The first symposium on water hammer was held in 1933 by the American Society of Mechanical Engineers [5]. In addition to the papers dealing mainly with case studies, the organizing committee on water hammer prepared a report which reviewed the state of the art on the use of this flow model contained a comprehensive bibliography, and recommended the use of a standard set of symbols. The one-dimensional inviscid compressible flow model was well understood by this time, and further advances in the model related to improvements in the methods of its application to practical problems. Notable contributions include the graphical techniques developed independently by Schnyder and Bergeron (see Jaeger [6]), which were more direct and simple to use than Allievi's method. Bergeron [7] extended his work to a general theory, which was applicable to elasticity theory and electrical transmission problems. Formulae based on these graphical construction methods have been used more recently as the basis for computer simulation techniques by Wylie [8] and Streeter [9].
Even though the model does not account for viscous effects within the conduit, it is a very useful tool in the design and analysis of fluid transmission systems, as it can be used to predict maximum pressures occurring in a system as a result of unsteady flow conditions. A common approach, for example Weng [10], to account for friction losses within the pipe is to lump the resistance at a junction in the transmission line. or depending on the load, to include the effects of line resistance in with the terminal impedance. Techniques based upon the application of this model to fluid transmission lines are presented in a number of general texts on the subject, for example Bergeron [7], Parmakian [11], Jaeger [6].
Researchers involved in water hammer analysis had observed viscous effects, but it was not until 1937, when Wood [13] applied Heaviside's operational calculus to the general theory of water hammer, that a frictional term was introduced as a distributed parameter. Wood considered the transient response of a line due to instantaneous valve closure and used several arbitrary friction coefficients to account for the dissipation effects along the line.
This form of model has been used by a number of researchers to determine the frequency and transient response of fluid transmission lines. In determining the value of the resistance coefficient, it is either assumed that the instantaneous velocity profile within the tube is the same as in steady laminar flow. in which case the Poiseuille flow coefficient 8n0/r02 is used (Schuder and Binder [14]) or an experimentally determined friction factor is used, (Oldenburger and Goodson [15]). Either way, if the resistance coefficient is assumed to be constant, the results are found to be valid over limited frequency ranges only.
In an attempt to predict theoretically the observed attenuation of disturbances propagating along a fluid filled tube, this model was used by Walker, Kirkpatrick and Rouleau [16], but they found that it gave poor agreement with experimental results. Because it is assumed that there is plane wave propagation, the solution underestimates the actual viscous losses which arise mainly due to the axial viscous shear associated with the complex velocity profile within the tube.
A solution identical to the one found by Grace - was developed independently by Sexl [20] in 1930, and again by Lambossy [21] in 1952. This work was later extended by Uchida [22] to include the effect of a superimposed steady flow component.
Womersley [23] used Lambossy's form of solution to develop a method by which instantaneous blood flows in circulatory systems could be determined from a known pressure gradient. In a series of papers, Womersley [23][24][25] developed the theory to account for non-linear terms due to steady flow, the effects of thin-walled elastic tubes, reflections from junctions and rigid inserts, and the effects of a liquid whose viscosity exhibits frequency-dependent characteristics. By introducing an extra restriction based upon the finite propagation velocity in the tube, Womersley obtained expressions for the propagation operator and characteristic impedance of the line. Experimental studies by Cotton and Gabe [26], and Linford and Ryan [38] have shown that the theory is adequate for most biological applications.
An early theoretical investigation of wave propagation in a viscous fluid enclosed within an elastic tube was made by Witzig )see Karreman [28]) in 1914. Assuming incompressible flow, he obtained solutions for the pressure and the axial and radial velocity components by simultaneously solving the axial and radial equations of motion. Witzig's sophisticated treatment of the problem involved some assumptions and simplifications, which were later criticised by Morgan and Kiely [29]. They showed from order-of-magnitude considerations that Witzig's basic mode - reduced to the two-dimensional viscous incompressible flow model, and in limiting cases - did not produce accurate results due to invalid approximations. Their own model was developed further [30] to include the effects of a superimposed steady flow.
The problem of oscillating flow near the mouth of a circular tube was considered by Atabek and Chang [31] who used the model to obtain velocity profiles at various distances from the inlet. This work was experimentally verified by Florio and Mueller [32] in 1968.
An interesting application of the two-dimensional, incompressible model was presented in a paper by Sarpkaya [33] in 1966. He used the model to identify the occurrence of inflection points in the instantaneous velocity profiles and hence, used this to predict the onset of turbulence. Sarpkaya concluded that, unlike steady flow, the presence of one or more inflection points was necessary but not sufficient to cause instability in oscillatory flows.
Ignoring the effects of heat conduction, the discriminant between the two cases was recognized to be the shear wave number rather than viscosity alone, and the solution presented by Crandall [34] provided the transition between the two limiting conditions. This solution of the axial equation of motion expresses the axial velocity as a function of pressure gradient and radial position, and is identical to the solution found later by Grace and Sexl under the assumption of incompressible flow. Assuming that the fluid is compressible allows the continuity and state equations to be combined with axial equation of motion so that the propagation operator and the characteristic impedance can be determined in addition to the series impedance. Crandall presented his solution in terms of real and imaginary parts, corresponding respectively to the effective resistance and inertial coefficients. Substituting these coefficients back into the equation of motion produces a frequency-dependent form of the linear resistance model in which the imaginary part of the viscous term is lumped with the inertial term as an effective change in density. This approach has been used by Rohmann and Grogan [35] in predicting the frequency response of transmission lines, and by Foster and Parker [36] in the analysis of an oil-filled transmission line in an alternating flow hydraulic system.
D'Souza and Oldenburger [42] extended Brown's results to liquid-filled conduits and included the effects line vibration and provided experimental support for their theoretical results. Because the ratio of specific heats is close to unity for most liquids, this model effectively reduces to the two-dimensional, viscous, compressible flow model when applied to liquid-filled lines.
Using a similar approach to that described above, a full solution to the exact first-order model was obtained by Gerlach [44]. Using numerical methods to extract a set of solutions, Gerlach predicted the existence of an infinite number of radial modes of propagation in addition to the fundamental longitudinal model. The theoretical treatment of the problem also included the effects of different wall and end conditions, the effects of non-linear terms, and line and system vibration. Experimental evidence of the existence of these higher modes was presented in a paper by Gerlach and Parker [45].
Gerlach showed that for most engineering applications, only the zeroth or fundamental mode is important, and that the solution for this mode was effectively equivalent to the exact two-dimensional, viscous, compressible solution, with the exception that the zeroth mode solution contained extra terms which accounted for attenuation of plane waves. For most practical applications the effects of this term were shown to be negligible.
The theoretical development of the solution, and the conclusion drawn from it were verified by Urata [46] and later in dimensionless form by Ohmi et al. [47]. The model was also considered by Inaba et al. [48], who used a perturbation analysis to examine the effect of a steady flow component on the velocity profiles and on the transfer matrix elements relating the pressure and velocity in the pipe.
In 1970, Scarton [49] showed that the higher order modes of propagation found by Gerlach were in error except at low frequencies. Using the method of eigenvalues, he was able theoretically to show the existence of two families of higher order propagation modes corresponding to the previously identified radial modes and to a new set of 'rotation-dominated' modes.
Laminar flow Whereas the criterion for the occurrence of turbulence in steady flow is simply the Reynolds number, in unsteady flows neither the criteria used to predict flow instability, nor the manner in which it occurs is well understood. This problem has been investigated by Sarpkaya [33], Clarion and Pelissier [50], Clamen and Minton [51], Hershey and Im [52], but the results were not conclusive and some results appear contradictory. A note in the paper by Scarton and Rouleau [53] suggests that an experimental relationship based on the shear wave number and a modified form of Reynolds number be used as the criterion. Within experimental limits, this expression is in close agreement with the work by Sergeev [54]. In the case of an oscillating flow component, which is superimposed on a mean turbulent flow, experimental evidence, Karam and Franke [55], Regetz [56] suggests that the laminar flow solutions are still applicable over a limited turbulent flow range.
Disturbances propagate isentropically The use of this assumption implies that the walls of the tube are perfectly rigid. Effects due to the elasticity of the conduit wall can be accounted for by using a propagation velocity based upon the effective bulk modulus of the fluid and tube, instead of using the isentropic speed of sound in the fluid.
Axisymmetric flow This assumption carries the implication that the conduit is straight, but experimental evidence of D'Souza [57] has shown that the models can be accurately applied to pipes with relatively small radii of curvature.
No bulk viscosity effects Theoretical and experimental work with liquids and gases Karim and Rosenhead [58], Herzfel [59] and Libermannn [60] have shown that excess absorption due to bulk viscosity is only of importance at ultrasonic frequencies and hence is beyond the range of most transmission line problems.
Non-linear convective acceleration terms are negligible Any time-varying flow can be considered to be made up of an unsteady flow component superimposed on a steady mean flow. As long as the magnitude of these components is much less than the speed of sound in the fluid, the non-linear terms are negligible; however, when either of these components have magnitudes comparable to the speed of sound, these terms are significant. These non-linear effects, together with a linearization procedure for large steady flows are treated and discussed in Gerlach [61].
Thermal effects are negligible Zwikker and Kosten [37] and Iberall [39] have solved Kirchoff's equations to show that thermal effects are significant compared to viscous effects in the dynamic response of gas-filled lines. In liquid lines, however, the effect of heat transfer on the equations of motion is negligible Brown [41].