Tuned Generalised k-ω (GEKO) Turbulence Model Parameters for Predicting Transitional Flow Through Stenosis Geometries of Various Degrees

Abstract

Stenosis geometries are constrictions of a biological tube that can be found in many forms in the human body. Capturing the flow field in such geometries is important. For this purpose, simulations were performed using the generalised k-$\omega$ (GEKO) turbulence model to study flow through stenosis geometries with throat constrictions of 75, 50 and 25% area reduction. Laminar flow conditions of Re = 2000 and 1000 were applied and the results were compared with experimental data. The effect of four GEKO parameters ($C_{SEP}$, $C_{NW}$, $C_{JET}$ and $C_{MIX}$) on flow in the post-stenotic region was investigated by simulating a wide range of parameter values. Results showed that the $C_{MIX}$ parameter, combined with a modified GEKO blending function, had the greatest effect on axial velocity, velocity fluctuations and the location of the jet breakdown region. A $C_{MIX}$ value of 0.4 closely matched the experimental results for a 75% area reduction stenosis at $Re=2000$ and showed significant improvements over existing Reynolds-averaged Navier–Stokes models. The GEKO model was also able to closely match the axial velocity results predicted by previously published large-eddy simulation models under the same flow conditions. Furthermore, the GEKO model was applied to a realistic oral-to-tracheal airway model for a Reynolds number of 2000 and produced results consistent with the idealised stenotic tube.

1. Introduction

Stenosis is a constriction of a biological tube, occurring throughout the body in various forms. In the cardiovascular system, conditions like atherosclerosis [1] and carotid stenosis [2] restrict blood flow, increasing stroke risk. In the respiratory tract, diseases such as tracheomalacia [3] and chronic obstructive pulmonary disease (COPD) [4] cause abnormal narrowing and impair breathing. These variations in geometry change the flow field and impact patients’ health. Some stenoses, like the glottis, are normal and serve functional roles, such as increasing airflow velocity through the vocal folds [5]. Accurate prediction of flow physics through these narrowed regions is essential for early detection and disease assessment.

It is often difficult, if not impossible, to obtain in vivo measurements of flow physics in these types of stenosis geometries. However, the rapid and continuous development of computational fluid dynamics (CFD) has allowed researchers to characterise flow through complex geometry. By using patient-specific geometries obtained from magnetic resonance imaging (MRI) and computed tomography (CT) scans, it is possible to solve the governing equations of fluid flows to produce quantitative and qualitative data for a wide range of flow properties. CFD has been used extensively in the diagnosis of stenosed blood vessels [6] and investigations on the effect of their severity [7,8] and eccentricity [9] on blood flow physics. CFD has also been used to assess airflow physics through the nasal cavity [10,11], upper respiratory tract [12,13], lungs [14] and complete respiratory system models [15,16]. The glottis, characterised as a stenosis, forms a narrow constriction in the larynx and generates a highly turbulent flow field through the trachea [17,18,19,20]. Furthermore, the size and shape of the glottis and larynx impact airflow characteristics and particle deposition in the trachea at identical inhalation flow rates [21,22]. In abnormal respiratory stenoses, Gosman et al. [23] simulated airflow and particle deposition in five different respiratory tracts from patients suffering from laryngotracheal stenosis, while Yang et al. [24] investigated subglottic stenoses in neonates, finding that stenoses with a higher area reduction drastically increased the work of breathing.

Validation of turbulence models in CFD is essential because comparing numerical predictions with experimental data provides researchers and clinicians with greater confidence in the results. The experimental work of Ahmed and Giddens [25,26] on stenotic flow has been used by researchers to validate a variety of turbulence models for internal flows. Their work assessed steady flow for upstream Reynolds numbers between 500 and 2000 through three stenotic geometries with cross-sectional area reductions of 25, 50 and 75% relative to the non-stenosed tube, as shown later in Figure 1. It was found that the breakdown of the stenosis jet was highly dependent on the combination of stenotic area reduction and the upstream Reynolds number. Turbulence modelling approaches are grouped into three distinct families: direct numerical simulation (DNS), large eddy simulation (LES) models and Reynolds-averaged Navier–Stokes (RANS) models. DNS solves the Navier–Stokes equations directly, making it too time- and resource-intensive for clinical use. For example, Sherwin and Blackburn [27] and Varghese et al. [28,29] used DNS to investigate steady and pulsatile flow at low Reynolds numbers through a stenosis and compared the results with the experimental data. DNS was generally in good agreement provided a suitable turbulence level was prescribed at the inlet, with these studies suggesting that upstream turbulence noise contributed to downstream jet breakdown in the work of Ahmed and Giddens [25,26,30]. LES resolves large turbulent eddies while modelling smaller eddies below the grid size, often providing more accurate results compared with RANS models but at increased computational cost. Luo et al. [31] assessed the performance of LES and RANS models for the same stenotic tube, finding that the LES model predicted the axial velocity downstream of the stenosis more accurately than the RANS results. Cui [32] found that the Smagorinsky model showed improvements in axial velocity results compared with the low Reynolds number k-$\omega$ and k-$\omega$ SST models. While LES models typically produce more accurate results, the high mesh requirement (e.g., $x^{+}\approx y^{+}\approx z^{+}$ at the near wall) and computational costs limit its widespread adoption. In many cases, LES models that use wall functions are employed without reporting the details of the wall treatment. These generally result in the use of the log-law condition at the wall and are unsuitable for low-Reynolds-number flows.

In contrast, RANS models offer moderate accuracy for a wide variety of flows at a significantly reduced computational cost. Zhang et al. [33] and Zhang and Kleinstreuer [34] investigated low-Reynolds-number variations of the k-$\omega$ and k-$\epsilon$ models alongside the Menter k-$\omega$ model in a stenosis with 75% area reduction at the throat and inlet conditions of $Re=2000$. Their work showed that RANS models’ low-Reynolds-number variants performed better than the Menter k-$\omega$ model. Elcner et al. [35] used a 75% area reduction stenosis model at $Re=2000$ and investigated a range of LES and RANS models. Their findings showed that LES models captured the velocity field accurately, while RANS models showed great variation in the results. Variants of the k-$\omega$ model over-predicted the axial velocity values downstream from the stenosis throat, while k-$\epsilon$ variants both over- and under-predicted velocity, depending on the k-$\epsilon$ model used. This variability demonstrates the challenge of accurately predicting flow through stenosis geometries using RANS models.

Despite these challenges, RANS simulations remain favourable among researchers for capturing averaged flow variables where time-dependent structures are less critical. In an effort to improve RANS model performance, a recently developed turbulence model, the generalised $k-\omega$ (GEKO) model, has the potential to accurately capture the downstream stenosis flow field without the high computational costs of LES and DNS approaches. The GEKO model is based on the $k-\omega$ formulation and allows users to tune model parameters without affecting the underlying empirical turbulence constants [36]. The GEKO model has been validated against a wide range of flow types [37,38,39]. Yüksekdağ et al. [40] investigated tuning of the GEKO parameters for the flow of internal jet impingement and found good agreement with the experimental data. Similarly, He et al. [41] found that GEKO parameter tuning leads to closer agreement with experimental data in a pump turbine. Ganatra et al. [42] demonstrated that GEKO parameter tuning led to closer agreement with the experimental data in a pump turbine, and showed that GEKO tuning yielded velocity results comparable to those of LES in free and impinging jets. However, the applicability of the GEKO model to stenosis geometries is unexplored in the literature.

Previous publications have demonstrated variability in the results produced by RANS models for stenotic flow [35,43]. To address this issue, this study aims to extend previous research by improving the RANS turbulence modelling for flow through a stenosis geometry at relatively low Reynolds numbers using the GEKO model. Specifically, this study examines the impact of inlet conditions for two Reynolds numbers (1000 and 2000) and three degrees of stenosis area reduction (25%, 50% and 75%) using the GEKO turbulence model, which allows for flexible adjustment of parameters to control flow behaviour. The results are validated through comparison with experimental data from Ahmed and Giddens [25,26].

2. Materials and Methods

2.1. Geometry

The three-dimensional stenosis geometry, shown in Figure 1a, was created using the same equation from Varghese et al. [28], which matches the geometry used in the experimental work of Ahmed and Giddens [25]. The geometry was created in Ansys SpaceClaim 2024R2 using the equation function to define the shape of the stenosis. The stenosis shape in the x-direction (the flow direction) is calculated by the function $S\left(x\right)$, defined as

$$\left\{\begin{array}{cc}\hfill S\left(x\right)& ={\displaystyle \frac{1}{2}}D\left[1-s_0\left(1+cos\left(2\pi {\displaystyle \frac{x-x_0}{L}}\right)\right)\right],\hfill \\ \hfill y& =S\left(x\right)cos\theta , z=S\left(x\right)sin\theta ,\hfill \end{array}\right.$$

(1)

where D is the diameter of the non-stenosed tube, $s_o=0.25$ corresponds to a 75% reduction in area at the stenosis throat, $s_o=0.1464$ corresponds to a 50% reduction and $s_o=0.068$ to a 25% area reduction, L is the length of the stenosis region in the x-direction ($2D$), $x_o$ is the centre location of the stenosis throat and the cross-stream coordinates in the y- and z- are calculated as a function of $S\left(x\right)$. The upstream, non-stenosed section of the pipe was extended one diameter length (equal to $x/D=1$) from the stenosis section, based on a systematic study presented later in the results. The downstream section was extended 35 diameter ($x/D=35$) lengths from the stenosis section to ensure adequate flow stabilisation prior to the outlet.

2.2. Mesh

All stenosis geometries were meshed using a poly-hexcore mesh, which applies polyhedral elements at the surface and hexahedral cells through the bulk flow region. The application of equally sized hexahedral elements to the bulk flow region ensured a high mesh quality throughout areas of interest, reducing numerical diffusion due to their perfect orthogonal quality. Ten prism layers were applied to all models with a first layer cell height of 0.08 mm. All k-$\omega$ models in Ansys Fluent use a $y^{+}$ insensitive wall treatment; however, the first layer satisfied an overall $y^{+}\approx 1$ with a maximum $y^{+}$ value of 1.4. One peel layer of polyhedral elements connected the prism layer to the hexahedral elements. Three bodies of influence (BOI) were applied to ensure that the hexahedral elements are equal in size throughout each specific region. A small BOI was created for the stenosis region to capture fine details of the flow field, while two larger BOI’s were created downstream from the stenosis towards the outlet, where flow physics is no longer critical.

Table 1. Mesh parameters used for the four resolution levels (coarse to finest) applied to the 75% stenosis geometry. The minimum hexahedral cell length refers to the finest elements applied to the stenosis region via a body of influence (BOI). Identical meshing strategies were used for the 50% and 25% stenosis cases, with slightly different final cell counts. All meshes include 10 prism layers at the wall, with a first-layer height of 0.08 mm.

Geometry	Cells (Millions)	Min Hex Length (mm)	Mesh Level
Stenosis 75%	0.236	3.0	Coarse
	0.305	2.5	Medium
	0.458	2.0	Fine
	0.830	1.5	Finest

summarises the total number of cells for the 75% stenosis geometry and the size of the finest hexahedral elements.

2.3. Description of the Generalised $k-\omega$ Model

The basis of the GEKO formulation for a steady, incompressible flow is given by

$${\displaystyle \frac{\partial \left(\rho u_{j}k\right)}{\partial x_j}}=P_k-C_{\mu }\rho k\omega +{\displaystyle \frac{\partial }{\partial x_j}}\left[\left(\mu +{\displaystyle \frac{{\mu }_t}{{\sigma }_k}}\right){\displaystyle \frac{\partial k}{\partial x_j}}\right]$$

(2)

$${\displaystyle \frac{\partial \left(\rho u_j\omega \right)}{\partial x_j}}=C_{\omega 1}{F}_1{\displaystyle \frac{\omega }{k}}{P}_k-C_{\omega 2}{F}_2\rho {\omega }^2+\rho F_{3}CD+{\displaystyle \frac{\partial }{\partial x_j}}\left[\left(\mu +{\displaystyle \frac{{\mu }_t}{{\sigma }_{\omega }}}\right){\displaystyle \frac{\partial \omega }{\partial x_j}}\right]$$

(3)

where $\rho$ is the fluid density, k is the turbulence kinetic energy, $u_j$ is the velocity vector component in the j direction, $C_{\mu }$, $C_{\omega 1}$ and $C_{\omega 2}$ are constants, $\omega$ is the specific dissipation rate, $\mu$ is the fluid viscosity and ${\sigma }_k$, ${\sigma }_{\omega }$ are constants. $F_1$, $F_2$ and $F_3$ are functions of the tunable free coefficients. The turbulence viscosity, ${\mu }_t$, is calculated by

$${\mu }_t=\rho {\nu }_t=\rho {\displaystyle \frac{k}{\max (\omega ,S/C_{\mathrm{Realize}})}}$$

(4)

where ${\nu }_t$ is the turbulent kinematic viscosity, S is the strain rate magnitude and $C_{realize}$ is an eddy viscosity limiter. Production of turbulence kinetic energy k is calculated by the equation

$$P_k=-{\tau }_{ij}{\displaystyle \frac{\partial u_i}{\partial x_j}}$$

(5)

where ${\tau }_{ij}$ is the Reynolds stress term, defined as

$${\tau }_{ij}=-\rho \overline{u_i^{\prime }{u}_j^{\prime }}.$$

(6)

The cross-diffusion term is defined as

$$CD={\displaystyle \frac{2}{{\sigma }_{\omega }}}{\displaystyle \frac{1}{\omega }}{\displaystyle \frac{\partial k}{\partial x_j}}{\displaystyle \frac{\partial \omega }{\partial x_j}}$$

(7)

The GEKO model uses a blending function, $F_{blend}$, which automatically detects near-wall and free-shear regions within the domain. $F_{blend}$ equates to 1 in the near-wall region and 0 in the free-shear region. In near-wall regions where $F_{blend}=1$, the effects of the $C_{SEP}$ and $C_{NW}$ parameters are active. In free-shear regions where $F_{blend}=0$, the $C_{MIX}$ and $C_{JET}$ parameters activate and impact spreading rates of free-shear flows. The exact details of the formulation of the blending function, $F_{blend}$, and the functions of the free coefficients ($F_1$, $F_2$ and $F_3$) have not been published at the time of writing this paper.

2.4. Physical Significance of the GEKO Parameters

The GEKO model offers flexibility by allowing users to adjust six model coefficients ($C_{SEP}$, $C_{NW}$, $C_{MIX}$, $C_{JET}$, $C_{CORNER}$ and $C_{CURV}$) to tune the model for various flow scenarios. These coefficients act as “tuning knobs” for different flow characteristics, enabling the model to be adapted to a wider range of applications than traditional two-equation models. The $C_{SEP}$ parameter alters eddy viscosity throughout the entire domain, primarily affecting boundary layer separation while preserving the near-wall log-layer calibration. Higher values of $C_{SEP}$ reduce the likelihood of boundary-layer separation. $C_{NW}$ impacts wall shear-stress values and heat transfer rates in the boundary layer with minimal influence on free-shear flows. The $C_{MIX}$ parameter affects eddy-viscosity levels in free-shear flows and has no impact on boundary layers through $F_{blend}$. Higher $C_{MIX}$ values will typically decrease velocity profiles by increasing turbulence within these regions. By default, the $C_{MIX}$ parameter is automatically adjusted depending on the $C_{SEP}$ value by the correlation equation

$$C_{MixCor}=0.35\mathrm{sign}(C_{SEP}-1)\sqrt{\left(\left|C_{SEP}-1\right|\right)}$$

(8)

which ensures that free shear flows are unaffected by the influence of $C_{SEP}$. The value of $C_{MIX}$ can also be manually adjusted if this automatic correlation is undesired. The $C_{JET}$ parameter impacts the spreading rate of jet flows and is a sub-function of $C_{MIX}$. It acts to reduce the impact of $C_{MIX}$ (if $C_{MIX}$≠ 0) on jet-like flows and typically does not require adjustment. $C_{CORNER}$ is a non-linear stress–strain term that corrects secondary flows in rectangular channels. $C_{CURV}$ is a pre-existing curvature correction model that improves the prediction of swirling and rotating flows.

For this study, the impact of parameters $C_{SEP}$, $C_{NW}$, $C_{JET}$ and $C_{MIX}$ were investigated, as they would have the most impact on flows of this type. The recommended range and default values of these GEKO parameters are shown in

Table 2. Generalised $k-\omega$ (GEKO) parameter ranges used across all stenosis geometries and flow conditions. These values lie within the recommended bounds specified by Menter et al. [36].

Parameter	Min	Max	Default
$C_{SEP}$	$0.80$	$2.50$	$1.75$
$C_{CNW}$	$-2.00$	$2.00$	$0.50$
$C_{JET}$	$0.00$	$1.00$	$0.90$
$C_{MIX}$	$0.00$	$1.00$	$C_{\mathrm{MixCor}}$

. Both $C_{CORNER}$ and $C_{CURV}$ remained deactivated as in the GEKO default settings, as they are not relevant for this particular geometry.

2.5. Numerical Simulation

The flow was simulated in steady state using the conservation of mass and momentum equations for incompressible flow, defined as

$${\displaystyle \frac{\partial u_i}{\partial x_i}}=0$$

(9)

$$u_j{\displaystyle \frac{\partial u_i}{\partial x_j}}=-{\displaystyle \frac{1}{\rho }}{\displaystyle \frac{\partial p}{\partial x_i}}+{\displaystyle \frac{\partial }{\partial x_j}}\left[\left(\nu +{\nu }_T\right){\displaystyle \frac{\partial u_i}{\partial x_j}}\right]$$

(10)

where p is the pressure, $\nu$ is the kinematic viscosity of the fluid and ${\nu }_t$ is the turbulent viscosity. The fluid used had a dynamic viscosity of $\mu =0.008$ kg/$\mathrm{m} · \mathrm{s}$ and density of $\rho =1156.5$ kg/m³, which matches the properties of the 63% glycerol mix at T = 33 °C used in Ahmed and Giddens [25].

Two flow conditions of $Re=1000$ and 2000, which match the inlet flow conditions of Ahmed and Giddens [25], were investigated and applied to the inlet as a parabolic velocity profile. Details of the turbulence parameters applied at the inlet are discussed in the Results section. Due to sufficient extension of the downstream pipe, an outflow condition was applied to prevent reversed flow at the outlet.

All simulations were performed using Ansys Fluent version 2024R2 using the coupled pressure–velocity scheme. The spatial discretization gradient was calculated with the least squares cell-based method and pressure was calculated using a second-order scheme. Second-order upwind discretization schemes were used for the momentum, turbulence kinetic energy and specific dissipation rate equations. Each simulation was run in steady state until the convergence criteria of $1\times {10}^{-10}$ for all residuals were met. Velocity data were monitored at various locations throughout the domain to ensure stabilisation of the flow field had occurred.

3. Results

3.1. Mesh Independence

Figure 2 presents the results of the mesh independence study, using the four different mesh sizes, outlined in Table 1, for an inlet flow condition of $Re=2000$ and a 75% area reduction at the stenosis throat.

The centreline axial velocity, normalised by the average inlet velocity, ${\overline{V}}_{in}$ is nearly identical across all meshes up to $x/D=3$ downstream. Minor discrepancies appear in the jet breakdown region but remain small. The normalised velocity fluctuations, $u\prime /{\overline{V}}_{in}$, are based on an isotropic turbulence assumption for the GEKO turbulence model, where $u\prime =\sqrt{2k/3}$, and k is the turbulence kinetic energy. Similar to the velocity results, the predicted normalised axial velocity fluctuations agree closely among all meshes until $x/D=3$, after which differences are still minimal. Consequently, the fine mesh (0.458 million cells) was used for the remaining simulations in this study.

3.2. Blending Function

The GEKO model is a generalisation of the k-$\omega$ model, which was constructed by blending the k-$\omega$ and k-$\epsilon$ models so that the best features of the two models are retained. Its blending function enforces use of the k-$\omega$ model in the wall region and a k-$\epsilon$ model in the free stream while limiting the influence of the $C_{MIX}$ and $C_{JET}$ parameters to areas outside the boundary layer. Under the built-in blending settings, the entire non-stenosed section was being treated as a boundary layer, and, thus, $C_{MIX}$ and $C_{JET}$ were not activated. To confine this effect to the near-wall region, we created a user-defined function (UDF) that forces the blending function to equal 1 only in the first 10 prism layer cells. Since the flow is laminar, the near-wall height in this study is arbitrary. Figure 3a shows the GEKO blending function being restricted to the first 10 prism layer cells. Figure 3a also compares downstream axial velocity contours for the built-in and modified blending functions. With the default blending function, the jet extends six diameters downstream, whereas the modified version shortens it to four diameters.

The effect of the blending function is shown in Figure 3b with a comparison of normalised axial velocity and velocity fluctuation along the centreline with the experimental data of Ahmed and Giddens [25]. Both blending functions predict similar axial velocity from the stenosis throat ($x/D=0$) until $x/D\approx 3.5$, where the modified blending function shows earlier jet breakdown, aligning more closely with the experiment. Although the centreline axial velocity is slightly higher than the experimental data from $x/D=3$ to $x/D=5$, it provides better overall agreement compared with the built-in blending function. In contrast, the built-in blending function causes a gradual decrease in centreline axial velocity, followed by a sharp drop at $x/D=4$ coinciding with the jet breakdown (see Figure 3a). At $x/D=6$, the built-in blending function predicts an axial velocity around 2 $V_x/{\overline{V}}_{in}$ whereas experimental data showed values around 1.5 $V_x/{\overline{V}}_{in}$.

Normalised centreline velocity fluctuations remain close to zero for both blending functions until $x/D=3$, where the axial velocity begins to diverge. With the built-in blending function, velocity fluctuations increase slowly after $x/D=3.5$ and peak at $x/D=5$. In contrast, the modified blending function produces a sharp increase at $x/D=3$ and a higher peak at $x/D=5$. Despite the improvement in location and value of the velocity fluctuations, neither case fully matches the experimental data.

3.3. Influence of Inlet Length

The stenosis geometry was extended by only one diameter upstream from the stenosis section for all preceding GEKO simulations. In numerical work, it is standard practice to extend the inlet domain to ensure that fully developed flow conditions enter the region of interest. Under laminar flow ($Re=2000$), however, the parabolic velocity profile evolved into a turbulent, power-law profile when the upstream geometry was significantly longer. This phenomenon is shown in Figure 4a, where the normalised axial velocity profile is plotted across the pipe diameter $y/D$ at various upstream distances from the stenosis throat.

At $30D$ upstream, the inlet velocity profile is distinctly parabolic, but at $14D$ upstream, it has lost its parabolic curved shape. At $10D$ upstream, the centreline velocity is flat while near-wall gradients grow steeper, and by $1D$, the profile has become a fully developed turbulent profile—an unphysical result for $Re=2000$. In comparison, when the inlet is placed just $2D$ upstream, the velocity profile at $1D$ remains parabolic (Figure 4a). This indicates that the RANS assumption of a fully turbulent flow will gradually reshape an initially laminar profile into a power-law turbulent profile over a sufficiently long development length. Figure 4b highlights this evolution with a contour plot of normalised axial velocity.

3.4. Influence of Inlet Turbulence Parameters

For turbulent flows, turbulence parameters must be specified at the inlet of numerical simulations. The work of Ahmed and Giddens [25] only considered laminar Reynolds numbers; therefore, no turbulence parameters were provided. However, this does not mean that there was no unsteadiness present within the upstream flow that may occur from the experimental setup (e.g., pump vibrations). In this case, the turbulence parameters were defined as the turbulence intensity (TI) percentage, which is a measure of random velocity fluctuations, and the hydraulic diameter, which limits the formation of eddies larger than the geometry itself and is set as a constant value. The plots in Figure 5 provide results with five different inlet conditions for normalised centreline axial velocity and velocity fluctuations. The inlet parameters range from extremely low turbulence (${10}^{-10}$% TI) to moderate turbulence (5% TI). At the stenosis throat ($x/D=0$), the choice of TI does not affect the axial centreline velocity, but further downstream, higher TI reduced the axial velocity values. At low turbulence (${10}^{-10}$% TI to 2% TI), similar axial velocity results are produced from $x/D=3$ onwards. Increasing the turbulence (5% TI) caused a sharper drop in axial velocity along the centreline.

Increasing turbulence at the inlet influenced velocity fluctuations, including at the stenosis throat. Lower turbulence parameters exhibited low velocity fluctuations from $x/D=0$ to $x/D=2.5$, but a high turbulence intensity greatly increased these velocity fluctuations through this region. Compared with the experimental data, the numerical results agree better at lower turbulence intensities at the stenosis throat. Further downstream, velocity fluctuations peak much later compared with the experimental results. Although a higher turbulence intensity compared well between $x/D=2.5$ and $x/D=6$, velocity fluctuations are too high compared with the experimental data. Consequently, a TI of 1% was chosen for the remaining GEKO simulations, as the axial velocity predictions along the centreline agreed well, and velocity fluctuations through the throat are consistent with the experimental data.

Figure 5b further illustrates the effect of inlet TI on the normalised axial velocity contours. Low inlet TI generated a long jet core extending to $x/D\approx$ 4.5 diameters downstream; increasing TI to 2% reduced the jet core length to $x/D\approx$ 4; while 5% TI reduced it further to $x/D\approx$ 3.5 diameters. The velocity fluctuation contours also show that increasing inlet TI significantly increased the level of velocity fluctuations, particularly through the stenosis throat. This is expected, as increasing TI increases the level of random velocity fluctuations at the inlet and the stenosis wall, and an increase in velocity fluctuations leads to faster jet breakdown.

3.5. Effect of the GEKO Parameters

3.5.1. Impact of $C_{SEP}$

Figure 6a plots the normalised centreline axial velocity and velocity fluctuations for various $C_{SEP}$ values within the range given in Table 2. All $C_{SEP}$ values yield the same velocity of $4.3 V_x/{\overline{V}}_{in}$ at $x/D=0$, indicating no upstream influence. The centreline velocity remains similar among all $C_{SEP}$ until $x/D=3$. Experimental data show a velocity spike at $x/D=1.5$ followed by a sharp decrease at $x/D=2.5$, whereas the computational results predict a gradual decline up to $x/D=3$. Beyond $x/D=3$, all $C_{SEP}$ values exhibit a steep drop from $4 V_x/{\overline{V}}_{in}$ to $1.5 V_x/{\overline{V}}_{in}$ by $x/D=6$. Here at the jet breakdown region, all $C_{SEP}$ values slightly over-predict velocity compared with experimental data, except for $C_{SEP}=0.8$, which under-predicts it. As $C_{SEP}$ primarily affects boundary-layer separation, altering this parameter has only a minimal impact on the overall flow.

Due to the stenosis geometry, the boundary layer separates as the constriction expands and pressure rises. Interestingly, changing the $C_{SEP}$ value does not affect the results uniformly: $C_{SEP}$ values of 1 and 2.5 produce similar centreline velocities, while 1.6 and 2 yield higher velocities. Figure 6 shows the normalised centreline velocity fluctuations. All $C_{SEP}$ values predict low fluctuations from the stenosis throat, diverging at $x/D=2.5$. The computational results agree with the experimental data up to $x/D=1$, where experimental data show a sharp rise in velocity fluctuations. At $x/D=2.5$, all $C_{SEP}$ settings predict increased fluctuations, peaking between $x/D=3.5$ and $x/D=4$, whereas experiments show a maximum at $x/D=3.5$.

In Figure 6b, normalised axial velocities are plotted radially at the downstream planes indicated in Figure 2. A value of 0 on the y-axis represents the pipe centre and 1 the wall. At $x/D=0$, velocity increases abruptly off the wall as fluid is forced through the constriction, with all $C_{SEP}$ values slightly exceeding the experimental results. At $x/D=1$, there is a region of low velocity at the near-wall extending 0.4R inwards due to fluid moving into the expanded non-stenosed pipe section; here, a small recirculation zone appears in the computational results but not in the experimental data. Nevertheless, the GEKO model captures the ‘S’ curve profile at $x/D=1$ and agrees well with experimental data for all $C_{SEP}$ values. By $x/D=2.5$, the recirculation zone disappears in the CFD results, whereas experimental data show a small region. Beyond $x/D=4$, the jet diffuses into a flatter, turbulent profile. Overall, changing $C_{SEP}$ has only a minor influence on the post-stenosis flow field, as axial velocities remain similar downstream across all parameter values.

3.5.2. Impact of $C_{NW}$

Figure 7 compares the normalised axial velocity and velocity fluctuations for $C_{NW}$ values ranging from $-2$ to 2 with experimental data. Adjustment of the $C_{NW}$ parameters has a minimal influence on the centreline axial velocity and velocity fluctuations. Normalised axial velocity along the y-direction at the six downstream planes was also minimally impacted by $C_{NW}$ and omitted. Despite the broad range of $C_{NW}$ values tested, there is no significant change in both the normalised axial velocity and velocity fluctuations. When all other parameters are set to their defaults, the results for all $C_{NW}$ values are effectively identical to the default GEKO option. Thus, adjusting $C_{NW}$ does not improve the GEKO model’s predictive capabilities in this stenosis flow configuration.

3.5.3. Impact of $C_{JET}$

Figure 8 compares the normalised axial velocity and velocity fluctuations with the experimental data of Ahmed and Giddens [25] for a range of $C_{JET}$ values. Similar to $C_{NW}$, changes in $C_{JET}$ had a negligible influence on the centreline axial velocity and velocity fluctuations at all measured downstream locations. Moreover, $C_{JET}$ had a minimal impact on normalised axial velocity in the y-direction at the six downstream locations and these plots are omitted.

3.5.4. Impact of $C_{MIX}$

Figure 9a shows that varying $C_{MIX}$ alters the normalised centreline axial velocity significantly. At $x/D=0$, velocity values are equal for all $C_{MIX}$ values and begin to deviate from $x/D=0.5$ onward. Lower values of $C_{MIX}$ (0 to 0.3) maintain high velocities (∼4.3 $V_x/{\overline{V}}_{in}$) while higher values (>0.4) drop below 4.2 $V_x/{\overline{V}}_{in}$ at $x/D=1$. Further downstream, the value of $C_{MIX}$ impacts the location and onset of the jet breakdown: $C_{MIX}=0$ delays jet breakdown until $x/D=5$, while $C_{MIX}=1$ triggers it by $x/D=2$, and the axial velocity reduces rapidly. Compared with experimental data, $C_{MIX}=0.4$ best reproduces the jet breakdown from $x/D=3$ onward and generally agrees with velocities upstream. Normalised centreline velocity fluctuations provide insight into how $C_{MIX}$ affects the jet breakdown region. Higher $C_{MIX}$ values produce a sharp increase in velocity fluctuations at an earlier location downstream and produce velocity fluctuations at $x/D=2$. Comparatively, lower $C_{MIX}$ values produce centreline velocity fluctuations further downstream at a gradual rate. However, none of the tested $C_{MIX}$ values reproduce the experimental observation of fluctuations beginning at $x/D=1$ [25].

In Figure 9a, normalised axial velocity contours on the centre plane show that $C_{MIX}=0$ maintains a jet core beyond $x/D=6$, while $C_{MIX}=1$ reduces it to $x/D=2.5$. Contours of normalised velocity fluctuations further reveal that $C_{MIX}$ influences both the magnitude and onset of turbulence through velocity fluctuations build-up from the pipe wall. All $C_{MIX}$ values generate fluctuations just downstream of the stenosis throat, but $C_{MIX}=0$ develops them gradually, converging near the centre at $x/D=5$. In comparison, a value of $C_{MIX}=1$ leads to rapid turbulence generation at the wall at a high magnitude before converging at the centre before $x/D=2.5$.

Figure 9b indicates the minimal sensitivity of normalised axial velocity across radial planes until $x/D=2.5$. At this location, only $C_{MIX}=0$ captures the small recirculation zone near $y/R=0.9$. Overall, velocities tend to be over-predicted for $0.6<y/R<1$ and under-predicted for $0<y/R<0.6$. Interestingly, $C_{MIX}=0$ matches experiments at $x/D=2.5$ but diverges downstream. Lower $C_{MIX}$ values produce flatter velocity profiles at $x/D=4$, mirroring experimental data, whereas higher values still exhibit a jet core. At $x/D=6$, all $C_{MIX}$ cases yield flat profiles except for a value of $C_{MIX}=0$.

3.6. Different Stenosis Area Reductions

To further examine the effect of the $C_{MIX}$ parameter on the stenosis flow, simulations at $Re=2000$ were performed with stenosis area reductions of 50% and 25%. Due to limited experimental data, only the normalised centreline axial velocity and velocity fluctuations were compared. Figure 10 presents the normalised axial velocity and velocity fluctuations along the centreline of a 50 and 25% stenosis for different values of $C_{MIX}$. The experimental results for the 50% stenosis show an axial velocity value of approximately 2.8 $V_x/{\overline{V}}_{in}$ that remains constant until $x/D=2.5$ then gradually decreases to about 1.5 $V_x/{\overline{V}}_{in}$ at $x/D=6$. Compared with the 75% stenosis, the axial velocity at the throat is lower due to the larger cross-sectional area. A higher value of $C_{MIX}$ triggers an earlier jet breakdown, which is consistent with results shown for a 75% stenosis. However, it is observed that no value of $C_{MIX}$ is capable of matching experimental axial velocity values through a 50% stenosis geometry.

The experimental results show that the axial velocity remains constant downstream from stenosis with an area reduction of 25%, suggesting that the geometry is not constricted enough to have a significant impact on the flow field. A $C_{MIX}$ value of 0 predicts a relatively constant velocity value of ∼2.0 $V_x/{\overline{V}}_{in}$ along the centreline, best matching the experimental results.

For a flow condition of $Re=2000$ through a 50% stenosis, the experimental data exhibited a sharp increase at $x/D=2$, whereas all values of $C_{MIX}$ predict that velocity fluctuations begin to increase further downstream. Increasing $C_{MIX}$ shortens the distance at which fluctuations appear, causing a sharp increase at $x/D=3$ when $C_{MIX}=1$. A $C_{MIX}$ value of 1 improves velocity fluctuation predictions, but they are still under-predicted compared with experimental data. For the 25% stenosis case, minimal velocity fluctuations are present in the experimental work, aside from a gradual increase at $x/D=4$. Increasing the $C_{MIX}$ value leads to higher velocity fluctuation levels, with $C_{MIX}$ values above 0.4 over-predicting velocity fluctuations.

3.7. GEKO Model for Stenotic Flow at $Re=1000$

To assess the GEKO model’s accuracy at lower Reynolds numbers, simulations were performed for flow through a 75% stenosis at $Re=1000$. Only $C_{MIX}$ was varied since previous results showed that it primarily influences stenotic flow among the other parameters. Figure 11a shows that normalised centreline velocity fluctuations exhibit a similar sensitivity to $C_{MIX}$, with higher values producing an earlier, more pronounced increase. Furthermore, lower $C_{MIX}$ values delay jet breakdown, suggesting they may be more suitable for lower Reynolds numbers. The experimental data indicate that velocity fluctuations occur farther downstream at $Re=1000$ compared with $Re=2000$.

Radial profiles of axial velocity at various downstream planes confirm that $C_{MIX}$ affects the flow similarly at $Re=1000$ and $Re=2000$. Higher $C_{MIX}$ values flatten the axial velocity profile sooner, starting at $x/D=2.5$, whereas lower values preserve the jet core for a longer distance. A value of $C_{MIX}=1$ aligns well with experimental data up to $x/D=6$, beyond which the experimental profile flattens while the $C_{MIX}=1$ profile preserves the jet core. These findings indicate that no single $C_{MIX}$ value accurately predicts flow at $Re=1000$ for this geometry.

3.8. Comparison with Other Turbulence Models

A variety of commonly used two-equation RANS turbulence models were simulated on the same mesh under identical flow conditions for a 75% area reduction stenosis at $Re=2000$ and compared against the best-fit GEKO model. These models include the realizable k-$\epsilon$, k-$\omega$ SST, low-Reynolds-number (LRN) k-$\omega$ and LRN k-$\omega$ SST variants. Figure 12 shows the normalised axial velocity and velocity fluctuation results for each RANS model. The realizable k-$\epsilon$ model underestimates axial velocity compared with the experimental data, particularly just downstream of the stenosis throat. In contrast, the k-$\omega$ SST model retains a higher velocity through the throat before over-predicting it from $x/D=3$ onward. The LRN k-$\omega$ and k-$\omega$ SST variants maintain a high velocity downstream and break down much later in comparison with experimental results; this is particularly so for the k-$\omega$ model. The GEKO model with $C_{MIX}=0.4$ agrees generally well with the experimental data downstream of the stenosis throat and converges well beyond $x/D=3$.

Normalised centreline velocity fluctuations in Figure 12 show that the realizable k-$\epsilon$ model predicts fluctuations at the stenosis throat, a feature absent in all k-$\omega$ variants. The GEKO model ($C_{MIX}=0.4$) shows no centreline fluctuations until $x/D\approx 2.5$, where they increase sharply. The k-$\omega$ SST model captures velocity fluctuations slightly further downstream, coinciding with a drop in axial velocity. The LRN k-$\omega$ variants predict fluctuations even further downstream and match experimental data poorly. Overall, none of the RANS models fully align with the experimental observations, highlighting the inherent challenges of RANS models for these flows.

Figure 12 also compares the normalised centreline axial velocity for the GEKO model ($C_{MIX}=0.4$) against LES results from Zhang and Kleinstreuer [44] and Elcner et al. [35]. At the stenosis throat, all models predict higher axial velocities than the experimental data. The GEKO model exhibits a sharper reduction in centreline velocity compared with the LES results, which maintain high velocity until $x/D\approx 2.5$. Both LES simulations accurately capture jet breakdown from $x/D=2.5$ onward, agreeing closely with the experimental data. The GEKO model also follows the experimental trends, although the breakdown occurs slightly further downstream at $x/D=3$. While LES provides more detailed three-dimensional turbulence structures and generally higher accuracy, these results suggest that the GEKO model is a viable alternative.

3.9. Application of the GEKO Model to a Realistic Stenosis Geometry

To assess the consistency of the GEKO results presented within this study, a further CFD simulation was run using a realistic oral cavity of a 30-year-old healthy female subject, with the model shown in Figure 13a. Details of the model acquisition and processing can be found in [45]. An area of minimal cross-sectional area was identified where the oral cavity posterior enters the oropharynx, creating a stenosis-like geometry. The length of the oral cavity was reduced to remain consistent with the shorter inlet length used in the previous simulations. The stenosis has an approximate 50% area reduction in relation to the inlet cross-sectional area. An inlet condition of $Re=2000$ was applied, equivalent to ∼25 L/min, with a final mesh count of 1.8 million polyhedral cells and six prism layers at the wall with a first-layer cell height of 0.04 mm for an overall $y^{+}<1$. The GEKO blending function was forced to a value of 1 at the near-wall prism layers and 0 value through the bulk flow region as applied in the previous stenosis simulations. $C_{MIX}$ values of 0.4 and 1, which agreed best with a 75% and 50% stenosis, respectively, were assessed and a simulation was run with the $k-\omega$ SST model for comparison.

The contours of normalised velocity plotted on the mid-sagittal plane in Figure 13b show the jet length shortening as $C_{MIX}$ increases, consistent with results found in the idealised stenotic tube. The normalised velocity plotted along the centreline of the jet, shown in Figure 13c, further highlights this phenomenon. Jet velocity predicted by the $k-\omega$ SST model remains higher further downstream in comparison with the GEKO model. As $C_{MIX}$ increases, the jet velocity reduces at a faster rate, as seen in the stenotic tube. The normalised velocity fluctuation contours plotted in Figure 13b show that velocity fluctuations build off the wall downstream from the stenosis throat and converge towards the location of jet breakdown. This fluctuation build-up is consistent with previous results, highlighted by the contour plots of velocity fluctuation shown in Figure 9a. Velocity fluctuation build-up increases more rapidly and peaks at a higher magnitude as $C_{MIX}$ increases, shown in the line plot in Figure 13c. Similar to results shown in Figure 12, the $k-\omega$ SST model predicts an increase in velocity fluctuations further downstream in comparison with the GEKO model.

4. Discussion

Ensuring that turbulence models are reliable for predicting a wide range of flow regimes is important in both respiratory and haemodynamic applications. Thus, the GEKO model’s ability to predict velocity fields and transition to turbulence in stenosis geometries is important for researchers simulating similar flows. Here, we investigated a range of values for the four GEKO parameters ($C_{SEP}$, $C_{NW}$, $C_{JET}$ and $C_{MIX}$) in three stenosis geometries and two flow conditions.

The GEKO parameters were tuned to assess their effect on flows at $Re<2000$ for a 75% stenosis geometry. Modification of the GEKO blending function to restrict a value of 1 to the near-wall region significantly improved agreement with the experimental data for a 75% stenosis geometry, without the need for parameter tuning. The $C_{SEP}$ term, which influences wall boundary-layer separation by altering eddy-viscosity levels, had a minimal impact on the flow due to the boundary layer being forced to separate after the stenosis throat. Adjusting $C_{SEP}$ typically affects boundary-layer separation when separation is not forced, such as flow over airfoils and flat plates [36]. The $C_{NW}$ and $C_{JET}$ terms, which affect wall-shear stress and jet spreading rates, respectively, showed a negligible impact on the axial velocity and velocity fluctuations (Figure 7 and Figure 8), suggesting they do not require calibration for these flows. The $C_{MIX}$ parameter, which affects turbulence levels for free shear flows, had the biggest influence on the velocity field and jet breakdown location (Figure 9). Lower $C_{MIX}$ values moved the jet breakdown location further downstream, while higher values reduced it. The optimal $C_{MIX}=0.4$ matched the experimental jet breakdown region and axial velocity for a 75% stenosis.

As realistic stenoses can present with various area reductions, 50 and 25% stenoses at $Re=2000$ (Figure 10) were also simulated. Similar to the 75% stenosis case, only the $C_{MIX}$ parameter affected the centreline axial velocity and velocity fluctuations. Higher $C_{MIX}$ values caused earlier jet breakdown; however, no value of $C_{MIX}$ was able to match the experimental results for axial velocity through a 50% stenosis. Normalised centreline velocity fluctuations (Figure 10) demonstrated that a higher $C_{MIX}$ value of 1 improved velocity fluctuation predictions for a 50% stenosis, but was, again, unable to match the experimental data. It has been stated in previous studies that the experimental work of Ahmed and Giddens [25,26] may have been compromised by unintentional upstream turbulence, which may impact the validity of their results. Moreover, advancements in machining and experimental measurement tools have improved drastically since these studies were published, suggesting that this work should be revisited to obtain more accurate data.

The impact of $C_{MIX}$ was further investigated in a realistic oral-to-trachea airway model for a Reynolds number of 2000, shown in Figure 13a. A section of minimum cross-sectional area was identified, and the GEKO results were shown to be consistent in the realistic airway stenosis, with higher values of $C_{MIX}$ reducing the downstream jet length and increasing velocity fluctuations at the wall. Numerical work involving stenotic flow in the respiratory airway and circulatory system may benefit from the application of the GEKO model, with the use of the modified blending function and tuning of the $C_{MIX}$ value.

Compared with the other two-equation RANS models, the GEKO model shows better predictions of the flow field in the stenosis geometries for a flow condition of $Re=2000$ through a 75% stenosis geometry (Figure 12). The widely used $k-\omega$ SST and realizable $k-\epsilon$ models fail to match experimental centreline velocity and fluctuations, and the low-Reynolds-number variants of $k-\omega$ perform significantly worse [46]. Turbulence fluctuations from all of the RANS models diverge from the experiments, indicating difficulty in predicting turbulence at this Reynolds number, which is consistent with results from Elcner et al. [35]. However, a tuned GEKO model with $C_{MIX}=0.4$ can accurately capture the jet breakdown location, demonstrating its suitability for low-Reynolds-number stenosis flows. Comparison with the published LES data [35,44], shown in Figure 12b, further highlights the capability of the GEKO model. Computationally, a steady-state simulation using both the GEKO and other two-equation RANS models took approximately 3 min to complete, utilising 104 cores. Comparatively, the work reported in Elcner et al. [35] had a runtime of ∼100 h using eight cores for the WALE and WMLES models. Both the GEKO and LES models show a small velocity bump immediately downstream of the stenosis throat at $x/D=0$, which is absent in the experimental data. Similar behaviour was reported by Varghese et al. [28], who used DNS for $Re=1000$ in a 75% stenosis geometry, implying that the measurements of Ahmed and Giddens [25] were not resolved with adequate spatial resolution to capture this flow effect.

Since the flow in a pipe is considered laminar at $Re<2300$, these conditions remain nominally laminar. Yet, even a small prescribed turbulence level at the inlet, necessary for RANS models, can induce a flat velocity profile (Figure 4) when a sufficient entry length is used. With a reduced inlet length, the velocity profile remains laminar, which is realistic at $Re=2000$. Low turbulence at the inlet leads to an over-prediction of the jet breakdown distance, shown in Figure 5, suggesting that a small level of turbulence needs to be introduced to reflect the experimental data. DNS data from Varghese et al. [28] suggest that no turbulence breakdown occurs at $Re=1000$ without inlet turbulence. Yet, experimental data from Ahmed and Giddens [25,26] show that jet breakdown occurs shortly after the stenosis throat, suggesting that turbulence may have been unintentionally introduced in the experiments; or, more likely, as $Re$ approaches the critical threshold of 2300, even nominally laminar flows can exhibit physical turbulence due to experimental sensitivity to flow disturbances.

As this study focuses on the GEKO model’s ability to predict flows through stenoses, consideration must be taken when applying the GEKO model to more complex geometries. The GEKO parameters significantly impact different flow physics, and as such, must be tuned and validated for different flow configurations. For example, the $C_{NW}$ parameter affects heat transfer and wall-shear stress, which were not investigated in this study and may have an impact on the flow field. Future studies will include using machine learning models to optimise the GEKO coefficients based on experimental data.

5. Conclusions

The GEKO model offers significant improvements over commonly used two-equation RANS models in predicting the velocity and velocity fluctuations for flow through a stenosis with 75% area reduction at $Re=2000$. A tuned GEKO model also produced velocity results similar to published LES data under the same flow conditions. The GEKO parameter $C_{MIX}$, which alters the spreading rate of free shear flows, had the largest impact on the flow velocity and velocity fluctuations, which impacted the jet breakdown location. A $C_{MIX}$ value of 0.4 produced results that were in agreement with experimental data, for a 75% stenosis geometry and flow condition of $Re=2000$. The three remaining GEKO parameters, $C_{SEP}$, $C_{NW}$ and $C_{JET}$, had a minimal impact on the flow. Alteration of the GEKO blending function was required via a user-defined function to ensure that the GEKO parameters behaved correctly, leading to a significant improvement in the results when compared with the experimental data. While tuning of the GEKO model improved the velocity and velocity fluctuation predictions for a 50% stenosis at $Re=2000$, the model struggled to match the experimental results. A similar finding was observed in the 75% stenosis geometry at a lower Reynolds number of 1000. Higher $C_{MIX}$ values reduced the jet length and increased the velocity fluctuations in a realistic oral-to-trachea airway, which is consistent with the results produced by GEKO in an idealised stenotic tube. Overall, a tuned GEKO model can provide an improved description of the flow physics through stenotic geometries and should be considered for future work involving stenotic tubes.