Numerical Dissipation Compensation in Liquid Column Separation: An Improved DVCM Approach

Abstract

Accurate water hammer mitigation simulation is crucial for designing and protecting pipeline systems. This study draws on the principles of the Muskingum method to develop an improved discrete vaporous cavity model (DVCM) that enhances computational accuracy. The key improvements include significantly reducing numerical dissipation by optimizing the Courant number (C_n) and adjusting the friction coefficient (f) to balance numerical and physical dissipation. Specifically, the predictive accuracy of the node water head was improved by 69.93%, and the accuracy of the liquid–column separation time was enhanced by 77.21%. These enhancements were achieved by proposing an equivalent treatment method for numerical and physical dissipation, ensuring that the model’s total dissipation matches actual physical conditions. The validation process involved simulating the liquid column separation process using data from the Simpson experiment. The results demonstrated a high degree of consistency with the experimental data, confirming the effectiveness of the proposed method.

1. Introduction

Liquid column separation refers to certain parts of the pipeline (especially the raised parts) of the pressure drop below the saturated steam pressure; the water in the pipeline begins to vaporize and form a steam cavity. When the pressure near the steam cavity of the pipeline rises above the steam pressure, steam liquefaction again, and the steam cavity gradually decreases until the collapse; the collapse of the moment will produce a considerable rise in pressure, the occurrence of a pipe burst. In the pumping station system, the maximum elevated pressure [1,2] will even exceed the maximum theoretical water hammer pressure [3]. It can reach 7 times the normal operating pressure of the water column [4]; improper protection will cause piping system [5] major water hammer accidents, such as pumps, piping, valves, and other equipment damage, and even flooding of the pumping station, water supply interruption, and other serious consequences.

Liquid–column separation is a complex phenomenon involving the variation in two-phase flow, which has attracted many scholars to propose various models to study its mechanisms over the years. In 1969, Streeter [6] first proposed the Discrete Vaporous Cavity Model (DVCM), laying the foundation for studying liquid–column separation. Subsequently, Wiggert and Sundquist [7] discovered in 1979 that pressure changes in transient flow could cause the release of dissolved gasses in water and established the dispersed bubble model. Wylie E. Reference [8] proposed the discrete gaseous cavity model (DGCM) based on the ideal gas law, which assumes that the volume of gas bubbles changes with pressure. In addition, Fu You [9] developed a numerical model for calculating liquid–column separation water hammer based on the two-fluid mathematical model (TFM), using the Steger–Warming flux vector splitting (SW-FVS) and non-oscillatory nonlinear dissipation (NND) schemes to solve the model. Among these models, the DVCM performs well in simulating the growth and collapse of vapor cavities but suffers from numerical instability and potential non-convergence issues. The DGCM introduces the concept of free gas, which helps suppress pressure spikes, but the presence of free gas may introduce uncertainties in the results. The two-fluid model (TFM) describes the two-phase flow characteristics by coupling the continuity and momentum equations of the gas and liquid phases. Although it offers higher computational accuracy, it requires more computational resources. Based on the research of predecessors, Yu Jianping, Hu Liren, and others [10] used the DVCM and other models in 2023 to simulate the pressure rise in a single pipeline due to liquid–column separation, achieving good prediction results that closely matched experimental values. This indicates that the DVCM has a high potential for application in simulating liquid–column separation. Therefore, resolving the numerical instability and dissipation issues of the DVCM would facilitate its broader application in engineering practices.

Satisfying the Courant–Friedrich–Lewy (CFL) [11,12] condition, i.e., C_n < 1, is crucial for the numerical stability of water strike calculations. In interpolating the DVCM using the eigenline method, as shown in Figure 1, the nodes on both sides often cannot be precisely located simultaneously at the endpoints due to the combined effect of wave speed and flow velocity. C_n = 1 cannot be guaranteed, which is particularly significant in open channel flow where the wave speed and the flow velocity are very close. The irrational selection of C_n may cause numerical oscillations or significant numerical dissipation in the bridging water strike computation, significantly reducing the prediction accuracy of DVCM [13]. This paper uses the measured data from Simpson’s experiment [14] as a benchmark, and three sets of simulations with different C_n are performed. The simulation results are shown in Figure 2, which clearly shows the effect of C_n on the DVCM prediction results.

When C_n is close to 1 (e.g., 0.99), the model prediction accuracy is high; when C_n is exactly 1, the prediction results will have violent numerical oscillations, which will affect the prediction effect; and when C_n is low (e.g., 0.90), the numerical dissipation generated in the process of simplifying the differential equation into a difference equation is too large, which has a more noticeable effect on the time and intensity of the occurrence of the bridging water strike [15].

Balancing numerical stability and computational accuracy is often a dilemma in numerical computations. This paper uses the Muskingum method [16] to address this contradiction and improve the traditional discrete vaporous cavity model (DVCM). As a widely adopted river flow evolution algorithm, the Muskingum method determines parameter factors through trial calculations. It utilizes the error of the finite difference numerical solution of the kinematic wave equation to compensate for the physical diffusion term of the diffusive wave. This ensures that the numerical solution is the exact solution of the diffusive wave equation. Inspired by this method, this paper analyzes the simulation process’s relationship between numerical and physical dissipation. It innovatively proposes adjusting the physical dissipation in the water hammer program to compensate for numerical dissipation in the calculations. This approach effectively improves the predictive accuracy of the DVCM and provides a new idea for enhancing computational accuracy in numerical simulations in other fields.

2. Research Methodology

2.1. MOC Algorithm Overview

For the small fluid element in the pipeline shown in Figure 3, a force analysis is conducted to derive the motion and continuity differential equations. These equations are then simplified to form a pair of quasilinear hyperbolic differential equations, denoted as L₁ and L₂:

$$\left\{\begin{array}{c}{L}_1={\displaystyle \frac{\partial v}{\partial t}}+g{\displaystyle \frac{\partial h}{\partial x}}+{\displaystyle \frac{fv\left|v\right|}{2D}}=0\\ L_2={\displaystyle \frac{\partial h}{\partial t}}+{\displaystyle \frac{a^2}{g}}{\displaystyle \frac{\partial v}{\partial x}}=0\end{array}\right.$$

(1)

By introducing the characteristic factor x and simplifying L₁ and L₂, the characteristic line equations can be obtained as follows:

$$\left\{\begin{array}{c}{\displaystyle \frac{g}{a}}{\displaystyle \frac{dh}{dt}}+{\displaystyle \frac{dv}{dt}}+{\displaystyle \frac{fv\left|v\right|}{2D}}={\left.0\right|}_{dx/dt=+a}\\ -{\displaystyle \frac{g}{a}}{\displaystyle \frac{dh}{dt}}+{\displaystyle \frac{dv}{dt}}+{\displaystyle \frac{fv\left|v\right|}{2D}}={\left.0\right|}_{dx/dt=-a}\end{array}\right.$$

(2)

Through the above transformation, the original two partial differential equations are converted into two ordinary differential equations using the method of characteristics. Each is accompanied by a constraint condition and restricted to the specific paths described by the corresponding characteristic line equations, as shown in Figure 4.

Assuming that the dependent variables h and q are known at points A and B on a given characteristic line, integrating along the two characteristic lines allows the solution of the characteristic line equations (Equation (3)):

$$\left\{\begin{array}{c}{h}_P-h_A+{\displaystyle \frac{a}{gA}}\left(q_P-q_A\right)+{\displaystyle \frac{f\Delta x}{2gD{A}^2}}{q}_A\left|q_A\right|=0\\ h_P-h_B-{\displaystyle \frac{a}{gA}}\left(q_P-q_B\right)-{\displaystyle \frac{f\Delta x}{2gD{A}^2}}{q}_B\left|q_B\right|=0\end{array}\right.$$

(3)

Letting $\mathrm{B}=a/gA$, $\mathrm{R}=f\Delta x/\left(2gD{A}^2\right)$, the above equation can be rewritten as

$$\left\{\begin{array}{c}{h}_P-h_A+B\left(q_P-q_A\right)+R{q}_A\left|q_A\right|=0\\ h_P-h_B-B\left(q_P-q_B\right)-R{q}_B\left|q_B\right|=0\end{array}\right.$$

(4)

Then, using the equations above, the values of H_P and Q_P at point P can be determined:

$$\left\{\begin{array}{c}{H}_P=(C_p+C_M)/2\\ Q_P=(C_p-C_M)/2\end{array}\right.$$

(5)

2.2. DVCM Algorithm Overview

DVCM assumes [17,18] that there is no gas in the initial stability of the pipeline; in the process of hydraulic transition, when the pressure at a node of the pipeline is lower than the saturation vapor pressure of the liquid, it is considered that the vapor generated by the vaporization of the liquid is concentrated in the cross-section of the calculation node, the head in the vapor cavity remains unchanged, and the pressure at the boundary is equal to the pressure inside the vapor cavity; when the vapor cavity collapses, the head and flow at the node can still be solved using the method of characteristic lines.

The simulation process of water–column separation and its reconnection boundary conditions can be carried out through the following specific computational steps.

• Determine whether water column separation occurs.

The absolute pressure at the cross-section of node i can be expressed as

$$H_{ai}={H_{Pi}-Z}_i+H_a$$

(6)

Among them, the condition for liquid column separation to occur is $H_{ai}\le H_v$.

2.

Calculate liquid column separation.

When node i is in a state of liquid column separation, the transient parameters can be calculated using the following formulas:

$$\left\{\begin{array}{c}{H}_{pi}=Z_i-H_a+H_v\\ Q_{pi,in}=\left(C_p-H_{pi}\right)/B\\ Q_{pi,out}=\left({H_{pi}-C}_m\right)/B\\ Q_g\left(i,j\right)=\sum (Q_{pi,in}-Q_{pi,out})\end{array}\right.$$

(7)

Furthermore, the flow rate of the cavity formed by liquid column separation can be expressed by the rate of change in the cavity’s volume, as shown in Equation (8):

$$Q_g\left(i,j\right)={\displaystyle \frac{V_g\left(i,j\right)}{\Delta t}}$$

(8)

Furthermore, the volume formula for the cavity formed by liquid column separation can be simplified to Equation (9):

$$V_g\left(i,j\right)=V_g\left(i,j-1\right)+0.5\Delta t\left(Q_{i,out}{-Q}_{i,in}+Q_{pi,out}-Q_{pi,in}\right)$$

(9)

3.

Increase the pressure after reconnection.

If $V_g\left(i,j\right)\le 0$ during the calculation process, it is considered that the water column reconnects at that instant, and the transient parameters can be calculated using the following formulas:

$$\left\{\begin{array}{c}{H}_{pi}=\left(C_p+C_m\right)/2\\ Q_{pi,out}=\left(C_p-H_{pi}\right)/B\\ Q_{pi,in}=Q_{pi,out}\end{array}\right.$$

(10)

When calculating node i, first determine the gas volume ${\mathrm{V}}_g(i,j-1)$ at the previous time step. If gas is present, the node is considered a liquid column separation boundary, and the node head, inlet flow rate, outlet flow rate, and gas cavity volume at time step j can be directly calculated using Equations (7)–(9). If no gas is present, further assess whether the node head H_pi is below the cavitation pressure. If it is below the cavitation pressure, the node is transformed into a boundary point with the pressure headset as the airhead. Subsequently, the inlet flow rate, outlet flow rate, and gas cavity volume at time step j are calculated using the continuity and characteristic line equations. If the node head is above the cavitation pressure, liquid column separation and reconnection are deemed to occur, and Equation (10) is employed for the calculation.

In all the above calculations, the wave velocity of water strikes in the piping system is unaffected by the gas volume change. When the liquid column separation phenomenon occurs, the wave velocity of the water column is considered to remain constant between the inner node and the outer node because all the generated gas is concentrated at the inner node. The specific calculation steps are shown in Figure 5.

2.3. Numerical Dissipation Compensation Methods

In engineering, only physical dissipation, such as along-travel head loss and local head loss, exists in the pipeline where liquid column separation occurs. In the hydraulic transition process, the physical dissipation is mainly reflected in the along-travel head loss [19,20]; according to the formula, the along-travel head loss is positively correlated with the resistance coefficient f. When using DVCM to simulate liquid column separation, the program will generate additional numerical dissipation, which leads to distortion of the prediction results. To solve the above problems, in this paper, the numerical dissipation is equivalently converted to the physical dissipation, and the total dissipation calculated by the program is the same as the actual total dissipation by adjusting the physical dissipation (i.e., the drag coefficient f) downward to improve the accuracy of the model calculation.

2.3.1. Evaluation of Error Model (EAM)

This paper introduces two metrics to assess the degree of influence of different parameters on the prediction results, namely the numerical dissipation error metric X_n and the physical dissipation error metric X_m. The peak head lift and the column separation time of [21] are chosen as the constituent factors in the metrics.

To calculate the numerical dissipation error index X_n, for example, taking into account that when the value of C_n is 1, the prediction results are prone to numerical oscillation phenomenon, C_n tends to 1, the differential equation is simplified to the difference equation generated by the truncation error caused by the numerical dissipation of close to negligible, this paper takes the prediction results when C_n is 0.99 as a benchmark and calculates the relative error of model prediction with different C_n, as shown in Equation (11):

$$X_n={\omega }_1\times \left({\displaystyle \frac{1}{n}}{\sum }_{i=1}^n\left(H_{C_n=0.99,i}-H_{C_n=x,i}\right)/H_{C_n=0.99,i}\right)+{\omega }_2\times \left({\displaystyle \frac{1}{n}}{\sum }_{i=1}^n\left(T_{C_n=0.99,i}-T_{C_n=x,i}\right)/T_{C_n=0.99,i}\right)$$

(11)

The physical error index X_m is calculated in the same way as above. In this paper, the relative error of the model at different f is calculated as shown in Equation (12). After taking the drag coefficient f of 0.0245 (experimental value) as the benchmark:

$$X_m={\omega }_1\times \left({\displaystyle \frac{1}{n}}{\sum }_{i=1}^n\left(H_{f=0.0245,i}-H_{f=x,i}\right)/H_{f=0.0245,i}\right)+{\omega }_2\times \left({\displaystyle \frac{1}{n}}{\sum }_{i=1}^n\left(T_{f=0.0245,i}-T_{f=x,i}\right)/T_{f=0.0245,i}\right)$$

(12)

where ${\omega }_1$ is the head error weight, ${\omega }_2$ is the time error weight, n is the number of times liquid column separation occurs at the node, H is the nodal head, T is the nodal liquid column separation time, and x is the value of C_n or f corresponding control group.

2.3.2. Assessment of the Level of Numerical Dissipation

Taking a single simple pipe as an example, when C_n is 1, there will be apparent numerical oscillations, and when it is 0.99, the numerical dissipation is minimal and no numerical oscillations occur. Five different numerical errors X_n are obtained when the C_n values are 0.99, 0.95, 0.9, 0.85, and 0.8, respectively. The X_n value decreases gradually as the C_n decreases, and the numerical dissipation increases. The calculation results will be plotted on the graph, which can be X_n and Coulomb number C_n of the change rule. Take a simple pipe as an example; this paper simulates five groups of different f values corresponding to the calculation of working conditions. As the resistance coefficient f decreases, the physical dissipation gradually decreases, X_m gradually increases, and when f is 0, the predicted value of the overestimation level reaches the peak and X_m reaches the maximum value. The results of the calculations are plotted on a graph to obtain the pattern of change of X_m concerning the drag coefficient f (refer to the following example for verification).

2.3.3. Parameter Optimization

When the sum of the numerical and physical error factors is close to zero (X_n + X_m ≈ 0), the numerical dissipation is compensated by reducing the physical dissipation, which improves the accuracy of the model calculations and brings the results closer to the real solution.

This strategy can be extended to series and bifurcated complex piping systems. Before the numerical simulation, trial calculations are first carried out for each pipeline separately to obtain its numerical dissipation error factor X_n. By adjusting the physical dissipation error factor X_m of each pipeline network, the additional numerical dissipation is compensated separately, which improves the accuracy of the model calculations and ensures numerical stability.

3. Instance Validation

This paper validates the liquid–column separation experiments carried out with the experimental setup built by Bergant and Simpson from the University of Adelaide, Australia. The model sketch is shown in Figure 6.

In the experiment, water flowed from water tank 1 through the upstream pipeline to water tank 2. The upstream water tank’s normal storage level was 22 m, and the downstream water tank’s storage level was 17.76 m. The water tank diversion pipeline’s inlet elevation was 0 m and its outlet elevation was 2.03 m. The experiments measured the steady-state pipeline flow velocity V₀ as 1.4 m/s, corresponding to a pipeline resistance coefficient f of 0.0245. The pipeline computational model parameters are as follows (

Table 1. Parameters of the pipeline calculation model.

ID	L/m	D/m	a/(m/s)	dt/s	dx/m	(Cn a·dt/dx)
P1	10	0.022	1319	0.001072	1.428	0.99
P1	27.23	0.022	1319	0.001072	1.512	0.93

).

The upstream and downstream Coulomb numbers C_ns of the valves were 0.99 and 0.93, respectively, at the time of calculation. During the operation, the downstream valve was suddenly closed, and the closure time was 0.006 s. The measured results at the valves and the predicted results of the DVCM are shown in Figure 7 below. The calculation results show that the model predicts that the peak head at the node at the valve is low compared with the experimental value, and the phase front shift is too significant. It indicates that the numerical dissipation of the program in the calculation is relatively large, causing error accumulation which affects the DVCM prediction results.

• Analysis of the effect of C_ns

The C_n = 0.99 corresponds to P1 in the calculation process. It can be assumed that the numerical dissipation of P1 can be neglected during the transient flow simulation and only the numerical dissipation of P2 needs to be evaluated. Keeping the resistance coefficient consistent with the experimental value, five different C_n combinations are taken for P2 from large to small for trial calculation. The calculation to obtain the node pressure fluctuation at the valve is shown in Figure 8a, and Equation (11) is used to calculate the numerical error factor corresponding to different C_ns, respectively. The trend change in Figure 8b is shown in Figure.

According to the analysis of the above figure, when the Coulomb number C_n corresponding to P2 is not equal to 1, the internal interpolation will cause additional numerical dissipation, resulting in low head boost and column separation time in the simulation results as well as low peak predicted head and phase forward shift. As the C_n decreases, the resulting numerical dissipation gradually increases, leading to a faster rate of energy decay within the system and a gradual increase in the errors in the head rise and column separation times.

2.

Impact analysis of the drag coefficient f

To assess the influence of physical dissipation in P2 on the model’s prediction accuracy, this paper sets the calculation results, when the Coulomb number corresponding to P1 and P2 is 0.99, as the baseline and adopts different P2 to calculate the interference of numerical dissipation on the simulation results. It is calculated to obtain the node pressure fluctuation at the valve Figure 9a, as shown in Figure 9. Then, Equation (12) calculates the physical error factor corresponding to different fs, respectively, and plots the trend change of Figure 9b, as shown in Figure 9.

Through the analysis of Figure 9a,b, it is found that when the resistance coefficient f set in the program is smaller than the actual value, the physical dissipation is lower than the actual value, which causes the head boosting and liquid–column separation time in the simulation to lag behind the experimental value. As the resistance coefficient f corresponding to the pipeline set in the program gradually decreases, the resulting physical dissipation also gradually decreases, the internal energy decay rate of the system slows down, and the overestimation level of the simulated head boost and liquid column separation time gradually increase.

3.

Analysis of the relationship between C_n and f parameters

Based on the compensation strategy of numerical dissipation and physical dissipation, when X_n + X_m ≈ 0, the best accuracy prediction result will be obtained. Combining the error trend data in Figure 8b and Figure 9b, the correlation between C_n and f can be plotted as seen in Figure 10.

4.

Numerical dissipation compensation

When C_n2 takes the value of 0.93, there will be a significant numerical dissipation phenomenon. To effectively reduce the adverse effect of numerical dissipation on the model prediction accuracy, this study uses the parameter correspondence shown in Figure 10 to take f₂ as 0.013. Through this artificial adjustment, the additional loss introduced by numerical dissipation is compensated by reducing physical dissipation so that the total dissipation level in the numerical computation is close to the same as the actual physical total dissipation level, and the prediction accuracy of the DVCM can be significantly improved. The results are shown in Figure 11a,b below.

The change in head at the valve before and after optimization is compared in Figure 11a,b. Before the parameter optimization, the numerical dissipation generated by the program calculation is more significant, resulting in higher energy dissipation within the system than in the actual situation. The peak head values of the four bridging boosts in the predicted results are lower than the actual head. The first two predicted heads have more significant relative errors, 12.38% and 16.32%, respectively. The four-times liquid column separation time is lower than the actual time, and the relative error of the predicted values gradually increases, resulting in the gradual forward shift in the expected phase. After improvement, the accuracy of the peak head prediction of the four bridging boosts in the calculation results is significantly improved, and the relative error of the maximum bridging water strike is controlled within 2.34%. The separation time of the four liquid columns is similar to the actual time, and the relative error of the predicted value is stabilized within 1.19%. The numerical dissipation introduced to ensure the computational stability is significantly compensated, the change in water head tends to be consistent with the actual situation, and the model’s prediction accuracy is improved considerably.

4. Conclusions and Future Work

4.1. Conclusions

In this paper, the method of mutual compensation between numerical and physical dissipation is proposed to improve the simulation of the liquid–column separation process of DVCM in complex pipelines, and good results are achieved. The following experiences are obtained in the research process for the reference of related researchers:

• When the characteristic line calculation is used when the Coulomb number C_n deviates from 1; the prediction results of the model will significantly deviate, resulting in lower predicted values of the nodal head rise and liquid–column separation time compared with the actual values. The Courant number in pipeline calculations is an important parameter to ensure the accuracy and reliability of numerical simulations. The model’s energy dissipation significantly impacts the calculation accuracy of the bridging water hammer.
• This paper observes that the total dissipation during numerical calculations can be finely tuned by adjusting the resistance coefficient f of the pipeline without affecting the numerical stability.
• Drawing on Muskingum’s method, reversing the operation to adjust the physical dissipation to compensate for the numerical dissipation can significantly improve the prediction accuracy of the numerical computation of the bridging water strikes in this paper. When the numerical dissipation is balanced with the physical dissipation, the relative errors of the peak head and liquid–column separation time at the DVCM prediction nodes are reduced to under 6%. The strategy of mutual compensation between physical and numerical dissipation not only improves the prediction accuracy of DVCM but also provides new ideas for other numerical simulations to improve computational accuracy and stability.

4.2. Future Work

Building on the research presented in this paper, future directions can be explored to further enhance the accuracy and practicality of simulating liquid–column separation phenomena:

• Application and optimization in real-world operational environments

Our ultimate goal is to collaborate closely with municipal water supply departments and industry partners to apply the methods proposed in this study to real-world operational settings and test their applicability in actual pipeline systems. We believe such collaboration will provide valuable insights to help us refine and optimize our models to address larger-scale and more complex pipeline networks. This interdisciplinary cooperation will not only advance theoretical research but also strongly support the safe operation and optimal design of municipal water supply systems.

2.

Consideration and response to unsteady friction in future simulations

In future simulation studies, we will consider steady-state friction and further investigate the impact of unsteady friction on liquid–column separation phenomena, as well as explore corresponding countermeasures. Unsteady friction can significantly influence pressure fluctuations and the dynamic behavior of liquid—column separation during transient flow. By incorporating unsteady friction models, we can more accurately simulate the complex flow conditions in real pipeline systems. This provides a more reliable theoretical basis for water hammer protection and pipeline system design.