**4. Accelerated Turbulent Pipe Flow—TULF Model by García García and Alvariño**

In this section, the model that extends from steady to unsteady turbulent pipe flow will be discussed. The model is based on Pai's [87] idea of decomposing the turbulent velocity profile to the sum of the Hagen–Poiseuille parabola and a purely turbulent component. The starting point of García García and Alvariño model (TULF model; TULF is an abbreviation for the theory of underlying laminar flow) is the non-homogeneous dimensionless Reynolds-averaged Navier–Stokes partial differential equation (RANSE) [84,88–90]:

$$\frac{\partial v}{\partial \hat{t}} - \left(\frac{\partial^2 v}{\partial \hat{t}^2} + \frac{1}{\hat{r}} \frac{\partial v}{\partial \hat{t}}\right) = -\frac{\partial \hat{p}}{\partial \hat{x}} - \frac{1}{\hat{r}} \frac{\partial (\hat{r}v)}{\partial \hat{r}} = II(\hat{t}) + \Sigma(\hat{t}, \hat{r})\tag{42}$$

with the following initial and boundary conditions:

$$v(0, \dot{\mathbf{r}}) = v\_0(\dot{\mathbf{r}}), \ v(\hat{\mathbf{f}}, 1) = 0, \ \frac{\partial v(\hat{\mathbf{f}}, 0)}{\partial \hat{\mathbf{r}}} = 0 \tag{43}$$

and knowledge of the dimensionless Reynolds-averaged continuity equation:

$$\frac{\partial v}{\partial \hat{\mathfrak{K}}} = 0 \tag{44}$$

In the above equations *Π* ˆ*t* is the mean (minus) pressure gradient, *σ* ˆ*t*,*r*ˆ is the Reynolds Shear Stress (RSS) field, satisfying the condition *σ* ˆ*t*, 0 = *σ* ˆ*t*, 1 = 0 and the function *Σ* ˆ*t*,*r*ˆ <sup>=</sup> <sup>−</sup><sup>1</sup> *r*ˆ *∂*(*r*ˆ*σ*) *<sup>∂</sup>r*<sup>ˆ</sup> is defined by the authors of this model as the Weighted Reynolds Shear Stress Gradient (WRSSG).

The new semi-analytical solution was obtained by García García and Alvariño [84,88–90] in the following steps:


The terms (*Π*0, *Π*∞; *χ*0, *χ*<sup>∞</sup> and *q*0, *q*∞) that govern RSS, mean velocity and WRSSG are defined in [84] as Spatial Degrees of Freedom (SDoF) that need to be calculated for the initial and final times. They define the radial dependence of the relevant flow fields. Coefficient *χ*<sup>0</sup> is the initial centreline turbulent dissipation being the ratio *vL* ˆ*t* = 0,*r*ˆ = 0 /*v* ˆ*t* = 0,*r*ˆ = 0 of underlying laminar flow to the mean velocity at the centerline of the pipe. The turbulence field can be switched off if one assumes *χ*<sup>0</sup> = 1. The other initial coefficient *q*<sup>0</sup> is a bestfitting integer power. The final values of the coefficients *χ*∞ and *q*∞ need to be defined from the experimental velocity profile of the final turbulent flow.

**Figure 13.** TULF model component-wise block diagram of detailed velocity field decomposition.

If the accelerated flow is started from rest then the initial values of RSS, velocity field and WRSSG are: *σ*0(*r*ˆ) = 0; *v*<sup>0</sup> ˆ*t* = 0,*r*ˆ = 0 and *Σ*0(*r*ˆ) = 0, while the final values can be calculated from the following equations [84]:

$$\begin{Bmatrix} \sigma(\hat{t}, \hat{r})\\ \Sigma(\hat{t}, \hat{r}) \end{Bmatrix} = \begin{Bmatrix} 0\\ 2\eta^2(\frac{3}{2} - \eta) \begin{Bmatrix} \sigma\_{\infty}(\hat{r})\\ \Sigma\_{\infty}(\hat{r}) \end{Bmatrix} \text{ if } \hat{t} < \hat{t}\_{0\prime}\\ \begin{Bmatrix} \sigma\_{\infty}(\hat{r})\\ \Sigma\_{\infty}(\hat{r}) \end{Bmatrix} \text{ if } \hat{t} > \hat{t}\_{2\prime} \end{Bmatrix} \tag{45}$$

where *<sup>η</sup>* = <sup>ˆ</sup>*t*−ˆ*t*<sup>0</sup> <sup>ˆ</sup>*t*2−ˆ*t*<sup>0</sup> ∈ [0, 1], <sup>ˆ</sup>*t*<sup>0</sup> is the mean time of transition to turbulence; <sup>ˆ</sup>*t*<sup>2</sup> is the mean turbulence-settling time. These times are not easily controlled by any experimenter (the values may change from one experiment to the other). They must be defined in the TULF model too, they are external parameters that cannot be determined from analytical theory. The relation between mentioned times ˆ*t*0, ˆ*t*<sup>2</sup> and Δˆ*t* defined as Temporal Degrees of Freedom (TDoF) can be a source of division of the flows into four classes [84]:


The above four types of accelerated pipe flows are the subject of intensive research discussed in [84];

(e) the use of formulas discussed in earlier points made it possible to determine the final solution for the terms *vL* ˆ*t*,*r*ˆ and *vT* ˆ*t*,*r*ˆ for the analyzed case of accelerated flow for appropriate ranges of dimensionless time. 

The final solution [84] for *vL* ˆ*t*,*r*ˆ is:

$$\text{(a)}\quad\text{for }0\le\dagger\le\Delta\h?$$

$$w\_L(\hat{\mathbf{f}},\hat{\mathbf{f}}) = \frac{12\Pi\left(\hat{\mathbf{f}}\right)}{\Delta\hat{\mathbf{f}}^3} \sum\_{n=1}^{\infty} \frac{j\_0(\lambda\_n \hat{\mathbf{r}})}{\lambda\_n^9 j\_1(\lambda\_n)} \left[ -\frac{\lambda\_n^6 \hat{\mathbf{f}}^3}{3} + \lambda\_n^4 \left(\frac{\lambda\_n^2 \Delta\hat{\mathbf{f}}}{2} + 1\right) \hat{\mathbf{f}}^2 + \left(\lambda\_n^2 \Delta\hat{\mathbf{f}} + 2\right) \left(1 - \lambda\_n^2 \hat{\mathbf{f}} - e^{-\lambda\_n^2 \hat{\mathbf{f}}}\right) \right] \tag{46}$$

(b) while for ˆ*t* > Δˆ*t*:

$$v\_L(\hat{\mathbf{f}},\hat{\mathbf{f}}) = \frac{12II(\hat{\mathbf{f}})}{\Delta\mathbf{f}^3} \sum\_{n=1}^{\infty} \frac{f\_0(\lambda\_n \hat{\mathbf{r}})}{\lambda\_n^9 I\_1(\lambda\_n)} \left[ \left( 2 - \lambda\_n^2 \Delta\hat{\mathbf{f}} \right) e^{-\lambda\_n^2 (\hat{\mathbf{f}} - \Delta\hat{\mathbf{f}})} - \left( 2 + \lambda\_n^2 \Delta\hat{\mathbf{f}} \right) e^{-\lambda\_n^2 \hat{\mathbf{r}}} + \frac{\lambda\_n^6 \Delta\hat{\mathbf{f}}^3}{6} \right] \tag{47}$$

And the final solution [84] for *vT* ˆ*t*,*r*ˆ :

$$\begin{aligned} \text{(a)} \quad \text{for } 0 \le \hat{t} \le \hat{t}\_0;\\ v\_T(\hat{t}, \theta) = 0 \end{aligned} \tag{48}$$

(b) for <sup>ˆ</sup>*t*<sup>0</sup> < <sup>ˆ</sup>*<sup>t</sup>* ≤ <sup>ˆ</sup>*t*2:

$$\begin{split} \boldsymbol{\upsilon}\_{T}\{\boldsymbol{\hat{t}},\boldsymbol{\hat{t}}\} &= \frac{\boldsymbol{II}(\boldsymbol{\hat{t}})}{\left(\hat{t}\_{2}-\hat{t}\_{0}\right)^{2}\boldsymbol{\hat{X}}\boldsymbol{\hat{t}}\boldsymbol{\hat{t}}-\boldsymbol{1}} \sum\_{n=1}^{\infty} \frac{l\_{0}(\boldsymbol{\lambda},\boldsymbol{\hat{t}})}{\lambda\_{n}^{4}(f\_{1}(\boldsymbol{\lambda}\_{n}))^{2}} \left[\boldsymbol{1}\,\boldsymbol{F}\_{2}\left(\boldsymbol{q};\boldsymbol{q}+\boldsymbol{1},\boldsymbol{1};-\frac{\lambda\_{n}^{2}}{4}\right) \\ & -\frac{2l\_{1}(\boldsymbol{\lambda}\_{n})}{\lambda\_{n}}\right] \Big{{}^{3}\Big{{}}\Big{{}}\Big{{}\left(\left(\hat{t}-\hat{t}\_{0}\right)\lambda\_{n}^{2}-{}1\right)^{2}+1-2e^{-\lambda\_{n}^{2}\left(\hat{t}-\hat{t}\_{0}\right)}} \Big{{}^{3}\Big{{}}\Big{{}^{3}\hat{t}} \\ & -\frac{2}{\left(\hat{t}\_{2}-\hat{t}\_{0}\right)\lambda\_{n}^{2}} \Big{{}^{3}\Big{{}}\Big{{}^{3}\hat{t}}\_{n} -3\left(\hat{t}-\hat{t}\_{0}\right)^{2}\lambda\_{n}^{4} +6\left(\left(\hat{t}-\hat{t}\_{0}\right)\lambda\_{n}^{2}-1+e^{-\lambda\_{n}^{2}\left(\hat{t}-\hat{t}\_{0}\right)}\right)\Big{{}^{3}\hat{t}}\Big{{}^{3}\hat{t}} \\ & \text{(c)}\quad\text{ for }\boldsymbol{f}>\boldsymbol{t}\_{2}:\end{split} \tag{4}$$

$$\begin{split} \upsilon\_{T}\left(\hat{\mathfrak{f}},\hat{\mathfrak{f}}\right) &= \frac{II\left(\hat{\mathfrak{f}}\right)}{\left(\hat{\mathfrak{f}}\_{2}-\hat{\mathfrak{f}}\_{0}\right)^{2}\overline{\chi(q-1)}} \sum\_{n=1}^{\infty} \frac{f\_{0}(\lambda\_{n}\boldsymbol{\mathfrak{f}})}{\lambda\_{n}^{\boldsymbol{\mathfrak{f}}}(\boldsymbol{f}\_{1}(\lambda\_{n}))^{2}} \left[{}\_{1}F\_{2}\left(\boldsymbol{q};q+1,1;-\frac{\lambda\_{n}^{2}}{4}\right) - \frac{2f\_{1}(\lambda\_{n})}{\lambda\_{n}}\right] \left\{\lambda\_{n}^{\boldsymbol{\mathfrak{f}}}\left(\hat{\mathfrak{f}}\_{2}-\hat{\mathfrak{f}}\_{0}\right)^{2} \\ & - 6\epsilon^{-\lambda\_{n}^{2}\left(\hat{\mathfrak{f}}-\hat{\mathfrak{f}}\_{0}\right)} \left(\frac{2}{\left(\hat{\mathfrak{f}}\_{2}-\hat{\mathfrak{f}}\_{0}\right)\lambda\_{n}^{2}}+1\right) + 6\epsilon^{-\lambda\_{n}^{2}\left(\hat{\mathfrak{f}}-\hat{\mathfrak{f}}\_{2}\right)} \left(\frac{2}{\left(\hat{\mathfrak{f}}\_{2}-\hat{\mathfrak{f}}\_{0}\right)\lambda\_{n}^{2}}-1\right) \right\} \end{split} \tag{50}$$

In Equations (49) and (50), <sup>1</sup>*F*2(*a*; *b*, *c*; *x*) is the Generalized Hypergeometric function, which generally is calculated in the following way:

$$\,\_1F\_2(a;b,c;x) = \sum\_{n=1}^{\infty} \frac{(a)\_n}{(b)\_n (c)\_n} \frac{x^n}{n!} \tag{51}$$

In Equation (51), (*z*)*<sup>n</sup>* is the Pochhammer's symbol, which is defined as:

$$(z)\_n = z(z+1)(z+2)\cdots(z+n-1) = \prod\_{k=1}^n (z+k-1) = \frac{\Gamma(z+n)}{\Gamma(z)}, \ (z)\_0 = 1, \ z \in \mathbb{N} \tag{52}$$

The complete accelerated solution is a sum of laminar and turbulent components:

$$\upsilon(\hat{t},\hat{r}) = \upsilon\_L(\hat{t},\hat{r}) + \upsilon\_T(\hat{t},\hat{r}) \tag{53}$$

An example of simulation with this model for a case of accelerated pipe flow from rest to *Re* = 56677 is presented in Figure 14. Input data for this early and slow class of accelerated flow are: *Π* ˆ*t* = 4.0912·106; <sup>Δ</sup>ˆ*<sup>t</sup>* = 0.006; *<sup>q</sup>* = 45; *<sup>χ</sup>* = 29.3758; <sup>ˆ</sup>*t*<sup>0</sup> = 0.0042; ˆ*t*<sup>2</sup> = 0.012.

**Figure 14.** Results of calculation using the García García and Alvariño TULF model: (**a**) early stage; (**b**) late stage.

In summary, the TULF solution presented by García García and Alvariño is an interesting addition to the theory of analytical methods for solving this type of accelerated flow. With its help, it was possible to theoretically justify many phenomena occurring during accelerated flows (e.g., the hyperlaminar jet effect [91], the lone concavity [92], or the annular jet effect [40,85,93]), which until now had only been observed experimentally. These findings show the role of analytical solutions in the continuous scientific progress regarding the issues of transient liquid flows through pressurized pipes. They were derived assuming a pressure gradient change as a flow generator. This means that an equivalent solution could also be derived for forced flow by means of a piston.
