Three-dimension losses and correlation in turbomachinery refers to the measurement of flow-fields in three dimensions, where measuring the loss of smoothness of flow, and resulting inefficiencies, becomes difficult, unlike two-dimensional losses where mathematical complexity is substantially less.
Three-dimensionality takes into account large pressure gradients in every direction, design/curvature of blades, shock waves, heat transfer, cavitation, and viscous effects, which generate secondary flow, vortices, tip leakage vortices, and other effects that interrupt smooth flow and cause loss of efficiency. Viscous effects in turbomachinery block flow by the formation of viscous layers around blade profiles, which affects pressure rise and fall and reduces the effective area of a flow field. Interaction between these effects increases rotor instability and decreases the efficiency of turbomachinery.
In calculating three-dimensional losses, every element affecting a flow path is taken into account—such as axial spacing between vane and blade rows, end-wall curvature, radial distribution of pressure gradient, hup/tip ratio, dihedral, lean, tip clearance, flare, aspect ratio, skew, sweep, platform cooling holes, surface roughness, and off-take bleeds. Associated with blade profiles are parameters such as camber distribution, stagger angle, blade spacing, blade camber, chord, surface roughness, leading- and trailing-edge radii, and maximum thickness.
Two-dimensional losses are easily evaluated using Navier-Stokes equations, but three-dimensional losses are difficult to evaluate; so, correlation is used, which is difficult with so many parameters. So, correlation based on geometric similarity has been developed in many industries, in the form of charts, graphs, data statistics, and performance data.
Types of losses.
Three-dimensional losses are generally classified as:
Three-dimensional profile losses.
The main points to consider are:
Three-dimensional shock losses.
The main points to consider are:
Secondary flow.
The main points to consider are:
 ζs = (0.0055 + 0.078(δ1/C)1/2)CL2 (cos3α2/cos3αm) (C/h) (C/S)2 ( 1/cos ά1)
where ζs = average secondary flow loss coefficient; α2, αm = flow angles; δ1/C = inlet boundary layer; and C,S,h = blade geometry.
Endwall losses in axial flow in turbomachinery.
The main points to consider are:
 ζ = ζp + ζew
 ζ = ζp[ 1 + ( 1 + ( 4ε / ( ρ2V2/ρ1V1 )1/2 ) ) ( S cos α2 - tTE )/h ]
where ζ=total losses, ζp=blade profile losses, ζew=endwall losses.
 η = ή ( 1 - ( δh* + δt*)/h ) / ( 1 - ( Fθh + Fθt ) / h )
where η=efficiency in absence of endwall boundary layer, where h refers to the hub and t refers to the tip. The values of Fθ and δ* are derived from the graph or chart.
Tip-leakage flow losses.
The main points to consider are:
 QL = 2 ( ( Pp - Ps ) / ρ )1/2
 a/τ = 0.14 ( d/τ ( CL )1/2 )0.85
 ζL ~ ( CL2 * C * τ * cos2β1 ) / ( A * S * S * cos2βm )
 ζW ~ ( δS* + δP* / S ) * ( 1 / A ) * ( ( CL )3/2) * ( τ / S )3/2Vm3 / ( V2 * V12 )
References.
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