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Modeling of fluids with complex rheology in the lattice Boltzmann method (LBM) is typically realized through the introduction of an effective viscosity. For fluids with a yield stress behavior, such as so-called Bingham fluids, the effective viscosity has a singularity for low shear rates and may become negative. This is typically avoided by regularization such as Papanastasiou’s method. Here we argue that the effective viscosity model can be re-interpreted as a generalized equilibrium in which no violation of the stability constraint is observed. We implement a Bingham fluid model in a three-dimensional cumulant lattice Boltzmann framework and compare the direct analytic effective viscosity/generalized equilibrium method to the iterative approach first introduced by Vikhansky which avoids the singularity in viscosity that can arise in the analytic method. We find that both methods obtain similar results at coarse resolutions. However, at higher resolutions the accuracy of the regularized method levels off while the accuracy of the direct method continuously improves. We find that the accuracy of the proposed direct method is not limited by the singularity in viscosity indicating that a regularization is not strictly necessary.

A popular way to model the stress of non-Newtonian fluids is by imposing an effective local viscosity. Unlike a Newtonian fluid with constant viscosity, this effective viscosity cannot be drawn in front of the gradient of the stress tensor of the Navier-Stokes equation. Implementing non-Newtonian fluids in Navier-Stokes solvers hence either requires storing the stress tensor explicitly or applying the chain rule to the effective viscosity field which can result in rather complicated differential operators. Being a method derived from kinetic theory, the cumulant LBM intrinsically stores the stress tensor separately from the primitive variables as second-order cumulants. Implementing non-Newtonian behavior through an effective viscosity thus comes naturally in the LBM with all differential operators applied in a consistent order.

Many substances of industrial interest are subject to non-Newtonian stress-strain relationships. One such example is fresh concrete [

In this work we discuss the implementation of the Bingham fluid [

In recent years, the LBM saw some significant evolution towards improved accuracy and stability. This improvement is due to the usage of tensor product lattices (i.e. using 27 discrete velocities in 3D), moment matching equilibria incorporating terms of higher than second-order in velocity, multiple relaxation times, Galilean invariant moment transformations (i.e. central moments or cumulants) and a better understanding of statistical independence of the different moments. These novel lattice Boltzmann methods include entropic schemes which typically implement tensor product lattices and tensor product equilibria and are primarily aimed at improving stability [

In this paper we present an implementation of the Bingham fluid in the context of the cumulant LBM. Our approach starts similar to the one of Vikhansky [

We test our model with two simple planar flows: the flow between two infinite plates and the flow between a rotating and a stationary cylinder. Even though these cases are relatively simple, they both cover the singularity problem of the effective viscosity and include extended areas in which the fluid is expected to behave as a solid.

In the following, we will briefly introduce the cumulant LBM, discuss the implementation of the Bingham fluid and the regularization including a fixed-point iteration scheme which removes the singularity. Next we will discuss an alternative interpretation of the effective viscosity model as a generalized equilibrium model in which the singularity disappears. This is followed by a numerical comparison between the two possibilities and conclusions.

The effective viscosity of non-Newtonian fluids can vary substantially such that it is imperative to use a model base with a large range of attainable viscosities. Early lattice Boltzmann models based on single relaxation time collision operators [

The cumulant collision operator assigns an individual relaxation rate ω a b c to each cumulant C a b c such that the post collision cumulants C a b c ∗ can be computed as:

C a b c ∗ = C a b c + ω a b s ( C a b c e q − C a b c ) (1)

where the equilibrium cumulant C a b c e q is typically zero for non-conserved cumulants but can also be chosen otherwise in certain circumstances, for example if energy conservation is not considered as is usually the case for the incompressible Navier-Stokes equation. Asymptotic analysis [

The viscosity is related to the relaxation rate of second-order cumulants (i.e. ω = ω 110 = ω 101 = ω 011 and the corresponding diagonal terms of second-order of the tensor ω a b c with the exception of the trace):

ν = 1 3 ( 1 ω − 1 2 ) Δ x 2 Δ t (2)

The shear stress in the LBM can be written as:

τ = ν ρ γ ˙ (3)

For a known viscosity the shear rate can be locally computed from the second-order cumulants:

γ ˙ = ∂ u i ∂ x j + ∂ u j ∂ x i (4)

∂ u 1 ∂ x 2 + ∂ u 2 ∂ x 1 = ω 3 ρ C 110 (5)

∂ u 1 ∂ x 1 − ∂ u 2 ∂ x 2 = 2 ω 3 ρ ( C 200 − C 020 ) (6)

Here we omitted the other permutations of Equations (5) and (6).

The LBM does not require any additional finite differencing for the calculation of the shear rate. However, it is of note that the relaxation time, which is a function of viscosity, appears in the equations for the shear rate such that an implicit problem arises if the viscosity depends on the shear rate.

The Bingham fluid [

τ B i n g = τ 0 + ν ∞ ρ γ ˙ (7)

Our aim is to impose Equation (7) onto the LBM. This is done through equating Equations (3) and (7):

τ 0 + ν ∞ ρ γ ˙ = ν ρ γ ˙ (8)

τ 0 ρ + γ ˙ 3 ( 1 ω ∞ − 1 2 ) Δ x 2 Δ t = γ ˙ 3 ( 1 ω − 1 2 ) Δ x 2 Δ t (9)

According to Equation (5) γ ˙ is a function of ω . To make this relationship explicit, we define:

γ ˙ ∞ = γ ˙ ( ω ∞ ) (10)

γ ˙ ( ω ) = γ ˙ ∞ ω ω ∞ (11)

With this Equation (9) can be stated as:

τ 0 ρ + γ ˙ ∞ 3 ( 1 ω ∞ − 1 2 ) Δ x 2 Δ t ω ω ∞ = γ ˙ ∞ 3 ( 1 ω − 1 2 ) Δ x 2 Δ t ω ω ∞ (12)

This can be solved for the relaxation rate to give explicitly:

ω a = ω ∞ ( 1 − 3 ω ∞ τ 0 Δ t γ ˙ ∞ ρ Δ x 2 ) (13)

We will call the relaxation rate in Equation (13) the analytic relaxation rate ω a . Apparently Equation (13) allows us to simulate a Bingham fluid using only local operations as in a standard Newtonian LBM solver. However, we observe that ω a remains in the range for linear stability ω a ∈ { 0 ⋯ 2 } only for γ ˙ ∞ ρ < 3 ω ∞ τ 0 . According to Equation (2) viscosity goes to infinity when ω = 0 and viscosity will be negative for ω < 0 . Regularization procedures like the one by Vikhansky [

From Equation (8) it is not obvious why the effective viscosity ν should become negative before the shear rate γ ˙ reaches zero. The reason for the problem encountered in the last subsection is in the implicit dependence of γ ˙ on the relaxation rate when computed from the second-order cumulants. If we ignore this implicit dependence and solve Equation (9) for ω we obtain:

ω = ω ∞ γ ˙ ρ Δ x 2 Δ x 2 γ ˙ ρ + 3 ω ∞ τ 0 Δ t (14)

This solution is always positive, but it requires γ ˙ which in lattice Boltzmann is only known through γ ˙ = γ ˙ ∞ ω ω ∞ − 1 . Instead of using the apparently unfeasible analytic solution, we can solve Equation (14) through a fixed-point iteration as follows [

ω 0 = ω ∞ (15)

ω n + 1 = ω ∞ γ ˙ ∞ ω n ρ Δ x 2 Δ x 2 γ ˙ ∞ ω n ρ + 3 ω ∞ 2 τ 0 Δ t (16)

Since all terms on the right hand side of Equation (16) are positive and since we start the iteration from a positive value ω n + 1 is always guaranteed to be positive.

In order to accelerate the execution of the fixed-point iteration we rewrite Equation (16):

ω ∞ ω n + 1 = 1 + 3 ω ∞ τ 0 Δ t γ ˙ ∞ ρ Δ x 2 ω ∞ ω n = 1 + A ω ∞ ω n (17)

We note that Equation (17) has a simple analytic solution for ω n + 1 = ω n which can easily be verified to be identical with Equation (13) which we are trying to avoid. In fact a solution of Equation (17) exists only for ω n ≠ 0 which is the mathematical reason for the analytic solution appearing to be unfeasible.

The obtained iteration can be easily unrolled for a fixed iteration depth. For example, the iteration of depth five reads:

ω 5 = ω ∞ 1 + A ( 1 + A ( 1 + A ( 1 + A ( 1 + A ) ) ) ) (18)

In his original paper [

ω n + 1 ω n = 1 1 + A (19)

The rate of change is constant implying linear convergence. However, we should remind ourselves that we deliberately replaced the exact solution by this approximation for the purpose of regularization and avoiding the singularity of the exact solution. The use of faster converging methods like Aitken extrapolation or Newton iteration would reintroduce the possibility for the relaxation rate to become negative and could hence eliminate the advantage of regularization. That is, of course, if regularization is really as advantageous as implied through the avoidance of the singularity in viscosity.

Naively, one could assume that a singularity in viscosity and the violation of the linear stability regime should result in unstable simulations. However, in the current case, as demonstrated below, this problem did not manifest in our simulations. A simple reason why the violation of the linear stability constraint might not be relevant in the current case is seen from the fact that according to Equation (13) a problem occurs only if the shear rate is small enough. Instability would necessarily manifest itself in a locally increased shear rate which would force the relaxation rate back into its stable range. Thus, even though the linear stability constrained is violated, this violation is self-limiting and consequently non-linearly stable.

We also have to take the nature of the LBM into consideration and how it compares to a classical Navier-Stokes solver. In a classical Navier-Stokes solver the stress tensor is constructed by computing the shear rate and multiplying it with the viscosity. A singularity in viscosity is numerically unmanageable in this context and obviously has to be avoided. However, in the LBM, viscosity is an emergent property of the relaxation rate. The singularity occurs when ω = 0 which is not necessarily a problem for the lattice Boltzmann algorithm itself as it does not divide by ω . The problem of a singular viscosity is hence a very serious numerical difficulty for the Navier-Stokes equation, but it is not necessarily relevant for the LBM.

It is instructive to recall that the effective viscosity ansatz is only an algebraic trick to express the yield stress through the relaxation rate. Alternatively we could specify the yield stress in the equilibrium second-order cumulants and leave the relaxation rates untouched. A similar dual approach exists in the correction for cubic error terms in viscosity of the LBM. Dellar [

Asymptotic analysis as, for example, demonstrated in appendix G of [

∂ u 1 ∂ x 2 + ∂ u 2 ∂ x 1 = ω 3 ρ ( C 110 − C 110 e q ) (20)

∂ u 1 ∂ x 1 − ∂ u 2 ∂ x 2 = 2 ω 3 ρ ( C 200 − C 020 − C 200 e q + C 020 e q ) (21)

Note that Equation (20) and (21) are identical to Equation (5) and (6), respectively (up to an unspecified equilibrium cumulant each). The equilibrium here is understood as a generalized equilibrium as introduced by Asinari [

∂ u 1 ∂ x 2 + ∂ u 2 ∂ x 1 = ω a 3 ρ C 110 (22)

= ω ∞ 3 ρ ( C 110 − ω ∞ − ω a ω ∞ C 110 ) (23)

C 110 e q = ω ∞ − ω a ω ∞ C 110 (24)

It is hence seen that the equivalence between the modified relaxation rate in Equation (22) and the generalized equilibrium at fixed relaxation rate in Equation (22) is established by choosing the specific equilibrium Equation (24). It is of particular note here that this relationship is independent of how ω a was obtained in the first place, implying that any method using an effective relaxation rate can be rewritten as a method with fixed relaxation rate and generalized equilibrium. This hence applies to Vikhansky’s model for Bingham fluids [

In the generalized equilibrium form, the linear stability constraint is not necessarily violated if ω a < 0 . It hence seems to be admissible to use the analytic solution of the effective relaxation rate even if it might become negative.

Being a Cartesian grid based method, the LBM is applicable to real world industrial problems only because effective methods to incorporate grid refinement and curved boundary conditions have been incorporated. Most of these methods are dependent on the local viscosity and their application to non-Newtonian fluids requires some adjustment. We leave the important topic of grid refinement to future work and discuss the implementation of an off-grid velocity boundary conditions applicable to moving curved walls.

In lattice Boltzmann, velocity boundary conditions are conveniently implemented via a bounce back scheme with modified velocity. Bounce back is applied on links entering the fluid domain during the streaming step. In its simplest implementation, bounce back returns the population leaving the domain as the population entering the domain in opposite direction. This results in a zero velocity boundary condition approximately halfway between grid nodes. The no-slip (i.e. bounce back) boundary is turned into a velocity boundary condition by adding twice the momentum of the moving wall [

f i ¯ j ¯ k ¯ B B ( x → , t + Δ t ) = f i j k ∗ ( x → , t ) + 2 w i j k ρ e → i j k ⋅ u → B C c s 2 (25)

Here f ∗ ( x → , t ) is the post-collision state of the boundary node at the previous time step. The over bar denotes the direction opposite to the boundary. Further we used the lattice weights w i j k ( w 100 = 2 / 27 , w 110 = 1 / 54 , w 111 = 1 / 216 ) and so on by permuting indices) and the link direction e → i j k .

For industrial applications it would be unacceptable that boundaries had to be set halfway between grid nodes. For this reason, interpolation bounce back schemes have been popular for quite some time [

f i j k ( x → + e → i j k Δ x , t + Δ t ) = f i j k ∗ ( x → , t ) (26)

Instead of interpolating in space only, as done by the classical interpolated bounce back scheme, it is possible to interpolate in space and time towards the boundary:

f B C ( x → + q e → i j k Δ x , t + q Δ t ) = q f i j k ( x → , t ) + ( 1 − q ) f i j k ∗ ( x → , t ) (27)

At the boundary the bounce back or velocity bounce back method according to Equation (25) can be applied to recover f i ¯ j ¯ k ¯ B B . Next to the boundary the incoming distributions are recovered by interpolation from the distributions leaving the boundary node in the direction opposite to the boundary:

f i ¯ j ¯ k ¯ ( x → , t + 2 q Δ t / ( q + 1 ) ) = f i ¯ j ¯ k ¯ B B ( x → + q e → i j k Δ x , t + q Δ t ) q + 1 + q f i ¯ j ¯ k ¯ ∗ ( x → , t ) q + 1 (28)

It is observed that the population arrives at the correct location. However it appears to arrive at the wrong time instance t + 2 q Δ t / ( q + 1 ) instead of at t + Δ t . This is of minor concern as the LBM recovers the Navier-Stokes equation only in diffusive scaling Δ t ∝ Δ x 2 . A first-order scheme in time is as accurate as a second-order scheme in space. Since the linear interpolation is already second-order in space, the first-order interpolation in time does not increase the error asymptotically compared to the classical interpolation method using several nodes. By using pre- and post-collision states, the boundary condition either requires the presence of both states or it has to compute and apply the collision operator in the boundary. Modern implementations of the LBM using in-place streaming methods like EsoTwist [

f i j k ∗ = f i j k + ( 1 − ω ) ( f i j k − f i j k e q ) (29)

The non-Newtonian fluid model enters the boundary condition only in the determination of ω . This can be implemented for the iterative as well as for the analytic model of the Bingham fluid’s viscosity.

The Bingham fluid model with unrolled iterative regularization according to Equation (18) and the method based on the analytic solution are implemented in the massively parallel lattice Boltzmann solver Virtual Fluids [

The lattice Boltzmann algorithm including the computation of the effective relaxation rate for boundary conditions and the collision operator is given in Algorithm 1.

Algorithm 1. Lattice Boltzmann algorithm for Bingham uids.

The main purpose of the current paper is to demonstrate that the viscosity singularity which has to be avoided in Navier-Stokes implementations of the effective viscosity model does not arise in the LBM. For demonstration purpose we chose two canonical test cases which are both popular and relevant for Bingham fluids: Poiseuille flow and Taylor-Couette flow. Yet, even for these comparatively simple setups proper convergence studies for yield stress fluids are rarely found in literature.

A classical and simple test for the implementation of a Bingham fluid is flow between two infinite plates driven by a constant force. This Poiseuille type flow results in a parabolic flow near the boundaries and a plug flow in the center of the channel for a Bingham fluid as described in [

u x = { 1 2 ( F x μ ) 1 2 ( H − z 0 ) 2 , 0 ≤ z ≤ z 0 1 2 ( F x μ ) 1 2 ( ( H − z 0 ) 2 − ( z − z 0 ) 2 ) , z 0 ≤ z ≤ H (30)

where F x is the force amplitude driving the flow, z 0 is the yield point given by z 0 = τ 0 F x , H is the half height of the channel and μ = ρ ν is the dynamic

viscosity. Driving the flow by a force F x is typically preferred in benchmark simulations over the more physically correct pressure gradient as the latter cannot be implemented with periodic boundaries and would hence introduce boundary effects from the pressure boundary conditions.

We used a simulation domain of size 2 × 2 H × 2 . The viscosity at the lowest resolution μ was set to 0.005, the yield stress τ 0 = 3 × 10 − 8 and the force was F x = 6 × 10 − 9 . As usual in LBM literature, all quantities here are given in normalized lattice units where we assume that grid spacing, time step and mass element are all unity, i.e. Δ x = 1 , Δ t = 1 and Δ m = 1 . For studying the convergence we varied the resolution of the problem for fixed Mach, Reynolds and Bingham numbers defined respectively as:

M a = U c s (31)

R e = ρ U H μ (32)

B m = τ 0 H μ U (33)

The speed of sound c s = 1 / 3 Δ x Δ t − 1 is a constant in the LBM such that consistency between the dimensionless numbers is established by scaling μ ⋆ = μ Δ x / Δ x ⋆ and F x ⋆ = F x Δ x ⋆ / Δ x . This scaling implies that the time step Δ t is scaled proportionally to the grid spacing Δ x which also keeps the velocity scale U constant. The disadvantage of this so-called acoustic scaling is that a finite error in Mach number persists such that absolute convergence is not expected. In order for this to be small we chose rather small values for the force and the yield stress.

To quantify the error we compute the following L2 norm

‖ u x − u x a ‖ L 2 = ∑ ( u x − u x a ) 2 ∑ ( u x a ) 2 (34)

Our next test is a planar Taylor-Couette flow depicted in

A semi-analytic solution for the laminar Taylor-Couette flow of a Bingham fluid is usually derived in dimensionless form scaled with the Reynolds number of the inner cylinder [

0 = 1 r 2 ∂ ∂ r ( r 2 τ r θ ) (35)

The stress in the Bingham fluid is given in cylindrical coordinates as:

τ r θ = ( μ + τ 0 | γ ˙ r θ | ) γ ˙ r θ (36)

γ ˙ r θ = { 0 , | τ r θ | ≤ τ 0 d V ( r ) d r − V ( r ) r , | τ r θ | > τ 0 (37)

where V ( r ) is the tangential velocity at radius r. For the purpose of this derivation we assume | τ r θ | > τ 0 and plug Equation (36) into Equation (35). This gives rise to a 2nd order ordinary differential equation which we supplement

with two boundary conditions. Unlike Bird et al. [

V ( r ) = { μ Ω R α 2 + τ 0 R β 2 ln ( r R y ) − τ 0 R γ 2 ln ( R i R y ) μ r ( R i 2 − R y 2 ) , R i < r ≤ R y r Ω , r > R y (38)

With

R α 2 = ( R i 2 − r 2 ) R y 2 (39)

R β 2 = ( R y 2 − R i 2 ) r 2 (40)

R γ 2 = ( R y 2 − r 2 ) R i 2 (41)

The yield radius R y is unknown and it is recovered from the condition of continuity for Equation (37). Basically we need to solve

0 = d V ( R y ) d R y − V ( R y ) R y (42)

for R y . This is done here numerically such that the final solution is only semi-analytic. As there are multiple solutions we have to pick the one for which R y > R i .

both in lattice units. The same scaling as in the Poiseuille case is applied holding Ma, Re and Bm constant.

We observe that a plug flow attached to the outer cylinder develops for sufficiently low angular velocities and that the different resolutions and the analytic and iterative methods agree well with respect to the velocity profiles.

In this work we presented a cumulant LBM for the simulation of Bingham fluids. The ansatz used is similar to the one proposed by Vikhansky, however, in contrast to his work we solve the implicit problem analytically. The resulting model

implies the existence of negative viscosities at shear rates below the yield threshold. At higher shear rates the viscosity returns to a positive value such that the method remains stable through self-limiting even though it appears to be linearly unstable. The singularity in viscosity occurring between the yielded and non-yielded state does not affect the performance of the lattice Boltzmann method much since viscosity is never explicitly used in the method and appears only as an emerging property derived from the relaxation rate. This is in stark contrast to an effective viscosity model implemented in a Navier-Stokes solver where the viscosity has to be explicitly specified such that the singularity would be fatal to the simulation.

We showed that an effective viscosity model can always be translated to a generalized equilibrium model with fixed viscosity. This result can also be applied in other contexts, most notably for Dellar’s Galilean correction of viscosity using a modified relaxation rate [

Using the analytic solution for the relaxation rate has two important advantages over Vikhansky’s method. First, it is simpler and more efficient since no iteration is required and the relaxation rate can be obtained by evaluating a simple analytic expression. Second, the analytic method is more accurate. The latter could appear obvious as we are evidently comparing the exact solution to a slowly converging approximation. However we have to recall that the necessity of some sort of regularization for the singular viscosity is almost universally accepted in literature. While the loss of convergence due to regularization in the case of high resolution is not surprising, we could show that regularization can be avoided altogether due to the way of how the LBM deals with viscosity and that avoiding regularization improves convergence and therefore overall accuracy. Finally, we note that more complex fluids will still require iterative function solvers to determine the effective viscosity or the generalized equilibrium.

Even using the analytic relaxation rate, the convergence properties of our method are below the expectation of a second order accurate method. Simulating a yield stress fluid includes the modeling of a quasi-solid phase which, for an explicit method on a Cartesian mesh, is not a well posed problem. Convergence studies are rarely shown for the simulation of such complex fluids. As the solid phase remains to be an approximation even with the analytic relaxation rate, it is essential for the underlying numerical method to support a high viscosity contrast. The cumulant LBM has been established for fluids with very low viscosity. In this paper we showed that the simulation of the solid domain is not limited by the requirement of a finite viscosity in this method.

We acknowledge financial support by the German Research Foundation (DFG) project number 414265976-TRR 277. Computational resources have been provided by the North-German Supercomputing Alliance (HLRN).

The authors declare no conflicts of interest regarding the publication of this paper.

Geier, M., Kutscher, K. and Krafczyk, M. (2021) A Direct Effective Viscosity Approach for Modeling and Simulating Bingham Fluids with the Cumulant Lattice Boltzmann Method. Open Journal of Fluid Dynamics, 11, 34-54. https://doi.org/10.4236/ojfd.2021.111003