DOI: 10.3303/CET24111085 

 
 

 

 

 

 

 
 
 

 
 
 

 
 

 

 
 

 
 
 

 
 
 

 
 
 

 
 
 

 
 

 
 
 

 
 
 

Paper Received: 17 January 2024; Revised: 2 June 2024; Accepted: 28 July 2024 
Please cite this article as: León-Cardona D., Rosso-Cerón A.M., Rodriguez-Ferreira J., Del Jesús Matínez M., Sanguino-Barajas P.A., 
2024, Assessment of RANS Turbulence Models for the Simulation of Turbulent Compressible Flows in Convergent-Divergent Nozzles, 
Chemical Engineering Transactions, 111, 505-510  DOI:10.3303/CET24111085 

CHEMICAL ENGINEERING TRANSACTIONS 

VOL. 111, 2024

A publication of 

The Italian Association 
of Chemical Engineering 
Online at www.cetjournal.it 

Guest Editors: Valerio Cozzani, Bruno Fabiano, Genserik Reniers 

Copyright © 2024, AIDIC Servizi S.r.l. 

ISBN 979-12-81206-11-3; ISSN 2283-9216 

Assessment of RANS Turbulence Models for the Simulation of

Turbulent Compressible Flows in Convergent-Divergent Nozzles 

Daniel León-Cardonaa, Ana M. Rosso-Cerónb,*, Julian Rodriguez-Ferreiraa, Manuel 

del Jesús Matíneza, Paola A. Sanguino-Barajasc

The main objective is to analyse gas flow dynamics in a Convergent-Divergent (CD) nozzle and predict its 

behaviour under various conditions. The model assumes adiabatic, impermeable walls with bidimensional,

compressible, steady, subsonic, and turbulent flow. Conservation equations for mass, energy, and momentum

form the basis of the mathematical model, which is solved using Reynolds-Averaged Navier-Stokes turbulent 

models including Standard k-ε, Realizable k-ε, Standard k-ω, and Shear-Stress Transport k-ω. Evaluation of 

these models across different Nozzle Pressure Ratios (NPR) shows that Shear-Stress Transport k-ω 

correlates best with experimental data on static pressure. Three ANSYS products: DesignModeler, Meshing, 

and Fluent, are used for numerical simulation. The methodology reliably predicts phenomena such as flow 

separation, Mach disc formation, and nozzle exit regime, crucial for safe and efficient system operation. 

1. Introduction

By 2026, a century will have passed since Robert Goddard introduced the CD nozzle in a liquid-fuelled rocket,

a milestone replicated in almost all rocket motors to date (NASA, 2023). Technological advancements in

manufacturing processes, materials, and numerical analysis methods have influenced the evolution of CD 

nozzles, incorporating these advances into their geometry. This has led to the emergence of alternatives to

Gustav de Laval's 1888 nozzle design (Krehl, 2009). 

This study analyses the impact of atmospheric pressure on the performance of rocket engines using

conventional contoured or bell-shaped CD nozzles, crucial components in the contemporary space race

(Mishra, 2017). As the rocket ascends vertically, the supersonic fluid expelled at the nozzle outlet undergoes 

an expansive regime (El-Sayed, 2016), determined by the relationship between the nozzle exit pressure and 

atmospheric pressure during ascent (Sutton and Biblarz, 2016). The expansion dynamics significantly

influence key performance variables such as thrust and impulse. If Pex≪Patm, overly expanded exhaust gases

result in instability and thrust loss, leading to mechanical issues (Baars et al., 2012). Conversely, if Pex≫Patm, 

the nozzle is under-expanded, leading to thrust loss by wasting outlet pressure. Optimal performance,

generating maximum thrust, occurs when Pex=Patm (Bahamon and Martinez, 2023). Variations from 

atmospheric pressure result in efficiency losses relative to optimal expansion, and since nozzle geometry is 

fixed (Ande and Yerraboina, 2018) while atmospheric pressure changes with altitude, the rocket motor spends

limited time within the optimal performance range during ascent. 

Pressure variations at the nozzle outlet, along its length and width, and their correlations with atmospheric 

pressure, as well as interactions with temperature, density, gas velocity, and fluid dynamic phenomena from

compressible fluid interacting with the geometry, have been thoroughly studied at analytical, numerical, and 

experimental levels. An illustrative example is Hunter's (2004) investigation into flow separation in a flat CD 

nozzle across various Nozzle Pressure Ratios (NPR), incorporating experimental, theoretical, and 

505

ana.rosso@unipamplona.edu.co

a Faculty of Physicomechanical Engineering, Universidad Industrial de Santander, Cra 27 Cll 9; Bucaramanga-Colombia
b Chemical Engineering Program, Universidad de Pamplona, km 1 vía Bucaramanga; Pamplona-Colombia
c Environmental Engineering Department, Unidades Tecnológicas de Santander, Calle de los Estudiantes # 9 - 82; 
Bucaramanga-Colombia



computational analyses. Detailed reports include experimental static pressure measurements on the nozzle 

wall and results from a two-dimensional numerical simulation using the PAB3D computational fluid dynamics 

code. This simulation incorporates two-equation turbulence closure and Reynolds Stress Algebraic models 

such as Shih-Zhu-Lumley (1995), Gatski-Speziale (1993), and Girimaji (1996). Fluid density variations for 

these NPR are depicted through both experimental and computational Schlieren images. Conclusively, 

controlled separation, as demonstrated in this work, ensures the entire over-expanded nozzle performance 

range stays within 10% of maximum thrust efficiency. Moreover, it is established that, across the analysed 

NPR range, the simulated turbulence models effectively approximate experimental values, with the Shih-Zhu-

Lumley model showing notable reliability. 

Hunter’s experimental data has become a foundational resource for validation in similar studies, where 

numerical simulations have been conducted for identical geometries, employing diverse turbulence models. 

For instance, Balabel et al. (2011) analysed the performance using specific NPR with turbulence models such 

as Standard k-ε (Launder and Spalding, 1972), Extended k-ε (Chen and Kim, 1987), k-ε-v2-f (v2-f-1) (Lien and 

Kalitzinm, 2001), Realizable k-ε-v2-f (v2-f-2) (Durbin, 1995), SST k-ω (Menter, 1994), and Reynolds Stress 

(Launder et al., 1975). Results indicate that SST k-ω (high-Reynolds) and realizable v2-f models outperform 

others in predicting shock wave positions and separation points, with SST k-ω being particularly notable. 

However, the v2-f model requires about a 20% increase in computation time. Similarly, Tolentino (2019) 

employed the same flat nozzle and experimental data to assess five turbulence models: SST k-ω, Standard k-

ω and k-ε, Transitional k-kl-ω, and Reynolds Stress. Conclusively, based on static pressure profiles and shock 

wave shapes, SST k-ω is identified as the optimal model, aligning most accurately with observed shapes and 

experimental data. 

In this work, the experimental data provided by Hunter are used as a bibliographic means of validation for the 

four selected turbulence models to describe and analyse the physical phenomena inside the flat nozzle. Three 

values of NPR were selected from the available list: 2.008, 2.412, and 3.413. The obtained static pressure 

profiles and density contours inside the nozzle are shown and compared with the experimental data. 

2. Computational Fluid Dynamic simulation 

Four RANS turbulence models are employed in computational fluid mechanics to simulate turbulence 

behaviour in a two-dimensional, stable, compressible, turbulent, and subsonic inlet flow within a flat nozzle 

with impermeable and adiabatic walls. This evaluation aims to establish the most suitable model for this 

specific application by comparing their strengths and weaknesses against experimental data. 

2.1 Turbulence models 

The mathematical model is based in the conservation of mass, energy, and momentum equations. The 

equation system is numerically solved using Reynolds-Averaged Navier-Stokes (RANS) turbulent models 

such as (ANSYS, 2021a): 

Standard k-ε 

This is one of the simplest and widely used RANS turbulence models. It is suitable for low and medium 

intensity turbulent flows but may not be suitable for high intensity turbulent flows or flow separation. It is based 

on two transport equations for turbulent kinetic energy and turbulent kinetic energy dissipation: 

𝜕

𝜕𝑡
(𝜌𝑘) +

𝜕

𝜕𝑥𝑖
(𝜌𝑘𝑢𝑖) =

𝜕

𝜕𝑥𝑗
[(𝜇 +

𝜇𝑡

𝜎𝑘
)
𝜕𝑘

𝜕𝑥𝑗
] + 𝐺𝑘 + 𝐺𝑏 − 𝜌𝜀 − 𝑌𝑀 + 𝑆𝑘  (1) 

𝜕

𝜕𝑡
(𝜌𝜀) +

𝜕

𝜕𝑥𝑖
(𝜌𝜀𝑢𝑖) =

𝜕

𝜕𝑥𝑗
[(𝜇 +

𝜇𝑡

𝜎𝜀
)

𝜕𝜀

𝜕𝑥𝑗
] + 𝐶1𝜀

𝜀

𝑘
(𝐺𝑘 + 𝐶3𝜀𝐺𝑏) − 𝐶2𝜀𝜌

𝜀2

𝑘
+ 𝑆𝜀  (2) 

Realizable k-ε 

This model is an improvement over the standard k-ε model, which considers the effects of turbulence 

production and anisotropy in the dissipation of turbulent kinetic energy. It is more accurate than the standard 

k-ε and is suitable for high-intensity turbulence flows and flow separation. The modelled transport equations 

for k and ε in this model are: 

𝜕

𝜕𝑡
(𝜌𝑘) +

𝜕

𝜕𝑥𝑗
(𝜌𝑘𝑢𝑗) =

𝜕

𝜕𝑥𝑗
[(𝜇 +

𝜇𝑡

𝜎𝑘
)
𝜕𝑘

𝜕𝑥𝑗
] + 𝐺𝑘 + 𝐺𝑏 − 𝜌𝜀 − 𝑌𝑀 + 𝑆𝑘  (3) 

𝜕

𝜕𝑡
(𝜌𝜀) +

𝜕

𝜕𝑥𝑗
(𝜌𝜀𝑢𝑗) =

𝜕

𝜕𝑥𝑗
[(𝜇 +

𝜇𝑡

𝜎𝜀
)

𝜕𝜀

𝜕𝑥𝑗
] + 𝜌𝐶1𝑆𝜀 − 𝜌𝐶2

𝜀2

𝑘+√𝜈𝜀
+ 𝐶1𝜀

𝜀

𝑘
𝐶3𝜀𝐺𝑏 + 𝑆𝜀  (4) 

506



Note that equation Eq(3) is the same as Eq(1) in the standard k-ε model, except for the model constants. 

However, the form of the ε equation is quite different from Eq(2). One of the notable features in this model is 

that the production term in Eq(4) does not involve the production of k. 

Standard k-ω 

The turbulent kinetic energy and the turbulent specific dissipation rate are obtained from the following 

transport equations: 

𝜕

𝜕𝑡
(𝜌𝑘) +

𝜕

𝜕𝑥𝑖
(𝜌𝑘𝑢𝑖) =

𝜕

𝜕𝑥𝑗
[(𝜇 +

𝜇𝑡

𝜎𝑘
)
𝜕𝑘

𝜕𝑥𝑗
] + 𝐺𝑘 − 𝑌𝑘 + 𝑆𝑘 + 𝐺𝑏  (5) 

𝜕

𝜕𝑡
(𝜌𝜔) +

𝜕

𝜕𝑥𝑖
(𝜌𝜔𝑢𝑖) =

𝜕

𝜕𝑥𝑗
[(𝜇 +

𝜇𝑡

𝜎ω
)
𝜕𝜔

𝜕𝑥𝑗
] + 𝐺𝜔 − 𝑌𝜔 + 𝑆𝜔 + 𝐺𝜔𝑏  (6) 

This model is more suitable for high-intensity turbulent flows and flow separation than the standard k-ε model 

but, may not be as accurate in low and medium-intensity turbulent flows. 

Shear Stress Transport k-ω 

This model combines the standard k-ω model and the standard k-ε model to take advantage of the strengths 

of both. It is suitable for low and high turbulence flows and is more accurate than individual models. Includes 

all the refinements of the Baseline k-ω (Menter, 1994) model and considers the transport of turbulent shear 

stress in the definition of turbulent viscosity: 

𝜕

𝜕𝑡
(𝜌𝑘) +

𝜕

𝜕𝑥𝑖
(𝜌𝑘𝑢𝑖) =

𝜕

𝜕𝑥𝑗
[(𝜇 +

𝜇𝑡

𝜎𝑘
)
𝜕𝑘

𝜕𝑥𝑗
] + 𝐺𝑘 − 𝑌𝑘 + 𝑆𝑘 + 𝐺𝑏  (7) 

𝜕

𝜕𝑡
(𝜌𝜔) +

𝜕

𝜕𝑥𝑖
(𝜌𝜔𝑢𝑖) =

𝜕

𝜕𝑥𝑗
[(𝜇 +

𝜇𝑡

𝜎ω
)
𝜕𝜔

𝜕𝑥𝑗
] + 𝐺𝜔 − 𝑌𝜔 + 𝐷𝜔 + 𝑆𝜔 + 𝐺𝜔𝑏  (8) 

2.2 Geometry 

Figure 1a illustrates the Hunter flat nozzle profile, and Figure 1b displays the dimensions of the computational 

domain employed in this study. Both are symmetrical and two-dimensional representations reconstructed in 

ANSYS using the DesignModeler tool. 

 

 

a. Hunter’s flat nozzle b. Dimensions of the computational domain 

Figure 1: Geometry 

2.3 Meshing and Computational domain 

Figure 2 shows the boundary conditions, as well as the 15 bodies into which the computational domain was 

divided. The SST k-ω turbulence model was used within a mesh independence analysis, from which it was 

concluded that for this study the best structured mesh to use was the one with 3,120 elements in the 

subdomain (bodies 2-5) and 131,441 for the rest of the domain (bodies 1, 6-15). Mesh that presents a 

maximum skewness of 0.378, which is classified as “very good” within the ANSYS skewness mesh metrics 

spectrum (ANSYS, 2021b). A smooth transition inflation was applied to the nozzle wall, with default transition 

ratio (0.272), 2 maximum layers and 1.2 growth rate. For each turbulence model, 12,000 iterations were 

executed, which took an average of 2 hours to finish. For the analysis, ANSYS 2019 R3 was used on a Dell 

Precision T7500 workstation with an Intel Xeon E5620 2.4 GHz processor and 12 GB RAM.  

507



 

Figure 2: Part of computational domain and mesh 

3. Results and analysis 

Figure 3 shows the comparison of Hunter's experimental data for the selected NPR with the static nozzle wall 

pressure profiles obtained according to the applied turbulence model. The total sums of the relative errors of 

each of these experimental values with respect to their computational equivalent, the maximum relative error 

and the computational time of each model are also included. It’s noted that the static pressure decreases as 

the fluid flows through the geometry of the nozzle due to the section gradually reduces, resulting in an 

increase in fluid velocity. Continuing from the throat area, as the flow expands, the boundary layer thickens, 

and the flow velocity begins to gradually decrease. This decrease will continue until the velocity is low enough 

for the detachment of the boundary layer to occur due to frictional forces, at which time the static pressure rise 

occurs. 

      

  
a. NPR = 2.008 b. NPR = 2.412 

  

 

 

 

 

 

 

 

 

c. NPR = 3.413 d. RANS models errors 

Figure 3: Comparison of experimental and computational static wall pressure profiles of the nozzle at different 

NPR and turbulence models 

 2      

  

     2

      

 utlet   

 nlet   

Symmetry   

 diabatic wall   

  .2  .  .  .   .2  .  .  . 2

x x
t

 

 . 

 .2

 . 

 . 

 . 

 . 

 . 

 . 

 . 

 

 
  

 

Experimental

S   k e

R   k e

S   k w

SS  k w

           

  .2  .  .  .   .2  .  .  . 2

x x
t

 

 . 

 .2

 . 

 . 

 . 

 . 

 . 

 . 

 . 

 

 
  

 

Experimental

S   k e

R   k e

S   k w

SS  k w

           

  .2  .  .  .   .2  .  .  . 2

x x
t

 

 . 

 .2

 . 

 . 

 . 

 . 

 . 

 . 

 . 

 

 
  

 

Experimental

S   k e

R   k e

S   k w

SS  k w

           

               

Error sum   .  22 .     .2 

S  k   aximum error  .    .    .2 

 omp. time  minutes          

Error sum2  . 2   . 2 22.  

R  k   aximum error 2.    .    .  

 omp. time  minutes        2 

Error sum2  .     .   2 .  

S  k   aximum error  .    .    .  

 omp. time  minutes        22

Error sum   . 2  .    .  

SS k   aximum error  . 22 .   .  

 omp. time  minutes  2  2  2 

 ozzle computational mesh

508



Figure 4 and Figure 5 show the Mach disks generated by the presence of a shock wave given by the 

supersonic expansion of the fluid in the divergent region of the flat nozzle. Its location and size depend, 

among other things, on the selected turbulence model and the relationship between the velocity and the static 

pressure of the fluid. 

 

   
a. NPR = 2.008 b. NPR = 2.412 c. NPR = 3.413 

Figure 4: Experimental Schlieren flow visualization at different NPR, Hunter (2004) 

   
a. NPR = 2.008 b. NPR = 2.412 c. NPR = 3.413 

Figure 5: Density contours at different NPR for the SST k-ω model 

4. Conclusions 

This study has provided valuable insights into how various turbulence models perform when simulating a flat 

nozzle under different Nozzle Pressure Ratios (NPR). The analysis focused on specific NPR values, revealing 

that the turbulence models aligned well with experimental data until encountering complex turbulence related 

to boundary layer separation. Among the models assessed, the SST k-ω model stood out as the most 

effective. It demonstrated superior accuracy in representing experimental data, particularly regarding static 

pressure on the flat nozzle's wall and the precise location of the Mach disk in the expanding gas zone. This 

highlights the SST k-ω model's reliability in capturing intricate turbulence phenomena, making it suitable for 

future studies in similar fluid dynamics applications. Nevertheless, this greater accuracy comes at a slightly 

higher computational cost than the other models. The RLZ k-ε model also shows acceptable accuracy with 

reduced errors compared to the STD k-ε and k-ω models, making it a viable option when it’s essential to 

balance accuracy and computational efficiency. In contrast, the STD k-ε and k-ω models have higher errors 

but are more computationally efficient, making them suitable for scenarios where computational resources are 

limited and a trade-off between accuracy and computational cost is acceptable. 

The significance of the SST k-ω model in accurately characterizing the dynamic features of the flat nozzle 

holds substantial implications for fluid dynamics and aerospace engineering. These findings not only advance 

our understanding of turbulence modelling but also emphasize the critical role of selecting appropriate models 

for precise simulations. Future research in rocket propulsion systems and nozzle design can leverage this 

study's methodology and results to choose turbulence models for enhanced predictive accuracy. 

Nomenclature

CD – Convergent-Divergent 

C1, C ε, C2, C2ε, C ε – constants, - 

Dω – cross-diffusion term, Pa/m2 

Gk – production of turbulence kinetic energy, Pa/s 

Gb – gen. of turbulence due to buoyancy, Pa/s 

Gω – production of ω, Pa/m2 

Gωb – buoyancy source for the ω equation, Pa/m2 

k – turbulent kinetic energy, m2/s2 

NPR – Nozzle Pressure Ratios 

Patm – atmospheric pressure, Pa 

Pex – nozzle exit pressure, Pa 

RANS – Reynolds-Averaged Navier-Stokes 

RLZ – Realizable 

S – modulus of the mean rate-of-strain tensor, s-1 

               

               

                                

               

               

                                

               

               

                                

               

               

                                

               

               

                                

               

               

                                

509



Sk – user-defined source terms, Pa/s 

Sε – user-defined source terms, Pa/s2 

Sω – user-defined source terms, Pa/m2 

SST – Shear-Stress Transport 

STD – Standard 

ui, uj – fluid velocity component, m/s 

Yk – dissipation of k due to turbulence, Pa/s 

YM – dilatation dissipation term, Pa/s 

Yω – dissipation of ω due to turbulence, Pa/m2 

ε – turbulent kinetic energy dissipation rate, m2/s3 

μ – molecular viscosity, kg/(m·s) 

μt – turbulent (or eddy) viscosity, kg/(m·s) 

ν – kinematic viscosity, m2/s 

ρ – density, kg/m3 

σk, σε, σω – turbulent Prandtl number, - 

ω – turbulent specific dissipation rate, s-1 

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	Assessment of RANS Turbulence Models for the Simulation of Turbulent Compressible Flows in Convergent-Divergent Nozzles