffusivity defined as D′′ = (Dt*x)/(Vo*Y902) for a plane shear layer and D′′ = (Dt*x)/(Vo*r902) for a circular jet;

D* dimensionless turbulent diffusivity : D* = Dt/(V**Y90) for self-aerated flow and D* = Dt/(V**Y90) for non-aerated flow;

D# dimensionless turbulent diffusivity : D# = Dt/(Vo*yo) for two-dimensional shear flow and D# = Dt/(Vo*ro) for circular jet (see App. E);

d 1- characteristic flow thickness (m) measured normal to the flow direction;
2- flow depth (m) measured perpendicular to the channel bottom;
3- clear water flow depth defined as : =∫0Y901‐C*dy ;

dDS downstream flow depth (m) or tailwater depth (m);

dI flow depth (m) at the inception point of air entrainment in open channel (i.e. at point of apparition of ‘white’ waters);

dT transition depth (m) as deflned by STRAUB and ANDERSON (1958) (see App. G);

dab air bubble size (m);

abmin minimum air bubble size (m) detected by a probe;

(dab)max 1- maximum air bubble size (m) in turbulent shear layer;
2- maximum air bubble size (m) in self-aerated open channel flow (Chap. 13);

(dab)mean mean air bubble size (m) in turbulent shear layer;

(dab)o initial air bubble size (m);

db1 flow depth (m) at intersection of free-surface with outer edge of developing boundary layer;

dc critical flow depth (m): for a rectangular channel: c=qW2/g3;

dimpact jet thickness at nappe impact (fig. 17-5);

do 1- intake flow thickness (m);
2- nozzle diameter (m);
3- initial flow thickness (m) of free-shear layer;

dS characteristic sediment size (m);

dvc flow depth (m) at vena contracta (e.g fig. 7-2);

dl 1- upstream flow thickness (m);
2- upstream flow depth (m) (e.g. of a hydraulic jump);
3- jet thickness (m) at plunge point
4- flow thickness (m) of free-shear layer at nozzle edge;

d2 conjugate flow depth (m) of a hydraulic jump;

d50 mean sediment size (m);

d* clear water flow depth (m) at the start of the gradually-varied flow region (Chap. 10);

d′ dimensionless flow depth : d′ = d/d*;

E aeration efficiency defined in terms of dissolved gas content as : E = (CDS - CUS)/(Csat - CUS);

Eb bulk modulus of elasticity (Pa): Eb = ρ*∂P/∂ρ ;

Ec kinetic energy correction coefficient (also called Coriolis coefficient);

Eco compressibility (1/Pa): co=1ρ*∂ρ∂P;

F frequency (Hz) of oscillation of thin free-falling nappe;

Fb buoyancy force (N);

F* Froude number defined in terms of the roughness height: *=qW/g*sinα*kS3;

Fr Froude number; for a rectangular channel, it is defined as : r=qW/g*d3;

Fre Froude number at onset of air entrainment: re=Ve/g*d1;

Frimpact Froude number at nappe impact: r=qW/g*dimpact3 (see App. J);

Frvc Froude number at vena contracta;

Fro initial Froude number; for a rectangular channel: ro=qW/g*do3;

Fr1 upstream Froude number ; for a rectangular channel: r1=qW/g*d13;

Fr* Froude number defined as : r*=qW/g*d*3;

f 1- Darcy friction factor;
2- friction...