# New Optimized Equal-Area Mesh Used in Axisymmetric Models for Laminar Transient Flows

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Numerical Models

#### 2.1. The Vardy and Hwang (1991) Model: Basic Equations, Assumptions and Numerical Scheme

_{C}concentric and hollow cylinders, extended along the whole pipe length (Figure 1). The pipe length, L, is divided into ${N}_{X}$ equal reaches, such that $\Delta x=L/{N}_{X}$, and each hollow cylinder has $\Delta x$ length. Each cylinder is defined by the respective index, j, increasing from the centerline ($j=0$) to the pipe wall ($j={N}_{C}-1$). The wall thickness of the j

^{th}cylinder is denoted by $\Delta {r}_{j}$ ($={r}_{j}-{r}_{j-1}$), the external radius by ${r}_{j}$ ($={\displaystyle \sum}_{0}^{j}\Delta {r}_{j}$) and the cylinder center radius is $\overline{{r}_{j}}$ $=\left({r}_{j-1}+{r}_{j}\right)/2$. The time step is determined according to the Courant condition, $\Delta t=\Delta x/c$. Each conduit section, referred to as x

_{i}, is located at a distance $x=\Delta x.i$ from the upstream end of the pipe.

#### 2.2. The Complete Zhao and Ghidaoui (2003) Model

_{C}× 2N

_{C}) sparse matrix. Zhao and Ghidaoui (2003) [24] proposed an improved higher efficiency numerical model, developed by algebraic manipulation of Equations (3) and (4) into two smaller systems with tridiagonal matrices. The use of tridiagonal matrices allows the use of a faster form of Gaussian elimination, named the Thomas algorithm [35]. This algorithm obtains the solution via N

_{C}operations instead of N

_{C}

^{3}operations needed by the original Vardy and Hwang (1991) model, where N

_{C}is the matrix dimension.

#### 2.3. The Simplified Zhao and Ghidaoui (2003) Model

#### 2.4. Radial Mesh Discretization

**N**

_{CR}**”, in which N**

_{C}_{C}represents the total number of cylinders. This radial geometry has been widely used by several researchers [24,25,27].

**N**, that is “EAC N

_{C}**”.**

_{C}_{C}, as “ETC N

**”, as follows:**

_{C}## 3. Experimental Facility Description

^{3}and a kinematic viscosity ν = 1.01 × 10

^{−6}m

^{2}/s.

## 4. Numerical Results for the Instantaneous Valve Closure

#### 4.1. Introduction

#### 4.2. Standard Equal-Area Cylinder Mesh: Effect of the Mesh Discretization

_{C}, is observed immediately after the first pressure wave arrival. The 1D model shows a high wall shear stress peak after each pressure surge, followed by an exponential decay (Figure 5a). Low N

_{C}meshes show a lower peak and a linear decay with time. Increasing the N

_{C}value brings the Q2D model simulation closer to Zielke’s 1D solution. The N

_{C}= 20 mesh reduces the Zielke’s peak more than 17 times; the N

_{C}= 150 mesh increases the previous value almost eight times but is still approximately half the 1D model peak.

_{C}= 20 mesh, the Q2D model piezometric-head results (Figure 5b–d) do not depict the correct wave shape (the round-shape), showing a square shape geometry, and present a poor calculation of the wave damping (e.g., for t/T = 0.25, the difference to the 1D model is noticeable). Increasing the number of cylinders allows progressive approximation of Zielke’s results.

_{0}is the velocity profile immediately after the pressure wave arrival, and each new time adds t/16T to the previous one (t

_{0}= 0, t

_{1}= t/16T, t

_{2}= t/8T, t

_{3}= 3/16T and t

_{4}= 1/4T).

_{0}to t

_{5}, the velocity profile gradient near the wall reduces as the inflection point moves into the pipe core. The transient flow changes occur close to the pipe wall. At t

_{4}, just before the next pressure surge, the inflection point maximum displacement from the pipe wall is less than 4% of the pipe radius. Therefore, a high percentage of the pipe profile is unaffected and the Q2D model’s cylinders maintain a steady-state profile axial velocity.

#### 4.3. Optimized Equal-Area Cylinder Mesh

#### 4.3.1. Mesh Description

_{C}; (ii) the limit of the high-resolution grid, r, defined as a percentage of the pipe radius and measured from the pipe wall; and (iii) the number of cylinders in the high-resolution region, N

_{HR}. Thus, the optimized equal area cylinder (OEAC) meshes are expressed as follows, OEAC

**N**Nr = N

_{C}_{HR}. For instances, OEAC40 N

_{5%}= 20 stands for a mesh that has a total of 40 cylinders (N

_{C}= 40); the limit of the high-resolution mesh occurs at r = 9.5 mm (r/R = 5%) and has 20 cylinders in this high-resolution region.

_{C}= 40, because the EAC mesh with 40 cylinders does not lead to an accurate simulation of the piezometric head (Figure 5b), representing a good comparison with optimized mesh.

#### 4.3.2. Shear Stress Results

_{HR}between 10 to 30, r = 5% and N

_{C}= 40 are presented in Figure 8. The r = 5% corresponds to the inflection point maximum displacement determined in the previous analysis (Figure 6). All simulations’ accuracy has significantly improved compared to the EAC results (Figure 4), for the same total number of cylinders (N

_{C}= 40). The new mesh with more than 20 cylinders in the wall region (OEAC40 N

_{5%}= 20) also presents better results than the 150 equal area cylinder meshes (EAC150). These results are compared with those obtained by the Zielke formulation used in the 1D model.

_{5%}= 30 mesh allows an almost perfect adjustment to Zielke’s results (Figure 8a), whereas the OEAC40 N

_{5%}= 10 mesh leads to a maximum wall shear stress four times lower than the Zielke’s value. This shows that the higher the number of cylinders in the high-resolution area, the more accurate the prediction of the maximum wall stress and of the subsequent decay. A second aspect concerns the relation between the wall shear stress peak values in the deceleration (t/T = 0 and t/T = 0.25) and the acceleration (t/T = 0.50 and t/T = 0.75) phases. For the meshes with higher resolution near the wall (N

_{HR}≥ 20), the peaks decay with time, as in the Zielke model; on the contrary, for the low resolution mesh (Figure 8c), the peaks increase compared with the previous surge for t/T = 0.25 and t/T = 0.75, as observed for the standard meshes (Figure 8a).

_{HR}reduction. For the same number of cylinders, the EAC40 mesh results have an error almost 10 times higher than that of the best OEAC mesh (OEAC40 N

_{5%}= 30). The EAC150 results present the same MAE as the OEAC40 N

_{5%}= 15 and a higher error than when using a higher N

_{HR}value. This analysis highlights that a higher accuracy can be achieved with one-quarter of the number of standard mesh cylinders (40/150). OEAC mesh’s results are clearly better than those obtained for the EAC grid, considering the same (EAC40), or even a higher (EAC150), value of the total number of cylinders.

#### 4.3.3. Mean Velocity and Piezometric Head Results

_{1D}is the steady-state mean velocity obtained by Hagen–Poiseuille solution and U

_{2D}is the Q2D model steady-state flow approximation for each mesh. Firstly, OEAC meshes have a worse steady-state velocity representation than the EAC meshes (i.e., the EAC40 mesh leads to a lower error). Secondly, this error increases exponentially with the N

_{HR}increase. The OEAC40 N

_{5%}= 10 mesh has a 0.08% error, which is close to EAC40 (0.06%), but the OEAC40 N

_{5%}= 30 error rises to 0.64% (i.e., 8 to 10 times higher).

_{HR}(Figure 11b; see black and green curves for r = 9.5 mm). This discontinuity occurs in the transition between the two regions of the OEAC meshes (i.e., at r= 9.5 mm), being more noticeable for the OEAC40 N

_{5%}= 30 mesh.

_{HR}values allow a considerable improvement in the wall shear stress simulation results without increasing N

_{C}; however, a balance with higher steady-state error must be accomplished. The MAE of the mean velocity simulation is calculated in the first wave period (t/T < 1) using Zielke’s model as benchmark, as depicted in Figure 12a. The EAC meshes (EAC40 and EAC150) and OEAC meshes with N

_{C}= 40, and varying the N

_{HR}values from 30 to 10, are considered. OEAC40 N

_{5%}= 20 has the lowest error (MAE = 0.15), close to that of the standard EAC150 (MAE = 0.16) and five times lower than that for the standard EAC40 mesh (with the same number of cylinders). Increasing the number of cylinders in the wall region (N

_{HR}from 20 to 30) increases the MAE value because a higher wall shear stress accuracy does not compensate for a lower steady velocity profile accuracy; decreasing the number of cylinders in the wall region also increases the error for the opposite reason. The piezometric-head results lead to the same conclusions (Figure 12b).

## 5. Numerical Results for the Calibrated Valve Closure

#### 5.1. Introduction

_{HR}/N

_{C}= 0.5 ratio is considered, as it presented the best results for an instantaneous valve closure. First, results obtained with traditional radial meshes (i.e., EAC, ETC and GS) are compared and discussed for both steady and unsteady flow conditions. Second, the Q2D results with an OEAC20 mesh are compared with the experimental data and with results from 1D models.

#### 5.2. Comparison with Traditional Meshes’ Results

#### 5.2.1. Steady-State Mesh Assessment

_{C}), and presented in Figure 13. The relative error is given by $\left({U}_{2D}-{U}_{1D}\right)/{U}_{1D}$ in which ${U}_{1D}$ is the input mean velocity and ${U}_{2D}$ is the Q2D model steady-state mean velocity (i.e., after the convergence process).

_{C}increase. The equal thickness cylinders mesh (ETC) has the best performance. The ETC mesh error drops 25 times with the $\Delta r$ reduction to 1/5 (increasing N

_{C}from 20 to 100), as expected by the finite difference scheme truncation error, O($\Delta {r}^{2}$); see Equation (10). The cylinders have a uniform thickness along the pipe radius, and there is no error associated with the grid non-uniformity.

_{C}increase, because the cylinder thickness reduces faster in the wall area than in the pipe core. The OEAC meshes give overall worse results. Only with N

_{C}= 60 does this geometry present a higher accuracy than the ETC20 (i.e., with three times the N

_{C}value and computer effort). The equal area cylinder mesh (EAC) presents better results than those of the OEAC meshes. The GS mesh does not show the same accuracy improvement with N

_{C}increase; for N

_{C}= 20, this mesh is close to the best results (ETC20); on the contrary, for N

_{C}= 100, it tends to the mesh with the worst results (OEAC100). With the N

_{C}increase, the GS geometry focuses its refinement on the wall area, where the axial velocity has lower values. The mesh refinement in the wall area explains the GS slower error reduction with N

_{C}and the lower overall performance of OEAC.

#### 5.2.2. Unsteady-State Mesh Assessment

_{C}= 300 is considered for benchmark comparison. The number of cylinders for each geometry varies from 20 to 100 (N

_{C}= 20, 40, 60, 80 and 100). Figure 14 shows the piezometric head results for the different meshes, plotting only results with significant differences from those of the benchmark mesh.

_{C}value (GS40) to ensure the same global accuracy (Figure 14b). The EAC mesh requires 80 cylinders (i.e., four times the OEAC effort) to achieve the same accuracy (Figure 14c). Only the ETC grid with the maximum number of 100 cylinders (ETC100) and five times the OEAC20 mesh computational effort can ensure the same simulation accuracy (Figure 14d). The wave damping is correctly calculated by most meshes, except for non-OEAC meshes with the lowest N

_{C}.

_{C}values (N

_{C}≤ 60). Compared with the second-best results (EAC20, GS40 and GS60), this mesh allows an overall error reduction of 86%, 88% and 66% for N

_{C}= 20, 40 and 60, respectively. Traditional meshes with a low number of cylinders are not capable of correctly describing the ${\mathit{\tau}}_{\mathrm{wall}}$ peak and its subsequent exponential decay. For higher Nc values (N

_{C}≥ 80), the GS meshes have higher accuracy than the OEAC meshes. EAC and ETC meshes have the worst results: for instance, the OEAC20 has a similar MAE value to the EAC100 and is three times smaller than ETC100, both with five times the new mesh computer effort. The quasi-steady 1D model results are incompatible with an accurate energy dissipation and with double the worst Q2D mesh error (i.e., ETC20). Trikha’s 1D results are comparable with those of low resolution meshes but do not match the best Q2D model’s results.

#### 5.3. Comparison with Experimental Data

_{5%}= 10). The quasi-steady and Trihka’s 1D models are also included, for comparison purposes. The analysis excludes Zielke’s 1D model, as it produces similar results to those obtained by the Q2D model, requiring higher computation time. All model simulations consider the calibrated valve closure presented in Figure 4a.

_{C}value (OEAC20) ensures both the wave attenuation (compare the Q2D model and the test data for the middle point of each wave peak) and the front wave shape (see Figure 16c,d). However, the Q2D model cannot correctly describe the symmetric wave shape observed in the experimental data.

## 6. Complete Versus Simplified Q2D Models

_{1}= 0.5 and θ

_{1}= 1.0, respectively. The accuracy of the three models—complete Q2D, simplified Q2D (θ

_{1}= 0.5) and simplified Q2D (θ

_{1}= 1)—is assessed herein for the instantaneous and for the calibrated valve maneuvers.

#### 6.1. Instantaneous Valve Manoeuvre

_{C}values from 20 to 100 and compared with the Zielke’s analytical solution for ∆x = 0.01m. For N

_{C}= 100, a slight difference is registered between the three models during the high velocity-gradient immediately after the pressure surge and fades shortly after (Figure 17). The complete model has the best accuracy, and the simplified model with θ

_{1}= 1.0 the worst. Nevertheless, for the lowest N

_{C}meshes (N

_{C}= 60 and N

_{C}= 80), the differences between the three models are hardly observed because the period immediately after the pressure surge is not described so accurately.

_{C}values. The rationale is to combine these two aspects in a single parameter for best selection of the numerical model. The MAE value exponentially decreases with N

_{C}. Increasing the N

_{C}value ensures a better simulation accuracy independently of the numerical model selected. Increasing the model complexity is not justifiable for meshes with low N

_{C,}and the simplified model with θ

_{1}= 1.0 is the best option. For higher N

_{C}values the simplified model with θ

_{1}= 0.5 compensates for the additional computation time. The complete model is only recommended for a detailed evaluation in the period immediately after the pressure surge.

#### 6.2. Calibrated Valve Manoeuvre

_{C}= 100 meshes are considered, since the results for lower N

_{C}values were negligible for instantaneous valve closure.

_{1}= 0.5). On the contrary, the results of the high-resolution grids (OEAC and GS) are influenced by the model, and the conclusions are similar to those obtained for the instantaneous valve maneuver.

## 7. Conclusions

_{C}≤ 60). The other two radial meshes (ETC and EAC) do not provide the same accuracy. The new mesh correctly describes the experimental data with only 20 cylinders, which is not achieved by the fastest 1D models.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

A_{j}, B_{j}, C_{j}, D_{j} | quasi-2D model coefficients (-) |

C_{R} | geometric sequence common ratio (-) |

c | pressure wave speed (m/s) |

D | pipe inner diameter (m) |

g | acceleration due to gravity (m/s^{2}) |

H | piezometric head (m) |

L | pipe length (m) |

N_{C} | total number of cylinders of the radial mesh (-) |

N_{HR} | total number of cylinders in the high-resolution area (OEAC meshes) |

N_{X} | number of axial mesh points in the pipe direction (-) |

Q | flow-rate or discharge (m^{3}/s) |

R | pipe inner radius (m) |

Re | Reynolds number, Re = UD/ν (-) |

r | distance from the axis in the radial direction (m) |

${r}_{j}$ | external radius of the j^{th} cylinder measured from the pipe axis (m) |

${\overline{r}}_{j}$ | center radius of the j^{th} cylinder measured from the pipe axis (m) |

T | pressure wave period, T = 4L/c (s) |

t | time (s) |

U | mean velocity of the fluid in the pipe cross-section (m/s) |

u | axial velocity (m/s) |

$\upsilon $ | radial velocity (m/s) |

x | distance along the pipe (m) |

∆t | numerical time step (s) |

∆x | numerical spatial step (m) |

∆r_{j} | $\mathrm{wall}\mathrm{thickness}\mathrm{of}\mathrm{the}{j}^{\mathit{th}}\mathrm{cylinder}(\Delta {r}_{j}={r}_{j}-{r}_{j-1}$) |

ε | pipe wall roughness (m) |

${\theta}_{1}$$,{\theta}_{1}$ | $\mathrm{weighting}\mathrm{coefficient}({\theta}_{1}+{\theta}_{2}=1$) |

ν | kinematic viscosity of liquid (m^{2}/s) |

𝜚 | liquid density (kg/m^{3}) |

𝜏 | wall shear stress (Pa) |

## Abbreviations

CFD | Computation fluid dynamics |

CFL | Courant-Friedrich-Lewy |

EAC | Equal area cylinder mesh |

ETC | Equal thickness cylinder mesh |

GS | Geometric sequence cylinder mesh |

MAE | Mean absolute error |

MOC | Method of Characteristics |

Q2D | Quasi-2D model |

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**Figure 2.**Radial meshes with different geometries for N

_{C}= 40: (

**a**) GS5%40; (

**b**) EAC40; and (

**c**) ETC40.

**Figure 3.**Experimental test facility: (

**a**) upstream anchored copper pipe; (

**b**) downstream pipe end; and (

**c**) pneumatically actuated ball valve.

**Figure 4.**Calibrated valve maneuver for laminar flow: (

**a**) Q2D model flow rate variation; (

**b**) experimental and Q2D model piezometric head variation.

**Figure 5.**Q2D simulations for EAC meshes with increasing N

_{C}using Zielke’s 1D model as benchmark: (

**a**) wall shear stress; (

**b**) piezometric-head; (

**c**,

**d**) details of the piezometric-head.

**Figure 6.**Axial velocity for four instances equally spaced in time during the first pressure wave: (

**a**) complete pipe radius, 0 ≤ r (mm) ≤ 10; (

**b**) close to the pipe wall, 8 ≤ r (mm) ≤ 10.

**Figure 8.**Wall shear stress obtained for OEAC meshes and for the Zielke’s 1D model: (

**a**) OEAC40 N

_{5%}= 30; (

**b**) OEAC40 N

_{5%}= 20; (

**c**) OEAC40 N

_{5%}= 10.

**Figure 11.**Shear stress profile for EAC40, EAC150, OEAC40 N

_{5%}= 10 and OEAC40 N

_{5%}= 30: (

**a**) 0 ≤ r ≤ 10 mm; and (

**b**) 9.0 ≤ r ≤ 10 mm.

**Figure 12.**MAE for EAC and varying the N

_{HR}value in OEAC meshes: (

**a**) wall shear stress; (

**b**) piezometric head.

**Figure 14.**Piezometric head time history for the S-shape valve closure: (

**a**) OEAC, (

**b**) GS, (

**c**) EAC, and (

**d**) ETC.

**Figure 16.**Piezometric head time history at conduit midsection, comparing experimental data, quasi-2D and 1D models: (

**a**) 0.0 ≤ t/T ≤ 10.0; (

**b**) 0.0 ≤ t/T ≤ 1.0; (

**c**) 9.0 ≤ t/T ≤ 10.0.

**Figure 18.**MAE and CPU time product calculation for the three numerical schemes using 1D Zielke model as benchmark: (

**a**) for 20 ≤ N

_{C}≤ 100; and (

**b**) detail for N

_{C}= 80 and 100.

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**MDPI and ACS Style**

Ferreira, P.L.; Covas, D.I.C.
New Optimized Equal-Area Mesh Used in Axisymmetric Models for Laminar Transient Flows. *Water* **2023**, *15*, 1402.
https://doi.org/10.3390/w15071402

**AMA Style**

Ferreira PL, Covas DIC.
New Optimized Equal-Area Mesh Used in Axisymmetric Models for Laminar Transient Flows. *Water*. 2023; 15(7):1402.
https://doi.org/10.3390/w15071402

**Chicago/Turabian Style**

Ferreira, Pedro Leite, and Dídia Isabel Cameira Covas.
2023. "New Optimized Equal-Area Mesh Used in Axisymmetric Models for Laminar Transient Flows" *Water* 15, no. 7: 1402.
https://doi.org/10.3390/w15071402