# Direct Numerical Simulations on Jets during the Propagation and Break down of Internal Solitary Waves on a Slope

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Numerical Models

#### 2.1. Momentum Equations and Continuity Equation

_{i}is the spatial coordinate; u is the fluid velocity; p is the pressure; ν is the kinematic viscosity, and f is the body force.

#### 2.2. Scalar Transport Equation

_{low}+ (1 − C)ρ

_{up}, where ρ

_{low}and ρ

_{up}are the designed fluid density in the two-layer system, respectively. The other scalar concentration carried by the jet, also prescribed by Equation (3), has no influence on the flow density when the concentration is relatively small [18]. This means that it has no influence on the flow development that is governed by Equations (1) and (2).

#### 2.3. Numerical Methods

## 3. Model Verifications

_{0}= 2.0 m and z

_{0}= 0.25 m as corresponding to the coordinate systems in Figure 1. The diameter of the round jet d

_{0}is 10.0 mm. The dimensionless length scale of the nozzle is 0.2 when normalized with the water’s depth. The jet velocity V

_{jet}is 0.58 m/s in this case, and the background free stream velocity remains constant at 0.055 m/s during the simulation. The scalar concentration carried by the jet is small, and its effects on the flow density can be neglected according to Xu’s laboratory experiments [18]. The computational domain is discretized using a rectangular grid of 8192 × 128 × 256 points corresponding to the x × y × z directions, respectively, which makes sure that there are about 50 grid points in the jet inlet. The bottom and side walls are set as non-slip, and the rigid-lid assumption is used at the free surface. The inflow boundary condition is implemented on the left face of the computational domain, and an advection outflow condition is employed on the right face.

_{jet}); N is the number of the jet modes (set to 6 in this case); the f is the frequency determined by the Strouhal number (0.3 in this case); θ is the phase angle; all these parameters are set with reference to Xu and Chen’s study [18,24].

_{0}), which means that the center of the jet inlet is at X = 0 after the translation. The results of the mean streamwise velocity <U> and the mean vertical velocity <V> are shown, respectively, in Figure 2a,b. Velocities U0, V0 are nondimensionalized by the jet mean inlet velocity V

_{jet}, and the horizontal axis is the vertical length scale Y0 nondimensionalized by the diameter of the round diameter inlet D.

## 4. Numerical Results

_{0}and the position of the jet x

_{0}vary as listed. In addition, the Richardson and Reynolds numbers are based on the jet or plume characteristics (width and vertical velocity) before impingement on the ISW and the initial density difference across the interface. They can be calculated by the following equations:

**x**

**, Δ**

^{+}**y**

**). Furthermore, the maximum of Re is 10**

^{+}^{4}in these cases, which is noted as the low Reynold number cases. Camussi’s study showed that in the high Reynold number cases (where the Re is larger than 10

^{5}), there are many intermittent coherent structures in the near field, while it is not a significant phenomenon in the low Reynold number cases [28]. Actually, this paper is mainly focused on the effects between jets and ISWs, so the grid is fine enough to capture the jets with low Reynold numbers under the condition that turbulence of jets is not significant in our cases.

_{0}= 0.15 m) is carried out to ensure reliability of the numerical simulated results with the specific grid sizes. The parameters of different cases are listed in Table 2.

#### 4.1. Jets Impinging on an Interface

_{jet}/h

_{2}(where the h

_{2}is the depth of the lower layer). The contours of the fluid density are plotted in Figure 4.

_{jet}is the kinetic energy contributed by the jet at a t time. Consequently, the ΔKE

_{0}means the change in kinetic energy between the two time steps normalized by the contribution from the jet. Figure 5 plots ΔKE

_{0}variation with the dimensionless time scale T

_{0}.

_{0}reaches the minimum, and the interface absorbs about 45%, 40% and 80% energy from the jet flow for Cases P1, P2 and P3, respectively. In Cases P2 and P3, with the bolus breaking, the kinetic energy grows sharply, while in Case P1, there is only a little increase of kinetic energy after T = 3 because the bolus does not break in this case. In Cases P1 and P2, the interface absorbed a large amount of kinetic energy, with the development of the impinging jet. Meanwhile, in Case P3, the kinetic energy fluctuation results from a strong mixing at the interface. Due to the strong mixing, the energy transformation between the potential and kinetic energy frequently occurs during the impingement of the jet on the internal interface.

#### 4.2. The Effects on the ISWs by the Jet

#### 4.3. The Dilution and Transport of Jet by an ISW Encountering a Slope

_{s}is defined as the distance from the center of the jet’s nozzle to the toe of the slope, namely l

_{s}= 15 – x

_{0}.

_{1}= 13 m. On the other hand, the C2 scalar, which is transported by the bottom shedding vortices, contains more concentration than C1, and it decays at x

_{2}= 11.7 m.

_{1}and l

_{2}are defined as corresponding to x

_{1}and x

_{2}; that is l

_{1}= x

_{1}− x

_{0}, l

_{2}= x

_{2}− x

_{0}. In the case with specific Re numbers, Ri numbers and the amplitudes of the ISWs, the l

_{1}and the l

_{2}are 3.0 m and 1.7 m, respectively. The length l

_{1}can be seen as the farthest distance where the jet can transport with the ISW and the l

_{2}is the farthest distance where the bottom vortices induced by the ISW and the jet are able to transport from the jet.

_{s}= 0.2 m), the peak value of M is 9.7 times as much as that in the Case S2. The results show that at t = 42 s, the M increases with the jet’s nozzle approaching the slope. In Cases S3–S5, the M has a sharp increase at t = 32 s when the ISWs begin to run up (seen in Figure 11a). At the end of the ISWs process, it leaves a considerable scalar concentration on the slope, and the concentration within the interface accounts for most of it. On the basis of this analysis, Cases S2–S5 can be divided into two groups in terms of the l

_{s}. The first group is Case S2, where l

_{s}is larger than l

_{2}but is smaller than l

_{1}. When l

_{s}is less than l

_{2}, this is the second group.

_{s}is too large for C2 to be transported, and the same holds for the vortices. As a result, there is little scalar concentration left on the slope at the end.

## 5. Conclusions

- (1)
- When the horizontal length scale of the interface is far larger than the jet nozzle, the stratified interface can exist under the conditions of low jet velocity with strong differences of density, corresponding to Re < 10,000, Ri > 3.7. This is because most of the kinetic energy from the jet flows will be absorbed by the interface and transfers to potential energy of the interface.
- (2)
- There are vortices at the bottom behind the crest and at a distance from the center of the jet inlet. This observation implies that the vortices result from the joint effects of the curved jets and the flow velocities induced by the ISWs. When the ISWs propagate, the scalar concentrations carried by jet are transported in two ways: moved by the interface or moved by the bottom vortices.
- (3)
- Two transport lengths of the two ways were related to the scalar transport by the interface and by the bottom vorticity. In the period of the ISWs running up and down, the integrals of scalar concentration over the slope reach the maximum at the end of the ISWs running up the slope. That concentration integral also grows as the jet approaches the slope.
- (4)
- It can be concluded that when the distance from the center of the jet to the slope toe l
_{s}is smaller than the l_{2}, there are vortices induced jointly by the jet and the ISW on the slope, leading to a strong mixing of water. As a result, more scalar concentration is left on the slope.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**(

**a**) The distribution of normalized streamwise mean velocities along different vertical lines in the xy-plane of z = 0.25 m. (Dots are the experiment results, U0 = <U>/V

_{jet}and Y0 = y/D). (

**b**) The distribution of normalized vertical mean velocities along different vertical lines in the xy-plane of z = 0.25 m. (Dots are the experiment results, V0 = <V>/V

_{jet}and Y0 = y/D).

**Figure 4.**Instantaneous density contours of the mid-plane of the computational domain during the jet impinging on the internal interface in three cases. (

**a**) P1 (Re = 2500, Ri = 5.89); (

**b**) P2 (Re = 5000, Ri = 1.47); (

**c**) P3 (Re = 10,000, Ri = 0.37) (Unit of axes: meter).

**Figure 6.**Instantaneous density contour plots illustrating the jet effect on the ISW in two cases. (

**a**) I1 (Re = 2500, Ri = 5.89, a = 0.15 m); (

**b**) I4 (Re = 10,000, Ri = 0.37, a = 0.15 m); (Unit of axes is meter; Unit of time is second).

**Figure 7.**Instantaneous vorticity contour plots in four cases. (

**a**) I2 (Re = 5000, a = 0.15 m); (

**b**) I4 (Re = 10,000, a = 0.15 m); (

**c**) I6 (Re = 5000, a = 0.10 m); (

**d**) I7 (Re = 10,000, a = 0.10 m); (Unit of axes is meter).

**Figure 8.**Instantaneous scalar transport of the jet during the propagation of the ISWs in Case S1 (l

_{s}= 5 m). (Unit of axes is meter; Unit of time is second).

**Figure 11.**Results of the ISWs running up. (

**a**) the instantaneous density contour in Case S2; Scalar concentration contours and velocity vectors (

**b**) in Case S2 (l

_{s}= 3.8 m) and (

**c**) in Case S5 (l

_{s}= 0.2 m) at t =42 s. (Unit of axes is meter; Unit of time is second).

Family of Case | Re(10^{3}) | Ri | a_{0}(m) | x_{0}(m) | Slope |
---|---|---|---|---|---|

P | 2.5–10 | 0.37–5.89 | NO | 10 | No |

I | 2.5–10 | 0.37–5.89 | 0.1–0.15 | 10 | No |

S | 5.0 | 1.47 | 0.15 | 10–14.8 | 0.5 |

Case | Grids | Δx^{+} | Δy^{+} |
---|---|---|---|

T1(coarse) | 4096 × 128 × 16 | 3.4 | 4.6 |

T2(moderate) | 8192 × 256 × 32 | 1.7 | 2.3 |

T3(fine) | 16384 × 512× 16 | 0.9 | 1.2 |

y/D | T1 | T2 | T3 |
---|---|---|---|

0.07 | 0.18 | 0.18 | 0.18 |

1.01 | 2.03 | 1.92 | 1.90 |

2.03 | 9.13 | 8.03 | 8.12 |

3.05 | 12.54 | 10.44 | 10.39 |

4.06 | 52.26 | 49.17 | 49.07 |

5.07 | 62.33 | 59.86 | 59.87 |

Decay of Amplitude of the ISW | 1.98 | 1.98 | 1.98 |

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**MDPI and ACS Style**

Xu, J.; Avital, E.J.; Wang, L. Direct Numerical Simulations on Jets during the Propagation and Break down of Internal Solitary Waves on a Slope. *Water* **2020**, *12*, 671.
https://doi.org/10.3390/w12030671

**AMA Style**

Xu J, Avital EJ, Wang L. Direct Numerical Simulations on Jets during the Propagation and Break down of Internal Solitary Waves on a Slope. *Water*. 2020; 12(3):671.
https://doi.org/10.3390/w12030671

**Chicago/Turabian Style**

Xu, Jin, Eldad J. Avital, and Lingling Wang. 2020. "Direct Numerical Simulations on Jets during the Propagation and Break down of Internal Solitary Waves on a Slope" *Water* 12, no. 3: 671.
https://doi.org/10.3390/w12030671