C void fraction defined as the volume f air per unit volume of air and water

(1)

Notation

NOTATION

The following symbols are used in this thesis:

Aw water flow cross-section area [m²];

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

Cmean depth-average void fraction defined in terms of Y90: Cmean=1-d/Y90; ch chord size [m];

chab bubble chord length [m];

Dt turbulent diffusivity [m²/s] of air bubbles in air-water flow;

D0 dimensionless coefficient which is function of the mean void fraction Cmean=0.7622(1.0434-e^(-3.614D0) ;

d equivalent clear water flow depth defined as:

Y90

0

d  (1 C)dy ^;

F air bubble count rate [Hz] or bubble frequency defined as the number of detected air bubbles per unit time;

Fmax maximum count rate [Hz] at a given cross-section;

Fr Froude number;

f Darcy-Weisbach friction factor;

fe equivalent Darcy-Weisbach friction factor in air-water skimming flows;

g gravity constant g=9.80 [m/s²] in Brisbane, Australia;

h height of steps [m] measured vertically;

K’ dimensionless coefficient which is function of the mean void fraction K’=K^*+1/(2D0)-8/(81D0);

K^* a constant calculated as: K^*=tanh^-1(√0.1)=0.32745015….

lstep length of steps [m];

lcav cavity length l_cav  h² l²_step ^;

Nab number of air bubbles per record;

Nb/d number of ari bubbles or droplets which were recorded;

Pw wetter perimeter [m];

Qair air discharge [m³/s];

Qw water discharge [m³/s];

qw water discharge per unit width [m²/s];

Re Reynolds number;

Rxx normalised auto-correlation function (reference probe);

Rxy normalised cross-correlation function between two probe output signals;

(Rxy)max maximum cross-correlation between two probe output signals;

t scan time [s];

Tu turbulence intensity defined as: Tu=u’/V;

i

(2)

Notation

Txx auto-correlation integral time scale defined as:

(Rxx 0)

xx xx

0

T R ( )d

 



   ^;

Txy cross-correlation integral time scale defined as:

xx

xy xy max

(R 0)

xy xy

(R (R ) )

T R ( )d

 

   ^;

τRxymax the characteristic time lag τ for which Rxy is maximum;

(τ0.5)xx the characteristic time lag τ for which Rxx=0.5;

(τ0.5)xy the characteristic time lag τ for which Rxy=0.5(Rxy)max ; (τ0)xx the time lag for which Rxy=0 ;

(τ0) xy the time lag for which Rxy=0;

u’ root mean square longitudinal component of turbulence velocity [m/s];

V velocity [m/s];

Vc critical velocity [m/s] V_c³g q _w ;

V90 characteristic velocity [m/s] where the air concentration is 90%;

W channel width [m];

x distance along the channel bottom [m];

yFmx distance [m] normal to the pseudo-bottom where F=Fmax; Y90 characteristic depth [m] where the void fraction is 90%;

z transverse distance [m] from the channel centreline;

zmax transverse distance [m] where the cross-correlation coefficient tends to zero;

ρ density [kg/m³];

μ dynamic viscosity [Pa s];

θ slope of the channel;

τ time lag;

Ø diameter [mm];

Subscript air air flow;

c critical flow conditions;

max maximum values;

mean mean values;

min minimum values;

w water flow;

xx auto-correlation of reference probe signal;

xy cross-correlation;

90 flow conditions where C=90%;

Abbreviations

PDF probability distribution function;

RCC rolled compacted concrete

ii