A Well-balanced Finite Volume-Augmented Lagrangian Method for an Integrated Herschel-Bulkley Model

Abstract

We are interested in the derivation of an integrated Herschel-Bulkley model for shallow flows, as well as in the design of a numerical algorithm to solve the resulting equations. The goal is to simulate the evolution of thin sheet of viscoplastic materials on inclined planes and, in particular, to be able to compute the evolution from dynamic to stationary states. The model involves a variational inequality and it is valid from null to moderate slopes. The proposed numerical scheme is well balanced and involves a coupling between a duality technique (to treat plasticity), a fixed point method (to handle the power law) and a finite volume discretization. Several numerical tests are done, including a comparison with an analytical solution, to confirm the well balanced property and the ability to cope with the various rheological regimes associated with the Herschel-Bulkley constitutive law.

Appendices

Appendix A: The Regula-Falsi Method for q

In this appendix, we describe the regula-falsi method to solve the non linear problem in q (44):

$$\bigl(2^{\frac{\wp+3}{2}}\nu_1|q|^{\wp-1}+r \bigr)q=\bigl(\mu +r\mathcal{B}({\boldsymbol{V}})\bigr) \biggl(1-\frac{\tau_y \sqrt {2}}{|\mu+r\mathcal{B}({\boldsymbol{V}})|} \biggr)_+.$$

For the discrete problem (51), we have ${\mathcal{B}}({\boldsymbol{V}})= \frac{{\boldsymbol {V}}^{k}_{i+1}-{\boldsymbol{V}}^{k}_{i}}{\Delta x}$, $\mu=\mu_{i+1/2}^{k}$, $q=q_{i+1/2}^{k+1}$.

First we tackle the problem coming from the term S(q)=|q|^℘−1 q for 0<℘<1. We avoid the singularity at point q=0 from the numerical point of view by defining the following approximation

$$ S(q)=\left \{ \everymath{\displaystyle}\begin{array}{l@{\quad}l}\frac{q}{|q|^{1-\wp}} & \textrm{if }|q|>\epsilon, \\[10pt]\frac{q}{|\epsilon|^{1-\wp}} & \textrm{if }|q|\leq\epsilon.\end{array} \right .$$

(64)

In the numerical, test we set ϵ=10⁻⁷. If we define the function

$$ F(q)=2^{\frac{\wp+3}{2}}\nu_1 S(q)+rq-\bigl(\mu+r\mathcal {B}({\boldsymbol{V}})\bigr) \biggl(1-\frac{ \tau_y \sqrt{2}}{|\mu +r\mathcal{B}({\boldsymbol{V}})|} \biggr)_+$$

(65)

then, we search for a root of F(q). For simplicity, we denote:

$$\alpha_1=2^{\frac{\wp+3}{2}}\nu_1; \quad\quad d=\bigl(\mu+r\mathcal {B}({\boldsymbol{V}})\bigr);\quad\quad\alpha_2=\tau_y \sqrt{2};\quad\quad A= \biggl(1-\frac{\alpha_2}{|d|} \biggr)_+,$$

so

$$F(q)=\alpha_1 S(q)+rq-dA.$$

Observe that F(q) is monotone increasing then, F(q) has only one root. Looking at (65), if |d|≤α ₂ then this root is q=0. From now on, we thus assume that |d|>α ₂.

For the regula-falsi method, we construct a sequence of decreasing intervals containing a root of the function F(q). The algorithm is initialized with two points $x_{a}^{0}$ and $x_{b}^{0}$ such that $F(x_{a}^{0}) F(x_{b}^{0})<0$; then for k=0,…,k _max , the following iteration is computed:

• Step 1:

  $$x_c^{k}=x_a^k-\frac{x_a^k-x_b^k}{F(x_a^k)-F(x_b^k)}F\bigl(x_a^k\bigr).$$

• Step 2:

  

Then, the problem is to define the initial points $x_{a}^{0}$ and $x_{b}^{0}$. We propose the following choice:

• If d>α ₂ then $x_{a}^{0}=0$ and

  • if $d (1-\frac{\alpha_{2}}{|d|} )>\alpha_{1}+r$, then $x_{b}^{0}=\frac{d}{r} (1-\frac{\alpha_{2}}{|d|} )$;

  • if $d (1-\frac{\alpha_{2}}{|d|} )\leq\alpha_{1}+r$, then $x_{b}^{0}=1$.

• If d≤−α ₂ then $x_{b}^{0}=0$ and

  • if $d (1-\frac{\alpha_{2}}{|d|} )<-(\alpha_{1}+r)$, then $x_{a}^{0}=\frac{d}{r} (1-\frac{\alpha_{2}}{|d|} )$;

  • if $d (1-\frac{\alpha_{2}}{|d|} )\geq-(\alpha_{1}+r)$, then $x_{a}^{0}=-1$.

We can easily prove that these choices ensure that $F(x_{a}^{0})F(x_{b}^{0})<0$. This initialization completes the algorithm of the regula-falsi method.

Appendix B: Rheological Regimes of the Integrated Herschel-Bulkley Model

In this appendix, we describe the various constitutive laws used in this paper and the different regimes that can be exhibited.

More precisely, we will detail the constitutive law associated to the rheology of the integrated model (38). We are here in 1D; if we denote the shear stress by σ and the rate of shear by $\dot{\gamma}$, the constitutive law is:

(66)

(67)

Note that in this 1D case, we have $\dot{\gamma}$ which is given by ∂ _x V. The idea is to compare the associated curves for different values of the Herschel-Bulkley parameter ℘. To have a graphical view of such variety, let us suppose that the viscosity ν ₁=1, the yield stress $\tau_{y} = 6/\sqrt{2}$ and that we consider three types of fluid with respect to ℘, namely ℘=1 (which is actually the special case of a Bingham fluid), ℘=0.7 and ℘=0.4. The curves are shown on Fig. 10. Note that the three curves have the same intersection point at $\dot{\gamma} =2^{-1/2}$ (which is thus independent of ℘).

Fig. 10

Various Herschel-Bulkley constitutive models for ℘=1,0.7,0.4

The interesting point is to look at the derivatives of these three curves in order to have an idea of the viscosity (in the generalized sense). This will show which fluid is the more likely to flow faster for a given rate of shear. These three derivatives are shown on Fig. 11. On this Figure we put two vertical (dashed) lines to show three zones:

• on the left (denoted as Zone 1), a zone where the Bingham fluid is the less “viscous” of the three fluids, followed in this order by the Herschel-Bulkley fluids ℘=0.7 and ℘=0.4;

• on the right (denoted as Zone 3), a zone completely opposed to the previous one, where the Bingham fluid is the more “viscous” of the three fluids, followed in this order by the Herschel-Bulkley fluids ℘=0.7 and ℘=0.4;

• an intermediate zone (denoted as Zone 2) where there is no clear order in terms of the viscosity of the three fluids.

These three zones can be precised thanks to the fluid parameters ℘. If we denote 1≥℘₁>℘₂>℘₃>0 (thinking of ℘₁=1, ℘₂=0.7 and ℘₃=0.4):

• Zone 1 is [0,x ₁] where x ₁ is the abscissa of the intersection between the curves of ℘₂ and ℘₃, namely

  $$x_1 = \frac{1}{\sqrt{2}} { \biggl( \frac{\wp_3}{\wp_2} \biggr)}^{\frac{1}{\wp_2 - \wp_3}}.$$

• Zone 2 is [x ₁;x ₂] where x ₂ is the abscissa of the intersection between the curves of ℘₂ and ℘₁, namely

  $$x_2 = \frac{1}{\sqrt{2}} {\wp_2}^{-\frac{1}{\wp_2 - 1}}$$

  (note that, by the same computation, we see that x ₂ is greater than the abscissa of the intersection between the curves of ℘₃ and ℘₁, since ℘₂>℘₃, leading to a definition of x ₂ which is always given by the Herschel-Buckley fluid which has the bigger ℘<1).

• Zone 3 is [x ₂;+∞].

Fig. 11

Derivatives of various Herschel-Bulkley constitutive models for ℘=1,0.7,0.4. The vertical dashed lines separate three zones. The one on the left (Zone 1, $\dot{\gamma} \leq 0.11$) is used for the test in Sect. 5.3, whereas the one on the right (Zone 3, $\dot{\gamma} \geq0.22$) is used for the test in Sect. 5.2. See text

In the numerical tests we perform in this paper, we inspired ourselves by Fig. 11 by performing a test where the Bingham fluid is the most viscous (see Sect. 5.2): to do so, we need to design a test with high rates of shear, in the sense that a significant part of the fluid experiences rate of shear $\dot{\gamma}\geq x_{2} \sim0.22$ (with the choice of parameters given above), in such a way it corresponds to Zone 3.

On the other hand, the Zone 1 ($\dot{\gamma} \leq x_{1} \sim0.1$) is naturally explored for all tests where we check the stationary states where V and ∂ _x V are zero, or very close to zero; see for instance Sect. 5.3.

The results of the aforementioned sections show the different behaviors of the motion of the free surface H, in accordance with the different “viscous” regimes associated to Zone 1 and 3. This is a nice property of the numerical method proposed here.