# Wall Stresses in Cylinder of Stationary Piped Carriage Using COMSOL Multiphysics

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

_{1}< k

_{2}< k

_{3}. Moreover, the increase of the discharge had the greatest influence on the circumferential component of the principal stress of the cylinder, followed by the axis component, and the smallest influence on the wall shear stress of the cylinder, i.e., ${\overline{k}}_{{\sigma}_{c}}$ > ${\overline{k}}_{{\sigma}_{a}}$ > ${\overline{k}}_{{\sigma}_{r}}$ > ${\overline{k}}_{{\tau}_{c}}$.

## 1. Introduction

_{v}, pressure drop ratio R

_{p}and unit energy loss [13]. Michiyoshp and Nakajima [14] calculated the annular slit flow velocity, friction coefficient, mixing length and eddy diffusion coefficient of the crevice flow based on the Reichardt momentum eddy diffusion coefficient expression and Nikurads mixing length expression. Nouri et al. [15] measured the fluid velocity and Reynolds stress of the annular slit flow through experiments.

## 2. Theoretical Analysis

#### 2.1. Piped Carriage Structure

#### 2.2. Force Analysis

_{c}/D

_{c}, which is the length-diameter ratio of the piped carriage. The motion models of the stationary piped carriage were deduced from Equation (1), which can be simplified as:

_{c}is the pressure difference across the ends of the piped carriage, D

_{c}and L

_{c}are the diameter and length of the cylinder of the piped carriage, respectively. τ

_{c}is the average shear stress on the cylinder of the piped carriage, μ is the dynamic friction coefficient, F

_{N}is the supporting force at the contact of the piped carriage support feet and the pipe inner wall.

_{c}/D

_{p}, which is the diameter ration between the piped carriage and the pipe. Thus, the Equation (3) can be written as:

_{p}and τ

_{c}can be written as follows [36]:

_{c}/L

_{c}represent the actual pressure gradient. The right-hand side of the Equation (8) is the Darcy-Weisbach formula [7]. The λ

_{a}is the resistances factor for the annular slit flow. v

_{a}is the average velocity of the annular slit flow.

_{a}are related to the characteristic length, i.e., the hydraulic radius. In this article, the piped carriage is approximated as a cylinder, and the hydraulic radius is calculated by (1−k)D

_{p}[10]. Then, the Reynolds number of the annular slit flow between the pipe and the piped carriage can be calculated as follows.

_{a}, the value of λ

_{a}can be obtained by looking up the Moody diagram or by calculating the following formula [37]:

^{−6}≤ ∆/d ≤ 10

^{−2}, 5 × 10

^{3}≤ Re

_{a}≤ 10

^{8}, in which ∆ is the equivalent roughness of the pipe, the value of plexiglass is 0.005, D

_{c}is the diameter of the pipe, 100 mm, so the relative roughness ∆/D

_{c}is 5 × 10

^{−5}.

## 3. Methods

#### 3.1. Model Setup in COMSOL Multiphysics

^{−5}. Finally, a consortium was formed.

^{5}. Therefore, the high Reynolds model, i.e., the standard k-ε model, was used for the fluid calculation. The parameters of the standard k-ε model have been checked by the pipe flow test, and the calculation of turbulence in the pipe flow has good stability and reasonable accuracy. For the near-wall region, the turbulent boundary layer is very thin and the gradient of the variables is very large. The standard k-ε model is no longer applicable. In this article, the standard wall function method was used to solve the variables near the outer wall of the piped carriage and the inner wall of the pipe.

#### 3.2. Definitions

#### 3.2.1. Fluid Properties

_{f}, p

_{f}. The continuity equation and the Reynolds time-averaged Navier-Stokes momentum equation can be expressed respectively as [33,38]:

_{f}is the fluid density, I is the unit diagonal matrix, F is the volume force. $\nabla =\mathrm{i}\frac{\partial}{\partial x}+j\frac{\partial}{\partial y}+k\frac{\partial}{\partial z}$ is a Hamiltonian differential operator, $K=(\mu +{\mu}_{T})(\nabla {u}_{f}+{(\nabla {u}_{f})}^{\mathrm{T}})$.

_{t}, is estimated using the Boussinesq approximation.

_{k}is a term of turbulent kinetic energy k caused by the mean velocity gradient, ${P}_{\mathrm{k}}={\mu}_{T}[\nabla {u}_{f}\cdot (\nabla {u}_{f}+{(\nabla {u}_{f})}^{\mathrm{T}})]$; σ

_{k}and σ

_{ε}are Prandtl numbers corresponding to turbulent kinetic energy k and the dissipation rate ε, respectively. C

_{ε}

_{2}= 1.92, C

_{ε}

_{1}= 1.44, C

_{μ}= 0.09, σ

_{k}= 1, σ

_{ε}= 1.3.

^{+}and y

^{+}were used to represent the velocity and distance, respectively. u

^{+}= u

_{f}/u

_{τ}, y

^{+}= ∆yρ

_{f}u

_{τ}/μ = ∆y/ν × (τ

_{w}/ρ

_{f})

^{1/2}where u

_{f}represents the average velocity of water flow, u

_{τ}represents the wall friction velocity, u

_{τ}= (τ

_{w}/ρ

_{f})

^{1/2}, τ

_{w}represents the wall shear stress, and ∆y represents the distance from the wall.

_{f}

^{+}> 11.63 and the flow is in the logarithmic law layer, the velocity u

^{+}can be calculated according to Equation (17), i.e.,

^{+}is given by:

_{f}denotes the time-averaged velocity of the node, k denotes the turbulent kinetic energy of the node, ∆y denotes the distance from the node to the wall, and μ denotes the dynamic viscosity of the flow.

^{+}< 11.63, the flow is in the viscous layer, and the velocity is linearly distributed along the normal direction of the wall.

_{k}and ε are equal.

#### 3.2.2. Structure Properties

_{s}is solid density and σ

_{s}is the Cauchy stress tensor which is related to the second Piola–Kirchhoff stress as: σ

_{s}= J

^{−1}FSF

^{T}, where F(F = I + ∇u

_{s}) is the deformation gradient tensor and J = det(F). S is the second Piola–Kirchhoff stress tensor, S = λ

_{s}(tr

**E**)I + 2μ

_{s}

**E**,

**E**= (F

^{T}F−I)/2. Lamé coefficients λ

_{s}and µ

_{s}can be calculated by Formulas (24) and (25). f

_{s}is the volume force vector and ${\ddot{d}}_{s}$ is the local acceleration vector in the structural domain. In this study, the piped carriage is in a static state, so ${\ddot{d}}_{s}=0$.

_{s}are Young’s modulus and Poisson’s ratio of organic glass respectively. The specific values are shown in Table 1.

#### 3.2.3. Interaction Conditions

_{f}= d

_{s}and dynamic conditions (also known as force balance) $n\cdot {\tau}_{f}=n\cdot {\tau}_{s}$ on the cylinder wall of the piped carriage. d

_{f}and d

_{s}represent the displacement of the pipe flow and the piped carriage respectively. The piped carriage is in a static state, so d

_{f}= d

_{s}= 0. τ

_{f}and τ

_{s}represent the stress of the pipe flow and the wall of the piped carriage respectively.

^{d}is the displacement of the structural point.

#### 3.3. Boundary Conditions

_{0}(u(y,z), 0, 0). u(y,z) defines its exponential velocity distribution by the measured velocity at the inlet section. $u(y,z)={u}_{p}{((0.05-\sqrt{{y}^{2}+{z}^{2}})/0.05)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$8$}\right.}$ [26], where u

_{p}is the average velocity of the section in the pipe flow. The turbulence length scale L

_{T}and the turbulence intensity I

_{T}were obtained using Equations (27) and (28) respectively.

#### 3.4. Finite Elements Mesh

^{3}·h

^{−1}in the pipe, the influence of different grid sizes on the principal stress of the middle cylinder wall of the stationary piped carriage in the straight pipe was compared until the difference was less than 3%. Therefore, a reasonable grid size was selected. The different subregion densities were used to reduce the computation and simulation time. The pipe flow was gridded with the maximum unit size of 5 mm. The piped carriage was gridded with the maximum unit size of 2 mm. The number of layers for boundary layer was set to 5, the thickness of the first layer grid was set to 0.005 mm, and the scale factor between layers was set to 1.2. The final model consisted of 1,245,910~1,752,472 elements.

## 4. Experimental Setup and Conditions

#### 4.1. Laser Doppler Velocimetry (LDV)

#### 4.2. Piped Carriage Force Measuring System

#### 4.3. Selection of Cross Section and Layout of Measurement Points

#### 4.3.1. The Layout of Measuring Points in LDV Measuring Flow Field Near the Cylinder Wall of Piped Carriage

#### 4.3.2. The Layout of Measuring Points in Measuring the Principal Stress by Pipeline Force Measuring System

#### 4.4. Design of Conditions

_{s}were measured by the nanoindentation testing system of Nano Indenter G-2000. The relevant flow parameters were summarized in Table 1.

## 5. Validation of Simulated Results

_{c}× L

_{c}= 80 mm × 150 mm with discharge of 30 m

^{3}·h

^{−1}in pipe. Due to the existence of the support feet of the piped carriage, the distribution of the flow field around the cylinder of the piped carriage and the distribution of the principal stress on the cylinder wall of the piped carriage were symmetrical with respect to the Z axis, so only the cases of 0~150° were listed. In the horizontal coordinates of Figure 7 and Figure 8, 0 mm corresponded to the rear-end and 150 mm corresponded to the front-end of the cylinder.

#### 5.1. Velocity Distribution

#### 5.2. The Circumferential Component of the Principal Stress

## 6. Results and Discussion

#### 6.1. Velocity Distributions

_{c}× L

_{c}= 80 mm × 150 mm at a different discharge in the pipe. The inlet of the test section was on the left, hence, the flow direction was from left to right. From the figure, it can be seen that the existence of the piped carriage made the velocity distribution of the pipe flow change greatly. As compared to a single phase flow where it is known, the velocity distribution of the pipe flow is logarithmic, and the streamline is parallel to the pipe axis [47]. There was a vortex region (often called wake vortex region) extending backward along the downstream direction of the piped carriage. There were symmetrical vortices in this region, and the rotation direction was different. This is because the pipe flow from the upper and lower surfaces of the piped carriage met downstream of the piped carriage, and they retained the momentum of the spreading direction. Therefore, the fluid on both sides of the interface had opposite transverse velocity components. It was also a discontinuous section of tangential velocity. In essence, it was also a vorticity surface. The axis direction of the vortices on the vortices should be perpendicular to the velocity difference, so the wake region had different rotational directions along the flow direction [48]. By comparing the flow field around the piped carriage at a different discharge in the pipe, it can be seen that with the increase of the discharge around the piped carriage, the flow velocity around the piped carriage increased gradually, especially the region between the piped carriage and the pipe. This is mainly due to the gap region of water passing between the piped carriage and the pipe, and the change of flow velocity caused by the change of the unit discharge was larger than that in other regions. In the gap region between the piped carriage and the pipe, the velocity gradient varied greatly. The velocity of the flow near the outside wall of the piped carriage and the inside wall of the pipe was very small, and the velocity of the flow in the middle position was very large. This phenomenon conformed to the theory of viscous substratum [37]. Moreover, the low velocity region near the inner wall of the pipe was smaller than that near the outer wall of the piped carriage. This is because the pipe flow in the inner wall of the pipe was mainly affected by the frictional resistance caused by the fluid viscosity, while in the outer wall of the piped carriage, the flow was not only affected by the frictional resistance, but also by the pressure difference resistance caused by the flow separation [49]. The low velocity region near the outer wall of the piped carriage and the inner wall of the pipe decreased with the increase of the discharge in the pipe.

^{3}·h

^{−1}. By comparing the velocity distribution of different sizes of the piped carriage, it can be seen that when the length of the barrel was fixed, the bigger the diameter of the cylinder was, the greater the velocity of the flow around the piped carriage, and the longer length of the downstream wake vortex region of the piped carriage. When the diameter of the cylinder was a constant, the wake vortex region of the piped carriage was longer, but the velocity around the piped carriage did not change remarkably. Therefore, the diameter of the cylinder had greater influence on the flow field around the piped carriage than the length of the cylinder.

#### 6.2. Wall Shear Stress Distributions

_{c}× L

_{c}= 80 mm × 150 mm at a different discharge in the pipe. Compared the shear stress on the cylinder of the piped carriage with a different discharge in the pipe, it can be seen that the wall shear stress of the cylinder increased with the increase of the discharge in the pipe. This is because the shear stress on the cylinder wall of a piped carriage was proportional to the square of the gap velocity between the inner wall of the pipe and the outer wall of the piped carriage, as shown in Formula (9). The increase of the discharge in the pipe inevitably lead to the increase of flow velocity in this gap region, so the wall shear stress increased. The wall shear stress of the cylinder of the piped carriage was greater than zero, which showed that the wall shear stress along the direction of the pipe flow was consistent with the theoretical analysis of Section 2.2. It can also be seen from the figure that the shear stress on the cylinder wall of the piped carriage was symmetrical with respect to Z axis, and the maximum shear stress appeared in the middle and rear sections of the cylinder, and lied between the two groups of support feet, such as, between 0° and 120°. However, the wall shear stress between the front and back support feet was smaller. This is mainly affected by the support feet of the piped carriage, as shown in Figure 9, the flow velocity between the front and back support feet groups was smaller than that in other positions of the annular gap.

^{3}·h

^{−1}. The distribution of shear stress on the cylinder wall of different sizes of the piped carriage was basically the same. Compared with Figure 12a–d, it can be seen that the greater the diameter, the greater the wall shear stress. This is due to the increase of the diameter of the piped carriage, resulting in the reduction of the gap region around the piped carriage, thus increasing the flow velocity of the annular gap, and increasing the wall shear stress. However, the length of the cylinder had little effect on the wall shear stress.

#### 6.3. Principal Stress Distributions

_{a}parallel to the axial direction of the pipe carriage, the circumferential component σ

_{c}perpendicular to the axial direction of the pipe carriage, and the radial component σ

_{r}along the radius direction. The three component values are expressed as: σ

_{a}> σ

_{c}> σ

_{r}.

_{c}× L

_{c}= 80 mm × 150 mm at a different discharge. As can be seen from the Figure 13, with the increased of the discharge in the pipe, the three components of the principal stress on the cylinder wall of the piped carriage increased, and the influence of the discharge on the three components of the principal stress was described in detail in Section 6.4. The distribution of the circumferential and axial components of the principal stress was similar. From the rear-end to the front-end of the cylinder, the distribution of stress components shows the M type, first increased, then decreased, then increased and decreased. The minimum value appeared at the rear-end and front-end of the piped carriage, and was negative. The two peaks appeared near the position of 25 mm from the rear-end and the front-end of the cylinder respectively, and the maximum appeared between the two groups of support feet near 25 mm from the rear-end of the cylinder. This is mainly due to the separation of flow caused by the pipe flow passing through the rear-end of the piped carriage, which resulted in a reverse pressure zone, in which the direction of the flow changed, resulting in a negative circumferential and axial component of the principal stress. When the flow field was separated from the rear-end of the piped carriage, the pressure decreased rapidly, and then the separated fluid re-attached to the cylinder wall. When the fluid arrives at the front-end of the piped carriage, the moving fluid must not only overcome the viscous effect of the outer wall of the piped carriage, but also resist the pressure difference caused by the sudden expansion of the water-crossing section, so that the flow separates again. When some fluid particles are insufficient to overcome the two effects, there would be a reflux in the reverse pressure zone again, and negative values of the circumferential and axial components of principal stress appear again. The radial component of the principal stress on the cylinder wall of the piped carriage was very small compared with the other two components of the principal stress. The minimum value also appeared in the front and rear ends of the cylinder.

^{3}·h

^{−1}. By comparing the distribution of the three components of the principal stress on the cylinder wall of the piped carriage with different sizes, it can be seen that when the cylinder length L

_{c}of the piped carriage was a constant, the larger the diameter D

_{c}was, and the larger were the three components of the corresponding principal stress. However, when the diameter D

_{c}of the cylinder was a constant, the shorter the length L

_{c}of the cylinder, the greater the three components of the principal stress.

#### 6.4. Effect of Discharge on the Wall Stress of Piped Carriage

_{c}× L

_{c}= 80 mm × 150 mm) as an example when the θ = 0°, as shown in Figure 15. Variable k was used to express the effect of unit discharge on the wall stress of the piped carriage.

_{1}, k

_{2}and k

_{3}. The corresponding stress was added with the corresponding lower corner mark. The slopes of the unit discharge stress change of the three components of the principal stress and shear stress on the cylinder were expressed by ${\overline{k}}_{{\tau}_{c}}$, ${\overline{k}}_{{\sigma}_{c}}$, ${\overline{k}}_{{\sigma}_{a}}$ and ${\overline{k}}_{{\sigma}_{r}}$ respectively. As can be seen from Figure 15, the relation was ${\overline{k}}_{{\sigma}_{c}}$ > ${\overline{k}}_{{\sigma}_{a}}$ > ${\overline{k}}_{{\sigma}_{r}}$ > ${\overline{k}}_{{\tau}_{c}}$. For the three components of principal stress and wall shear stress, the stress change rate of smaller discharge was less than that of the larger discharge, that is, k

_{1}< k

_{2}< k

_{3}. It shows that the larger the discharge was, the greater was the influence of the unit discharge on the stress of the piped carriage.

## 7. Conclusions

- -
- With the increase of the discharge in the pipe, the flow velocity around the piped carriage increased obviously, especially in the region between the inner wall of the pipe and the piped carriage. The influence of the diameter of the cylinder D
_{c}on the flow field around the piped carriage was greater than that of the length of the cylinder L_{c}. - -
- The wall shear stress on cylinder of the piped carriage was greater than zero, that is, the wall shear stress along the direction of the pipe flow, and the maximum value appeared between the two groups of support feet in the middle and rear section of the cylinder. When the length of the cylinder L
_{c}was fixed, the larger the diameter of cylinder D_{c}, the greater the wall shear stress. - -
- The stress components on the cylinder wall of the piped carriage obeyed the rule as follow: σ
_{a}> σ_{c}> σ_{r}. From the rear-end to the front-end of the piped carriage, the distribution of stress components shows the M type, first increased, then decreased, then increased and decreased. The minimum value appeared at the rear-end and front-end of the piped carriage, and it was negative. The maximum value appeared between the two groups of support feet 25 mm away from the rear-end of the cylinder. When the length of cylinder L_{c}was fixed, the larger the diameter D_{c}was, the greater were the three components of the corresponding principal stress. When the diameter of the cylinder, D_{c}, was a constant, the shorter the cylinder length L_{c}, the greater the three components of the principal stress. - -
- The larger the flow, the greater the influence of unit flow on the wall shear stress and principal stress of the piped carriage, that is, k
_{1}< k_{2}< k_{3}. At the same time, the increase of the flow has the greatest influence on the circumferential component of the principal stress of the cylinder, followed by the axis component, and the smallest influence on the wall shear stress of the cylinder, i.e., ${\overline{k}}_{{\sigma}_{c}}$ > ${\overline{k}}_{{\sigma}_{a}}$ > ${\overline{k}}_{{\sigma}_{r}}$ > ${\overline{k}}_{{\tau}_{c}}$.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**The structural sketch of piped carriage. Note: 1—Cylinder; 2—Plug; 3—Support body; 4—Support foot.

**Figure 3.**Geometric model and mesh generation of the piped carriage. (

**a**) Geometric model, (

**b**) Mesh generation.

**Figure 4.**Layout sketch of experiment facilities. Note: 1—Centrifugal pump; 2—Gate valve; 3—Electromagnetic flowmeter; 4—Placement device; 5—Braking device; 6—Straight pipe segment; 7—Square water jacket; 8—LDV; 9—Piped carriage; 10—Flange; 11—Bend pipe; 12—Straight pipe segment; 13—Water tank; 14—Steady flow grid.

**Figure 5.**Experimental devices and instruments. Note: 1—Main engine of force measurement system; 2—Gateway of force measurement system; 3—System software TSTDAS V5.0; 4—Square water jacket; 5—Probes of LDV; 6—Coordinate frame.

**Figure 6.**The layout of measuring points near the cylinder wall of piped carriage. (

**a**) Layout of test section around cylinder; (

**b**) Distribution of measuring points on each test section.

**Figure 8.**Comparison of the simulation results and experiment results for circumferential component of the principal stress.

**Figure 9.**The velocity distribution cloud of the XZ plane of the piped carriage with different discharge. (D

_{c}× L

_{c}= 80 mm × 150 mm) Note: (

**a**) Q = 30 m

^{3}·h

^{−1}; (

**b**) Q = 40 m

^{3}·h

^{−1}; (

**c**) Q = 50 m

^{3}·h

^{−1}; (

**d**) Q = 60 m

^{3}·h

^{−1}.

**Figure 10.**The velocity distribution cloud of the XZ plane of different sizes piped carriage with the discharge of 50 m

^{3}·h

^{−1}. Note: (

**a**) D

_{c}× L

_{c}= 75 mm × 150 mm; (

**b**) D

_{c}× L

_{c}= 80 mm × 150 mm; (

**c**) D

_{c}× L

_{c}= 75 mm × 120 mm; (

**d**) D

_{c}× L

_{c}= 80 mm × 120 mm.

**Figure 11.**Wall shear stress distribution of the same piped carriage with different discharge (D

_{c}× L

_{c}= 80 mm × 150 mm). Note: (

**a**) Q = 30 m

^{3}·h

^{−1}; (

**b**) Q = 40 m

^{3}·h

^{−1}; (

**c**) Q = 50 m

^{3}·h

^{−1}; (

**d**) Q = 60 m

^{3}·h

^{−1}.

**Figure 12.**Wall shear stress distribution for different sizes of piped carriage with the discharge of 50 m

^{3}·h

^{−1}. Note: (

**a**) D

_{c}× L

_{c}= 75 mm × 150 mm; (

**b**) D

_{c}× L

_{c}= 80 mm × 150 mm; (

**c**) D

_{c}× L

_{c}= 75 mm × 120 mm; (

**d**) D

_{c}× L

_{c}= 80 mm × 120 mm.

**Figure 13.**Principal stress distribution of the same piped carriage with a different discharge (D

_{c}× L

_{c}= 80 mm × 150 mm).

**Figure 14.**Principal stress distribution for different sizes of the piped carriage with the discharge of 50 m

^{3}·h

^{−1}.

Runs | D_{c} × L_{c} (mm × mm) | Q (m^{3}·h^{−1}) | U_{p} (m/s) | E (Pa) | ν_{s} | Re |
---|---|---|---|---|---|---|

1 | 75 × 150 | 30/40/50/60 | 1.06/1.41/1.77/2.12 | 11.2 × 10^{9} | 0.49 | 105,366/140,488/175,610/210,731 |

2 | 80 × 150 | |||||

3 | 75 × 120 | |||||

4 | 80 × 120 |

_{p}is the mean velocity over the pipe cross-sectional area; E is the Young’s modulus of Piped carriage; ν

_{s}is the Poisson’s ratio of plexiglass at 20 °C. Re is the Reynolds number for each fully developed flow (Re = U

_{p}D/ν, where ν is the kinematic viscosity of water at 20 °C).

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Yang, X.; Ma, J.; Li, Y.; Sun, X.; Jia, X.; Li, Y.
Wall Stresses in Cylinder of Stationary Piped Carriage Using COMSOL Multiphysics. *Water* **2019**, *11*, 1910.
https://doi.org/10.3390/w11091910

**AMA Style**

Yang X, Ma J, Li Y, Sun X, Jia X, Li Y.
Wall Stresses in Cylinder of Stationary Piped Carriage Using COMSOL Multiphysics. *Water*. 2019; 11(9):1910.
https://doi.org/10.3390/w11091910

**Chicago/Turabian Style**

Yang, Xiaoni, Juanjuan Ma, Yongye Li, Xihuan Sun, Xiaomeng Jia, and Yonggang Li.
2019. "Wall Stresses in Cylinder of Stationary Piped Carriage Using COMSOL Multiphysics" *Water* 11, no. 9: 1910.
https://doi.org/10.3390/w11091910