# SPH Simulations of Solute Transport in Flows with Steep Velocity and Concentration Gradients

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. Shallow Water Equations

_{w}is the water depth, b is the bottom elevation, ${S}_{f}={n}^{2}v\left|v\right|/{d}_{w}{}^{4/3}$ is the bed friction, n is the Manning roughness coefficient, and g is the gravity acceleration. Because the fluid motion is assumed to be incompressible, the water density ${\mathsf{\rho}}_{w}$ is uniform and constant. Furthermore, the water depth and the water density are related by $\mathsf{\rho}={\mathsf{\rho}}_{w}\cdot {d}_{w}$ where $\mathsf{\rho}$ is defined as the density of particle [18]. Figure 1 illustrates the sketch of SPH with SWEs.

#### 2.2. Advection–Diffusion Equations

_{u}is the diffusion coefficient, and $\mathsf{\rho}$ is the density of particle. The concentration C is assumed to be dilute and the flow field is not affected by the concentration field.

#### 2.3. SPH-SWEs-ADEs Model

#### 2.3.1. Smoothed Particle Hydrodynamics (SPH) Formulation

#### 2.3.2. Approximated Momentum Equations

_{i}is the pressure wave speed (=$\sqrt{g{d}_{w,i}}$), ${\overline{\mathsf{\rho}}}_{ij}=0.5\left({\mathsf{\rho}}_{i}+{\mathsf{\rho}}_{j}\right)$, and $\mathsf{\eta}=0.01\sqrt{\Delta {x}_{0}^{2}+\Delta {y}_{0}^{2}}$.

_{bp}is equal to $\Delta {x}_{0}$ in 1D and $\Delta {x}_{0}\cdot \Delta {y}_{0}$ in 2D.

#### 2.3.3. Water Depth Evaluation

#### 2.3.4. Approximated Advection–Diffusion Equations

_{u}is discontinuous over the computational domain, it is replaced $({D}_{ui}+{D}_{uj})$ by

_{u}will be a tensor and has the elements of D

_{uxx}, D

_{uxy}, D

_{uyx}, and D

_{uyy}[30]. Most importantly the concentration field is computed at the same time with the flow field, which means that the SWEs and the ADE are solved together to obtain the modeling results. However, these two fields are not coupled because the solute concentration is assumed to be diluted; thus, the concentration field will not disturb the flow field.

#### 2.3.5. Time Integration Scheme

#### 2.4. Boundary Conditions

#### 2.4.1. In/Out-Flow Boundaries

#### 2.4.2. Wall Boundary

## 3. Results and Discussion

#### 3.1. Steep Concentration Gradients

#### 3.1.1. A Top Hat Tracer in a 1D Uniform Flow

^{3}and C(10,000, t) = 0 kg/m

^{3}, while a water depth of 0.5 m and a water velocity of 0.5 m/s are prescribed as the inflow boundary condition. The total number of fluid particles is 100 and the initial time step is 100 s. The exact solution is given from [14].

#### 3.1.2. A Unit-Step Tracer in a 1D Uniform Flow

^{2}/s, 2 m

^{2}/s, and 50 m

^{2}/s are adopted to give three various diffusion scenarios as advection-dominated, advection–diffusion, and diffusion-dominated, respectively. To solve ADEs, the initial conditions are provided as

^{3}and C(20,000, t) = 0 kg/m

^{3}are given as the boundary conditions.

^{2}/s, 2 m

^{2}/s, and 50 m

^{2}/s at t = 9600 s are respectively given in Figure 3. It can be found that the moving speed of the leading edge of Figure 3c owing to the effects of advection and diffusion is larger than the one of Figure 3a resulting from the effect of pure advection. Furthermore, the simulated results show good agreement against the exact solutions obtained in [33]. It can be concluded that the present model is capable of solving steep gradient or discontinuous solutions in concentration fields of advection-dominated, advection–diffusion, and diffusion-dominated problems. In Figure 3, the symbol Pe denotes the numerical Peclet number, which is defined as $u\Delta x/{D}_{u}$.

#### 3.1.3. A Circular Tracer in a 2D Uniform Flow

_{u}= 0 m

^{2}/s and 1 × 10

^{5}m

^{2}/s. The rough channel of length 1000 m and width 400 m has the channel slope of 0.001. The Manning roughness coefficient is 0.0316 s/m

^{1/3}representing the land use pattern of cultivated areas in floodplains. The water velocity u

_{in}of 2.929 m/s at the inflow boundary and the water depth of 5 m at the outflow boundary are given, while a circle region of radius of 50 m with material concentration 1 kg/m

^{3}centered at (x, y) = (250 m, 200 m) is set to be the initial concentration condition.

_{in}, 120 s × u

_{in}, and 200 s × u

_{in}at t = 50 s, 120 s, and 200 s, respectively, because of the absence of the material diffusion. Next, Figure 4b gives the locations of the leading edge are x = 300 m, 476 m, 702 m and 945 m at t = 0 s, 50 s, 120 s, and 200 s, respectively. Since the transport behavior of the tracer is subject to the coupled advection and diffusion reaction, the displacements of the leading edge in Figure 4b are significantly larger than those in Figure 4a. Furthermore, Figure 4 also demonstrates that the present model still has the ability to preserve the shape of the tracer in 2D cases.

#### 3.2. Steep Concentration Gradients Coupled with Steep Velocity Gradients

#### 3.2.1. A Periodic Top Hat Tracer in a 1D Transcritical Flow

#### 3.2.2. A Unit-Step Tracer in a 1D Dambreak Flow

_{u}of 100 m

^{2}/s is also conducted herein [4].

^{3}is initially placed at the upstream boundary [22]. The initial particle spacing of 0.5 m and the initial time step of 0.1 s are used. Figure 6 is the simulated profiles of water depth and concentration at t = 2 s. The numerical solutions of water depth shown in Figure 6a give good agreement with the exact solution. Figure 6b illustrates that the concentration of each particle remains constant corresponding to D

_{u}= 0 m

^{2}/s.

^{3}is imposed at the upstream, while a 1000 m long and 0.2 m high water with a concentration of 0.5 kg/m

^{3}is placed at the downstream. The initial particle spacing of 1 m and the initial time step of 1 s are used herein. The exact solution is given in [4]. Figure 7 shows the simulated profiles of water depth and concentration. In Figure 7, the proposed model provides good predictions in both water depth and concentration against the exact ones. Nevertheless, the entire simulated concentration profile given in [4] is smoother than the exact one. Thus, the present model can work well under the condition existing discontinuities both in the flow and concentration filed.

#### 3.2.3. A Circular Tracer in a 2D Dambreak Flow

^{2}/s and 10 m

^{2}/s are considered. A water column of a 10 m length, a 5 m width, and a 0.7 m depth is imposed initially, and a circular tracer with a radius of 1 m and a concentration of 1 kg/m

^{3}is released in the water column with its center located at (2.5 m, 5 m). The numerical test is performed using the initial particle spacing of 0.1 m in both x and y directions and the initial time step of 0.005 s.

^{2}/s and 10 m

^{2}/s, respectively. Since the solute transport is affected by the flow condition, the shape of the tracer is deformed as the flow approaches the mound and is also separated in Figure 10. By contrast, in Figure 11, besides the shape deformation and flow separation, the concentration of the tracer also occurs the diffusion phenomenon due to the non-zero diffusion coefficient. Based on the results in this study case, it shows that the present model can perform on solute transports with steep concentration gradients in steep velocity gradients flows.

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 3.**The simulated profiles of concentration for the second study case: (

**a**) D

_{u}= 0 m

^{2}/s, (

**b**) D

_{u}= 2 m

^{2}/s, and (

**c**) D

_{u}= 50 m

^{2}/s.

**Figure 4.**The simulated profiles of concentration for the third study case: (

**a**) D

_{u}= 0 m

^{2}/s and (

**b**) D

_{u}= 10

^{5}m

^{2}/s.

**Figure 5.**The simulated profiles of the fourth study case: (

**a**) water depth; (

**b**) water velocity; and (

**c**) concentration.

**Figure 6.**The simulated profiles for the case of a dry-bed dambreak flow in the fifth study case: (

**a**) water depth and (

**b**) concentration.

**Figure 7.**The simulated profiles for the case of a wet-bed dambreak flow in the fifth study case: (

**a**) water depth and (

**b**) concentration.

**Figure 9.**The simulated profiles for the sixth study case: (

**a**) water depth and (

**b**) water velocity vector.

**Figure 10.**The temporal simulated profiles of concentration for the case without material diffusion in the sixth study case: (

**a**) 0 s, (

**b**) 2 s, (

**c**) 4 s, (

**d**) 6 s, (

**e**) 8 s, and (

**f**) 10 s.

**Figure 11.**The temporal simulated profiles of concentration for the case with material diffusion in the sixth study case: (

**a**) 0 s, (

**b**) 2 s, (

**c**) 4 s, (

**d**) 6 s, (

**e**) 8 s, and (

**f**) 10 s.

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**MDPI and ACS Style**

Chang, Y.-S.; Chang, T.-J.
SPH Simulations of Solute Transport in Flows with Steep Velocity and Concentration Gradients. *Water* **2017**, *9*, 132.
https://doi.org/10.3390/w9020132

**AMA Style**

Chang Y-S, Chang T-J.
SPH Simulations of Solute Transport in Flows with Steep Velocity and Concentration Gradients. *Water*. 2017; 9(2):132.
https://doi.org/10.3390/w9020132

**Chicago/Turabian Style**

Chang, Yu-Sheng, and Tsang-Jung Chang.
2017. "SPH Simulations of Solute Transport in Flows with Steep Velocity and Concentration Gradients" *Water* 9, no. 2: 132.
https://doi.org/10.3390/w9020132