# Suppress Numerical Oscillations in Transient Mixed Flow Simulations with a Modified HLL Solver

^{*}

## Abstract

**:**

_{a}and P

_{b}, and their values can be determined easily. It can sufficiently suppress numerical oscillations under an acoustic wave speed of 1000 ms

^{−1}. Good agreement is found between simulation results and analytical results or experimental data. This paper can help readers to choose an appropriate oscillation-suppressing method for numerical simulations of flow regime transition under a realistic acoustic wave speed.

## 1. Introduction

## 2. Governing Equations and Discretization Method

_{b}is the bed slope and S

_{f}is the friction slope, which can be computed using the Manning relation:

_{sl}is slot width. In order to make the gravity wave speed inside the slot equal to the acoustic wave speed a, the slot width B

_{sl}= gA

_{f}a

^{−2}, where A

_{f}is the full cross-sectional area of the conduct [27]. Using the Godunov-type finite volume methods with first-order accuracy and assuming a piecewise constant data construction, the governing equations are discretized as

## 3. Review of Current Oscillation-Suppressing Methods

^{−1}and the slot width is 9.8 × 10

^{−6}m. Under the initial condition, 0.6 m-deep stagnant water is in the conduct while the water level inside the reservoir is constantly 4 m, as shown in Figure 2.

**U**

_{L}in the reservoir and

**U**

_{R}in the conduct are discontinuous:

_{L}is larger than h

_{R}, a shock wave (filling-bore) belonging to the second characteristic field is formed at the conduct inlet and propagates downstream. Consider 1–1 as a cross-section in the reservoir and 2–2 as a cross-section at conduct inlet: flow sates at 1–1 and 2–2 are

**U**

_{1}and

**U**

_{2}, respectively. Then

**U**

_{2}can be obtained by solving the following equations iteratively [24]:

**U**

_{1}=

**U**

_{L}by assuming that flow velocity inside the reservoir is negligible. In this case, h

_{2}and u

_{2}are 3.167 m and 4.0334 ms

^{−1}, a ghost cell is set at the upstream boundary adopting h

_{2}and u

_{2}. Since

**U**

_{2}is connected to

**U**

_{R}through a right shock, the flow states inside the conduct ultimately will take place by

**U**

_{2}. The propagation speed of the filling-bore is given by the Rankine–Hugoniot condition, which is 10.067 ms

^{−1}. Then we can construct the analytical result in the benchmark model at t

_{0}:

^{−1}, the magnitude of the numerical oscillations become so large that the simulated piezometric head become negative, and the simulation will not proceed [24]. In the remaining part of this section, the readers will see that only one method can get a satisfactory result under a high acoustic wave speed, while its performance rely on two parameters which must be well tuned. This shows the importance of devising an alternative method, which is stable and convenient.

#### 3.1. Numerical Filtering Method

**U**

_{i}

_{+1/2}at x

_{i}

_{+1/2}satisfy the following equations:

#### 3.2. Hybrid Flux Method

_{i}

_{+1/2}= 1, as L changes from 0 to 1, the eigenvalues switches from those obtained by the Roe solver to those adopted in the LxF scheme. At the other cell boundaries, Δc

_{i}

_{+1/2}is approximately 0, thus the eigenvalues in Equation (24) remain close to those obtained by the Roe solver. In this way, numerical viscosity is added at the cell boundary where flow condition transition happens, and its amount increases with L. The simulation results of the hybrid one with L = 0.6 are drawn in Figure 4. This method overestimates the spreading length of the filling-bore, and it fails to suppress the numerical oscillations under a high acoustic wave speed.

_{wR}and smallest wave speed S

_{wL}in the Riemann problem. The fluxes are computed based on the signs of S

_{wL}and S

_{wR}:

_{wL}and S

_{wR}follow Toro [31]:

_{i}

_{+1/2}is an estimate of the wetted area at x

_{i}

_{+1/2}; we adopt the one proposed by Leno et al. [32], which admits the minimum amount of numerical viscosity:

#### 3.3. Modified HLL Solver

_{wL}and S

_{wR}. In fact, the HLL fluxes equal the LxF fluxes when |S

_{wL}| and |S

_{wR}| equal Δx/Δt. To increase the amount of numerical viscosity, they proposed a modified HLL solver (referred to as the M_HLL solver). In the M_HLL solver, A

_{i}

_{+1/2}in Equation (29) is computed according to a reference depth h

_{G}:

_{G}is defined as K

_{a}, multiplying the highest piezometric head in the 2NS+1 cells surrounding cell i, while K

_{a}> 1 and NS ≥ 3. The solution of Equation (33) produces a larger magnitude of S

_{wL}and S

_{wR}; thus, increasing the numerical viscosity before the flow regime transition happens. The simulation results of M_HLL with K

_{a}= 1.4 and NS = 5 are drawn in Figure 6.

_{a}and NS can affect the diffusion and accuracy of the M_HLL solver to a great extent; see Figure 7.

_{a}and NS must be well-tuned. Meanwhile, the way to determine h

_{G}makes the HLL solver hard to use in parallelized computation. In the next section, the authors present an alternative method, which is equally efficient as the M_HLL solver.

## 4. A New Modified HLL Solver

_{i}

_{+1/2}depends on the water depths at cell i and i + 1. When h

_{i}and h

_{i}

_{+1}are below P

_{b}H, A

_{i}

_{+1/2}is computed using Equation (30) to admit the minimum amount of numerical viscosity, otherwise A

_{i}

_{+1/2}is computed according to a constant depth P

_{a}H:

_{b}must be smaller than one, and a preferable value is between 0.6 and 0.8. P

_{a}H must be larger than the piezometric head peak during the transition to admit enough numerical viscosity.

_{a}and P

_{b}, we study the Riemann problem at x

_{i}

_{+1/2}in the benchmark model. Suppose h

_{i}and h

_{i}

_{+1}is 3.167 m and 0.6 m, respectively, and u

_{i}and u

_{i}

_{+1}is 4.0334 ms

^{−1}and 0 ms

^{−1}, respectively. The solution of Equation (30) lies between A

_{i}and A

_{i}

_{+1}, and after substituting it into Equation (28), one will get S

_{wL}(noted as S

_{wL1}) as the speed of the left rarefaction wave head, and S

_{wR}(noted as S

_{wR1}) as the speed of right shock wave:

_{wL1}| nearly equals the acoustic wave speed. The entropy condition of right shock wave requires

_{wR1}equals the propagation speed of the filling-bore, which is 10.067 ms

^{−1}, and it is much smaller than the acoustic wave speed.

_{i}and A

_{i}

_{+1}, which produces two shock waves in the Riemann problem. Consequently, S

_{wR2}is the speed of the right shock wave, and S

_{wL2}is the speed of the left shock wave; see Figure 9.

_{i}<< c

_{i}; thus, |S

_{wL2}| > |S

_{wL1}| and they are both close to the acoustic wave speed. The speed of the right shock wave increases with A

_{i}

_{+1/2}; thus, S

_{wR2}> S

_{wR1}. This larger magnitude of S

_{wR}admits more mass and momentum into cell i + 1 before it becomes pressurized. The loci of

**U**

_{i}

_{+1}simulated by HLL and P_HLL (P

_{a}= 5, P

_{b}= 0.7) are drawn in Figure 10.

^{−1}velocity, which is very close to the flow states at the entrance of the conduct. The discrepancy in water depth is more pronounced due to the small value of the slot width. Therefore, P_HLL preserves the conservation in mass and momentum.

_{wR2}admits more mass and momentum into cell i + 1 before it is pressurized; thus, it increases the diffusion of the filling-bore.The magnitude of S

_{wL2}and S

_{wR2}are related to the value of A

_{i}

_{+1/2}, which consequently depends on the value of P

_{a}; see Figure 11 and Figure 12:

_{wR2}increases with P

_{a}, while|S

_{wL2}| barely changes with P

_{a}and stays close to the acoustic wave speed. However, S

_{wR2}does not increase significantly when P

_{a}changes from 1 to 10, which denotes that the diffusion of the filling-bore does not increase significantly when P

_{a}changes from 1 to 10. The simulation results using the P_HLL solver with P

_{b}= 0.7 and different values of P

_{a}are drawn in Figure 13.

_{a}= 10 produces a more diffusive filling-bore, the spreading length of the filling-bore does not increase significantly compared to that using P

_{a}= 5. During a realistic transition phenomenon, the piezometric head peak seldom exceeds 10 times the cross-sectional depth. Therefore, one can always start by adopting a large P

_{a}(for example 10) in the P_HLL solver and do not worry about significantly compromising the representation of the filling-bore.

_{a}, the value of P

_{b}has a more significant effect on the numerical oscillations, for it determines the threshold depth where numerical viscosity starts to increase. P

_{b}must be smaller than one so that the numerical viscosity is added before the flow regime transition happens. A smaller P

_{b}leads to more stable result, but it may cause more diffusion. The simulation results using P

_{a}= 5 and P

_{b}= 0.7 or 0.9 are drawn in Figure 14.

_{b}= 0.9, a smooth transition between the free-surface and pressurized flows cannot be guaranteed. Therefore, we suggest a P

_{b}between 0.6 and 0.8 to avoid causing two much diffusion of the filling-bore. This is also supported by the numerical tests in the next section.

_{b}H. Thus, a smooth transition from the free-surface flow to pressurized flow can be obtained. Meanwhile, P_HLL causes diffusion of the filling-bore. In realistic applications, a P

_{a}of 10 is large enough to suppress numerical oscillations without significantly increasing the spreading-length of the filling-bore. The value of P

_{b}is suggested to be between 0.6 and 0.8.

## 5. Numerical Tests

#### 5.1. Two Filling-Bores

^{−1}. At initial conditions, water in the conduct is stationary with a depth of 0.6 m, while water depth at the upstream and downstream reservoir is 4 m and 3 m, respectively. The model set up is sketched in Figure 15.

^{−1}, and the flow states behind it are 2.42 m and −3.3717 ms

^{−1}. Boundary conditions adopt ghost cells at the conduct inlet and outlet. Before the two filling-bores collide with each other, the analytical result at t

_{0}is

_{a}= 5 and P

_{b}= 0.8, optionally. In M_HLL, we choose K

_{a}= 1.4 and NS = 5 as suggested by Malekpour and Karney. The computational cell is 1 m, time step is 0.0008 s and the Courant number is 0.8. The simulation results in the two tests at t = 6 s are drawn in Figure 16. In this paper, an error indicator based on the L

_{2}-norm [34] is used to evaluate the accuracy of P_HLL and M_HLL. In the following equation, φ

_{i}stands for the simulation result at cell i, while φ

_{ref}stands for the analytical result.

_{2}in the piezometric head of P_HLL and M_HLL are 0.2963 and 0.2913, respectively; the L

_{2}in the velocity of P_HLL and M_HLL are 0.2879 and 0.2873, respectively. In P_HLL, the spreading length of the right filling-bore is slightly longer than that in M_HLL. This denotes that P_HLL is more diffusive than M_HLL in there. At the same time, P_HLL has eliminated some minor numerical oscillations while M_HLL does not. Both solvers are very robust and stable.

#### 5.2. Negative Pressure Flow

^{−1}. The pipe is horizontal and frictionless; a steady flow rate of 0.477 m

^{3}s

^{−1}is initially in it. It connects to a reservoir at the downstream end, and water depth inside it is 45 m; see Figure 17.

^{3}s

^{−1}, which triggers a water hammer phenomenon; the water hammer pressure is 48.05 m according to Kerger et al. [26]. In P_HLL, the values of P

_{a}and P

_{b}are 100 and 0.8. In M_HLL, the values of K

_{a}and NS are 1.2 and 12, as suggested by Malekpour and Karney. The computational cell is 1.2 m, the time-step is 0.0008 s and Courant number is 0.8. A ghost cell is set at the upstream boundary, and flow rate in it is constantly 0.4 m

^{3}s

^{−1}, while the piezometric head adopts the transmissive condition. Another ghost cell is set at the downstream boundary in which the piezometric head is constantly 45 m and the flow rate adopts the transmissive condition.

_{2}indicator is adopted to evaluate the accuracy of the two solvers; it is defined as

_{t}is the number of time step, φ

_{i}is the simulation result at the midpoint of the pipe, and φ

_{ref}is the analytical result, which is given as

_{2}in the piezometric head of P_HLL and M_HLL are 6.3965 and 6.3970, respectively, while L

_{2}in the velocity of P_HLL and M_HLL are 0.1332 and 0.1333, respectively.

_{a}; for example, the simulation result using P

_{a}= 1 is drawn in Figure 19.

_{a}will produce a wave speed that is close to the acoustic wave speed, provided that the cell is under pressurized flow condition; see Figure 12 for detail. In this test, all the computational cells are under a pressurized flow condition, which makes the simulation result of P

_{a}= 100 and P

_{a}= 1 almost the same. In the flow regime transition simulation, P

_{a}H must be larger than the highest piezometric head.

#### 5.3. Vasconcelos’s Experiment

^{−1}water into the fill box, and when water level inside the fill box reaches its top, water is simply spilled away. A gate is installed at the tunnel outlet; its opening is smaller than the initial water depth. When the filling bore collides with the gate, it triggers a water-hammer phenomenon. A ventilation tower is fixed upstream of the gate so that no air is trapped in the tunnel. The experiment setup is drawn in Figure 20.

^{1/6}, acoustic wave speed is 300 ms

^{−1}and head loss coefficient associated with the partially opened gate is 12, as suggested by Malekpour and Karney. In P_HLL, the values of P

_{a}and P

_{b}are 5 and 0.8. In M_HLL, the values of K

_{a}and NS are 1.2 and 12. The computational cell is 0.1 m, and the time-step is set for a Courant number of 0.8. At the upstream end, the three unknowns are the discharge, the wetted area at the tunnel inlet and the water level in the fill box. At the downstream end, the three unknowns are discharge, the wetted area at tunnel outlet and the water level in the surge tank. The continuity, energy and characteristic equations are applied to obtain the three unknowns at each boundary [36]. The loci of flow states at x = 9.9 m in the simulation results of P_HLL and M_HLL are shown in Figure 21 and Figure 22.

#### 5.4. Aureli’s Experiment

_{a}and P

_{b}are 4 and 0.7; in M_HLL, the values of K

_{a}and NS are 1.4 and 5. The computational cell is 0.04 m, acoustic wave speed is 200 ms

^{−1}and time-step is set for a Courant number of 0.8. A reflective boundary condition was set at the upstream end, while a transmissive boundary condition was set at the downstream end. At the wet/dry interface, the estimates of wave speed followed Leon et al. [27]. The loci of the flow states at x = 6.8 m simulated by P_HLL and M_HLL are drawn in Figure 24 and Figure 25.

_{b}H, while M_HLL adds numerical viscosity at any free-surface cells. Nonetheless, the simulation results of the two solvers are in good agreement with the experimental data.

## 6. Conclusions

_{a}and P

_{b}. P

_{a}needs to be larger than the highest piezometric head while P

_{b}needs to be between 0.7 and 0.9. This solver adds numerical viscosity when the water depth is above P

_{b}H so that a smooth transition between the free-surface and pressurized flows can be achieved. The amount of numerical viscosity increases with P

_{a}, while a large P

_{a}does not smear the simulation result significantly. Therefore, one can always start by adopting a P

_{a}that is large enough under realistic applications, for example 10.

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**A rectangular conduct with a slot on its top: (

**a**) free surface flow; (

**b**) pressurized flow.

**Figure 3.**Comparison of results simulated by the numerical filtering method (filtered) and the analytical result (AR): (

**a**) piezometric head; (

**b**) flow velocity.

**Figure 4.**Comparison of the results simulated by hybrid one and the analytical result (AR): (

**a**) piezometric head; (

**b**) flow velocity.

**Figure 5.**Comparison of the results simulated by hybrid two and the analytical result (AR): (

**a**) piezometric head; (

**b**) flow velocity.

**Figure 6.**Comparison of the results simulated by M_HLL, with K

_{a}= 1.4 and NS = 5, and the analytical result (AR): (

**a**) piezometric head; (

**b**) flow velocity.

**Figure 7.**Comparison of the results simulated by M_HLL, with K

_{a}= 1.4 and NS = 3, and the analytical result (AR): (

**a**) piezometric head; (

**b**) flow velocity.

**Figure 10.**(

**a**) Locus of the flow states in cell i + 1 simulated by HLL and P_HLL (P

_{a}= 5, P

_{b}= 0.7); (

**b**) history of the flow states in cell i + 1 simulated by P_HLL (P

_{a}= 5, P

_{b}= 0.7).

**Figure 13.**Comparison of the results simulated by P_HLL, with P

_{b}= 0.7 and P

_{a}= 5 or P

_{a}= 10, and the analytical result (AR): (

**a**) piezometric head; (

**b**) flow velocity.

**Figure 14.**Comparison of the results simulated by P_HLL, with P

_{a}= 5 and P

_{b}= 0.7 or P

_{b}= 0.9, and the analytical result (AR): (

**a**) piezometric head; (

**b**) flow velocity.

**Figure 16.**Comparison of the results simulated by P_HLL (P

_{a}= 5 and P

_{b}= 0.8), M_HLL (K

_{a}= 1.4 and NS = 5) and the analytical result (AR): (

**a**) piezometric head; (

**b**) flow velocity.

**Figure 18.**Locus of the flow states at the midpoint of the pipe simulated by P_HLL (P

_{a}= 100 and P

_{b}= 0.8), M_HLL (K

_{a}= 1.2 and NS = 12) and the analytical result (AR): (

**a**) piezometric head; (

**b**) flow velocity.

**Figure 19.**Locus of the flow states at the midpoint of pipe simulated by P_HLL (P

_{a}= 1 and P

_{b}= 0.8) and the analytical result (AR): (

**a**) piezometric head; (

**b**) flow velocity.

**Figure 21.**Locus of flow states at x = 9.9 m simulated by P_HLL (P

_{a}= 5 and P

_{b}= 0.8) and experimental data: (

**a**) piezometric head; (

**b**) flow velocity.

**Figure 22.**Locus of flow states at x = 9.9 m simulated by M_HLL (K

_{a}= 1.2 and NS = 12) and experimental data: (

**a**) piezometric head; (

**b**) flow velocity.

**Figure 23.**A sketch of the experimental setup, drawn by Aureli et al. [25].

**Figure 24.**Locus of flow states at x = 6.8 m simulated by P_HLL (P

_{a}= 4 and P

_{b}= 0.7) and experimental data: (

**a**) piezometric head; (

**b**) flow velocity.

**Figure 25.**Locus of flow states at x = 6.8 m simulated by M_HLL (K

_{a}= 1.4 and NS = 5) and experimental data: (

**a**) piezometric head; (

**b**) flow velocity.

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## Share and Cite

**MDPI and ACS Style**

Mao, Z.; Guan, G.; Yang, Z.
Suppress Numerical Oscillations in Transient Mixed Flow Simulations with a Modified HLL Solver. *Water* **2020**, *12*, 1245.
https://doi.org/10.3390/w12051245

**AMA Style**

Mao Z, Guan G, Yang Z.
Suppress Numerical Oscillations in Transient Mixed Flow Simulations with a Modified HLL Solver. *Water*. 2020; 12(5):1245.
https://doi.org/10.3390/w12051245

**Chicago/Turabian Style**

Mao, Zhonghao, Guanghua Guan, and Zhonghua Yang.
2020. "Suppress Numerical Oscillations in Transient Mixed Flow Simulations with a Modified HLL Solver" *Water* 12, no. 5: 1245.
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