Active open-loop control of elastic turbulence

We demonstrate through numerical solutions of the Oldroyd-B model in a two-dimensional Taylor–Couette geometry that the onset of elastic turbulence in a viscoelastic fluid can be controlled by imposed shear-rate modulations, one form of active open-loop control. Slow modulations display rich and complex behavior where elastic turbulence is still present, while it vanishes for fast modulations and a laminar response with the Taylor–Couette base flow is recovered. We find that the transition from the laminar to the turbulent state is supercritical and occurs at a critical Deborah number. In the state diagram of both control parameters, Weissenberg versus Deborah number, we identify the region of elastic turbulence. We also quantify the transition by the flow resistance, for which we derive an analytic expression in the laminar regime within the linear Oldroyd-B model. Finally, we provide an approximation for the transition line in the state diagram introducing an effective critical Weissenberg number in comparison to constant shear. Deviations from the numerical result indicate that the physics behind the observed laminar-to-turbulent transition is more complex under time-modulated shear flow.


Sine-wave driving
In this section, we present our results when a sine-wave modulated angular velocity is applied to the outer cylinder of the Taylor-Couette cell. Thus, we drive the system with only one frequency, δ −1 , compared to the case of square-wave driving, for which all higher harmonics of δ −1 are also present. Moreover, the shear rate changes gradually with a sine wave. Nevertheless, in the following, we show that the response is nearly identical to the one for square-wave driving.

Secondary flow strength
We start again with the secondary-flow strength where for the sinusoidal driving the base flow velocity is u 0 (r,t) = u 0 φ e φ with u 0 φ = Ar + Br i Ω 0 sin(2πt/δ ) and ... r,φ denotes the spatial average over coordinates r, φ . Analogous to the case for square-wave driving, the secondary flow strength is clearly reduced upon applying sine-wave modulations at the outer cylinder. In Fig. 1 we plot σ versus time for 5 different values for the period of the sine-wave modulaton, 10 s ≤ δ ≥ 30 s. Similar characteristics are observed: for large periods σ exhibits irregular peaks, while for small periods σ strongly tends to zero. It also shows oscillations with the driving period δ . Finally, at low δ the flow ultimately becomes laminar.
The order parameter, the time average of the secondary-flow strength Φ = σ , also shows comparable behavior to the square-wave driving and scales as Φ ∼ De −1 − De −1 c close to the transition. This is shown in Fig. 2, which plots Φ versus De −1 for several Wi together with the square-root fits (dashed lines). In contrast to the square-wave driving, Φ quickly reaches a maximum value for Wi=21.4 and Wi = 27.6. These findings are again independent of the initial condition as the test case for Wi = 21.4 shows, where we applied the modulated driving directly to the initial rest state (open symbols in Fig. 2). The critical inverse Deborah numbers for the different Wi, De −1 c,sin (Wi = 15.1) = 5.8, De −1 c,sin (Wi = 21.4) = 2.9, and De −1 c,sin (Wi = 27.6) = 2.2, shift to lower values compared to the square-wave driving and thus the transitions occur at smaller δ . A possible explanation could be that the transition is related to the larger amplitude of the modulated angular velocity, Ω 0 sin = π 2 s −1 > Ω 0 sq = 2πs −1 .

Flow resistance
As in the main text, the elastic nature of the transition can be illuminated by the flow resistance where we rescale the shear stress τ rφ (r o ) under constant driving with amplitude Ω 0 = π 2 s −1 with

2/11
For the non-turbulent laminar base flow oscillating with period δ we derive in section 3, Γ = Γ lin , where Γ lin reads As studied in the main text for square-wave driving, we start with a turbulent flow obtained for Wi = 21.4 for constant driving until time t = 250s, for which Γ > 1. Then the sine-wave driving is switched. Beyond the transition at De −1 = 2.9 or for periods larger than δ = 10s −1 , Γ develops irregular fluctuations (see Fig. 3). They are imposed on the regular oscillations, which follow from the linear or Maxwell model in section 3. The graphs in Fig. 3 show that the time evolution of Γ becomes increasingly irregular with increasing δ and elastic turbulence is again observed. This is further illustrated by taking the time average of Γ and plotting it versus De −1 in Fig. 4. Also for the sine-wave driving we compare our results to the analytic expression (38) derived in section 3 for the linear version of the Oldroyd-B model. We observe a strong increase in Γ beyond the transition, which is also observed from the inset of the figure, while below the transition Γ = Γ lin corresponding to the laminar the base flow.
Thus, we conclude the general behaviour is the same for square-wave and sine-wave driving. As also demonstrated in the state diagram of Fig. 4 in the main text, upon increasing the Deborah number, the transition to elastic turbulence shifts to a larger Weissenberg number Wi.

Oldroyd-B model in laminar flow: Square-wave modulation
In this section, we derive the stress components for the laminar flow under square-wave shear modulations using the Oldroyd-B model. For the laminar flow in our geometry the constitutive equation for the stress tensor, given in the main text, reduces to whereγ is the shear rate. Considering the laminar case, where τ rr starts at rest, we obtain τ rr = 0 for all times. In the next two sections we solve for the shear stress and the azimuthal normal stress component under square-wave driving and also obtain their time averages.

Mean polymeric shear stress
The shear stress component τ rφ can be obtained from Eq. (6), which is equivalent to the Maxwell model mentioned in the main text. Note that we skip here the superindex M in the symbol τ rφ introduced in the main text. The solution to this ODE for an arbitrary time evolution ofγ(t) is expressed using Green's function G c as where the Green's function reads We now evaluate τ rφ (t) for the square-and sine-wave modulated shear rate used in the main text and this supplement, respectively. We also calculate the time-average of its magnitude to obtain expressions for the time-averaged flow resistance Γ.
Here, we consider that the outer cylinder of the Taylor-Couette cell switches between clockwise and counter-clockwise constant rotation, each with a duration of δ /2. Thus the shear rate at time t is either positive or negative,γ(t) = ±γ 0 . Denoting the time of the latest switching event with t 1 and times of previous events with t n , the shear stress component of Eq. (8) becomes

4/11
Evaluating the integrals and restructuring the occurring sums gives Now, setting t n = t 1 − (n − 1) δ 2 we obtain where in the last line we have used the known expression for the sum of the geometric series. Finally, setting τ 0 = η pγ0 and introducing the Deborah number De = λ /δ , yields We calculate the mean polymeric stress by integrating τ rφ (t) over one full period. Since τ rφ (t) δ = 0, we consider the magnitude of the shear stress component |τ rφ | instead and integrate over half the period, from t = t 1 to t = t 1 + δ /2. The last switching event at t = t 1 changes the rate of strain from ∓γ 0 to ±γ 0 . Then the shear stress component τ rφ starts from negative/positive value at t 1 and relaxes towards a positive/negative value at t 1 + δ /2. In between it becomes zero at t c . To integrate over |τ rφ |, we need to know the stress values at the two boundaries of the integral but also the time t c , which we calculate now. For τ rφ (t 1 ) one finds and τ rφ (t 1 + δ /2) is given by Now, we determine t c from which leads to and

5/11
With this the integral to obtain the mean shear stress component |τ(t) rφ /τ 0 | δ /2 can be calculated, Inserting Eqs. (20) and (21) and using De = λ /δ finally gives the result used in the main text,

Mean azimuthal normal stress.
The azimuthal stress component τ φ φ is obtained by solving Eq.(7) using Green's function presented in Eq.(9). The time evolution ofγ and τ rφ are periodic, which implies τ φ φ is periodic. It follows from Using G c from Eq.(9), the time evolution of τ rφ from Eq. (16), defining c = 2/(1 + e −1/2De ), and splitting the integral into integrations over one half period as before, we obtain Performing manipulations similar to the previous subsection, we ultimately arrive at Finally, the time average over one half period gives

Oldroyd-B model in laminar flow: Sine-wave modulation
In this section, we derive the same quantities as in the previous section, the mean polymeric shear stress and the mean azimuthal normal stress, now for the case of sinusoidal shear-rate modulations.

Mean polymeric shear stress
Here, we consider the case of sine-wave driving withγ(t) =γ 0 sin(ωt) in Eq.(8), and obtain for the shear stress component,

6/11
where ω = 2π/δ . This is a sinusoidal variation with zeros at The mean of the absolute polymeric shear stress component then becomes, Now, using the identities sin arctan(x) = x/ √ 1 + x 2 and cos arctan(x) = 1/ √ 1 + x 2 the result simplifies to Finally, setting De = λ /δ leads to

Mean azimuthal normal stress
The azimuthal stress τ φ φ for the case of sinusoidal driving follows as in Sect. 2.2 but now withγ(t) =γ 0 sin(ωt). It is given by Using Green's function from Eq. (9) and the shear stress component from Eq. (33), we obtain after some rewriting Performing the integration and after some straightforward manipulations this leads to where ω = 2π/δ . Calculating the time average over half a period gives since the terms involving cos(2ωt) and sin(2ωt) give zero. Introducing the Deborah number, we obtain where we used the same normalization factor as for the square-wave driving in the previous section.

Effective Weissenberg number
In this section, we define an effective Weissenberg number and introduce a criterion that approximately determines the transition to elastic turbulence under time-modulated driving. The Weissenberg number was originally defined as the ratio of normal stress difference to shear stress 1, 2 . We thus introduce an effective Weissenberg number as where we used τ rr = 0 and . . .

Limitations of our model
Here, we discuss some limitations of our chosen model: the finite extensibility of the dissolved polymers and the twodimensional geometry. We first demonstrate how the polymer length remains finite. Secondly, we discuss how the instability in the same geometry develops when the full three dimensions are considered.

Polymer extension
We quantify the elongation of the polymer by the trace of the conformation tensor, which is directly proportional to the polymeric stress tensor 3 . Following Bird et al. 3 the trace of the conformation tensor C is a measure for the end-to-end distance R of the polymer trC ∼ R 2 . The trace of the conformation tensor in the base flow is given by since τ rr = 0 and the end-to-end distance of polymers in the base flow is denoted R 0 . Now a dimensionless measure for the extension of the polymers is given by where we normalize by the extension of the corresponding base flow trC 0 ∼ R 2 0 . As an example, we show the time trace of E in Fig. 6 for Wi = 21.4 and De = 0.17 in the turbulent regime, where instead of averaging over the entire cylinder we take the maximum value. On average the extension in the turbulent regime is about twice the extension in the laminar regime. The polymers are not getting unphysically long, assuming that in the base flow they are not fully stretched.

Instability in three-dimensional Taylor-Couette geometry
The goal of our work is to demonstrate a fundamental physical effect in wall-bounded flows at low Reynolds numbers. However, as is discussed in the main text, the restricted dimensions of our geometry might influence the results. Therefore, as a consistency check, we have performed a simulation in three dimensions of a wide-gap Taylor-Couette cell at low Re. Other than the spatial dimensions we keep all the parameters the same and choose Wi = Ωλ = 21.5. The geometry consists of N z = 40 mesh cells in the axial direction and is otherwise equal to our two-dimensional mesh. As is further pointed out in the main text, the aspect ratio of the flow has great influence on the stability when no-slip boundary conditions are chosen at the top and bottom walls.
To limit the effect of the sharp gradients in the corner of the moving and static walls, we set the top and bottom boundaries to slip conditions. The other boundary conditions are as mentioned in the methods section in the main text.
A key element is determining the nature of the first unstable mode. As indicated in the main text, we expect the nonaxisymmetric mode to be the first unstable mode in wide-gap Taylor-Couette flows. In Fig. 7   axial plane only two edge effects are visible. These edge effects are equivalent to the described behavior above, found in Ref. 4 and due to the boundary conditions. We furthermore present snapshots of the rr-component of the stress field τ rr in Fig. 8. The base flow rr-component of the stress tensor is τ 0 rr = 0, so instabilities arising in the coupled stress components are easily identified. Figure 8 clearly shows a non-axisymmetric instability and no axi-symmetric instability.
These results strongly indicate that our two-dimensional Taylor-Couette study is sensible, also from the perspective of the instabilities expected in the three-dimensional Taylor-Couette geometry.

Movie legends
Movie S1. Normalized radial component of the velocity field u r /u max , where u max is the maximum velocity of the base flow, under constant driving at Weissenberg number Wi = 21.4. The radial symmetry of the base flow, where u r = 0, is broken and elastic turbulence is observed. Movie S4. Magnitude of the polymeric stress tensor |τ| under constant rotation at Weissenberg number Wi = 21.4, which are the same conditions as Movie S1. The stress field displays a chaotic spiral-like instability and the flow is turbulent.
Movie S5. Magnitude of the polymeric stress tensor |τ| under square wave modulations, with driving period δ = 12 s, at Weissenberg number Wi = 21.4 and Deborah number De = 0.28, which are the same conditions as Movie S2. The magnitude of the polymeric stress field is suppressed compared to the case of constant rotation and |τ| is nearly regular. After switching the direction of rotation, the stress field quickly relaxes. Afterwards it slowly builds up again and some irregular behavior is still observed. The stress field displays a chaotic spiral-like instability, which is reduced compared to the case of constant rotation. After switching the direction of rotation, the stress field quickly relaxes. However, a new spiral-like instability is quickly formed. Elastic turbulence is still observed.