Journal of Computational Physics 520 (2025) 113499
Contents lists available at ScienceDirect
Journal of Computational Physics
journal homepage: www.elsevier.com/locate/jcp
Improving turbulence modeling for gas turbine blades: A novel
approach to address flow transition and stagnation point
anomalies
Ali Akbar Shahbazi a,∗ , Vahid Esfahanian b
a
b
Department of Mechanical Engineering, Qom University of Technology, Qom, Iran
Vehicle, Fuel and Environment Research Institute, School of Mechanical Engineering, College of Engineering, University of Tehran, Tehran, Iran
A R T I C L E
Keywords:
Gas turbine
Heat transfer simulation
Turbulence
Transition
V2F model
I N F O
A B S T R A C T
Accurate prediction of temperature and Heat Transfer Coefficient (HTC) distributions over gas
turbine blades is crucial for the design process and life assessment of these components. Numerical
studies of flow over gas turbine blades face significant challenges in accurately simulating two
complex phenomena: (1) the transition of flow from laminar to turbulent, and (2) stagnation
point flow at the leading edge. Many turbulence models tend to overpredict the temperature on
turbine blades, leading to incorrect identification of hot-spot regions and, consequently, erroneous
estimations of blade life. This paper investigates the performance of various turbulence models in
simulating flow and heat transfer over gas turbine vanes. The study includes three full turbulence
models, i.e., Spalart-Allmaras (SA), Shear Stress Transport 𝑘 −𝜔 (SST-kw), and 𝑣2 −𝑓 (V2F), as well
as two transitional models, i.e., Transition SST (Trans-SST) and 𝑘 − 𝑘𝐿 − 𝜔 (k-kl-w). Simulation
results indicate that the 𝑣2 − 𝑓 , Trans-SST, and 𝑘 − 𝑘𝐿 − 𝜔 models can detect flow transition.
However, the transition length and onset location predicted by the Trans-SST and 𝑘 − 𝑘𝐿 − 𝜔
models do not align with experimental data. Conversely, the 𝑣2 − 𝑓 model suffers from overpredictions at the leading edge due to stagnation point anomaly. To address these issues and due
to capacities of the V2F model, this study proposes two modifications to enhance the performance
of the V2F model. First, the production term of turbulent kinetic energy is redefined to mitigate
the stagnation point anomaly. Second, the model is recalibrated to improve the prediction of flow
transition. The new model, named the Production Modified V2F (PMV2F) model, shows promising
results in predicting temperature and heat transfer coefficients.
1. Introduction
Increasing the turbine inlet temperature in a gas turbine is well established to enhance both efficiency and output power. However,
this temperature increase significantly reduces the service life of hot section components. Proper blade cooling methods can allow
for higher turbine inlet temperatures while maintaining a reasonable lifespan for these hot sections. Therefore, developing accurate
* Corresponding author.
E-mail addresses: shahbazi@qut.ac.ir (A.A. Shahbazi), evahid@ut.ac.ir (V. Esfahanian).
https://doi.org/10.1016/j.jcp.2024.113499
Received 7 October 2024; Accepted 7 October 2024
Available online 18 October 2024
0021-9991/© 2024 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
Journal of Computational Physics 520 (2025) 113499
A.A. Shahbazi and V. Esfahanian
numerical tools to calculate temperature distribution over blades is essential for designing an optimized secondary air system for
cooling flows and for assessing the life of hot section components.
Two primary factors affect the accuracy of heat transfer simulations for turbine blades: the turbulence model and the coupling
approach. Historically, before the advent of increased computational power, simulations of turbine blades were performed in a
decoupled manner, where the governing equations of fluid zones and the conduction equation in solid zones were solved separately.
Boundary conditions at interface walls were derived from numerical or experimental correlations. However, heat transfer in gas
turbine blades is inherently a coupled/conjugate problem due to the effective thermal energy exchange between the hot gas flow,
solid blades, and cooling flows. Today, conjugate simulations of gas turbine blades are feasible due to significant advancements in
computational technology and power.
Nevertheless, selecting an appropriate turbulence model remains a challenge due to the simultaneous occurrence of two complex
phenomena:
1. Stagnation Point Flow at the Leading Edge: Eddy viscosity turbulence models often overpredict the heat transfer coefficient
at stagnation regions, a problem known as the “stagnation point anomaly”. This issue arises from an incorrect definition of the
turbulent kinetic energy production term, which is primarily designed for fully turbulent flows.
2. Flow Transition from Laminar to Turbulent: Flow transition is a complex phenomenon that is not yet fully understood and
remains a key focus of advanced fluid dynamics research. Although recent advancements have been made in simulating flow
transition, a mature method with low computational cost has not yet been developed. Most turbulence models are designed
to simulate fully turbulent flow regions and lack the capability to detect the transition from laminar to turbulent flow. This
deficiency results in significant errors in evaluating temperature distribution.
One of the early experimental investigations in the field of conjugate heat transfer of gas turbine vanes was conducted by Hylton
et al. (1983) [1]. This study involved a series of experiments on the well-known untwisted “C3X” and “Mark-II” vanes, commonly
referred to as Hylton’s C3X and Hylton’s Mark-II vanes. These vanes are characterized by ten radial cooling channels distributed
along the blade camber line. Another valuable study in this field involves the experimental and numerical investigations conducted
by Dees (2010) [2] and Dees et al. (2010) [3]. Dees pursued two primary objectives in his experiments: (1) to ensure that the vane
geometry closely resembled the realistic geometry of gas turbine vanes, and (2) to simplify the geometry as much as possible for use
in CFD validation. As a result, he conducted five separate experiments on the C3X (referred to as Dees’s C3X) vane. Bon et al. (1995)
[4] performed a two-dimensional simulation of Hylton’s Mark-II vane using an implicit finite-volume method and a simple oneequation turbulence model. Their findings demonstrated that the conjugate heat transfer method performed better than conventional
isothermal/heat flux boundary conditions. York et al. (2003) [5] simulated the Hylton’s C3X vane test-case using Fluent software with
standard 𝑘 − 𝜖 and realizable 𝑘 − 𝜖 models. Their results showed that these models could not predict the transition on the suction side.
To address this, they developed a nonlinear eddy viscosity turbulence model, which provided better results compared to the previous
𝑘 − 𝜖 models. However, the absence of HTC distribution data made it difficult to identify the cause of the improvement. According to
the authors’ knowledge, using nonlinear eddy viscosity models can overcome the stagnation point anomaly problem [6–8], but the
capability to predict transition remains uncertain. Facchini et al. (2004) [9] studied the Hylton’s C3X vane using standard, realizable,
and RNG 𝑘 − 𝜖 turbulence models in the STAR-CD code. All three models failed to predict the transition on the suction side and
exhibited the stagnation point anomaly problem, resulting in overpredictions of HTC and temperature distribution. Luo et al. (2007)
[10] conducted a numerical investigation of the Hylton’s C3X vane using three turbulence models: standard 𝑘 − 𝜖 , Quadratic 𝑘 − 𝜖 ,
and 𝑣2 − 𝑓 . The standard model showed the stagnation point anomaly problem and could not predict the transition. The 𝑣2 − 𝑓 model
predicted the transition occurrence but with incorrect length. The Quadratic 𝑘 − 𝜖 model, being a nonlinear eddy viscosity model,
did not exhibit the stagnation point anomaly but was unable to accurately predict the transition length and onset location. Qiang et
al. (2009) [11] simulated the Hylton’s Mark-II vane using the finite-difference method with three turbulence models: Baldwin-Lomax
algebraic model, 𝑞 − 𝜔 low-Re model, and BL & AGS1 model. They found that none of these models overpredicted at the leading
edge and could predict the transition with acceptable length and onset location. Dees et al. [3] simulated the Dees’s C3X vane using
Fluent with the 𝑘 − 𝜔 turbulence model and 21 million mesh volumes. Similar to other models mentioned, this model also exhibited
the stagnation point anomaly problem and failed to predict the transition. Dyson et al. (2013) [12] evaluated the performance of
four turbulence models (realizable 𝑘 − 𝜀, RNG 𝑘 − 𝜀, SST-kw, and Trans-SST) in simulating Dees’s C3X vane at two different inlet
turbulence intensities. Their findings indicated that the three full turbulence models–realizable 𝑘 − 𝜀, RNG 𝑘 − 𝜀, and SST 𝑘 − 𝜔–could
not predict the flow transition. Conversely, the Trans-SST model did not perform adequately in the fully turbulent region. Bak et al.
[13] studied the flow and heat transfer over Hylton’s C3X and Mark-II vanes, focusing on the effect of inlet turbulence conditions
and near-wall treatment methods. They investigated several turbulence models, including RNG 𝑘 − 𝜖 , SSG RSM, and SST + 𝛾 − Re𝜃 .
They showed that all models failed to predict HTC variations over the suction side. Yousefi et al. [14] investigated the Hylton’s C3X
vane using the SST-𝑘𝜔 turbulence model in CFX software, which was unable to detect flow transition on the suction side. Other
numerical simulations [15–18] have also struggled to accurately predict temperature and heat transfer coefficient distribution due to
the complex phenomena of flow transition and stagnation point anomalies.
Despite the extensive research conducted on turbulence models and their application to gas turbine blade simulations, the accurate
prediction of temperature distribution and heat transfer coefficients remains a significant challenge. Existing models frequently exhibit
1
Baldwin-Lomax & Abu-Ghannam and Shaw.
2
Journal of Computational Physics 520 (2025) 113499
A.A. Shahbazi and V. Esfahanian
deficiencies in handling the stagnation point anomaly and accurately predicting the transition from laminar to turbulent flow. The
stagnation point anomaly, in particular, results in overprediction of the heat transfer coefficient at stagnation regions, leading to
substantial errors in temperature prediction and life assessment. Moreover, while some models can detect flow transition, they often
fail to predict the transition length and onset location accurately, resulting in further inaccuracies.
To address these critical issues, this study proposes two modifications to the existing 𝑣2 − 𝑓 (V2F) turbulence model. First, the
production term of turbulent kinetic energy is redefined to mitigate the stagnation point anomaly. Second, the model is recalibrated
to improve the prediction of flow transition. The resulting Production Modified V2F (PMV2F) model aims to provide more accurate
predictions of temperature and heat transfer coefficients over gas turbine blades.
This paper is organized as follows: Section 2 provides a detailed review of the turbulence models considered in this study. Section 3
outlines the methodology and simulation setup used to evaluate these models. Section 4 presents the results and discussion, highlighting the performance improvements of the PMV2F model. Finally, Section 5 concludes with the main findings and implications
of this research, emphasizing the potential impact of the PMV2F model on the design and analysis of gas turbine blades.
2. Theory
In order to simulate the flow field over a gas turbine vane, this study employs the Reynolds-Averaged Navier-Stokes (RANS)
approach, wherein time-averaged operators are applied to the equations of continuity, momentum, and energy. Due to the timeaveraging process, the Reynolds stress terms appear in the equations. These terms are approximated using the Boussinesq eddy
viscosity assumption [19]:
2
−𝜌𝑢′𝑖 𝑢′𝑗 = 2 𝜇𝑡 𝑆𝑖𝑗∗ − 𝜌𝑘
3
(
)
1 𝜕𝑈𝑖 𝜕𝑈𝑗 2 𝜕𝑈𝑘
𝑆𝑖𝑗∗ =
+
−
𝛿𝑖𝑗
2 𝜕𝑥𝑗
𝜕𝑥𝑖
3 𝜕𝑥𝑘
(1)
(2)
Turbulence models try to approximate turbulent viscosity 𝜇𝑡 .
2.1. Turbulence modeling
In this section, we will provide a brief review to several turbulence models, starting with the Spalart-Allmaras (SA) model. This
model was originally proposed in 1992 [20], and it has been modified several times since then [21]. The SA model is a one-equation
turbulence model that solves a modeled transport equation for the turbulent kinematic viscosity, as follows:
)
𝜕
1
𝜕 (
𝜌𝑈𝑗 𝜈̃ = 𝐺𝜈 +
̃ +
(𝜌𝜈)
𝜕𝑡
𝜕𝑥𝑗
𝜎𝜈̃
{
(
[
]
)2 }
𝜕
𝜕 𝜈̃
𝜕 𝜈̃
− 𝑌𝜈 + 𝑆𝜈̃
̃
+ 𝐶𝑏2 𝜌
(𝜇 + 𝜌𝜈)
𝜕𝑥𝑗
𝜕𝑥𝑗
𝜕𝑥𝑗
(3)
where 𝐶𝑏2 and 𝜎𝜈̃ are constants, 𝐺𝜈 is the production term, 𝑌𝜈 is the destruction term, and 𝑆𝜈̃ is the source term [21]. The model
approximates the eddy viscosity using the following equation:
𝜇𝑡 = 𝜌𝜈𝑓
̃ 𝜈1
(4)
where 𝑓𝜈1 is the viscous damping function. The SA model was specifically designed for aerospace applications involving wall-bounded
flows and has been shown to yield good results for boundary layers subjected to adverse pressure gradients. It is also gaining popularity
in turbomachinery applications [21].
From dimensional analysis, it can be understood that:
𝜈𝑡 ∝  2 
(5)
𝜈𝑡 ∝  
(6)
𝜈𝑡 ∝  ∕
(7)
2
where  is the turbulent velocity scale,  is the turbulent time scale, and  is the turbulent length scale. Two-equation turbulence
models attempt to approximate two of these three turbulent scales by introducing transport equations for appropriate parameters. The
turbulent kinetic energy (𝑘), its dissipation rate (𝜀), specific dissipation rate (𝜔 = 𝜀∕𝑘), and turbulence stress normal to streamlines
(𝑣2 ) are examples of the parameters
of interest for modeling.
√
The 𝑘 − 𝜀 model employs 𝑘 as the turbulent velocity scale and 𝑘∕𝜀 as the turbulent time scale. This model was originally proposed
by Jones and Launder (1972) [22] and later improved by Launder and Sharma (1974) [23]. The model solves two equations for 𝑘
and 𝜀, as follows:
[(
)
]
𝜇
𝜕𝑘
𝜇+ 𝑡
+ 𝑃𝑘 − 𝜌𝜀 + 𝑆𝑘
𝜎𝑘 𝜕𝑥𝑗
[(
)
]
𝐶 ′ 𝑃𝑘 − 𝐶𝜀2 𝜀
𝜇
𝜕
𝜕
𝜕
𝜕𝜀
(𝜌𝜀) +
(𝜌𝜀𝑢𝑖 ) =
𝜇+ 𝑡
+ 𝜀1
+ 𝑆𝜀
𝜕𝑡
𝜕𝑥𝑖
𝜕𝑥𝑗
𝜎𝜀 𝜕𝑥𝑗

𝜕
𝜕
𝜕
(𝜌𝑘) +
(𝜌𝑘𝑢𝑖 ) =
𝜕𝑡
𝜕𝑥𝑖
𝜕𝑥𝑗
3
(8)
(9)
Journal of Computational Physics 520 (2025) 113499
A.A. Shahbazi and V. Esfahanian
Table 1
Coefficients of 𝑣2 − 𝑓 turbulence model [26].
𝐶𝜇
𝐶1𝜖
𝐶2𝜖
𝐶1
𝐶2
𝐶𝜂
𝐶𝐿
𝛼
𝜎𝑘
𝜎𝜖
0.22
1.4
1.9
1.4
0.3
70
0.23
0.6
1
1.3
where 𝑃𝑘 is the turbulent kinetic energy production term, obtained from
𝑃𝑘 = −𝜌𝑢′𝑖 𝑢′𝑗
𝜕𝑢𝑗
𝜕𝑥𝑖
= 2𝜇𝑡 𝑆𝑖𝑗 𝑆𝑖𝑗
(10)
The turbulent viscosity is modeled as:
𝜇𝑡 = 𝑓𝜇 𝐶𝜇 𝜌𝑘
(11)
where  = 𝑘∕𝜀 is the turbulent time scale, 𝑓𝜇 is the damping function, and 𝐶𝜇 is a constant. Other terms of Eqs. (8) and (9) are
described more in Ref. [24].
The 𝑘 − 𝜔 model proposes solving a transport equation for the specific dissipation rate 𝑤 = 𝜀∕𝑘 instead of 𝜀 as follows:
)
𝜕
𝜕
𝜕 (
𝜌𝜔𝑢𝑖 =
(𝜌𝜔) +
𝜕𝑡
𝜕𝑥𝑖
𝜕𝑥𝑗
[(
𝜇+
𝜇𝑡
𝜎𝑤
)
]
𝜕𝑘
+ 𝑃𝜔 − 𝑌𝜔 + 𝑆𝜔
𝜕𝑥𝑗
(12)
The turbulent eddy viscosity is obtained from
𝜇𝑡 = 𝛼 ∗ 𝜌𝑘
(13)
where  = 1∕𝜔. Other terms of Eq. (12) are explained more in [24]. The Shear-Stress Transport 𝑘 − 𝜔 (SST 𝑘 − 𝜔) model is a
combination of the 𝑘 − 𝜀 and 𝑘 − 𝜔 models. In the near-wall region, it behaves like the 𝑘 − 𝜔 model, while in the far field, it
transitions to the 𝑘 − 𝜀 model.
√
The V2F turbulence model, presented by Durbin (1995) [25], proposes using
𝑣2 as the turbulent velocity scale. Durbin noted
that the velocity scale 𝑣2 provides the appropriate scaling for representing the damping of turbulent transport close to the wall, a
feature that 𝑘 does not offer. This model formulates the turbulent viscosity as follows:
𝜇𝑡 = 𝐶𝜇 𝜌𝑣2 
(14)
In order to obtain 𝑣2 , the two transport equations (8) and (9) should be solved simultaneously with the following equations [25,26]:
{(
}
)
( )
(
)
𝜇𝑡 𝜕𝑣2
𝜌𝑣2 𝜖
𝜕
𝜕
𝜕
2
2
𝜌𝑣 +
𝜇+
+ 𝜌𝑘𝑓 − 6
𝜌𝑢𝑖 𝑣 = +
𝜕𝑡
𝜕𝑥𝑖
𝜕𝑥𝑗
𝜎𝑘 𝜕𝑥𝑗
𝑘
)
)
(
(
(
)
𝑃
𝐶
𝜕𝑓
1
𝜕
𝑣2 2
𝑣2 2
− 𝐶2 𝑘 −
−𝑓 = 1
2
6 −
−
𝜕𝑥𝑗 𝜕𝑥𝑗
𝑘
3
𝑘
𝑘
3


(15)
(16)
where 𝑓 is named elliptic relaxation factor and behaves like damping function for 𝑣2 . Other terms are obtained from following
equations:
𝑃𝑘 = 2𝜇𝑡 𝑆𝑖𝑗 𝑆𝑖𝑗
(17)
⎧
⎛
⎞ √ ⎫
⎪
𝑘
𝛼𝑘
⎟,6 𝜈 ⎪
 = max ⎨min ⎜ , √
⎜
𝜖
𝜖⎬
2
6𝐶𝜇 𝑣 |𝑆| ⎟⎠
⎪
⎪
⎝
⎩
⎭
(18)
⎫
⎧
⎞
⎛ 3∕2
3∕4
⎪
𝑘1.5
𝑘
⎟,𝐶 𝜈 ⎪
 = 𝐶𝐿 max ⎨min ⎜
,√
𝜂 1∕4 ⎬
⎜ 𝜖
𝜖 ⎪
6𝐶𝜇 𝑣2 |𝑆| ⎟⎠
⎪
⎝
⎭
⎩
√
(
)
′
𝐶𝜀1
= 𝐶𝜀1 1 + 0.050 𝑘∕𝑣2
(19)
(20)
The Coefficients of V2F model are presented in Table 1. More details about this model can be found in References [25] and [26].
2.2. Transition modeling
Flow transition from laminar to turbulence is of great importance in turbomachinery applications, as it occurs over compressor and
turbine blades. Therefore, accurate consideration of transition is essential in CFD modeling and simulations. Flow transition can occur
4
Journal of Computational Physics 520 (2025) 113499
A.A. Shahbazi and V. Esfahanian
through several modes, including natural transition, bypass transition, and separation-induced transition. Given the high turbulence
intensity typically present in turbomachines, bypass transition is the predominant mode. This study examines the performance of the
Trans-SST and 𝑘 − 𝑘𝐿 − 𝜔 models, both of which utilize the RANS approach to capture flow transition.
The Trans-SST model is a transitional turbulence model that integrates the two-equation SST-kw model with two additional
transport equations: one for the intermittency, 𝛾 , and the other for the momentum-thickness Reynolds number, Re𝜃 . given as follows
[27]:
{(
)
}
𝜇
𝜕𝛾
𝜇+ 𝑡
+ 𝑃𝛾1 − 𝐸𝛾1 + 𝑃𝛾2 − 𝐸𝛾2 +
𝜎𝛾 𝜕𝑥𝑗
{
}
(
)
(
)
̃ 𝜃𝑡
(
) 𝜕 Re
𝜕
̃ 𝜃𝑡 + 𝜕 𝜌𝑈𝑗 Re
̃ 𝜃𝑡 = 𝜕
𝜌Re
𝜎𝜃𝑡 𝜇 + 𝜇𝑡
+ 𝑃𝜃𝑡
𝜕𝑡
𝜕𝑥𝑗
𝜕𝑥𝑗
𝜕𝑥𝑗
)
𝜕
𝜕 (
𝜕
𝜌𝑢𝑗 𝛾 =
(𝜌𝛾) +
𝜕𝑡
𝜕𝑥𝑗
𝜕𝑥𝑗
(21)
(22)
The first equation, for intermittency (𝛾 ), is designed to identify regions in the flow where transition occurs. Intermittency is a measure
of the fraction of time the flow remains turbulent, and by solving for this variable, the model can dynamically adjust the production
of turbulent kinetic energy in transitional regions. This approach allows the model to more accurately reflect the physics of flow
transition, as opposed to relying solely on empirical data. The second transport equation is for the momentum-thickness Reynolds
number (Re𝜃𝑡 ), which serves as a criterion for the onset of transition. Re𝜃𝑡 is a dimensionless parameter that characterizes the state of
the boundary layer flow. By incorporating this variable, the model can predict the location and length of the transition region more
accurately. The combined use of intermittency and Re𝜃𝑡 enables the Trans-SST model to simulate the gradual process of transition from
laminar to turbulent flow, rather than treating it as an abrupt change. A notable feature of the Trans-SST model is its correlation-based
approach. While the model does not directly simulate the detailed physics of the transition process, it employs empirical correlations
derived from extensive experimental data to guide the behavior of the intermittency and Re𝜃𝑡 equations. These correlations effectively
bridge the gap between theoretical modeling and practical observations, enhancing the model’s reliability and accuracy. However, this
reliance on empirical data means that the model’s performance is highly dependent on the quality and relevance of the correlations
used.
The 𝑘 − 𝑘𝐿 − 𝜔 model, introduced by Walters and Cokljat (2008) [28], is a transitional model based on the 𝑘 − 𝜔 framework. The
𝑘 and 𝜔 equations govern the development of turbulence in fully turbulent regions. However, to address the transition from laminar
to turbulent flow, the 𝑘 − 𝑘𝐿 − 𝜔 model introduces a third equation for laminar kinetic energy (𝑘𝐿 ), which represents the energy
contained in laminar fluctuations within the boundary layer. The transport equations for this model are as follows:
)
𝜕𝑘𝑇
𝜕 (
𝑈𝑗 𝑘𝑇 =
+
𝜕𝑡
𝜕𝑥𝑗
𝑃𝑘𝑇 + 𝑅BP + 𝑅NAT − 𝜔𝑘𝑇 − 𝐷𝑇 +
)
𝜕𝑘𝐿
𝜕 (
+
𝑈𝑗 𝑘𝐿 =
𝜕𝑡
𝜕𝑥𝑗
𝑃𝑘𝐿 − 𝑅BP − 𝑅NAT − 𝐷𝐿 +
)
𝜕 (
𝜕𝜔
+
𝑈𝑗 𝜔 =
𝜕𝑡
𝜕𝑥𝑗
𝜕
𝜕𝑥𝑗
[(
[
]
𝜕𝑘
𝜕
𝜈 𝐿
𝜕𝑥𝑗
𝜕𝑥𝑗
𝜈+
𝛼𝑇
𝜎𝑘
)
𝜕𝑘𝑇
𝜕𝑥𝑗
]
(
)
)
𝐶𝜔𝑅
𝜔
𝜔 (
𝑃𝑘𝑇 +
−1
𝑅BP + 𝑅NAT − 𝐶𝜔2 𝜔2
𝑘𝑇
𝑓𝑊
𝑘𝑇
√
[(
)
]
𝑘𝑇
𝛼
𝜕
𝜕𝜔
2
𝜈+ 𝑇
+ 𝐶𝜔3 𝑓𝜔 𝛼𝑇 𝑓𝑊
+
𝜕𝑥𝑗
𝜎𝜔 𝜕𝑥𝑗
𝑑3
(23)
(24)
𝐶𝜔1
(25)
The 𝑘𝐿 equation captures the onset and progression of transition by modeling the amplification of laminar disturbances. This approach
is more physically grounded than purely empirical methods, as it incorporates the underlying mechanisms driving flow transition. The
model dynamically shifts from laminar to turbulent behavior based on local flow conditions, governed by the interactions between
𝑘𝐿 , 𝑘, and 𝜔.
2.3. Stagnation point anomaly
Numerical simulations of turbulent flows have shown that turbulence models that employs linear eddy viscosity models, overestimates the turbulent kinetic energy in stagnation point regions [29–31]. The Reynolds stress terms in linear eddy viscosity models
are obtained from Eq. (1) and the production term in the turbulent kinetic energy equation is obtained as follows:
𝑃𝑘 = −𝜌𝑢𝑖 𝑢𝑗 𝑈𝑖𝑗
(26)
Hence, for a 2-D flow, and considering the continuity equation, it can be written as below:
)
(
𝜕𝑈
𝜕𝑉
+ 𝑣2
𝑃𝑘 = − 𝜌 𝑢2
𝜕𝑥
𝜕𝑦
)
(
𝜕𝑈
= − 𝜌 𝑢2 − 𝑣2
𝜕𝑥
(27)
In linear eddy viscosity models substituting the equation (1) into the equation (26) yields:
5
Journal of Computational Physics 520 (2025) 113499
A.A. Shahbazi and V. Esfahanian
𝑃𝑘 = 2𝜇𝑡 𝑆𝑖𝑗 𝑆𝑖𝑗 = 2𝜇𝑡
(
= 4𝜇𝑡
𝜕𝑈
𝜕𝑥
[
(
𝜕𝑈
𝜕𝑥
(
)2
+
𝜕𝑉
𝜕𝑦
)2 ]
)2
(28)
Eq. (28) demonstrates that in stagnation point flow, the production term 𝑃𝑘 increases as the magnitude of 𝜕𝑈
( ∕𝜕𝑥 along
) the streamline
grows. However, this does not correspond to reality. In fact, in near-wall regions of stagnation, the term 𝑢2 − 𝑣2
of Eq. (27) tends
to approach zero, and in some areas, it may even become negative. This leads to very low values of the production term. Therefore,
eddy viscosity models tend to overpredict the turbulent kinetic energy at flow stagnation points. This issue, known as the “stagnation
point anomaly” in the literature, leads to the overprediction of heat transfer coefficients and skin friction coefficients. Consequently,
simulations at the leading edge of gas turbine vanes may not produce accurate results.
2.4. Production modified V2F (PMV2F) model
This section provides a detailed explanation of the PMV2F model. As mentioned in Section 1, the PMV2F model is introduced to
enhance the performance of the V2F model in flow transition and stagnation regions. Literature review indicates that V2F model has
some capability in detecting flow transition, However, the authors propose that its calibration can be further optimized to enhance
this ability. Furthermore, it is suggested that the issue of stagnation point anomalies can be resolved by redefining the production of
turbulent kinetic energy.
To address the stagnation point anomaly several idea is proposed in literature. Menter [32] and Kato and Launder [33] proposed
using the vorticity tensor Ω𝑖𝑗 in production term 𝑃𝑘 instead of strain tensor 𝑆𝑖𝑗 . Since |Ω𝑖𝑗 | ≈ 0 for irrotational flows, this replacement
reduces unwanted effects such that 𝑃𝑘 ≈ 0 at the stagnation points. Medic and Durbine [30] suggested using a “realizability” constraint
on the turbulent time scale for the V2F turbulence model to improve the overprediction of HTC on turbine blade leading edges. Basara
and Jakirlic [34] employed a hybrid approach combining the Reynolds Stress Model (RSM) and 𝑘 − 𝜀 to overcome the over-prediction
of 𝑘 in stagnation regions. They solved the Reynolds stress equations and calculated 𝑃𝑘 from Eq. (26). These Reynolds stresses are
not directly used in the RANS momentum equations but to compute a value for 𝐶𝜇 in the formulation of the eddy viscosity of the
standard 𝑘 − 𝜀 model. Some others such as Craft et al. [35] and Raisee et al. [7] employed non-linear eddy-viscosity models. Kalitzin
et al. [36] followed Durbin’s realizability approach [25,30], but applied the limiting constraints directly to the Reynolds stresses.
As the stagnation point anomaly problem still exist in simulation of gas turbine vane [31], this study modifies the V2F turbulence
model using Kato and Launder (1993) [33] approach, replacing the strain tensor with the vorticity tensor Ω𝑖𝑗 in the production term
as follows:
𝑃𝑘 =2𝜇𝑡 |𝑆||Ω|
(29)
√
𝑆𝑖𝑗 𝑆𝑖 𝑗
(30)
√
|Ω| = Ω𝑖𝑗 Ω𝑖 𝑗
(31)
where
|𝑆| =
To address flow transition prediction, a detailed explanation is necessary. The standard V2F model is a full turbulence model
designed to accurately resolve near-wall turbulence structures and flow separation. Although it does not explicitly include mechanisms
for predicting transition onset and length, some studies have demonstrated its potential in this area. For instance, Luo et al. [10] and
Esfahanian et al. [31] have shown its capability in simulating flow transition over the C3X gas turbine vane. Rahman [37] presented
a redefined version of the V2F model that accurately detects flow transition over flat plates. Rahman et al. [38] further indicated the
model’s potential for predicting natural and bypass transition over flat plates and airfoils. Liu et al. [17] highlighted the role of wallnormal fluctuation, 𝑣2 , in the transition process, noting that the rapid rise in wall-normal fluctuation kinetic energy correlates with
transition length. Wall-normal turbulence leads to Klebanoff-mode production and the initiation of the transition process. Additionally,
rapid rise in wall-normal fluctuation kinetic energy is correlated with transition length. Thus, it is reasonable to argue that the V2F
model inherently possesses transition prediction capabilities. However, the standard model is primarily calibrated and designed for
fully turbulent flows.
This study proposes a recalibration approach to enhance the model’s transition predictability. Recalibration is often necessary
when new data becomes available or when the model needs to be applied to different conditions. In this context, recalibration
involves modifying the production term and investigating the V2F model for the transition region, which differs from fully turbulent
conditions that were originally calibrated. To achieve this, the most sensitive coefficients of the V2F model were selected for study.
According to Rahman [37], 𝐶𝜇 , 𝐶2 , 𝐶1 , and 𝐶𝜂 are the most sensitive coefficients. These coefficients were varied within a range of
±20% from the original V2F model values to perform sensitivity analysis and optimization. The recalibration procedure is illustrated
in Fig. 1 and will be explained in the subsequent sections.
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Journal of Computational Physics 520 (2025) 113499
A.A. Shahbazi and V. Esfahanian
Fig. 1. Recalibration procedure.
3. Methodology and simulation
This section outlines the numerical procedure followed for simulation purposes. The experimental test cases selected are Hylton’s
C3X vane [1] and Dees’s C3X vane [2]. For these test cases, the steady-state Reynolds Averaged Navier-Stokes (RANS) equations and
energy equations in fluid zones, along with the conduction equation in solid zones, are solved simultaneously using Fluent software.
Additionally, three full turbulence models and two transitional models are examined: the Spalart-Allmaras (SA) model, the shear stress
transport 𝑘 − 𝜔 (SST-kw) model, the 𝑣2 − 𝑓 (V2F) model, the Trans-SST model, and the 𝑘 − 𝑘𝐿 − 𝜔 (k-kl-w) model. The details of the
simulation conditions for each case are provided in the subsequent subsections. To evaluate the effect of modifying the production
term of the V2F model in stagnation point flow, the heat transfer of an impingement jet flow over a flat plate test case is employed.
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Journal of Computational Physics 520 (2025) 113499
A.A. Shahbazi and V. Esfahanian
Fig. 2. Computational domain and types of boundary conditions for Case-I.
Table 2
Boundary condition imposed for the main gas path of Case-I [1].
𝑃T,in (Pa)
𝑇T,in (K)
Ma𝑖𝑛
TIin
TLSin
𝑃s,out (Pa)
321800
783
0.17
8.3
0.005
192500
In order to recalibrate the V2F model for the study of the transition region, flat plate test cases from the “ERCOFTAC2 T3” series are
considered, including the T3A, T3C2, and T3C5 test cases.
3.1. Simulations of C3X vane
Hylton et al. (1983) [1] constructed a linear cascade of three C3X vane and performed some experiment to understand flow and
heat transfer characteristics of flow over gas turbine vanes. According to their experimental setup, translational periodic boundary
condition can be used in numerical simulation. The computational domain is illustrated in Fig. 2 and boundary conditions of the main
gas flow are presented in Table 2 (corresponding to run 112). Cooling air flows are not considered in this study, and the boundary
condition of cooling channels are set using the HTC and temperature. The HTC of the cooling channels are acquired from the following
formula [1]:
Nu𝐷 = Cr(0.022Pr 0.5 Re0.8
𝐷 )
(32)
Table 3 presents the HTC and the temperature for each cooling channel. The Reynolds number in the table is obtained from cooling
mass flow that reported in Reference [1]. The material properties of the main gas path and 310 stainless steel are assumed to be the
same as those reported in Reference [9], as follows:
⎧ 𝐶 = 1.077 × 103 − 6.112 × 10−1 𝑇 + 1.632 × 10−3 𝑇 2 − 1.460 × 10−6 𝑇 3
⎪ 𝑝,𝑔
+5.994 × 10−10 𝑇 4 − 9.243 × 10−14 𝑇 5 [kJ∕(kg.K)]
⎪
⎪
⎪ 𝑘 = 2.911 × 10−3 + 1.193 × 10−4 𝑇 + 8.714 × 10−8 𝑇 2
⎨ 𝑔
−4.836 × 10−11 𝑇 3 − 9.651 × 10−15 𝑇 4 [W∕mK]
⎪
⎪
)
( )1.5 (
⎪ 𝜇𝑔
𝑇0 +𝑆
𝑇
Sutherland′ s law
⎪ 𝜇0 =
𝑇0
𝑇 +𝑆
⎩
2
European Research Community on Flow Turbulence and Combustion.
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Journal of Computational Physics 520 (2025) 113499
A.A. Shahbazi and V. Esfahanian
Table 3
Temperature and HTC of cooling channels correspond to Fig. 2 [1].
Chan. No.
Tsc
Cr
Pr
ReD
NuD
k
D
h
1
2
3
4
5
6
7
8
9
10
409.08
409.37
391.54
397.15
376.91
434.86
391.49
407.58
466.43
516.21
1.118
1.118
1.118
1.118
1.118
1.118
1.118
1.056
1.056
1.025
0.68
0.68
0.68
0.68
0.68
0.68
0.68
0.68
0.68
0.68
68780
57340
57040
59240
60250
56080
56910
39920
22100
16110
150.34
129.98
129.44
133.42
135.23
127.69
129.2
91.9
57.26
43.16
0.05067
0.05069
0.04966
0.04998
0.0488
0.05213
0.04965
0.05058
0.05389
0.05659
0.0063
0.0063
0.0063
0.0063
0.0063
0.0063
0.0063
0.0031
0.0031
0.00198
1209.2
1045.84
1020.3
1058.44
1047.52
1056.6
1018.24
1499.38
995.41
1233.54
Table 4
Grid independence study for Case-I.
Grid name
Cell numbers
Cell number ratio
𝑦+avr
𝑇avr,vane
Course
Medium
Fine
643,682
1,310,634
2,643,658
0.491
1.000
2.017
0.164
0.081
0.037
587.16
587.21
587.23
Fig. 3. Grid independence study for Case-I, comparing HTC distribution for 3 grid size.
⎧ 𝜌 = 7900 [kg∕m3 ]
⎪ 𝑠
⎨ 𝐶𝑝,𝑠 = 585.15 [J∕kg − K]
⎪ 𝑘𝑠 = 6.811 + 0.02017𝑇 [W∕m − K]
⎩
(𝑇0 = 273 K, 𝑆 = 110.4 K, 𝜇0 = 1.71 × 10−5 kg∕ms.) The grid independence of the results is investigated by comparing the heat
transfer coefficient distribution for three grid types: coarse, medium, and fine, with cell counts of 345,938, 699,811, and 1,457,269,
respectively (Table 4). Fig. 3 illustrates the heat transfer coefficient distribution for different turbulence models. The results demonstrate grid independence, leading to the selection of the medium grid for further analysis. Fig. 4 depicts the generated grid and the
boundary layer mesh around the airfoil.
Similar to Case-I, the experimental setup by Dees (2010) [2] allows for the use of translational periodic boundary conditions in
numerical simulations. Consequently, the computational domain is constructed as shown in Fig. 5. The domain comprises four zones:
the hot section gas path, solid vane, forward U-bend cooling, and aft radial cooling. In this case, air is modeled as an incompressible
ideal gas with constant specific heat and thermal conductivity, while viscosity is modeled using Sutherland’s formula. The vane is
made of castable epoxy resin ((Poly-cast PC-287)) with a thermal conductivity of 𝑘 = 1.03 W∕mK and a thickness of 𝑡 = 1.27 cm. The
boundary conditions for the hot gas path and coolant flows are listed in Table 5. For both coolant channels, the hydraulic diameter is
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Journal of Computational Physics 520 (2025) 113499
A.A. Shahbazi and V. Esfahanian
Fig. 4. Computational grid for Case-I.
Fig. 5. Computational domain for Case-II.
Table 5
Boundary conditions for Case-II [2].
Hot gas path
U-bend coolant
Aft radial coolant
𝑉𝑖𝑛
(m∕s)
𝑇𝑖𝑛
(K )
TI𝑖𝑛
5.8
2.31
4.88
300
250
250
20
5
5
%
TLS𝑖𝑛
(m)
𝐷ℎ𝑦𝑑
(m)
𝑃𝑜𝑢𝑡
(atm)
Re𝑖𝑛
0.037
-
-
0.098
0.046
1
1
1
7.5 × 105
2.0 × 104
2.0 × 104
-
determined from the Reynolds number Re = 20000, as provided in the experiment. The grid independence of the results is investigated
for all turbulence models using three unstructured prism meshes (Fig. 6) by comparing heat transfer coefficient distributions. Detailed
grid information is presented in Table 6. The heat transfer coefficient distribution of the V2F and Trans-SST models are shown in
Fig. 7. The results indicate grid independence, leading to the selection of the medium grid for further simulations.
3.2. Simulations of PMV2F model
As outlined in previous sections, the development of the PMV2F model follows a structured procedure, consisting of the following
steps:
1. The study of impingement jet flow over a flat plate as the stagnation point flow test case
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Journal of Computational Physics 520 (2025) 113499
A.A. Shahbazi and V. Esfahanian
Fig. 6. Computational grid for Case-II.
Table 6
Grid independence study for Case-II.
Grid name
Cell Num.
Cell Num. Ratio
𝑦+avr
𝑇avr,vane
Course
Medium
Fine
1,547,203
3,144,722
6,365,834
0.492
1.000
2.024
0.122
0.064
0.028
282.32
282.36
282.36
Fig. 7. Grid independence study for Case-II, comparing HTC distribution for 3 grid size.
2. The study of flow over a T3 flat plate seris as the test cases for transition from laminar to turbulence
The impingement jet flow is a well-established test case for studying turbulent heat transfer in stagnation point regions. This
phenomenon has been extensively explored in the literature (e.g., [7,39–43]). In the present study, we employ the test case described
by Cooper et al. with 𝐻∕𝐷 = 2 and Re𝐷 = 2300 (Fig. 8(a)). Given the axisymmetric nature of the geometry and boundary conditions,
the computational domain is generated as illustrated in Fig. 8(b).
For the recalibration procedure, we have selected the ERCOFTAC T3 flat plate series, which includes multiple test cases focused
on transitional and turbulent flows over flat plates and airfoils. These test cases are widely documented and frequently used to assess
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Journal of Computational Physics 520 (2025) 113499
A.A. Shahbazi and V. Esfahanian
Fig. 8. Computational domain and grid for impingement jet test-case.
Table 7
The boundary conditions of T3 flat plates [46,47,27].
Case
𝑉𝑖𝑛 (m∕s)
TIin
𝜇𝑡 ∕𝜇
𝜌 (kg∕m3 )
𝜇 (kg∕ms)
T3A
T3C2
T3C5
5.4
5.29
9
3.3
3
4
12
11
15
1.2
1.2
1.2
1.8 × 10−5
1.8 × 10−5
1.8 × 10−5
and compare the performance of turbulence models, particularly in their ability to predict the transition from laminar to turbulent
flow.
In this study, we specifically consider the T3A, T3C2, and T3C5 test cases. T3A is characterized by a zero pressure gradient, while
T3C2 and T3C5 exhibit pressure gradients, as documented in [44–47]. For the T3C test cases, the pressure gradient is induced by
implementing a variable height for the upper edge of the computational domain. Fig. 9 depict the computational domains, and the
boundary conditions are presented in Table 7.
The recalibration process, illustrated in Fig. 1, is aimed at improving the accuracy of the skin friction coefficient (𝐶𝑓 ) distribution
for the flat plate test cases (T3A, T3C2, and T3C5) and optimizing the heat transfer coefficient (HTC) for the impingement jet test case.
The computational model is developed as described earlier. A Design of Experiments (DOE) approach is applied, where parameter
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Journal of Computational Physics 520 (2025) 113499
A.A. Shahbazi and V. Esfahanian
Fig. 9. Computational domain and grid for flat plate test-cases.
ranges (±20%) are defined and samples are generated using Latin Hypercube Sampling (LHS). For each parameter combination, CFD
simulations are conducted, and the resulting 𝐶𝑓 and HTC distributions are recorded. To identify the most influential coefficients,
sensitivity analysis is performed using SALib Python library. Subsequently, a multi-objective optimization is carried out using a genetic
algorithm. The objective function is designed to minimize the error in 𝐶𝑓 and HTC, and the optimization algorithm is employed to
refine the coefficients (𝐶𝜇 , 𝐶2 , 𝐶1 , 𝐶𝜂 ). The optimized model predictions are then validated against experimental data. If further
refinement is required, the process is repeated; otherwise, the recalibration procedure is considered complete.
4. Results and discussion
This section details the simulation outcomes of the C3X vane using various turbulence models. Following this, we present the
results of the impingement jet test case for the PMV2F model, along with a thorough explanation of the recalibration results. Lastly,
we examine the performance of the PMV2F model in the context of flow over the C3X vane.
4.1. Simulations of C3X vane
Fig. 10 provides a detailed illustration of the non-dimensional pressure distribution and wall vorticity along the vane surfaces for
Case-I (Hylton’s C3X vane). Upon examination of the figure, it becomes evident that the sign of vorticity does not undergo any change
along either the suction side or the pressure side of the vane. This observation is crucial as it implies that flow separation does not
occur on any part of these surfaces. From the absence of flow separation, it can be inferred that the pressure distribution along the
vane surface remains consistent and is not significantly affected by variations in the chosen turbulence model. This finding highlights
that the turbulence model has a negligible effect on the pressure distribution.
Fig. 11 demonstrates the results of non-dimensional HTC and temperature around the vane for Case-I. According to experimental
results presented in Fig. 11(a) the laminar boundary layer begins to form as the flow encounters the leading edge of the vane. On the
suction side, the transition from laminar to turbulent flow initiates around 𝑥∕𝐶𝑥 ≈ 0.25 and becomes fully turbulent by approximately
𝑥∕𝐶𝑥 ≈ 0.65. The length of the transition region is considerable, indicating the critical importance of accurately predicting this region.
On the pressure side, the flow remains laminar from 𝑥∕𝐶𝑥 ≈ 0 to 𝑥∕𝐶𝑥 ≈ −0.3, after which the transition begins. However, due to
strong pressure forces, the flow does not achieve turbulence on this side and remains in a transitional state throughout.
According to the numerical results presented in Fig. 11(a), the SA and SST-kw can not predict transition on the suction side, since
they are full turbulence models and calibrated specifically for detecting fully turbulent flow regimes. Conversely, the V2F, k-kl-w,
and Trans-SST models successfully detect the transition occurrence on the suction side. Among these, the transition onset location
and transition length predicted by the V2F model are closer to the experimental results compared to the other models. Although both
the Trans-SST and k-kl-w models predict the transition occurrence, the transition onset location and length do not align with the
experimental data. One significant cause of this discrepancy is that these models do not account for the effects of compressibility, as
discussed in Reference [48].
It is evident from Fig. 11(a) that the SA, SST-kw, and V2F models overpredict the heat transfer coefficient at the leading edge,
a phenomenon previously introduced as the “stagnation point anomaly”. As explained earlier, this deficiency originates from the
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Journal of Computational Physics 520 (2025) 113499
A.A. Shahbazi and V. Esfahanian
Fig. 10. Distribution of (a) non-dimensional pressure and (b) non-dimensional vorticity around the vane for Case-I.
Fig. 11. Predictions of HTC and temperature around the vane for Case-I using different turbulence models.
incorrect definition of the production term for turbulent kinetic energy. Unlike the fully turbulent models, the Trans-SST and k-klw models treat the boundary layer as a laminar zone near the leading edge, effectively mitigating the overprediction of HTC. For
additional insight, consider the Trans-SST model which defines the effective turbulent kinetic energy production term as follows:
𝑃ef f = 𝛾𝑃
(33)
where 𝛾 is the intermittency function. This model accurately predicts that the transition from laminar to turbulent flow does not occur
at the leading edge, resulting in an intermittency (𝛾 ) of zero and, consequently, a 𝑃ef f of zero. Therefore, unlike the other models,
the turbulent kinetic energy is correctly predicted, and the stagnation point anomaly does not occur.
Comprehensive study of Figs. 11(a) and 11(b) depicts that V2F model predictions are more close to experimental results, due to
its capability in detecting transition region. Deeper study of this model shows that solving an equation for 𝑣2 helps this model to
performs better for complex flows [17].
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Journal of Computational Physics 520 (2025) 113499
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Fig. 12. Predictions of HTC and temperature around the vane for Case-II using different turbulence models.
Fig. 13. Distribution of Nusselt number versus radial coordinate.
Fig. 12 depicts the results of the HTC and overall effectiveness around the vane for Case-II (Dees’s C3X vane). The overall effectiveness is defined by the relation:
𝜙=
𝑇∞ − 𝑇
𝑇∞ − 𝑇𝑐
(34)
where 𝑇 is the temperature, 𝑇∞ is the free stream temperature and 𝑇𝑐 is the coolant flow temperature. In other words, overall
effectiveness measures the efficiency of the cooling channels. Fig. 12(a) shows the HTC distribution, where the SA and SST-kw models
fail to accurately predict the transition on the suction side. In contrast, the Trans-SST and k-kl-w models provide precise predictions
of the transition onset and length. The V2F model detects the transition occurrence but overpredicts values at the stagnation point,
resulting in higher readings compared to experimental results.
In Fig. 12(a), we observe the Heat Transfer Coefficient (HTC) distribution, where the SA and SST-kw models fail to accurately
predict the transition on the suction side. In contrast, the Trans-SST and k-kl-w models show precise prediction of the transition
onset and length. The V2F model detects the transition occurrence but overpredicts values at the stagnation point, resulting in higher
readings compared to experimental results.
Similarly, Fig. 12(b) presents the temperature distribution along the blade. The observations align with the HTC results: the SA
and SST-kw models again fail to capture the transition accurately, while the Trans-SST and k-kl-w models effectively predict the
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Journal of Computational Physics 520 (2025) 113499
A.A. Shahbazi and V. Esfahanian
Fig. 14. Skin friction versus Reynolds number for different flat plat test cases.
Fig. 15. Sobol index values of 𝐶𝑓 and HTC for various coefficients of V2F model.
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Journal of Computational Physics 520 (2025) 113499
A.A. Shahbazi and V. Esfahanian
Fig. 16. The effect each coefficients on skin friction distribution for T3A flat plat test case.
transition onset and length. The V2F model detects the transition but overestimates the values at the stagnation point, leading to
higher temperatures than observed experimentally.
4.2. Simulations of PMV2F model
This section presents the results of simulations used to define and develop the PMV2F model. It aims to demonstrate the effects
of recalibration and redefinition of the production term.
As discussed in Section 2.3, the production term of turbulent kinetic energy is redefined by Equation (29) in the PMV2F model.
The simulation of impingement jet flow over a flat plate serves as an ideal test case to investigate the stagnation point anomaly
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Journal of Computational Physics 520 (2025) 113499
A.A. Shahbazi and V. Esfahanian
Table 8
Coefficients of PMV2F turbulence model.
𝐶𝜇
𝐶1𝜖
𝐶2𝜖
𝐶1
𝐶2
0.3056
2.6802
1.7932
2.5451
0.3664
Fig. 17. HTC distribution for Case-I and Case-II with PMV2F model.
problem. Fig. 13 shows the distribution of the Nusselt number across the radial coordinate for V2F, PMV2F, and Uncalibrated PMV2F
models. The results indicate that the root mean square of the relative error is 28.83% for the V2F model, while it decreases to 7.26%
for the Uncalibrated PMV2F model. This 21.57% improvement originates from the redefinition of the production term by Equation
(29). The relative error further decreases to 0.88% for the PMV2F model, indicating the effect of recalibration, which is a significant
improvement in the numerical simulations.
The results of flow simulations over the T3A, T3C2, and T3C5 flat plates using the V2F and PMV2F models are presented in
Fig. 14. It can be observed that the PMV2F model aligns with the experimental results. Although these three simulations are part of
the optimization process, it is clear that recalibration has a significant influence on the simulation results.
The results of sensitivity analysis in terms of first-order Sobol index are shown in Fig. 15. It can be seen that all coefficients of V2F
model have an effect on the transition predictablility of V2F model. In addition it can be observed that the effect of model coefficients
on HTC is a bit more than skin friction (𝐶𝑓 ).
The effect of each coefficient on skin friction distribution is shown in Fig. 16. It can be observed that increasing 𝐶𝜇 , 𝐶𝜀2 , and 𝐶2
causes the transition to begin earlier and shortens its length. In contrast, increasing 𝐶𝜀1 and 𝐶1 extends the transition length and
delays the transition onset location. The optimum values of investigated coefficients are presented in Table 8.
The results of flow simulations over C3X gas turbine vanes (Case-I and Case-II) are presented in Fig. 17. It can be seen that the
PMV2F model’s prediction closely matches the experimental results. In the Case-I and Case-II the results are improved by 19.7%
and 38.1% respectively. It can be seen that the PMV2F model not only resolves the stagnation point anomaly but also accurately
predicts the transition onset and length. The uncalibrated version of the PMV2F model is also included in the figure to show that
the modification of the production term eliminates the stagnation point anomaly. The difference between the PMV2F model and the
uncalibrated version indicates the effect of calibration in improving the results. Fig. 17(a) shows that the first and second modifications
improves the predictions by 16.6% and 3.1%. Fig. 17(b) shows that the first and second modifications improves the predictions by
30.4% and 7.7%.
Fig. 17 presents the results of flow simulations over C3X gas turbine vanes for both Case-I and Case-II. The PMV2F model demonstrates a significant improvement in accuracy compared to experimental data. Notably, Case-I and Case-II results exhibit enhancements
of 19.7% and 38.1%, respectively. This highlights the PMV2F model’s ability to not only resolve the stagnation point anomaly but
also accurately predict both the onset and length of flow transition. Additionally, the figure includes the uncalibrated version of
the PMV2F model for comparison. This visually demonstrates the effectiveness of the modified production term in eliminating the
stagnation point anomaly. The difference between the PMV2F model and the uncalibrated version further emphasizes the impact
of calibration on improving results. Specifically, Fig. 17(a) reveals that the first and second modifications contribute to accuracy
improvements of 16.6% and 3.1%, respectively, while Fig. 17(b) shows enhancements of 30.4% and 7.7%, respectively.
18
Journal of Computational Physics 520 (2025) 113499
A.A. Shahbazi and V. Esfahanian
5. Conclusions
This study investigates the flow and heat transfer characteristics over a gas turbine vane, employing various turbulence and
transition models. The complex nature of the flow, characterized by the stagnation point anomaly at the leading edge and the
transition from laminar to turbulent flow on the suction side, precluded the satisfactory performance of existing models. Given the
potential of the V2F model for detecting flow transition, this paper proposes two modifications to enhance its capabilities. The
first modification involves a revised production term for turbulent kinetic energy, incorporating the vorticity tensor to address the
stagnation point anomaly. The effectiveness of this modification was validated through simulations of impingement jet flow over a
flat plate, demonstrating a significant improvement in the heat transfer predictions. The second modification focuses on recalibration
of the turbulence model coefficients using test cases involving impingement jet flow and the ERCOFTAC T3 flat plate series. The
final results of flow and heat transfer simulations over C3X gas turbine vanes reveal that the first modification significantly improves
the accuracy of the heat transfer coefficient distribution by 16.6-30.4% while eliminating the stagnation point anomaly. The second
modification contributes to an additional 3.1-7.8% improvement in overall performance.
CRediT authorship contribution statement
Ali Akbar Shahbazi: Investigation, Methodology, Software, Validation, Visualization, Writing – original draft, Writing – review
& editing. Vahid Esfahanian: Supervision, Writing – review & editing.
Declaration of competing interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to
influence the work reported in this paper.
Acknowledgements
The authors wish to express their gratitude for scientific supports of VFERI (Vehicle, Fuel and Environment Research Institute) of
University of Tehran.
Data availability
Data will be made available on request.
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