**5. Conclusions**

Two types of accelerated flows described in the literature were analyzed: (1) acceleration as a result of a change in pressure gradient (occurs in systems where a change in pressure occurs as a result of a sudden opening of the gate valve) and (2) acceleration resulting from the assumption of a specific change in the flow rate (such a situation occurs in hydraulic systems with forced movement due to the displacement of piston elements). A survey of the literature shows that the standard 1932 Szyma ´nski's solution has been independently derived much earlier by at least two scientists: (1) the Italian Roiti in 1871 and the Russian Gromeka in 1882, hence it should be quoted as the RGS solution, as adopted in this work from the letters of the authors' surnames. An interesting solution, which is the prototype of the 1992 Andersson and Tiseth solution, was derived in 1933 by Vogelpohl. This combines the universal solution for any change in the pressure gradient discussed in detail in Section 2.3 of this work with solutions from the second group, i.e., flows forced by a jump in the flow rate. Vogelpohl succeeded in determining the pressure gradient function

such that the average value of the flow velocity is kept constant (identical to that obtained almost sixty years later by Anderson and Tiseth). All the analytical solutions discussed in this work are of great practical value, as they give the opportunity to accurately analyze the variability in velocity profiles, mean velocity values (or pressure gradients), shear stresses on the pipe wall, dynamic friction coefficients, etc. Thanks to this, they are perfect for verifying commercial programs in the field of CFD (computational fluid dynamics). The completed review of known analytical models of accelerated flow indicates that Telionis was right when he wrote in his book that "impulsive fluid motions do not exist in reality". In real life the motion of a piston is never instantaneous as assumed in the Anderson and Tiseth solution; similarly, an instantaneous pressure gradient never takes place in real systems (as an effect of the valve opening time). So real flow is always somewhere between these two solutions defined and discussed in this paper. In the case of accelerated flows resulting from the impact of the pressure gradient at the inlet and outlet of the conduit, it seems that further research is necessary to show the real time variability of pressure gradient functions in laminar flows. This will determine whether it can actually be described by a function similar to the one observed in experimental studies by Avula in 1968, if so, it will still be necessary to choose a function that will give a complete analytical solution (and not, as now, the integral of this function should be determined numerically). There is a need to summarize all scientific papers describing experimental studies of accelerated flows—most are focused on the transition between laminar and turbulent flow. Experimental studies strictly concerning only laminar flows (*Re* < 2320) are almost non-existent in the literature.

A drawback of the presented models is their mathematical complexity (infinite series). They are based on Bessel functions and their zeros. For these zeros no analytical formula has been developed to date, hence they must be determined numerically. Bessel functions, on which all discussed solutions of velocity profile are based, are still simulation problems, especially for very small and large arguments. To overcome this problem in the computer programs used to carry out sample comparisons presented in several figures, this function was expanded in a Taylor series for small arguments, and asymptotic formulas for large arguments were used. According to this approach, this solution should be approximated with much simpler functions to be more useful in practice.

A significant strength of the presented analytical solutions is the possibility to define other related flow parameter-time dependences: mean velocity (in pressure-gradientdriven flows), the pressure gradient (in flow-rate-driven flows), friction factor and wall shear stress. These parameters are very helpful for a better understanding of the accelerated flow characteristics.

The semi-analytical solution for turbulent flow (TULF model developed by García García and Alvariño) is the most complex of all the analyzed solutions. Here, the turbulent velocity field is dependent on Bessel functions, their zeros and a generalized hypergeometric function. There is no easy and straightforward way to define the required spatial degrees of freedom coefficients (*Π*, *χ* and *q*—defining respectively: mean pressure gradient, initial centreline turbulent dissipation and best-fitting integer power). However, it is possible to theoretically investigate and better understand the behavior of experimentally discovered phenomena such as: the lone concavity, annular jet effect and hyperlaminar jet effect. This solution would be useful to study the behavior of other related phenomena not discovered experimentally yet. Finally, Table 2 summarizes the advantages and disadvantages of the reviewed analytical solutions.


**Table 2.** Short summary of the reviewed analytical solutions valid for laminar accelerating pipe flow (except TULF model).

In the present review, it was also noted that, to date, there is a lack of an analytical solution as well as experimental studies of the accelerated flow of Newtonian fluid in plastic conduits (HDPE, ABS, PVC, PP, PB). In these conduits, the water hammer theory implies that the additional damping is due to the delayed strains that occur during this type of flow. The influence of these deformations is taken into account in the equation of continuity and not momentum, so the question remains whether in accelerated flows the same profiles and development times will be obtained in pipes with identical parameters (diameter, roughness, wall thickness) but made of two different materials: plastic and metal.

**Author Contributions:** Conceptualization, K.U., A.B. and M.S.; methodology, K.U. and M.K.; software, K.U., M.K. and A.D.; validation, K.U., A.D. and M.S.; formal analysis, K.U., A.D. and M.S.; investigation, K.U., A.B, M.S. and M.K.; resources, K.U.; data curation, K.U. and A.D.; writing—original draft preparation, K.U. and A.B.; writing—review and editing, K.U., A.B., A.D. and M.S.; visualization, K.U. and M.K.; supervision, K.U. and A.B.; project administration, K.U. and M.S.; funding acquisition, K.U. and A.B. All authors have read and agreed to the published version of the manuscript.

**Funding:** A. Bergant gratefully acknowledges the support of the Slovenian Research Agency (ARRS) conducted through the research project L2-1825 and programme P2-0162.

**Data Availability Statement:** Codes generated during the study and experimental data are available from the corresponding author by request.

**Conflicts of Interest:** The authors declare no conflict of interest.
