Elastic Alfven waves in elastic turbulence

Abstract

Speed of sound waves in gases and liquids are governed by the compressibility of the medium. There exists another type of non-dispersive wave where the wave speed depends on stress instead of elasticity of the medium. A well-known example is the Alfven wave, which propagates through plasma permeated by a magnetic field with the speed determined by magnetic tension. An elastic analogue of Alfven waves has been predicted in a flow of dilute polymer solution where the elastic stress of the stretching polymers determines the elastic wave speed. Here we present quantitative evidence of elastic Alfven waves in elastic turbulence of a viscoelastic creeping flow between two obstacles in channel flow. The key finding in the experimental proof is a nonlinear dependence of the elastic wave speed c_el on the Weissenberg number Wi, which deviates from predictions based on a model of linear polymer elasticity.

Introduction

A small addition of long-chain, flexible, polymer molecules strongly affects both laminar and turbulent flows of Newtonian fluid. In the former case, elastic instabilities and elastic turbulence (ET)^1,2,3,4,5 are observed at Reynolds number ${\mathrm{Re}} \ll 1$ and Weissenberg number ${\mathrm{Wi}} \gg 1$, whereas in the latter, turbulent drag reduction (TDR) at ${\mathrm{Re}} \gg 1$ and ${\mathrm{Wi}} \gg 1$ has been found about 70 years ago but its mechanism is still under active investigation⁶. Here both Re = ρUD/η and Wi = λU/D are defined via the mean fluid speed U and the vessel size D, and ρ, η are the density and the dynamic viscosity of the fluid, respectively, and λ is the longest polymer relaxation time. ET is a chaotic, inertialess flow driven solely by nonlinear elastic stress generated by polymers stretched by the flow, which is strongly modified by a feedback reaction of elastic stresses⁷. The only theory of ET based on a model of polymers with linear elasticity predicts elastic waves that are strongly attenuated in ET, but elastic waves may play a key role in modifying velocity power spectra in TDR^7,8. Using the Navier-Stokes equation and the equation for the elastic stresses in uniaxial form of the stress tensor approximation, one can write the polymer hydrodynamic equations in the form of the magneto-hydrodynamic (MHD) equations⁸. Then, by analogy with the Alfven waves in MHD^9,10, one gets the elastic wave linear dispersion relation as $\omega = ({\bf{k}} \cdot \hat n)\left[ {tr\left( {\sigma _{ij}} \right){\mathrm{/}}\rho } \right]^{1/2}$ with the elastic wave speed^7,8 c_el = [tr(σ_ij)/ρ]^1/2, where ω and k are frequency and wavevector, respectively, σ_ij is the elastic stress tensor, and $\hat n$ is the major stretching direction, similar to the director in nematics. Such an evident difference between the elastic stress tensor characterized by the director and the magnetic field that is the vector, however, does not alter the similarity between the elastic and Alfven waves, since only uniaxial stretching independent of a certain direction is a necessary condition for the wave propagation determined by the stress value⁷.

A simple physical explanation of both the Alfven and elastic waves can be drawn from an analogy of the response of either magnetic or elastic tension on transverse perturbations and an elastic string when plucked. As in the case of elastic string, the director is sufficient to define the alignment of the stress. Thus, to excite either Alfven or elastic waves the perturbations should be transverse to the propagation direction, unlike longitudinal sound waves in plasma, gas, and fluid media¹¹. The detection of the elastic waves is of great importance for a further understanding of ET mechanism and TDR, where turbulent velocity power spectra get modified according to ref. ⁷. Moreover, c_el provides unique information about the elastic stresses, whereas the wave amplitude is proportional to the transversal perturbations, both of which are experimentally unavailable otherwise⁸.

Numerical simulations of a two-dimensional Kolmogorov flow of a viscoelastic fluid with periodic boundary conditions reveal filamented patterns in both velocity and stress fields of ET¹². These patterns propagate along the mean flow direction in a wavy manner with a speed c_el ≃ U/2, nearly independent of Wi. In subsequent studies, extensive three-dimensional Lagrangian simulations of a viscoelastic flow in a wall-bounded channel with a closely spaced array of obstacles show transition to a time-dependent flow, which resembles the elastic waves¹³. Further, the elastic stress field around the obstacles demonstrates similar traveling filamental structures^12,13 in ET, interpreted as elastic waves^7,8. However, in both studies neither the linear dispersion relation nor the dependence of wave speed c_el on elastic stress–primary signatures of the elastic waves–were examined. Moreover, c_el was found to be close to the flow velocity, contradicting the theory^7,8. Strikingly, an indication of the elastic waves, in numerical studies, originates from observed frequency peaks in the velocity power spectra above the elastic instability^12,13. Analogous frequency peaks in the power spectra of velocity and absolute pressure fluctuations above the instability were also reported in experiments of a wall-bounded channel flow in a creeping viscoelastic fluid, obstructed by either a periodic array of obstacles¹⁴ or two widely-spaced cylinders^15,16. These observations were in agreement with numerical simulations¹⁷ and were associated with noisy cross-stream oscillations of a pair of vortices engendered due to breaking of time-reversal symmetry.

Our early attempts to excite the elastic waves both in a curvilinear flow and in an elongation flow of polymer solutions at ${\mathrm{Re}} \ll 1$ were unsuccessful¹⁸. In the ET regime of the curvilinear channel flow, either an excitation amplitude was insufficient and/or an excitation frequency was too high. The reason we chose the elongation flow, realized in a cross-slot micro-fluidic device, is a strong polymer stretching in a well-defined direction along the flow. However, the elongation flow generated in the cross-slot geometry has the highest elastic stresses in a central vertical plane parallel to the flow in the outlet channels–analogous to a stretched vertical elastic membrane. The transverse periodic perturbations in the experiment were applied in a cross-stream direction from the top wall¹⁸, however a more effective method would be to perturb it in a span-wise direction that was difficult to realize in a micro-channel. A higher frequency range of perturbations, compared to that found in the current experiment, was used that lead to the wave excitation with wave numbers in the range of high dissipation.

Here we report evidence of elastic waves observed in elastic turbulence of a dilute polymer solution flow in a wake between two widely-spaced obstacles, hindering a channel flow. The central finding in the experimental proof of the elastic wave observation is a power-law dependence of c_el on Wi, which deviates from the prediction based on a model of linear polymer elasticity⁷. The distinctive feature of the current flow geometry is a two-dimensional nature of the ET flow, in the mid-plane of the device, in contrast to other flow geometries studied earlier.

Results

Flow structure and elastic turbulence

The schematic of the experimental setup is shown in Fig. 1, where two-widely spaced obstacles hinder the channel flow of a dilute polymer solution (see Methods section for the experimental setup, solution preparation and its characterization). The main feature of the flow geometry used is the occurrence of a pair of quasi-two-dimensional counter-rotating elongated vortices, in the region between the obstacles, as a result of the elastic instability¹⁵ at ${\mathrm{Re}} \ll 1$ and Wi > 1; ${\mathrm{Re}} = 2R\bar u\rho {\mathrm{/}}\eta$ and ${\mathrm{Wi}} = \lambda \bar u{\mathrm{/}}2R$, where obstacles’ diameter 2R and average flow speed $\bar u_{}^{}$ are defined in Methods section. The frequency power spectra of cross-stream velocity v fluctuations show oscillatory peaks at low frequencies^15,16 below λ⁻¹. Above the elastic instability, the main peak frequency f_p grows linearly with Wi, characteristic to the Hopf bifurcation¹⁵. The two vortices form two mixing layers with a non-uniform shear velocity profile and with further increase of Wi their dynamics become chaotic, exhibiting ET properties, with vigorous perturbations that intermittently destroy vortices¹⁶ and seemingly excite the elastic waves. The ET flow in the region between the obstacles is shown through long-exposure particle streaks imaging in Supplementary Movies 1–3 for three different Wi.

Fig. 1

Schematic of experimental setup. It consists of a linear channel of dimension L × w × h = 45 × 2.5 × 1 mm³ with two cylindrical obstacles (shown as two black dots), diameter 2R = 0.3 mm and separated by a distance between the obstacles centers e = 1 mm, embedded at the center line of the channel. The polymer solution is driven by Nitrogen gas and injected through the inlet into the channel. The fluid exiting the channel outlet is weighed instantaneously as a function of time. An absolute pressure sensor, marked as P, after the downstream cylinder is employed to detect pressure fluctuations

Characterization of low frequency oscillations

To investigate the nature of these oscillations we present time series of the streamwise u(t) and cross-stream v(t) velocity components and their temporal auto-correlation functions A(u) = 〈u(t)u(t + τ)〉_t/〈|u(t)|²〉_t and A(v) = 〈v(t)v(t + τ)〉_t/〈|v(t)|²〉_t in Fig. 2a–d. Distinct oscillations in v(t) contrary to weak noisy oscillations in u(t) indicate flow anisotropy. Further, the cross-stream velocity power spectra S_f (v) as a function of normalized frequency λf for five Wi values in the ET regime are shown in log-lin and log-log coordinates in Fig. 3a, b, respectively. The power spectra S_f (v) exhibit the oscillation peaks at low frequencies up to λf ≤ 40 with an exponential decay of the peak values (Fig. 3a). These low frequency oscillations look much more pronounced on a linear scale (Supplementary Fig. 1a). Further, these oscillations are also observed in the power spectra of pressure fluctuations S(P) versus λf, though not so regular (Supplementary Fig. 1b). The exponential decay of S_f (v) at λf ≤ 40 implies that only a single frequency (or time) scale is identified for each Wi (Fig. 3a). This frequency f_d, for each Wi, is obtained by an exponential fit to the data, i.e., S_f (v) ~ exp(−f/f_d). The variation of f_d with Wi is shown in the inset in Fig. 3b; it varies from 0.7 to 2.5 Hz in the range of Wi from 75 to 200, which is comparable to oscillation peak frequency f_p (Fig. 4) and larger than λ⁻¹. Strikingly, on normalization of f with f_d for each Wi, S_f (v) for all Wi collapse on to each other (Fig. 3b). At higher frequencies up to λf ≤ 100, S_f (v) decay as the power-law with the exponent α_f = −3.4 ± 0.1 typical for ET⁵ (Fig. 3c). Contrary to a general case, where the power-law decay of S_f (v) corresponding to ET^3,4,5 commences at λf ≈ 1, the low frequency oscillations cause the power-law spectra start to decay at higher frequencies 10 < λf < 40, perhaps due to an additional mechanism of energy pumping into ET associated with the low frequency oscillations. In addition, S(P) exhibit the power spectra decay in the high frequency range 10 < λf < 100 with the exponent close to −3 (see the bottom inset in Fig. 2 in ref. ¹⁶), characteristic to the ET regime¹⁹. It should be emphasized that the power spectra of the streamwise velocity S_f (u) do not show the low frequency oscillations and decays with a power-law exponent α ≤ 2.

Fig. 2

Streamwise and cross-stream components of velocity and corresponding autocorrelation functions. Time series of a streamwise velocity u and b cross-stream velocity v, obtained at (x/R, y/R) = (2.3, 0.03), corresponding to the location near the line connecting the centers’ of obstacles and close to the center region between the obstacles, for three values of Wi. c, d Their respective temporal autocorrelation functions A(u) and A(v)

Fig. 3

Cross-stream velocity power spectra versus normalized frequency in elastic turbulence. a Cross-stream velocity power spectra S_f (v) in log-lin coordinates to emphasize an exponential decay of the oscillation peak values at low frequencies λf ≤ 40. An exponential decay is shown by the dashed line, e.g., for the case of Wi = 197.5. b S_f (v) for different Wi collapse on to each other upon normalization of f with f_d. Inset: variation of f_d with Wi. The error bars on f_d are estimated based on standard deviation (s.d.) of exponential fit of S_f (v) versus f, and for Wi they are calculated based on the s.d. from the mean value of fluid discharge rate Q (see Methods section). c S_f (v) in log-log coordinates, for different Wi, to demonstrate the power-law decay at high frequencies ~10 < λf ≤ 100. The spectra are obtained at (x/R, y/R) = (5.2, 0.56), which is close to the downstream obstacle and to the center of the upper large vortex. The dashed line in c is a fit to the data at high frequencies with the power-law exponent α_f ≃ −3.4 ± 0.1, typical to the ET regime. S_f (v) of steady flow is shown by gray lines in a, c

Fig. 4

Dependence of oscillation peak frequency on Wi. Dashed line is a linear fit to the data in the regime above the elastic instability. Inset: f_p as a function of Wi_int. The error bars on f_p are estimated from the spectral width of the oscillatory peaks of S_f (v), and for Wi_int they are calculated based on the s.d. from the mean value of (∂u/∂y)

Figure 4 shows the dependence of f_p in a wide range of Wi. The first elastic instability, characterized as the Hopf bifurcation, occurs at low Wi, where f_p grows linearly with Wi–in accord with our early results¹⁵. At higher Wi in the ET regime, f_p(Wi) dependence becomes nonlinear at Wi ≥ 60. In the inset in Fig. 4, we present the same data for f_p as a function of Wi_int. Here, the Weissenberg number of the inter-obstacle velocity field is defined as ${\mathrm{Wi}}_{{\mathrm{int}}} = \lambda \dot \gamma$ and $\dot \gamma \left( { = \left\langle {\partial u{\mathrm{/}}\partial y} \right\rangle _t} \right)$ is the time-averaged shear-rate in the cross-stream direction in the inter-obstacle flow region. The parameter Wi_int is relevant to the description of elastic waves in ET flow between the obstacles’ region. The inset in Fig. 5b shows a linear dependence of Wi_int on Wi.

Fig. 5

Elastic wave speed versus Wi_int. a Cross-correlation functions of the cross-stream velocity C_v(Δx, τ) versus lag time τ for different values of Δx, obtained at y/R = 0.18 and for Wi = 148.4. Inset: Δx versus τ_p for Wi = 148.4, and a slope of linear fit to it (shown by dashed line) provides c_el. The error bars on Δx are determined by the spatial resolution of measurements, and for τ_p they are estimated based on the s.d. of gaussian fit of C_v(Δx, τ). b Dependence of c_el on Wi_int, where the dashed line is a fit of the form $c_{{\mathrm{el}}} = A\left( {{\mathrm{Wi}}_{{\mathrm{int}}} - {\mathrm{Wi}}_{{\mathrm{int}}}^{\mathrm{c}}} \right)^\beta$, where A = 8.9 ± 1.2 mm s⁻¹, β = 0.73 ± 0.12, and onset value ${\mathrm{Wi}}_{{\mathrm{int}}}^{\mathrm{c}} = 1.75 \pm 0.2$. Inset: Wi_int versus Wi. The error on c_el is estimated based on the s.d. of the linear fit of Δx versus τ_p

Dependence of elastic wave speed on Wi_int

Figure 5a shows a family of temporal cross-correlation functions C_v(Δx, τ) = 〈v(x, t)v(x + Δx, t + τ)〉_t/〈v(x, t)v(x + Δx, t)〉_t of v between two spatially separated points, with their distance being Δx, located on a horizontal line at y/R = 0.18 for Wi = 148.4. A gaussian fit to C_v(Δx, τ) in the vicinity of τ = 0 yields the peak value τ_p at a given Δx. A linear dependence of Δx on τ_p (e.g., Fig. 5a inset for Wi = 148.4) provides the perturbation propagation velocity as c_el = Δx/τ_p. The variation of c_el as a function of Wi_int is presented in Fig. 5b together with nonlinear fit of the form $c_{{\mathrm{el}}} = A\left( {{\mathrm{Wi}}_{{\mathrm{int}}} - {\mathrm{Wi}}_{{\mathrm{int}}}^{\mathrm{c}}} \right)^\beta$, where A = 8.9 ± 1.2 mm s⁻¹, β = 0.73 ± 0.12, and onset value ${\mathrm{Wi}}_{{\mathrm{int}}}^{\mathrm{c}} = 1.75 \pm 0.2$. The same data of c_el is plotted against Wi (see Supplementary Fig. 3) and fitted as c_el ~ (Wi − Wi_c)^β that yields the onset value Wi_c = 59.7 ± 1.8.

Discussion

In the light of the predictions⁷, it is surprising to observe the elastic waves in the ET regime due to their anticipated strong attenuation. An estimate of the wave number k = ω/c_el = 2πf_p/c_el from c_el (Fig. 5b) and f_p (Fig. 4) provides k in the range between 0.63 and 1.3 mm⁻¹ (Supplementary Fig. 2). The corresponding wavelengths (~2π/k) are significantly larger than the inter-obstacle spacing e − 2R = 0.7 mm. The spatial velocity power spectra S_k is limited by a size of the observation window of about 0.7 mm that gives k_x ≈ 9 mm⁻¹, much larger than the wave numbers calculated above. Thus, the low k_x part of S_k(v), where the elastic wave peaks can be anticipated, is not resolved by the spatial velocity spectra (Supplementary Fig. 4b). The power-law decay with α_k ≈ −3.3 is found at low k_x followed by a bottleneck part and a consequent gradual power-law decay with an exponent ~−0.5 at higher k_x (Supplementary Fig. 4b), unlike S_f (v), where the peaks appear at low f and the steep power-law decay with the exponent α_f = −3.4 at higher f (see Fig. 3b). The spatial streamwise velocity power spectra S_k(u), obtained at the same Wi and near the center line y/R = 0.01, are similar to S_k(v) at low k_x and decays gradually with exponent ~−0.3 at higher k_x (Supplementary Fig. 4a).

The observed nonlinear dependence of c_el on Wi_int differs from the theoretical prediction based on the Oldroyd-B model^7,8. The expression for the elastic wave speed in the model²⁰ gives c_el = [tr(σ_ij)/ρ]^1/2 ≈ (N₁/ρ)^1/2, where $N_1 = 2{\mathrm{Wi}}_{{\mathrm{int}}}^2\eta {\mathrm{/}}\lambda$ is the first normal stress difference. Then one obtains c_el = (2η/ρλ)^1/2Wi_int. First, c_el is proportional to Wi_int and second, the coefficient in the expression for the parameters used in the experiment is estimated to be (2η/ρλ)^1/2 = 4.5 mm s⁻¹. Taking into account that the model^7,8 and the estimate of elastic stress are based on linear polymer elasticity²⁰, whereas in experiments polymers in ET flow are stretched far beyond the linear limit²¹, thus it is not surprising to find the quantitative discrepancies between them. Indeed, the value of the coefficient found from the fit (8.9 mm s⁻¹) and estimated theoretical value (4.5 mm s⁻¹) differ almost by a factor of two (see Fig. 5b). Moreover, for the maximal value of c_el = 17 mm s⁻¹ (at Wi_int ≈ 4) obtained in the experiment, an estimate of elastic stress gives $\left\langle \sigma \right\rangle = c_{{\mathrm{el}}}^2\rho = 0.37\,{\mathrm{Pa}}$ that is lower but comparable with 〈σ〉 ≈ 1 Pa obtained from the experiment on stretching of a single polymer T4DNA molecule at similar concentrations²¹. Thus, both the c_el dependence on Wi_int and the coefficient value indicate that the Oldroyd-B model based on linear polymer elasticity cannot quantitatively describe the elastic wave speed and so the elastic stresses. Another aspect of this result is the Mach number ${\mathrm{Ma}} \equiv \bar u{\mathrm{/}}c_{{\mathrm{el}}}$; the maximum value achieved in the experiment is ${\mathrm{Ma}}_{{\mathrm{max}}} = \bar u_{{\mathrm{max}}}{\mathrm{/}}c_{{\mathrm{el}}} \approx 0.3$, contrast to what is claimed in refs.^22,23 due to a wrong definition based on the elasticity El = Wi/Re instead of elastic stress σ used for the estimation of c_el and Ma.

We discuss two possible reasons related to the detection of the elastic waves. As indicated in the introduction, the key feature of the current geometry is a two-dimensional nature of the chaotic flow, at least in the mid-plane of the device (see Fig. 4SM in Supplemental Material of ref. ¹⁶), that makes it analogous to a stretched elastic membrane. This flow structure is different from three-dimensional elastic turbulence in other studied flow geometries and thus may explain the failure in the earlier attempts to observe the elastic waves. Another qualitative discrepancy with the theory^7,8 is the predicted strong attenuation of the elastic waves in ET. Below we estimate the range of the wave numbers with low attenuation for the elastic waves and compare with the observed values.

There are two mechanisms of the elastic wave attenuation, namely polymer (or elastic stress) relaxation and viscous dissipation^7,8. The former has scale-independent attenuation λ⁻¹, which at the weak attenuation satisfies the relation ωλ > 1, and the latter provides low attenuation²⁴ at ηk²/ρω < 1. The first condition leads to ks > 1, where s = Wi_int(2ηλ/ρ)^1/2 that provides a minimum wave number in the ET regime as k_min > s⁻¹ = 6.3 × 10⁻³ mm⁻¹ for Wi_int = 4. The maximum value k_max follows from the second condition that gives kΛ < 1 at Λ = (Wi_int)⁻¹(ηλ/2ρ)^1/2. Thus, one obtains in the ET regime k_max < Λ⁻¹ = 0.2 mm⁻¹ for Wi_int = 4 and therefore, the range of the wave numbers with the low attenuation is rather broad 6.3 × 10⁻³ < k < 0.2 mm⁻¹ and lies far outside of the k-range of S_k(u) and S_k(v) presented in Supplementary Fig. 4, where the range of the wave numbers of the elastic waves is not resolved. However, the range of the observed wave number 0.63 ≤ k ≤ 1.3 mm⁻¹ of the elastic waves, shown in Supplementary Fig. 2, is sufficiently close to the estimated upper bound of k.

Methods

Experimental setup

The experiments are conducted in a linear channel of L × w × h = 45 × 2.5 × 1 mm³, shown schematically in Fig. 1. The channel is prepared from transparent acrylic glass (PMMA). The fluid flow is hindered by two cylindrical obstacles of 2R = 0.30 mm made of stainless steel separated by a distance of e = 1 mm and embedded at the center of the channel. Thus the geometrical parameters of the device are 2R/w = 0.12, h/w = 0.4 and e/2R = 3.3 (see Fig. 1). The longitudinal and transverse coordinates of the channel are x and y, respectively, with (x, y) = (0, 0) lies at the center of the upstream cylinder. The fluid is driven by N₂ gas at a pressure up to ~10 psi and is injected via an inlet into the channel.

Preparation and characterization of polymer solution

As a working fluid, a dilute polymer solution of high molecular weight polyacrylamide (PAAm, M_w = 18 MDa; Polysciences) at concentration c = 80 ppm (c/c^* ≃ 0.4, where c^* = 200 ppm is the overlap concentration for the polymer used²⁵) is prepared using a water-sucrose solvent with sucrose weight fraction of 60%. The solvent viscosity, η_s, at 20 °C is measured to be 100 mPa · s in a commercial rheometer (AR-1000; TA Instruments). An addition of the polymer to the solvent increases the solution viscosity, η, of about 30%. The stress-relaxation method²⁵ is employed to obtain longest relaxation time (λ) of the solution and it yields λ = 10 ± 0.5 s.

Flow discharge measurement

The fluid exiting the channel outlet is weighed instantaneously W(t) as a function of time t by a PC-interfaced balance (BA210S, Sartorius) with a sampling rate of 5 Hz and a resolution of 0.1 mg. The time-averaged fluid discharge rate $\bar Q$ is estimated as $\overline {{\mathrm{\Delta }}W{\mathrm{/\Delta }}t}$. Thus, Weissenberg and Reynolds numbers are defined as ${\mathrm{Wi}} = \lambda \bar u{\mathrm{/}}2R$ and ${\mathrm{Re}} = 2R\bar u\rho {\mathrm{/}}\eta$, respectively; here $\bar u = \bar Q{\mathrm{/}}\rho wh$ and fluid density ρ = 1286 Kg m⁻³.

Imaging system

For flow visualization, the solution is seeded with fluorescent particles of diameter 1 μm (Fluoresbrite YG, Polysciences). The region between the obstacles is imaged in the mid-plane via a microscope (Olympus IX70), illuminated uniformly with LED (Luxeon Rebel) at 447.5 nm wavelength, and two CCD cameras attached to the microscope: (i) GX1920 Prosilica with a spatial resolution 1000 × 500 pixel at a rate of 65 fps and (ii) a high resolution CCD camera XIMEA MC124CG with a spatial resolution 4000 × 2200 pixel at a rate of 35 fps, are used to acquire images with high temporal and spatial resolutions, respectively. We perform micro particle image velocimetry²⁶ (μPIV) to obtain the spatially-resolved velocity field U = (u, v) in the region between the cylinders. Interrogation windows of 16 × 16 pixel² (26 × 26 μm²) for high temporal resolution images and 64 × 64 pixel² (10 × 10 μm²) for high spatial resolution images, with 50% overlap are chosen to procure U.