# A Simple and Unified Linear Solver for Free-Surface and Pressurized Mixed Flows in Hydraulic Systems

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. Governing Equations

_{m}= Manning’s roughness coefficient; and R = hydraulic radius.

_{p}) when the water level is equal to or larger than the celling of the cross-section (z

_{c}).

#### 2.2. Computational Grid and Description of Mixed Flows

_{i}represents the length of the cell; B

_{i}is the wet surface width and is related to the cell water level that varies with respect to time; A

_{i}represents the conveyance area of the cross-section located at the center of the cell. For side i + 1/2 which connects cells i and i + 1, Δx

_{i}

_{+1/2}is the distance between the centers of cells i and i + 1; A

_{i}

_{+1/2}and R

_{i}

_{+1/2}are, respectively, the conveyance area and the hydraulic radius of side i + 1/2, which are interpolated using those of cells i and i + 1.

#### 2.3. Numerical Formulation

#### 2.3.1. Discretization of Governing Equations

_{bt}” is the number of substeps and $\stackrel{\rightharpoonup}{u}$ is the velocity vector at the starting point of each substep. When a tracking point has been found, the velocity at this point is interpolated based on the velocity field of time level n, for calculating the next displacement and searching the next tracking point. Once the location of the foot of the trajectory at time level n is found, the initial condition at that time step is interpolated and recorded as u

_{bt}. The cell-face velocity is then directly updated with u

_{bt}, which means that the advection term is solved.

_{bt}and u

^{n}

^{+1}to enhance computational stability.

^{n}

^{+1}) emerge and Equation (4) is then transformed into the following form:

_{i}is assumed to be constant within the cell. Moreover, the wet surface width of cell i (B

_{i}) is explicitly represented by the value of time level n (${B}_{i}^{n}$) within a time step (Δt). From time level n to n + 1, a finite-volume discretization of the continuity equation, Equation (3), is given by:

_{i}and A

_{i}

_{±1/2}) are present, respectively, on the left hand side (LHS) and the right hand side (RHS) of Equation (6). However, when the B

_{i}on the LHS and the A

_{i}

_{±1/2}on the RHS have been explicitly represented by ${B}_{i}^{n}$ and ${A}_{i\pm 1/2}^{n}$ (within the Δt, from time level n to n + 1), the solution problem associated with the nonlinear terms does not exist any longer.

#### 2.3.2. Solution of Velocity–Pressure Coupling

^{n}

^{+1}) as unknowns. When all the dry cells have been excluded from the system, for wet cells we have ${C}_{i}^{a}<0$, ${C}_{i}^{c}<0$ and ${C}_{i}^{b}>0$. As a result, this linear system has a symmetric and positive definite coefficient matrix, and can be solved using a direct solution method without iterations, to obtain the final solutions (η

^{n}

^{+1}). Once the η

^{n}

^{+1}have been obtained, the velocity u

^{n}

^{+1}defined at a side center is calculated by Equation (5), and then the A

_{i}and the R

_{i}are updated.

#### 2.3.3. Considerations of Abrupt Change of Wet Surface Width in Mixed Flows

_{i}) has a positive finite value for free-surface flows, and is equal to 0 for pressurized flows in pipes. Hence, abrupt changes of the B

_{i}may occur in mixed flows, especially when the cross-section of a closed pipe has a wide top (e.g., the rectangular cross-section in Figure 2). In Figure 2, the B

_{i}changes abruptly from the width of W

_{i}(representative width of the cross-section) in the free-surface part to 0 in the pressurized part, at the transition point. However, if the pipe has a closed circular cross-section, the B

_{i}is a decreasing function of the water level and gradually approaches 0 when the water level approaches the ceiling of the pipe. Such a cross-section shape is defined as the “shrinking-top” here. In case of a shrinking-top cross-section, variation of the B

_{i}shows a gradual change near the pipe ceiling instead of an abrupt change, at a transition point of different flow regimes (see Figure 1).

_{i}has a positive value and the coefficient matrix of the linear system arising from the velocity–pressure coupling is diagonally dominant. The larger the B

_{i}is, the larger the main-diagonal coefficient is. When the model is used to simulate pure pressurized flows, with B

_{i}= 0, the coefficient matrix is no more diagonally dominant. When the model is used to simulate mixed flows in a closed pipe with wide-top cross-sections, abrupt changes of the B

_{i}at a transition point will result in a linear system with discontinuous main-diagonal coefficients (DMDC) (see Figure 2). First, a model with a DMDC linear system is apt to produce relatively larger errors than that with a diagonally-dominant linear system, and is more sensitive to disturbances of the errors from the time-space discretization and the floating point operation. Second, if the simulated mixed flows are strongly dynamic (e.g., with frequent flow-regime conversions), the errors of a model with a DMDC linear system are possibly magnified, resulting in nonphysical oscillations of solutions at the transition points or even spurious flow-regime conversions.

_{i}).

## 3. Results

#### 3.1. Test Case 1

^{2}. In the middle of the pipe, there is a valve that is closed. The water levels of the two reservoirs (η

_{1}and η

_{2}) are kept constant, and their difference is η

_{1}− η

_{2}= 1 m. At time t = 0, the valve is opened suddenly and fully. Under the assumption that the pipe is frictionless, the water velocity and the pressure in the pipe can be described by analytical solutions as follows:

_{0}is calculated to be 4.43 m/s when η

_{1}− η

_{2}= 1 m.

#### 3.2. Test Case 2

^{3}/s and a constant pressure head of 0.554 m are imposed, respectively, at the upstream and downstream boundaries. The steady flow of the test falls into the category of TYPE-II mixed flows. A 1D grid of 0.01 m is used, and the implicit factor θ is set to 1.0. The wall friction in horizontal reaches of the pipe is neglected, which is the same as those in existing simulations [23,33]. Moreover, in the downward inclined reach, energy loss associated with air pockets [32,33] is assumed to act as an additional wall friction and be imposed on the free-surface parts, where the corresponding n

_{m}is determined by calibration.

^{3}/s within a prescribed period (6 s), and is preserved at 0.03 m

^{3}/s after this period. The initial profile (at t = 0) and the simulated profiles (at t = 3, 6, 9, 12, 66 s) of the pressure are plotted in Figure 5. In the final steady state of the simulated mixed flow, the hydraulic jump occurs at about 0.3 m from the last measuring station. The results (at t ≥ 66 s) simulated by the current model agree well with the measured data and the simulation results reported by [23].

_{bt}) is, respectively, set to 20, 10, 4, 2, and 1. Using these time steps, a time-step sensitivity study of the current model is carried out, in terms of stability and accuracy.

#### 3.3. Test Case 3

^{2}. The initial conditions are mathematically defined by:

_{0}is a reference level, and L is the horizontal length of the tube (L = 32 m), as shown in Figure 7. A 1D grid of Δx = 1 m is used and Δt is set to 0.01 s. Three scenarios of oscillating flows are considered below.

_{0}= 0.011, the oscillating flow in the horizontal section of the tube is fully pressurized and its oscillation frequency is given by $\sqrt{2g/L}$. The flow of Scenario 1 falls into the category of TYPE-I mixed flows, and the A-slot is not needed. In Scenario 2, by setting z

_{0}= –0.011, the oscillating flow is free-surface everywhere and its oscillation frequency is given by $\pi \sqrt{gH}/L$, where H is the average water depth (0.989 m). The flow of Scenario 2 has only one flow regime and does not fall into any category of the TYPE-I, -II or -III mixed flows, and the A-slot is also not needed. In simulations of Scenarios 1 and 2, water-level histories (at x = 0) produced by the linear solver (LS) are plotted in Figure 8, and they are almost the same as the analytical solutions.

_{0}= 0, the horizontal section of the tube is filled with a partially pressurized and partially free-surface flow, which falls into the category of TYPE-III mixed flows. Moreover, the advection of the flow in this case is so weak that its solution provides little help to suppress the possible nonphysical oscillations in the solution of a linear system with the DMDC. As a result, the oscillating flow of Scenario 3 forms a challenging test case of TYPE-III mixed flows, to which the theoretical solutions are not available. The solutions of Scenario 3 have been only reported in studies by numerical methods, and the simulation results of Casulli and Stelling [23] is taken here as a reference. The new model is tested using six A-slots whose ε are sequentially 0.02, 0.03, 0.04, 0.05 0.1, and 0.2, to clarify its performance in simulating TYPE-III mixed flows.

## 4. Discussion

#### 4.1. Differences between the New Model and Existing Models

#### 4.2. The DMDC Problem and the A-Slot Approach

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Notation

A | Wetted cross-sectional area (conveyance area); |

A_{p} | The area of the cross-section; |

A-slot | Artificial slot for coping with the discontinuous main-diagonal coefficients problem |

B | =$\partial A/\partial \eta $, the wet surface width of a cross-section for the free-surface parts and is equal to 0 for the pressurized parts of a closed pipe; |

g | Gravity acceleration; |

ne | The quantities of cells; |

ns | The quantities of sides; |

n_{m} | Manning’s roughness coefficient; |

N_{bt} | The number of substeps of the ELM |

R | Hydraulic radius; |

t | Time; |

u | Averaged velocity of cross-section; |

u_{bt} | Solution of the advection term uisng the ELM; |

W_{i} | Representative width of the cross-section; |

x | Longitudinal distance along the channel; |

z_{c} | The celling of the cross-section; |

t | Time; |

Δt | Time step |

Δx_{i} | The length of cell i; |

Δx_{i}_{+1/2} | The distance between the centers of cells i and i + 1; |

θ | Implicit factor |

η | Water level measured from an undisturbed reference water surface (for free-surface flows); piezometric head measured from an undisturbed reference water surface (for pressurized flows); |

ε | The percent that the width of the A-slot account for the representative width |

ASE | A single set of equations; |

CLO | A mark which is used to distinguish closed and nonclosed reaches; |

DMDC | Discontinuous main-diagonal coefficients |

ELM | Eulerian–Lagrangian method; |

PRE | A mark which is used to distinguish free-surface and pressurized reaches; |

SVE | Saint–Venant equations |

TSE | Two sets of equations; |

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**Figure 2.**Description of discontinuous distribution of the main-diagonal coefficients of linear system (using the case of a closed pipe with rectangular cross-sections).

**Figure 4.**Simulated histories at the first grid point inside the pipe (using different grid scales): (

**a**) the velocity; (

**b**) the pressure head.

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**MDPI and ACS Style**

Hu, D.; Li, S.; Yao, S.; Jin, Z. A Simple and Unified Linear Solver for Free-Surface and Pressurized Mixed Flows in Hydraulic Systems. *Water* **2019**, *11*, 1979.
https://doi.org/10.3390/w11101979

**AMA Style**

Hu D, Li S, Yao S, Jin Z. A Simple and Unified Linear Solver for Free-Surface and Pressurized Mixed Flows in Hydraulic Systems. *Water*. 2019; 11(10):1979.
https://doi.org/10.3390/w11101979

**Chicago/Turabian Style**

Hu, Dechao, Songping Li, Shiming Yao, and Zhongwu Jin. 2019. "A Simple and Unified Linear Solver for Free-Surface and Pressurized Mixed Flows in Hydraulic Systems" *Water* 11, no. 10: 1979.
https://doi.org/10.3390/w11101979