A Simple and Unified Linear Solver for Free-Surface and Pressurized

water

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

Dechao Hu 1 , Songping Li 1, Shiming Yao 2 and Zhongwu Jin 2,*

1 School of Hydropower and Information Engineering, Huazhong University of Science and Technology, Wuhan 430074, China; [email protected] (D.H.); [email protected] (S.L.) 2 Yangtze River Scientiﬁc Research Institute, Wuhan 430010, China; [email protected] * Correspondence: [email protected]; Tel.: +86-027-8282-9873

Received: 19 August 2019; Accepted: 19 September 2019; Published: 23 September 2019

Abstract: A semi–implicit numerical model with a linear solver is proposed for the free-surface and pressurized mixed flows in hydraulic systems. It solves the two flow regimes within a unified formulation, and is much simpler than existing similar models for mixed flows. Using a local linearization and an Eulerian–Lagrangian method, the new model only needs to solve a tridiagonal linear system (arising from velocity-pressure coupling) and is free of iterations. The model is tested using various types of mixed flows, where the simulation results agree with analytical solutions, experiment data and the results reported by former researchers. Sensitivity studies of grid scales and time steps are both performed, where a common grid scale provides grid-independent results and a common time step provides time-step-independent results. Moreover, the model is revealed to achieve stable and accurate simulations at large time steps for which the CFL is greater than 1. In simulations of a challenging case (mixed flows characterized by frequent flow-regime conversions and a closed pipe with wide-top cross-sections), an artificial slot (A-slot) technique is proposed to cope with possible instabilities related to the discontinuous main-diagonal coefficients of the linear system. In this test, a slot-width sensitivity study is also performed, and the suitable slot-width ratio (ε) for the linear solver is suggested to be 0.05–0.1.

Keywords: free-surface flow; pressurized flow; mixed flow; semi–implicit; numerical model; linear solver

1. Introduction Hydraulic systems are often filled with free-surface and pressurized mixed flows. For example, the urban drainage systems are generally designed to convey water under free-surface conditions. However, during intense rain events, water may overflow from storm sewers through manholes onto the land surface and then cause inundation. In this case, the sewer is often filled with mixed flows (characterized by the simultaneous occurrence of free-surface and pressurized flows). Mixed flows are highly dynamic [1], with frequent transitions of flow regimes. Moreover, characteristics of the two regimes are quite different, e.g., the celerity of acoustic waves of pressurized flows may be 2–3 orders of magnitude larger than that of free-surface flow waves. In practice, the fluid dynamics of the mixed flows are usually simulated using one-dimensional (1D) hydrodynamic models, where accurate and fast simulations are expected to provide real-time data for storm management. However, a unified mathematical description of the free-surface and the pressurized parts of the mixed flows, as well as a simple solution, are often challenging [2]. The existing models for mixed flows can be divided into two categories, which, respectively, describe the flows using two sets of equations (TSE) and a single set of equations (ASE). A TSE model solves the free-surface and the pressurized parts of mixed flows separately, mainly including the rigid

Water 2019, 11, 1979; doi:10.3390/w11101979 www.mdpi.com/journal/water Water 2019, 11, 1979 2 of 15 column approach [3–5] and the shock-fitting approach [6–11]. This kind of model explicitly tracks the pressurization front, and their advantages and limitations have been well reviewed in Bousso et al. [12]. Using a single set of equations, an ASE model builds a unified formulation applied to different regimes of the mixed flow. In ASE models, the Saint–Venant equations (SVE) or their equivalents are often used as the mathematical descriptions. In terms of the form of the ASE, two sub kinds of these models (ASE-1 and ASE-2) have been identified in references. In ASE-1 models, the pressure gradient in the momentum equation is divided into a gradient of water depth and a source term associated with bed slope, while the governing equations are written in their conservative form [2,13–18]. The ASE-1 models often adopt Godunov-type schemes and Riemann solvers (e.g., Bourdarias and Gerbi [19]), so they are good at capturing physical discontinuousness and are able to automatically track the pressurization fronts in mixed flows. As a result, the ASE-1 models are often called the “shock-capturing” family of models. However, the governing equations used in ASE-1 models may not be directly used to simulate pressurized flows, because the free surface term is not explicitly included. Hence, additional techniques are needed for the ASE-1 models to quantify the piezometric head when the pipe is pressurized. Two popular techniques are the Preissmann slot technique [20] and the two-component pressure approach [17,21]. The first technique assumes a narrow slot is in the ceiling of the pipe, while the second decouples the hydrostatic pressure from surcharged pressures. In ASE-2 models, the pressure term is explicitly expressed in the momentum equation instead of the two terms of water depth and bed slope, and a nonconservative form of equations is used [22–24]. The pressure represents either the water level for free-surface parts or the piezometric head for pressurized parts, so a unified description of mixed flows is reached. Moreover, benefiting from the nonconservative form of momentum equations, the advection term can be calculated explicitly using an Eulerian–Lagrangian method (ELM). Using such kind of governing equations, Casulli and Stelling [23] proposed a semi–implicit formulation that naturally applies to the two regimes of mixed flows, where the pressure (water levels for free-surface parts and piezometric heads for pressurized parts) is synchronously solved by a coupling solution of the continuity and momentum equations. Because the cross-sectional conveyance area is a nonlinear function of water level, the velocity–pressure coupling in their model results in a nonlinear system which requires iterative solutions. Benefiting the high stability and large time steps, the ASE-2 model is reported to be tens of times faster than conventional models for mix flows in [25]. On the other hand, the linearization technique (based on the solutions of time level n) is often used in existing models for mixed flows, and should be noticed. For example, in Fuamba [9] and SWMM [26], the ∂A/∂t (A is cross-sectional conveyance area) in the continuity equation is linearized. In Ji [15], all nonlinear coefficients for governing variables are linearized based on the solutions of time level n. In Bourdarias and Gerbi [19], the numerical flux at cell-faces and the source term are linearized using the solutions of time level n, to construct a linear system. The numerical formulation and solution are sharply simplified by various linearization techniques in these models. In this study, using a linearization [9,26], a semi–implicit numerical model with a linear solver is proposed for the free-surface and pressurized mixed flows in hydraulic systems, which solves the two flow regimes within a unified formulation and very simple. The new model may be considered as a variant of the ASE-2 model of [23]. In the new model, the velocity–pressure coupling is solved by constructing and solving a linear system instead of the nonlinear system in [23]. Characteristics of representative types of mixed flows in pipes are analyzed, and the applicability of the new model to them is discussed. The model is then tested using three classical problems of mixed flows.

2. Methods The governing equations, the description of mixed ﬂows, and the numerical formulation of the proposed new model for mixed ﬂows are introduced. Water 2019, 11, 1979 3 of 15

2.1. Governing Equations The interest of this paper is confined to 1D modeling of incompressible mixed flows in closed inelastic pipes and connected open channels. Assuming hydrostatic flow, the governing equations (including the continuity and momentum equations) for such mixed flows are taken to be:

∂A ∂Au + = 0 (1) ∂t ∂x

∂u ∂u ∂η n2 u u + u = g g m | | (2) ∂t ∂x − ∂x − R4/3 where u = averaged velocity of cross-section; A = wetted cross-sectional area (conveyance area); x = longitudinal distance along the channel; t = time; η = the pressure representing either the water level (for free-surface parts) or the piezometric head (for pressurized parts); g = gravity acceleration; nm = Manning’s roughness coefficient; and R = hydraulic radius. According to Casulli and Stelling [23], using Equations (1) and (2), a unified description of the free-surface parts and the pressurized parts of mixed flows is reached. In the free-surface parts of mixed flows, the pressure η represents the water level, and the pressure varies synchronously with respect to the water level. In the pressurized parts of mixed flows, the pressure η represents the piezometric head which includes two parts, respectively, the hydrostatic part and the surcharged part. In fact, when the flow in a pipe is pressurized, the hydrostatic part can be determined using the information of height and geometry of the pipe, after the water compressibility and wall elasticity have been neglected. As a result, only the surcharged part of the piezometric head is left unknown. In [23] and this study, for the pressurized parts of the mixed flows, the piezometric head is used directly (namely the pressure η is not further divided) for simplicity. Using the unified description of the mixed flows based on Equations (1) and (2), the pressure (including the water levels for free-surface parts and the piezometric heads for pressurized parts) is universally quantified by solving the velocity-pressure coupling, in the nonlinear solver of Casulli and Stelling [23]. In the new model, this idea of [23] is also adopted, but the velocity–pressure coupling is solved by constructing and solving a linear system instead of a nonlinear system. The A in Equation (1) is a nonlinear function of water levels. If the continuity equation with a nonlinear A is directly coupled with the momentum equation, the resulting system is to be nonlinear and needs to be solved iteratively. To avoid the complex solutions of a nonlinear system, the term ∂A/∂t in Equation (1) is linearized in this study in a similar way of Fuamba [9] and SWMM [26]. The resulting continuity equation after a local linearization is given by:

∂η ∂Au B + = 0 (3) ∂t ∂x where B (=∂A/∂η) represents 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. In Equations (2) and (3), for a cross-section located in a closed pipe, A is a nonnegative nonlinear function of the water level when the pipe is not fully filled; A is equal to the area of the cross-section (Ap) when the water level is equal to or larger than the celling of the cross-section (zc). It must be pointed out that: the local linearization of the continuity equation does not break the unified description and solution of the mixed flows using Equations (1) and (2). Based on Equations (2) and (3), the pressures of the free-surface and the pressurized parts of the mixed flows can also be universally quantified by solving the velocity-pressure coupling.

2.2. Computational Grid and Description of Mixed Flows A channel with both closed and nonclosed reaches is considered here, which is divided by a set of 1D cells (elements). The ne and ns are used to denote, respectively, the quantities of cells and sides in Water 2019, 11, 1979 4 of 15 Water 2019, 11, x FOR PEER REVIEW 4 of 16 thein the computational computational grid, grid, and and every every cell cell/side/side has has a a unique unique global global cell cell/side/side index.index. A staggered grid grid of of variable arrangement is used. The velocity u is defined at side centers (i 1/2, i + 1/2, ... ) and the variable arrangement is used. The velocity u is defined at side centers (i− − 1/2, i + 1/2, …) and the pressure η is defined at cell centers (i 1, i + 1, ... ), as shown in Figure1. A cross-section with general pressure η is defined at cell centers (−i − 1, i + 1, …), as shown in Figure 1. A cross-section with general topographicaltopographical data data is is arranged arranged at at the the center center of of each each cell, cell, and and the the geometry geometry of of a cell a cell is describedis described by by the followingthe following variables. variables. For cellFor celli, ∆ xi,i Δrepresentsxi represents the the length length of theof the cell; cell;Bi isBi theis the wet wet surface surface width width and and is relatedis related to the to the cell cell water water level level that that varies varies with with respect respect to time; to time;Ai representsAi represents the the conveyance conveyance area area of theof cross-sectionthe cross-section located located at the at center the center of the of cell. the cell. For sideFor sidei + 1 i/ 2+ which1/2 which connects connects cells cellsi and i andi + 1,i + ∆1,x Δi+x1/i+1/22 is theis the distance distance between between the the centers centers of cellsof cellsi and i andi + i1; + 1;A iA+1i+1/2/2 and andR Ri+i+1/21/2 are,are, respectively, respectively, the the conveyance conveyance areaarea and and the the hydraulic hydraulic radius radiusof of sidesidei i+ + 11/2,/2, whichwhich are interpolated using those of of cells cells ii andand ii ++ 1.1. TheThe cells cells in in closed closed and and nonclosednonclosed reachesreaches areare distinguisheddistinguished by a mark (CLO). CLO CLO is is set set to to 1 1 for for cellscells in in closed closed reaches reaches and and setset toto 00 forfor thosethose inin nonclosednonclosed reaches. In closed closed reaches, reaches, cells cells are are further further distinguisheddistinguished according according toto theirtheir flowflow regimesregimes (denoted(denoted byby a mark “PRE”). PRE PRE is is set set to to 1 1 for for cells cells in in thethe pressurized pressurized parts parts of of the the closed closed pipe pipe reach reach and and set set to 0to for 0 for those those with with a free-surface a free-surface flow flow regime regime (see Figure(see Figure1). The 1). cells The in cells all the in free-surface all the free-surface and the pressurizedand the pressurized parts of the parts channel of the together channel contribute together tocontribute the construction to the construction of an algebraic of an system. algebraic system.

FigureFigure 1. 1.Sketch Sketch ofof mixedmixed flowflow inin aa closedclosed sewersewer pipe with manholes and computational grids. grids. 2.3. Numerical Formulation 2.3. Numerical Formulation 2.3.1. Discretization of Governing Equations 2.3.1. Discretization of Governing Equations The momentum equation is discretized using an operator-splitting technique, and the model solvesThe the governingmomentum equations equation inis threediscretized steps. using First, allan theoperator-splitting explicit termsin technique, the momentum and the equation model aresolves computed the governing to obtain equations provisional in th velocities.ree steps. First, Second, all the the explicit velocity–pressure terms in the momentum coupling is equation solved to are computed to obtain provisional velocities. Second, the velocity–pressure coupling is solved to obtain new pressures. Third, a back substitution of the new pressures into the momentum equation is obtain new pressures. Third, a back substitution of the new pressures into the momentum equation performed to obtain the final velocities. is performed to obtain the final velocities. The advection term in the momentum equation is solved by a pointwise ELM. The ELM is an The advection term in the momentum equation is solved by a pointwise ELM. The ELM is an explicit advection scheme, combining the stability and accuracy of Lagrangian approaches with the explicit advection scheme, combining the stability and accuracy of Lagrangian approaches with the convenience of the fixed computation grid of Eulerian approaches. The inherently upwind property of convenience of the fixed computation grid of Eulerian approaches. The inherently upwind property the ELM eliminates the Courant–Friedrichs–Lewy number (CFL) restriction associated with explicit of the ELM eliminates the Courant–Friedrichs–Lewy number (CFL) restriction associated with Eulerian methods and allows large time steps. Solution of the advection term using an ELM requires explicit Eulerian methods and allows large time steps. Solution of the advection term using an ELM a backtracking of cell faces. Because the velocity varies along a channel (as a result of the varying requires a backtracking of cell faces. Because the velocity varies along a channel (as a result of the cross-sectional topography, the friction, etc.), a multistep tracking technique [27] is used to ensure the varying cross-sectional topography, the friction, etc.), a multistep tracking technique [27] is used to accuracy of the ELM tracking. ensure the accuracy of the ELM tracking. The backtracking of the ELM starts at a side center, and a multistep backward Euler technique is The backtracking of the ELM starts at a side center, and a multistep backward Euler technique performed to backtrack from the known location at time level n + 1 to the unknown foot of the backward is performed to backtrack from the known location at time level n + 1 to the unknown* * foot of the trajectorybackward at timetrajectory level nat. Thetime tracking level n displacement. The tracking of eachdisplacement substep is givenof each by ∆substepx = u ∆ist/ Ngivenbt, where by   *  “NΔbt”= is theΔ number of substeps and u is the velocity vector at the starting point of each substep. When x u t / Nbt , where “Nbt” is the number of substeps and u is the velocity vector at the starting a tracking point has been found, the velocity at this point is interpolated based on the velocity field of 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

Water 2019, 11, 1979 5 of 15 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 ubt. The cell-face velocity is then directly updated with ubt, which means that the advection term is solved. To eliminate the stability restrictions imposed by rapid gravity waves, the θ semi-implicit method [28–31] is used to discretize the gradient of the pressure in two parts, where the explicit and the implicit parts have a weight factor of 1 θ and θ, respectively. For both the two regimes of mixed − ﬂows, the momentum equation can be universally discretized as (side i + 1/2):   2 n n n n+1 n+1  nm ubt i+  η η η η  + , 1/2  n+1 = n ( ) i+1 − i i+1 − i 1 g∆t n u + ubt,i+1/2 1 θ g∆t θg∆t (4)  R 4/3  i 1/2 − − ∆x + − ∆x +  i+1/2  i 1/2 i 1/2 where θ is the implicit factor; ∆t is the time step; superscript n indicates time level n; and the riverbed n+1 friction term is discretized using ubt and u to enhance computational stability. When all the explicit terms in the discretized momentum equations are incorporated, the unknowns (the pressures at time level n + 1, ηn+1) emerge and Equation (4) is then transformed into the following form:

Gn n+1 n+1 i+1/2 1 ηi+ ηi un+1 = θ∆tg 1 − (5) i+1/2 Fn Fn x i+1/2 − i+1/2 ∆ i+1/2

n 4/3 Fn = + g tn2 un R Gn where i+1/2 1 ∆ m bt,i+1/2 / i+1/2 ; i+1/2 is the incorporated explicit terms and Gn Fn i+1/2/ i+1/2 can be regarded as the provisional solution by solving all the explicit terms, ηn ηn where Gn = un (1 θ)g∆t i+1− i . i+1/2 1 ∆xi+1/2 bt,i+ 2 − − Before the introduction of the discretization of the continuity equation, we want to explain that the different derivates (related to the term A in Equation (1)) have different roles in terms of their physics meanings. The term ∂A/∂t describes the change of the cross-sectional area (or the water n+1 volume of a cell) with respect to time. Hence, in the discretization of the term ∂A/∂t, the Ai and n the Ai of a cell, respectively, at time levels n and n + 1 should be both involved. When the ∂A/∂t is n+1 n transformed into the B∂η/∂t in Equation (3), it means that ηi and ηi of a cell should be both involved in the discretization of the B∂η/∂t. At the same time, in the discrete continuity equation, the resulting discretized expression of the term ∂(Au)/∂x accounts for the calculation of the water fluxes through (Au) x An An+1 cell faces. For the discretization of the ∂ /∂ , we can use either the i 1/2 or the i 1/2 (at cell ± An± faces), which may only influence the order of the accuracy of solutions. In practice, the i 1/2 is often used for simplicity (such as [23]), and it is also used in our model. ± The continuity equation is discretized using a finite-volume method, where the mass of water is conserved both locally and globally by the unique flux at a cell face. For cell i, the discrete pressure ηi is assumed to be constant within the cell. Moreover, the wet surface width of cell i (Bi) is explicitly n represented by the value of time level n (Bi ) within a time step (∆t). From time level n to n + 1, a finite-volume discretization of the continuity equation, Equation (3), is given by: Bn x n+1 n = t An un+1 An un+1 ( ) t An un An un i ∆ i ηi ηi θ∆ i+1/2 i+1/2 i 1/2 i 1/2 1 θ ∆ i+1/2 i+1/2 i 1/2 i 1/2 (6) − − − − − − − − − −

The nonlinear terms (Bi and Ai 1/2) are present, respectively, on the left hand side (LHS) and ± the right hand side (RHS) of Equation (6). However, when the Bi on the LHS and the Ai 1/2 on the ± Bn An t n n RHS have been explicitly represented by i and i 1/2 (within the ∆ , from time level to + 1), the solution problem associated with the nonlinear terms± does not exist any longer. Our linear solver and the nonlinear solver in [23] use diﬀerent discrete continuity equations. h i In [23], the LHS of the discrete continuity equation may written as ∆x A ηn+1 A ηn , where the i i i − i i function A(η) is a nonlinear function of water level. In the LHS of the discrete continuity equation Water 2019, 11, 1979 6 of 15 in [23], existence of the nonlinear function A(η) potentially leads to a complex formulation for the construction and solution of the velocity-pressure coupling.

2.3.2. Solution of Velocity–Pressure Coupling With the advection term explicitly calculated (by the ELM) and the ∂A/∂t linearized, the coupling solution of the continuity and momentum equations in the new model only needs to solve a linear system. The velocity–pressure coupling is performed by substituting Equation (5) into Equation (6). The resulting equation is given by (i = 1, 2, ... , ne):

a n+1 + b n+1 + c n+1 = n n + n Ci ηi 1 Ci ηi Ci ηi+1 Bi ∆xiηi ri (7) − gθ2∆t2An gθ2∆t2An a i 1/2 c i+1/2 b n a c n = − = = = where Ci Fn ∆x , Ci Fn ∆x , Ci Bi ∆xi Ci Ci , ri − i 1/2 i 1/2 − i+1/2 i+1/2 − − h − − i h i t An Gn Fn An Gn Fn ( ) t An un An un θ∆ i+1/2 i+1/2/ i+1/2 i 1/2 i 1/2/ i 1/2 1 θ ∆ i+1/2 i+1/2 i 1/2 i 1/2 . − − − − − − − − − − Equation (7) forms a tridiagonal linear system of ne equations with the cell pressures (ηn+1) as a unknowns. When all the dry cells have been excluded from the system, for wet cells we have Ci < 0, c b Ci < 0 and Ci > 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 (ηn+1). Once the ηn+1 have been obtained, the velocity un+1 defined at a side center is calculated by Equation (5), and then the Ai and the Ri are updated. Relative to the solution of a linear system, the solution of a nonlinear system (e.g., [23]) which requires iterations will be much more complex. Casulli and Stelling [23] found that the convergence of most Newton-type methods is often problematic for the solution of mixed flows, and they used a nested Newton-type method to solve their nonlinear system. Equation (7) is applicable to both free-surface and pressurized flow regimes, based on which a 1D unified numerical model is developed for mixed flows. However, in preliminary tests of certain mixed flows, it is found that the main-diagonal coefficients of the matrix of the linear solver may be discontinuous in value, arising from the potentially abrupt change of wet surface widths of cross-sections. The problem may leads to nonphysical oscillations in the solutions, or even unstable simulations. In the next section, characteristics of several typical mixed flows are analyzed, and the applicability of the new model to each type of them is discussed.

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

The wet surface width of a cross-section (Bi) has a positive finite value for free-surface flows, and is equal to 0 for pressurized flows in pipes. Hence, abrupt changes of the Bi 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 Figure2). In Figure2, the Bi changes abruptly from the width of Wi (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 Bi 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 Bi shows a gradual change near the pipe ceiling instead of an abrupt change, at a transition point of different flow regimes (see Figure1). When the current model is used to simulate pure free-surface flows, the Bi has a positive value and the coefficient matrix of the linear system arising from the velocity–pressure coupling is diagonally dominant. The larger the Bi is, the larger the main-diagonal coefficient is. When the model is used to simulate pure pressurized flows, with Bi = 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 Bi at a transition point will result in a linear system with discontinuous main-diagonal coefficients (DMDC) (see Figure2). First, a model with a DMDC linear system is apt to produce Water 2019, 11, x FOR PEER REVIEW 7 of 16

diagonally dominant. When the model is used to simulate mixed flows in a closed pipe with Waterwide-top2019, 11 cross-sections,, 1979 abrupt changes of the Bi at a transition point will result in a linear system7 of 15 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 relativelysystem, and larger is more errors sensitive than that to disturbances with a diagonally-dominant of the errors from linear the time-space system, and discretization is more sensitive and the to disturbancesfloating point of theoperation. errorsfrom Second, the time-spaceif the simulated discretization mixed andflows the are floating strongly point dynamic operation. (e.g., Second, with iffrequent the simulated flow-regime mixed conversions), flows are strongly the errors dynamic of a model (e.g., with with a frequent DMDC linear flow-regime system conversions),are possibly themagnified, errors of resulting a model in with nonphysical a DMDC oscillations linear system of so arelutions possibly at the magnified, transition resultingpoints or ineven nonphysical spurious oscillationsflow-regime of conversions. solutions at the transition points or even spurious flow-regime conversions.

FigureFigure 2. 2.Description Description ofof discontinuousdiscontinuous distribution of the main-diagonal coefficients coeﬃcients of of linear linear system system (using(using the the case case of of a a closed closed pipe pipe with with rectangularrectangular cross-sections).cross-sections).

TheThe possible possible DMDC DMDC problemproblem ofof thethe currentcurrent linearlinear solver is analyzed using using three three kinds kinds of of mixed mixed flowsflows which which are are di differentiatedfferentiated withwith twotwo characteristics.characteristics. The first first is “with flow-regime flow-regime conversions conversions (C1)”(C1)” and and the the second second is is “in “in a a closed closed pipepipe withwith wide-topwide-top cross-sectionscross-sections (C2)”.(C2)”. TYPETYPE II (without (without C1C1 oror C2):C2): PressurizedPressurized flowsflows meet free-surface flows flows in in nonclosed nonclosed channels channels (manholes(manholes and and open open channels,channels, e.g.,e.g., cellcell ii ++ 4 in Figure 22).). InIn thisthis case,case, althoughalthough the DMDC may be significant,significant, thethe regimesregimes ofof thethe free-surfacefree-surface flowflow in nonclosed channels and and the the pressurized pressurized flow flow in in closedclosed pipes pipes are are both both kept, kept,without without flow-regimeflow-regime conversions.conversions. TYPE II (with C1): Pressurized Pressurized flows flows meetmeet free-surface free-surface flows flows in in aa closedclosed pipepipe withwith shrinking-topshrinking-top cross-sections (e.g., (e.g., the the transition transition point point in in FigureFigure1). 1). Despite Despite flow-regime flow-regime conversions, conversions, the the main-diagonal main-diagonal coe coefficientsfficients may may be be still still assumed assumed to to be graduallybe gradually varying varying around around the transition the transition points points (the DMDC (the DMDC is not significant)is not significant) if the unsteady if the unsteady property ofproperty the mixed of flowthe ismixed not strong.flow is TYPE not strong. III (with TYPE C1 and III C2): (with Pressurized C1 and C2): flows Pressurized meet free-surface flows meet flows infree-surface a closed pipe flows with in wide-topa closed pipe cross-sections with wide-top (e.g., cross-sections the transition (e.g., point the in Figuretransition2), where point in flow-regime Figure 2), conversionswhere flow-regime and the conversions DMDC are bothand the significant. DMDC are both significant. InIn preliminary preliminary tests,tests, thethe newnew model is is revealed revealed to to be be able able to to simulate simulate the the mixed mixed flows flows of ofTYPE TYPE I Iand and II stably. When the model is is used to simulate TYPE-III TYPE-III mixed mixed flows, flows, it it produces produces nonphysical nonphysical oscillationsoscillations oror becomesbecomes unstable.unstable. The instability of the linear solver, solver, arising arising from from the the simulation simulation of of thethe mixed mixed flows flows with with frequent frequent flow-regime flow-regime conversions conversions in closed in pipes closed with wide-toppipes with cross-sections, wide-top iscross-sections, called the DMDC is called problem. the DMDC It is inferred problem. that: It inis inferred the nonlinear that: in solver the nonlinear of [23], the solver nonlinear of [23], system the hasnonlinear been continually system has updated been contin usingually new updated flow variables using new in theflow iterative variables solutions in the iterative and the solutions potential instabilityand the potential (from similar instability discontinuous (from similar problems) discontinu is eliminatedous problems) in a is timely eliminated manner. in a timely manner. ForFor simulating simulating TYPE TYPE III III mixed mixed flows, flows, an artificialan artificial slot (A-slot)slot (A-slot) is proposed is proposed to alleviate to alleviate the possible the DMDCpossible problem DMDC ofproblem the linear of solverthe linear and enhancesolver and its enhance stability. its Because stability. the Because wet surface the widthwet surface of the cross-section is expressed explicitly in Equation (7), applying the A-slot to Equation (7) can be done straightforwardly: a n+1 + b n+1 + c n+1 = ( n ) n + n Ci ηi 1 Ci ηi Ci ηi+1 Max Bi , εWi ∆xiηi ri (8) − Water 2019, 11, x FOR PEER REVIEW 8 of 16 width of the cross-section is expressed explicitly in Equation (7), applying the A-slot to Equation (7) can be done straightforwardly:

an+++111 bn cn n n n CCCMaxBWxrηηη−+++=(, εη ) Δ+ (8) Water 2019, 11, 1979 ii11 ii ii i i ii i 8 of 15 bn=Δ−−ε ac where CMaxBWxCCiiiiii(, ) ; and ε is the percent that the width of the A-slot account b n a c wherefor the Crepresentative= Max(B , ε Wwidthi)∆x iof theC cross-sectionC ; and ε is the (W percenti). that the width of the A-slot account for the i i − i − i representativeEquation width(8) is coded of the cross-sectioninstead of Eq (Wuationi). (7) in the model, with the ε being an adjustable parameterEquation according (8) is coded to the instead types ofof Equationmixed flows. (7) in However, the model, the with DMDC the ε beingis not anobvious adjustable in simulations parameter ofaccording many mixed to the flows types (e.g., of mixed TYPE flows.I and -II However, mixed flows), the DMDC when isthe not A-slot obvious is not in necessary simulations and of εmany is set tomixed 0. Hence, flows the (e.g., A-slot TYPE is I not and an -II essential mixed flows), part of when the thecurrent A-slot model. is not That necessary is to say: and theε is model set to 0. without Hence, anthe A-slot A-slot can is not be an applied essential to partall type of the of current mixed model.flows except That is the to say:TYPE the III model mixed without flows; anthe A-slot A-slot can is onlybe applied needed to for all typethe simulation of mixed flows of TYPE except III mixed the TYPE flows. III mixed flows; the A-slot is only needed for the simulation of TYPE III mixed flows. 3. Results 3. Results The new model is demonstrated by tests of various mixed flows, including the aforementioned mixedThe flows new of model TYPE is I, demonstrated II and III. The by simulation tests of various results mixed using flows,the new including model theare aforementionedcompared with analyticalmixed flows solutions, of TYPE experiment I, II and III. data The an simulationd results reported results by using other the researchers. new model are compared with analytical solutions, experiment data and results reported by other researchers. 3.1. Test Case 1 3.1. Test Case 1 This test describes a pressurized flow in a horizontal pipe, connecting two reservoirs (see Figure This test describes a pressurized flow in a horizontal pipe, connecting two reservoirs (see Figure3). 3). The pipe length (L) is 400 m and its square cross-section has an area of A = 1 m2. In the middle of The pipe length (L) is 400 m and its square cross-section has an area of A = 1 m2. In the middle of the pipe, there is a valve that is closed. The water levels of the two reservoirs (η1 and η2) are kept the pipe, there is a valve that is closed. The water levels of the two reservoirs (η1 and η2) are kept constant, and their difference is η1 − η2 = 1 m. At time t = 0, the valve is opened suddenly and fully. constant, and their difference is η η = 1 m. At time t = 0, the valve is opened suddenly and fully. Under the assumption that the pipe1 − is frictionless,2 the water velocity and the pressure in the pipe can 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: be described by analytical solutions as follows: = () uxt(,) u00 tanh/ t t (9) u(x, t) = u0tanh(t/t0) (9) Lx− − ηηηη=+ − 2 () (,)xt212 ( )L x cosh2 t / tM (10) η(x, t) = η + (η η ) − cosh− (t/tM) (10) 2 1 − 2 L p where uug==−2(2g(ηηη η )), andand t tLu==22/L/u . The. The analytical analytical solutions solutions are are independent independent from from shape, shape, size, 00121 − 2 0 000 and depth of the pipe, as long as the pipe is fully filled. The u is calculated to be 4.43 m/s when η size, and depth of the pipe, as long as the pipe is fully filled. The0 u0 is calculated to be 4.43 m/s when1 − η12 –= η12 m.= 1 m.

Figure 3. TwoTwo reservoirs connected by a horizontal pressurized pipe.

The unsteady flow ﬂow of this test falls into the categorycategory of TYPE-I mixed ﬂows.flows. The present model neglects water compressibility and pipe elasticity, which means the same discharge through the pipe from upstream upstream to to downstream. downstream. The The model model is istest testeded on on gradually gradually reduced reduced uniform uniform grids grids (Meshes (Meshes 1– 1–5)5) whose whose grid grid scales scales are are sequentially sequentially 40, 40, 20, 20, 16, 16, 10, 10, and and 5 5 m, m, to to establish establish the the grid independence of solutions. InIn thethe simulations,simulations,∆ tΔ=t 1= s1 ands andθ = θ1, = which 1, which are commonare common in existing in existing simulations. simulations. The model The modelcompletes completes all the all simulations the simulations stably. stably. When When the grid the scale grid becomesscale becomes smaller, smaller, the simulation the simulation result resultgradually gradually converges converges at the analyticalat the analytical solution, solution, as shown as shown in Figure in4 Figure. The grid 4. The scale grid equal scale to equal or smaller to or than 16 m (Mesh 3) provides grid-independent results, when the errors in the simulated histories of the velocity and the pressure are all minor. Water 2019, 11, x FOR PEER REVIEW 9 of 16

Watersmaller2019 than, 11, 1979 16 m (Mesh 3) provides grid-independent results, when the errors in the simulated9 of 15 histories of the velocity and the pressure are all minor.

1.0 1.0 dx=5m 0.8 0.8 dx=5m dx=10m dx=10m dx=16m 0 0.6 (m) 0.6

2 dx=20m dx=16m η u/u

− dx=40m 0.4 dx=20m 1 0.4 η dx=40m 0.2 0.2 analytical solution analytical solution 0.0 0.0 (a) 0123(b) 0123 t/t0 t/t0 Figure 4. Simulated histories at the first first grid point inside the pipe (using didifferentfferent gridgrid scales): (a) the velocity; (b)) thethe pressurepressure head.head. 3.2. Test Case 2 3.2. Test Case 2 The laboratory experiment in a closed pipe [32] is tested to demonstrate the model in simulating The laboratory experiment in a closed pipe [32] is tested to demonstrate the model in simulating the free-surface and the pressurized flows connected by extreme forms (hydraulic jump). The pipes the free-surface and the pressurized flows connected by extreme forms (hydraulic jump). The pipes have a circular cross-section with an inner diameter of 0.22 m and consist of three reaches. The upstream have a circular cross-section with an inner diameter of 0.22 m and consist of three reaches. The and downstream reaches (L1 and L3) are both horizontal and 2 m long, which are connected by a upstream and downstream reaches (L1 and L3) are both horizontal and 2 m long, which are downward inclined reach (L2) at an angle α = 10◦ having a length of 4 m. In an experiment, a constant connected by a downward inclined reach (L2)− at an angle α = −10° having a length of 4 m. In an water discharge of 0.03 m3/s and a constant pressure head of 0.554 m are imposed, respectively, at the experiment, a constant water discharge of 0.03 m3/s and a constant pressure head of 0.554 m are upstream and downstream boundaries. The steady flow of the test falls into the category of TYPE-II imposed, respectively, at the upstream and downstream boundaries. The steady flow of the test falls 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 into the category of TYPE-II mixed flows. A 1D grid of 0.01 m is used, and the implicit factor θ is set horizontal reaches of the pipe is neglected, which is the same as those in existing simulations [23,33]. to 1.0. The wall friction in horizontal reaches of the pipe is neglected, which is the same as those in Moreover, in the downward inclined reach, energy loss associated with air pockets [32,33] is assumed existing simulations [23,33]. Moreover, in the downward inclined reach, energy loss associated with to act as an additional wall friction and be imposed on the free-surface parts, where the corresponding air pockets [32,33] is assumed to act as an additional wall friction and be imposed on the free-surface nm is determined by calibration. parts, where the corresponding nm is determined by calibration. In the first group of simulations (∆t = 0.003 s), to observe the evolution of mixed flows and the In the first group of simulations (Δt = 0.003 s), to observe the evolution of mixed flows and the formation of the hydraulic jump, the discharge at the inlet of the pipe is gradually increased from 0 to formation of the hydraulic jump, the discharge at the inlet of the pipe is gradually increased from 0 0.03 m3/s within a prescribed period (6 s), and is preserved at 0.03 m3/s after this period. The initial to 0.03 m3/s within a prescribed period (6 s), and is preserved at 0.03 m3/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 Figure5. profile (at t = 0) and the simulated profiles (at t = 3, 6, 9, 12, 66 s) of the pressure are plotted in Figure In the final steady state of the simulated mixed flow, the hydraulic jump occurs at about 0.3 m from the 5. In the final steady state of the simulated mixed flow, the hydraulic jump occurs at about 0.3 m last measuring station. The results (at t 66 s) simulated by the current model agree well with the from the last measuring station. The results≥ (at t ≥ 66 s) simulated by the current model agree well measuredWater 2019 data, 11 and, x FOR the PEER simulation REVIEW results reported by [23]. 10 of 16 with the measured data and the simulation results reported by [23]. 1.0 Measured 0.9 Casulli and Stelling (2013) 0.8 free surface

0.7 L1 piezometric head 0.6

(m) 0.5 Z 0.4 L2 linear solver (t=0 s) 0.3 linear solver (t=3 s) linear solver (t=6 s) 0.2 linear solver (t=9 s) linear solver (t=12 s) 0.1 linear solver (t=66 s) L3 0.0 012345678 x(m)

FigureFigure 5. Evolution 5. Evolution of mixedof mixed ﬂows flows in in closed closed circularcircular pipe pipe and and formation formation of hydraulic of hydraulic jump. jump.

In the second group of simulations, the model is tested on gradually reduced time steps which are sequentially 0.01, 0.005, 0.002, 0.001, and 0.0005 s, to establish the time-step independence of the solutions. Correspondingly, the number of substeps for the ELM backtracking (Nbt) 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. On the one hand, the largest velocity of the simulations is found to be 3.28 m/s and occurs in the downward inclined reach. The CFL is calculated to be 3.38, 1.64, 0.66, 0.33, and 0.16, for the simulations using a Δt from 0.01 to 0.0005 s. The linear solver completes all the simulations stably. Moreover, it is found that the model begin to produce oscillated solutions when the Δt is equal to or larger than 0.1 s (CFL ≥ 3.38). In this test (a TYPE-II mixed flow), a gradually variation of the wet surface width at the circular cross-section can be observed in Figure 6. It means that the value of the main-diagonal coefficients of the linear system is gradually varying around the transition point (namely the DMDC problem is minor), so the A-slot is not needed. On the other hand, the time step, which is less than or equal to 0.005 s, provides time-step-independent results.

Water 2019, 11, 1979 10 of 15

In the second group of simulations, the model is tested on gradually reduced time steps which are sequentially 0.01, 0.005, 0.002, 0.001, and 0.0005 s, to establish the time-step independence of the solutions. Correspondingly, the number of substeps for the ELM backtracking (Nbt) 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. On the one hand, the largest velocity of the simulations is found to be 3.28 m/s and occurs in the downward inclined reach. The CFL is calculated to be 3.38, 1.64, 0.66, 0.33, and 0.16, for the simulations using a ∆t from 0.01 to 0.0005 s. The linear solver completes all the simulations stably. Moreover, it is found that the model begin to produce oscillated solutions when the ∆t is equal to or larger than 0.1 s (CFL 3.38). In this test (a TYPE-II mixed ﬂow), a gradually variation of the wet surface width at ≥ the circular cross-section can be observed in Figure6. It means that the value of the main-diagonal coeﬃcients of the linear system is gradually varying around the transition point (namely the DMDC problem is minor), so the A-slot is not needed. On the other hand, the time step, which is less than or equalWater 2019 to 0.005, 11, x FOR s, provides PEER REVIEW time-step-independent results. 11 of 16

1.0 Measured 0.9 free surface Casulli and Stelling (2013) linear solver (dt=0.01 s) 0.8 linear solver (dt=0.005 s) L1 linear solver (dt=0.002 s) 0.7 0.6 Measured linear solver (dt=0.001 s) 0.6 C&S (2013) linear solver (dt=0.0005 s) LS (dt=0.01 s) LS (dt=0.005 s) piezometric head

(m) 0.5

0.5 LS (dt=0.002 s) Z LS (dt=0.001 s) 0.4 LS (dt=0.0005 s) L (m) 2 Z

0.3 0.4 0.2

0.1 0.3 4.2 4.3 4.4 4.5 x(m) L3 0.0 012345678 x(m) Figure 6. Simulated profiles of pressure head and hydraulic jump using different ∆t. Figure 6. Simulated profiles of pressure head and hydraulic jump using different Δt. 3.3. Test Case 3 3.3. Test Case 3 The frictionless oscillating flows within a U-shaped tube are simulated. The tube has a square cross-sectionThe frictionless area of 1oscillating m2. The initial flows conditions within a U-shaped are mathematically tube are simulated. defined by: The tube has a square cross-section area of 1 m2. The initial conditions are mathematically defined by: u(x, 0) = 0 (11) ux(,0)0= (11)

ηπη(x, 0) ==+z0 + 0.01 cos(()πx/L) (12) (xz ,0)0 0.01cos xL / (12) where z0 is a reference level, and L is the horizontal length of the tube (L = 32 m), as shown in Figurewhere7 z.0 A is 1Da reference grid of ∆ level,x = 1 and m is L used is the and horizontal∆t is set tolength 0.01 of s. Threethe tube scenarios (L = 32 ofm), oscillating as shown flowsin Figure are considered7. A 1D grid below. of Δx = 1 m is used and Δt is set to 0.01 s. Three scenarios of oscillating flows are consideredIn Scenario below. 1, by setting z = 0.011, the oscillating flow in the horizontal section of the tube is 0 p fully pressurizedIn Scenario and1, by its setting oscillation z0 = 0.011, frequency the oscillating is given by flow2g /inL .the The horizontal flow of Scenario section 1 of falls the into tube the is categoryfully pressurized of TYPE-I and mixed its oscillation flows, and frequency the A-slot is is given not needed. by 2/gL In Scenario. The flow 2, byof Scenario setting z 1 =falls–0.011, into 0 p thethe oscillatingcategory of flowTYPE-I is free-surfacemixed flows, everywhere and the A-slot and is itsnot oscillation needed. In frequency Scenario 2, is by given setting by zπ0 = –0.011,gH/L, wherethe oscillatingH is the averageflow is free-surface water depth everywhere (0.989 m). The and flow its oscillation of Scenario frequency 2 has only is onegiven flow by regimeπ gH and/ L , does not fall into any category of the TYPE-I, -II or -III mixed flows, and the A-slot is also not needed. 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.

Figure 7. U-shaped tube with a square cross-section.

Water 2019, 11, x FOR PEER REVIEW 11 of 16

(m) 0.5

0.5 LS (dt=0.002 s) Z LS (dt=0.001 s) 0.4 LS (dt=0.0005 s) L (m) 2 Z

0.3 0.4 0.2

0.1 0.3 4.2 4.3 4.4 4.5 x(m) L3 0.0 012345678 x(m)

Figure 6. Simulated profiles of pressure head and hydraulic jump using different Δt.

3.3. Test Case 3 The frictionless oscillating flows within a U-shaped tube are simulated. The tube has a square cross-section area of 1 m2. The initial conditions are mathematically defined by:

ux(,0)0= (11) ηπ=+ () (xz ,0)0 0.01cos xL / (12) where z0 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. In Scenario 1, by setting z0 = 0.011, the oscillating flow in the horizontal section of the tube is fully pressurized and its oscillation frequency is given by 2/gL. 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 z0 = –0.011, the oscillating flow is free-surface everywhere and its oscillation frequency is given byπ gH/ L , Water 2019 11 where H, is ,the 1979 average water depth (0.989 m). The flow of Scenario 2 has only one flow regime11 and of 15 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)0) produced produced by by the the linear linear solver solver (LS) (LS) are plotted in Figure8 8,, andand theythey areare almostalmost thethe samesame as as the the analytical analytical solutions. solutions.

Water 2019, 11, x FOR PEER REVIEW 12 of 16 Figure 7. U-shaped tube with a square cross-section. Figure 7. U-shaped tube with a square cross-section. 0.010 Pressrized (Analytical)

Pressrized (LS) 0.005 (m) 0 0.000 z

η−

-0.005 Free-surface (Analytical) Free-surface (LS) -0.010 0 2 4 6 8 101214161820222426283032 x(m)

Figure 8. Simulated histories of water levels in scenarios of pure pressurizedpressurized/free-surface/free-surface ﬂows.flows.

In Scenario 3, by setting z0 = 0, the horizontal section of the tube is filled with a partially pressurized In Scenario 3, by setting z0 = 0, the horizontal section of the tube is filled with a partially andpressurized partially and free-surface partially free-surface flow, which flow, falls intowhich the falls category into th ofe category TYPE-III of mixed TYPE-III flows. mixed Moreover, flows. theMoreover, advection the ofadvection the flow of in the this flow case in is this so weakcase is that so weak its solution that its provides solution littleprovides help little to suppress help to thesuppress possible the nonphysical possible nonphysical oscillations oscillations in the solution in the ofsolution a linear of system a linear with system the with DMDC. the AsDMDC. a result, As thea result, oscillating the oscillating flow of Scenario flow of 3Scenario forms a 3 challenging forms a challenging test case of test TYPE-III case of mixed TYPE-III flows, mixed to which flows, the to theoreticalwhich the theoretical solutions are solutions not available. are not The available. solutions The of solutions Scenario 3of have Scenario been only3 have reported been only in studies reported by numericalin studies by methods, numerical and methods, the simulation and the results simulation of Casulli results and of Stelling Casulli [23 and] is Stelling taken here [23] as is ataken reference. here Theas a newreference. model The is testednew model using is six tested A-slots using whose six A-slotsε are sequentially whose ε are 0.02,sequentially 0.03, 0.04, 0.02, 0.05 0.03, 0.1, 0.04, and 0.05 0.2, to0.1, clarify and 0.2, its performanceto clarify its performanc in simulatinge in TYPE-III simulating mixed TYPE-III flows. mixed flows. For the sixsix simulationssimulations ofof ScenarioScenario 3,3, thethe water-levelwater-level historieshistories (at(at xx == 0) simulated by the linear solver (LS)(LS) usingusing didifferentfferent A-slots A-slots are are plotted plotted in in Figure Figure9 and 9 and compared compared with with that that in the in reference the reference [ 23]. The[23]. slots The slots of ε < of0.05 ε < are0.05 found are found to be to unable be unable to fully to fully eliminate eliminate the nonphysicalthe nonphysical oscillations. oscillations. When Whenε = 0.05–0.1,ε = 0.05 − simulation 0.1, simulation results results of the of current the current model model achieve achieve the best the agreement best agreement with those with in those [23]. in When [23]. theWhenε is the larger ε is than larger 0.1, than solutions 0.1, solutions of the linear of the solver linear begin solver to becomebegin to di becomeffusive. Thediffusive. suitable The value suitable of ε, whichvalue of helps ε, which the current helps modelthe current achieve model stable achieve and accurate stable and simulations accurate ofsimulations the TYPE-III of mixedthe TYPE-III flows, ismixed suggested flows, to is besugges 0.05–0.1.ted to be 0.05–0.1.

0.010 Casulli and Stelling (2013) LS (slot=0.02) LS (slot=0.03) 0.005 LS (slot=0.04) LS (slot=0.05) LS (slot=0.1) LS (slot=0.2) (m) 0 0.000 z

η−

-0.005

-0.010 0 2 4 6 8 101214161820222426283032 x(m)

Figure 9. Simulated histories of water levels in scenarios of mixed flows with an A-slot.

Water 2019, 11, x FOR PEER REVIEW 12 of 16

0.010 Pressrized (Analytical) Pressrized (LS) 0.005 (m) 0 0.000 z

η−

-0.005 Free-surface (Analytical) Free-surface (LS) -0.010 0 2 4 6 8 101214161820222426283032 x(m)

Figure 8. Simulated histories of water levels in scenarios of pure pressurized/free-surface flows.

In Scenario 3, by setting z0 = 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. For the six simulations of Scenario 3, the water-level histories (at x = 0) simulated by the linear solver (LS) using different A-slots are plotted in Figure 9 and compared with that in the reference [23]. The slots of ε < 0.05 are found to be unable to fully eliminate the nonphysical oscillations. When ε = 0.05 − 0.1, simulation results of the current model achieve the best agreement with those in [23]. When the ε is larger than 0.1, solutions of the linear solver begin to become diffusive. The suitable value of ε, which helps the current model achieve stable and accurate simulations of the TYPE-III Water 2019 11 mixed flows,, , 1979 is suggested to be 0.05–0.1. 12 of 15

0.010 Casulli and Stelling (2013) LS (slot=0.02) LS (slot=0.03) 0.005 LS (slot=0.04) LS (slot=0.05) LS (slot=0.1) LS (slot=0.2) (m) 0 0.000 z

η−

-0.005

-0.010 0 2 4 6 8 101214161820222426283032 x(m)

Figure 9. Simulated histories of water levels in scenarios of mixed ﬂowsflows with an A-slot.A-slot. 4. Discussion

4.1. Differences between the New Model and Existing Models In the introduction, two categories of models for the mixed flows have been reviewed, respectively, the TSE model (using two sets of equations) and the ASE model (using a single set of equations). The algorithms of the TSE models are often case-specific so that it is difficult to apply them to real applications. The ASE models can be further divided into two sub kinds, respectively, the ASE-1 (the pressure is often not explicitly included in the governing equations) and the ASE-2 (the pressure is often explicitly included). The new model solves two regimes of mixed flow using a unified formulation and the pressure term is explicitly included in the governing equations, so it is simpler than most of the existing TSE and ASE-1 models. The new model can be classified to the ASE-2 models and may be considered as a variant of the ASE-2 model of Casulli and Stelling [23]. In the new model, the velocity–pressure coupling is solved by constructing and solving a linear system instead of the nonlinear system in [23]. The resulting linear system can be solved using a direct method and no iterations are needed. The new model is different with the model of [23], in terms of the continuity equation and its discretization (see Section 2.3.1), the construction of velocity-pressure coupling and the solution of algebraic equations (see Section 2.3.2). Generally, construction and solution of a nonlinear system is much more complex than those of a linear system, and the new model is obviously simpler than that of [23]. Moreover, the new model is revealed in tests to be applicable to various types of mixed flows, where stable and accurate simulations can be achieved at large time steps (CFL > 1). The ability of the new model is attractive for real applications. Moreover, the simplicity of the model allows that other researchers can develop a similar mixed-flow model easily and quickly.

4.2. The DMDC Problem and the A-Slot Approach The main disadvantage of the new model is the possible DMDC problem, and this problem is clarified as follows: We analyze the characteristics of various typical mixed flows (see Section 2.3.3), and only the simulation of TYPE-III mixed flows (mixed flows with frequent conversions of flow regimes in a closed pipe with wide-top cross-sections) may trigger the DMDC problem of the linear solver. At the same time, the DMDC problem of the linear solver is not obvious in the simulations of other mixed flows (e.g., the TYPE I and -II mixed flows). The A-slot technique is proposed to alleviate the possible DMDC problem of the linear solver (eliminate the nonphysical oscillations of solutions), when the model is used to simulate the TYPE-III mixed flows. On the one hand, the A-slot technique and the traditional Preissmann slot technique [20] can be differentiated in two aspects. First, these two slot techniques have quite different physical meanings. Water 2019, 11, 1979 13 of 15

The A-slot technique is proposed to alleviate the possible instabilities of the linear solver related to the DMDC problem. The traditional Preissmann slot is used to help realizing “only one flow type (free-surface flow) throughout the whole pipe [9]” in the solution of mixed flows. Second, the model without an A-slot can be applied to all types of mixed flows except TYPE III mixed flows, and the A-slot is only needed for the simulation of TYPE III mixed flows. As a result, the A-slot is not an essential part of the new model except that a special case (the TYPE-III mixed flows) is simulated. To be different, the Preissmann slot technique is an essential part of the model no matter which kind of mixed flows is simulated. On the other hand, an interesting finding is that the A-slot and the Preissmann slot favor quite similar slot widths. Although the slot width (ε) of a Preissmann slot should be sized to match the assumed celerity in the pressurized part of the flow, in most cases the ε is arbitrarily chosen based on numerical considerations [17]. In existing studies, the ε of a Preissmann slot is often set to 0.02 [13,18], 0.05 [2,34], 0.1 [14,16], etc. In tests of this study, the suitable value of ε, which enables the linear solver to achieve stable and accurate simulations of TYPE-III mixed flows, is suggested to be 0.05–0.1, and they are quite similar to the widths of the Preissmann slots used in existing models.

5. Conclusions A semi–implicit numerical model with a linear solver is proposed for the free-surface and pressurized mixed flows in hydraulic systems. It solves the two flow regimes within a unified formulation and is much simpler than existing similar models for mixed flows. Using a local linearization and an ELM, the new model only needs to solve a tridiagonal linear system arising from the velocity-pressure coupling. The linear system has a symmetric and positive definite coefficient matrix, and can be directly solved without iterations. Characteristics of three representative types (TYPE I, II, and III) of mixed flows are analyzed, and the applicability of the new model to each type of them is discussed. TYPE-III mixed flow, characterized by both flow-regime conversions and a closed pipe with wide-top cross-sections, is the most challenging case. In simulations of TYPE-III mixed flows using the linear solver, an artificial slot (A-slot) technique is proposed to cope with the possible instabilities related to the discontinuous main-diagonal coefficients of the linear system. The new model is tested using various mixed flows of TYPE I, II, and III, where the simulation results agree with the analytical solutions, experiment data, and the simulation results reported by former researchers. Sensitivity studies of grid scales and time steps are both performed, where a common grid scale provides grid-independent results and a common time step provides time-step-independent results. Moreover, the model is revealed to achieve stable and accurate simulations at large time steps for which the CFL can be greater than 1. In simulations of a TYPE-III mixed flow using an A-slot, a slot-width sensitivity study is also performed. To ensure stable and accurate simulations of TYPE-III mixed flows, the suitable slot-width ratio (ε) for the linear solver is suggested to be 0.05–0.1.

Author Contributions: Conceptualization: D.H. and Z.J.; methodology: D.H. and Z.J.; software: D.H. and S.L.; validation: D.H., S.L. and Z.J.; formal analysis: D.H., S.L. and Z.J.; investigation: D.H. and Z.J.; resources: S.Y., and Z.J.; data curation: S.Y. and D.H.; writing—original draft preparation: D.H. and Z.J.; writing—review and editing: S.L. and D.H.; visualization: S.L.; supervision: S.Y. and Z.J.; project administration: S.Y. and Z.J.; funding acquisition: D.H., S.Y. and Z.J. Funding: Financial support from Science and Technology Program of Guizhou Province, China (grant no. 2017-3005-4), the Fundamental Research Funds for the Central Universities (2017KFYXJJ197), China’s National Natural Science Foundation (51339001, 51779015, and 51479009) are acknowledged. Conﬂicts of Interest: The authors declare no conﬂicts of interest.

Notation The following symbols and abbreviations are used in this paper: Water 2019, 11, 1979 14 of 15

A Wetted cross-sectional area (conveyance area); Ap The area of the cross-section; A-slot Artificial slot for coping with the discontinuous main-diagonal coefficients problem =∂A/∂η, the wet surface width of a cross-section for the free-surface parts and is equal to 0 for B the pressurized parts of a closed pipe; g Gravity acceleration; ne The quantities of cells; ns The quantities of sides; nm Manning’s roughness coefficient; Nbt The number of substeps of the ELM R Hydraulic radius; t Time; u Averaged velocity of cross-section; ubt Solution of the advection term uisng the ELM; Wi Representative width of the cross-section; x Longitudinal distance along the channel; zc The celling of the cross-section; t Time; ∆t Time step ∆xi The length of cell i; ∆xi+1/2 The distance between the centers of cells i and i + 1; θ Implicit factor Water level measured from an undisturbed reference water surface (for free-surface flows); η 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;

References

1. Yen, B.C. Hydraulics of sewer systems. In Stormwater Collection Systems Design Handbook; McGraw-Hill: New York, NY, USA, 2001. 2. Kerger, F.; Archambeau, P.; Erpicum, S.; Dewals, B.J.; Pirotton, M. A fast universal solver for 1D continuous and discontinuous steady flows in rivers and pipes. Int. J. Numer. Meth. Fluids 2011, 66, 38–48. [CrossRef] 3. Hamam, M.A.; McCorquodale, J.A. Transient conditions in the transition from gravity to surcharged sewer flow. Can. J. Civ. Eng. 1982, 9, 189–196. [CrossRef] 4. Li, J.; McCorquodale, A. Modeling mixed flow in storm sewers. J. Hydraul. Eng. 1999, 125, 1170–1180. [CrossRef] 5. Zhou, F.; Hicks, F.E.; Steffler, P.M. Transient flow in a rapidly filling horizontal pipe containing trapped air. J. Hydraul. Eng. 2002, 28, 25–634. [CrossRef] 6. Song, C.S.S.; Cardle, J.A.; Leung, K.S. Transient mixed flow models for storm sewers. J. Hydraul. Eng. 1983, 109, 1487–1504. [CrossRef] 7. Cardle, J.A.; Song, C.S.S. Mathematical modeling of unsteady flow in storm sewers. Int. J. Eng. Fluid Mech. 1988, 1, 495–518. 8. Guo, Q.; Song, C.S.S. Surging in urban storm drainage systems. J. Hydraul. Eng. 1990, 116, 1523–1537. [CrossRef] 9. Fuamba, M. Contribution on transient flow modelling in storm sewers. J. Hydraul. Res. 2002, 40, 685–693. [CrossRef] 10. Wang, K.H.; Shen, Q.; Zhang, B. Modeling propagation of pressure surges with the formation of an air pocket in pipelines. Comput. Fluids 2003, 32, 1179–1194. [CrossRef] 11. Politano, M.; Odgaard, A.J.; Klecan, W. Case study: Numerical evaluation of hydraulic transients in a combined sewer overflow tunnel system. J. Hydraul. Eng. 2007, 133, 1103–1110. [CrossRef] Water 2019, 11, 1979 15 of 15

12. Bousso, S.; Daynou, M.; Fuamba, M. Numerical Modeling of Mixed Flows in Storm Water Systems: Critical Review of Literature. J. Hydraul. Eng. 2013, 139, 385–396. [CrossRef] 13. Garcia-Navarro, P.; Priestley, A.; Alcrudo, S. Implicit method for water flow modeling in channels and pipes. J. Hydraul. Res. 1994, 32, 721–742. [CrossRef] 14. Capart, H.; Sillen, X.; Zech, Y. Numerical and experimental water transients in sewer pipes. J. Hydraul. Res. 1997, 35, 659–672. [CrossRef] 15. Ji, Z. General hydrodynamic model for sewer/channel network systems. J. Hydraul. Eng. 1998, 124, 307–315. [CrossRef] 16. Trajkovic, B.; Ivetic, M.; Calomino, F.; D’Ippolito, A. Investigation of transition from free surface to pressurized flow in a circular pipe. Water Sci. Technol. 1999, 39, 105–112. [CrossRef] 17. Vasconcelos, J.G.; Wright, S.J.; Roe, P.L. Improved Simulation of Flow Regime Transition in Sewers: Two-Component Pressure Approach. J. Hydraul. Eng. 2006, 132, 553–562. [CrossRef] 18. León, A.S.; Ghidaoui, M.S.; Schmidt, A.R.; García, M.H. Application of Godunov-type schemes to transient mixed flows. J. Hydraul. Res. 2009, 47, 147–156. [CrossRef] 19. Bourdarias, C.; Gerbi, S. A conservative model for unsteady flows in deformable closed pipes and its implicit second-order finite volume discretisation. Comput. Fluids 2008, 37, 1225–1237. [CrossRef] 20. Cunge, J.A.; Wegner, M. Numerical integration of Barre de Saint-Venant’s flow equations by means of implicit scheme of finite differences. Houille Blanche 1964, 19, 33–39. [CrossRef] 21. Sanders, B.F.; Bradford, S.F. Network implementation of the two-component pressure approach for transient flow in storm sewers. J. Hydraul. Eng. 2011, 137, 158–172. [CrossRef] 22. Aldrighetti, E. Computational Hydraulic Techniques for the Saint Venant Equations in Arbitrarily Shaped Geometry. Ph.D. Thesis, University of Trento, Trento, Italy, 2007. 23. Casulli, V.; Stelling, G.S. A semi-implicit numerical model for urban drainage systems. Int. J. Numer. Meth. Fluids 2013, 73, 600–614. [CrossRef] 24. Dumbser, M.; Iben, U.; Ioriattia, M. An efficient semi-implicit finite volume method for axially symmetric compressible flows in compliant tubes. Appl. Numer. Math. 2015, 89, 24–44. [CrossRef] 25. Leskens, J.G.; Brugnach, M.; Hoekstra, A.Y.; Schuurmans, W. Why are decisions in flood disaster management so poorly supported by information from flood models? Environ. Model. Softw. 2014, 53, 53–61. [CrossRef] 26. Storm Water Management Model (SWMM). Storm Water Management Model Reference Manual Volume II-Hydraulics; EPA/600/R-17/111; Rossman: Los Angeles, CA, USA; National Risk Management Laboratory: Cincinnati, OH, USA, 2017. 27. Dimou, K. 3-D Hybrid Eulerian–Lagrangian/Particle Tracking Model for Simulating Mass Transport in Coastal Water Bodies. Ph.D. Thesis, Department of Civil Engineering, Massachusetts Institute of Technology, Cambridge, MA, USA, 1992. 28. Casulli, V.; Zanolli, P. Semi-implicit numerical modeling of nonhydrostatic free-surface flows for environmental problems. Math. Comput. Model. 2002, 36, 1131–1149. [CrossRef] 29. Zhang, Y.L.; Baptista, A.M.; Myers, E.P.A cross-scale model for 3D baroclinic circulation in estuary-plume-shelf systems, I. Formulation and skill assessment. Cont. Shelf Res. 2004, 24, 2187–2214. [CrossRef] 30. Hu, D.C.; Zhong, D.Y.; Zhang, H.W.; Wang, G.Q. Prediction–Correction Method for Parallelizing Implicit 2D Hydrodynamic Models I Scheme. J. Hydraul. Eng. 2015, 141, 04015014. [CrossRef] 31. Hu, D.C.; Zhong, D.Y.; Zhu, Y.H.; Wang, G.Q. Prediction–Correction Method for Parallelizing Implicit 2D Hydrodynamic Models II Application. J. Hydraul. Eng. 2015, 141, 06015008. [CrossRef] 32. Pothof, I. Co–Current Air–Water Flow in Downward Sloping Pipes; Gildeprint Drukkerijen BV: Enschede, The Netherland, 2011; ISBN 978-90-8957-018-5. 33. Lubbers, C.L. On Gas pockets in Wastewater Pressure Mains and Their Effect on Hydraulic Performance. Ph.D. Thesis, Department of Civil Engineering and Geosciences, University of Technology, Delft, The Netherlands, 2007. 34. Ferreri, G.B.; Freni, G.; Tomaselli, P. Ability of Preissmann slot scheme to simulate smooth pressurisation transient in sewers. Water Sci. Technol. 2010, 62, 1848–1858. [CrossRef]

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