A lattice Boltzmann method (LBM) is utilized to solve single-phase transient flow in pipes. In order to eliminate grid limitation related to the method of characteristics, governing equations are modified using appropriate coordinate transformation. The introduced modification removes connection between Courant number and spatial disposition of the computational nodes, forming a more flexible and robust mathematical base for numerical simulations. The computational grid is configured independently of the wave speed, significantly decreasing the demand for computational resources and maintaining the required accuracy of the method. Thereafter, the appropriate equilibrium distribution function for the D1Q3 lattice has been defined. In order to give a comprehensive base for modeling transient flow in complex pipeline systems, detailed elaboration of the corresponding boundary conditions has been given. Two benchmark problems with the corresponding error analysis are used to validate the proposed procedure.

INTRODUCTION

Transition between two steady states in pipeline systems is a common phenomenon in engineering practice. Caused by sudden change in flow regime (pump failure, instantaneous valve closure), transient flow is characterized as highly unsteady flow with intensive pressure and velocity fluctuations, classically known as water hammer. In order to describe this phenomenon, a set of two hyperbolic partial differential equations, mass and momentum equation is utilized (Fox 1977; Wylie & Streeter 1978; Chaudhry 2014). Due to their hyperbolic nature, these equations are transformed to finite differential equations by using the method of characteristics (MC) (Wylie & Streeter 1978). Hartree (1958) was the first to introduce spatial interpolation in the case where the Courant number (Cr) was less than 1.0, while Wiggert & Sundquist (1977) and Vardy (1976) solved the pipeline transients for characteristics projecting over the fundamental grid size. Their analysis shows the effects of interpolation, spacing, and grid size on numerical attenuation and dispersion. Instead of widely used spatial interpolation, Goldberg & Wylie (1983) used interpolations in time, while Sibetheros et al. (1991) investigated the MC with spline polynomials for interpolations. Besides the MC, finite-difference (FD) and finite-volume (FV) methods have also been used in order to solve transient equations. Chudhury & Hussaini (1985) solved water hammer equations by MacCormack, Lambda, and Gabutti explicit FD schemes, while Verwey & Yu (1993) applied a space-compact high-order implicit scheme. Furthermore, Zhao & Ghidaoui (2004) formulated, applied, and analyzed first- and second-order explicit FV Godunov-type schemes for water hammer problems, while Leon et al. (2007, 2008) and Leon & Oberg (2013) applied FV method on two-phase water hammer flows. Application of the MC is proven to be simple to code, is accurate and efficient. However, major drawbacks related to the MC arise in practical applications. Since the spatial disposition of computational points is limited to relation , where a is the propagation (wave) speed and is time step, frequently a single value for is adopted for the whole system, as being representative. This distance is often prescribed by the system configuration; distance between two inner boundary conditions (for example, two air chambers). Consequently, a long pipe section can impose a large number of computational points, which can greatly affect the efficiency of the method. On the other hand, if spatial or time interpolation is used, additional procedures are inevitably introduced, which again influence the stability and accuracy of the method (Vardy 1976; Wiggert & Sundquist 1977; Goldberg & Wylie 1983).

In order to somehow overcome deficiencies related to the previously described methods, a lattice Boltzmann method (LBM) (Wolf-Gladrow 2005) is offered as an alternative approach. The first application of the LBM on transient flow was considered by Cheng et al. (1998), while only a case of rather simple practical implementation was done by Wu et al. (2008). In this paper, further development of the LBM is considered. In order to enhance the efficiency of the method, adaptive grid procedure is introduced. With this modification, restriction between Cr and spatial step is removed, and arbitrary grid configuration approach is established. Hence, the number of computation points now can be significantly reduced, which is a desirable feature for long pipe sections. The corresponding grid size accuracy analysis is also conducted. Furthermore, mathematical formulation of most used boundary conditions is presented in detail. Implementation of the basic pipeline elements to the lattice Boltzmann model, such as pump, air chamber, valve, expansion/contraction of pipes, branching of pipes, is elaborated from the distribution function point of view, and some simplifications are introduced. This will much enhance the applicability of the method, with the opportunity to efficiently simulate complex pipeline systems. In contrast to the few lattice Bolzmann applications on transient flows available in the literature, where only the basics of the LBM are presented, the adaptive grid approach followed by detailed and comprehensive elaboration of the boundary condition and practical implementation offers scientists and engineers a robust and efficient tool for managing the transient flow in complex pipeline systems. The presented method was tested and verified on two different examples, while the MC was used for comparison.

TRANSIENT FLOW EQUATIONS

In general, one-dimensional transient flow is described by a set of two partial differential equations, namely the mass and momentum equation (Fox 1977; Chaudhry 2014): 
formula
1
 
formula
2
In the above equations, t is time, x is Cartesian coordinate, is hydraulic head, V is longitudinal mean velocity, g is gravitational acceleration, and is friction factor. Wave speed a is defined as: 
formula
3
where K is bulk modulus, E is Young's modulus, is volumetric mass density and e is wall thickness. Depending on the pipe anchors, parameter is calculated as:
  1. – pipe anchored upstream,

  2. – pipe anchored throughout (no axial dilatation),

  3. – expansion joints (axial dilatation allowed),

where is Poisson's coefficient.
In order to establish a base for adaptive grid procedure, Equations (1) and (2) are further transformed to the alternative coordinate system using the basic rules of coordinate transformation (Simmonds 1994): 
formula
4
 
formula
5
In Equations (4) and (5), represents a new coordinate, while term denotes corresponding basis. This formulation enables calculation conducted in equidistant grid frame, imposed by the symmetry of the LBM.

LATTICE BOLTZMANN MODEL

In this paper, a D1Q3 lattice Boltzmann Bhatnagar–Gross–Krook model (LBGK) (Bhatnagar et al. 1954) is utilized. The evolution equation is defined as: 
formula
6
where is the particle distribution function along the link, is the local equilibrium distribution function, is the position vector in the 1D domain, t is time, is the time step, is the force term, and is relaxation time. It should be noted that Equation (6) is defined using the new -domain, therefore the required symmetry for discrete particle velocities is ensured. For the three-velocity lattice, particle velocities along the direction are defined as and . Accordingly, is lattice size in direction.
In order to model transient flow equations using the corresponding LB method, appropriate equilibrium function is required. Using the form derived for 1D shallow water equations, introduced by Van Thang et al. (2010), the following formulation is proposed here: 
formula
7
Term G in Equation (7) actually denotes connection between the physical and computational (lattice) domains, and it is calculated prior to the LBM computations. Furthermore, discrete formulation takes the form , where is physical distance between computational nodes.
The last term in Equation (6) is the force term, and it is formulated as: 
formula
8
In order to obtain second-order accuracy of the method, the force term is evaluated using the centered-scheme (Zhou 2004). Values are calculated at mid-point between the lattice points and its neighboring lattice points as: 
formula
9
Finally, the zeroth and first statistical moment is used to derive the corresponding hydraulic head and longitudinal velocity V: 
formula
10

Deduction of transient flow equations

To develop Equations (4) and (5), the Chapman–Enskog analysis (Wolf-Gladrow 2005) will be applied. By applying a Taylor series expansion in time and space around point to the left side of Equation (6), and assuming , Equation (6) takes the form 
formula
11
Further, distribution function is expressed as: 
formula
12
while the centered scheme proposed by Zhou (2004) is used for the force term 
formula
13
which can also be written – via a Taylor expansion – as: 
formula
14
Substituting Equations (12) and (14) in Equation (11), the equation to order is: 
formula
15
to order it is: 
formula
16
and to order it is: 
formula
17
Inserting Equation (16) into Equation (17) and then adding it to Equation (16) yields 
formula
18
Enforcing conditions and for , and taking the sum about , Equation (18) takes the following form: 
formula
19
Evaluation of terms in the above equation using Equations (7), (8), and (10) results in the second-order accurate continuity equation (Equation (4)).
From Equation (18) about we have 
formula
20
Again, using Equations (7), (8), and (10) with the , results in the second-order accurate momentum equation (Equation (5)). Since the first term on the RHS actually denotes the second-order velocity derivative (diffusion), the parameter acts like an artificial viscosity which controls dispersion (oscillations) in the vicinity of the shocks.

Boundary conditions

Water supply systems are often characterized by a variety of elements (fittings, pumps, and storage tanks, etc.), hence adequate implementation of the boundary conditions represents one of the major tasks when transient pipe flow modeling is considered. Contrary to open channel flow cases, where formally only two types of boundary conditions are used, in transient pipe flow every type of element requires appropriate and specific LB formulation. Therefore, the detailed presentation of boundary conditions in the case of the transient LBM is presented in the following section.

Contraction/expansion of a pipe

In order to define contraction/expansion of a pipe as a boundary condition, equality of hydraulic heads (energy losses are neglected) and discharges Q between two boundary sections of the pipes and are used (Figure 1): 
formula
21
 
formula
22
Figure 1

Contraction/expansion of the pipe.

Figure 1

Contraction/expansion of the pipe.

Index j denotes the corresponding pipe, while index marks the computational node along pipe j. It is obvious from Figure 1 that and are the unknown distribution functions related to pipes and , respectively. Introducing Equation (10) into Equations (21) and (22) and then expressing them in terms of unknown distribution functions and , respectively, gives 
formula
23
 
formula
24
where A is the cross section area of the corresponding pipe. Finally, substituting Equation (24) into Equation (23) leads to final expressions for calculating the unknown distribution functions in the forms 
formula
25
 
formula
26
It should be noted that Equations (25) and (26) are equally applicable for both cases, i.e., contraction and expansion, which makes them rather universal when change of pipe diameter is considered.

Valve

Opposite to pipe contraction/expansion, where energy losses are considered to be small (therefore neglected), a valve located at a particular node along a pipe (valve installed to the end of pipe is considered separately) can impose significant energy loss. Hence, in order to connect pipes and (Figure 2), the continuity and energy equation is utilized, respectively: 
formula
27
 
formula
28
where is loss coefficient and is cross-sectional area of the pipe at the location of the valve. It should be noted that ‘±’ in Equation (28) refers to direction of flow. Hence, the ‘+’ sign denotes the case when flow is directed from pipe to pipe , while ‘−’ refers to the opposite case. To define unknown distribution functions and corresponding to boundaries of pipes and , respectively, a similar procedure used in the contraction/expansion case will be applied. Furthermore, if the second term in Equation (28) is ignored, Equation (21) is obtained. Thus, in order to derive distribution functions at the location of the valve, the already defined form of Equation (25) is further modified, introducing the energy loss term 
formula
29
After some algebra, Equation (29) takes the form of a quadratic equation with solution in the form 
formula
30
where 
formula
31
Next, the second unknown distribution function is obtained applying Equation (26).

Valve located at the end of the pipe

A special case of boundary condition is a valve located at the end of the pipe (Figure 3). Since the hydraulic head at the endpoint of the pipe coincides with the level of the outlet, Equation (28) consequently transforms into 
formula
32
Introducing Equation (10) into the above equation, the following form is obtained: 
formula
33
Since in this case is the only unknown distribution function (Equation (30)), Equation (33) again takes the form of a quadratic equation with the corresponding terms formulated as: 
formula
34
Figure 3

Valve at the end of the pipe.

Figure 3

Valve at the end of the pipe.

Branching pipes

In the case of branching pipes, a branch of four pipes with corresponding flow direction (see Figure 4) will be used as an example for derivation of the boundary condition. Hence, and are the unknown distribution functions related to the pair of pipes and , respectively. In order to form a boundary condition for a branch, a continuity equation along with hydraulic head equality between the pipes is utilized: 
formula
35
 
formula
36
Further, introducing the first statistical moment 
formula
37
implementation of the corresponding zeroth moment into Equation (36) yields 
formula
38
Finally, Equations (37) and (38) are written in matrix form: 
formula
39
while the solution is obtained using simple matrix inversion.

Reservoir located at the end of the pipe

Energy equation between the left reservoir, T, providing constant hydraulic head, and cross section of the pipe is introduced (Figure 5): 
formula
40
Applying zeroth moment for hydraulic head and first moment for discharge (Equation (10)), the following equation is derived: 
formula
41
It is evident that Equation (41) is quadratic; hence the solution for the unknown distribution function is defined as: 
formula
42
where 
formula
43
Furthermore, for a reservoir located at the right end of the pipe, the same procedure is utilized. Using the energy equation in the form 
formula
44
the solution of the corresponding quadratic equation is defined as: 
formula
45
where 
formula
46
Figure 5

Reservoir.

Figure 5

Reservoir.

Surge tank

To minimize and possibly eliminate surges produced by the water hammer effect, the surge tank is often used as a possible solution (Figure 6). Water from the system enters and leaves the tank according to the pressure difference between the tank and the pipe, hence relieving the system from the sharp and intensive pressure surges. In order to introduce the surge tank as a boundary condition, the following set of equations is utilized: 
formula
47
 
formula
48
 
formula
49
where is flow in the tank, is hydraulic head inside the tank, is cross-sectional area of the tank, is loss coefficient for the pipe–tank connection, and is cross-sectional area of the pipe connecting the tank and the system. Equations (47) and (48) represent continuity and energy equation between the tank and sections and , respectively, while Equation (49) describes the connection between the flow and the hydraulic head inside the tank. Introducing the first statistical moment into Equation (47), and then eliminating the unknown distribution functions and from the equation using the zeroth statistical moment, connection between flow in the tank and hydraulic head in the pipe is derived: 
formula
50
Figure 6

Surge tank.

Figure 6

Surge tank.

Further, combining Equation (50) with Equation (48), the quadratic equation is obtained: 
formula
51
Finally, Equation (51) is solved using Equation (45), where 
formula
52
For the hydraulic head in the tank , Equation (49) is utilized in the form 
formula
53

Air chamber

Since surge tanks are mostly used as a solution in larger systems (power plants), for smaller water supply systems a more practical and profitable solution in the form of an air chamber is utilized (Figure 7). Practically, both approaches are based on the same procedure. In comparison with the surge tank, the air chamber uses air pressure to compensate pressure head in the open tank, therefore reducing dimensions and overall cost of the chamber. Hence, for the derivation of boundary conditions, a procedure similar to that used in the case of the surge tank has been applied. In order to introduce air chamber as a boundary condition, the following set of equations is utilized: 
formula
54
 
formula
55
 
formula
56
 
formula
57
where is flow in the chamber, is hydraulic head inside the chamber, is water level, is air volume, is cross-sectional area of the chamber, is loss coefficient for the pipe–chamber connection, and is cross-sectional area of the pipe connecting the chamber and the system. Again, Equations (54) and (55) represent continuity and energy equation between the chamber and sections and , respectively, while Equation (56) describes connection between the flow and the water level inside the chamber. Temporal change of air volume inside the chamber is defined by Equation (57). The hydraulic head in the chamber is formulated as: 
formula
58
where is air pressure, relationship between the pressure and air volume is required in order to close the system. For this, polytropic relation for ideal gas is used: 
formula
59
where is absolute pressure and m is the polytropic index. Introducing the finite difference form of Equations (56) and (57) along with Equation (59) in to Equation (58), relation for the pressure head inside the chamber is obtained: 
formula
60
where n denotes time level. Further, combining Equation (50) – representing relationship between flow in the chamber and hydraulic head in the pipe – and Equation (60) with Equation (55), a quadratic equation similar to Equation (51) is derived: 
formula
61
Figure 7

Air chamber.

Figure 7

Air chamber.

Finally, solution is obtained using Equation (52) with the following replacements , and .

Pump

Pump failure is seen as the most common reason for inducing water hammer effect, hence pump has been incorporated into the transient LB model as the last possible form of boundary condition. For this purpose, the configuration in Figure 8 is utilized. In order to define unknown distribution functions and , continuity and energy equations are defined: 
formula
62
 
formula
63
In the above equation represents total head developed by the pump, defined by the well-known four quadrants Suter relations (Wylie & Streeter 1978): 
formula
64
where and are dimensionless flow and speed of rotation of the pump, respectively. Subscript ‘0’ refers to rated conditions, usually those at the best efficiency point. Function of the considered pump can be determined accomplishing measurements on the corresponding pump, utilizing the relationship 
formula
65
stating the condition of identical capacity of two pumps. If rotational speed of the pump is known (regular pump regime), Equations (62)–(64) close the system of equations. However, when pump failure occurs, is unknown, hence an additional equation is required. For this purpose a torque balance equation is used: 
formula
66
where is the torque of the pump, is the hydraulic torque, l is the moment of inertia of the pump and entrained liquid, and is the change in angular speed with time. It should be noted that in the case of pump failure the torque of the pump becomes zero instantaneously. By introduction of relations and , Equation (66) can be discretized using the finite difference method as: 
formula
67
where is derived applying the four quadrant relation in the form 
formula
68
The set of Equations (62), (63), (64), (67), and (68) close the system of required equations, which is then solved iteratively. The procedure is described here:
  1. In the first step, functions and are determined using assumed values for v and .

  2. In the second step, dimensionless speed of rotation and torque is calculated using Equations (67) and (68), respectively.

  3. Introducing the zeroth statistical moment in the form into Equation (63), and using assumed value in the first iteration for the second unknown distribution function , in the third step the following equation is solved 
    formula
    69
  4. In the fourth step, the second unknown distribution function is calculated using the combination of Equation (62) and the first statistical moment 
    formula
    70
  5. Finally, dimensionless flow is calculated using the distribution function from the previous step.

  6. The entire procedure is repeated from stage 2 using tolerance .

THE NUMERICAL SIMULATION AND THE VALIDATION OF THE MODEL

In order to test the proposed form of the LBM on more practical transient problems, we focus our test analysis mainly on the pump failure cases. Being the most common reason for inducing transient flows in pipeline systems, pump failure is, at the same time, complex enough and representative enough for testing the novel numerical method.

Single pipe system

For the first example a rather simple pipeline system is chosen (Figure 9). At the front end of the pipe, having length of m, diameter m, and friction factor λ = 0.022, one centrifugal pump (KRTK 500-630/908UNG-S) with actual flow rate m3/h, actual developed head m, efficiency , and speed of rotation rpm is set up (Amarex 2014). From the open reservoir, having water level m, water is pumped along the horizontal pipe vertically positioned at level m. Furthermore, two valves are implemented in the system. However, primarily the role of valves in this example is not flow regulation using an adapted closing dynamics, but rather boundary condition implementation on one hand (free outflow at the end of the pipe), and enforcement of the MC to define the computational grid along the pipe according to the smallest distance limitation (valve located m from the pump) on the other hand. As stated in the Introduction, MC utilizes an approach where the governing equations are solved along the characteristics (trajectories), which can connect two computational points (), or it can be projected outside () or inside () the computational cell. In order to avoid additional computational work and reduce numerical error (caused by interpolation) as much as possible, especially when nonlinear definition for characteristic is used, most commercial softwares utilize a standard model, i.e., when . However, this approach also introduces a certain type of limitation if the computational grid is defined according to the smallest distance in the system. This can significantly increase dimension of the computational grid, which in turn influences overall computational efficiency. For the purpose of testing and verification of the transient LB method, a commercial software AFT Impulse 4.0 (Applied Flow Technology 2011) is used in our investigation.
Figure 9

Single pipeline system.

Figure 9

Single pipeline system.

Prior to the transient flow simulation, a steady flow state is established. For this purpose a classical energy equation is adopted. Applying the condition where the pump torque is instantaneously reduced to zero, pump failure is induced and transient flow regime is obtained. As a consequence, intensive pressure and velocity variations are developed. In order to test the LBM for solving the transient equations (Equations (1) and (2)), comparison against the MC (AFT Impulse 4.0) is provided. For a control section, location at 1.500 m from the front end of the pipe is chosen. Comparison of the pressure and flow variations between the two methods is presented in Figure 10. Very good agreement between the compared models is achieved. Furthermore, in order to test stability of the LB procedure, related to the relaxation time , hydraulic grade-line corresponding to the extreme values () of hydraulic head is presented in Figure 11. It is evident from the presented figure that the change of the relaxation parameter does not affect the overall stability of the procedure, but small deviations between the compared results as well as between the two methods are present. Increase of the relaxation time parameter introduces a second-order velocity derivative in the model through Equation (20), therefore differences in results of cases with different values may be explained as dissipation effects caused by this additional term acting like a generator of artificial diffusion. Intensity of dissipation is presented in Figure 12. On the other hand, maximal difference between the LBM and MC, relative to the maximal drop of hydraulic head in the vicinity of the pump, is .
Figure 10

Pump failure in a single pipeline system: (a) comparison of hydraulic head and (b) discharge obtained by the LBM and MC (AFT Impulse).

Figure 10

Pump failure in a single pipeline system: (a) comparison of hydraulic head and (b) discharge obtained by the LBM and MC (AFT Impulse).

Figure 11

hydraulic grade-lines: comparison of the LBM with different relaxation time values to the MC (AFT Impulse).

Figure 11

hydraulic grade-lines: comparison of the LBM with different relaxation time values to the MC (AFT Impulse).

Figure 12

Dissipation intensity caused by .

Figure 12

Dissipation intensity caused by .

The second part of the validation procedure includes grid-size analysis. In order to examine how the grid density affects the basic LBM indicators (in relation to the MC), primarily, efficiency and accuracy, four different cell size configurations and m have been adopted. This further produced one-dimensional computational grids with and 26 computational points, while for time step, and have been used, respectively. For the MC, however, condition is imposed by the system, which resulted in a grid size cell of m (distance between the pump and valve is adopted as minimal length). Comparison of the results obtained for four different grid configurations with the MC is presented in Figure 13. It is evident from the figure that significant deviations between the compared results are notable only for m, while for the finer grid sizes accuracy obtained for the finer grid ( m) is maintained.
Figure 13

hydraulic grade-lines: comparison of the LBM with different grid sizes to the MC (AFT Impulse).

Figure 13

hydraulic grade-lines: comparison of the LBM with different grid sizes to the MC (AFT Impulse).

A small pipeline system

To investigate water hammer effect using the proposed transient LB model in the case of complex pipeline configurations, for the second example the system presented in Figure 14 is utilized. Representing a ‘real’ practical problem, for which water hammer analysis is required, the chosen system is composed of five pipe sections (P), three branches (B), one air chamber (AC), one valve (V), and two centrifugal pumps () in parallel connection (same type of pump is used as in the previous example). All necessary parameters regarding the chosen system are depicted in Figure 14. After setting up the pipeline system, a computational grid with corresponding cell sizes is established. In order to install the optimal number of computational points for each pipe section, different cell sizes are set for each pipe (see Figure 14). Hence, for the longest pipe cell size of m is chosen, while for the shortest pipe m is adopted. However, for MC the constant cell size of is used. As a result, one-dimensional grids with and computational points are set for the LBM and MC, respectively.
Figure 14

Pipeline system configuration.

Figure 14

Pipeline system configuration.

Two separate cases, with and without air chamber, are investigated. Similar to the previous example, initial conditions for both cases are derived using the classical energy equation. First, the case without air chamber is examined. For time step and relaxation time, values s and are adopted, respectively. Again, transient regime is induced by pump failure, i.e., instantaneously reducing torque of both pumps to zero. This produces water hammer effect followed by intensive flow and pressure variations along the system. In order to compare results from the LBM and MC method, a control section located in the middle of pipe is chosen (CS). Comparison of the obtained results in the form of hydraulic head variations and hydraulic grade-lines related to the extreme values () of hydraulic head is presented in Figures 15 and 16. Similar to the single pipe system, very good agreement between the two models is achieved. For the time variations of the hydraulic head (Figure 15), small deviations are present only at the local minimum, starting at s. Relative to the maximal drop of hydraulic head, this digression falls in the range of . However, for the minimal and maximal extrema, i.e., for extreme pressure drops instantaneously after pump fails ( s in Figure 15), excellent agreement is obtained. Furthermore, this trend is relatively maintained along the whole system, which is shown in Figure 16. Only at a few computational points, located near the end of the system, is some divergence noted. Also, a 33.22% decrease in simulation time when the LB method is used is achieved. However, this model did not include additional procedures and coding for high performance computing (CUDA, OpenCL), which can further accelerate overall procedure metrics.
Figure 15

Hydraulic head, comparison between LBM and MC (AFT Impulse).

Figure 15

Hydraulic head, comparison between LBM and MC (AFT Impulse).

Figure 16

hydraulic gradeline, comparison between LBM and MC (AFT Impulse).

Figure 16

hydraulic gradeline, comparison between LBM and MC (AFT Impulse).

Protection from high pressure oscillations using an air chamber is adopted and analyzed in the second step. To prevent severe damage of the pipeline system, caused by intensive pressure variations along the pipeline, an air chamber is installed 90.61 m downstream to the pump station (Figure 14). Installation of an air chamber into the LB model, setting the basic chamber parameters is required. The chamber is of cylindrical shape with diameter of m, height m. The initial water level of the chamber is set to m, yielding initial air volume m3. The chamber is connected to the system via a short pipe having loss coefficient , according to Equation (52). The same procedure of transient regime induction and analysis is applied here as in the previous example. Comparison of the results obtained by the LBM and MC in terms of hydraulic head variations and envelopes is given in Figures 17 and 18, respectively. Once again, very good agreement between the compared results is achieved in both cases. Minimal deviations noted in Figure 18, ranging from relative to the maximal head drop, demonstrate that the LBM model is more than capable for solving a variety of complex practical problems.
Figure 17

Hydraulic head, comparison between LBM and MC (AFT Impulse) with an air chamber included.

Figure 17

Hydraulic head, comparison between LBM and MC (AFT Impulse) with an air chamber included.

Figure 18

hydraulic gradeline, comparison between LBM and MC (AFT Impulse) with an air chamber included.

Figure 18

hydraulic gradeline, comparison between LBM and MC (AFT Impulse) with an air chamber included.

CONCLUSIONS

To overcome grid limitation typical to the MC, requiring Cr equal to one for stability in solving transient equations of pressurized flow, the same problem is solved by the LBM. To provide a higher level of flexibility in grid use, an adaptive grid approach is exploited. Sectional autonomy in the application of specific grid density to each pipe section significantly increases the efficiency of the overall procedure, maintaining the basic stability and accuracy. Introducing the LB method as an alternative to the MC in water hammer calculations provides the opportunity for parallel processing, which further increases the efficiency of calculations and cuts down calculation time. This is of high importance in large and complex pipeline systems. A detailed overview and elaboration of internal and external boundary conditions, i.e., pipeline elements and components is given. A real pipeline system with pump failure is modeled for the verification of the proposed LBM. Very good agreement with the MC is achieved.

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