A numerical study on nonlinear dynamics of three-dimensional time-depended viscoelastic Taylor-Couette flow

Abstract

In this paper, three-dimensional viscoelastic Taylor-Couette instability between concentric rotating cylinders is studied numerically. The aim is to investigate and provide additional insight about the formation of time-dependent secondary flows in viscoelastic fluids between rotating cylinders. Here, the Giesekus model is used as the constitutive equation. The governing equations are solved using the finite volume method (FVM) and the PISO algorithm is employed for pressure correction. The effects of elasticity number, viscosity ratio, and mobility factor on various instability modes (especially high order ones) are investigated numerically and the origin of Taylor-Couette instability in Giesekus fluids is studied using the order of magnitude technique. The created instability is simulated for large values of fluid elasticity and high orders of nonlinearity. Also, the effect of elastic properties of fluid on the time-dependent secondary flows such as wave family and traveling wave and also on the critical conditions are studied in detail.

Appendix

PISO algorithm

The PISO algorithm is consisted of one predictor step and two corrector steps. This algorithm can be summarized as follows (Versteeg and Malalasekera 2007):

• i. Assigning the initial values and boundary conditions for variables;

• ii. Solving the discretized equations of momentum and constitutive equation in order to determine the velocity components and the stress tensor;

• iii. Calculating the mass fluxes at the cell faces;

• iv. Solving the pressure equation;

• v. Correcting the mass fluxes at the cell faces;

• vi. Correcting the velocity components based on the new pressure;

• vii. Updating the boundary conditions and repeat from iii for a certain number of times;

• viii. Changing the time step and repeat from ii.

In the finite volume method, vector variables are saved on the control volume faces while scalar variables are stored at the grid points. Thus, for an arbitrary dependent variable, the transport equation can be written as follows (Norouzi et al. 2013):

$$ \frac{\partial }{\partial t}\left(\kappa \xi \right)+\nabla .\left(\kappa u\xi -D\nabla \xi \right)={f}_{\mathrm{p}}\xi +{f}_{\mathrm{c}} $$

(A1–1)

where (κuξ − D∇ξ) is the flux term in which D represents the diffusivity. f _p ξ + f _c is the source term for the variable ξ. Furthermore, the mentioned source term in Eq. (A1–1) is a combination of linear and constant terms in an arbitrary choice. Thus, over a control volume, integration of Eq. (A1–1) leads to the following equation:

$$ \iiint \frac{\partial }{\partial t}\left(\kappa \xi \right) dv+\iiint \nabla .\left(\kappa u\xi -D\nabla \xi \right) dv=\iiint \left({f}_{\mathrm{p}}\xi +{f}_{\mathrm{c}}\right) dv $$

(A1–2)

By using an approximation scheme for Eq. (A1–2) over the control volume, the discretized equation for the dependent variable ξ can be symbolically written as follow:

$$ {a}_{\mathrm{p}}{\xi}_{\mathrm{p}}=\Sigma {a}_{\mathrm{n}\mathrm{b}}{\xi}_{\mathrm{n}\mathrm{b}}+{f}_{\mathrm{c}}\varDelta V+{a}_{\mathrm{p}}^0{\xi}_{\mathrm{p}}^{\mathrm{n}} $$

(A1–3)

where

$$ {a}_{\mathrm{p}}^0=\kappa \frac{\varDelta V}{\delta t} $$

(A1–4)

$$ {a}_{\mathrm{p}}=\sum \limits_{\mathrm{nb}}{a}_{\mathrm{nb}}+{a}_{\mathrm{p}}^0-{f}_{\mathrm{c}}\varDelta V $$

(A1–5)

Order of magnitude analysis

Determination of the effect of various parameters on the formation of secondary flow is very crucial in understanding the instability conditions. Here, the order of main flow becomes greater than the order of secondary flow. Thus, the order of velocity components and their derivations can be written as follows:

$$ {\displaystyle \begin{array}{l}O\left({u}_{\theta}\right)=O\left(\frac{\partial }{\partial r}\left(\right)\right)=O\left(\frac{\partial }{\partial z}\left(\right)\right)\approx O(1)\\ {}O\left({u}_z\right)=O\left({u}_r\right)=O\left(\frac{\partial }{\partial \theta}\left(\right)\right)\approx O\left(\varepsilon \right)\end{array}} $$

(A2–1)

The order of magnitude for polymeric stress components by assuming negligible value for the mobility factor is presented in Eq. (A2–2). This equation is calculated by simplifying the constitutive equation and neglecting the terms of order ε:

$$ \left\{\begin{array}{l}{S}_{\mathrm{r}\mathrm{r}}^{\mathrm{p}}\approx O\left(\varepsilon \right)\\ {}{S}_{\mathrm{r}\mathrm{z}}^{\mathrm{p}}\approx O\left(\varepsilon \right)\\ {}{S}_{\mathrm{zz}}^{\mathrm{p}}\approx O\left(\varepsilon \right)\\ {}{S}_{\mathrm{r}\theta}^{\mathrm{p}}\approx \frac{r}{2}\frac{\partial }{\partial r}\left(\frac{u_{\theta }}{r}\right)+O\left(\varepsilon \right)\\ {}{S}_{\theta \mathrm{z}}^{\mathrm{p}}\approx \frac{\partial {u}_{\theta }}{\partial z}+O\left(\varepsilon \right)\\ {}{S}_{\theta \theta}^{\mathrm{p}}\approx Wi\left(2r\frac{\partial }{\partial r}\left(\frac{u_{\theta }}{r}\right){S}_{\mathrm{r}\theta}^{\mathrm{p}}+\frac{\partial {u}_{\theta }}{\partial z}{S}_{\theta \mathrm{z}}^{\mathrm{p}}\right)+O\left(\varepsilon \right)\end{array}\right. $$

(A2–2)

On the other hand, Eq. (A2–3) expresses the momentum equation by considering the order of magnitude analysis. By substituting the stress components into Eq. (A2–3), the resulted equation is valid in the core region (the annulus region between the cylinders and far from the walls) and can be helpful in identifying the effect of various factors on the flow stability:

$$ \left(\frac{u_{\theta}^2}{r}\right)=\frac{1}{Ta}\left(\frac{\partial p}{\partial r}+\frac{S_{\theta \theta}^{\mathrm{p}}}{r}\right) $$

(A2–3)

$$ \left(\frac{u_{\theta}^2}{r}\right)=\frac{1}{Ta}\left(\frac{\partial p}{\partial r}\right)+ En\left({\left(r\frac{\partial }{\partial r}\left(\frac{u_{\theta }}{r}\right)\right)}^2+{\left(\frac{\partial {u}_{\theta }}{\partial z}\right)}^2\right) $$

(A2–4)

In the core region, the radial pressure gradient is balanced with the centrifugal and elastic forces. In comparison with Newtonian fluid, the hoop stress leads to different behavior in viscoelastic fluids. In addition, Eq. (A2–4) states the importance of elastic number which expresses the qualitative deviation from Newtonian flow behavior. These equations are used in the paper to discuss the results of CFD simulations.