Comparative Study of RANS Models for Simulating Turbulent Flow and Heat Transfer in Corrugated Pipes

Abstract

Corrugated pipes are extensively used in engineering applications that require flexibility and enhanced heat exchange, such as drainage and compact heat exchangers, and recently as inner layers in cryogenic flexible hoses for offshore liquid ship-to-ship transfer. The great flexibility of these hoses makes them well-suited for deployment in dynamic and harsh marine environments. However, the corrugated geometry also induces flow separation, elevated turbulence, and intricate heat transfer behaviors. This study focuses on the flow and heat transfer characteristics in corrugated pipes with various geometries, addressing the current lack of systematic comparative studies on the performance of different Reynolds-Averaged Navier–Stokes (RANS) models in such configurations. Despite their limitations in accuracy compared to high-fidelity methods, RANS models remain the workhorse for engineering analysis due to their computational efficiency. This study employs several RANS models to simulate flow and heat transfer in three corrugated pipe geometries—sinusoidal (Sin), C-type, and U-type—over a Reynolds number range of O(10⁴) to O(10⁵) and assesses their performance against high-fidelity Large Eddy Simulation benchmarks. The results show that prediction accuracy decreases with increasing corrugation depth, with the most significant errors in trough regions where reverse flow dominates, and that the choice of turbulence model has a strong influence on the predicted flow and heat transfer behavior. Among all models, the $k-ϵ$ models overall provide the most consistent and accurate predictions for friction factor, velocity distribution, and Nusselt number, while the $k-\omega$ models perform the worst. The Reynolds Stress Model improves friction factor prediction accuracy at high Reynolds numbers and provides marginally better accuracy in mean Nusselt number prediction, but its advantages are limited relative to its substantially higher computational cost. The Standard $k-ϵ$ model with Enhanced Wall Treatment demonstrates robust and balanced performance across geometries and flow regimes, making it a practical choice for engineering use. This work provides engineers and researchers guidance for choosing RANS models that balance accuracy and computational efficiency in simulations of LNG ship-to-ship transfer, compact heat exchangers, and other industrial systems that employ corrugated pipes.

1. Introduction

Corrugated pipes are widely employed in engineering systems such as drainage and sewage infrastructure, compact heat exchangers, and metal corrugated hoses for flexible fluid transport, owing to their high flexibility, enhanced heat exchange ability, and mechanical robustness. An emerging application of corrugated pipes is their use as the inner layer of corrugated cryogenic flexible hoses (see Figure 1), which are employed for ship-to-ship transfer of cryogenic liquids. The corrugated metal pipe serves as the innermost structural layer, providing the hose with both the flexibility required to accommodate dynamic motions and the strength necessary to withstand internal pressure. These hoses have been successfully utilized in LNG offloading operations between Floating Liquefied Natural Gas (FLNG) platforms and LNG carriers [1] (see Figure 2), where their enhanced flexibility offers significant advantages over rigid loading arms, particularly in terms of motion compensation and ease of deployment [2]. However, the same corrugated geometry that grants flexibility also introduces challenges: it induces strong boundary layer separation, complex secondary flows, and enhanced turbulence near the wall, resulting in higher frictional pressure losses. Meanwhile, the enlarged surface area and turbulence level increase convective heat transfer, potentially raising the temperature of LNG and generating boil-off gas (BOG), which reduces transport efficiency and causes economic losses. Therefore, to ensure reliable and efficient operation in offshore cryogenic transport, corrugated cryogenic flexible hoses must be optimized to strike a balance between mechanical flexibility and internal flow performance—minimizing pressure drop and thermal loss without compromising structural adaptability. This calls for accurate prediction and analysis of flow and heat transfer behavior in corrugated pipes.

The inner wall of corrugated cryogenic flexible hoses exhibits a characteristic periodic contraction–expansion structure, with continuously varying wall curvature and inclination angles along the axial direction. As the mainstream flow encounters the sudden expansion downstream of each corrugation crest, boundary layer separation occurs and forms a recirculating region inside the corrugation groove. Flow reattachment typically takes place within or just after the groove, completing a cyclic separation–reattachment process along the hose. These trapped vortices and shear-layer instabilities promote strong three-dimensional secondary flows [3,4,5,6]. According to Morris’ classification, rough-walled turbulence can be divided into isolated roughness flow, wake-interference flow, and skimming flow; corrugated pipes generally operate in the wake-interference regime, where overlapping wakes from successive corrugations produce abnormally intense turbulence near the wall [7]. The associated disturbances inject turbulent kinetic energy into the near-wall region [8,9], thereby disrupting the viscous sublayer that would otherwise be present in smooth straight pipes. Consequently, the transition to turbulence occurs much earlier, with the critical Reynolds number reduced to below 1000 [10], significantly lower than the 2100–2300 threshold for smooth straight pipes. As a result of the intensified turbulence and persistent secondary motions, the overall pressure dropped along the pipe, with the friction factor elevated by approximately 10% to 60% compared to smooth pipes, depending on dimensionless geometric and flow parameters such as corrugation depth, pitch, and velocity [11]. Smaller corrugation pitch and larger corrugation height both intensify flow disturbances and frictional resistance: reduced pitch amplifies shear and vortex generation, while increased height is more prone to triggering separation [9,12,13]. Bruno et al. [14,15] showed through LES of 2D Gaussian roughness and irregular triangular roughness that no single geometric parameter can universally predict the roughness function; instead, improved correlations arise from combining effective slope with additional descriptors such as areal coverage or distribution-based parameters. Their results highlight that turbulence modification by rough walls depends on multiple geometrical features, with parameters like the effective area or effective distribution function offering better predictive capability than height-based measures alone.

For smooth pipes, empirical friction factor correlations such as the Blasius empirical formula (valid for turbulent flow in smooth circular pipes) have been extensively validated and widely applied. In contrast, the Moody chart provides a graphical representation of friction factor variations with Reynolds number and relative roughness, accounting for both smooth and uniformly rough pipes. However, the Moody chart was originally developed for smooth pipes and pipes with uniformly distributed roughness, and it thus fails to accurately characterize the complex periodic geometry and typical flow features inside corrugated pipes, especially in cases involving turbulent reattachment, flow separation, and localized vortex formation. In comparison, the Morris curve was specifically developed for corrugated pipes based on experimental datasets [7], and therefore provides improved agreement with experimental results compared to the Moody chart. However, as a purely empirical model, the Morris method inherently lacks universality and may produce considerable prediction errors when corrugation geometries, such as depth, pitch, or waveform, deviate significantly from the tested configurations. Calomino et al. [8] demonstrated that, particularly at low Reynolds numbers, the use of either the Morris curve or the Moody chart to estimate friction factor in corrugated pipes introduces errors ranging from 10% to 30%. No unified and reliable analytical model exists for accurately predicting the friction factor of corrugated pipes under varying geometric and flow conditions.

In addition to their impact on flow resistance, the unique periodic geometry of corrugated cryogenic flexible hoses also has a significant influence on heat transfer performance. Compared to smooth straight pipes, the corrugated structure not only increases the heat transfer area per unit length but also effectively disrupts the stability of the thermal boundary layer through periodic geometric disturbances, thereby enhancing convective heat transfer and significantly increasing the Nusselt number [4,16,17,18]. Existing studies have further highlighted the pronounced effect of corrugation geometry on heat transfer performance. For example, Poredoš et al. [17] demonstrated that sinusoidal corrugated pipes exhibit a 55% to 85% enhancement in Nusselt numbers compared to smooth pipes within the Reynolds number range of 4000 to 20000. Numerical simulations by Zhong et al. [9] demonstrated that the periodic corrugations in heat exchangers significantly enhance turbulence intensity and thermal boundary layer disruption, with the Nusselt number increasing nearly fivefold as the corrugation height ratio (H/D) increases from 1/8 to 3/8 at Re = 1 × 10³. Bilen et al. [19] experimentally investigated the influence of groove geometry, including ribbed and corrugated shapes, on heat transfer characteristics, demonstrating that groove design plays a decisive role in heat exchanger performance. Accordingly, in corrugated cryogenic flexible hoses, a larger corrugation height ratio—while beneficial for mechanical flexibility—often results in greater heat leakage. As a result, while corrugations in heat exchangers are typically designed to maximize heat transfer efficiency, the design priorities shift in corrugated cryogenic flexible hoses, where it becomes essential to optimize the corrugation geometry to suppress excessive heat transfer and enhance thermal insulation performance [20,21]. This trade-off between mechanical flexibility and thermal efficiency highlights the necessity of geometry-based optimization for corrugated cryogenic flexible hoses.

In engineering applications, the Gnielinski [22] empirical formula is sometimes applied to estimate the Nusselt number for corrugated pipes [23]. However, this formula was originally developed based on simplified assumptions and conventional pipe geometries, and its applicability to complex periodic structures, such as corrugated cryogenic flexible hoses, has not been rigorously validated. In recent years, high-fidelity numerical simulations have provided valuable insight into the complex flow behavior in corrugated pipes. Calomino et al. [8] experimentally studied a sinusoidal corrugated pipe and validated the results against Large Eddy Simulation (LES) predictions within the Reynolds number range of 1.5 × 10⁵ to 3.4 × 10⁵. The maximum prediction error of friction factor is within 3% under wall-adjacent grid resolution y⁺ ≤ 2, confirming the accuracy of the LES approach. Similarly, Hu et al. [24] conducted LES simulations for helical corrugated pipes in heat exchangers, showing that under y⁺ ≈ 0.5 grids, the mean velocity and dimensionless temperature distributions agreed with DNS benchmarks [25] within 5%. Compared with the experimental data by Hu et al. [11], Zhong et al. [10] conducted LES simulations on corrugated hoses with customized geometries at Reynolds numbers ranging from 1 × 10⁴ to 2 × 10⁴, predicting friction factors and Nusselt numbers with maximum deviations of 1.9% and 10.4%, respectively. Jaiman et al. [26] applied both the Delayed Detached-Eddy Simulation (DDES) model and the $k-\omega$ SST RANS model in a sinusoidal corrugated pipe. Results showed that at Re = 1 × 10⁵~1 × 10⁸, DDES predicted friction factor within 5% of experimental data, while $k-\omega$ SST deviated by 12–24%. DDES also accurately captured unsteady features such as recirculation vortices, vortex shedding, and Reynolds-number-dependent separation point migration, demonstrating its advantage in resolving the complex flow mechanisms in corrugated hoses.

Aforementioned studies demonstrate that high-fidelity models such as LES or DDES provide irreplaceable advantages in resolving detailed flow structures, but their high computational cost limits their applicability for engineering design problems. In contrast, RANS models offer much higher computational efficiency but suffer from reduced accuracy in predicting localized turbulence structures, leading to unreliable friction factor and Nusselt number predictions for corrugated pipes. Kadivar et al. [27] compared four RANS models in smooth and rough channels, showing that the Realizable $k-ϵ$ model with Enhanced Wall Treatment yielded the smallest deviations in velocity profile variation, skin friction coefficient, and Nusselt number (6.9%, 2.8%, 9.5%) for rough walls. Keshmiri et al. [28] conducted numerical simulations of flow and heat transfer in rib-roughened ducts at Re = 30,000 using two refined low-Reynolds-number RANS closures: the Lien–Chen–Leschziner $k-ϵ$ model and a Durbin $v^2-f$ variant. The study showed that the $v^2-f$ model yielded more accurate predictions of Nusselt number distributions, while the low-Re $k-ϵ$ closure consistently overestimated heat transfer and underestimated the length of separation zones. Nevertheless, both models displayed notable limitations in predicting pressure losses and secondary flow structures. Elsaadawy et al. [29] investigated two-dimensional ribbed channels with close rib spacing, a configuration associated with the skimming flow regime where vortices remain stably trapped between ribs. They compared four turbulence models (Standard $k-ϵ$, RNG $k-ϵ$, $k-\omega$ SST, and RSM) and found that the SST and RSM closures achieved significantly better agreement with DNS and experimental data than the $k-ϵ$ based models. However, for corrugated pipe, Jayakumar et al. [30] revealed that $k-\omega$ SST model predictions exhibited large errors for U-type corrugated hoses, with the maximum error of friction factor reaching 32.2% at Re = 1.42 × 10⁴ and improved despite exceeding 20% error at Re ≥ 2 × 10⁴. Calomino et al. [31] employed standard $k-\omega$ and $k-\omega$ SST models in combination with the Volume of Fluid method to simulate two-phase free surface flow in an inclined sinusoidal corrugated pipe. Their results indicated that the standard $k-\omega$ model generally predicted slightly lower velocity magnitudes compared to the $k-\omega$ SST model. Nevertheless, both models provided satisfactory agreement with the experimental velocity profiles. Liao & Lian [32,33] employed the Realizable $k-ϵ$ model to investigate the effect of compound corrugation on heat transfer performance (Re = 5 × 10³~3 × 10⁴), comparing tubes with simple concave corrugations and compound concave–convex corrugations. Their results showed maximum deviations of 15% in Nusselt number and friction factor compared to experimental data [33], indicating the model’s limited accuracy. Han et al. [34] highlighted the limitations of the $k-ϵ$ model in capturing secondary flows and turbulence anisotropy in outward convex corrugated pipe heat exchangers. They demonstrated that the Reynolds Stress Model (RSM) reduced errors in separation point prediction and friction factor to within 3%, using approximately 1.6 times more grid points than the Chen–Kim $k-ϵ$ model. Córcoles-Tendero et al. [35] simulated helical corrugated pipes using the Realizable $k-ϵ$ model with Enhanced Wall Treatment tailored to the specific geometry, but the predicted friction factor still deviated by up to 9% from the experimental results of Vicente et al. [36].

Overall, no universally accepted RANS model has yet been established that can reliably replace high-fidelity simulations for the engineering analysis of flow and convective heat transfer behavior in corrugated pipes. The lack of comprehensive and systematic comparisons among RANS models for predicting non-isothermal turbulent flows in corrugated geometries often leaves engineers uncertain when selecting suitable modeling approaches. During the design phase of corrugated cryogenic hoses, rapid yet accurate estimation of turbulence-induced friction factors and heat transfer characteristics is critical; however, full three-dimensional scale-resolving simulations remain computationally prohibitive for routine design purposes. To bridge this gap, this study systematically assesses several widely used RANS turbulence models, including the Standard $k-ϵ$, Realizable $k-ϵ$, RNG $k-ϵ$, Standard $k-\omega$, $k-\omega$ SST, and Reynolds Stress Model (RSM). The primary objective is to evaluate the predictive performance and underlying modeling mechanisms of each RANS turbulence closure under varying Reynolds number conditions, thereby providing practical guidance and valuable technical insights for the efficient design and optimization of corrugated cryogenic flexible hoses.

The remainder of this paper is structured as follows: Section 2 introduces the corrugated pipe geometries investigated in this study, detailing their specific geometric parameters and associated modeling considerations. Section 3 systematically compares the performance of six widely used RANS turbulence models in predicting friction factors, velocity fields, and flow patterns across different geometries and Reynolds numbers, using validated Large Eddy Simulation (LES) results as benchmarks. Subsequently, Section 4 evaluates these RANS models’ ability to accurately predict convective heat transfer characteristics in corrugated pipes. Finally, conclusions drawn from the comparative analysis, along with practical recommendations regarding model selection for engineering applications, are provided in Section 5.

2. Geometry Models of the Corrugated Pipe

Given the potentially different flow and heat transfer behavior associated with varying corrugation geometries, this study evaluates the turbulence modeling performance of several RANS models applied to Sinusoidal (Sin), C-type, and U-type corrugated pipes, as illustrated in Figure 3a. For the Sin-type corrugated pipe, experimental data from Calomino et al. [8] are available; thus, the geometric configuration is kept identical to theirs, featuring an inner radius of R = 85.5 mm, a sinusoidal wall profile with a wave period of $\lambda$ = 25.4 mm, and a wave amplitude of $ϵ$ = 3.0 mm. In contrast, the C-type and U-type corrugated pipes, which are more representative of cryogenic hose designs, adopt a realistic industrial-scale configuration: the C-type pipe has an inner diameter of 4 inches (101.6 mm), a corrugation radius of r = 3.5 mm, and a period of $\lambda$ = 14 mm. The U-type pipe shares the same parameters but includes an additional corrugation height of h = 5 mm. For both C-type and U-type geometries, a wall thickness of t = 0.8 mm is considered, which influences the inner and outer half-circle radii, as illustrated in Figure 3b.

Despite the differences in geometry, the inlet velocity for each case was adjusted based on the pipe diameter to maintain a consistent Reynolds number. This approach ensures that any observed variations in flow and heat transfer behavior can be primarily attributed to the corrugation shape rather than differences in flow regime.

3. Comparative Evaluation of RANS Models for Predicting Flow Characteristics

This section presents the flow simulation results of the selected RANS models for different corrugated geometries under various Reynolds number conditions. The corresponding friction factors are computed and compared against both LES results and empirical correlations, while the internal velocity distributions are evaluated relative to LES data. The objective is to identify the most suitable RANS model for predicting flow behavior in different pipe configurations and to discuss the mechanisms underlying the discrepancies observed among the RANS model predictions.

3.1. RANS Models

To comprehensively assess the predictive capabilities of Reynolds-Averaged Navier–Stokes (RANS) models for the flow behavior in corrugated pipes, six classical turbulence models are selected in this study. In RANS formulations, the mean turbulent flow field of an incompressible Newtonian fluid is governed by the continuity equation (Equation (1)) and the momentum equation (Equation (2)).

$$\nabla \cdot \overline{\mathit{u}}=0$$

(1)

$$\rho \left(\overline{\mathit{u}}\cdot \nabla \right)\overline{\mathit{u}}=-\nabla P+\mu {\nabla }^2\overline{\mathit{u}}-\nabla ·\rho \langle {\mathit{u}}^{\prime }{\mathit{u}}^{\prime }\rangle$$

(2)

where the turbulence effect is considered through the Reynolds stress tensor $\rho \langle {\mathit{u}}^{\prime }{\mathit{u}}^{\prime }\rangle$ in Equation (1). The mean velocity vector is denoted as $\overline{u}$. In Equation (2), $\rho$ is the fluid density, $P$ is the flow pressure, and $\mu$ is the fluid viscosity.

Two-equation models such as $k-ϵ$ and $k-\omega$ families close Equation (2) by invoking the Boussinesq hypothesis, which assumes that the additional turbulent momentum transport can be represented by a scalar (isotropic) turbulent viscosity ${\mu }_T$:

$$\rho \langle {\mathit{u}}^{\prime }{\mathit{u}}^{\prime }\rangle =2{\mu }_T{\overline{S}}_{ij}-{\displaystyle \frac{2}{3}}\rho k{\delta }_{ij}$$

(3)

where ${\overline{S}}_{ij}$ is the mean strain rate tensor (${\overline{S}}_{ij}={\displaystyle \frac{1}{2}}\left({\displaystyle \frac{\partial {\overline{u}}_i}{\partial x_k}}+{\displaystyle \frac{\partial {\overline{u}}_k}{\partial x_i}}\right)$), $k$ is the turbulent kinetic energy, and ${\delta }_{ij}$ is the Kronecker delta. ${\mu }_T$ is obtained from additional transport equations for the turbulence scales$-k$ and $ϵ$ in the Standard, Realizable, and RNG $k-ϵ$ models, and $k$ and $\omega$ in the Standard and $k-\omega$ SST models. Equation (2) is therefore converted to

$$\rho \left(\overline{\mathit{u}}\cdot \nabla \right)\overline{\mathit{u}}=-\nabla \left(P+{\displaystyle \frac{2}{3}}\rho k\right)+\nabla \cdot \left(\mu +{\mu }_T\right)(\nabla \overline{\mathit{u}}+\left(\nabla \overline{\mathit{u}}{)}^T\right)$$

(4)

Unlike the two-equation models that rely on the Boussinesq hypothesis and assume an isotropic turbulent viscosity, the Reynolds Stress Model (RSM) belongs to the second-moment closure class. It solves a transport equation for each independent component of the Reynolds-stress tensor $\rho \langle u^{\prime }{u}^{\prime }\rangle$, plus one for the dissipation rate $ϵ$. This approach captures the anisotropy of turbulence generated by strong curvature, separation, and secondary motion that commonly arise in corrugated pipes at the expense of higher computational cost.

The models assessed in later sections are summarized in

Table 1. Analysis of the advantages and disadvantages of the assessed RANS models.

Models	Advantages	Disadvantages
Standard $k-ϵ$ with Enhanced Wall Treatment (EW)	Robust, cheap, and widely validated; EW allows integration to the wall.	Based on isotropic eddy viscosity assumption; typically weak in modeling strong separation, rotation/curvature and adverse pressure gradients.
Realizable $k-ϵ$ with EW	Better behavior in swirling/strained flows than Standard $k-ϵ$ due to variable $C_{\mu }$ and realizability constraints.	Can over-predict turbulence in free shear layers; isotropy assumption retained.
RNG $k-ϵ$ with EW	Extra $ϵ$-equation term improves response to high strain rates and moderate swirl; often slightly better in adverse pressure gradient modeling than Standard $k-ϵ$.	Benefits are problem-dependent; still isotropic, may over-damp/over-predict eddy viscosity in stagnation or adverse pressure gradient regions; accuracy strongly depends on near-wall resolution.
Standard $k-\omega$	Accurate near walls and suitable for low-Re integration without wall functions; often good for attached boundary layers.	Highly sensitive to freestream  $\omega$ (or turbulent viscosity ratio); can mispredict free shear layers and separation length; still isotropic; requires low y+.
$k-\omega$ SST	Combines the advantages of both $k-ϵ$ and $k-\omega$ models; better suited for separated flow problems than the Standard $k-\omega$.	Can be too dissipative (over-predict eddy viscosity/heat transfer) in stagnation or strong adverse pressure gradient regions; slightly higher CPU cost; still isotropic; requires low y+.
$\mathrm{RSM}$ with EW	Resolves turbulence anisotropy and curvature/rotation effects.	5~10 times (or more) computational cost compared to other RANS models.

, which presents their key advantages and disadvantages.

3.2. Numerical Model Setup

The two-dimensional axisymmetric form of the RANS equations is solved using Ansys Fluent, a widely adopted and well-validated CFD package, which enhances the reproducibility of the study and facilitates comparison with existing research. To reduce computational cost while ensuring physical representativeness, the domain is reduced to a single corrugation period by applying axial periodic boundary conditions, representing a fully developed flow through a periodically corrugated pipe, as illustrated in Figure 4. A no-slip condition is imposed on the pipe wall. The computational domain is defined in the radial–axial ($r-x$) plane, with symmetry imposed along the pipe centerline.

The computational domain is discretized using an unstructured quadrilateral mesh. For each corrugated pipe configuration, a mesh independence study was performed at the highest Re number, with details provided in Appendix A. An inflation layer consisting of 20 prism layers with a growth rate of 1.2 was applied along the corrugated wall (Figure 5), resulting in a total thickness of less than 2 mm. For near-wall modeling, the $k-ϵ$ family models and the RSM model adopt the Enhanced Wall Treatment, whereas the $k-\omega$ family models are directly resolved to the wall. The computational mesh is configured to ensure y⁺ $\approx$ 1 at the highest Reynolds number.

A coupled algorithm is used for pressure–velocity coupling to enhance convergence in strongly coupled flows. Spatial discretization is performed using a second-order upwind scheme for all convective terms, including those in the momentum and energy equations. Viscous terms are discretized using second-order central differences. A pseudo-transient steady-state solver is employed, with the time-scale factor set to 0.1. Convergence is considered achieved when all scaled residuals fall below 10⁻⁸.

Four representative flow cases are simulated, corresponding to Reynolds numbers ranging from 1.63 × 10⁴ to 1.16 × 10⁵, based on the hydraulic diameter and bulk velocity. The working fluid is water at 293 K, with a density of 998.2 kg/m³ and a dynamic viscosity of 1.003 × 10⁻³ Pa·s. The Re number assessed in this study is selected according to the experimental study of Calomino et al. [8], as summarized in

Table 2. Sampled Re numbers and flow rates.

Case	Re	Flow Rates in Sin Pipes [L/s]	Flow Rates in C and U Pipes [L/s]
1	1.63 × 104	2.20	1.31
2	3.18 × 104	4.29	2.55
3	7.55 × 104	10.20	6.05
4	1.16 × 105	15.60	9.32

, to ensure consistency and enable validation of the turbulence models.

3.3. LES Model Validation

To establish a reliable reference for subsequent RANS model evaluation, the benchmark Large Eddy Simulation (LES) model, employing the dynamic Smagorinsky subgrid-scale closure, is first validated. A three-dimensional model of a corrugated pipe consisting of 32 periodic units is used, with axial periodic boundary conditions applied to ensure fully developed turbulence in the streamwise direction.

Experimental data for the friction factor and axial velocity distribution in Sin-type corrugated pipes at representative Reynolds numbers (listed in Table 1) are available from [8] and are used here to validate the LES results. The friction factor is calculated using the following equation:

$$f=H_f{\displaystyle \frac{D}{L}}{\displaystyle \frac{2g}{U^2}}$$

(5)

where $H_f$ is the head loss (in meters) from the inlet to the outlet of the pipe, $D$ is the pipe diameter, $L$ is the pipe length, $g$ is the gravitational acceleration, and $U$ is the average flow velocity in the pipe.

After the flow reaches a statistically steady state, the friction factor and velocity profiles are time-averaged over several seconds until convergence is achieved. Comparisons between the experimental and LES-predicted friction factors and axial velocity profiles at the corrugation crest are presented in Figure 6 and Figure 7, respectively. The LES predictions of the friction factor show good agreement with the experimental results, with a mean relative difference (MRD) below 5% (defined in Equation (6)). This metric, later referred to as the mean relative error (MRE), is used to assess the accuracy of RANS models. It is worth noting that friction factor measurements at low flow rates and in short laboratory-scale pipes are inherently less accurate due to limited pressure drop and instrument resolution, as highlighted in [8], which explains the relatively larger deviations observed at the lower two Reynolds numbers. In contrast, the velocity distributions at Re = 7.55 × 10⁴ and Re = 1.16 × 10⁵ exhibit excellent agreement, with mean differences below 2% (velocity data for the other two Reynolds numbers are not available).

These results confirm the accuracy of the LES model, supporting its use as a reference baseline for evaluating RANS model performance in the subsequent sections.

$$MRD={\displaystyle \frac{1}{n}}{\displaystyle \sum }_{i=1}^n{\displaystyle \frac{\left|f_{LES}-f_{expriment}\right|}{\left|f_{expriment}\right|}}$$

(6)

3.4. Comparison of Friction Fator

This section evaluates the predictive performance of various RANS turbulence models for the friction factor, using LES results as a benchmark and the Morris empirical correlation as a reference. Simulations were conducted over a range of Re numbers (listed in Table 1) and corrugated pipe geometries, with all cases using a mesh validated through convergence analysis. Figure 8 compares the friction factors predicted by six RANS models, LES, and the Morris correlation for three geometries: sinusoidal (Sin-type), C-type, and U-type corrugated pipes. For the Sin-type pipe, both LES and the Morris correlation exhibit a monotonic increase in friction factor with Re number. In contrast, all RANS models fail to capture this trend, instead predicting a decreasing friction factor as Re number increases. This divergence persists with increased corrugation height, progressing from Sin-type to C-type and U-type geometries. The empirical Morris correlation closely matches the LES trends for Sin- and U-type corrugated pipes but shows degraded performance for C-type corrugations. This limitation likely stems from its lack of explicit consideration of corrugation height in the formulation, introducing uncertainty when the corrugation depth varies. Among the RANS models, the Reynolds Stress Model (RSM) consistently predicts the highest friction factors, followed by the $k-ϵ$ family, while the $k-\omega$ models yield the lowest predictions. These differences highlight systematic variations in how turbulence models respond to the complex flow structures induced by wall corrugations.

Figure 9 presents the mean relative error of each model. For the Sin-type geometry (Figure 9a), the Morris correlation achieves the lowest error, while all RANS models show notable deviations. Among the RANS models, the Standard and $k-\omega$ SST models perform reasonably well at the lower two Re numbers, as both are designed for low-Re applications. However, their accuracy deteriorates at higher Re numbers, likely due to assumptions inherent to their low Re number formulation. The SST model exhibits marginally better performance than the Standard $k-\omega$, benefitting from its hybrid approach that combines near-wall accuracy with greater robustness in inertia dominated regions. The three $k-ϵ$ variants, which employ Enhanced Wall Treatment (EW), tend to overpredict friction factors at lower Re numbers due to their reliance on high Re number assumptions. However, as the Re number increases, their predictive accuracy improves significantly. This favorable performance at higher Re number can be attributed not only to the alignment with their underlying high Re formulations, but also to the capability of the EW approach to account for the near-wall viscous sublayer, thereby enhancing their applicability across a broader range of flow regimes. The RSM model shows the largest error at lower Re numbers, possibly due to inadequate near-wall damping. However, as the Re number increases, its capacity to resolve turbulence anisotropy improves prediction accuracy, making it particularly effective under high Re conditions.

For C-type and U-type geometries, the deeper corrugations in the U-type pipe amplify near-wall viscous effects, which partially compensate for the underprediction of the $k-\omega$ models in inertia-dominated regimes, thus enhancing their accuracy. The $k-ϵ$ models demonstrate consistent performance across different geometries, with the Realizable and RNG variants outperforming the Standard model at higher Re numbers. This improvement is likely due to their enhanced capabilities in modeling separated flows and high-strain-rate regions, which are characteristic of corrugated pipe flows. Notably, the accuracy of the RSM model degrades with increasing corrugation height, indicating that while it excels at capturing turbulence anisotropy in high-Re conditions, it remains limited by insufficient near-wall damping in geometrically complex, low-inertia regions.

In summary, the $k-ϵ$ models exhibit the most accurate and consistent prediction of the friction factor across different Re numbers and geometries. The $k-\omega$ models systematically underpredict friction factors, particularly at high Re, and the RSM model—despite its advantages in high Re flows—does not justify its increased computational cost except at the highest Re number examined. From a practical standpoint, the Morris correlation provides sufficiently accurate estimates for engineering applications involving Sin-type and C-type pipes. However, it does not provide detailed flow field characteristics, limiting its applicability in scenarios involving more complex phenomena simulation such as convective heat transfer or multiphase flow.

3.5. Comparison of Velocity Distribution

While the friction factor provides a global measure of overall flow resistance, it does not reveal the underlying flow physics—particularly in pipes with complex internal geometries such as corrugated walls. In applications where local flow behavior plays a critical role, such as solid particle transport or localized vaporization and bubble dynamics, accurate resolution of the local velocity field becomes essential. To better understand the sources of model prediction errors and evaluate the capability of different RANS models in capturing such detailed flow phenomena, a more refined analysis of local flow structures is necessary. This section examines the velocity field, with emphasis on cross-sectional velocity profiles, to assess how well each turbulence model predicts the velocity distribution under single-phase flow conditions.

Two critical cross-sectional locations are selected—one at the crest (C-1) and the other at the trough (C-2) of the corrugation, as illustrated in Figure 10. These locations represent regions of local flow acceleration and deceleration, respectively, where the velocity field is highly sensitive to geometric perturbations. At each location, the axial velocity ($Ux$) is extracted along the radial direction $r$ to assess the model’s capability in capturing local flow behavior in the corrugated pipe. To systematically evaluate the performance and limitations of various turbulence models in capturing complex internal flows—particularly under conditions involving adverse pressure gradients, high shear regions, and flow recirculation—this study compares local axial velocity profiles at C-1 and C-2 across the range of Reynolds numbers listed in Table 2. By performing a cross-comparison of velocity distributions across different geometries and flow conditions, this analysis highlights the strengths and weaknesses of each model in representing key flow features. These results provide a critical foundation for attributing model errors and guiding the selection of appropriate turbulence models for complex internal flow simulations.

In the velocity distribution of the Sin-type corrugated pipe (Figure 11a–d and Figure 12a–d), the LES model predicts significantly steeper velocity gradients compared to all RANS models. In the near-wall region of the trough location (Figure 12a–d), the LES result captures distinct reverse flow structures. In contrast, all RANS models underestimate the magnitude of this reverse velocity to varying degrees, highlighting their limited capability in resolving the strong recirculation induced by the corrugated geometry. From a physical perspective, LES captures large-scale coherent structures and resolves instantaneous shear layers caused by surface curvature, allowing it to sustain stronger reverse flow regions. In contrast, conventional RANS models employ isotropic eddy-viscosity assumptions that inherently dampen flow reversal under adverse pressure gradients, leading to a consistent underestimation of reverse velocities. Across different corrugation geometries, a consistent trend is observed: RANS models generally underestimate velocity gradients relative to LES predictions, as shown in the C-type and U-type (Figure 13a–d, Figure 14a–d, Figure 15a–d and Figure 16a–d) corrugated pipes. At the C-2 location of the U-type pipe (Figure 16a–d), some RANS models begin to overpredict the reverse flow velocity.

To further evaluate the performance of turbulence models across varying Reynolds numbers and corrugation geometries, the axial velocity ($Ux$) prediction errors were analyzed at two representative cross-sections: the crest (C-1) and the trough (C-2). The RANS model error, denoted as ${ϵ}_{RANS}$, is defined as the mean absolute error across 1000 uniformly sampled points, normalized by the bulk velocity, as expressed in Equation (7).

$${ϵ}_{RANS}={\displaystyle \frac{{{\displaystyle \sum }}_i^n\left|U{x}_{LES}^i-U{x}_{RANS}^i\right|/n}{U{x}_{bulk}}}$$

(7)

where $U{x}_{LES}^i$ is the axial velocity predicted by the LES model, $U{x}_{RANS}^i$ is that predicted by the RANS model, $U{x}_{bulk}$ is the bulk flow velocity, and $n$ is the number of sampled points. For the Sin-type corrugated pipe (Figure 11e–h and Figure 12e–h), results show that velocity prediction errors generally increase with Reynolds number across all models. However, the inter-model differences remain small, suggesting that mild geometric perturbations do not significantly impact RANS model performance. In these cases, the prediction errors are maintained below 10%. As the corrugation depth increases—from Sin- to C- to U-type—the magnitude of velocity prediction error rises correspondingly, reaching up to 15% for C-type and 18% for U-type corrugations. The most pronounced errors occur at the C-2 location. This degradation in predictive accuracy is primarily due to intensified recirculation and the development of larger, more energetic vortices, which challenge the assumptions inherent in conventional RANS models. These models often fail to capture the correct structure and strength of recirculating flows, leading to a shift from consistent underprediction to occasional overprediction of reverse velocity, and ultimately to larger errors in deeper corrugation geometries.

Among the RANS models, the $k-\omega$ family models consistently yield the largest velocity prediction errors at both the crest (C-1) and trough (C-2) locations, whereas the $k-ϵ$ family and the RSM models provide more accurate and stable results. This disparity becomes increasingly pronounced as the corrugation geometry transitions from C-type to U-type, reflecting the greater challenge posed by deeper and more complex surface features. Notably, the relative ranking of $Ux$ prediction errors among the RANS models generally aligns with the trend observed in friction factor discrepancies. This correlation underscores the critical role of accurately resolving local flow transport phenomena, particularly wall separation and recirculation, in predicting global flow resistance.

To further evaluate the performance of different RANS models, streamline patterns at the highest Reynolds number (Re = 1.2 × 10⁵) are compared. Figure 17 presents the streamlines from the time-averaged LES results for various corrugation geometries. In all three geometries, the LES predicts a distinct separation vortex, with the vortex elongating as the corrugation depth increases. Flow separation occurs near the crest and reattaches at the subsequent crest. The corresponding RANS model predictions are shown in Figure 18, Figure 19 and Figure 20. For the sinusoidal corrugation, all RANS models capture a single continuous vortex within the groove, though the reattachment locations differ slightly. As shown in Figure 18, the $k-ϵ$ family models predict reattachment points closest to the LES, while the $k-\omega$ family models predict delayed reattachment, and the RSM model shows earlier reattachment. This explains the relatively good performance of the $k-ϵ$ models. In the case of the C-type corrugation (Figure 19), the agreement between the $k-ϵ$ models and the LES results remains consistent. However, for the U-type corrugation (Figure 20), none of the RANS models successfully reproduce the continuous separation vortex observed in the LES. The $k-\omega$ models, in particular, predict a significantly larger secondary vortex compared to the $k-ϵ$ and RSM models, resulting in greater deviation from the LES solution. These discrepancies account for the increasing errors in predicted friction factor and velocity distribution as the corrugation shape transitions from sinusoidal to C-type and then to U-type with increasing corrugation depth.

In general, this section systematically evaluates friction factor and flow velocity predictions from six mainstream RANS models over three corrugation geometries and four Re number regimes, using validated LES as the reference. The key observations are as follows: (1) as corrugation depth increases, all RANS models lose fidelity in both friction factor and velocity profile, with the largest discrepancies concentrated in the trough region; (2) overall, the $k-ϵ$ family models demonstrate superior performance compared to the RSM model, which in turn outperforms the $k-\omega$ models. The latter consistently fail to accurately capture the recirculation zone and the reattachment point of the separated flow; (3) although friction factor predictions may deviate by as much as 60%, the normalized velocity distribution errors of the RANS models remain below 20%, highlighting their practical utility for capturing detailed flow behavior in corrugated pipes; and (4) streamline comparisons reveal that RANS models increasingly deviate from LES predictions as the corrugation depth increases, with $k-ϵ$ models showing the closest agreement and $k-\omega$ models exhibiting the largest discrepancies, particularly in capturing reverse flow structures.

4. Comparative Evaluation of RANS Models for Predicting Heat Transfer Characteristics

The intensity of convective heat transfer in corrugated cryogenic pipes plays a critical role in determining their thermal performance. During the precooling phase, understanding the level of convective heat transfer allows designers to evaluate the risk of thermal stress caused by steep temperature gradients as cryogenic refrigerants flow through the pipe. In the subsequent transportation phase, the convective heat transfer coefficient—commonly expressed by the dimensionless Nusselt (Nu) number—serves as a key parameter for assessing the risk of vaporization, which may arise from excessive heat ingress during cryogenic liquid transport. Therefore, the accuracy of Nusselt number predictions by various RANS models is evaluated using the U-type corrugated pipe in this section, a geometry widely adopted in practical corrugated cryogenic hose designs.

4.1. Model Setup and Boundary Conditions

Unlike the flow simulations presented in the previous section, which utilized a single corrugation period, the computational domain in this section is extended to include 18 consecutive corrugation periods along the axial direction. This extension allows for the observation of the Nusselt number variation as the thermal boundary layer develops, thereby providing insights into the convective heat transfer behavior during the precooling phase. To impose more realistic flow and thermal boundary conditions, the previously applied axial periodic boundaries are replaced with a mass-flow inlet and a pressure outlet. The inlet mass flow rate is specified to match the target Reynolds numbers used in the flow analysis (Section 2). Water is used as the working fluid, with an inlet temperature of 26.85 °C and a constant wall temperature of 56.85 °C.

At the high Reynolds numbers considered, heat transfer is treated solely as forced convection, with buoyancy effects from natural convection neglected due to their relative weakness; under this assumption, the governing RANS equations remain in the 2D axisymmetric form. Therefore, all RANS simulations are conducted using a two-dimensional axisymmetric framework, whereas the Large Eddy Simulation employs a three-dimensional setup consistent with that in Section 2. This modeling approach ensures a robust foundation for evaluating the predictive performance of various RANS turbulence models in capturing the heat transfer characteristics of corrugated cryogenic flexible hoses.

4.2. Comparison of Nusselt Number Distributions

First, the distributions of Nusselt numbers ($Nu$) predicted by various RANS models across a range of Reynolds numbers are compared against the LES simulation results, which serve as the benchmark. In addition, the applicability of simplified engineering methods is evaluated using the Gnielinski [22] empirical correlation for convective heat transfer, expressed as follows:

$$\begin{gathered} Nu={\displaystyle \frac{\left(f/8\right)\left(Re-1000\right)P{r}_f}{1+12.7\sqrt{f/8}\left({P{r}_f}^{2/3}-1\right)}}\left[1+{\left({\displaystyle \frac{d}{l}}\right)}^{2/3}\right]c_t \\ {c}_t={\left({\displaystyle \frac{P{r}_f}{P{r}_w}}\right)}^{0.01},{\displaystyle \frac{P{r}_f}{P{r}_w}}=0.05\sim 20 \end{gathered}$$

(8)

where $f$ is the friction coefficient $\mathrm{Re}$ is the Reynolds number, $P{r}_f$ is the Prandtl number calculated based on bulk fluid temperature ($P{r}_f$= 5.91), and $P{r}_w$ is the Prandtl number of water calculated based on wall temperature ($P{r}_f$= 3.17). To allow a comparison under roughly developed heat transfer conditions, all Reynolds numbers listed in Table 2 are evaluated except for the first case, where the weak flow perturbation induced by the corrugated wall leads to an extended thermal development length, resulting in excessive computational cost. The comparative results for RANS models, LES, and the Gnielinski correlation are presented in Figure 21, Figure 22 and Figure 23, offering a clear visual assessment of the differences in heat transfer prediction among the three approaches.

Along the axial direction, all models display a distinct periodic variation in the local $Nu$ number, characterized by peaks near the corrugation crests and troughs at the valleys. This spatial pattern closely corresponds to the velocity field features discussed in Section 3. Near the crests, the acceleration of the main flow and the presence of strong velocity gradients near the wall intensify local shear and reduce the thermal boundary layer thickness, thereby enhancing convective heat transfer and resulting in elevated $Nu$ values. In contrast, the trough regions are dominated by recirculating vortices that lead to significantly reduced, and in some areas reversed, near-wall velocities. This weakens the local velocity gradients and thickens the thermal boundary layer, thereby suppressing convective heat transfer and resulting in sharp reductions in $Nu$.

Across all cases, the RANS models consistently underestimate the absolute Nusselt numbers, with the degree of underprediction increasing with Reynolds number. Although RANS and LES produce similar mean velocity profiles in the corrugated geometry (as shown in Section 3), they diverge significantly in predicting wall heat transfer. This discrepancy arises because RANS models, particularly two-equation formulations, rely on isotropic eddy-diffusivity assumptions and fixed turbulent Prandtl numbers. These simplifications over-smooth the temperature field, suppress secondary flows, and fail to resolve vortex-induced mixing between the recirculating flow in the corrugation cavities and the bulk core flow. In contrast, LES captures the unsteady vortex dynamics and wall-normal turbulent heat fluxes that promote strong convective transport and steepen the near-wall temperature gradient, resulting in higher $Nu$ values. The inability of RANS to capture these key physical mechanisms leads to systematic underprediction of convective heat transfer in corrugated pipes. For comparison, the Gnielinski correlation provides a single, spatially averaged $Nu$ value and consistently falls between the LES-predicted maximum and minimum $Nu$ values along the axial direction. While useful for smooth, fully developed turbulent pipe flows, this empirical model cannot account for the localized heat transfer variations caused by complex wall geometries, limiting its fidelity in the present application.

Among the RANS models, the $k-ϵ$ family, particularly the RNG $k-ϵ$ model and the Reynolds Stress Model (RSM), demonstrate relatively good agreement with the LES-predicted $Nu$ distribution. In contrast, the $k-\omega$ models, including the SST variant, consistently underpredict Nusselt numbers, with the largest discrepancies occurring in the trough regions. This underperformance is attributed to their tendency to generate overly strong secondary vortex structures near the bottom of the U-shaped corrugations, which lead to exaggerated local flow stagnation. The resulting suppression of wall-normal velocity gradients and weakened near-wall convection reduce the predicted heat transfer, causing systematic underestimation of $Nu$ in these regions.

To quantitatively assess the performance of each turbulence model, the spatially averaged $Nu$ number over all 18 corrugation periods is calculated for each RANS model, and the relative error is computed using the LES results as the benchmark. These comparisons are presented in Figure 24. In the entrance region, where the flow remains laminar, both RANS and LES predict similar $Nu$ values, resulting in minimal error across all models. However, as the flow transitions to a fully developed turbulent regime, discrepancies emerge. The $k-\omega$ family models exhibit the largest errors across all Reynolds numbers, followed by the Realizable $k-ϵ$, Standard $k-ϵ$, and RSM models. The RNG $k-ϵ$ model consistently yields the smallest error, aligning well with LES predictions. The trend in $Nu$ prediction error closely mirrors the velocity field discrepancies shown in Figure 15 and Figure 16, reinforcing the link between accurate flow modeling and reliable convective heat transfer prediction. Overall, RANS model performance degrades with increasing Reynolds number, with relative errors exceeding 35% at Re = 1.2 × 10⁵. As for Gnielinski empirical correlation, the error is large near the pipe inlet and gradually as the turbulence flow becomes more developed, with mean $Nu$ number prediction error outperforming all the RANS models as the Re number increases.

Since an 18-period computational domain does not fully guarantee convergence of the Nusselt number distribution, the single-period corrugation model was again adopted, with both thermal and hydraulic periodic boundary conditions applied. By contrast, the 3D LES simulations retained the 18-period domain length but also applied periodic boundary conditions, effectively reproducing a fully developed thermal boundary layer and yielding a Nusselt number distribution representative of a very long corrugated tube. The Nusselt number distributions obtained at four Reynolds numbers (Re = 1.6 × 10⁴, 3.2 × 10⁴, 7.6 × 10⁴, and 1.2 × 10⁵) are shown in Figure 25, Figure 26, Figure 27 and Figure 28. In general, the RANS models increasingly underpredict the Nusselt number with rising Reynolds number, a trend consistent with the earlier observations in the pipe entrance region. To quantitatively assess overall predictive performance, Figure 29 reports the mean relative error (MRE) of the period-averaged Nusselt number from each RANS model against the LES benchmark. The results indicate that errors in the $k-ϵ$ family models increase with Reynolds number, with the RNG $k-ϵ$ performing better than the Standard $k-ϵ$, while the Realizable $k-ϵ$ shows the poorest accuracy. The RSM also shows mild error growth with the Re number but consistently achieves the lowest MRE across all conditions, slightly outperforming the other models. In contrast, the $k-\omega$ models yield the largest overall errors, although their weak sensitivity to Re indicates a persistent inability to represent vortex-induced mixing and wall-normal heat exchange. Considering the high computational demand of the RSM, the RNG $k-ϵ$ model—showing the second-best accuracy—emerges as a practical compromise for obtaining stable Nusselt number predictions in corrugated pipes. Meanwhile, the Gnielinski correlation shows improved accuracy in predicting the mean Nusselt number at higher Reynolds numbers. This can be attributed to the fact that the boundary layer profile of wake-interference flows (such as those over corrugated walls) progressively resembles that of fully developed turbulence in smooth pipes as Reynolds number increases [7], thereby reducing the deviation from the underlying assumptions of the Gnielinski correlation. However, when the local Nusselt number distribution is of interest, empirical correlations like Gnielinski remain insufficient, and predictive accuracy requires the use of numerical approaches such as RANS or LES.

5. Conclusions

This study presents a comprehensive comparative analysis of six widely used RANS turbulence models—Standard $k-ϵ$, Realizable $k-ϵ$, RNG $k-ϵ$, Standard $k-\omega$, $k-\omega$ SST, and Reynolds Stress Model (RSM)—for predicting turbulent flow and convective heat transfer in corrugated pipes with varying geometries, including sinusoidal, C-type, and U-type profiles, and Reynolds numbers ranging from O(10⁴) to O(10⁵). High-fidelity LES simulations and available experimental data are employed as benchmarks to assess the accuracy of each model in terms of friction factor, velocity distribution, and Nusselt number predictions.

The results reveal that all RANS models exhibit reduced prediction accuracy as corrugation depth increases, with the most significant discrepancies occurring in the trough regions where reverse flow and recirculation dominate. Among the models evaluated, the $k-ϵ$ family models implemented with Enhanced Wall Treatment demonstrate the most consistent and accurate predictions of both global friction factor and local velocity fields, particularly at higher Reynolds numbers. However, the relative performance of the Standard $k-ϵ$, Realizable $k-ϵ$, and RNG $k-ϵ$ variants is not consistent across different corrugation configurations and flow regimes, reflecting their distinct empirical adjustments and sensitivity to flow features. The RSM model shows improved accuracy in high-Re regimes due to its ability to resolve turbulence anisotropy but does not consistently outperform simpler two-equation models, and its performance gains do not justify the substantially higher computational cost compared to these models. The $k-\omega$ models, while suitable for near-wall flows, consistently underpredict friction factor and velocity gradients in inertia-dominated regions and are the least accurate across all flow metrics.

In terms of convective heat transfer, RANS models systematically underestimate Nusselt numbers compared to LES, especially in the high-Re turbulent regime. This underestimation stems from their inability to resolve unsteady vortex dynamics and wall-normal turbulent heat fluxes, which are critical for accurately capturing thermal transport in corrugated geometries. Among the models assessed, the $k-ϵ$ family and RSM models exhibit relatively better agreement with LES results, with RNG $k-ϵ$ and RSM yielding the most accurate predictions, while the $k-\omega$ models show the poorest performance.

In parallel, two empirical correlations were assessed: the Morris model for friction factor and the Gnielinski correlation for Nusselt number. The Morris model provides acceptable accuracy for Sin-type and C-type corrugated pipes but fails to account for geometric variability and flow separation in U-type corrugated pipes with deeper corrugation, resulting in increased deviation for U-type geometries. The Gnielinski correlation offers a single averaged value and thus cannot capture spatially varying heat transfer rates, yet it remains useful for preliminary engineering estimates under fully developed turbulent conditions.

Overall, while the $k-ϵ$ family models with Enhanced Wall Treatment (EW) consistently outperform other RANS models for flow and heat transfer simulations, the relative performance among the Standard, Realizable, and RNG variants is not universal. Although the RSM model captures heat transfer with relatively high accuracy, it performs less reliably for flow predictions, and its advantages are offset by substantially higher computational cost and lower efficiency. The Realizable and RNG models incorporate empirical corrections and tuning strategies aimed at specific flow behaviors, which can introduce sensitivity and occasional overprediction in certain regimes. In contrast, the Standard $k-ϵ$ model demonstrates robust and balanced accuracy across a wide range of Reynolds numbers and corrugation geometries. Given its computational efficiency and stability, the Standard $k-ϵ$ with EWT offers a practical and reliable modeling strategy for general engineering use. These findings provide valuable guidance for selecting appropriate RANS turbulence models for modeling flow and heat transfer behavior in corrugated pipes.