# Numerical Investigation of the Stress on a Cylinder Exerted by a Stratified Current Flowing on Uneven Ground

^{1}

^{2}

^{3}

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^{5}

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## Abstract

**:**

## 1. Introduction

## 2. Numerical Models

#### 2.1. Governing Equations

_{i}stands for the spatial coordinate; u

_{i}stands for the flow velocity; p stands for the pressure; µ stands for the kinematic viscosity; and f

_{i}stands for the body force.

#### 2.2. Scalar Transport Equation

_{2}+ (1 − C)ρ

_{1}, (ρ

_{1}and ρ

_{2}represent the upper layer density and the lower layer density, respectively); k stands for the diffusion coefficient; and S stands for the source term or sink term.

#### 2.3. Turbulence Model

_{s}stands for the Smagorinsky constant. In the IWs environment, C

_{s}, changing with space-time, can be calculated by a dynamic procedure [10].

#### 2.4. Establishment of a Numerical Tank

_{1}= 0.2 m is placed on the centerline in the spread direction (Z). The cylinder bottom center is located at (x, y, z) = (2.0, 0, 0.15) m. By using the gravity collapse approach to motivate the IWs [11], the numerical tank is separated into the IW generation region (x = 0–0.23 m) and the IW propagation region (x = 0.23–4 m), where Δh is the thickness difference in the pycnocline between the two regions, and η

_{0}is the initial amplitude.

_{1}= 0.998 g/cm

^{3}and depth l

_{1}= 0.075 m, and the lower layer is defined as the higher-density water layer with density ρ

_{2}= 1.017 g/cm

^{3}and depth l

_{2}= 0.225 m, respectively. Therefore, the total water depth of the tank H is kept at 0.3 m.

#### 2.5. Numerical Model Verification

#### 2.5.1. Verification by Physical Model Test

^{3}, and the thicknesses of the lower saline layer are 0.4 m with density of 1.030 g/cm

^{3}. In the initial phase, there is a thickness difference controlled by a gate in the pycnocline between the IW generation region and the IW propagation region, which is similar to Figure 1 above. By lifting the gate, a leading IW of depression is motivated by the gravity collapse (caused by the thickness difference) method and propagates toward the tank’s left end [19], as shown in Figure 2. Two ultrasonic probes fixed in the tank top were adopted to capture the IWs spatial distribution in the propagation region of the numerical tank. A triangular bar was fixed between the probes, and the basis length Lw was equal to the wavelength.

#### 2.5.2. Grid Independence Test

_{Fn}in T

_{2}almost coincides with that in T

_{3}over time. Therefore, the grid independence test results show the convergence of the grids. Consequently, the moderate gird density (case T

_{2}) is sufficiently fine enough to discretize the computational domain, which can be adopted in the rest numerical simulations.

## 3. Result and Analysis

_{1}), the bottom-step terrain model (Case N

_{2}), the flat-top-knoll terrain model (Case N

_{3}), and the flat-top-platform model (Case N

_{4}).

_{1}/l

_{2}of the two water layers, and the density difference Δρ between the two water layers) and the dimensionless IW amplitude η

_{0}/H are consistent: the upper water depth l

_{1}= 0.075 m, the lower water depth l

_{2}= 0.225 m, making l

_{1}/l

_{2}= 0.33; Δρ = 0.019 g/cm

^{3}, η

_{0}/H = 0.0575. The diameter of the cylinder is 0.05 m, the length of the cylinder h

_{1}= 0.2 m, and the coordinates of the center of the cylinder bottom are (2, 0.1, 0.15) XYZ. The height difference between the cylinder bottom and the tank bottom is h

_{2}= 0.1 m. The lengths of the terrain platform in Cases N

_{2}, N

_{3}, and N

_{4}are 0.3 m, 0.3 m, and 2.15 m, respectively, and the slope angles α in Cases N

_{3}and N

_{4}are 45°. Schematic diagrams of numerical tank for the four cases are shown in Figure 6a–d, and the cases introduction is presented in Table 2.

#### 3.1. Coupled Influence of Terrain and IWs on the Forces on the Cylinder

_{Fn}on the cylinder vs. time t for the SC case and the three terrain cases. The figure presents that the peak value of C

_{Fn}in the cylinder of the N

_{1}case is significantly larger than that of the other three terrain cases, indicating that the bottom topography significantly changes the IW forces on the cylinder and makes the IW forces peak value C

_{Fn-max}change from positive to negative. Obvious negative peaks of C

_{Fn}appear in the three terrain cases N

_{2}, N

_{3}, and N

_{4}when the IWs are close to the front edge of the terrain. The reduction in the lower layer depth of IW due to the topographical factors causes a shallow-water effect, and the strength of the fluid flow field in the lower layer around the bottom terrain is enhanced. Moreover, the negative forces applied to the cylinder over the terrain result in the negative peak value of C

_{Fn}. As shown in Figure 8, the negative forces on the cylinder in the lower layer in cases N

_{2}, N

_{3}, and N

_{4}are much greater than those in case N

_{1}, which can also explain the difference in the peak value between N

_{1}and N

_{2}–N

_{4}in Figure 7. As a result, the shallow-water effect enhances the strength of the flow field around the cylinder in the lower layer, thereby causing a greater negative force on the lower parts of the cylinder.

#### 3.2. Comparison of the Vertical Distribution of the Force on the Cylinder in Different Cases

_{Fn}= C

_{Fn-max}of all the cases (as shown in Figure 8, t ≈ 25 s, the time of the most unfavorable forces on the cylinder) are selected for analysis. By further analyzing the stress mechanism of the cylinder over different terrains, the cylinder body is divided into ten parts along the vertical direction, and a section is taken at a separation of 0.02 m. The parts in the upper layer are defined as “upper parts”, and the parts in the lower layer are called “lower parts”. The pycnocline between the upper parts and lower parts is at the depth of 0.225 m. The vertical distribution of the non-dimensional horizontal forces C

_{f}at various water depths in all the cases are shown in Figure 8. The calculation expression of C

_{f}is similar to Equation (8) for C

_{Fn}. C

_{f}on the upper parts of the cylinder is positive, and the value in case N

_{1}condition is significantly greater than that in the other three cases. C

_{f}on the lower parts of the cylinder becomes negative and the values in cases N

_{2}, N

_{3}, and N

_{4}are much greater than that in case N

_{1}.

_{max}are the point pressure on the cylinder periphery and the maximum velocity, respectively; P

_{o}is the point hydrostatic pressure on the cylinder periphery; and ρ is the fluid density. The subscript y denotes a certain depth y. In this paper, the values of y are 0.26 m and 0.14 m, which are the center heights of the upper and lower layers, respectively. The schematic diagram for the definition of the circumferential angle is illustrated in Figure 9, where the angel degrees ranging from 90° to 270° form the cylinder windward surface area (frontal side), and the angel degrees ranging from 270° to 90° from the lee side.

_{p}on the cylinder at Plane

_{y=0.26}(P

_{0.26}for short, y = 0.26 m) and Plane

_{y=0.14}(P

_{0.14}for short, y = 0.14 m) sections.

_{1}is significantly greater than that in cases N

_{2}–N

_{4}, resulting in the forces on the upper parts in case N

_{1}being significantly greater than those in cases N

_{2}–N

_{4}.

_{2}–N

_{4}are much greater than those in case N

_{1}. Therefore, the reverse horizontal forces on the cylinder in cases N

_{2}–N

_{4}are much greater than those in case N

_{1}.

#### 3.3. Variation in the Flow Field around the Cylinder under Different Cases

_{1}) with the terrain cases (N

_{2}–N

_{4}) in Figure 13a–d, it is found that, when the IWs reach the bottom terrain, vortexes can be found near the frontal side of the terrain, but no obvious vortex appears in case N

_{1}. The vortexes are generated by the coupled effects of the IWs, the topography, and the shallow-water effect. Compared with the SC case, when the IWs propagate over the terrain, the interactions between the IWs and the terrain make the flow field around the cylinder more complex and changeable, especially near the lower parts of the cylinder. As a result, the complex hydrodynamic environment caused by the IWs and the terrains compels the cylinder to experience larger forces.

#### 3.4. Influence of the Amplitudes on the IWs Forces and the Flow Field over the Terrain

#### 3.4.1. Influence on the IWs Forces

_{Fn}vs. time t between the case S (SC) and the case F (flat-top-platform terrain) under different amplitudes conditions are shown in Figure 14. On the whole, with the increase of the IW amplitude, the forces on the single cylinder increase, and the duration curve of the IW forces on cylinder are all similar in all the cases. Under the small amplitude condition of η

_{0}/H = 0.0275 (see Figure 14a), the IW forces peaks C

_{Fn}

_{-max}for the case S

_{1}and the case F

_{1}are both positive. C

_{Fn-max}for the cases F

_{2}–F

_{5}turn negative as the amplitude increases, while C

_{Fn-max}for the cases S

_{2}–S

_{5}still keep positive (see Figure 14b–e). Meanwhile, a percentage parameter R

_{Fn-max}is applied here to specify the differences of C

_{Fn-max}between the cases S and the cases F, and the expression can be defined as follow:

_{Fn-max}decreases as the IW amplitudes η

_{0}/H increase from 0.0275 to 0.0575 but it sharply increases to 34.5% when η

_{0}/H reaches 0.0674. This can be explained by the flow field and density distribution characteristics as illustrated in Figure 15e and Figure 16e. When IW with large amplitude propagates to the bottom terrain, the interaction between the IW and the terrain is intensified and the shallow-water effect occurs, which strengthens the flow field strength near the terrain in the lower layer and influences the force on the cylinder. A phenomenon of “elevation” simultaneously occurs because of the “blockage” effect caused by the bottom topography, which directly reflects the influence of the existing terrain on the flow field near the cylinder.

#### 3.4.2. Influence on the Flow Field

## 4. Conclusions

- (1)
- The topographic factors of the terrain significantly affect the IW forces on the cylinder. There is a strong distinction between the SC case and the three terrain cases: in the SC case, the maximum resultant forces on the cylinder are positive, and the maximum resultant forces are negative in the terrain cases.
- (2)
- Compared with the SC case, the shallow-water effect caused by the IW-terrain coupled environment enhances the strength of the flow field around the cylinder, so that the lower parts of the cylinder are subjected to larger forces in the reverse wave direction.
- (3)
- Compared with the SC case, when the IWs propagate over the terrain, the interactions between the IWs and the terrain make the flow field around the cylinder more complex and changeable. As a result, the complex hydrodynamic environment compels the cylinder to experience larger forces.
- (4)
- A percentage parameter R
_{Fn-max}is applied in this research to specify the differences of C_{Fn-max}between the SC case and the terrain case. R_{Fn-max}decreases as the IW amplitude increases when the amplitude is relatively small, but it sharply increases when amplitude is large enough. It is can be explained by the shallow-water effect. When IWs with large amplitude propagate to the bottom terrain, the interaction between the IW and the terrain is intensified and the shallow-water effect occurs, which strengthens the flow field strength near the terrain in the lower layer. - (5)
- With the increase of IW amplitude, the interaction between the IW and the terrain is enhanced. Vortices can be found on the bank slope in all the cases, but the size of the vortices is obviously different when amplitude changes. The vortex size increases with the amplitude, and more than one vortex appears when the amplitude is large enough.
- (6)
- With the increase of the IW amplitude, the IW pattern is more strongly disturbed by the terrain. IW propagating over the bank slope is partially reflected, causing a “blockage” near the terrain and a “elevation” in the reverse wave propagation direction. Therefore, the intensification of the interaction strength between the IWs and the terrain could not only cause greater horizontal forces on the lower parts of the cylinder, but also make the flow field around the terrain more complex.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 2.**Schematic diagram of the physical model test [18].

**Figure 5.**Numerical model verification results of the IWs forces on a cylinder of different mesh densities, ƞ

_{0}/H = 0.0575.

**Figure 8.**Comparison diagram of layered horizontal resultant force under four working conditions, C

_{Fn}= C

_{Fn-max}.

**Figure 10.**Comparison of the pressure distribution between N1 and N

_{2}–N

_{4}. (

**a**) y = 0.26 m, (

**b**) y = 0.14 m.

**Figure 11.**Plots of vorticity around the cylinder at P

_{0.26}: (

**a**) N

_{1}, (

**b**) N

_{2}, (

**c**) N

_{3}, (

**d**) N

_{4}, when C

_{Fn}= C

_{Fn-max}.

**Figure 12.**Plots of vorticity around cylinder at P

_{014}: (

**a**) N

_{1}, (

**b**) N

_{2}, (

**c**) N

_{3}, (

**d**) N

_{4}, when C

_{Fn}= C

_{Fn-max}.

**Figure 13.**Plots of the flow field in Z direction, when C

_{Fn}= C

_{Fn-max}: (

**a**) N

_{1}, (

**b**) N

_{2}, (

**c**) N

_{3}, (

**d**) N

_{4}.

**Figure 14.**Comparison of C

_{Fn}vs. time t between the S case and the F case: (

**a**) η

_{0}/H = 0.0275, (

**b**) η

_{0}/H = 0.0384, (

**c**) η

_{0}/H = 0.0494, (

**d**) η

_{0}/H = 0.0575, (

**e**) η

_{0}/H = 0.0674.

**Figure 15.**Plots of the flow field in Z direction, when C

_{Fn}= C

_{Fn-max}: (

**a**) η

_{0}/H = 0.0275, (

**b**) η

_{0}/H = 0.0384, (

**c**) η

_{0}/H = 0.0494, (

**d**) η

_{0}/H = 0.0575, (

**e**) η

_{0}/H = 0.0674.

**Figure 16.**Plots of the density distribution in Z direction, when C

_{Fn}= C

_{Fn-max}: (

**a**) η

_{0}/H = 0.0275, (

**b**) η

_{0}/H = 0.0384, (

**c**) η

_{0}/H = 0.0494, (

**d**) η

_{0}/H = 0.0575, (

**e**) η

_{0}/H = 0.0674.

No. | Case | ∆t (s) | C_{Fn-max} | Elements Number |
---|---|---|---|---|

1 | T_{1} (low density) | 0.02 | 0.0810 | 525,454 |

2 | T_{2} (moderate density) | 0.01 | 0.0857 | 2,384,640 |

3 | T_{3} (high density) | 0.006 | 0.0862 | 3,318,278 |

No. | Case | h_{1}/h_{2} | η_{0}/H | C_{Fn-max} |
---|---|---|---|---|

1 | Single cylinder (N_{1}) | 0.33 | 0.0575 | 0.0857 |

2 | Bottom-step terrain (N_{2}) | 0.33 | 0.0575 | −0.0683 |

3 | Flat-top-knoll terrain (N_{3}) | 0.33 | 0.0575 | −0.0692 |

4 | Flat-top-knoll terrain (N_{4}) | 0.33 | 0.0575 | −0.0753 |

No. | Case | h_{1}/h_{2} | η_{0}/H | C_{Fn-max} | R_{Fn-max} |
---|---|---|---|---|---|

1 | S_{1} | 0.33 | 0.0275 | 0.0245 | 17.6% |

2 | F_{1} | 0.33 | 0.0275 | 0.0202 | |

3 | S_{2} | 0.33 | 0.0384 | 0.0428 | 17.5% |

4 | F_{2} | 0.33 | 0.0384 | −0.0353 | |

5 | S_{3} | 0.33 | 0.0494 | 0.0664 | 13.9% |

6 | F_{3} | 0.33 | 0.0494 | −0.0572 | |

7 | S_{4} | 0.33 | 0.0575 | 0.0857 | 12.1% |

8 | F_{4} | 0.33 | 0.0575 | −0.0753 | |

9 | S_{5} | 0.33 | 0.0674 | 0.132 | 34.5% |

10 | F_{5} | 0.33 | 0.0674 | −0.0864 |

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## Share and Cite

**MDPI and ACS Style**

Wang, Y.; Xu, M.; Wang, L.; Shi, S.; Zhang, C.; Wu, X.; Wang, H.; Xiong, X.; Wang, C.
Numerical Investigation of the Stress on a Cylinder Exerted by a Stratified Current Flowing on Uneven Ground. *Water* **2023**, *15*, 1598.
https://doi.org/10.3390/w15081598

**AMA Style**

Wang Y, Xu M, Wang L, Shi S, Zhang C, Wu X, Wang H, Xiong X, Wang C.
Numerical Investigation of the Stress on a Cylinder Exerted by a Stratified Current Flowing on Uneven Ground. *Water*. 2023; 15(8):1598.
https://doi.org/10.3390/w15081598

**Chicago/Turabian Style**

Wang, Yin, Ming Xu, Lingling Wang, Sha Shi, Chenhui Zhang, Xiaobin Wu, Hua Wang, Xiahui Xiong, and Chunling Wang.
2023. "Numerical Investigation of the Stress on a Cylinder Exerted by a Stratified Current Flowing on Uneven Ground" *Water* 15, no. 8: 1598.
https://doi.org/10.3390/w15081598