Effects of Low Reynolds Number on Loss Characteristics in

Effects of Low Reynolds Number on Loss Characteristics

in Transonic Axial Compressor

By Minsuk CHOI,1Þ Je Hyun BAEK,2Þ Jun Young PARK,3Þ Seong Hwan OH4Þ and Han Young KO4Þ

1ÞMyongji University, Yongin, Korea2ÞPohang University of Science and Technology, Pohang, Korea

3ÞKorea Institute of Machinery & Materials, Daejeon, Korea4ÞAgency for Defense Development, Daejeon, Korea

(Received June 11th, 2009)

A three-dimensional computation was conducted to understand effects of low Reynolds numbers on loss character-

istics in a transonic axial compressor, Rotor 67. As a gas turbine becomes smaller and it operates at high altitude, the

engine frequently operates under low Reynolds number conditions. This study found that large viscosity significantly

affects the location and intensity of the passage shock, which moves toward the leading edge and has decreased intensity

at low Reynolds number. This change greatly affects performance as well as internal flows, such as pressure distribution

on the blade surface, tip leakage flow and separation. The total pressure ratio and adiabatic efficiency both decreased by

about 3% with decreasing Reynolds number. At detailed analysis, the total pressure loss was subdivided into four loss

categories such as profile loss, tip leakage loss, endwall loss and shock loss.

Key Words: Low Reynolds Number, Loss, Transonic Axial Compressor, Rotor 67

Nomenclature

Aw: endwall area

C: chord length

Cc: contraction coefficient

Cd: dissipation coefficient

Cv: specific heat capacity

Cx: axial chord length

h: blade height

M: Mach number

m: mass flow rate

mj: mass flow rate across tip clearance

p: static pressure

pt: total pressure

s: pitch

S: entropy generation

Sew: entropy generated on endwall

t: blade thickness

T : static temperature

V : absolute velocity on inertial coordinate

V�: surface velocity relative to endwall

W: relative velocity

W�: surface velocity on blade surface

W�p: surface velocity on pressure surface

W�s: surface velocity on suction surface

Greek

�: angle between tip leakage flow and main flow

�: flow angle

�: specific heat ratio

�: increment or decrement

��: displacement thickness

�: air density

�bl: boundary layer loss coefficient

�el: endwall loss coefficient

�tl: tip leakage loss coefficient

�tr: trailing edge loss coefficient

�: height of tip clearance

Subscripts

1: inlet in computational domain

2: outlet in computational domain

1. Introduction

The Reynolds number based on operating conditions of

gas turbines has been decreasing recently, because aero-

engine designers are trying to reduce gas turbine size and

the need to operate gas turbines at high altitudes is increas-

ing for high-altitude UAVs (Unmanned Air Vehicles),

mainly used in military surveillance and reconnaissance. It

is known that wall boundary layer thickens and separation

occurs frequently on blade surfaces with decreasing Rey-

nolds number. Because these degraded internal flows have

a negative effect on axial turbomachinery performance,

many researchers are investigating the effects of low Rey-

nolds number.

From the design viewpoint, a low Reynolds number con-

dition is often attributed to low air densities at high altitudes.

Weinberg and Wyzykowski1) investigated the effects of low

Reynolds number on a gas turbine with NASA. The PW545

jet engine, designed to operate at 13,700m altitude, was

tested above 18,300m. They reported that the engine effi-

ciency and performance were reduced by the low air density� 2010 The Japan Society for Aeronautical and Space Sciences

Trans. Japan Soc. Aero. Space Sci.

Vol. 53, No. 181, pp. 162–170, 2010

at high altitude. Castner et al.2) compared their numerical

results with the experimental data of Weinberg and

Wyzykowski.1) There was a good agreement between the

test data and the calculated efficiency and performance within

Reynolds numbers from 30,000 to 295,000. Schreiber et al.3)

researched transition phenomena on the blade surface in a

compressor cascade by changing the Reynolds number

and turbulence intensity, finding a laminar separation bub-

ble with reattachment on the suction surface at relatively

low Reynolds number. Van Treuren et al.4) ran wind tunnel

tests at Reynolds numbers from 25,000 to 50,000 and found

that a large separation near the trailing edge of a turbine

cascade degraded total pressure loss. After wind tunnel tests

on a turbine cascade with tip clearance, Matsunuma5) ana-

lyzed the relationship between tip leakage flow, Reynolds

number and turbulence intensity, reporting that the Rey-

nolds number and free-stream turbulence intensity don’t

affect tip clearance loss although they change the distribu-

tion of the internal flow. Using experiments on total pressure

loss change caused by low Reynolds numbers from 32,000

to 128,000 in a small one-stage axial turbine, Matsunuma

and Tsutsui6) showed that the region with high turbulence

intensity due to the wake and the flow fluctuation due to

the stator-rotor interaction increase with decreasing Rey-

nolds number.

Although there are some studies on the effect of low

Reynolds numbers on the internal flow in turbomachinery,

most of them have focused on flow phenomena but not on

loss characteristics. After extensive numerical simulations,

Choi et al.7) analyzed the effects of Reynolds number on

profile loss, tip leakage loss and endwall loss in a subsonic

axial compressor, reporting that the tip leakage loss drops

with low Reynolds number due to weaker mixing of the

tip leakage and mainstream flows. However, the internal

flow in a transonic compressor is more complex due to

the presence of shocks.8–13) This study investigated the

effects of low Reynolds numbers on loss characteristics

in a transonic compressor using numerical simulations.

A comprehensive analysis of the loss mechanism was

attempted by applying Denton’s loss model14) to the com-

putational result.

2. Test Configuration

2.1. Geometric specifications

We conducted a numerical study on the effect of low

Reynolds number on internal flows and loss characteristics

using a transonic axial compressor, Rotor 67, which has

been tested previously by Strazisar et al.15) Figure 1 shows

the schematic diagram of Rotor 67 and measurement posi-

tions. This rotor rotates about its axis at 16,043 rpm with a

relative tip Mach number of 1.38. The mass flow rate and

pressure ratio between the inlet and outlet are 33.25 kg/s

and 1.63 at the design condition respectively. The Reynolds

number is about 1:17� 106 based on the inlet velocity at the

design condition and meridional blade chord length on the

hub. This rotor has 22 blades with an aspect ratio of 1.56.

Detailed design parameters are summarized in Table 1.

The spanwise distribution of the static and total pressures,

the total temperature and the flow angle were measured by

Strazisar et al.15) at aero-station 1 and 2 located upstream

and downstream of the rotor.

2.2. Computational grid

Using the blade profile given by Strazisar et al.,15) the

computational domain was fixed in the region including

aero-station 1 and 2, and a multi-block hexahedral mesh

was generated as shown in Fig. 2. Each passage consists

of 117 nodes in the streamwise direction, 57 nodes in the

pitchwise direction, and 53 nodes in the spanwise direction.

To capture the tip leakage flow accurately, the region of the

tip clearance was filled with an embedded H-type grid,

which has 52 nodes from the leading edge to the trailing

edge, 9 nodes across the blade thickness and 9 nodes from

the blade tip to the casing. Therefore, the computational

domain has about 350,000 nodes.

2.3. Numerical method

Simulations of the three-dimensional flow were con-

ducted using the in-house flow solver called TFlow, which

has been improved to calculate internal flow in turbomachin-

ery since its development in the mid-1990s.16) This flow

Fig. 1. Schematic diagram of Rotor 67 showing measurement positions

(Strazisar et al.15)).

Table 1. Design parameters of Rotor 67.

Number of rotor blades 22

Rotational speed (rpm) 16,043

Mass flow rate (kg/s) 33.25

Pressure ratio 1.63

Rotor tip speed (m/s) 429

Tip clearance (cm) 0.061

Inlet tip relative Mach number 1.38

Rotor aspect ratio 1.56

Rotor solidityHub 3.11

Tip 1.29

Tip diameter (cm)Inlet 51.4

Exit 48.5

Hub/tip ratioInlet 0.365

Exit 0.478

Nov. 2010 M. CHOI et al.: Effects of Low Reynolds Number on Loss Characteristics in Transonic Axial Compressor 163

solver has been pre-validated by a series of calculations in a

transonic axial compressor, a subsonic axial compressor and

a subsonic axial turbine, and has also been used to simulate

rotating stall in a subsonic rotor.13,17–19) TFlow uses com-

pressible RANS (Reynolds Averaged Navier-Stokes) equa-

tions as the governing equations, discretized in space by the

finite volume method. An upwind TVD (Total Variational

Diminishing) scheme based on Roe’s flux difference split-

ting method was used to discretize inviscid flux terms and

the MUSCL (Monotone Upstream Centered Scheme for

Conservation Law) technique was used to discretize

viscous flux terms. The equation was solved using the Euler

implicit time marching scheme with first-order accuracy

to obtain a steady solution. The laminar viscosity was calcu-

lated by Sutherland’s law and the turbulent viscosity was

obtained by the algebraic Baldwin-Lomax model.20)

2.4. Boundary conditions and Reynolds number

At internal flow simulations of turbomachinery, there are

four types of boundaries such as inlet, outlet, wall, and

periodic conditions. The total pressure, total temperature,

whirl, and meridional flow angle were fixed at the inlet con-

dition and the outgoing Riemann invariant was taken from

the interior domain. The inlet total pressure was specified

as a constant in the core flow and was reduced in the endwall

regions with the 1/7 power law velocity profile. The inlet

boundary layer thickness on the hub and the casing was

set to be 12mm, which was estimated from the experimental

data. The inlet total temperature was also specified as a

constant. The whirl velocity was set as zero, and the radial

velocity was chosen to make the flow tangential to the

meridional projection of the inlet grid lines. For the outlet

condition, the static pressure on the hub was specified and

the local static pressures along the span were given by using

the SRE (Simplified Radial Equilibrium) equation. Other

flow variables such as density and velocity were extrapo-

lated from the interior. To calculate velocity components

on the wall, the relative velocity was set to be zero on the

blade surface and rotating part of the hub, and the absolute

velocity on the casing and stationary part of the hub was set

to be zero as the no-slip condition. The surface pressure and

density were obtained using the normal momentum equation

and the adiabatic wall condition respectively. Since only one

blade passage was used here, the periodic condition was

implemented using the ghost cell next to the boundary cell

and enabled flow variables to be continuous across the

boundary.

To study the effect of Reynolds number on loss charac-

teristics in a transonic axial compressor, the boundary

conditions at each operating point were fixed and only the

Reynolds number was changed by varying the kinematic

viscosity. Referring to the U.S. standard atmosphere,21) the

density and viscosity at 20,000m altitude are reduced by

the ratio of 1 to 0.07 and 1 to 0.77 respectively in compar-

ison to the values at sea level. The Reynolds number at

20,000m is 10% of that at sea level, if other operating con-

ditions except for the density and viscosity are the same.

Therefore, simulations were conducted at five different

Reynolds numbers between 1:17� 105 and 1:17� 106,

based on the inlet velocity and meridional blade chord

length on the hub. We described the lowest Reynolds num-

ber (1:17� 105) as Low Reynolds number or Low Re, and

the highest Reynolds number (1:17� 106) as Reference

Reynolds number or Ref. Re.

3. Computational Results

3.1. Performance curves

The total pressure ratio and adiabatic efficiency were

calculated at the design speed to estimate the performance

of Rotor 67. The results are shown in Fig. 3, where the hor-

izontal axis is represented by the normalized mass flow rate,

defined as the ratio of the mass flow rate to the choked mass

flow rate. It can remove some uncertainties in the mass flow

rate measurement in an experiment. The choked mass flow

rate was 34.96 kg/s in the experiment and 34.57 kg/s in

the simulation at Ref. Re, where the difference was within

1%. The total pressure ratios from the computation agreed

well with the experimental values at Ref. Re. The values

at Low Re have a decrement of about 3% at the same flow

condition, compared to Ref. Re. The numerical surge at

Low Re occurs at higher normalized mass flow rate than

Ref. Re, resulting in the severe decrease in the compressor

operating range. The computed adiabatic efficiency agrees

well with the experimental value, although the peak position

slightly deviates from the experimental one. There is also a

3% efficiency drop with decreasing Reynolds number. This

decrement is the same as the total pressure ratio drop. Under

the same simulation conditions except for Reynolds number,

the operating point moves along lines A and B on the

performance curves. The Reynolds number affects the mass

flow rate as well as the total pressure ratio and adiabatic

efficiency. It should be noted that the decrement in the

mass flow rate is also the result of the change in Reynolds

number.

3.2. Internal flows

From here on, all flow data and loss characteristics are

calculated at the peak efficiency with Reynolds number

change. Figure 4 shows the spanwise distribution of the

Fig. 2. Computational grid.

164 Trans. Japan Soc. Aero. Space Sci. Vol. 53, No. 181

mass-averaged static pressure, total pressure, total tempera-

ture and exit flow angle with experimental data. The com-

puted values at Ref. Re are in good agreement with the

experimental values, although the static and total pressures

are slightly under-predicted near the casing. At Ref. Re,

the static pressure increases along the span from the hub

to the casing. The flow angle decreases along the span up

to the 90% span, but it increases rapidly due to blockage

of the tip leakage flow above the 90% span. The total pres-

sure and temperature are almost constant in the core flow

region, but the former decreases and the latter increases near

the casing, meaning that the tip leakage flow generates large

loss in this region. At Low Re, the static pressure distribu-

tion is similar to its counterpart, but it becomes smaller

above the mid-span. The flow is over-turning from the hub

to the 90% span but under-turning near the casing. The total

pressure and temperature become small in the core flow

region but large near the casing unlike the reference case,

meaning that the total pressure loss increases from the hub

to the 90% span but decreases near the casing with decreas-

ing Reynolds number.

Figure 5 compares the experimental and computed rela-

tive Mach number contours at the 30%, 70% and 90% spans

from the hub. The calculated values at Ref. Re agree well

with the experimental values and a shock structure is clearly

observed. There are two shocks above the 70% span, such as

a bow shock at the leading edge and a passage shock across

the flow passage. The boundary layers on the blade surface

grow rapidly downstream of the passage shock. The flow

at the 30% span from the hub is well-behaved without a

passage shock although the flow field has a small supersonic

region at the leading edge. At Low Re, relatively large

viscosity adds dissipative effects throughout the flow field

but it does not affect the bow shock significantly because

the flow field upstream of it is nearly inviscid. However,

there are considerable changes in the location and intensity

of the passage shock. The passage shock at the 70% and

90% spans moves toward the leading edge, enabling the

boundary layer to develop more upstream than for Ref. Re

and a separation to occur at the trailing edge on the suction

surface. The Mach number contours also show that the in-

tensity of the passage shock becomes weak. The pressure

jump across this weak shock decreases and makes the static

pressure above the 60% span become smaller than the refer-

ence case (Fig. 4(a)). The decrease of the normalized mass

flow rate caused by low Reynolds number leads to these

different shock structures, because the shock location in a

transonic compressor depends on the upstream Mach

number, which is equivalent to mass flow rate in this case.

The overall flow field at the 30% span has a similar feature

regardless of Reynolds number, but the flow is more dif-

fused through the passage due to the large viscosity.

Figure 6 shows the static pressure distribution on the

blade surface at several spanwise positions, where it was

non-dimensionalized by the inlet total pressure. At Ref.

Static pressure ratio

Span

(%

)

0.9 1 1.1 1.2 1.30

20

40

60

80

100

Exp.Ref. ReLow Re

Exit flow angle (degree)

Span

(%

)

30 40 50 600

20

40

60

80

100

(a) Static pressure ratio (b) Flow angle

Total pressure ratio

Span

(%

)

1 1.2 1.4 1.6 1.8 20

20

40

60

80

100

Total temperature ratio

Span

(%

)

1.05 1.1 1.15 1.2 1.250

20

40

60

80

100

(c) Total pressure ratio (d) Total temperature ratio

Fig. 4. Exit flow data along span at aero-station 2.

0.92 0.94 0.96 0.98 1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

15))Ref. Re7/10 Ref. Re5/10 Ref. Re3/10 Ref. Re1/10 Ref. ReLine A

Numerical surge

(a) Total pressure ratio

Normalized mass flow rate0.92 0.94 0.96 0.98 1

0.7

0.8

0.9

1.0

Exp. (Strazisar et al.15))Ref. Re7/10 Ref. Re5/10 Ref. Re3/10 Ref. Re1/10 Ref. ReLine B

Peak efficiency

(b) Adiabatic efficiency

Normalized mass flow rate

Tota

l pre

ssur

e ra

tioPeak efficiency

Exp. (Strazisar et al.

Adi

abat

ic e

ffic

ienc

y

Fig. 3. Performance curves.


Re, the distribution clearly shows the passage shock position

and the load variation in the streamwise direction at each

span. The pressure jump occurs at the pressure side leg of

the passage shock near the 30% chord and at the suction side

leg near the 80% chord above the 70% span, while there is

no shock below the 50% span. The static pressure distribu-

tion near the casing is important because pressure difference

in this region drives tip leakage flow. At the 99% span from

the hub, the static pressure on the pressure surface increases

rapidly due to the passage shock and it is nearly constant

behind the pressure side leg of the shock. The static pressure

on the suction surface decreases continuously until it in-

creases behind the suction side leg of the shock. The pres-

sure difference starts increasing just behind the pressure side

leg and it has the maximum value just before the suction

side leg of the passage shock. Therefore, the strong roll-up

of the tip leakage flow starts near the 30% chord at Ref.

Re. At Low Re, each leg moves upstream so the pressure

side leg is located at the 20% chord and the suction side

leg at the 70% chord from the leading edge. The pressure

jump becomes smaller because the passage shock is weaker

than that at Ref. Re. At the 70% span, the effect of the pas-

sage shock is so small that it is not easy to recognize its

location. Near the casing, the pressure difference has large

values between the pressure and suction legs but its value

is much smaller than that at Ref. Re. This small pressure dif-

ference weakens the tip leakage flow and retards the vortex

formation.

Tip leakage flow is generated by the pressure difference

between the pressure and suction surfaces near the casing.

(a) Experiment (Strazisar et al. 15) )

0.95

1.41.35

1.051

0.9

1.3

1.35

1.2

at 90% span from the hub

1.2

1.2

1.15

1.3

1.15

1.1 1

0.8

1.2

1.2

1.25


0.951.05

0.90.85

0.8

0.7 0.6

1.15


(b) Ref. Re

0.95

1.41.35

1.051

1.3

1.35

1.21.1

1.3


1.2

1.2

1.15

0.75

1.1

1.25

1

1.3

0.95

0.9

1.15


0.951.05

0.85

0.75

0.7

0.8

1.050.6


(c) Low Re

Fig. 5. Relative Mach number contours.


Figure 7 shows the static pressure distribution near the

casing and the particle traces placed just before and after

the pressure side leg of the passage shock. Clearly, the pas-

sage shock moves upstream toward the leading edge with

decreasing Reynolds number and the particle traces coincide

with the static pressure trough in both cases. The pressure

trough at Ref. Re is deeper than that at Low Re, meaning that

the tip leakage flow is stronger in the former case. Accord-

ing to particle motion, the tip leakage flow at Ref. Re is more

vortical than that at Low Re due to strong roll-up caused by

the large pressure difference across the tip. As a result of

different tip leakage flow intensities, the passage shock is

severely disturbed by the strong interaction with the tip leak-

age flow at Ref. Re while it is hardly disturbed at Low Re.

To inspect the structure of separations on the blade

surfaces, limiting streamlines are shown in Fig. 8. Similar

streamlines are found on the pressure surface regardless of

Reynolds number. The limiting streamlines near the tip

are more disturbed by tip leakage flow at Ref. Re, because

it has strong roll-up and larger spiral motion than its count-

erpart. However, the limiting streamlines on the suction

surface are significantly affected by Reynolds number. At

Ref. Re, three separations are found at the leading edge near

the hub (A), at the trailing edge near the hub (B) and near

the tip (C). The first is caused by the large incidence angle

near the hub. The second is generally called hub-corner-

separation and is frequently developed by large deflection

of the blade in a subsonic flow regime. The last is caused

by the passage shock and located just behind the suction side

leg of the shock. At Low Re, some different features in each

separation may be observed, although the three separations

are still found on the blade surface. The separation at the

leading edge (A0) is larger than its counterpart (A). The

hub-corner-separation expands to become the full-span

separation (B0) at the trailing edge. In the results of

Van Treuren et al.4) and Choi et al.,7) similar separations

originated on the suction surface of the turbine cascade

and the subsonic compressor rotor, resulting in large total

pressure loss. The separation line (C0) is also caused by

x/Cx

p/p t

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.830% span from hub50% span from hub70% span from hub90% span from hub99% span from hub

Passage shock

Passage shock

(a) Ref. Re

x/Cx

p/p t

0 0.25 0.5 0.75 10.4

0.6

0.8

1.0

1.2

1.4

1.6

1.830% span from hub50% span from hub70% span from hub90% span from hub99% span from hub

Passage shock

Passage shock

(b) Low Re

0 0.25 0.5 0.75 1

Fig. 6. Non-dimensional static pressure distribution.

M=1M=1

(a) Ref. Re (b) Low Re

Fig. 7. Static pressure contours and particle traces ejected from pressure

side leg of passage shock.

TELE TELE

(a) PS, Ref. Re (b) PS, Low Re

LETE

B

A

C

LETE

B'

A'

C'

(c) SS, Ref. Re (d) SS, Low Re

Fig. 8. Limiting streamlines on blade surfaces.


the passage shock and is located further upstream because

the shock moves upstream with decreasing Reynolds

number.

3.3. Loss characteristics

Figure 9 shows the mass-averaged total pressure loss of

the transonic rotor with Reynolds number. The calculated

loss is almost constant from Ref. Re to 350,000, but it

increases rapidly near 100,000 as described by Fielding.22)

For reference, Choi et al.7) found that the total pressure loss

in a subsonic single rotor is distributed between the �0:35

and �0:2 powers of Reynolds numbers below 250,000.

These two results suggest that a compressor might have a

critical Reynolds number near 200,000 and 300,000 like

a turbine, but we need more test cases to confirm this. In

addition, the total pressure loss change with Reynolds

number is less significant in a transonic compressor than

in a subsonic compressor, meaning there are some differ-

ences in the loss mechanism between transonic and subsonic

compressors. To analyze the effects of low Reynolds num-

ber on total pressure loss, we divide total pressure loss into

four loss categories using Denton’s loss model:14) profile

loss, tip leakage loss, endwall loss and shock loss. The

surface velocity, which is frequently used to calculate each

loss, means the velocity at the outer edge of the boundary

layer and the method for obtaining this from the numerical

result has been explained in the previous study.7)

The loss generated by the blade itself is called profile loss

and it is composed of two parts: loss generated in boundary

layers on the blade surface and loss caused by wake mixing.

The former is proportional to the cube of the surface veloc-

ity and is calculated using Eq. (1).

�bl ¼1

0:5s cos �1

Z C

0

Cd

W�

W1

� �3

dx ð1Þ

where the dissipation coefficient, Cd, is equal to 0.002 as

suggested by Denton and Cumpsty.23) The trailing edge loss

can be calculated using the following equation when block-

age is dominant due to separation on the blade surface.

�tr ¼�� þ t

s cos �2

� �2

ð2Þ

The boundary layer loss and trailing edge loss calculated

using Eqs. (1) and (2) are shown in Fig. 10. The boundary

layer loss is closely related to inlet velocity profile because

this loss is proportional to the cube of the surface velocity on

the blade as mentioned above. The overall distributions

along the span are similar because an identical inlet velocity

profile was imposed, but the magnitude of the loss decreases

due to reduction of mass flow rate with decreasing Reynolds

number. The trailing edge loss at Ref. Re has a large value

from the hub to the 20% span due to the hub-corner-separa-

tion, but it diminishes steadily above the 40% span because

there is no separation. This loss has a non-physical value

above the 90% span because this region is included in the

region where tip leakage flow has a major effect. At Low

Re, the large trailing edge loss is generated by the full-span

separation at the trailing edge from the hub to the 80% span,

and it is much larger than that at Ref. Re over the full span.

The tip leakage flow mixes into the main flow and com-

plicates the internal flow, resulting in large loss. The loss

caused by tip leakage flow is calculated using the following

Eq. (3).

�tl ¼2Cc

s cos �1

Z C

0

�

h

W�s

W1

� �2

1�W�p

W�s

� � ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1�

W�p

W�s

� �2s

dx

ð3Þwhere the contraction coefficient, Cc, is set to be 0.3. The

calculated tip leakage loss is shown in Fig. 11 and decreases

with decreasing Reynolds number. This trend is the same

as the subsonic rotor in the previous study,7) while

Matsunuma5) has shown that tip leakage loss in a turbine

cascade is almost constant regardless of Reynolds number.

This tendency might be explained by the original form used

in deriving Eq. (3).

�pt

0:5�W�s2¼

mj

m1

2�W�p

W�s

� �ð4Þ

In this equation, the total pressure defect caused by tip leak-

age flow is proportional to mj=m1 and 2�W�p=W�s, where

W�p=W�s indicates flow angle between the tip leakage flow

Span (%)

ζ bl

0.00

0.01

0.02

0.03Ref. ReLow Re

(a) Boundary layer loss

Span (%)

ζ tr

0 20 40 60 80 1000.00

0.02

0.04

0.06Ref. ReLow Re

(b) Trailing edge loss

0 20 40 60 80 100

Fig. 10. Profile loss along span.

Reynolds Number0 500000 1000000

0.20

0.25

0.30

0.35

Re-0.2

Calculated LossTrend line

Re-0.2

Tota

l Pre

ssur

e L

oss

Fig. 9. Total pressure loss with Reynolds number.


and main flow as cos� ¼ W�p=W�s. The mass flow ratio at

the tip clearance and inlet, mj=m1, and the surface velocity

ratio on the pressure and suction surfaces, W�p=W�s, are cal-

culated and the results are shown in Fig. 12. The value of

W�p=W�s increases as Reynolds number decreases, while

the mj=m1 is almost constant regardless of Reynolds

number. This means that the tip leakage loss drops with

decreasing Reynolds number because mixing between the

tip leakage flow and main flow weakens.

The loss generated in the boundary layer on the hub and

casing is called endwall loss. The endwall loss coefficient

is calculated using a similar method to the boundary layer

loss with the surface velocity relative to the endwall, and

non-dimensionalzed by the inlet dynamic head.

�el ¼

T1�Sew

0:5mV12

(for stationary wall)

T1�Sew

0:5mW12

(for moving wall)

8>>>><>>>>:

ð5Þ

�Sew ¼Z Aw

0

Cd�V�3

TdA

Because the loss on the casing is severely influenced by tip

leakage flow, it is difficult to classify into tip leakage loss

and endwall loss. Therefore, only endwall loss on the hub

is shown in Fig. 13. The endwall loss is almost constant

from Ref. Re to about 350,000 but drop slightly around a

Reynolds number of 100,000. Like the boundary layer loss

on the blade surface, it is probably due to the relatively small

surface velocity near the endwall at Low Re.

Assuming that the passage shock is a single shock wave

and it is sufficiently weak, the entropy generation across

the shock is represented by a function of the static pressure

as Eq. (6).

�S � Cv

�2 � 1

12�2

�p

p1

� �3

þO�p

p1

� �4

ð6Þ

Because the passage shock exists above about the 60% span

from the hub, the entropy generation is calculated in this re-

gion and the result is shown in Fig. 14. The entropy gener-

ation is almost constant over 7/10 Reynolds number but de-

creases significantly with further decreases in Reynolds

number, because the intensity of the passage shock becomes

weak as shown in Fig. 5. The weak shock diminishes the

pressure increment across the shock, causing the reduced

entropy generation.

4. Conclusion

Three-dimensional numerical simulations were con-

ducted at five different Reynolds numbers between

1:17� 105 and 1:17� 106 to study the effects of low Rey-

nolds numbers on loss characteristics in a transonic axial

compressor. Compared with Ref. Re, the performance at

Low Re dropped by about 3% and mass flow rate also dimin-

ished significantly. The passage shock at Low Re moved

toward the leading edge and its intensity also decreased

Reynolds Number

End

wal

l Los

s,ζ el

0 500000 10000000.010

0.012

0.014

0.016

0.018

Fig. 13. Integrated endwall loss according to Reynolds number.

Span (%)∆S

(J/

g-k)

10080602.0

3.0

4.0 Ref. Re7/10 Ref. Re5/10 Ref. Re3/10 Ref. Re1/10 Ref. Re

Fig. 14. Entropy generation caused by passage shock along span.

Reynolds Number

Tip

Lea

kage

Los

s,ζ tl

0 500000 10000000.002

0.003

0.004

0.005

Fig. 11. Integrated tip leakage loss according to Reynolds number.

Reynolds Number

mj/m

1

Wδp

/ Wδs

0 500000 10000000.003

0.004

0.005

0.006

0.007

0.008

0.7

0.75

0.8

0.85mj/m1

Wδp/ Wδs

Fig. 12. Mass flow ratio on tip clearance and inlet and surface velocity

ratio.


compared to Ref. Re. This change had large effects on the

pressure distribution, separation on the blade surface and

tip leakage flow. The pressure difference across the blade

tip decreased with decreasing Reynolds number and it weak-

ened tip leakage flow. The passage shock movement and

large viscosity at Low Re caused a full-span separation on

the suction surface at the trailing edge. These changes in

internal flows also affected the loss characteristics. As the

Reynolds number decreased, the trailing edge loss increased

while the other losses decreased. The overall loss was al-

most constant from Ref. Re to 350,000, but increased rapidly

around 100,000. However, the increased loss in a transonic

compressor with decreasing Reynolds number was smaller

than in a subsonic compressor, because the former had a

relatively larger Reynolds number than the latter and also

because the decrease in boundary layer loss, tip leakage loss,

endwall loss and shock loss in a transonic compressor

compensated for the increase in trailing edge loss caused

by full-span separation.

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