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Numerical model for calculation of hydraulic transients with two-phase flow and fluid–structure interaction

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Abstract

Fluid transport systems such as pipelines are subject to loads whenever changes in fluid momentum or in pipeline structure occur. These loads can generate extremely harmful hydraulic transients which may be responsible for several major accidents. This paper presents a model for the solution of these hydraulic transients, considering two-phase flow and fluid–structure interaction. Mathematical and numerical solutions are proposed and analyzed for the proper capture of the physical phenomena associated with the fluid compressibility and fluid celerity, which are variable in two-phase fluid, together with the disturbances generated by the fluid–structure interaction. The proposed solution for the model considers the simultaneous action of these phenomena. The developed numerical model is based on the solution of the mathematical model formed by a system of four partial differential equations, in which the necessary adaptations are integrated in fluid–structural equations and in the nonlinear mathematical coefficients for the solution of the compressible and two-phase flow in question. Classical formulation is selected for the implementation of friction between fluid and pipe in the model. For the solution, it is applied the method of characteristics and finite difference, with subsequent numerical integration. The validation of the results is carried out based on comparisons with experimental and analytical data. The model presented, in general, was quite adherent to the experimental and analytical results, mainly in relation to the first pressure peak, which is one of the main focuses of the transient analyses.

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Abbreviations

A :

Previous point, pipe transversal area

f :

Darcy–Weisbach friction coefficient

t :

Time

V :

Average fluid speed [m/s]

\(\forall\) :

Volume

P :

Average pressure [Pa]

L :

Pipe length [m]

c :

Celerity

Δ :

Variation of a given entity

ρ :

Density [kg/m3]

g :

Gravity acceleration [m/s2]

K ut :

Time adjustment coefficient

K ux :

Length adjustment coefficient

R :

Pipe internal radius

D :

Pipe internal diameter [m]

e :

Pipe thickness [m]

E :

Pipe Young modulus (elasticity modulus)

\(\nu\) :

Poisson coefficient

\(p\) :

Absolute pressure

K :

Bulk modulus

\(\gamma\) :

Pipe angle

\(\tau_0\) :

Friction term

i,cur:

Counters

j :

Time counter

\(\alpha \) :

Void fraction

\(K_v\) :

Valve pressure loss coefficient

\(Cd\) :

Outlet valve coefficient

m :

\(Cd\) Adjustment coefficient

val:

Valve

\(u,U\) :

Pipe displacement

\(\sigma\) :

Axial stress

N :

Number of pipe sub-elements

λ :

Wavelength

z :

Axial coordinate

t :

Time coordinate

\(\frac{{\text{d}}}{{{\text{d}}t}}\) :

Time total derivative

\(\frac{\partial }{\partial t}\) :

Partial derivative in relation to t coordinate

\(\frac{\partial }{\partial z}\) :

Partial derivative in relation to z coordinate

1,2,3,4:

Characteristics curves

o :

Reference to beginning

F,f :

Fluid

t :

Pipe

g :

Gaseous phase

l :

Liquid phase

out:

External

max:

Maximum

min:

Minimum

rel:

Relative

c :

Valve closure

d,D :

At right

e :

At left

\((i,j)\) :

Position and time indicators in computational mesh

cur, i :

Counter

cond:

Condition

points:

Number of points

z :

Axial direction

m :

Mixture

 ~ :

Adjusted value

.:

First derivative in time

-:

Average value at transversal section

|:

Taken at

IAB:

Instantaneous acceleration based

FSI:

Fluid–structure interaction

MOC:

Method of characteristics

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Acknowledgements

The authors thank for the Brazilian Navy support and for the data kindly shared by Eindhoven University of Technology (Netherlands) and Georgia Institute of Technology (USA).

Funding

This research did not receive any specific grant from funding agencies in the public, commercial or not-for-profit sectors.

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Authors and Affiliations

  1. Instituto de Pesquisas Energéticas e Nucleares—IPEN, University of São Paulo (USP), São Paulo, Brazil

    Pedro Henrique do Nascimento Rocha, Josie Aguila Alves de Campos, Micelli Rodrigues Camargo & Marcelo da Silva Rocha

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  1. Pedro Henrique do Nascimento Rocha
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  2. Josie Aguila Alves de Campos
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  3. Micelli Rodrigues Camargo
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  4. Marcelo da Silva Rocha
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Each author contributed to the research presented in this manuscript, approved the contents now presented and agreed to the compliance with ethical standards.

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Correspondence to Pedro Henrique do Nascimento Rocha.

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do Nascimento Rocha, P.H., de Campos, J.A.A., Camargo, M.R. et al. Numerical model for calculation of hydraulic transients with two-phase flow and fluid–structure interaction. J Braz. Soc. Mech. Sci. Eng. 46, 206 (2024). https://doi.org/10.1007/s40430-024-04818-w

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