Frictional loss in bend pipes: numerical simulation and data driven modeling

Abstract

Here, we present a quasi-universal correlation for the friction factor in 90-degree and 180-degree bend pipes using the ensemble of metamodels (EOM). Using computational simulations, we obtain frictional losses at curvature ratios from 0.01 to 0.2 and Reynolds numbers up to 0.1 million. With the help of this data, we develop the EOM based correlation. The extensive window of parameters considered in the present study encompasses a vast design and operating conditions of 90-degree and 180-degree bend pipes. The large parametric space considered here essentially represents a variety of hydrodynamic scenarios. We use a sufficient yet nominal set of parameters representative of the variation and perform numerical simulations over the sampled set. Using the simulation data, we then train and validate the EOM to develop the required correlation. The k-fold cross-validation technique is employed to the EOM to optimize the model’s predictive capabilities over the changing flow regimes. We believe that this modeling strategy, along with the database, may act as a base framework towards the further development of data augmentation techniques in pipes of various configurations, in general. Using this EOM based correlation, we further generate a large database for the friction factor in the two smooth bend pipe configurations at various curvature ratio and different Reynolds number. This approach can be viewed as an alternative to the Moody chart, however, for pipe bends.

Appendix A: Reynolds Stress Model theory

The steady RANS based Reynolds Stress closure model (RSM) is used in this analysis. The theory of this model is quite extensive. Therefore, we present only a brief overview of the model equations. Since the Fluent solver is used for our simulations, we greatly refer to the ANSYS documentation for this model. We also refer to the NASA turbulence modeling resource [70], for a more verified and up to date representation of the model. Furthermore, we attempt to represent the equations here in a structured manner, in tune with the NASA turbulence modeling resource. RSM is used in turbulent flow cases where anisotropy predominantly effects mean flow. This may include unbalances in axial flow due to curvature, pressure differences, mixing, and secondary flow. Modeling transport equations using RSM includes five additional equations in two-dimensional and seven additional equations in three-dimensional flows [71, 72]. The exact Reynolds stress transport equations for Reynolds stresses $-\rho \overline{{u }_{i}^{\prime}{u}_{j}^{\prime}}$ is given by

$$ \begin{aligned} & \frac{\partial }{\partial t}\left( {\rho \overline{{u_{i}^{{\prime }} u_{j}^{{\prime }} }} } \right) + \frac{\partial }{{\partial x_{k} }}\left( {\rho u_{k} \overline{{u_{i}^{{\prime }} u_{j}^{{\prime }} }} } \right) \\ & \quad = - \frac{\partial }{{\partial x_{k} }}\left[ {\rho \overline{{u_{i}^{{\prime }} u_{j}^{\prime} u_{k}^{{\prime }} }} + \overline{{p\left( {\delta_{kj} u_{i}^{{\prime }} + \delta_{ik} u_{j}^{{\prime }} } \right)}} } \right] + \frac{\partial }{{\partial x_{k} }}\left[ {\mu \frac{\partial }{{\partial x_{k} }}\left( {\overline{{u_{i}^{{\prime }} u_{j}^{{\prime }} }} } \right)} \right] \\ & \quad - \rho \left( {\overline{{u_{i}^{{\prime }} u_{k}^{{\prime }} }} \frac{{\partial u_{j} }}{{\partial x_{k} }} + \overline{{u_{j}^{{\prime }} u_{k}^{{\prime }} }} \frac{{\partial u_{i} }}{{\partial x_{k} }}} \right) + \overline{{p\left( {\frac{{\partial u_{i}^{{\prime }} }}{{\partial x_{j} }} + \frac{{\partial u_{j}^{{\prime }} }}{{\partial x_{i} }}} \right)}} - 2\mu \overline{{\frac{{\partial u_{i}^{{\prime }} }}{{\partial x_{k} }}\frac{{\partial u_{j}^{{\prime }} }}{{\partial x_{k} }}}} \\ & \quad - 2\rho \Omega_{k} \left( {\overline{{u_{j}^{{\prime }} u_{m}^{{\prime }} }} \smallint_{ikm} + \overline{{u_{i}^{{\prime }} u_{m}^{{\prime }} }} \smallint_{jkm} } \right) \\ \end{aligned} $$

(A.1)

Where, $k$ is the turbulent kinetic energy and its dissipation is $\epsilon$. The first term on the right-hand side $-\partial /\partial {x}_{k}\left[\rho \overline{{u }_{i}^{\mathrm{^{\prime}}}{u}_{j}^{\mathrm{^{\prime}}}{u}_{k}\mathrm{^{\prime}}}+\stackrel{-}{p\left({\delta }_{kj}{u}_{i}^{\mathrm{^{\prime}}}+{\delta }_{ik}{u}_{j}\mathrm{^{\prime}}\right)}\right]$ describes turbulent diffusion ${(D}_{T,ij})$, which is modeled using a gradient-diffusion model [73] given by

$${D}_{T,ij}={C}_{s}\frac{\partial }{\partial {x}_{k}}\left(\rho \frac{k\overline{{{u }_{k}}^{\prime}{u}_{i}^{\prime}}}{\epsilon }\frac{\partial \overline{{u }_{i}^{\prime}{u}_{j}^{\prime}}}{\partial {x}_{l}}\right)$$

(A.2)

Following the lead of Launder and Spalding [74], Fu et al [75] and Launder [76], the pressure strain term $p\left(\partial {u}_{i}^{\prime}/\partial {x}_{j}+\partial {u}_{j}^{\prime}/\partial {x}_{i}\right)$ may be denoted as ${\phi }_{ij}$ given by

$${\phi }_{ij}={\phi }_{ij,1}+{\phi }_{ij,2}+{\phi }_{ij,w}$$

(A.3)

Where, ${\phi }_{ij,1}$ and ${\phi }_{ij,2}$ are pressure strain terms where 1 denotes slow and 2 denotes rapid. They are solved using Eqs. (A.4) and (A.5). ${\phi }_{ij,w}$ represents wall reflection.

$${\phi }_{ij,1}\equiv -{C}_{1}\rho \frac{\epsilon }{k}\left[\overline{{u }_{i}^{\prime}{u}_{j}^{\prime}}-\frac{2}{3}{\delta }_{ij}k\right]$$

(A.4)

$${\phi }_{ij,2}\equiv -{C}_{2}\left[\left({P}_{ij}+{F}_{ij}+\frac{5}{6}{G}_{ij}-{C}_{ij}\right)-\frac{2}{3}{\delta }_{ij}(P+\frac{5}{6}G-C)\right]$$

(A.5)

Where, $P=0.5{P}_{kk}$, $G=0.5{G}_{kk}$ and $C=0.5{C}_{kk}$. The equation for wall reflection is

$$ \begin{aligned} \phi_{ij,w} & \equiv \frac{{C_{1}^{\prime} \epsilon }}{k}\left( {\overline{{u_{k}^{\prime} u_{m} {^{\prime}}}} n_{u} n_{m} \delta_{ij} - \frac{3}{2}\overline{{u_{i}^{\prime} u_{k} {^{\prime}}}} n_{j} n_{u} - \frac{3}{2}\overline{{u_{j}^{\prime} u_{k}^{\prime} }} n_{i} n_{u} } \right)\frac{{C_{l} k^{\frac{3}{2}} }}{{\epsilon d_{n} }} \\ & \quad + C_{2}^{{\prime }} \left( {\phi_{km,2} n_{u} n_{m} \delta_{ij} - \frac{3}{2}\phi_{ik,2} n_{j} n_{u} - \frac{3}{2}\phi_{jk,2} n_{i} n_{u} } \right)\frac{{C_{l} k^{\frac{3}{2}} }}{{\epsilon d_{n} }} \\ \end{aligned} $$

(A.6)

Where, ${n}_{u}$ unit normal of wall, ${d}_{n}$ is distance normal of wall and ${C}_{l}={C}_{\mu }^{3/4}/k$, where ${C}_{\mu }=0.09$. A two-layer model with enhanced wall treatment is considered here. This approach divides the domain into two regions. Within these two regions, one may be dominated by turbulence and the other by viscosity. This division is determined by turbulent Reynolds number ($R{e}_{t}$), given by

$$R{e}_{t}\equiv \frac{\rho y\sqrt{k}}{\mu }$$

(A.7)

Launder and Shima [77] state that ${C}_{1},{C}_{2},{C}_{1}^{\prime}$ and ${C}_{2}^{\prime}$ are functions of the resultant invariants and $R{e}_{t}$. They are equated by

$${C}_{1}=1+2.58A{A}_{2}^{0.25}(1-exp[-{\left(0.0067R{e}_{t}\right)}^{2}])$$

(A.8)

$${C}_{2}=0.75\sqrt{A}$$

(A.9)

$${C}_{1}^{\prime}=-\frac{2}{3}{C}_{1}+1.67$$

(A.10)

$${C}_{2}^{\prime}=max\left[\frac{\frac{2}{3}{C}_{2}-\frac{1}{6}}{{C}_{2}},0\right]$$

(A.11)

Where, ${A}_{2}$ and ${A}_{3}$ are the tensor invariants and $A$ is the flatness invariant.

$$A\equiv \left[1-\frac{9}{8}({A}_{2}-{A}_{3})\right]$$

(A.12)

$${A}_{2}\equiv {a}_{ik}{a}_{ki}$$

(A.13)

$${A}_{3}\equiv {a}_{ik}{a}_{kj}{a}_{ji}$$

(A.14)

The anisotropy tensor (${a}_{ij}$) is given by

$${a}_{ij}=-\left(\frac{-\rho \overline{{u }_{i}^{\prime}{u}_{j}^{\prime}}+\frac{2}{3}\rho k{\delta }_{ij}}{\rho k}\right)$$

(A.15)