Numerical simulation of the transition of a Newtonian fluid to a viscoplastic state in a turbulent flow

2.1 Bingham-Schwedoff rheology model

Waxy crude oil rheology can be modeled as the generalized Newtonian fluids (Vinay et al., 2005; Salehi-Shabestari et al., 2016). The effective (apparent) viscosity of a BS fluid is defined using a rheology model (Barnes, 1999; Beverly and Tanner, 1992; Bingham, 1922). The Bingham-Schwedoff model is the simplest model of viscoplastic non-Newtonian fluids (Barnes, 1999; Bingham, 1922):

The singularity of a numerical solution of the Eq. (1) is appeared in zones at τ < τ₀. The singular property of the effective viscosity can be solved by improving the original rheological model (Barnes, 1999). Effective viscosity can be represented as a smooth function using the approach (Papanastasiou, 1987):

2.2 The set of governing equations for a non-Newtonian turbulent fluid

The system of stationary RANS equations for the flow and heat transfer of an incompressible fluid has the form

Here, S is averaged strain rate tensor, $S=\sqrt{2S_{\mathit{ij}}{S}_{\mathit{ij}}}$ is modulus of averaged strain rate tensor. The turbulent characteristics $-\rho 〈{\mathbf{u}}^{/}{\mathbf{u}}^{/}〉$ and $-\rho C_p〈{\mathbf{u}}^{/}{t}^{/}〉$ was written according to Boussinesq hypothesis. The expression $\mathrm{\nabla }·〈2{\mu }_{\mathit{eff}}^{/}{\mathbf{S}}^{/}〉$ in equation (4) is found according to representation (Gavrilov and Rudyak, 2016b). The last term in equation (5) takes into account the dissipation of kinetic energy and has the form (Vinay et al., 2005). In the recent paper (Pakhomov and Zhapbasbayev, 2021) the system of equations (3)–(5) was solved jointly with k– $\stackrel{\sim }{\epsilon }$ model (Hwang and Lin, 1998).

The elliptical relaxation RSM (Manceau and Hanjalic, 2002) considers anisotropy of complex turbulent flows and is computationally more difficult than ARSM. These models give better predictable results than the k- $\stackrel{\sim }{\epsilon }$ turbulence isotropic model. The ARSM from (Rokni, 2000) is a version of the well-known explicit Reynolds algebraic stress model (Speziale and Xu, 1996). The Reynolds stress components are derived from a system of partial differential equations, and the system of basic equations of the second-moment closure model of (Manceau and Hanjalic, 2002) is given: $$\mathrm{\nabla }·(\rho \mathbf{U}\langle {\mathbf{u}}^{/}{\mathbf{u}}^{/}\rangle )=\rho (P_{\mathit{ij}}+\mathrm{\Phi }-\epsilon )+\mathrm{\nabla }·\left[\rho \frac{C_{\mu }{T}_T}{{\sigma }_k}\langle {\mathbf{u}}^{/}{\mathbf{u}}^{/}\rangle \mathrm{\nabla }(\langle {\mathbf{u}}^{/}{\mathbf{u}}^{/}\rangle )\right]$$ $$\mathrm{\nabla }·\left(\rho \mathbf{U}\epsilon \right)=\frac{1}{T_t}\left(C_{\epsilon 1}{P}_2-C_{\epsilon 2}\epsilon \right)+\mathrm{\nabla }·\left[\rho \frac{C_{\mu }{T}_T}{{\sigma }_{\epsilon }}〈{\mathbf{u}}^{/}{\mathbf{u}}^{/}〉\mathrm{\nabla }\epsilon \right]+\mathrm{\nabla }·\left(\mu \mathrm{\nabla }\epsilon \right)+C_{\epsilon 3}\frac{\mu k}{\epsilon }〈{\mathbf{u}}^{/}{\mathbf{u}}^{/}〉·{\mathrm{\nabla }}^2\mathbf{U}·{\mathrm{\nabla }}^2\mathbf{U}$$

Here, P_ij is the intensity of the energy transfer from the average velocity to the pulsating one; P₂ = 0.5P_kk, T_T is the turbulent time macroscale; ϕ is the redistribution term (Manceau and Hanjalic, 2002), ε is the dissipation rate. $$\mathrm{\nabla }·\left(\rho \mathbf{\text{U}}k\right)=\mathrm{\nabla }·\left[\left(\mu +\frac{C_{\mu }\rho k}{{\sigma }_k\epsilon }\right)\mathrm{\nabla }k\right]+P_{\mathit{kk}}-\rho \epsilon$$ $$\mathrm{\nabla }·\left(\rho \mathbf{\text{U}}\epsilon \right)=\mathrm{\nabla }·\left[\left(\mu +\frac{C_{\mu }\rho k^2}{{\sigma }_{\epsilon }\epsilon }\right)\mathrm{\nabla }\epsilon \right]+\frac{1}{T_T}\left(f_{\epsilon 1}{C}_{\epsilon 1}{P}_{\mathit{kk}}-f_{\epsilon 2}{C}_{\epsilon 2}\rho \epsilon \right)$$

2.3 Boundary conditions

All boundary conditions for a non-isothermal flow are given in details (Pakhomov and Zhapbasbayev, 2021). No-slip conditions are set on the wall surface for fluid velocity. The heat transfer coefficient from fluid flow (WCO) to cold soil through pipe surface is determined by the formula (Zhapbasbayev et al., 2021). The symmetry conditions are set on the pipe axis. The uniform distributions of velocity and temperature are set over the cross-section of pipe in inlet section. At the outlet edge, gradients of all flow and heat transfer parameters in axial direction.

4.1 Isothermal flow in a pipe

At the first stage, the comparisons with data of DNS predictions for the turbulent flow of Newtonian fluid in pipe were performed. DNS simulations (Singh et al., 2017) on the transverse distributions of axial and radial velocity fluctuations for Newtonian turbulent fluid were used. These results are given in Fig. 3. The maximum differences in the intensity of the axial pulsations of the Newtonian fluid velocity do not exceed 10% when predicted using the RSM model (see Fig. 3a). The largest discrepancy between the data of our numerical RANS simulations and DNS data (Singh et al., 2017) does not exceed 15% and is observed for radial pulsations (see Fig. 3b) at y₊ ≈ 80.

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Fig. 3

Radial profiles of axial (a) and radial (b) velocities pulsations of a Newtonian fluid. Solid symbols are the DNS results (Singh et al., 2017) and lines are authors’ RANS predictions. Solid lines are the k– $\stackrel{\sim }{\epsilon }$ model (Hwang and Lin, 1998), dashed curves are the RSM (Manceau and Hanjalic, 2002) and dotted lines are the ARSM (Rokni, 2000). Re = 1.3 × 10⁴, Re_τ = 323.

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Fig. 4 shows the profiles of (a) axial and (b) radial velocity fluctuations of the Bingham–Schwedoff fluid over the pipe section in the wall units y₊. The dots are the DNS data (Singh et al., 2017), the lines are the authors’ RANS predictions. The solid lines are the k– $\stackrel{\sim }{\epsilon }$ model (Hwang and Lin, 1998), the dashed lines are the RSM model (Manceau and Hanjalic, 2002) and the dotted lines are the full explicit ARSM (Rokni, 2000). In the calculations of the k– $\stackrel{\sim }{\epsilon }$ model (Hwang and Lin, 1998), the individual components of the Reynolds stresses are determined in the framework of the isotropic representation: $u^{\prime }=v^{\prime }=2k/3$ . Satisfactory agreement between our simulations and the data of DNS predictions is obtained only when using the RSM approach.

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Fig. 4

Profiles of average axial velocity (a) and turbulent kinetic energy (b) of a NNF. Solid symbols are DNS results (Singh et al., 2017) and lines are authors’ RANS predictions. Solid lines are the k– $\stackrel{\sim }{\epsilon }$ model (Hwang and Lin, 1998), dashed curves are the RSM (Manceau and Hanjalic, 2002) and dotted lines are the ARSM (Rokni, 2000). Re = 1.3 × 10⁴, Re_τ = 323, τ₀/τ_W = 1.2.

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The significant anisotropy of profiles velocity pulsations of a non-Newtonian fluid is obtained in predictions using RSM and ARSM, which is consistent with the data of (Singh et al., 2017) (see Table 1). The positions of the maxima in the distributions of axial and radial pulsations are in satisfactory agreement with the positions of the maxima for DNS (Singh et al., 2017). An exception is the distribution of radial fluctuations $v^{\prime }$ predicted using the isotropic k– $\stackrel{\sim }{\epsilon }$ model (Hwang and Lin, 1998) (see Table 2). The radial velocity fluctuations are significantly less than the intensity of axial fluctuations. The (Hwang and Lin, 1998) model describes well distribution of axial pulsation component along pipe radius (difference does not exceed 15%). In this case, there is a significant excess of radial fluctuations in comparison with the data [22] and our calculations using anisotropic turbulence models (Manceau and Hanjalic, 2002; Rokni, 2000) (up to 3 times).

The comparison of the DNS data (Singh et al., 2017) and our calculations of the axial velocity and kinetic energy of turbulence (KET) in wall units for the BS fluid are presented in Fig. 4. Here dots in Fig. 4b are the DNS data (Singh et al., 2017), curves are authors’ RANS calculation by using k– $\stackrel{\sim }{\epsilon }$ (Hwang and Lin, 1998), the RSM (Manceau and Hanjalic, 2002) and the ARSM (Rokni, 2000) turbulence models. Fig. 4a shows the velocity profiles U₊ for NNF. Difference between the three turbulence models is insignificant in the regions (y₊ < 5 and (5 < y₊ < 30). These results quantitative agree with the DNS data (Singh et al., 2017) (see Fig. 4a). Yield stress has practically no effect within the viscous sublayer and the differences between the various models are minimal. It qualitatively agrees with the conclusions of (Wilson and Thomas, 1985). For all three models, the difference is no more than 15% and the velocity profile is similar for a NF. Our RANS predictions and DNS at 30 < y₊ < 200 differ by up to 15%.

4.2 Turbulent non-isothermal flow structure

The distribution of the action zone of yield stress τ₀/τ_NF = 10 along the axial coordinate depending on the ambient soil temperature is shown in Fig. 5. The zone height of yield stress τ₀/τ_NF = 10 increases gradually along the streamwise coordinate. It tends to the coordinate y/R ≈ 0.6 at x/D = 40 (x = 8 m) for temperature T_S = 273 K and y/R ≈ 0.06 for temperature T_S = 293 K.

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Fig. 5

Evolution of radial coordinates of yield stress values presence τ₀/τ_NF = 10 along the pipe length. Re = 8200, T₁ = 298 K.

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The effect of ambient soil temperature on the radial profiles of yield stress y/R is shown in Fig. 6. The viscoplastic non-Newtonian behavior of WCO is clearly shown at T_S = 273 K. In this case, the fluid with the manifestation of NNF properties occupies the most part of pipe cross-section (y/R ≤ 0.75). The yield stress value exceeds the stresses in a turbulent Newtonian fluid flow by several orders of magnitude. The magnitude of yield shear stress is minimal at T_S = 293 K. The non-Newtonian behavior of waxy crude oil appears only in the pipe near-wall part (y/R ≤ 0.2).

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Fig. 6

Effect of Reynolds number on distributions of maximal values of the Reynolds stress. x/D = 20, T₁ = 298 K, T_Soil = 273 K.

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4.3 Wall friction and heat transfer

In Fig. 7 distributions of wall friction and Nusselt numbers are represented at x/D = 10. Here wall friction coefficient is $C_f=2{\tau }_W/\left(\rho U_1^2\right)$ and the Nusselt number is Nu = hD/λ_W, heat transfer coefficient is h. Abbreviation FD is a fully developed Newtonian fluid and subscript W is a wall. The results are given in Fig. 7 are conducted for the developing flow of Newtonian waxy crude oil (NF line), and for the hydrodynamically fully developed Newtonian waxy crude oil (FD line). Therefore, the magnitude of heat transfer in the developing flow is higher than in the developed flow under all other conditions being equal. The coefficients of friction and heat transfer increase with an increase in the Reynolds number at the pipe inlet (see Fig. 7). It can be seen that the decrease in surrounding soil temperature causes a more rapid manifestation of the non-Newtonian properties of a Bingham-Schwedoff fluid and this confirms the conclusions from our previous predicted results. This leads to an increase in resistance (Fig. 7a) and a weakening of heat transfer (see Fig. 7b) of turbulent flow in pipe. Fluid velocity increase leads to a slow cooling of the turbulent flow, as well as an increase in the pipe length, where the Newtonian property of the fluid preserved.

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Fig. 7

Effect of Reynolds numbers on wall friction and Nusselt number at x/D = 10.

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The normalized distributions of wall friction C_f/C_f,fd (a) and heat transfer Nu/Nu_fd (b) on the wall during the turbulent fluid flow along streamwise coordinate are given in Fig. 8. A decrease in the environment temperature leads to a significant reduction in wall friction C_f and heat transfer Nu. Close to the wall, yield stress and a fluid stagnation zone appear (a medium behaves like a solid). The magnitudes of C_f, Nu are equal to zero in this region. In the case of the manifestation of the yield stress in the near-wall region of the pipe, heat transfer is possible only due to thermal conductivity. It qualitatively agrees with the conclusions of (Aljerf, 2016; Aljerf and Al Masri, 2018).

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Fig. 8

Influence of soil temperature T_S on wall friction (a) and transfer (b) distributions of WCO along pipe length. Re = 8200, T₁ = 298 K.

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