Analysis of a curvature corrected turbulence model using a

90 degree curved geometry modelled after a centrifugal

compressor impeller

K. J. Elliott1, E. Savory

1, C. Zhang

1, R. J. Martinuzzi

2 and W. E. Lin

1

1Department of Mechanical and Materials Engineering

The University of Western Ontario, London, ON, Canada N6A 5B9

2Department of Mechanical and Manufacturing Engineering

University of Calgary, Calgary, AB, Canada T2N 1N4

Email: kellio9@uwo.ca

ABSTRACT

The effects of curvature on turbulence quantities and

the performance and functionality of a curvature

corrected SST (SST-CC) model are investigated.

Steady state simulations are run using a simplified

geometry that is similar in curvature and Reynolds

number to a typical centrifugal compressor design.

Velocity, turbulence kinetic energy and Reynolds

normal stress profiles, as well as production

multiplier, , and eddy viscosity contours are

compared between the RSM-SSG, SST-CC and SST

models. Overall, the SST-CC model showed an

appropriate sensitivity to curvature, however there

are still questions to be answered regarding the

effects of the term in certain regions.

1. INTRODUCTION

Flow and surface curvature are always present in

turbomachinery components. Some examples are the

curved blades of an axial machine, or the axial to

radial transition in a centrifugal machine. An

important part of understanding the flow physics in

these machines is identifying and knowing how to

deal with curvature. Curvature introduces an extra

level of complexity that can greatly affect the flow

structure and turbulence quantities. This adds to the

complexity of the flow field that is present in a

turbomachine. Thus, to fully understand and quantify

the effects of curvature on a turbulent flow, it is

important to limit as many intricacies as possible to

be able to isolate curvature effects.

To eliminate other effects, this study considers a

simplified geometry modelled after a centrifugal

compressor impeller. The simplified geometry is

designed with a similar curvature and operated at a

similar Reynolds number to a centrifugal compressor

currently being studied. With the simplified

geometry, complexities such as developing flow,

three dimensional blade curvature, high rotation rate,

and a converging cross section are eliminated, which

allows the focus to be directed towards the surface

curvature.

Curvature effects become particularly relevant when

using numerical modelling techniques. This is the

basis of “curvature corrected” turbulence models,

which use various methods to account for the

changes that arise due to curvature. The present

analysis will focus on a specific curvature corrected

model, specifically the SST-CC model of Smirnov

and Menter [1], and examine how it performs in

regards to curvature as compared to two other

uncorrected models. The eventual intentions are to

compare the simplified geometry, which resembles a

blade passage of a centrifugal compressor, to the

more complex compressor flow in terms of curvature

parameters.

2. BACKGROUND

A classical review by Bradshaw [2] describes the

effects of curvature as an “extra rate of strain” to the

already present principal strain. From a review of the

literature, it has been found that the effect of this

“extra rate of strain” depends on many factors that

include: the magnitude of the curvature, the

directionality of the curvature (convex (CVX) or

concave (CCV)), the Reynolds number and the

presence of pressure gradients [3-5]. All of these

factors must be taken into consideration when

analyzing a flow with curvature. Directionality of the

curvature is a dominant factor. It has been well

documented that a convex curvature will suppress (or

stabilize) the effects of turbulence, showing

decreased shear stress, turbulence kinetic energy and

turbulent mixing, whereas a concave curvature has

the opposite, destabilizing effect, showing increases

in those turbulence quantities [3].

The effects of curvature on turbulence and flow

structure has been extensively studied using various

simplified configurations such as 90 degree ducts,

180 degree U-turn ducts or rotating ducts using either

numerical or experimental techniques. A summary of

completed experiments using ducts can be found in

[6]. In general, the experiments focus on

investigating the flow characteristics relating to

curvature, while the numerical studies have

investigated the abilities of the most common

turbulence models available today ( , ,

SST, RSM) to predict the behavior of a curving flow.

More recently, the numerical studies tend to focus on

turbulence models that have been altered with

“curvature corrections”. The and

models have been corrected in different ways and

have shown improvements that are competitive with

more complex models such as Reynolds Stress

Models (RSM), while still maintaining the simplicity

of eddy viscosity models. Some examples of these

corrections can be found in [7,8].

A curvature corrected version of the SST model has

been recently developed [1] and has performed well

for various test cases, including some curved ducts.

This particular curvature correction is more attractive

than the and corrections mentioned

above since the SST model has been shown to

perform well in centrifugal compressor flows [9],

which ties in with relating these results to a

compressor geometry. Smirnov and Menter [1] have

effectively demonstrated the performance of the

developed SST-CC (curvature corrected) model,

however they do not provide an in-depth analysis of

the different quantities affected by curvature,

focusing primarily on mean flow field quantities.

Thus, the present research attempts to further study

the SST-CC model and improve the understanding of

the flow physics by analyzing various turbulence and

curvature related quantities in the SST turbulence

model and comparing them to the SST-CC results.

3. TURBULENCE MODELLING

Three different turbulence models are considered in

this analysis: the SST, SST-CC and RSM-SSG

models. Since there is no experimental data

available, a comparison is made between the SST and

SST-CC models to determine where differences

occur, or in other words, where the curvature

correction is applied. The RSM-SSG model acts as a

guideline for this comparison due to its anisotropic

nature and increased sensitivity to curvature as

compared to eddy viscosity based models.

The SST model uses the formulation in the

freestream and the formulation in the near wall

region, in combination with blending functions to

connect the two domains. The transport equations for

the SST model are given as [10]:

( )

( )

((

)

)

(1)

( )

( )

((

)

)

(2)

Where:

(

) (3)

Constants and details relating to Eqs. (1) – (3) can be

found in [10].

The SST-CC turbulence model was developed based

on a correction [11] to the Spalart-Allmaras (S-A)

one equation model. It consists of a multiplier to the

production term, , in the and equations of the

SST model (Eqs. (1) – (2), respectively [10]) that is

given by [1]:

* ( ) + (4)

Where:

( )

, ( )- (5)

The constants and are equal to 1.0, 2.0

and 1.0, respectively [1]. Note that the magnitude of

is greater than 1 for concave curvatures (enhanced

production) and less than 1 for convex curvatures

(decreased production).

In Eq. (5), the terms and are dependent on the

strain rate tensor, , the rotation tensor, , the

rotation rate of the system, , and a variable, ,

dependent on the strain and the turbulence eddy

frequency, , given by [1]:

(

)

(

)

(6)

[

( )

]

(7)

Where:

( ) (8)

√ (9)

√ (10)

The RSM-SSG model does not use the eddy viscosity

assumption and instead solves transport equations

(Eq. (11)) for the six individual Reynolds stresses,

given, where is the production term and is the

pressure-strain correlation term, given by Eqs. (12) –

(15) [12].

( )

((

)

)

(11)

(12)

(13)

[ (

)] (14)

√

(

)

( )

(15)

The constants for Eqs. (11) – (15) can be found in

[12].

4. GEOMETRY AND NUMERICAL METHOD

The geometry studied in this work, shown in Fig. 1,

was modelled after a centrifugal impeller. The full

model is shown on the left, with the 10 degree curved

portion of the full geometry shown on the right. A 10

degree section was chosen to roughly match the pitch

of the centrifugal impeller passage, but also to

drastically reduce the mesh complexity and the

computational time involved in running the case.

Periodic boundary conditions (shown in green in Fig.

1) were used to connect the full 360 degree model.

Since the basis of this work was to isolate curvature

effects, a long straight section was added to the inlet

of the curved region (see Fig. 2) to ensure that the

flow entering the curved region was fully developed.

The required entrance length was approximated using

the equation for turbulent flow through a pipe, given

by [13]:

(

)

(16)

Figure 1: Left: Full geometry, Right: 10 degree

section

Figure 2: Full computational domain

To enforce similar flow conditions, the inlet

boundary condition was set as a velocity inlet with a

Reynolds number (

) that

matched that of the centrifugal compressor blade

passage. The traditional Re formulation for

turbomachinery is , where is the

rotational speed, however the simplified geometry

does not rotate, so this alternate Re was used. The

outlet was set as a static pressure outlet, as was the

outlet in the centrifugal compressor case.

The geometry was meshed with a hexahedral mesh,

using y+ values close to 1 adjacent to the walls. The

mesh was found to be fully independent after

gradually increasing the number of elements over 5

different meshes in the curved region of the

geometry. The meshes were refined until the

differences in velocity, total pressure and TKE were

less than 1%.

Steady state simulations were run using the

commercial software ANSYS CFX 13 [14], in which

a coupled solver and a finite volume method are

used. The default advection schemes are a first order

accurate scheme for turbulence quantities and a

second order accurate scheme for the continuity,

momentum and energy equations.

5. RESULTS AND DISCUSSION

The SST-CC model is analyzed from different

perspectives: first, the velocity and turbulence kinetic

energy profiles are considered; second, the Reynolds

normal stresses are examined, and finally, the

parameter and eddy viscosity are investigated. The

performance of the SST-CC model is measured

relative to the RSM-SSG model, since the latter is

more sensitive to curvature than the eddy viscosity

based SST models. Throughout this section, the

vertical axis, , represents the traverse from concave

(zero) to convex (unity) curvature in the geometry,

and all plots were taken in the periodic boundary

condition plane. Fig. 3 shows a schematic of the plot

locations, and the coordinate system used.

Figure 3: Schematic of plot locations and

coordinate system used

5.1 Velocity and Turbulence Kinetic

Energy (TKE)

The velocity profiles at = 45° and 90° along the

curve are shown in Fig. 4. Locations before 45°

show minimal differences between the SST and SST-

CC models, and, thus, are not shown for conciseness.

In the 45° plot (Fig. 4a), it can be seen that even at

this location, there is only a small difference between

the SST and SST-CC models, and overall both match

well with the RSM-SSG results. At 90° on the other

hand (Fig. 4b), significant differences appear

throughout the entire section, with the SST-CC

matching the RSM-SSG velocity more closely than

the SST model. That being said, on the concave side

( = 0), fairly sizeable differences are still seen

between the SST-CC and RSM-SSG models.

(a)

(b)

Figure 4: Streamwise velocity profiles at (a) 45°

and (b) 90°

The turbulent kinetic energy profiles at the same two

locations (45° and 90°) are presented in Fig. 5. In the

45° plot (Fig. 5a), the SST-CC model matches the

RSM-SSG model very well, indicating that it is

behaving appropriately as compared to the original

SST (uncorrected) model, based on the known

curvature effects (that there is enhanced TKE near the

concave side and suppressed TKE near the convex

side). In the 90° plot (Fig. 5b), the SST-CC model is

reacting to the curvature accordingly by showing the

same trends as in the 45° case, however the

differences are not as drastic. The RSM-SSG model

is predicting a very high peak towards the convex

side, which may be due to a potential onset of

separation in this region.

Thus, in terms of velocity and TKE, the SST-CC

model appears to be behaving similarly to the RSM-

SSG model. This is with the exception of the TKE at

90°, where the RSM-SSG model shows unusual

results on the convex side. All in all, these plots

suggest that the SST-CC is correctly accounting for

the effects of curvature.

(a)

(b)

Figure 5: TKE profiles at (a) 45° and (b) 90°

5.2 Reynolds Normal Stresses

Eddy viscosity models, such as the SST models,

assume local isotropy of the turbulent length scale

and for this reason are known to perform poorly in

flows with sudden changes in the mean strain rate, or

when the flow and strain principal axes are not

aligned, for example in flows with streamline

curvature. The RSM-SSG model does not suffer

from this problem because it does not make the local

isotropy assumption. The poor performance in eddy

viscosity models is particularly apparent in the

Reynolds normal stresses [15].

In Fig. 6, the Reynolds normal stresses are plotted at

the 45° section. From these plots it can be seen that

the SST-CC model tends to show an improvement on

the convex side in terms of peaks and/or curve shape

and is consistent with predicted curvature trends,

showing an increase in turbulent stresses near the

concave side and a corresponding decrease on the

convex side as compared to the SST model. It is also

noteworthy that at the centre of the geometry, at =

0.5, the SST and SST-CC models match well,

suggesting that there is no curvature correction

occurring here. In terms of matching the RSM-SSG

results, the SST-CC model generally shows good

agreement near the convex side, however near the

concave side the it underpredicts the RSM-SSG peak

for the in-plane ( and ) stresses and overpredicts

the RSM-SSG peak for the out of plane ( )

stresses.

(a)

(b)

(c)

Figure 6: Reynolds normal stresses at 45°: (a) ,

(b) and (c) .

The same trends continue into the 90° section

Reynolds normal stresses shown in Fig. 7; the SST-

CC model shows the correct effects of curvature

relative to the SST model. The stresses are

unusual for the RSM-SSG model, showing a large

peak on the convex side, which is clearly responsible

for the same large peak on the TKE plot in Fig. 5b.

Again, there is a potential onset of separation in this

region, which could be causing this peak. The

stresses predicted by the SST-CC model are generally

closer to the RSM-SSG model on both the convex

and concave sides. The stresses are predicted

well by both the SST and SST-CC models, with both

models predicted similar profiles.

In a general sense, the SST-CC model is performing

as expected, showing higher stresses near the

concave side and lower stresses near the convex side,

relative to the SST model. The SST-CC seems to

match the RSM-SSG results better at the 45° location

than the 90° location, which could be due to an

interesting distribution in the curvature correction

parameter, which will be discussed in the next

section. Overall, the SST-CC model is predicting the

correct trends in Reynolds stresses due to curvature

effects as compared to the SST model, which

suggests an improvement with the curvature

correction addition.

5.3 Curvature Correction Parameter,

As stated previously, the SST-CC model uses a

production correction multiplier ( ) to either

increase or decrease the production around concave

and convex curvatures. Fig. 8 demonstrates that,

qualitatively, the curvature correction is functioning

as expected as it shows a large region of increased

production near the concave surface, a region of

decreased production near the convex surface and a

multiplier near 1 (i.e. No correction) prior to the

curved section of the geometry.

There are several interesting areas to focus on in the

contours. First, there are very sharp gradients in

in the transition region from concave to convex

curvature at the centre of the geometry. In this area,

there is a rapid change from to

side by side. Although this sharp gradient is not

physically realistic, its appearance is likely due to the

formulation of . Second, there is a large region

near the concave side with maximum . This

region is interesting since the limiter of 1.25 in the

definition of has a strong effect in this region. It

would be interesting to investigate the sensitivity of

with respect to the limiter, specifically in these

regions and in other similar regions. Third, there is

an interesting region on the concave side towards the

90° section, where the value quickly changes

from maximum to eventually reducing to a

production reduction ( ).

(a)

(b)

(c)

Figure 7: Reynolds normal stresses at 90°: (a) ,

(b) and (c) .

Looking back at the TKE and Reynolds stresses in

the previous sections, the SST-CC did not match the

RSM-SSG at the 90° section as well as at the 45°

section. This sudden change in could have an

effect on these differences. One unanswered

question is the reasoning behind this sudden change

in in terms of flow field.

Figure 8: Production multiplier fr1 (SST-CC)

One other evaluation of the general qualitative

performance of the curvature correction is in the

prediction of the eddy viscosity, as shown in Fig. 9.

This has been used previously to evaluate the

curvature corrected S-A model [16] which is the

basis of the correction for the SST-CC model.

(a)

(b)

Figure 9: Eddy viscosity contours for the (a) SST

and (b) SST-CC models.

From Fig. 9b, it can be seen that the SST-CC model

is suitably responding to the curved walls, in that

there is an increased eddy viscosity region appearing

near the concave wall, and a decreased region near

the convex wall that appears roughly halfway up the

curve. This is contrary to the SST prediction of the

eddy viscosity in Fig. 9a, which does not show any

sensitivity to curvature by predicting roughly the

same eddy viscosity across the entire span.

6. CONCLUSIONS AND FUTURE WORK

A simplified geometry, based on the impeller of a

centrifugal compressor stage, was investigated to

determine the ability of the SST-CC model to predict

the effects of curvature. The model was evaluated by

considering streamwise velocity, TKE, Reynolds

normal stresses, the production multiplier, , and

the eddy viscosity at two different locations along the

curve: 45° and 90°. The evaluation was made based

on a comparison with the RSM-SSG model, which

has an increased sensitivity to curvature. The

following conclusions were made:

The SST-CC mean streamwise velocity

profile at 90° matched the RSM-SSG results

better than the SST model; minimal

differences were found at 45°.

SST-CC TKE profiles showed appropriate

sensitivity to curvature, with increased TKE

on the concave side and decreased TKE on

the convex side, and matched well with the

RSM-SSG results at 45°.

Reynolds normal stresses were predicted

reasonably well by the SST-CC model in

terms of curvature effects, however the SST-

CC model tends to underpredict the in-plane

normal stresses and overpredict the out of

plane normal stresses on the concave side,

with respect to the RSM-SSG model.

Better agreement was found between the

SST-CC and RSM-SSG models at 45° than

at 90°, which could be attributed to a rapid

change in the parameter near the 90°

section.

Both the and eddy viscosity plots

qualitatively showed the appropriate effects

of the curvature correction, but there are

some interesting regions in the plot that

require further investigation.

Overall, the SST-CC model showed

sensitivity to curvature that is consistent

with the literature.

Future work will include relating this simplified

geometry to experimental results of the centrifugal

compressor stage after which it is modelled.

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