*3.1. Rapid Instantaneous Increment of Flow Rate*

In the second group of models, the equation from which the final solution is derived is the very same equation Equation (1) that was discussed in the previous section. The pipe is also treated as long in this case, so that the influence of the formation of the flow profile in the entrance section (additional resistance) can be neglected. The fluid is assumed to be incompressible and the pipe is assumed to be horizontal (ignoring gravitational forces). The original authors of this solution are the Norwegians Helge I. Andersson and Knut L. Tiseth [32]. In this model, a sudden imposition of a constant flow rate is assumed. Such flow can be treated as generated by a piston that is suddenly set in motion with a constant speed. Mathematically, this particular kind of flow is subject to the integral constraint:

$$Q \equiv \int\_0^R 2\pi r v dr = \text{constant} \tag{29}$$

where *Q* is the flow rate.

During the start-up of this flow, the uniform motion is set initially into the pipe and the time scale for viscous diffusion is finite and of the order *<sup>R</sup>*<sup>2</sup> *<sup>ν</sup>* , which means that only an infinitesimally thin viscous layer exists at *t* = 0.

The complete boundary and initial conditions are:

(a) uniform distribution of velocity at the cross-section:

$$
v(r,0) = \frac{1}{2}v\_{\max} \tag{30}$$


The main equations in the Anderson and Tiseth paper [32] were scaled and presented in dimensionless form. The dimensionless pressure gradient solution (in this case variable in time) tends to four when fluid approaches its steady state after acceleration. The dimensionless velocity solution was split into steady and transient deviation terms made up of two separate functions, one being related to the radial position *S*(*r*ˆ) and other to time *T*(ˆ*t*). Applying a boundary condition and using the integral of the momentum equation helped Anderson and Tiseth to derive a partial differential equation that was next separated into two ordinary differential equations, the solution of which was found in a form of an infinite Bessel function series:

$$v(r,t)\_{AT} = 2v\_{\infty} \left[ \left( 1 - \left(\frac{r}{R}\right)^2 \right) + 2 \sum\_{n=1}^{\infty} \frac{J\_0\left(a\_n \frac{r}{R}\right) - J\_0(a\_n)}{a\_n^2 J\_0(a\_n)} \cdot \exp\left(-a\_n^2 \hat{t}\right) \right] \tag{31}$$

where *α<sup>n</sup>* is the nth zero of the Bessel function *J*2(*αn*) and *v*<sup>∞</sup> is the final mean velocity of the flow.

Based on the solution of Equation (31), formulas can be derived for the basic parameters of this type of accelerated unsteady flow [4]:

$$\begin{cases} \pi\_{w,AT} = \frac{2\mu v\_{\infty}}{R} \left[ 2 + \sum\_{n=1}^{\infty} e^{-a\_n^2 t} \right] \\ \left( \frac{\partial p}{\partial x} \right)\_{AT} = \frac{-4\mu v\_{\infty}}{R^2} \left[ 2 + \sum\_{n=1}^{\infty} e^{-a\_n^2 t} \right] \\\ f\_{AT} = \frac{64}{\mathcal{R}\varepsilon\_{\infty}} \left[ 1 + \frac{1}{2} \sum\_{n=1}^{\infty} e^{-a\_n^2 t} \right] \end{cases} \tag{32}$$

The course of the AT solution over time is shown in Figures 9 and 10. Figure 9 shows the dynamics of the development of the velocity profile, while Figure 10 compares the change in the maximum velocity value (in the axis of the pipe) obtained with the AT and RGS models. The last comparison shows that in the AT solution, the final profile similar to the Hagen–Poiseuille flow is obtained much faster (for <sup>ˆ</sup>*<sup>t</sup>* ≈ 0.2) than with the use of the RGS solution.

**Figure 9.** Development of velocity profile in AT accelerated flow driven by an abruptly imposed constant volume flux.

**Figure 10.** Normalized axial velocity comparison of RGS and AT solutions.

The experimental and numerical validations of this model have been performed by Chaudhury et al. [41]. The conclusion was that the Andersson and Tiseth [32] analytical model is also valid for finite-length tubes at locations beyond the entrance flow development length. Chaudhury et al. additionally write that: "This has been demonstrated by observing the same flow at *x*/*D* = 55 downstream of the inlet. The developed transient event is insensitive to the position of the piston provided the piston is more than two piston diameters away from the tube entrance. Under these conditions, results apply for constant volume flux start-up flows in physically similar piston pumps. The flow development region is significantly shorter spatially and temporally than in constant pressure gradient-driven flows."

Other experimental runs that are frequently cited were performed by Kataoka et al. [40]. Kataoka et al. reported that during their experiment an "annular jet effect" (AJE) was observed. Anderson and Tiseth [32] concluded that this AJE results from the unintentional pressure oscillations induced in the early stage of the start-up period. García García and Alvariño were motivated by these experimental results and this very untypical AJE. They found [84] that in another experimental paper by Maruyama et al. [85], the authors showed the initial stage of an AJE, thus confirming the Kataoka et al. discovery. García García and Alvariño write that: "the origin of the AJE is the result of a partial or local accordion effect, which only involves the flow away from the centreline. During an interval around *τ* ≈ 0.02, the mean velocity is greater in a region midway between the core and the wall. Later, Uprofiles, *τ* - 0.06, show a more conventional accordion effect, affecting the complete profile (global deformation). Now, the qualitative difference between early and late turbulence should emerge: the former does not present the accordion effect but AJE, whereas the latter manifests varying degrees of local accordion effects that translate into lone concavities and AJE" and "With slow turbulence, the mid-section experiences a greater increase, even yielding an AJE if the acceleration of first stage is high enough, whereas the core velocity tends to decrease".

Sparrow et al. [76] derived an equation for velocity formulation in a pressure inlet from a reservoir. Interestingly it has the same mathematical form as the AT solution for accelerating pipe flow:

$$v(r,t)\_S = 2v\_{\infty} \left[ \left( 1 - \left( \frac{r}{R} \right)^2 \right) + \sum\_{n=1}^{\infty} \frac{2}{a\_n^2} \left\{ \frac{J\_0\left(a\_n \frac{r}{R}\right)}{J\_0(a\_n)} - 1 \right\} e^{-a\_n^2 X^\*} \right] \tag{33}$$

The difference is that in place of AT dimensionless time, the Sparrow et al. analytical solution has the dimensionless distance function *X*∗ calculated as the ratio of a stretched axial coordinate *x*∗ multiplied by the kinematic viscosity divided by the mean velocity multiplied by a square of the pipe radius *X*<sup>∗</sup> = *<sup>x</sup>*∗*<sup>ν</sup> <sup>R</sup>*2*v*<sup>∞</sup> .
