Three-dimensional losses are generally classified as:

1. Three-dimensional profile losses
2. Three-dimensional shock losses
3. Secondary flow
4. Endwall losses in axial turbomachinery
5. Tip leakage flow losses
6. Blade boundary layer losses

The main points to consider are:

• Profile losses that occur due to the curvature of blades, which includes span-wise mixing of flow field, in addition to two-dimensional mixing losses (which can be predicted using Navier-Stokes equations).
• Major losses in rotors that are caused by radial pressure gradient from midspan to tip (flow ascending to tip).
• Reduction in high losses between annulus wall and tip clearance region, which includes the trailing edge of a blade profile. This is due to flow mixing and flow redistribution at the inner radius as flow proceeds downstream.
• Between the hub and annulus wall, losses are prominent due to three-dimensionality.
• In single-stage turbomachinery, large radial pressure gradient losses at exit of flow from rotor.
• Platform cooling increases the endwall flow loss and coolant air increases profile loss.
• Navier-Stokes identifies many of the losses when some assumptions are made, such as unseparated flow. Here correlation is no longer justified.

The main points to consider are:

• Shock losses continuously increase from the hub to tip of the blade in both supersonic and transonic rotors.
• Shock losses are accompanied by shock-boundary-layer interaction losses, boundary-layer losses in profile secondary flow, and tip clearance effects.
• From the Mach number prospective, fluid inside rotor is in supersonic phase except at initial hub entry.
• The Mach number increases gradually from midspan to tip. At the tip, the effect is less than secondary flow, tip clearance effect, and annulus wall boundary-layer effect.
• In a turbofan, shock losses increase overall efficiency by 2% because of the absence of tip clearance effect and secondary flow being present.
• Correlation depends on many parameters and is difficult to calculate.
• Correlation based on geometric similarity is used.

The main points to consider are:

• The rotation of a blade row causes non-uniformity in radial velocity, stagnation pressure, stagnation enthalpy, and stagnation temperature. Distribution in both tangential and radial directions generates secondary flow.
• Secondary flow generates two velocity components V_y, V_z, hence introducing three-dimensionality in the flow field.
• The two components of velocity result in flow-turning at the tailing end of the blade profile, which directly affects pressure rise-and-fall in turbomachinery. Hence efficiency decreases.
• Secondary flow generates vibration, noise, and flutter because of unsteady pressure field between blades and rotor–stator interaction.
• Secondary flow introduces vortex cavitation, which diminishes flow rate, decreases performance, and damages the blade profile.
• The temperature in turbomachinery is affected.
• Correlation for secondary flow, given by Dunham (1970), is given by:

                   ζs = (0.0055 + 0.078(δ1/C)1/2)CL2 (cos3α2/cos3αm) (C/h) (C/S)2 ( 1/cos ά1)

where ζ_s = average secondary flow loss coefficient; α₂, α_m = flow angles; δ₁/C = inlet boundary layer; and C,S,h = blade geometry.

The main points to consider are:

• In a turbine, secondary flow forces the wall boundary layer toward the suction side of the rotor, where mixing of blade and wall boundary takes place, resulting in endwall losses.
• The secondary flow carries core losses away from the wall and blade boundary layer, through formation of vortices. So, peak loss occurs away from endwall.
• Endwall losses are high in stator (Francis turbine/Kaplan turbine) and nozzle vane (Pelton turbine), and the loss distribution is different for turbine and compressor, due to flows being opposite to each other.
• Due to the presence of vortices, large flow-turning and secondary flow result to form a complex flow field, and interaction between these effects increases endwall losses.
• In total loss, endwall losses form the fraction of secondary losses given by Gregory-Smith, et al., 1998. Hence secondary flow theory for small flow-turning fails.
• Correlation for endwall losses in an axial-flow turbine is given by:

                  ζ = ζp + ζew
     ζ = ζp[ 1 + ( 1 + ( 4ε / ( ρ2V2/ρ1V1 )1/2 ) ) ( S cos α2 - tTE )/h ]

where ζ=total losses, ζ_p=blade profile losses, ζ_ew=endwall losses.

• The expression for endwall losses in an axial-flow compressor is given by:

                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 F_θ and δ^* are derived from the graph or chart.

The main points to consider are:

• The rotation of a rotor in turbomachinery induces a pressure differences between opposite sides of the blade profile, resulting in tip leakage.
• In a turbomachinery rotor, a gap between the annulus wall and the blade causes leakage, which also occurs in the gap between the rotating hub and stator.
• Direct loss through clearance volume, as no angular momentum is transferred to fluid. So, no work is done.
• Leakage, and its interaction with other losses in the flow field, is complex; and hence, at the tip, it has a more pronounced effect than secondary flow.
• Leakage-flow induced three-dimensionality, like the mixing of leakage flow with vortex formation, entrainment process, diffusion and convection. This results in aerodynamics losses and inefficiency.
• Tip leakage and clearance loss account for 20–40% of total losses.
• The effects of cooling in turbines causes vibration, noise, flutter, and high blade stress.
• Leakage flow causes low static pressure in the core area, increasing the risk of cavitation and blade damage.
• The leakage velocity is given as:

                QL = 2 ( ( Pp - Ps ) / ρ )1/2

• The leakage flow sheet due to velocity induced by the vortex is given in Rains, 1954:

               a/τ = 0.14 ( d/τ  ( CL )1/2 )0.85

• Total loss in clearance volume is given by two equations-

               ζL ~ ( CL2 * C * τ * cos2β1 ) / ( A * S * S * cos2βm )

               ζW ~ ( δS* + δP* / S ) * ( 1 / A ) * ( ( CL )3/2) * ( τ / S )3/2Vm3 / ( V2 * V12 )

• Axial compressor
• Centrifugal
• Centrifugal compressor
• Centrifugal fan
• Centrifugal pump
• Francis turbine
• Kaplan turbine
• Mechanical fan
• Secondary flow
• Turbomachinery

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