Dynamic thick-wall extension of the fluid–structure-interaction water-hammer model: radial inertia, external-medium coupling, and validation for very thick pipes

The one-dimensional four-equation model for water hammer with fluid–structure interaction (FSI) in liquid-filled pipes was re-derived by Tijsseling (2007) for thick-walled pipes by cross-sectional averaging of the axisymmetric two-dimensional liquid and pipe-wall equations, with hoop and radial stresses given by the quasi-static Lamé distribution of a pressurized ring; exact solutions showed that the thick-wall corrections matter only for pipes with radius/thickness ratio $R/e<2$. The model deliberately neglects radial inertia of the liquid and wall (the long-wave approximation), so local short-wavelength effects near sharp wave fronts and the validity range of the quasi-static through-thickness stress assumption remain unquantified; it treats the fluid outside the pipe as a constant pressure with no dynamics, assumes axial strain uniform through the wall thickness, and — being inviscid, gravity-free and undamped — admits closed-form exact solutions only for instantaneous excitations without lumped end masses. The open problem is to construct and validate the dynamic thick-wall counterpart: retain radial inertia and its dispersion, couple the outer wall to a dynamically responding external medium, quantify the through-thickness variation of axial strain, and establish whether, when and how the quasi-static assumption ceases to be adequate for very thick pipes. As a further extension, the dissipation and numerical machinery developed since 2007 for thin-wall models is to be merged into the thick-wall setting. Since no transient measurements exist for pipes with $R/e<2$, validation of that counterpart means agreement with a full axisymmetric reference computation and recovery of the established thin-wall and quasi-static thick-wall limits, not agreement with experiments.

Background

Water hammer is the pressure-surge phenomenon generated in a liquid-filled pipeline when a valve closes, a pump trips, or the flow is otherwise rapidly changed: the decelerating liquid launches steep-fronted pressure waves that travel at the coupled liquid–pipe wave speed (the Korteweg celerity, set by the liquid bulk modulus and the distensibility of the pipe wall), reflect at boundaries, and can produce extreme loads. When the pipe is not rigidly anchored, these pressure waves excite axial and radial motion of the pipe wall, and that motion in turn generates additional pressure waves; this two-way coupling is fluid–structure interaction (FSI). In the standard one-dimensional (1D) theory FSI enters through three mechanisms: Poisson coupling (axial stress waves in the wall breathing radially and compressing the liquid, and vice versa), junction coupling (forces exchanged at moving bends, valves and dead ends), and friction coupling (the moving wall shearing the liquid). The reference mathematical model for straight pipes is the linear hyperbolic four-equation FSI system — two equations for liquid pressure and velocity and two for axial pipe stress and velocity — originating in R. Skalak's 1956 extension of water-hammer theory. It assumes a thin-walled pipe, i.e. wall thickness $e$ small relative to radius $R$, so that hoop stress is uniform across the wall and the radial (breathing) inertia of wall and liquid can be replaced by a quasi-static relation between pressure and hoop strain. Many practical pipes — high-pressure lines in chemical and nuclear plants, concrete-lined ductile iron, thick plastic pipes — violate $e/R\ll 1$. Tijsseling (2007) therefore re-derived the 1D four-equation model by cross-sectionally averaging the axisymmetric two-dimensional liquid and pipe-wall equations while keeping the wall thickness finite: the hoop and radial stresses across the wall are taken as the quasi-static Lamé stress distribution — the classical elasticity solution for a pressurized thick-walled ring — which produces small correction terms and factors in the four equations. Exact time- and frequency-domain solutions showed that these corrections are important only for very thick pipes, roughly $R/e<2$, and the model reproduced measured data from an impact-test rig at the University of Dundee (a steel pipe with $e/R=0.152$, i.e. outside the very-thick regime where the corrections matter). The derivation rests on idealizations that the paper itself enumerates: (i) the long-wave approximation — radial inertia of liquid and wall is dropped, so the through-thickness stress state is quasi-static and local short-wavelength effects near sharp wave fronts are not represented; (ii) an inviscid liquid without convective or gravity terms and an undamped structure; (iii) the fluid outside the pipe reduced to a constant pressure $P_{\text{out}}$, its dynamics (a non-zero outer-wall radial velocity $(v_r)_{\text{out}}$) not modelled; (iv) axial strain $\varepsilon_z$ taken as nearly independent of the radial coordinate $r$, i.e. cross-sections remain almost plane when stretched; and (v) closed-form exact solutions only for instantaneous excitations without lumped end masses or dissipation. Which of these restrictions have since been removed — and by whom — delimits the question below.

Problem Statement

The problem is to construct and validate a dynamic counterpart of the quasi-static thick-wall FSI water-hammer model of Tijsseling (2007) for very thick pipes, and thereby to establish whether, when and how that quasi-static model ceases to be adequate there. Concretely, the problem comprises the following jointly linked results, whose common objective is to determine whether, when and how the quasi-static thick-wall model — and with it all current 1D FSI water-hammer practice — ceases to be adequate for pipes with $R/e<2$:

1. Derive a tractable reduced (1D or equivalent) thick-wall FSI model of finite and explicitly stated dimension — a one-dimensional system of evolution equations or an equivalent low-order reduction — that retains the radial inertia of the pipe wall and the liquid, so that the through-thickness stress distribution is no longer imposed as the quasi-static Lamé solution, and exhibit its wave speeds and dispersion relation, including the local short-wavelength behavior near sharp wave fronts.
2. Establish the validity range of the long-wave (quasi-static through-thickness stress) assumption as a function of wavelength-to-radius ratio and $R/e$, in particular for genuinely thick pipes with $R/e<2$, where the published thick-wall corrections matter but no dynamic analysis and no experimental transient data exist.
3. Quantify the consequences of the through-thickness variation of axial strain ($r$-dependence of $\varepsilon_z$ and $\sigma_z$) for FSI wave speeds and for predicted pressure and strain histories.
4. Provide well-posed boundary conditions coupling the outer pipe wall to a dynamically responding external medium (an external fluid or elastic surroundings) in place of the constant $P_{\text{out}}$, and characterize the effect of this coupling, including any radiation damping (energy carried away by waves radiating into the external medium).
5. Validate the reduced model of item 1 against a full axisymmetric (two- or three-dimensional) reference computation, and show that it recovers the established thin-wall limit and the quasi-static thick-wall limit.
6. Incorporate the dissipative and non-ideal-excitation physics developed since 2007 for thin-wall models (unsteady wall friction, structural and viscoelastic damping, non-instantaneous valve excitations, lumped end masses) into the thick-wall setting, with solution methods that go beyond the exact closed forms (which cease to exist in that setting). Validation of the extended model against transient measurements on a pipe thick enough ($R/e<2$) to discriminate thick-wall and dynamic effects remains an open objective of the field, for which no data and no documented test rig currently exist.

Item 6 (dissipative and non-ideal-excitation physics) exists in the thin-wall literature but has not been merged with the thick-wall model or validated in the thick-pipe regime.

References

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