**1. Introduction**

Fluid flow in hydraulic pipes is often time-varying flow. This means that the basic parameters of the flow, i.e., the average value of velocity and pressure in the analyzed pipe cross-section, dynamically change over time. Time-varying flows can be accelerated, decelerated, reverse, pulsating, oscillating and water hammer-type flows. This paper reviews analytical models for accelerating pipe flows.

Starting with the work of Navier [1], the basic hydrodynamic equations (according to the work of Darrigoll [2], these have been re-derived at least four times chronologically by the following well-known researchers: Cauchy, Poisson, Saint-Venant and Stokes), are the subject of numerous studies. Analyzing the literature on analytical solutions of laminar accelerated flows, it can be seen that two groups of flows can exist [3,4]: the first is flows forced by the occurrence of a step pressure gradient change, while the second group of flows is those in which the fluid movement is forced by a step change in the flow rate.

Much earlier, solutions from the first group were analyzed, i.e., when the flow is forced by a change of the pressure gradient along the length of the pipe. The oldest paper found for this review was published by the Italian scientist Roiti [5] in 1871, who worked at the University of Pisa. He studied accelerated flow in a simple vertical water system. The

**Citation:** Urbanowicz, K.; Bergant, A.; Stosiak, M.; Deptuła, A.; Karpenko, M. Navier-Stokes Solutions for Accelerating Pipe Flow—A Review of Analytical Models. *Energies* **2023**, *16*, 1407. https://doi.org/10.3390/en16031407

Academic Editors: Artur Bartosik and Dariusz Asendrych

Received: 30 November 2022 Revised: 5 January 2023 Accepted: 20 January 2023 Published: 31 January 2023

**Copyright:** © 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

pressure change was obtained by rapidly opening the valve. Apart from experimental research, Roiti's most important achievement was the presentation of an analytical theory of this flow. This referred to solutions previously described by another researcher from the University of Pisa, namely Betti [6], who analyzed heat flows. Another important work was written in 1882 by Gromeka [7]. Unfortunately, due to the fact that it was published only in Russian, it remains known mainly among authors of Russian origin [8–11], who mostly know it from an important reprint of all Gromeka's works in a book form, published on the 100th anniversary of his birth [12]. It was only thanks to the work of Szyma ´nski [13] published in 1932 in a respected French mathematical journal (Journal de Mathématiques Pures et Appliquées—still existing and new papers being published) that this solution was noticed and reproduced in many books on the flow of viscous fluids [14–16]. The accelerated solution forced by a rapid, strictly defined pressure gradient was then analyzed by a number of researchers who extended its applicability. Among these studies, the work of Gerbes [17], was the first to apply the Laplace transform to its derivation, should be noted, as well as that of Ito [18], who extended the scope of this solution to make it suitable for the analysis of unsteady transient flows occurring between one steady flow (of the Hagen–Poiseuille type) and another characterized by a higher average velocity.

For short pipes, the entrance length plays an important role. Solutions for this unsteady acceleration problem devoted to the development of the velocity profile along the pipe length and in time have been derived by Atabek [19] and Avula [20]. Fan [21] solved the problem of accelerated flow in rectangular ducts. Solutions in ducts with an arbitrary geometry were the subject of research by Laura [22]. This issue has been recently analyzed by Muzychka-Yovanovich [23,24] as well. In the following years, similar solutions were developed for non-Newtonian fluid flows [25–28] for which the shearing stress is not linearly related to the rate of shearing strain [29].

The second group of models, i.e., those in which the function of velocity or flow rate is defined in the main equation of motion, has been analyzed since 1933, when Vogelpohl's short technical note was published [30]. In his note, Vogelpohl also mentioned the experimental research carried out at the University of Berlin by Prof. Föttinger on accelerated flows. In addition, he derived a formula (using the Whittaker method [31]) for the time-varying pressure gradient for this type of flow (later re-derived by Andersson and Tiseth [32]) for a constant value of average velocity. Other researchers who were interested in this type of flow were Weinbaum and Parker [33], although their interest was mostly focused on the decelerated flow resulting from the sudden closing of the gate valve. Weinbaum and Parker's research was the inspiration for subsequent works, including the work of Andersson and Tiseth [32], which is particularly important from the point of view of this review. It was in this work that for the first time a complete analytical solution for the flow velocity profile was derived for a constant mean velocity scenario that can be the effect of piston movement. It should be mentioned here that Andersson and Kristoffersen [34] had previously analyzed a wide range of flows forced by a pressure gradient (RGS type) as well as the correction to this solution proposed by Otis [35]. The Anderson and Tiseth solution has been improved by Das and Arakeri [36], who presented analytical solutions for a complete piston cycle (acceleration period, motion with constant velocity, deceleration period and final period). The Das and Arakeri solutions for the first two periods (i.e., acceleration and piston work at constant velocity) can be treated as a complete solution when imposing a ramp-type motion. This solution was further refined by Kannaiyan et al. [37] by introducing the possibility of determining the transition from one steady state to another.

The main experimental studies that confirmed the correctness of the solution based on a step change in pressure were carried out by van de Sande et al. [38] and Lefebvre and White [39]. These studies show that laminar flow is maintained during acceleration for a relatively long time—in the paper by van de Sande et al. [38] to Re = 57,500, and in the work of Lefebvre and White [39] to Re > 105. This delayed transition from laminar to turbulent flow was also mentioned by Goldberg in a discussion on van de Sande et al. at the Pressure Surges Conference in Canterbury [38]. Goldberg (p. 499 of Pressure Surges 1980 proceedings) wrote about a long distant oil transportation pipeline (105 miles long, twenty inches in diameter) located in the Gulf of Mexico in which, during the start-up, a delayed transition to turbulent flow was detected.

As opposed to pressure-driven flows, the other type of acceleration occurs in systems due to a step change in flow rate, e.g., piston-driven flow. In an early experiment by Kataoka et al. [40], the measured results did not agree well with the theory as an annular jet effect was reported. The research, which indicated the correctness of the theoretical solution for a fixed flow rate, was carried out by a team led by Chaudhury et al. [41]. Recently He et al. [42] have reported that the whole acceleration phase in pipes is a laminar–turbulent bypass transition, so even during the initial flow, some turbulent structures that initially occupy the near wall region are present in the flow.

Although this paper concerns only accelerated Newtonian liquid flows in pipes with circular cross-sections (typical, commonly used in practice, Figure 1), the scope of research is very extensive. Readers interested in further solutions (flows through other crosssections, accelerated flows of non-Newtonian fluids, etc.), which were developed on the basis of the accelerated flows theory discussed in this review, will be offered additional reading elsewhere.

**Figure 1.** Layout of the pipe considered in the present review.

In Sections 2 and 3 of the present review paper, accelerating laminar flow solutions will be discussed. Laminar flow means that the orderly movement of the fluid along parallel paths occurs in which the fluid elements do not mix with each other, and there is, therefore, a purely viscous mechanism of momentum and energy exchange. Section 4 discusses a turbulent semi-analytical solution. Turbulent flow is understood as a chaotic motion of a fluid. During this flow the elements of the fluid mix with each other, which leads to the intensification of mass, momentum and energy exchange.

#### **2. Accelerated Laminar Flows Driven by Pressure Gradient**
