Research on Peak Characteristics of Turbulent Flow in Horizontal Annuli with Varying Curvatures Based on Numerical Simulation

Abstract

Annular flow is a common flow configuration encountered in fields such as food engineering, energy and power engineering, and petroleum engineering. The annular space formed by the inner and outer pipes exhibits unique characteristics, with the distinct curvatures of the inner and outer pipes rendering the annulus fundamentally different from a circular pipe. The complexity of the annular structure complicates the rapid calculation of turbulent statistics in engineering practice, as modeling these statistics necessitates a comprehensive understanding of their peak characteristics. However, current research lacks a thorough understanding of the peak characteristics of turbulent flows in annuli with varying diameter ratios (the ratio of the inner tube’s diameter to the outer tube’s diameter) between the inner and outer pipes. To gain a deeper insight into the turbulent peak characteristics within annular flows, this study employs numerical simulation methods to investigate the first- and second-order turbulent statistics under different diameter ratios resulting from varying curvatures of the inner and outer pipes. These statistics encompass velocity distribution, the position and magnitude of maximum velocity, turbulence intensity, turbulent kinetic energy, and Reynolds stress. The research findings indicate that the contour plots of velocity, turbulence intensity, and turbulent kinetic energy distributions under different diameter ratio conditions exhibit central symmetry. The peaks of the first-order statistical quantities are located in the mainstream region of the annulus, and their positions gradually shift closer to the center of the annulus as the diameter ratio increases. For the second-order statistical quantities, peaks are observed near both the inner and outer walls, and their positions move closer to the walls as the diameter ratio rises. The peak values of turbulent characteristics show significant variations across different diameter ratios. Both the inner and outer wall surfaces exhibit peaks in their second-order statistical quantities. For instance, the maximum value of Reynolds stress near the inner tube is 101.4% of that near the outer tube, and the distance from the wall where the maximum Reynolds stress occurs near the inner tube is 97.2% of the corresponding distance near the outer tube. This study is of great significance for optimizing the diameter combination of the inner and outer pipes in annular configurations and for evaluating turbulent statistics.

1. Introduction

Annular flow, as a typical type of non-circular pipe flow, is widely encountered in key fields such as mixing and transportation in food engineering, nuclear reactor cooling systems in energy and power engineering, and drilling and oil production operations in petroleum engineering. Its flow characteristics are determined by the annular space geometry formed by the inner and outer pipes. The diameter ratio (curvature difference) between the inner and outer pipes, as a core parameter, directly affects the complexity of the flow field and the evolution law of turbulence. Compared with traditional circular pipe flow, annular flow exhibits significant differences in boundary layer development and turbulence characteristics due to the asymmetry of the curvature of the inner and outer walls. Therefore, annular flow holds significant value in engineering applications, particularly regarding the influence of curvature effects on turbulent statistics under different diameter ratios.

Cadot et al. and Min et al. pointed out in their studies that since the drag-reducing properties of polymers are highly dependent on wall effects, it is necessary to accurately understand how polymer additives vary with the transverse curvature of the annular space in order to characterize drag-reducing flow in the annular space [1,2]. Quadrio and Luchini [3] conducted a Direct Numerical Simulation (DNS) study on turbulent concentric annular pipe flow, with the radius ratio maintained between 0.33 and 0.5. They examined first-and second-order flow statistics and investigated the effect of transverse curvature on the peak Reynolds shear stress. Chung et al. [4,5] conducted a DNS of turbulent concentric annular pipe flow at a Reynolds number of 8900 with different diameter ratios. Their research mainly focused on the transverse curvature effects on near-wall turbulent structures. By calculating low- and high-order statistics, they carefully examined the near-wall turbulent structures near the inner and outer walls. The numerical results showed that the turbulent structures near the outer wall were more active than those near the inner wall, possibly due to different vortex regeneration processes between the inner and outer walls. Meanwhile, they extended their research to include turbulent heat transfer under a constant ratio of wall heat flux densities. Boersma & Breugem and Ghaemi et al. [6,7] studied Reynolds shear stress and maximum axial velocity in the high-Reynolds-number range at a small radius ratio (0.1). Xiong et al. [8,9] simulated fully developed turbulent flow in concentric annuli with two radius ratios (0.4 and 0.5) at three Reynolds numbers (8900, 26,600, and 38,700) using Computational Fluid Dynamics (CFD). They characterized the near-wall turbulent structures near the inner and outer walls by analyzing first-and second-order statistics. The simulation results indicated that the RANS model could capture the curvature effects induced by the annular geometry. Due to this effect, the mean axial velocity distribution of fully developed turbulence in the concentric annulus was asymmetric and inclined towards the inner wall. The shape of the velocity profile was further influenced by the Reynolds number and the ratio of the inner and outer radii. Changes in the velocity profile in the near-wall region suggested the attenuation of the viscous sublayer caused by an increase in the Reynolds number. This phenomenon was also observed when studying the near-wall structure of Reynolds stress. In addition, it was estimated that the Reynolds stress at the outer wall was higher than that at the inner wall because there was a larger surface area to support higher turbulent energy. In this study, numerical predictions showed a slight offset between the zero-Reynolds-shear-stress position and the maximum-axial-velocity position. Moreover, it was expected that these positions were less dependent on the Reynolds number but more sensitive to the radius ratio. In Bagheri’s study [10], a pseudo-spectral method was employed to perform DNS of turbulent flow in annular spaces. The objective was to investigate how computational domain dimensions affect turbulence statistics. The study concluded that both the scale and dynamic characteristics of turbulent structures vary with changes in concave–convex wall curvature. In a subsequent investigation [11], Bagheri again utilized DNS to analyze the impact of the radius ratio on fully developed flow characteristics and structural features in concentric annular pipes. The findings revealed that variations in the radius ratio modify the interaction between boundary layers formed on the inner and outer cylinder surfaces, thereby significantly influencing turbulent flow structures and dynamic behaviors.

In terms of the choice between experimental and numerical simulation methods, experiments are less precise than numerical simulations in terms of the resolution of peak characteristics. Among numerical simulations, DNS is more computationally demanding, while the Reynolds-Averaged Navier–Stokes (RANS) equations are more computationally efficient, which has also been demonstrated in some studies on annular turbulent flow characteristics.

Rahman et al. [12] analyzed the near-wall turbulent characteristics of drag-reducing polymer fluids flowing through a concentric annulus using CFD simulation. The simulation results showed that if the flow rate of the power–law fluid increased from 3.92 to 5.95 kg/s, the drag reduction in the annulus increased from 10% to 20% compared to the water case, indicating strong damping of turbulent dynamics and energy in the flow. CFD analysis using the Shear Stress Transport (SST) model had low computational costs and could thus be conveniently used to study the flow characteristics of drag-reducing polymer fluids in a concentric annulus. Rahman et al. [13] analyzed the near-wall turbulent characteristics of water flowing through a concentric annulus using CFD simulation. The annular velocity profiles obtained from the simulation study were in good agreement with the experimental data. This indicated that CFD analysis using the SST model, with its low computational costs, could be conveniently used to study the near-wall turbulent characteristics of flow in a concentric annulus. The SST model could be conveniently used to simulate fully developed turbulence in a concentric annulus without significant computational costs or instability. The presence of turbulent kinetic energy and vorticity very close to the wall suggested enhanced turbulent activity in the viscous sublayer and logarithmic law region of the wall. Xiong et al. [9] conducted a simulation using the RANS -SST model to study fully developed turbulent drag-reducing flow in a concentric annulus using CFD. The simulation results showed that a RANS model with only a power–law model could not capture the experimental data of turbulence with non-Newtonian fluids because the stress deficiencies present in this type of flow could not be captured using the RANS method. However, by adjusting the turbulence model constants, accurate simulation results for pressure drop, axial velocity, and Reynolds stress in fully developed turbulent non-Newtonian flow could be successfully obtained. This study provided a simple method for predicting the turbulent statistics of non-Newtonian shear-thinning flows in a concentric annulus. Xiong et al. [14] used the commercial code ANSYS–FLUENT to perform Reynolds-Averaged Modeling of polymer-induced drag-reducing fluids in a fully developed turbulent state in a concentric annulus. According to the simulation results, the two different turbulent boundary layers at the inner and outer walls tended to redistribute along the cross-section of the annulus with the presence of polymers. By studying the mean near-wall velocity, Reynolds stress, and turbulent kinetic energy of the flow, it was demonstrated that this change was mainly caused by the elongation properties of polymers. Near the outer wall, due to stronger turbulence, the polymers were more strongly stretched, attenuating more turbulence; while near the inner wall, due to weaker turbulence, the polymers were less strongly stretched, resulting in less turbulence attenuation. In addition, it was found that the redistribution of the turbulent field depended on the rheological properties. This study provided a benchmark work demonstrating how to use available RANS models for the numerical prediction of the evolution of the turbulent field between different drag-reducing fluids in an annulus. Sun et al. [15] used numerical simulation methods to study the microscopic characteristics of the flow field, considering the roughness of the inner and outer pipes and positive and negative eccentricities. The numerical model, which utilized the RANS approach, was validated by comparing its predictions with the pressure drop data measured in laboratory experiments. Furthermore, the microscopic variation laws of the flow field were explored through numerical simulation methods, and analyses were made from the perspectives of wall shear stress and turbulent characteristics. The results showed that an increase in the roughness of the inner pipe significantly increased the wall shear stress on the inner pipe and also increased the wall shear stress on the outer pipe, and vice versa. In the concentric case, the wall shear stress and turbulent characteristics exhibited central symmetry; eccentricity led to non-uniform distributions of flow velocity, turbulence intensity, turbulent kinetic energy, and shear stress; and this non-uniformity showed axial symmetry under positive and negative eccentricities.

In summary, the peak characteristics of annular turbulence near the wall have not been reported, especially in the main areas of turbulent activity near the inner and outer pipe walls. This study, based on the RANS-based numerical simulation method, analyzes the peak characteristics of first- and second-order turbulent statistics to provide support for research such as the construction of theoretical models.

2. Numerical Simulation Methods

When employing the RANS model to simulate complex turbulent flows, it is quite appealing as it can balance computational efficiency while ensuring a certain level of accuracy [16]. In this study, the RANS model is utilized to conduct a systematic investigation into turbulent flows within a concentric annulus. CFD simulations are employed to analyze the fully developed turbulent flow in a horizontal annulus.

2.1. Numerical Model

2.1.1. Governing Equations

RANS Equations: In the RANS formulation, turbulent variables are decomposed into a time-averaged component and a fluctuating component that varies with time.

$${\stackrel{\wedge }{\varphi }}_i={\varphi }_i+{\phi }_i\mathrm{with}{\varphi }_i={\displaystyle \frac{1}{\Delta t}}{\displaystyle {\int }_t^{t+\Delta t}{\varphi }_{i}dt}$$

(1)

By performing velocity decomposition and averaging operations on the original Navier–Stokes equations, the continuity equation and the momentum conservation equation can be transformed into the following expressions:

$$\nabla \cdot U=0$$

(2)

$${\displaystyle \frac{\partial \left(\rho U_i\right)}{\partial t}}+{\displaystyle \frac{\partial }{\partial x_j}}\left(\rho U_i{U}_j\right)=-{\displaystyle \frac{\partial P}{\partial x_i}}+{\displaystyle \frac{\partial }{\partial x_j}}\left[\mu \left({\displaystyle \frac{\partial U_i}{\partial x_j}}+{\displaystyle \frac{\partial U_j}{\partial x_i}}\right)-\rho \overline{u_i{u}_j}\right]$$

(3)

The term $\rho \overline{u_i{u}_j}$ in Equation (3) represents the Reynolds stress term, which accounts for turbulent effects. It is evident that within the RANS framework, only the averaged velocity field is solved explicitly. The influence of fluctuating turbulent velocities on the overall flow field is represented through Reynolds stresses, which fundamentally correspond to velocity covariances obtained from averaging the Navier–Stokes equations. To accurately capture turbulent structures using RANS models, an appropriate turbulence model must be selected. In CFD, the most widely used turbulence models are the Boussinesq hypothesis-based eddy viscosity models [17]. These models relate Reynolds stresses to mean velocity gradients through the gradient diffusion hypothesis and eddy viscosity concepts.

2.1.2. Turbulence Models

The SST model plays a significant role in CFD simulations. The selection of turbulence models has a substantial impact on the turbulent flow structures predicted by CFD codes. In this study, the standard SST model based on the Boussinesq hypothesis was adopted to estimate Reynolds stresses [18]. This model was chosen because it combines the advantages of both the k-ε and k-ω models. The SST model employs a specially designed blending function to achieve a smooth transition from the low Reynolds number (LRN) formulation in near-wall boundary layers (found in the standard k-ω model) to the k-ε formulation in free shear layers. The development of this model was based on the following considerations: in the k-ω model, the near-wall treatment for LRN requires lower resolution requirements, eliminates the need for specific non-linear damping functions used in k-ε models, and avoids the sensitivity of k-ε models to free-stream boundary conditions [19].

The SST model introduces two independent transport equations [20], one of which is specifically dedicated to describing the turbulent kinetic energy k (which reflects the magnitude of velocity fluctuation variance), while the other characterizes the turbulence frequency ω. The intrinsic relationship between k, ε, and ω can be expressed through the following equation:

$$\epsilon =C_{\mu }k\omega$$

(4)

Here, ε denotes the dissipation rate of velocity fluctuations, while C_μ represents the constant specific to the model.

$$\begin{array}{cc}k-equation& \\ & {\displaystyle \frac{\partial }{\partial x_j}}\left(\rho U_{j}k\right)={\displaystyle \frac{\partial }{\partial x_j}}\left[\left(\mu +{\displaystyle \frac{{\mu }_t}{{\sigma }_k}}\right){\displaystyle \frac{\partial k}{\partial x_j}}\right]+\widetilde{P_k}-{\beta }^{\text{'}}\rho k\omega \end{array}$$

(5)

where

$$\tilde{P}=\min \left(P_k,C_{\lim }\rho \epsilon \right)$$

(6)

$$\begin{array}{l}\omega -equation\\ {\displaystyle \frac{\partial }{\partial x_j}}\left(\rho U_j\omega \right)=\alpha \rho {\left({\displaystyle \frac{\partial U_i}{\partial x_j}}+{\displaystyle \frac{\partial U_j}{\partial x_i}}\right)}^2+{\displaystyle \frac{\partial }{\partial x_j}}\left[\left(\mu +{\displaystyle \frac{{\mu }_t}{{\sigma }_{\omega }}}\right){\displaystyle \frac{\partial \omega }{\partial x_j}}\right]-\rho \beta {\omega }^2+{\displaystyle \frac{2\rho \left(1-F_1\right)}{\omega {\sigma }_{\omega }^2}}{\displaystyle \frac{\partial k}{\partial x_j}}{\displaystyle \frac{\partial \omega }{\partial x_j}}\end{array}$$

(7)

$$\alpha ={\alpha }_1{F}_1+{\alpha }_2\left(1-F_1\right)$$

(8)

$$F_1=\tanh \left({\arg }_1^4\right)$$

(9)

$${\arg }_1=\min \left[\max \left({\displaystyle \frac{\sqrt{k}}{{\beta }^{\prime }\omega y}},{\displaystyle \frac{500\upsilon }{y^2\omega }}\right),{\displaystyle \frac{4\rho k}{{\sigma }_{\omega 2}C{D}_{k\omega }{y}^2}}\right]$$

(10)

$$C{D}_{k\omega }=\max \left({\displaystyle \frac{2\rho }{{\sigma }_{\omega 2}\omega }}{\displaystyle \frac{\partial k}{\partial x_j}}{\displaystyle \frac{\partial \omega }{\partial x_j}},{10}^{-10}\right)$$

(11)

$$F_2=\tanh \left({\arg }_2^2\right)$$

(12)

$${\arg }_2=\max \left({\displaystyle \frac{2\sqrt{k}}{{\beta }^{\prime }\omega y}},{\displaystyle \frac{500\upsilon }{y^2\omega }}\right)$$

(13)

$$S=\sqrt{2S_{ij}{S}_{ij}}$$

(14)

Among them, S_ij can be expressed as follows:

$$S_{ij}={\displaystyle \frac{1}{2}}\left({\displaystyle \frac{\partial U_i}{\partial x_j}}+{\displaystyle \frac{\partial U_j}{\partial x_i}}\right)$$

(15)

The formulation for eddy viscosity can be presented as follows:

$${\upsilon }_t={\displaystyle \frac{{\alpha }_1\kappa }{\max \left({\alpha }_1\omega ,S{F}_2\right)}}$$

(16)

The mathematical formulation representing Reynolds stress is as follows:

$$-\rho \stackrel{\_\_\_\_\_}{u_i{u}_j}=2{\mu }_t{S}_{ij}$$

(17)

The correlation coefficients pertinent to the aforementioned system of equations are elaborated upon in [8].

2.2. Model Validation

Prior to conducting research using a numerical model, it is imperative to validate the numerical model first, and then proceed with subsequent studies. The validation of the numerical model is performed by simulating under the same conditions as the experiment [15], as illustrated in Figure 1a. The outer tube of the annulus experimental setup has a diameter of 110 mm, while the inner tube has a diameter of 63 mm and a length of 8 m. The inner tube is centrally positioned, with a rotational speed of zero, and the flow velocity is adjustable within the range of 8–13 m/s, and the validation is carried out through the pressure drop gradient.

The computational domain used in this study comprises an annular space with an outer pipe diameter of 110 mm, an inner pipe diameter of 63 mm, and a total length of 6 m. For numerical modeling, the SIMPLE algorithm was employed to decouple the pressure–velocity coupling relationship during the solution process. Regarding numerical discretization, second-order upwind schemes were applied for solving the momentum equations, turbulence kinetic energy equations, and specific dissipation rate equations. It is particularly important to note that a unified residual convergence criterion of 10⁻⁵ was consistently applied throughout all calculations. The boundary conditions consisted of velocity inlet and pressure outlet specifications. The domain was discretized using structured hexahedral meshing with a grid resolution corresponding to a y⁺ value of 0.9. As demonstrated in Figure 2a, showing mesh independence verification, the pressure drop gradient stabilized with negligible fluctuations when the mesh point count reached 2048, 160. Based on this validation, subsequent simulations were conducted using this mesh configuration (2048, 160 elements) to balance computational accuracy and efficiency.

As shown in Figure 2b, the current simulation results are in high agreement with the experimental results, which validates the reliability of the RANS model in predicting the near-wall turbulent characteristics of wall-bounded flows.

On the basis of verifying the reliability of the numerical model, the inlet velocity for the annular numerical simulation was set to 8 m/s, with air under standard conditions used as the working fluid. Without altering the outer tube, the diameter ratio was varied by replacing the inner tube, and the range of the diameter ratio was set from 0.4 to 0.71.

3. Results and Discussion

Following the validation of the numerical model mentioned above, the numerical simulation method is adopted to study the impact of the diameter ratio on turbulent flow characteristics under various curvatures. These characteristics include velocity distribution, maximum velocity characteristics, turbulence intensity, turbulent kinetic energy, and Reynolds stress. The numerical simulation results are thoroughly analyzed through a combination of qualitative and quantitative approaches.

3.1. Velocity Distribution

As can be observed from Figure 3, the contour plots of velocity distribution under different diameter ratio conditions exhibit central symmetry. As the diameter ratio decreases, the velocity distribution within the annulus becomes more non-uniform. In the scenario where the ratio of the inner pipe diameter to the outer pipe diameter is at its lowest, there is a significant increase in the low-velocity regions near both the inner and outer walls.

From Figure 4, it is evident that as the diameter ratio increases, the velocity distribution curves gradually contract. The slopes near the walls become steeper, indicating a larger velocity gradient. Moreover, there is a slight increase in the peak velocity. Further analysis of the maximum velocity characteristics is necessary to delve into the underlying microscopic changes.

3.2. Maximum Velocity Characteristics

An analysis is carried out on the characteristics of the maximum velocity within the velocity distribution of the annulus. As depicted in Figure 5 and Figure 6, with the increase in the diameter ratio, the position of the maximum velocity in the annulus gradually shifts closer to the outer pipe. In other words, relative to the center of the annulus, the location of the maximum velocity is nearer to the outer pipe. This indicates that the velocity distribution within the annulus is not symmetrical, which is one of the main differences between the velocity distributions in annulus and circular pipes.

From a fluid mechanics perspective, the following explains this phenomenon: In annulus flow, the fluid is simultaneously constrained by both the inner and outer pipes. When the diameter ratio is small (i.e., the inner pipe is smaller compared to the outer pipe), the inner pipe exerts a significant obstructive effect on the fluid. This leads to substantial suppression of the fluid velocity near the inner pipe region, while the fluid near the outer pipe region can flow more easily. As a result, the position of the maximum velocity is biased towards the outer pipe side. As the diameter ratio increases, the relative size of the inner pipe with respect to the outer pipe gradually increases, and its obstructive effect on the fluid weakens. The flow space for the fluid within the annulus relatively expands, and the flow becomes more uniform. Therefore, the position of the maximum velocity gradually moves towards the center of the annulus, though it remains closer to the outer pipe (due to the continuous constraint of the outer pipe on the fluid). Research by Chung et al. [4] points out that a notable feature of concentric annular pipe flow is that the smaller the radius ratio, the more asymmetric the flow. Since the radius ratio is essentially related to the diameter ratio, this research conclusion is consistent with the phenomena and viewpoints observed in this paper. From a quantitative analysis standpoint, compared to the minimum diameter ratio of 0.4, the positions of the maximum velocity for diameter ratios of 0.48, 0.58, 0.69, and 0.71 increase by 6%, 13%, 21%, and 22%, respectively. This data change trend further supports the aforementioned mechanistic analysis, that is, as the diameter ratio increases, the position of the maximum velocity nears the center of the annulus (in a relative sense; in fact, it is still nearer to the outer pipe). It is not difficult to see that a circular pipe can be regarded as a special case of an annulus, where the diameter ratio equals 1. In this case, with the absence of the inner pipe, the fluid flow is completely symmetrical, and the position of the maximum velocity is exactly at the center of the circular pipe.

As shown in Figure 6, the maximum velocity value generally increases with the rise in the diameter ratio. However, a critical value appears around the medium diameter ratio of 0.48, which means that the maximum velocity value does not always keep increasing. Compared to the case when the diameter ratio is 0.4, the maximum velocity values for diameter ratios of 0.48, 0.58, 0.69, and 0.71 increase by −0.12%, 0%, 0.23%, and 0.23%, respectively. The changes in the maximum velocity value are very small, all below 1%.

3.3. Reynolds Stress Distribution

As depicted in Figure 7, Figure 8 and Figure 9, critical values of Reynolds stress are observed near both the inner and outer walls. With the increase in the distance from the walls, the Reynolds stress first increases and then decreases as it moves from the walls towards the middle of the annulus. On the whole, the maximum Reynolds stress occurs near the walls. The distribution of Reynolds stress from the inner and outer walls to the middle of the annulus exhibits 180-degree symmetry. This is mainly due to the opposite directions of the velocity gradients in the calculations. That is, when viewed from the inner pipe towards the outer pipe, the velocity gradually increases from the inner pipe to the middle of the annulus, resulting in a positive velocity gradient. Conversely, the velocity gradually decreases from the middle of the annulus to the outer pipe wall, leading to a negative velocity gradient. Consequently, the Reynolds stresses from the inner and outer pipes to the middle of the annulus are in opposite directions.

From a quantitative analysis perspective, the maximum value near the inner pipe is 101.4% of that near the outer pipe. The distance from the position of the maximum value near the inner pipe to the wall is 97.2% of the corresponding distance near the outer pipe. This indicates that the maximum value near the inner pipe is higher, and that the position of the maximum Reynolds stress near the inner pipe is closer to the wall.

3.4. Turbulence Intensity Distribution

As shown in Figure 10, Figure 11, Figure 12 and Figure 13, the turbulence intensity exhibits central symmetry under different diameter ratio conditions, and critical values of turbulence intensity are observed near both the inner and outer walls. With the increase in the distance from the walls, the turbulence intensity first increases and then decreases as it moves from the walls towards the middle of the annulus. On the whole, the maximum turbulence intensity occurs near the walls, while the minimum turbulence intensity appears in the middle of the annulus. The underlying mechanism for this phenomenon can be explained as follows: Near the walls, the fluid experiences significant shear forces due to the no-slip condition (the fluid velocity at the wall is zero). This creates a large velocity gradient in the immediate vicinity of the walls. The large velocity gradient leads to intense mixing and eddy formation, as the fluid layers with different velocities interact vigorously. These eddies and turbulent fluctuations contribute to the high turbulence intensity near the walls. As we move away from the walls towards the middle of the annulus, the velocity gradient gradually decreases. The fluid flow becomes more uniform, and there is less intense interaction between different fluid layers. With reduced shear and mixing, the generation of new eddies slows down, and the existing eddies start to dissipate. This dissipation process, which involves the transfer of turbulent kinetic energy to smaller-scale motions and eventually to heat through viscous effects, causes the turbulence intensity to decrease. In the mainstream region, the turbulence intensity exhibits a parabolic quadratic distribution pattern. This relationship is completely different from that of the Reynolds stress. The reason for this difference lies in the fundamental factors controlling these two quantities. The Reynolds stress is controlled by the velocity gradient. Near the walls, although the velocity gradient is large, the fluid motion is highly constrained by the wall, and the correlation between the fluctuating velocity components that contribute to the Reynolds stress is affected by the wall’s presence. In the mainstream region, where the flow is more developed and less influenced by the wall, the velocity gradient is relatively stable compared to the near-wall region. However, the turbulence intensity is more related to the overall energy transfer and dissipation processes within the turbulent flow. The parabolic quadratic distribution of turbulence intensity in the mainstream region is a result of the balance between the production of turbulent kinetic energy (due to the remaining velocity gradients and flow instabilities) and its dissipation. In summary, the high turbulence intensity near the walls is due to the large velocity gradients and intense shear, while the decrease in turbulence intensity towards the middle of the annulus is a consequence of the reduced velocity gradient and increased dissipation. The distinct distribution patterns of turbulence intensity and Reynolds stress are a reflection of the different physical mechanisms governing these two important turbulent flow characteristics.

From a quantitative analysis perspective, the distance from the position of the maximum turbulence intensity near the inner pipe to the wall is 73.8% of the corresponding distance near the outer pipe. The maximum turbulence intensity near the inner pipe is 101.9% of that near the outer pipe. This indicates that the inner pipe has a higher turbulence intensity, and that the position of the maximum turbulence intensity near the inner pipe is closer to the inner pipe.

3.5. Turbulent Kinetic Energy Distribution

As shown in Figure 14, Figure 15, Figure 16 and Figure 17, in the scenario of annular flow, critical values of turbulent kinetic energy are observed near both the inner and outer walls, and the turbulent kinetic energy exhibits central symmetry. This phenomenon is rooted in the principles of fluid mechanics. From the perspective of fluid mechanics, at the wall, due to the no-slip condition (the fluid velocity at the wall is zero), a large velocity gradient is formed between the wall and the adjacent fluid layer. The existence of a velocity gradient implies relative motion between fluid particles, which in turn generates strong shear forces. These shear forces are one of the key factors driving the fluid to produce turbulent motion. In the region close to the wall, although the shear force is large, the fluid motion is strongly constrained by the wall. The interaction and energy transfer between fluid particles are limited, so the growth of turbulent kinetic energy is relatively slow.

As the distance from the wall increases, the fluid gradually moves away from the confinement of the wall. The energy generated by the shear force can be more effectively input into the fluid. At this time, the vortices and turbulent fluctuations within the fluid can develop and interact fully, and the turbulent kinetic energy also increases accordingly. However, when the fluid continues to move towards the center of the annulus, the situation changes. On the one hand, moving away from the wall weakens the shear force, reducing the energy input. On the other hand, the vortices within the fluid collide and merge with each other during their motion. Meanwhile, they gradually dissipate energy under the action of fluid viscosity, converting the turbulent kinetic energy into other forms of energy such as heat. These two factors work together, causing the turbulent kinetic energy to start decreasing gradually. Therefore, in summary, near the wall, due to the relatively strong shear force and the relatively small limitation on fluid motion, the turbulent kinetic energy reaches its maximum value. In the center of the annulus, because the shear force is weak and energy dissipation dominates, the turbulent kinetic energy drops to its minimum value.

In the mainstream region, the turbulent kinetic energy exhibits a parabolic quadratic distribution pattern, which is extremely similar to the distribution characteristic of turbulence intensity. This is because the fluid motion in the mainstream region is influenced by both the wall and the overall flow characteristics of the annulus. The production and dissipation processes of turbulent kinetic energy reach a relatively balanced state in space, thus forming this specific distribution pattern.

Further analysis of the quantitative analysis results shows that the position of the maximum turbulent kinetic energy near the inner tube is 65.1% of the distance from the wall of the outer tube, and that the maximum turbulent kinetic energy near the inner tube is 103.7% of that near the outer tube. This clearly indicates that the turbulent activity near the inner tube in the annular flow is more intense, with higher turbulent kinetic energy. This is mainly due to the more significant disturbance effect of the inner tube on the fluid flow. The inner tube has a relatively small size. When the fluid flows around the inner tube, it experiences more severe velocity changes and direction shifts, which leads to a further increase in the shear force. A larger shear force prompts the fluid to generate more vortices and turbulent fluctuations, thereby increasing the turbulent kinetic energy. At the same time, because the shear force near the inner tube is stronger, the position of the maximum turbulent kinetic energy is also closer to the wall of the inner tube.

4. Conclusions

This paper has reviewed the current research status of horizontal annuli and found that there is a scarcity of studies on the diameter ratios resulting from different curvatures of horizontal annuli, and that the understanding of turbulent flow characteristics in annuli is still insufficient. Therefore, based on numerical simulation methods and after validating the turbulence model through experiments, this study has investigated the effects of different diameter ratios on the first- and second-order statistical quantities of turbulence. The following conclusions have been drawn:

The contour plots of velocity, turbulence intensity, and turbulent kinetic energy distributions under different diameter ratio conditions exhibit central symmetry.

The velocity distribution in annuli with different diameter ratios does not exhibit symmetry. Quantitative analysis reveals that the velocity peak is closer to the inner pipe. However, as the diameter ratio increases, the velocity peak gradually moves towards the center of the annulus, showing a tendency towards symmetry.

Reynolds stresses reach their maximum values near the inner and outer walls, and exhibit an approximately 180-degree morphological symmetry from the inner and outer walls towards the center of the annulus. This is determined by the vector direction of the velocity gradient.

Turbulence intensity peaks near both the inner and outer walls, with the peak near the inner pipe wall being the maximum and closer to the wall. The distance from the wall to the location of the maximum turbulence intensity near the inner tube is 73.8% of that near the outer tube, and the maximum turbulence intensity near the inner tube is 101.9% of that near the outer tube. The turbulence intensity in the middle of the annulus follows a quadratic distribution pattern, with a minimum value occurring near the center of the annulus.

The distribution of turbulent kinetic energy is similar to that of turbulence intensity, also exhibiting peaks near the inner and outer walls. Quantitative analysis reveals that the distance from the wall to the location of the maximum value near the inner tube is 65.1% of that near the outer tube, and that the maximum turbulent kinetic energy near the inner tube is 103.7% of that near the outer tube. The peak close to the inner tube represents the maximum turbulent kinetic energy in the annulus and is located even closer to the tube wall.

Although this study has conducted an in-depth analysis of the turbulent flow characteristics in annuli with different diameter ratios and confirmed the significant impact of diameter ratios on turbulence characteristics, for practical engineering applications, it is necessary to conduct research on interactions by considering the application scenarios and the mutual influences among parameters.