Flow Characteristics of a Fully Developed Concentric Annular Turbulent Flow

Abstract

The hydraulic transportation technology of piped vehicles is a new type of pipeline transportation mode. A concentric annular turbulent flow with different boundaries is formed between the barrel of the piped vehicle and the pipe wall. The study on the annular turbulent flow can provide basic support for the application and promotion of this technology. Therefore, in this paper, the PIV technique was utilized to experimentally investigate the statistical characteristics of the annular turbulent flow in a fully developed smooth concentric annular pipe. The results showed that the position of the maximum velocity in the annular turbulent flow was not at the center but biased towards the barrel wall. Moreover, the smaller the radius ratio, the more it shifted towards the barrel wall. The position of the maximum velocity was independent of the Reynolds number and was a univariate function of the radius ratio; it was obtained by fitting experimental data that $r_{mt}^{*}={\displaystyle \frac{k^{0.349}}{1+k^{0.349}}}$. The resistance coefficient of annular turbulence was independent of the radius ratio and was a univariate function of the Reynolds number; it was obtained by fitting experimental data that $\lambda ={\displaystyle \frac{0.3183}{{R{e}_a}^{0.2487}}}$. The shear stress on the barrel wall was greater than that on the pipe wall in annular turbulent flow. Moreover, as the radius ratio increased, the shear stress on the barrel wall decreased, while that on the pipe wall increased. The velocity distribution in annular turbulent flow was divided into an inner region and an outer region. In the inner region, the $u_c^{+}-y_c^{+}$ curves were greatly affected by the Reynolds number, and the average gradient increased with the increase in the Reynolds number, while in the outer region, the average gradient of the $u_p^{+}-y_p^{+}$ curves decreased with the increase in the Reynolds number. The velocity distribution in annular turbulent flow cannot be expressed by a unified relationship. However, at high Reynolds numbers, there existed a region where the velocity distribution satisfied the logarithmic law in the outer region, and the slope of the logarithmic region was greater than that in circular pipe flow and parallel-plate flow.

1. Introduction

Different from circular pipe flow, annular flow involves the combination of two boundary layers. Moreover, the difference in the shapes of the two walls makes it different from the flow between two flat plates, that is, the characteristics of the two boundary layers are different. For annular laminar flow, many scholars have obtained its theoretical solutions [1,2,3]. However, for annular turbulence, due to the non-linear variation in the total shear stress of the fluid in the radial direction, its flow characteristics are much more complex than those of annular laminar flow, circular pipe turbulence, and turbulence between two parallel plates. The research on turbulent flow in annular pipes can provide a theoretical basis for solving the energy consumption problem of the hydraulic transportation technology of pipeline trains.

In the mid-20th century, scholars conducted some experimental studies on fully developed concentric annular turbulence. Carpenter [4] and Rothfus [5] concluded through measuring the pressure drop that the resistance coefficient of annular turbulence was roughly consistent with that of circular pipe. Rehme [6] and Kjellstrom [7] found that the position of the maximum average velocity in annular turbulence was significantly different from that in laminar flow. Rothfus [8,9] summarized the literature on the friction coefficient of annular turbulence before 1948, and measured the resistance coefficient and velocity distribution of annular air flow at various Reynolds numbers in the viscous, transitional, and lower-turbulence flow. The results showed that the turbulent resistance coefficient was affected by the diameter ratio and was larger than that of the circular pipe. He believed that when the Reynolds number was greater than 10,000, the position of the maximum velocity in annular turbulence was the same as that in laminar flow. Knudsen and Katz [10] measured the average velocity distribution of annular flow with a radius ratio of 0.272 and Reynolds numbers of 9000, 36,000, and 70,000, and reached the same conclusion that the position of the maximum velocity in annular turbulence was “basically” the same as that in laminar flow. For annular pipes with a radius ratio ranging from 0.0625 to 0.562 and Reynolds numbers from 46,000 to 327,000, Brighton [11] presented the average velocity distribution and the position of the maximum average velocity under each flow condition based on the experimental results. Quarmby [12,13] predicted the relationship between the resistance coefficient and the Reynolds number, as well as the velocity distribution of concentric annular flow with a radius ratio from 0.02 to 0.95 and Reynolds numbers from 6000 to 450,000, and conducted experimental measurements. Brighton and Quarmby believed that the resistance coefficient of annular turbulence is independent of the radius ratio and is about 7–8% higher than that of circular pipes. Brighton thought that the wall logarithmic law is not applicable to annular flow, especially when the diameter is relatively large. Quarmby found that the $u^{+}-y^{+}$ correlation of the velocity profile largely depended on the diameter ratio and the Reynolds number. Jonsson and Sparrow [14] presented the research results of eccentric annular flow, including some findings about concentric annular flow, that the resistance coefficient depended on the radius ratio and decreased as the radius ratio increased. Steven [15] and Shigechi [16] obtained analytical solutions for turbulent flow in smooth wall and rough wall based on the modified turbulence models proposed by Van Driest and Reichardt.

In recent years, research on fully developed annular flow has mainly focused on the field of microchannels, where the gap width is less than 3 mm. Zhou [17] found that only when the annular gap width was very small (≤0.01 mm) and the pressure was low (<6.3 MPa), the flow of pressurized water in the gap was laminar. However, in practical engineering applications, the flow of water in the annular gap under common working pressure conditions was close to turbulent flow. Li [18] proposed a new model to predict the pressure gradient of annular flow in microchannels. Chang [19] provided the theoretical basis for flow friction and pressure drop in microchannels, and used ANSYS Fluent software (Version 2021) to numerically simulate the pressure drop and friction of annular laminar flow, mainly studying the velocity and pressure distributions in the inlet section and the fully developed section. Zhu [20] adopted a method combining numerical simulation and experimental research. On the basis of verifying the reliability of the numerical simulation through experiments, he completed the two-dimensional and three-dimensional simulations of the flow characteristics of the annular microchannel and gave the expression of the resistance coefficient when the annular gap width was 0.5 mm. Miu [21] conducted an experimental study on the fluid resistance characteristics in an annular channel with an annular gap width of 0.540–2.685 mm. He proposed that an annular gap width of 2.5 mm was the critical point of annular flow characteristics, when the annular gap width was greater than 2.5 mm, whether it was laminar flow or turbulent flow, the friction resistance coefficient could be calculated according to the theoretical formula for circular pipes, while when the annular gap width was less than 2.5 mm, there were deviations between the experimental results and the theoretical calculation values, and the friction resistance coefficient increased with the increase in the annular gap width. Zhou [22], Fan [23] et al. also obtained the same conclusion in their experiments. Sun [2] conducted an experimental study on the resistance characteristics of an annular flow channel with an annular gap width of 0.57–3.08 mm and found that the experimental values of the resistance coefficient in the laminar flow region were higher than the theoretical calculation values for circular pipes, while the resistance coefficient in the turbulent flow region could be well predicted by the theoretical formula for circular pipes.

Previous computational and experimental results on annular turbulent flow had shown significant discrepancies. However, the above research findings revealed that the primary factors influencing annular turbulent flow characteristics were the Reynolds number and the annular gap width (the ratio of the inner wall radius to the outer wall radius). Therefore, this paper, in conjunction with the National Natural Science Foundation project “Energy Consumption Study of Hydraulic Transportation of Piped Vehicles (51179116)”, conducts experiments in a fully developed smooth concentric annular pipe to investigate the statistical characteristics of annular turbulence using Reynolds number and radius ratio as parameters.

2. Theoretical Analysis

In turbulent motion, adjacent flow layers not only exhibit relative motion but also exchange particles laterally. Therefore, turbulent shear stress is regarded as a combination of viscous shear stress generated by the relative motion of average velocities between adjacent layers and additional shear stress produced by pulsating velocities. Based on Boussinesq hypothesis, the additional shear stress induced by turbulent pulsations is similar to laminar flow, and the method of establishing the relationship between viscous stress and velocity gradient in molecular kinetic theory can be used to study the relationship between additional stress and average velocity in turbulent flow. Fully developed concentric annular turbulence can be regarded as an axisymmetric steady uniform flow, where the shear stress can be expressed as:

$$\tau =\rho \left(\nu +{ϵ}_m\right){\displaystyle \frac{du}{dy}},$$

(1)

written in dimensionless form:

$${\displaystyle \frac{\tau }{{\tau }_w}}=(1+{\displaystyle \frac{{ϵ}_m}{\nu }}){\displaystyle \frac{d{u}^{+}}{d{y}^{+}}},$$

(2)

where ${\tau }_w$ is the wall shear stress, ${ϵ}_m$ is the momentum transfer coefficient.

In the region immediately adjacent to the wall, pulsating flow velocities are minimal, resulting in negligible additional shear stresses from pulsations and a highly concentrated velocity gradient. Consequently, viscous shear stresses dominate, and the fluid motion remains in a laminar state—referred to as the viscous sublayer. Outside the viscous sublayer, the additional shear stress exceeds the viscous shear stress by tens of thousands of times, so the effect of viscous stress is negligible, and the flow enters a fully turbulent state, known as the turbulent core region.

For the viscous sublayer, the dimensionless expression for the momentum transfer coefficient is:

$${\displaystyle \frac{{ϵ}_m}{\nu }}=n^2{u}^{+}{y}^{+}\left[1-exp\left(-n^2{u}^{+}{y}^{+}\right)\right],$$

(3)

where $n$ is the damping coefficient, with an empirical value of 0.124.

For the turbulent core region, Prandtl’s mixing length theory and Von Karman’s similarity assumption yield:

$${ϵ}_m=l^2{\displaystyle \frac{du}{dy}},$$

(4)

$$l=K{\displaystyle \frac{du/dy}{d^{2}u/d{y}^2}},$$

(5)

where $l$ is the Prandtl mixing length, $K$ is a constant.

As shown in Figure 1, assuming the concentric annular flow formed between the outer wall of the barrel and the inner wall of the pipe is fully developed, the annular flow is radially divided into two zones—the inner zone and the outer zone—using the position of maximum velocity ($r_m$) as the boundary. The fluid near the outer wall of the barrel constitutes the inner zone, while the fluid near the inner wall of the pipe constitutes the outer zone. Denoting the inner and outer zones by subscripts c and p, respectively, and labeling the viscous sublayer and turbulent core regions by 1 and 2, respectively. The dimensionless velocity distribution of the annular flow satisfies the following relationships.

Inner zone ($r_c<r<r_m$):

(a)

For the viscous sublayer ($0<y_c^{+}<y_{c1}^{+}$)

$${\displaystyle \frac{d{u}_{c1}^{+}}{d{y}_c^{+}}}={\displaystyle \frac{\tau /{\tau }_c}{1+n_c^2{u}_{c1}^{+}{y}_c^{+}\left[1-exp\left(-n_c^2{u}_{c1}^{+}{y}_c^{+}\right)\right]}},$$

(6)

when $y_c^{+}=0$, $u_{c1}^{+}=0$.

(b)

For the turbulent region ($y_{c1}^{+}<y_c^{+}<y_{cm}^{+}$)

$${\displaystyle \frac{d^2{u}_{c2}^{+}}{d{y}_c^{+2}}}={\displaystyle \frac{-K{\left(d{u}_{c2}^{+}/d{y}_c^{+}\right)}^2}{{(\tau /{\tau }_c-d{u}_{c2}^{+}/d{y}_c^{+})}^{1/2}}},$$

(7)

when $y_c^{+}=y_{c1}^{+}$, $u_{c1}^{+}=u_{c2}^{+}$ and ${\displaystyle \frac{d{u}_{c1}^{+}}{d{y}_c^{+}}}={\displaystyle \frac{d{u}_{c2}^{+}}{d{y}_c^{+}}}$; when $y_c^{+}=y_{cm}^{+}$, $u_{c2}^{+}=u_{cm}^{+}$.

Outer zone ($r_m<r<r_p$):

(a)

For the viscous sublayer ($0<y_p^{+}<y_{p1}^{+}$)

$${\displaystyle \frac{d{u}_{p1}^{+}}{d{y}_p^{+}}}={\displaystyle \frac{\tau /{\tau }_p}{1+n_p^2{u}_{p1}^{+}{y}_p^{+}\left[1-exp\left(-n_p^2{u}_{p1}^{+}{y}_p^{+}\right)\right]}},$$

(8)

when $y_p^{+}=0$, $u_{p1}^{+}=0$.

(b)

For the turbulent region ($y_{p1}^{+}<y_p^{+}<y_{pm}^{+}$)

$${\displaystyle \frac{d^2{u}_{p2}^{+}}{d{y}_p^{+2}}}={\displaystyle \frac{-K{\left(d{u}_{p2}^{+}/d{y}_p^{+}\right)}^2}{{(\tau /{\tau }_p-d{u}_{p2}^{+}/d{y}_p^{+})}^{1/2}}},$$

(9)

when $y_p^{+}=y_{p1}^{+}$, $u_{p1}^{+}=u_{p2}^{+}$ and ${\displaystyle \frac{d{u}_{p1}^{+}}{d{y}_p^{+}}}={\displaystyle \frac{d{u}_{p2}^{+}}{d{y}_p^{+}}}$; when $y_p^{+}=y_{pm}^{+}$, $u_{p2}^{+}=u_{pm}^{+}$.

At the position of maximum velocity, the velocity gradient is zero. By continuity, the velocities in the outer region and the inner region are equal at the position of maximum velocity, expressed in dimensionless form as:

$${\displaystyle \frac{u_{pm}^{+}}{u_{cm}^{+}}}={\displaystyle \frac{\sqrt{{\tau }_c}}{\sqrt{{\tau }_p}}},$$

(10)

Set $k={\displaystyle \frac{r_c}{r_p}}$, $\eta ={\displaystyle \frac{r_m}{r_p}}$, $\epsilon ={\displaystyle \frac{r}{r_p}}$.

Since

$$y_{pm}=r_p-r_m,$$

(11)

$$y_{cm}=r_m-r_c,$$

(12)

then

$$y_{pm}^{+}=\left(1-\eta \right)r_p^{+},$$

(13)

$$y_{cm}^{+}=\left(\eta -k\right)\sqrt{{\displaystyle \frac{{\eta }^2-k^2}{k\left(1-{\eta }^2\right)}}}{r}_p^{+},$$

(14)

The average velocity of the annular flow ($\overline{u_a}$) can be expressed as:

$$\overline{u_a}={\displaystyle \frac{{{\displaystyle \int }}_0^{y_{cm}}2\pi u_{c}rdr+{{\displaystyle \int }}_0^{y_{pm}}2\pi u_{p}rdr}{\pi \left(r_p^2-r_c^2\right)}},$$

(15)

then the Reynolds number of the annular flow ($R{e}_a$) is:

$$R{e}_a={\displaystyle \frac{4k}{1+k}}\left[{\displaystyle \frac{1}{r_c^{+}}}{{\displaystyle \int }}_0^{y_{cm}^{+}}{u}_c^{+}\left(r_c^{+}+y_c^{+}\right)d{y}_c^{+}+{\displaystyle \frac{1}{k{r}_p^{+}}}{{\displaystyle \int }}_0^{y_{pm}^{+}}{u}_p^{+}(r_p^{+}-y_p^{+})d{y}_p^{+}\right],$$

(16)

where

$$r_c^{+}=k\sqrt{{\displaystyle \frac{{\eta }^2-k^2}{k\left(1-{\eta }^2\right)}}}{r}_p^{+},$$

(17)

$$r_p^{+}={\displaystyle \frac{r_p\sqrt{\frac{{\tau }_p}{\rho }}}{\nu }},$$

(18)

For an annular flow segment with length $L$, the force analysis yields:

$$∆P·\pi \left(r_p^2-r_c^2\right)=2\pi \left(r_p+r_c\right)·\overline{\tau }·L,$$

(19)

so, the average wall shear stress of annular flow can be expressed as:

$$\overline{\tau }={\displaystyle \frac{∆P}{2L}}\left(r_p-r_c\right),$$

(20)

Using Shigechi’s assumption [16] that the location of maximum velocity in annular turbulence coincided with the location of zero shear stress, we perform force analysis on the fluid between $r_m$ to $r_p$ and $r_c$ to $r_m$, respectively.

Then, the shear stress on the pipe wall can be expressed as:

$${\displaystyle \frac{{\tau }_p}{\overline{\tau }}}={\displaystyle \frac{\left(r_p^2-r_m^2\right)}{r_p\left(r_p-r_c\right)}}={\displaystyle \frac{1-{\eta }^2}{1-k}},$$

(21)

and the shear stress on the barrel wall can be expressed as:

$${\displaystyle \frac{{\tau }_c}{\overline{\tau }}}={\displaystyle \frac{\left(r_m^2-r_c^2\right)}{r_c\left(r_p-r_c\right)}}={\displaystyle \frac{{\eta }^2-k^2}{k\left(1-k\right)}},$$

(22)

Hence,

$${\displaystyle \frac{{\tau }_c}{{\tau }_p}}={\displaystyle \frac{r_p\left(r_m^2-r_c^2\right)}{r_c\left(r_p^2-r_m^2\right)}}={\displaystyle \frac{{\eta }^2-k^2}{k\left(1-{\eta }^2\right)}},$$

(23)

Similarly, the tangential stress at any radius can be expressed as:

$${\displaystyle \frac{{\tau }_r}{{\tau }_p}}={\displaystyle \frac{r_p\left(r^2-r_m^2\right)}{r\left(r_p^2-r_m^2\right)}}={\displaystyle \frac{{\epsilon }^2-{\eta }^2}{\epsilon \left(1-{\eta }^2\right)}},$$

(24)

$${\displaystyle \frac{{\tau }_r}{{\tau }_c}}={\displaystyle \frac{r_c\left(r_m^2-r^2\right)}{r\left(r_m^2-r_c^2\right)}}={\displaystyle \frac{k({\eta }^2-{\epsilon }^2)}{\epsilon \left(({\eta }^2-k^2\right)}},$$

(25)

From Equations (22) and (25), it follows that:

$${\displaystyle \frac{{\tau }_r}{\overline{\tau }}}={\displaystyle \frac{\left(r_m^2-r^2\right)}{r\left(r_p-r_c\right)}}={\displaystyle \frac{\left({\eta }^2-{\epsilon }^2\right)}{\epsilon \left(1-k\right)}},$$

(26)

The resistance coefficient of annular flow is expressed as:

$$\lambda ={\displaystyle \frac{8\overline{\tau }}{\rho {\overline{u_a}}^2}},$$

(27)

According to Equations (20) and (27), we obtain:

$$\lambda ={\displaystyle \frac{4\left(r_p-r_c\right)}{\rho {\overline{u_a}}^2}}·{\displaystyle \frac{∆P}{L}},$$

(28)

Since

$$R{e}_a={\displaystyle \frac{2\overline{u_a}\left(r_p-r_c\right)}{\nu }},$$

(29)

Hence,

$$\lambda ={\displaystyle \frac{16{\left(1-k\right)}^3{r}_p}{\rho {R{e}_a}^2{\nu }^2}}·{\displaystyle \frac{∆P}{L}},$$

(30)

3. Experimental Scheme

3.1. Experimental System

The experimental system mainly consisted of several parts, such as the power device, water pipeline, test section, and receiving device, as shown in Figure 2. The power device was a centrifugal pump and the flow rate was controlled using a regulator valve and an electromagnetic flow meter. The water pipeline was a PPR round pipe with a length of 15 m, arranged in an L-shape. The test section was a plexiglass pipe with an inner diameter of 50 mm and a length of 10.6 m. The receiving device was a stainless steel tank, which was also the water-supplying device for the whole system.

3.2. Measuring Device

The measurement devices of this experiment mainly included a pressure measurement device and a flow velocity measurement device. The pressure measurement device was mainly composed of two parts: data acquisition equipment (Kaifeng Kaiyi Automation Instruments Co., Ltd., Kaifeng, China) and dynamic signal continuous monitoring system software V3.0 (Chengdu Taisite Electronic Information Co., Ltd., Chengdu, China), as shown in Figure 3. The measurement range of the pressure measurement device is 0–200 KPa, and its uncertainty is ±0.5% FS.

A particle image velocimetry (PIV) system (Dantec Dynamics A/S, Copenhagen, Denmark) was selected for the flow velocity measurement, as shown in Figure 4. The velocity field in the PIV measurements is calculated based on the cross-correlation method. Specifically, consecutive image pairs are divided into overlapping interrogation windows, and the displacement of tracer particles is obtained through cross-correlation method, followed by velocity derivation using the time interval between frames. The actual size of the interrogation window was 32 mm × 32 mm. The maximum pulse frequency of the laser is 50 Hz, and the frequency set for image acquisition during the experiment was 45 Hz to prevent laser energy insufficiency. For each operating condition, 100 instantaneous flow field images were captured and processed for turbulence statistics and averaging. An illuminated plane used for image capture was shown in Figure 5.

In order to minimize light refraction caused by the round pipe wall, a rectangular water jacket was installed around the test section. Additionally, fluorescent paint was applied to the outer surface of the barrel to effectively eliminate the interference caused by wall scattering. The tracer particles (Beiting Measurement Technology (Beijing) Co., Ltd., Beijing, China) used in the experiment were hollow glass microbeads with a density of 1.1 g/cm³ and a particle size of 10 μm.

3.3. Experimental Program Design

3.3.1. Test Conditions

The theoretical analysis in Section 2 indicated that the Reynolds number of annular flow and radius ratio are the primary factors influencing its hydraulic characteristics. The Reynolds number of annular flow depends on the Reynolds number of pipe flow and radius ratio, that is:

$$R{e}_a={\displaystyle \frac{Re}{k}},$$

(31)

where $Re$ is the Reynolds number of pipe flow, achieved by controlling the pipe flow rate, $k$ is the radius ratio, determined by the outer diameter of the barrel and the inner diameter of the pipe. Therefore, the key design parameters in this study were as follows:

The radius ratios ($k$) were set with 7 values: 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, and 0.8, corresponding to the outer diameters of barrels of 5 mm, 7.5 mm, 10 mm, 12.5 mm, 15 mm, 17.5 mm, and 20 mm, respectively.

The Reynolds numbers of pipe flow (Re) were 17,378, 34,757, 49,652, 69,513, 84,335, 104,270, and 139,027, corresponding to the flow rates of 2.5 m³/h, 5 m³/h, 7 m³/h, 10 m³/h, 12 m³/h, 15 m³/h, and 20 m³/h respectively.

The Reynolds numbers of annular flow ($R{e}_a$) were shown in

Table 1. Reynolds numbers of annular flow.

Re	$\mathit{k}$ = 0.2	$\mathit{k}$ = 0.3	$\mathit{k}$ = 0.4	$\mathit{k}$ = 0.5	$\mathit{k}$ = 0.6	$\mathit{k}$ = 0.7	$\mathit{k}$ = 0.8
17,378	14,482	13,368	12,413	11,586	10,861	10,223	9655
34,757	28,964	27,031	24,826	23,171	21,723	20,445	19,309
49,652	41,377	38,194	35,466	33,102	31,033	29,207	27,585
69,513	57,928	53,472	49,652	46,342	43,446	40,890	38,619
84,335	70,279	64,930	60,292	56,224	52,756	49,609	46,894
104,270	86,892	80,208	74,479	69,513	65,169	61,335	57,928
139,027	115,856	106,944	99,305	92,685	86,892	81,780	77,237

.

3.3.2. Arrangement of Test Section and Test Points

According to hydrodynamic theory, for circular pipes, it is generally accepted that fully developed turbulent flow can only be achieved when the length-to-diameter ratio is at least 20 to 40 [24]. Taking the annular diameter as the characteristic dimension, the minimum radius ratio requires a development section length of at least 800 mm. The developing section length designed for this test was 900 mm, indicating that the annular flow downstream of the developing section was fully developed turbulent flow. Experiments revealed that due to the influence of the outlet, velocity distribution variations occurred within approximately 100 mm of the annular outlet. Consequently, the test sections were positioned between the developing section and the section 100 mm from the rear end of the barrel. Five pressure measurement points were arranged at 100 mm intervals within the test section, as shown in Figure 6.

4. Results and Discussion

4.1. Position of the Maximum Velocity

Analysis has indicated that an error of 0.1 in the maximum velocity radius ($r_m$) could cause a 15–20% variation in the predicted resistance coefficient. The dimensionless parameter for the maximum velocity radius ($r_m^{*}$) is defined as the ratio of the radial distance from the position of maximum velocity to the barrel wall to the total width of the annular flow.

Table 2. Empirical formulas for $r_m^{*}$.

Proposers	Empirical Formulas
Kays [25]	${\displaystyle \frac{k^{0.343}}{1+k^{0.343}}}$
Quarmy [13]	${\displaystyle \frac{k^{0.366}}{1+k^{0.366}}}$
Barrow [26]	${\displaystyle \frac{k^{0.027}\sqrt{\frac{1-k^2}{2ln\left(1/k\right)}}-k}{1-k}}$
Steven [15]	$0.5017+0.062lnk$

lists several empirical formulas for $r_m^{*}$ in turbulent flow in concentric annular gap. For laminar flow, the velocity distribution in annular flow was given by:

$$u_l={\displaystyle \frac{1}{4\mu }}(r_p^2-r_c^2+{\displaystyle \frac{r_p^2-r_c^2}{ln\frac{r_p}{r_c}}}ln{\displaystyle \frac{r}{r_p}})·{\displaystyle \frac{∆P}{∆L}},$$

(32)

Hence [27],

$$r_{ml}^{*}={\displaystyle \frac{\sqrt{\frac{1-k^2}{2ln\left(1/k\right)}}-k}{1-k}},$$

(33)

Figure 7 showed the variation in the dimensionless value of the maximum velocity radius for annular turbulence as a function of the annular Reynolds number under different radius ratios. The figure indicated that, for any given radius ratio, the positions of the maximum velocity were independent of the annular Reynolds number. Therefore, $r_{mt}^{*}$ was a univariate function of $k$. The relationship between $r_{mt}^{*}$ and $k$ obtained by fitting the experimental results was as follows:

$$r_{mt}^{*}={\displaystyle \frac{k^{0.349}}{1+k^{0.349}}},$$

(34)

As shown in Figure 8, the sum of squared residuals between the fitted curve and the experimental data was 1.98 × 10⁻⁴, meeting the requirement. Figure 8 also presented findings from previous studies. Compared with their results, the exponent of the formula fitted from experimental data lied between those of Quramby’s and Leung’s empirical formula, leaning closer to Leung’s. This might be because Quarmby believed that the position of maximum velocity became independent of Reynolds number only when the annular Reynolds number exceeded 4 × 10⁴–5 × 10⁴, whereas the results in this paper indicated that $r_m^{*}$ was solely a function of the radius ratio within the Reynolds number range of the present experiments (1 × 10⁴ $<R{e}_a<$ 1.16 × 10⁵). The predictions by Barrow, Lee and Roberts, and Steven were less accurate.

Figure 8 also showed the theoretical values for laminar flow. It was found that regardless of whether in laminar or turbulent flow, the maximum velocities did not occur at the center but were shifted toward the cylinder wall. Furthermore, the positions of maximum velocity in turbulent flow were shifted even closer to the cylinder wall than that in laminar flow. $r_{mt}^{*}$ increased as the radius ratio increased, and gradually approached 0.5. That is, the larger the radius ratio, the closer the position of maximum velocity approached the center. It was because when the radius ratio was sufficiently large, the curvature of the pipe wall and barrel wall had almost no effect on the annular flow. When $k=1$ (the limiting case of annular flow), the annular flow transformed into the parallel-plate flow, with the maximum velocity occurring at the midpoint between the two plates. Furthermore, as the radius ratio increased, the position of maximum velocity in turbulent flow was closer to laminar theory value. That is, the differences between the positions of maximum velocity in turbulent flow and laminar flow decreased with the increase in radius ratio. This occurred because when the annular width was exceptionally small, viscous forces remained dominant in the annular turbulence even at high Reynolds numbers, thereby exhibiting laminar characteristics.

4.2. Resistance Characteristic

At present, there are relatively accurate empirical formulas for calculating the resistance characteristics of turbulent flow in circular pipes.

When $4000<Re<{10}^5$:

$$\lambda ={\displaystyle \frac{0.316}{R{e}^{0.25}}},$$

(35)

When $Re<{10}^6$:

$${\displaystyle \frac{1}{\sqrt{\lambda }}}=2lg\left(Re\sqrt{\lambda }\right)-0.8,$$

(36)

For ordinary non-circular pipe channels, due to the lack of sufficient experimental data, the above formulas are often used for rough estimation in engineering.

For annular laminar flow, an analytical solution for the resistance coefficient has been obtained as follows:

$${\lambda }_{la}={\displaystyle \frac{\left(1-k^2\right)lnk}{\left(1+k^2\right)lnk+\left(1-k^2\right)}}·{\displaystyle \frac{64}{R{e}_a}},$$

(37)

Experiments in References [21,23,28] demonstrated that the resistance coefficient of annular laminar flow was independent of the radius ratio. The fitted empirical formulae were given by:

$${\lambda }_{la}={\displaystyle \frac{97.7657}{{R{e}_a}^{-1.0028}}},$$

(38)

and

$${\lambda }_{la}={\displaystyle \frac{95}{R{e}_a}},$$

(39)

The results calculated from these formulae were higher than those of laminar flow in circular pipes.

For annular turbulent flow, based on Equation (30), the variation curves of the resistance coefficient with the annular flow Reynolds number at different radius ratios were shown in Figure 9. It can be seen from Figure 9 that within the tested Reynolds number range, the resistance coefficient of annular flow had a weak correlation with the radius ratio. The resistance coefficients at different radius ratios showed the same variation trend with the annular flow Reynolds number, which can be expressed by the general formula $\lambda ={\displaystyle \frac{C}{{R{e}_a}^m}}$. The constants C and m in the formula were obtained by regression analysis based on experimental data, as listed in

Table 3. The constants C and m.

k	C	m	$\mathit{\alpha }$
0.2	0.3175	0.2496	0.9967
0.3	0.3192	0.2489	0.9978
0.4	0.3193	0.2503	0.9959
0.5	0.3165	0.2483	0.9971
0.6	0.3176	0.2481	0.9959
0.7	0.3180	0.2478	0.9952
0.8	0.3198	0.2480	0.9931
Average	0.3183	0.2487

, with correlation coefficients $\alpha$ all above 0.99. Taking the average of all constants, the optimal fitted relation was:

$$\lambda ={\displaystyle \frac{0.3183}{{R{e}_a}^{0.2487}}},$$

(40)

Within the tested Reynolds number range, the empirical formulas for the resistance coefficients of turbulent flow in circular pipes and between two parallel plates [29] were also plotted in Figure 9. The resistance coefficient of annular flow was between the two and close to that of circular pipe flow, about 2% higher than that of circular pipe flow and 3–10% lower than that of parallel-plate flow. Therefore, it is reasonable to use the empirical formula of circular pipe flow for estimation in practical engineering.

4.3. Shear Stress

Taking the dimensionless radius ($r^{*}={\displaystyle \frac{r-r_c}{r_p-r_c}}$) as the abscissa and the shear stress (${\displaystyle \frac{{\tau }_r}{\overline{\tau }}}$) at any radius (r) calculated by Equation (26) as the ordinate, Figure 10a was plotted. Taking the radius ratio as the abscissa, ${\displaystyle \frac{{\tau }_p}{\overline{\tau }}}$, ${\displaystyle \frac{{\tau }_c}{\overline{\tau }}}$, and ${\displaystyle \frac{{\tau }_c}{{\tau }_p}}$ calculated by Equations (21)–(23) were plotted in Figure 10b.

It can be seen from Figure 10a that at small radius ratios, the variation curves of shear stress were curved and steep in the inner region but relatively flat in the outer region. As the radius ratio increased, the curve became flatter, and when $k$ = 0.8, it was almost linearly distributed. It can be seen from Figure 10b that for any radius ratio, ${\displaystyle \frac{{\tau }_p}{\overline{\tau }}}<1$, ${\displaystyle \frac{{\tau }_c}{\overline{\tau }}}>1$, ${\displaystyle \frac{{\tau }_c}{{\tau }_p}}>1$, that is, the shear stress on the barrel wall was always greater than that on the pipe wall. It can be seen that the barrel wall contributed more to the frictional resistance of annular turbulent flow. As the radius ratio increased, the shear stress on the barrel wall decreased while the shear stress on the pipe wall increased, so ${\displaystyle \frac{{\tau }_c}{{\tau }_p}}$ decreased and approached 1. That is to say, the increase in the radius ratio weakened the resistance effect of the barrel wall on annular turbulent flow and enhances the resistance effect of the pipe wall, making the two walls tend to be consistent and close to the flow between two parallel plates.

4.4. Velocity Distribution

The dimensionless average velocity profiles ($u_c^{+}-y_c^{+}$ and $u_p^{+}-y_p^{+}$) normalized by the inner region friction velocity ($u_c^{*}$) and the outer region friction velocity ($u_p^{*}$) respectively were plotted in Figure 11. From the velocity distribution at a single Reynolds number, it was found that both in the inner and outer regions, the $u^{+}-y^{+}$ distribution was related to both the Reynolds number and the radius ratio. In the inner region, the influence of the Reynolds number was more obvious than that of the radius ratio, and the average gradient of the $u_c^{+}-y_c^{+}$ curves increased with the increase in the Reynolds number. In the outer region, the average gradient of the $u_p^{+}-y_p^{+}$ curves decreased with the increase in the Reynolds number, and the Reynolds number effect was minimized with the decrease in the radius ratio.

The velocity distribution in the turbulent core flow in circular pipes and between parallel plates satisfies the logarithmic law [30]:

$$u^{+}={\displaystyle \frac{1}{K}}ln\left(y^{+}\right)+B,$$

(41)

where $K$ is a “universal constant” for turbulent exchange, later known as the Von Karman constant. To determine whether the velocity distribution in annular flow satisfies the logarithmic law, a more sensitive method is adopted in this paper by introducing a diagnostic function [31]:

$$\gamma =y^{+}{\displaystyle \frac{d{u}^{+}}{d{y}^{+}}},$$

(42)

In the logarithmic region, $\gamma$ is a constant equal to ${\displaystyle \frac{1}{K}}$. The variation curves of the diagnostic function ($\gamma$) with $y^{+}$ in the inner and outer regions were shown in Figure 12. In the outer region, there was a constant range of $\gamma$ at high Reynolds numbers, with a value around 2.9, that is, $k\approx 0.34$. As the radius ratio increased, the logarithmic law in the outer region weakened. A large number of experiments and direct numerical simulations have shown that the values of $\gamma$ in the logarithmic region of circular pipe flow and parallel-plate flow were between 2.5 and 2.8. It can be seen that the slope of the logarithmic region in the outer region of annular flow was greater than that of circular pipe flow and parallel-plate flow. There was almost no flat segment of $\gamma$ in the inner region. Therefore, the $u^{+}-y^{+}$ distribution in the inner region had no logarithmic law region, which corresponded to the distribution law of shear stress.

5. Conclusions

There are two states of the piped vehicle during the hydraulic transportation of the piped vehicles: static and moving. When the piped vehicle is stationary in the pipeline, the fluid around it is in a fully turbulent state. The water flows through the gap between the outer wall of the barrel and the inner wall of the pipe, forming concentric annular turbulent flow. Previous calculation and experimental results on annular turbulent flow have been obviously inconsistent. Therefore, this paper conducted an experimental study on the statistical characteristics of annular turbulent flow in fully developed smooth concentric annular pipes, and the following conclusions were obtained:

(1)

For both laminar and turbulent flows, the maximum velocity did not occur at the center of the annular flow but biased towards the barrel wall. The smaller the radius ratio, the more it shifted towards the barrel wall. The position of the maximum velocity in turbulent flow was more biased towards the barrel wall than that in laminar flow. As the radius ratio increased, the position of the maximum velocity in turbulent flow approached that in laminar flow.

In annular turbulent flow at any radius ratio, the position of the maximum velocity was independent of the Reynolds number, that is, $r_{mt}^{*}$ was a univariate function of the radius ratio (k). The relationship between $r_{mt}^{*}$ and k fitted from the experimental results was

$$r_{mt}^{*}={\displaystyle \frac{k^{0.349}}{1+k^{0.349}}}$$

(2)

For both laminar and turbulent flows, the resistance coefficient of annular flow was independent of the radius ratio, that is, $\lambda$ was a univariate function of the annular Reynolds number ($R{e}_a$). The relationship between the resistance coefficient of annular turbulent flow and the annular Reynolds number fitted from the experimental results was

$$\lambda ={\displaystyle \frac{0.3183}{{R{e}_a}^{0.2487}}}$$

It was found that the resistance coefficient of annular turbulent flow was about 2% higher than that of circular pipe flow and 3–10% lower than that of parallel-plate flow. Therefore, it is reasonable to use the empirical formula of circular pipe flow to estimate the resistance coefficient of annular flow in practical engineering.

(3)

At small radius ratios, the variation curves of shear stress in annular turbulent flow were curved and steep in the inner region but relatively flat in the outer region. As the radius ratio increased, the curves became flatter. The shear stress on the barrel wall was always greater than that on the pipe wall. As the radius ratio increased, the shear stress on the barrel wall decreased while the shear stress on the pipe wall increased.

(4)

The dimensionless velocity distribution in annular turbulent flow was related to both the Reynolds number and the radius ratio. In the inner region, the influence of the Reynolds number was more obvious than that of the radius ratio, and the average gradient of the $u_c^{+}-y_c^{+}$ curves increased with the increase in the Reynolds number. In the outer region, the average gradient of the $u_p^{+}-y_p^{+}$ curves decreased with the increase in the Reynolds number, and the Reynolds number effect was minimized with the decrease in the radius ratio. The velocity distribution in annular turbulent flow cannot be expressed by a unified relation, but there was a logarithmic region in the velocity distribution at high Reynolds numbers in the outer region, and the slope of the logarithmic region was greater than that in circular pipe flow and parallel-plate flow.