Effects of Mach number and shock losses in turbomachines

For any kind of turbomachine operation at very high ranges of speed, it is mandatory to bear substantial compressibility effects. High compressibility effects lead to high variation in density of the flow medium. Although high Mach number signifies high mass flow per unit area as well as high pressure ratios across the stage, but excessive increase can result in shock wave generation which decreases the efficiency of the Turbo-machine owing to entropy generation. As we are dealing with compressors which majorly concerns with pressure variation across the stage, so for that analysis, the empirical formulation to be used is

p₂/ p₁ = 1+(C_p*γ*M²/2)

where
p = Static pressure
C_p = Pressure drop or rise coefficient
γ = Specific heat of gas
M = Mach number

For high subsonic flows, the Critical Mach number (M_1cr) is a characteristic value of considerable importance. It is the Mach number for which sonic conditions are reached locally in flow field i.e. there will be no shocks. So as to minimize shock losses and profile losses, the turbo-machine should be operated below M_1cr. The Mach number range can also be extended beyond M_1cr by carefully designing blade shapes.^[1]

As the relative Mach number increases, so does the value of C_p, leading to an increase in static pressure in compressors, and, in turn, rises in the boundary-layer thickness and losses. So for a given incidence, off-design losses will increase with increasing Mach number and there will be an drastic increase close to a critical Mach number, resulting in shock waves inside passage.

Shock loss coefficient vs Mach number

Estimation of shock losses

For consideration of two-dimensional shock losses for a compressor, three major contributing factors have been considered:

1. Bluntness of the leading edge with Supersonic upstream Mach Number.
2. Location and strength of passage shock.
3. Losses from boundary growth and shock-boundary-layer interaction. (Very small for weak shocks)

Koch and Smith in 1976 were the first to develop some correlation for estimation of shock loss coefficient (ζ_sh). The models used for estimation were empirical correlations for leading-edge losses and passage shock loss model. They assumed that the passage shock loss is equivalent to entropy rise of oblique shock that reduces passage inlet Mach number to unity. The results shown by their experiments is shown in the following figure:

This graph shows relationship between shock loss coefficient and Rotor inlet Mach number

^[2]

Formation of separation bubble and trailing edge shock structure

The formation of separation bubble

^[3] The adverse pressure gradient which can be caused by the blade loading factor may result in the separation of the boundary layer. The layer may become separated at the trailing edge and reattach depending on the flow separation, causing the formation of a separation bubble. The flow becomes fully turbulent after the reattachment and the boundary layer will separate near the trailing edge as the incident shocks strike the suction surface.

Boundary layers in turbines are thinner than in compressors, and increasing the Mach number along the suction surface up to the trailing edge makes the layer thinner. A laminar boundary layer tends to become separated due to the adverse pressure gradient, while the turbulent has less of a tendency to separate. However, the turbulent boundary layer will have higher viscous loss compares to laminar one.^[4]

^[5] Shock is formed in transonic turbines at the trailing edge, when the Mach number reaches unity. When the Mach number is increased further, the normal shocks transform to oblique shocks. The shock formed on the pressure surface will impinge onto the suction surface of the blade and is reflected back as a shock. The incident shock on the suction surface will produce pressure rise. The viscous layer near the impingement point increases its momentum and thickness to overcome the pressure rise in that region, resulting in separation of the localized boundary layer.

Key Losses in Turbine Blades

^[6]

1. Profile loss associated with boundary layer growth.
2. Shock loss arising from normal or oblique shocks at trailing edge.
3. Mixing loss due to rapid dissipation of the wake and shock-boundary layer interaction.

The profile loss in a transonic turbine consists of (1) the loss due to the boundary layer and wake (ζ_p), (2) the loss associated with the trailing edge shock system (ζ_sh)

Trailing edge flow in a turbine and loss model

In transonic flow, the trailing edge shock system and its interaction with blade boundary layer and wake may surpass the losses due to the blade boundary layer alone. The trailing edge and mixing losses account for the large portion for the total losses. Mee et al. (1990) recorded 70-90% of the total losses associated with the trailing edge shock,wake mixing and separation bubble. In some cases,the total loss can be 100% due to the major key losses at transonic flow, exceeding mach number unity. The trailing edge losses are dependent on blade profile, including curvature, near the trailing edge and thickness.

Previous Researches

Experimental curves of individual losses

Mee et al. (1992)

^[7] Mee et al. (1992) carried out a systematic experimental program to identify contributions to the loss from various sources in a turbine blade row. The investigations were carried out at a blow-down wind tunnel.

Cascade parameters taken are as follows:

α₁= 42.8°

M₁= 0.31

α₂= -68.0°

M₂= 0.92

Chord (C) = 230.7 mm

Span (S) = 252.1 mm

• The profile loss (ζ_p) is due to boundary layer alone. This was calculated from the wake boundary layer survey. The loss is estimated by subtracting the actual kinetic energy from that which would be present in the absence of the boundary layer.
• The shock losses (ζ_sh) are estimated from inviscid consideration. This only includes shock losses in the inviscid region and does not account for the mixing or shock-boundary layer interaction losses.
• The mixing losses, given by: ζ_m = ζ_T - ζ_p - ζ_sh
• The total losses ζ_T are determined from a survey of the flow field downstream, from which values of ζ_p and ζ_sh are calculated, where (ζ_T)_ref is the loss at the design conditions (shock free).^[8]

It is the evident that boundary layer profile loss dominates at subsonic exit mach numbers and the downstream wake mixing is about 30% of the total loss. But when the shock wave develops, the shock and the mixing losses dominates with nearly 100% increase in total losses at M_exit = 1.2. Part of mixing losses can be attributed to either the shock (which brings about sudden increase in the thickness of boundary layer) or the wake mixing losses. Wake width measured downstream increases rapidly with an increase in the mach number, while the wake width is nearly identical at the trailing edge.^[9]

Martelli and Boretti (1985)

Effect of mach number on total losses in a turbine cascade (Martelli and Boretti,1985)

^[10] They developed a method for transonic turbine cascade. For Mach number (M)>1.2 at exit conditions, a two oblique shock structure is formed at the trailing edge. Trailing edge shock is produced at the downstream of the shock due to flow acceleration. The mixing of two supersonic jets, results in a re-attachment shock.

Following calculations are to be considered to calculate the Losses:

1. Pressure distribution:^[11]
   • Pressure-correction method (PCM), developed by Pratap and Spalding (1976).
   • Pressure substitution method (PSM), developed by Hobson and Lakshminaranaya (1991).
2. Trailing edge flow: Analysis of the supersonic exit.^[12]
3. Boundary layer calculations:
   • Nature of boundary layers.^[13]
   • Numerical solution of Boundary layer Equations.
4. One-dimensional analysis is carried out in a control volume to estimate all the losses.^[14]

Maritelli and Boretti computed and compared measured losses for various types of blading.

In the case of Mee et al. (1992), the sudden increase in Mach number from 0.9-1.0 increases the losses due to shock and substantially the total loss. While in case of Maritelli and Boretti, the maximum losses occur beyond M₂ = 1.0. In the later case, as the mach number (M) is increased, the shock wave emanating from one vane swings downstream and this impinges further downstream on the suction surface resulting in lower losses.

See also

• Three-dimensional losses and correlation in turbomachinery
• Centrifugal compressor
• Centrifugal pump
• Flow through cascades
• Centrifugal Fan Design

Notes

1. section 3.4 , Lakshminarayana B. (1996)
2. section 6.4.1 , Lakshminarayana B. (1996)
3. section 2.2.1 , Teik Lin Chu (1999)
4. section 2.2.1 , Teik Lin Chu (1999)
5. section 2.2.2 , Teik Lin Chu (1999)
6. section 2.3 , Teik Lin Chu (1999)
7. section 6.4.2, pg 567,Lakshminarayana B. (1996)
8. section 6.4.2, pg 567,Lakshminarayana B. (1996)
9. section 6.4.2, pg 568,Lakshminarayana B. (1996)
10. section 6.4.2, pg 569,Lakshminarayana B. (1996)
11. section 5.7 ,Lakshminarayana B. (1996)
12. section 3.5.1.4, Lakshminarayana B. (1996)
13. section 5.5, Lakshminarayana B. (1996)
14. section 6.4, Lakshminarayana B. (1996)

References

• Lakshminarayana, B. Fluid Dynamics and Heat Transfer of Turbomachinery; New York: John Wiley & Sons, 1996. ISBN 978-0-471-85546-0
• Effects of Mach Number and Flow Incidence on Aerodynamic Losses of Steam Turbine Blades; Copyright 1999, Teik Lin Chu.