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.
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
– pipe anchored upstream,
– pipe anchored throughout (no axial dilatation),
– expansion joints (axial dilatation allowed),
LATTICE BOLTZMANN MODEL
Deduction of transient flow equations
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
Valve located at the end of the pipe
Reservoir located at the end of the pipe
Finally, solution is obtained using Equation (52) with the following replacements , and .
In the first step, functions and are determined using assumed values for v and .
- 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
- In the fourth step, the second unknown distribution function is calculated using the combination of Equation (62) and the first statistical moment
Finally, dimensionless flow is calculated using the distribution function from the previous step.
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
A small pipeline system
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.