only fishing for values in well-defined cells. With the estimates for velocity gradient components in place, we can evaluate the cell-centred rate-of-strain magnitude and apparent viscosity directly.
7.3
Validation studies
Before evaluating the methodology for genuinely three-dimensional viscoplastic flows, we need to ensure that the underlying incompressible flow solver works for a problem with non-trivial domain and known solution. Furthermore, we verify that viscoplastic effects are captured by computing the solution to a problem which has an analytical solution for the Bingham model.
Newtonian Taylor-Couette flow
To start off with, we consider the case of Taylor-Couette flow between two concentric cylinders. The setup is extensively used in rheometry, since it is straightforward to set up experimentally and gives consistent measurements of viscous drag as a function of applied strain-rate. The two cylinders can be rotating in the same or opposing directions at given speeds, and since we can derive an analytical solution for the Newtonian case, it is an ideal case for validation of prescribed tangential velocities on the no-slip boundaries. In figure 7.5, we illustrate the problem conceptually.
The inner and outer cylinders are denoted by 1 and 2, respectively. We thus take R1 as the radius of the inner cylinder, while Ω2 is the angular velocity of the outer one, etc. Note that the sign of the angular velocity is positive if the rotation is counter-clockwise as viewed from above. The fluid motion is driven entirely by the rotating cylinders, so the pressure gradient is zero and the flow is purely rotational. In cylindrical coordinates, the flow field is therefore
u= urˆer+ uθˆeθ+ uzˆez = ωrˆeθ. (7.26)
The steady-state solution is governed by ∇2u
R
1R
2Ω
1Ω
2Figure 7.5: Schematic illustration of the Taylor-Couette cell, with rotational flow induced by prescribed inner and outer wall speeds.
which in terms of the angular velocity can be written r∂
2ω ∂r2 + 3
∂ω
∂r = 0. (7.28)
The solution to (7.28) which satisfies the boundary conditions ωi(Ri) = Ωi is
ω(r) = Ω1 1 − ξ2 (1 − χ) R2 1 r2 + χ − ξ 2 ! , (7.29)
where we have introduced the dimensionless quantities ξ = R1/R2 and χ = Ω2/Ω1. Taking the radius and speed of the inner cylinder as characteristic values for length and velocity, the analytical solution in dimensionless form is
ˆω(ˆr) = 1 − ξ1 2 1 − χˆr2 + χ − ξ2
. (7.30)
We take ξ = −χ = 1/4, which means that the outer cylinder is four times as wide as the inner one, and that they are rotating in opposite directions with the same speed U = −Ω1R1 = Ω2R2. Figure 7.6 shows the velocity profile for N = 200 cells per direction. The fluid near the inner cylinder is rotating clockwise with maximum speed U, while the outer one rotates at the same speed but counter-clockwise. Figure 7.7 shows that our numerical solution converges to the analytical one for increased mesh
7.3 Validation studies 123
resolution. The error is greater near the inner cylinder than the outer one, which is to be expected since there is a higher degree of curvature per cell when the cell size is equal throughout the domain.
Figure 7.6: Velocity distribution in a ˆxˆy-slice of the Taylor-Couette flow. The colormap corresponds to the polar velocity component ˆuθ(ˆr).
Poiseuille flow of Bingham plastic in a cylinder
In order to verify that the embedded boundaries work with viscoplastic rheology, we again consider Poiseuille flow of a Bingham fluid like in section 2.3, but now on the interior of a circular cylinder with radius R, aligned along the z-axis. Since the problem is both axisymmetric and independent of z, the variation in velocity is a function of r = q
(x − xc)2+ (y − yc)2 alone, where the cylinder centre axis has
coordinates (xc, yc, z). The analytical Bingham solution is then the cylindrical version
of (2.42), i.e. with z replaced by r. Our simulations were performed in a domain of size Ω = [0, 2.5R]2×[0, R] with the cylinder centred at ˆx
c = ˆyc= 1.25. We set physical
1 2 3 4 ˆ r −1.0 −0.5 0.0 0.5 1.0 ˆω N = 50 N = 100 N = 200 Analytical 1 2 3 4 ˆ r 0.02 0.04 0.06 0.08 Error N = 50 N = 100 N = 200
Figure 7.7: Comparison of our numerical results with the analytical angular velocity profile for Taylor-Couette flow.
Figure 7.8a shows the resulting velocity distribution in the xy-plane for a simulation with N = 2R/∆x = 64 cells over the cylinder diameter, and Papanastasiou number P a= 100. In the middle of the cylinder, we can see the plug of unyielded fluid travelling with constant velocity. Surrounding it, an annulus of constant shear rate leads to the parabolic profile we expect between the wall and the yield surface. The yield surface is characterised by the stress contour kτ k = τ0. However, as discussed previously in the literature, there is instability near this stress value which means that a better measure of the fully converged yield surface is the contour kτ k = (1 + δ)τ0[155], where δ is some small parameter of the order 10−3. This is because the solution converges much faster in the yielded region than in the unyielded ones. On the other hand, Treskatis argues that a better visual investigation of the yield surface is obtained by plotting the relative deviation from the yield surface, kτ k /τ0−1, restricted to some small range around zero[104]. In this manner, we avoid the introduction of systematic error through overestimation of the unyielded regions. Note that this is done using a colormap which changes abruptly at zero, as seen in figure 7.8b.
From figure 7.8b, we can see that there is a sharp transition from the yielded to the unyielded regions which is in agreement with the analytical solution for the yield surface, ˆr = r0. This validates that our model accurately captures the Bingham properties of the fluid, and that the regularisation parameter is small enough at Pa = 100 for the