Water Hammer Equations Research Articles

Multi-leak diagnosis in pipelines based on Extended Kalman Filter

A model-based approach to detect and isolate non-concurrent multiple leaks in a pipeline is proposed, only using pressure and flow sensors placed at the pipeline ends. The approach relies on a nonlinear modeling derived from Water–Hammer equations, and related Extended Kalman Filters used to estimate leak coefficients. This extends former results developed for the single leak case, but with the difficulty that the model is modified at each new leak occurrence. A model adaptation strategy is thus proposed, allowing us to monitor indeed each new leak, and no matter where it appears. Experimental results illustrate the performance of the proposed algorithm.

A mixed MOC/FDM numerical formulation for hydraulic transients

A mixed numerical formulation, based on the method of characteristics (MOC) and finite difference method (FDM), is proposed for the computational simulations of hydraulic transients. To include the effects of unsteady friction, i.e. the contribution of unsteady behavior of wall shear stress, the numerical algorithm includes the Brunone friction model. The governing water hammer equations are discretized by MOC and the discretization of the local and convective acceleration term in the unsteady friction model is conducted by FDM. The procedure results in an explicit-implicit time marching scheme used for predicting the unknown values of piezometric head and discharge from the known initial conditions. The resulting system of equations is resolved in an iterative fashion. Due to its relatively low complexity, the obtained numerical algorithm is attractive for computational implementation. At the end of the paper a few illustrative numerical examples are presented, compared with available experimental data, and discussed.

Noninvasive calculation of the aortic blood pressure waveform from the flow velocity waveform: a proof of concept.

Estimation of aortic and left ventricular (LV) pressure usually requires measurements that are difficult to acquire during the imaging required to obtain concurrent LV dimensions essential for determination of LV mechanical properties. We describe a novel method for deriving aortic pressure from the aortic flow velocity. The target pressure waveform is divided into an early systolic upstroke, determined by the water hammer equation, and a diastolic decay equal to that in the peripheral arterial tree, interposed by a late systolic portion described by a second-order polynomial constrained by conditions of continuity and conservation of mean arterial pressure. Pulse wave velocity (PWV, which can be obtained through imaging), mean arterial pressure, diastolic pressure, and diastolic decay are required inputs for the algorithm. The algorithm was tested using 1) pressure data derived theoretically from prespecified flow waveforms and properties of the arterial tree using a single-tube 1-D model of the arterial tree, and 2) experimental data acquired from a pressure/Doppler flow velocity transducer placed in the ascending aorta in 18 patients (mean ± SD: age 63 ± 11 yr, aortic BP 136 ± 23/73 ± 13 mmHg) at the time of cardiac catheterization. For experimental data, PWV was calculated from measured pressures/flows, and mean and diastolic pressures and diastolic decay were taken from measured pressure (i.e., were assumed to be known). Pressure reconstructed from measured flow agreed well with theoretical pressure: mean ± SD root mean square (RMS) error 0.7 ± 0.1 mmHg. Similarly, for experimental data, pressure reconstructed from measured flow agreed well with measured pressure (mean RMS error 2.4 ± 1.0 mmHg). First systolic shoulder and systolic peak pressures were also accurately rendered (mean ± SD difference 1.4 ± 2.0 mmHg for peak systolic pressure). This is the first noninvasive derivation of aortic pressure based on fluid dynamics (flow and wave speed) in the aorta itself.

Discharge evaluation from pressure measurements by a genetic algorithm based method

This paper presents an alternative to the Gibson method, for discharge estimation of a turbine inside a hydropower plant. The method proposes a genetic algorithm which includes an integration procedure for the water hammer equations using the method of characteristics. The decision variables are represented by the steady-state regime discharge before the valve closure and the pipe friction factor. The recordings of pressure/differential pressure in measuring sections are used.The method was verified by a numerical experiment and then validated with the measured data in a laboratory experiment at NTNU, Norway.

Carotid haemodynamics during sympathetic nervous system stimulation via handgrip and cold pressor testing in young healthy subjects: A feasibility study

IntroductionAssessing arterial and haemodynamic properties in response to a physiological perturbation might provide additional information on an individual's “vascular health”. The aim of this study was to assess the feasibility of measuring changes in carotid stiffness and haemodynamics in response to sympathetic nervous system stressors, such as the Hand Grip Test (HGT) and the Cold Pressor Test (CPT). MethodsA non-invasive protocol, consisting of two HGT's at 30% and 40% of maximal voluntary contraction and a CPT, was performed in 12 young healthy volunteers (6 males/6 females). Measurements included continuous finger blood pressure recordings (NexFin; non-dominant hand) and ultrasound measurement of common carotid diameter distension and flow velocity at discrete moments in time during the protocol (GE Vivid 7 system). Carotid distension (ΔD/D), local wave speed using the waterhammer equation (PWVWH) and reflection magnitude (RM; ratio of backward to forward diameter wave) were derived from the data. ResultsConsistent with the overall increase in blood pressure, carotid diameter increased while ΔD/D decreased. PWVWH and RM showed high variability during both HGT's and CPT. ConclusionIt is feasible to monitor the carotid haemodynamic response to selected sympathetic nervous system stimulus. Moreover, the analysis showed supporting evidence of specific physiological response to regulate adequately each stress stimulus.

Analysis of Coupled Water Hammer Vibration Equation of Improvement

This paper further analyzes some existent problems of coupling vibration equations of water hammer, based on the improved continuity equation, it is derived simply for calculating coupled water hammer vibration, comparison with continuity equation that is to be used widely, the new continuity equation is basically consistent with commonly used continuity equations, so, the improved continuity equation can be used to calculate water hammer based on fluid-structure interaction (FSI).

Finite Volume Method for Computation of Water Hammer in Conical Tubes

The most traditional way to calculate water hammer in the conical tube is using some small discrete equivalent uniform tubes to replace it. this approximate treatment can not show the much accurate results of the conical tube,but also not reflect the actual physical discontinuities of the system.This paper use finite volume method to integrate water hammer equations in conical tubes on spatial and temporal scales.Compare the results of FVM discrete equations with MOC. Conclusion shows that: the new discrete equations not only has high accuracy and stability in the calculation of water hammer in conical tubes,but also has a simple derivation process and clear physical meanings.This method provides a new way of thinking in water hammer calculation of conical tubes.

Further Developments in Rapidly Decelerating Turbulent Pipe Flow Modeling

In the last two decades, energy dissipation in unsteady-state pressurized pipe flow has been examined by various authors, where the instantaneous wall shear stress is split into a quasi-steady and an unsteady shear stress component. The focus of most past studies is on formulating expressions for the unsteady wall shear stress, but there has been less work on the key parameters governing the dominance of unsteady friction in transient flows. This paper derives an expression for the head envelope damping for turbulent flows in smooth and rough pipes and provides new and carefully measured field data for the initial (i.e., pretransient) Reynolds number, R0, that ranges from 97,000 to 380,000. The analytical solutions is derived on the basis of one-dimensional (1D) water hammer equations in which the unsteady component is represented by existing convolutional unsteady friction formulas for both smooth and rough turbulent subregimes. The analytical solution is used to formulate general, encompassing and theoretically-based dimensionless parameters to assess the importance of unsteady friction in comparison to the quasi-steady component. In addition, the analytical solution furnishes the similitude relations that allowed the damping behavior from existing laboratory tests, the field tests conducted as part of this research, and the weighting function-based (WFB) models to be investigated and compared in a coherent manner in a single graph. The analysis confirms that the magnitude of R0 has a significant impact on the damping for transients generated by flow stoppage. In addition, the results show that convolutional unsteady friction model that uses the frozen eddy viscosity hypothesis and R0 has accuracy that decreases with time. An improvement for this shortcoming is proposed and verified and involves the use of the instantaneous Reynolds number in lieu of the pretransient Reynolds number in the evaluation of the WFB models. The result is a modified unsteady friction model that provides improved matches for both laboratory and field data compared with the original model.

Approximation of the Friction Integral in Water Hammer Equations

A method is proposed to improve the accuracy of the approximation commonly used with the friction term in the water hammer equations. This method partially corrects the truncation error of the friction integral by incorporating the truncation error associated with a linear flow profile into the solution algorithm. The effectiveness and utility of this accuracy enhancement are demonstrated by examples.

Predictive Control With Guaranteed Stability for Water Hammer Equations

We study the application of the receding horizon optimal control (RHOC) for hydraulic pipeline systems described by the so-called water hammer equations. Sufficient conditions to guarantee an asymptotic stability to an equilibrium state are first introduced and then integrated in the RHOC scheme. For the implementation, calculus of variations is employed to characterize the optimal solution in terms of the adjoint state and the recently proposed Lattice Boltzmann method is used to solve both direct and adjoint partial differential equations. This approach is finally validated in simulation.

Sensitivity Analysis for Water Hammer Problem in Pipelines

Water hammer is a transient flow in pipes that was created by sudden changesof velocity in pipe lines. This phenomenon can cause strong positive and negative pressures in water conveyance pipes and usually it poses pipeline to danger. Overall, water hammer creates by rapidly closing valves, shutting off or suddenly restarting pumps. It has destructive hydrodynamic effects in pressurized pipelines. In this study, governing equations of water hammer is numerically simulated using MATLAB software. Then, the sensitivity analysis in negative and positive pressures by changing some variables such as pipe diameter, pipe length and also wave velocity in pipewas investigated. Numerical simulation is based on characteristic method. Sensitivity analysis help designers to have well understand of water hammer phenomenon.

Carotid hemodynamics during sympathetic nervous system stimulation via handgrip and cold pressure testing in young healthy subjects

Introduction. Assessing arterial properties and hemodynamics in response to a physiological perturbation might provide additional information on an individual’s “vascular health”. The aim of this study was to assess the feasibility of measuring changes in carotid stiffness and hemodynamics in response to sympathetic nervous system stressors (Hand Grip Test (HGT) or Cold Pressor Test (CPT)). Methods. A non-invasive protocol, consisting of two HGT (30% (HGT1) and 40% (HGT2) of maximal voluntary contraction) and CPT, was performed in 12 young healthy volunteers (6 males/6 females). Measurements included continuous finger blood pressure recordings (NexFin; non-dominant hand) and ultrasound measurement of common carotid diameter distension and flow velocity at discrete moments in time during the protocol (GE Vivid 7 system). Carotid distensibility (D/D), local wave speed using the waterhammer equation (PWVWH) and reflection magnitude (RM; ratio of backward to forward diameter wave) were derived from the data. Results. Consistent with the overall increase in blood pressure, carotid diameter increased while D/D decreased. PWVWH and RM showed an increase during both HGT and a decrease during CPT (see Figure). Conclusion. It is feasible to monitor the carotid hemodynamic response to a sympathetic nervous system stimulus. In this young, healthy population, the net result of the increased diameter and decreased distensibility on local PWVWH was similar for all tests (increase) and on the magnitude of reflections was different for HGT (increase) and CPT (decrease).

Chebyshev super spectral viscosity method for water hammer analysis

In this paper, a new fast and efficient algorithm, Chebyshev super spectral viscosity (SSV) method, is introduced to solve the water hammer equations. Compared with standard spectral method, the method's advantage essentially consists in adding a super spectral viscosity to the equations for the high wave numbers of the numerical solution. It can stabilize the numerical oscillation (Gibbs phenomenon) and improve the computational efficiency while discontinuities appear in the solution. Results obtained from the Chebyshev super spectral viscosity method exhibit greater consistency with conventional water hammer calculations. It shows that this new numerical method offers an alternative way to investigate the behavior of the water hammer in propellant pipelines.

Application of a heterogenous multiscale method to multi-batch driven pipeline

The problem of simulating pipelines that are used for transporting different fluids is addressed in the paper. The model of the multi-batch pipeline is obtained by extending the classical “water hammer equations” (dealing with pressure and velocity of the medium) with fluid density. In such way a system of nonlinear partial differential equations is derived and solved by the method of characteristics. However, the ordinary differential equations resulting from the method of characteristics are defined on domains with very different slopes in the (x,t) space. A heterogenous multiscale method using two grids is capable of coping with associated numerical problems as shown by comparison of simulated and measured data on a real pipeline.

Energy dissipation in unsteady turbulent pipe flows caused by water hammer

Energy dissipation and turbulent kinetic energy production and its dissipation in unsteady turbulent pipe flows due to water hammer phenomena are numerically studied. For this purpose, the two-dimensional governing equations of water hammer are solved using the method of characteristics. A k–ω turbulence model which is accurate for two-dimensional boundary layers under adverse and favorable pressure gradients is applied. The numerical results are in good agreement with the experimental data. Through an order of magnitude analysis, two dimensionless parameters have been identified which can be used for the evaluation of viscous and turbulent shear stress terms. The influence of these non-dimensional parameters on pressure oscillations, wall-shear-stress, dissipation rate as well as profiles of velocity, turbulent production and dissipation are investigated. The non-dimensional parameter P, which represents time scale ratio of turbulence diffusion in the radial direction to the pressure wave speed, is used to study the structure and strength of turbulence. It is found that for the case of P≈1, for which the values of the non-dimensional groups are larger, the peaks of turbulence energy production and dissipation move rapidly away from the wall and turbulence structure is significantly changed. For the case of P≫1, for which the values of non-dimensional parameters are smaller, these variations are found to be small.

Well-Balanced Scheme for Modeling Open-Channel and Surcharged Flows in Steep-Slope Closed Conduit Systems

AbstractThe model presented in this paper preserves lake at rest conditions in sloped prismatic conduits. These schemes are based on the two-governing equation model in which open-channel flows are simulated using the Saint-Venant equations and pressurized flows using the compressible water hammer equations. The model in this paper preserves lake at rest conditions (horizontal still water) regardless of the conduit slope, resolves moving jump discontinuities over dry beds in sloped conduits, and resolves small perturbations from steady states, even when adjacent to dry regions. The preserving steady-state capability of this model is of particular importance in continuous long simulations when the conduits are relatively steep (|So|>∼0.5%. Two main contributions are presented in this paper, namely, (1) a new method for water stage reconstruction is presented that preserves lake at rest conditions regardless of the pipe slope, and (2) a horizontal system of coordinates, instead of the commonly used inclined...

Simplified Approach for the Optimal Sizing of Throttled Air Chambers

AbstractAmong water hammer damping devices, air chambers are often used in pumping plants to reduce pressure surges to acceptable values. The chamber is more effective if a throttling device is used, resulting in a reduction of the required volume. Design charts for a simple and fast sizing of air volume and orifice diameter are available in the literature using the rigid column theory (incompressible flow) and the De Sparre rule. Nevertheless, in many cases the pressure pattern is far from being constant during the first quarter of the transient period and lower pressures can be attained; furthermore, the rigid column model is not able to reproduce elastic phenomena arising from throttling, and so water hammer equations should be used instead. Although orifice induced pressure waves are evident only in the first part of the transient, differences between air chamber pressure and pipe pressure can be quite significant. Negative pressure surge should be limited because column separation and cavitation coul...

Water hammer prediction and control: the Green’s function method

By Green’s function method we show that the water hammer (WH) can be analytically predicted for both laminar and turbulent flows (for the latter, with an eddy viscosity depending solely on the space coordinates), and thus its hazardous effect can be rationally controlled and minimized. To this end, we generalize a laminar water hammer equation of Wang et al. (J. Hydrodynamics, B2, 51, 1995) to include arbitrary initial condition and variable viscosity, and obtain its solution by Green’s function method. The predicted characteristic WH behaviors by the solutions are in excellent agreement with both direct numerical simulation of the original governing equations and, by adjusting the eddy viscosity coefficient, experimentally measured turbulent flow data. Optimal WH control principle is thereby constructed and demonstrated.

Non-invasive single slice estimate of aortic distensibility from phase-contrast MRI: application to hypoplastic left heart syndrome

Background Knowledge of aortic distensibility (D) is important in several congenital scenarios. For instance, palliation of hypoplastic left heart syndrome (HLHS) often involves surgical arch reconstruction with a patch. Patch calcification is likely to impinge on aortic arch D, with a detrimental effect on ventriculo-arterial coupling. D can thus be a useful measure to derive. In order to estimate it, pressure change data from invasive measurements is combined with area (A) change information. Alternatively, distensibility can be derived from wave speed. This study proposes a method for calculation of wave speed from a single phase-contrast MRI measurement and consequently single-point estimate of D, applying the method to a cohort of HLHS patients. Methods Ten patients with HLHS and aortic arch reconstruction (3.4±1.0 years, BSA = 0.6±0.1 m2) were assessed with MRI pre-total cavil pulmonary connection (TCPC) completion and compared with another ten, non-HLHS patients (4.7±1.5 years, BSA = 0.7±0.1 m2) also imaged pre-TCPC. Their ascending aortic flow sequence was analysed with an in-house written plugin (Osirix). Care was taken that, for the HLHS group, the flow sequence was acquired after the Damus-Kaye-Stansel, ensuring that the reconstructed portion of the arch was captured (Figure 1). Area (A) and velocity (V) were extracted with a previously validated automatic propagation algorithm based on nonrigid registration. These data were interpolated and smoothed. Then, a loop was obtained plotting lnA versus V. The fractional change in A and changes in V are related by the waterhammer equation, dV=cdlnA, where c=wave speed. The slope of the linear part of the loop (early systole) thus yields c. D can finally be derived using the Bramwell-Hill equation, c2=1/rD, where r=blood density. Results

Rapid filling of pipelines with the SPH particle method

The paper reports the development and application of a SPH (smoothed particle hydrodynamics) based simulation of rapid filling of pipelines, for which the rigid-column model is commonly used. In this paper the water-hammer equations with a moving boundary are used to model the pipe filling process, and a mesh-less Lagrangian particle approach is employed to solve the governing equations. To assign boundary conditions with time-dependent (upstream) and constant (downstream) pressure, the SPH pressure boundary concept proposed recently in literature is used and extended. Except for imposing boundary conditions, this concept also ensures completeness of the kernels associated with particles close to the boundaries. As a consequence, the boundary deficiency problem encountered in conventional SPH is remedied. The employed particle method with the SPH pressure boundary concept aims to predict the transients occurring during rapid pipe filling. It is validated against laboratory tests, rigid-column solutions and numerical results from literature. Results obtained with the present approach show better agreement with the test data than those from rigid-column theory and the elastic model solved by the box scheme. It is concluded that SPH is a promising tool for the simulation of rapid filling of pipelines with undulating elevation profiles