Mechanism of vorticity amplification by elastic waves in a viscoelastic channel flow

Significance

In numerous viscoelastic flow geometries with curvilinear streamlines, instability mechanism is suggested and verified though it becomes ineffective in parallel streamlines flows. The latter reveals instability directly from laminar to chaotic flow witnessing its nonnormal mode nature. At higher velocities, chaotic flow regimes, transition, elastic turbulence, and drag reduction, accompanied by elastic waves are observed and characterized. Here, we show experimentally that elastic waves play a key role in amplification of rms vorticity fluctuations at increasing flow velocity by presenting strong correlation of flow resistance and rms vorticity fluctuations with elastic wave energy. Growing elastic wave frequency with increasing flow velocity results in strong attenuation of wave energy. The latter causes decay of rms vorticity fluctuations and flow resistance.

Abstract

Inertia-less viscoelastic channel flow displays a supercritical nonnormal mode elastic instability due to finite-size perturbations despite its linear stability. The nonnormal mode instability is determined mainly by a direct transition from laminar to chaotic flow, in contrast to normal mode bifurcation leading to a single fastest-growing mode. At higher velocities, transitions to elastic turbulence and further drag reduction flow regimes occur accompanied by elastic waves in three flow regimes. Here, we demonstrate experimentally that the elastic waves play a key role in amplifying wall-normal vorticity fluctuations by pumping energy, withdrawn from the mean flow, into wall-normal fluctuating vortices. Indeed, the flow resistance and rotational part of the wall-normal vorticity fluctuations depend linearly on the elastic wave energy in three chaotic flow regimes. The higher (lower) the elastic wave intensity, the larger (smaller) the flow resistance and rotational vorticity fluctuations. This mechanism was suggested earlier to explain elastically driven Kelvin–Helmholtz-like instability in viscoelastic channel flow. The suggested physical mechanism of vorticity amplification by the elastic waves above the elastic instability onset recalls the Landau damping in magnetized relativistic plasma. The latter occurs due to the resonant interaction of electromagnetic waves with fast electrons in the relativistic plasma when the electron velocity approaches light speed. Moreover, the suggested mechanism could be generally relevant to flows exhibiting both transverse waves and vortices, such as Alfven waves interacting with vortices in turbulent magnetized plasma, and Tollmien–Schlichting waves amplifying vorticity in both Newtonian and elasto-inertial fluids in shear flows.

Inertia-less polymer solution flow s with curvilinear streamlines exhibit either supercritical or subcritical normal mode elastic instability and further transition to elastic turbulence (ET) at sufficiently large elastic stress. An elastic stress emerges due to polymers stretched by a velocity gradient along curvilinear streamlines (hoop stress). The latter generates a radial bulk force toward the curvature center that triggers an elastic instability at Weissenberg number, Wi ≫ 1, and Reynolds number, Re <  1 (1–5). Here, Wi = λU/L is the Weissenberg number that is the ratio of elastic stress and elastic stress dissipation via relaxation, the Reynolds number, Re = ρUL/η, is the ratio of inertial stress to viscous dissipation (6), U is the fluid mean velocity, λ is the longest polymer relaxation time, η and ρ are the solution’s kinematic viscosity and density, respectively, and L is the characteristic length scale. Moreover, this mechanism remains valid also above the instability onset in transition and ET flow regimes causing amplification of velocity fluctuations with growing Wi. It occurs due to energy pumping, withdrawn from the main shear flow, via the hoop stress into velocity fluctuations resulting in a friction factor growth. The instability mechanism is tested and verified in numerous geometries of viscoelastic flow with curved streamlines, though it becomes ineffective in parallel shear flows with zero curvature streamlines, such as pipe, channel, and plane Couette flows (3–5, 7–10). As a result, those flows are linearly stable at all Wi, as rigorously proved in refs. 8 and 9 that is similar to linear stability of Newtonian pipe and plane Couette shear flow at all Re (11).

Linear stability of parallel shear flows of both Newtonian and viscoelastic fluids does not imply their global stability. Indeed, in his seminal experiment, Reynolds discovered that a Newtonian pipe flow became unstable at Re ≈ 2200, despite its linear stability at all Re (12, 13). Recently, experiments in straight pipe (14), square microchannel (15–17), and planar macrochannel flows of dilute polymer solutions with prearranged strong (18) and weak (19, 20) finite-size perturbations at the inlet reveal purely elastic supercritical instabilities at Wi ≫ 1 and Re <  1. In the viscoelastic planar channel flows, the elastically driven instability is quantitatively characterized in refs. 18–21. A remarkable common feature of all recent experiments conducted in viscoelastic parallel shear flows is a direct transition from laminar to chaotic flow above the instability onset (14–21). Indeed, continuous velocity power spectra with an algebraic decay at higher frequencies are at odds with a single, fastest-growing normal mode predicted and observed above the normal mode instability onset (22). Thus, an infinite number of exciting modes above the instability onset together with exponent values of scaling dependencies of flow resistance and rms velocity and pressure fluctuations on Wi distinctly differ from well-known and verified experimentally characteristic features of linear normal mode instability, suggesting that this is the nonnormal mode bifurcation (18–21). The appearance of nonnormal mode instability is attributed to the non-Hermitian nature of governing equations. Thus, amplification of nonnormal unstable modes cannot be described by the normal mode decomposition of classical hydrodynamic stability analysis (22, 23). In contrast to the normal modes growing exponentially above the instability onset, the nonnormal unstable modes increase algebraically due to background disturbances and experimental imperfections. However, the latter can be amplified considerably at sufficiently long times, compared with exponentially growing normal modes, and reach a regime, where nonlinear interactions may lead to their saturation (23–26).

In spite of the discussed similarities of the nonnormal mode instability in both Newtonian and viscoelastic channel flows, a distinct difference between these two flows is revealed. Indeed, in the case of weak finite-size perturbations at the inlet considered in ref. 20 and the current paper, the remarkable and unexpected observation reveals that the elastic waves and coherent structures (CSs) exist in three flow regimes over the entire channel length. It means that only supercritical absolute instability occurs in inertia-less viscoelastic channel flow, without a prior convective (or advective) instability, in contrast to Newtonian parallel shear flows at large Re, where the subcritical instability caused by finite-size perturbations leads to the coexistence of turbulent and laminar phases followed first by convective and further on absolute instabilities (22). In the convectively unstable region, a pulse perturbation propagates and spatially spreads as a turbulent spot downstream without spreading upstream and temporally decays locally. At further Re increase, in the absolute instability region, the pulse spreads upstream and downstream leading eventually to sustained turbulence (22, 27–34). Thus, the direct transition to sustained chaotic flow distinctly differs from the involved transition region in Newtonian parallel shear flow (22, 29). Moreover, both the convective and absolute instabilities take place generally in all Newtonian open flows at Re > 1 without relation to transition to turbulence (27, 28, 30–34). The discrepancy occurs due to the distinct difference of nonlinear terms in Navier–Stokes and elastic stress equations. The former, which describes nonlinear dynamics of the velocity vector field, has only a single nonlinear advective term, whereas the latter describes the dynamics of elastic stress tensor field and contains three nonlinear terms: one advective and two key terms depicting polymer stretching by fluctuating velocity gradient over the entire channel length. The perturbations of elastic stress propagate downstream with sound velocity due to direct, exact relation between pressure and elastic stress fluctuations in chaotic flows derived in ref. 35, independent of its advection with much slower flow velocity. Furthermore, due to linear feedback introduced by an additional term of the elastic stress gradient in the Stokes equation at Re ≪ 1, the velocity field is affected by the growing elastic stress leading to absolute instability. Another distinct common feature of the nonnormal mode instability of the planar macrochannel polymer solution flows with prearranged strong and weak experimental finite-size perturbations at the inlet is the existence of the pronounced peak at low frequencies in the spanwise velocity power spectrum that appears above the instability threshold (18–21, 36). The Wi-dependence of the peak frequency indicates the emergence of elastic waves (36). The elastic waves, predicted in ref. 37, are detected for the first time in a flow between two obstacles obstructing a channel flow in ET by observing distinct peaks in the spanwise velocity power spectra at different Wi (36), and in particular, by a study of their propagation velocity dependence on Wi. The elastic wave speed is obtained from a temporal cross-correlation function of the spanwise velocity calculated at two stream-wise spatially separated points with a variable interval (36). In spite of different perturbation intensities and spectra, the stream-wise elastic wave speed exhibits universal scaling law as c_el = A(Wi − Wi_c)^δ with A = 0.5 ± 0.05 mm/s and δ = 0.73 ± 0.03, where Wi_c is the elastic wave onset (18, 20, 36). The elastic waves are nondispersive transverse waves, where the wave speed depends on elastic stress instead of medium elasticity, analogous to the Alfven wave in magnetized plasma (36). Furthermore, the stream-wise elastic waves are found in a wake between two obstacles (36) and past an obstacle obstructing a channel flow (38), solely in the random flow of ET and drag reduction (DR) regimes. However, no elastic waves are observed above the normal mode elastic instability in a normal mode transition regime at different Wi in both flows. On the other hand, stream-wise propagating elastic waves are also found in channel flows with prearranged strong and weak instrumental perturbations due to a nonsmooth inlet in all three chaotic regimes: transition, ET, and DR emerged due to a nonnormal mode bifurcation (18–20). Moreover, recently the spanwise propagating elastic waves are detected in a channel flow with a smooth inlet and very small perturbations due to a tiny cavity in a top plate at the middle of the channel with more than two orders of magnitude smaller propagating speed (19). Thus, the elastic waves in a channel flow with finite-size perturbations are discovered exclusively in chaotic flows above the nonnormal mode instability onset in the transition, ET, and DR regimes (18–21).

The main questions we address in this paper are the following: i) What is the mechanism of amplification of velocity fluctuations with increasing Wi >  Wi_c, which replaces the mechanism of the hoop stress valid in flows with curved streamlines? ii) Are velocity fluctuations mostly wall-normal vorticity fluctuations? iii) Do the friction factor and wall-normal vorticity fluctuations grow in the transition and ET and decay in DR regimes correlate with the elastic wave intensity behavior in these flow regimes?

Here, we present experimental results of a linear dependence of the flow resistance and rotational part of the wall-normal rms vorticity fluctuations on the energy of the elastic waves in three chaotic flow regimes. It proves the key role of the elastic waves in the amplification of the wall-normal rms vorticity fluctuations by energy pumping, withdrawn from the main flow, into the wall-normal fluctuating vortices that strongly supports the suggested mechanism of the vorticity amplification by the elastic waves.

Experimental Results

Elastic Instability and Velocity and Pressure Fluctuations.

We conduct the experiment in a long channel with a nonsmoothed inlet and two small cavities in the top plate close to the inlet and outlet used for pressure drop measurements (SI Appendix, Fig. S1). Finite-size perturbations produced by the nonsmoothed inlet and two cavities are expected to be much weaker than strong prearranged perturbations used in refs. 18 and 39 and slightly smaller than those in the channel discussed in ref. 20, where six holes are employed. Polymer solution and channel dimensions are the same as used in ref. 20 (Materials and Methods). Particle image velocimetry (PIV), pressure drop, ΔP, and flow discharge measurement techniques are discussed in Materials and Methods.

In Fig. 1, normalized rms stream-wise velocity, (u_rms − u_{rms, lam})/u_mean, and pressure fluctuations, p_rms/p_{rms, lam} − 1, as a function of Wi are shown in lin–log scales. Here, u_{rms, lam} and p_{rms, lam} denote velocity and pressure fluctuations in laminar regime, respectively, caused by experimental noise, which grows also at Wi <  Wi_c, where Wi_c is the critical value of Wi, and u_mean is the mean flow velocity obtained from the discharge rate (Materials and Methods). The vertical three gray dashed lines separate four flow regimes, three of which are chaotic that follow from velocity power spectra (SI Appendix, Fig. S2). At Wi >  Wi_c, a direct transition from laminar to a chaotic flow, observed in the transition regime, is characterized by growing (u_rms − u_{rms, lam})/u_mean and p_rms/p_{rms, lam} − 1 and fitted algebraically by (Wi/Wi_c − 1)^β with β = 1.10 ± 0.05 and β = 0.85 ± 0.05, respectively. Both exponents remarkably coincide with those presented in ref. 20. Thus, both the observed direct transition to a chaotic flow and scaling exponent values, significantly different from the exponent 0.5, associated with a linear normal mode bifurcation (22), indicate nonnormal mode elastic instability. This conclusion was already made in our early publications in similar channel geometries but with different perturbation intensities at the inlet causing different Wi_c values (18–20, 39).

Fig. 1.

Characterization of elastic instability and three flow regimes: transition, ET, and DR. Normalized rms pressure, p_rms/p_{rms, lam} − 1, and normalized rms stream-wise velocity fluctuations, (u_rms − u_{rms, lam})/u_mean, versus Wi in lin–log scales. The measurements are conducted far downstream at l/h = 380 at the channel center. The vertical three gray dashed lines separate four flow regimes determined by different scaling of the flow parameters as a function of Wi. Three flow regimes are chaotic: transition at 150 ± 10 <  Wi <  1000 ± 100, ET at 1000 ± 100 <  Wi <  3000 ± 300, and DR at Wi >  3000 ± 300. In the transition regime, both p_rms/p_{rms, lam} − 1 and (u_rms − u_{rms, lam})/u_mean are fitted by power-law (Wi/Wi_c − 1)^β with β = 1.1 ± 0.05 and β = 0.85 ± 0.05, respectively.

Rotational Part of Wall-Normal Vorticity Fluctuations.

In Fig. 2A, we present an instantaneous image of streamlines of two-dimensional (2D) velocity fluctuations field in the x–z plane with already subtracted mean velocity profile overlapped with the absolute value of the vortical part of the wall-normal vorticity of single vortex fluctuations, |Ω_R|. It is obtained by using the recently developed Liutex (or Rortex) criterion to calculate |Ω_R|. Here, $|{\mathrm{\Omega }}_R|=\mathbf{\Omega }·\mathbf{n}-\sqrt{{(\mathbf{\Omega }·\mathbf{n})}^{\mathbf{2}}-\mathbf{4}{\mathbf{\Lambda }}_{\mathbf{ci}}^{\mathbf{2}}}$, where Ω is the vorticity, n is the unit vector of the vortex axis direction, and ${\mathbf{\Lambda }}_{\mathbf{ci}}$ is the imaginary part of the eigenvector of the velocity gradient tensor (40). The wall-normal vortex is aligned along the y axis, perpendicular to the x–z plane. |Ω_R| is obtained first by differentiation of the velocity fluctuations field in the x–z plane and further using the Liutex (or Rortex) criterion defined above, whose strengths are marked by color (40). We want to emphasize that the streamlines and |Ω_R| fields are shown with less than 2 μm spatial resolution (Fig. 2 A and B, SI Appendix, Fig. S3, and Movies S1 and S2), significantly higher than the spatial resolution used in earlier published papers (18–21, 39) presenting images of stream-wise streaks with about 30 times lower resolution (Materials and Methods).

Fig. 2.

Instantaneous streamlines for identification of wall-normal vortices, their strength, and core size vs Wi in three flow regimes. (A) An instantaneous snapshot of the 2D image of streamlines of velocity fluctuations field (u′,w′) in x–z plane at the channel center overlapped with the field of the absolute value of the vortical part of the wall-normal vorticity of single vortex fluctuations, |Ω_R|, obtained via the Liutex (or Rortex) criterion (40), whose strengths are marked by color at Wi= 1,035 in ET at l/h = 380. The color bar of the Liutex (or Rortex) |Ω_R| is shown at the Bottom. (B) Six examples of wall-normal vortices identified by the closed or spiral streamlines at Wi = 210, 410, 784, 1,265, 2,297, and 4,811 overlapped with six color spots of the Liutex (or Rortex), |Ω_R|, in the transition, ET, and DR flow regimes. Both (A) and (B) share the same legend. (C) The absolute value of circulation, |Γ|, is calculated by integration of azimuthal velocity fluctuations u_r over the circular streamlines in (B) at different radii, r, of the integrated circuit, following the formula $|\mathrm{\Gamma }|=\oint u_r(r)·dl=|{\mathrm{\Omega }}_c|2\pi r^2$ due to the Stokes’ theorem, where |Ω_c| is the absolute value (or strength) of the vorticity of the vortex tube. The dashed lines are the fits to the data at different Wi using the expression for |Γ|. (D) The spatially and temporally averaged absolute values of the vortical part of the wall-normal vorticity as a function of Wi are computed by two approaches: Ω_{c, rms} (red) obtained from the circulation |Γ|, and the Liutex Ω_{R, rms} (green) followed from the Liutex criterion (40). (E) The core radius of the wall-normal vortex, r₀, as a function of Wi. r₀ is defined by the edge of the |Γ| parabolic dependence on increasing r. Each point for r₀ versus Wi is the result of averaging on over 20 samples. The dashed line and gray region define the average value r₀ ≈ 7.9 ± 0.2 μm.

Here, we employ and present in Fig. 2D two approaches of a proper and reliable way to quantify the spatially and temporally averaged vortical part of the wall-normal vorticity fluctuations over the region of interest

$$\begin{array}{c}{\text{Ω}}_{rms}={\displaystyle \sum _{i=1}^p{\displaystyle \sum _{j=1}^q|\text{Ω}{|}_{ij}}}\end{array}$$

as a function of Wi, where p is the number of vortices in a given instantaneous image (spatial averaging), and q is the number of instantaneous images (temporal averaging) taken at a given Wi. The first approach is based on the relation between the vortical part of the wall-normal vortex, |Ω_c|, and its circulation absolute value, |Γ|. We calculate the circulation only in the core of the vortex tubes characterized by circular or spiral streamline patterns in the 2D plane. As well known (22), the vorticity of the core of the vortex tube provides exclusively the vortical part of the wall-normal vorticity Ω_c calculated from Γ. A wall-normal vortex is identified via roughly closed or spiral streamline patterns, which characterize a rigid rotation or swirling motion with constant angular velocity shown in Fig. 2B, where six typical examples of the wall-normal instantaneous vortices at Wi= 210, 410, 784, 1,265, 2,297, and 4,811 overlapped with six color spots of the |Ω_R| in the transition, ET, and DR flow regimes. (See also SI Appendix, Fig. S3 A and B with two single vortices at Wi= 1,035 and SI Appendix, Fig. S3 C and D with two single vortices at Wi= 1,912 at higher spatial resolution. All images are presented in ET regime. See also Materials and Methods about the spatial and temporal resolution of imaging in the experiment.) By integrating the azimuthal velocity fluctuations, u_r, at different radii r from the center, we get the absolute value of the circulation, $|\mathrm{\Gamma }|=\oint u_r·dl=2\pi r^2|{\mathrm{\Omega }}_c|$, using the Stokes’ theorem. The way to compute Γ instead of the vorticity directly helps to avoid artificial errors due to differentiation of the velocity field with limited experimental resolution in PIV, whereas the calculation of the circulation does not involve differentiation. As found from the fit, the ranges of the parabolic relations in r are limited for all Wi values and define the core of the vortex tubes, r₀. By plotting r₀ as a function of Wi, one finds the averaged value of the core of the vortex tubes, r₀ ≈ 7.9 ± 0.2 μm, which is constant inside of the gray region defining rms range in the three flow regimes as shown in Fig. 2E.

The second approach employs the Liutex criterion, presented above, to calculate |Ω_R|, the absolute value of the vortical part extracted from the full wall-normal vorticity of single vortex (40). As the result, an excellent agreement between the spatially and temporally averaged vortical part of the wall-normal vorticity fluctuations obtained via its circulation, Ω_{c, rms}, and via the Liutex criterion, Ω_{R, rms}, in a wide range of Wi in three flow regimes is found. The agreement between two approaches can be explained by quasi-2D velocity fluctuations field in viscoelastic channel flow, since the divergence of (u, w) in 2D (x, w) plane, namely (∂u/∂x + ∂w/∂z)≈0 (18). Indeed, in this case, the axis direction of the vortices is fixed and normal to the x–z plane. Furthermore, the circulation of the vortex tube core contains only the vortical part without the contribution of shear strain. Thus, it should coincide with the vortical part of the wall-normal vorticity calculated from the Liutex criterion, where the shear strain is subtracted by definition (40), and both values Ω_{c, rms} and Ω_{R, rms} can be used equally well to provide the strength of the vortical part of the wall-normal vorticity averaged temporally and spatially.

Correlation between Friction Factor, Liutex Vorticity, and Energy of Elastic Waves.

In Fig. 3A we present the measurements of the friction factor as a function of Wi in high-resolution presentation, f/f_lam, calculated from the flow resistance f = 2D_hΔP/ρU²L_p normalized by the laminar one f_lam ∼ Re⁻¹. Here, D_h = 2wh/(w + h)=0.875 mm is the hydraulic length, L_p = 487.5 mm is the distance between two pressure measurement holes, ΔP is the pressure difference on L_p, and rms pressure fluctuations, u_mean is the mean flow velocity obtained from the flow discharge rate (Materials and Methods). It is rather remarkable that the scaling exponents of algebraic fits of f/f_lam as a function of Wi in three flow regimes are equal to those found in ref. 20, where six holes are used, whereas the instability onset in both experiments differs significantly: Wi_c = 150 in the current experiment versus Wi_c = 120 in early one (20).

Fig. 3.

Main flow properties in three flow regimes. (A) Friction factor, f/f_lam, (B) spatially and temporally averaged rms of vortical part of wall-normal vorticity fluctuations, Ω_{R, rms}, (C) spatially and temporally averaged elastic wave energy normalized by rms squared spanwise velocity fluctuations, $I_{\mathit{el}}/w_{\mathit{rms}}^2$, obtained via PIV measurements in the central channel region in x–z plane at l/h=380 versus Wi in log–log scales. The dependencies of f/f_lam on Wi are fitted algebraically at different flow regimes: at transition f/f_lam − 1 is fitted by (Wi/Wi_c − 1)^δ with δ = 0.68 ± 0.05; at ET and DR, f/f_lam is fitted by Wi^γ with γ = 0.05 ± 0.01 and −0.15 ± 0.05, respectively. Four different flow regimes are separated by three dashed lines at Wi = 150, 1,000, and 3,000. (D) Schematic of the interaction of a wall-normal vortex (in red) with a transverse elastic wave propagating stream-wise (in black) along the flow direction.

Continuous stream- and span-wise velocity power spectra characterize chaotic flow above the instability onset, whereas the spanwise velocity power spectrum exhibits also a significant peak in the low-frequency range, indicating the elastic waves with a frequency depending on Wi (SI Appendix, Fig. S2). The dimensionless elastic wave energy, $I_{\mathit{el}}/w_{\mathit{rms}}^2$, is defined by the peak value in the spanwise velocity power spectra above the rms of squared spanwise velocity fluctuations, $w_{\mathit{rms}}^2$ (SI Appendix, Fig. S2). Another way to get I_el, discussed in ref. 19, is directly from elastic wave visualization after removing the background noise, which is low due to smaller external perturbations. Similar visualization of stream-wise propagating elastic waves is presented in the current paper (SI Appendix, Fig. S4). By the way in this case, the elastic wave speed is obtained directly from their visualization in x − t presentation thanks to low background perturbations.

Fig. 3 presents the Wi dependence of the friction factor, f/f_lam, normalized elastic wave energy, $I_{\mathit{el}}/w_{\mathit{rms}}^2$, and spatially and temporally averaged vortical part of rms wall-normal vorticity fluctuations, Ω_{R, rms}. The results are shown in three flow regimes above the instability threshold, Wi_c: transition, ET, and DR. The trends of the data shown in Fig. 3 A and C are similar to those obtained in a flow past an obstacle obstructing the viscoelastic channel flow, where the physical mechanism of the recently observed DR and relaminarization (21) is explained (38). Moreover, the new central result presented in Fig. 3B is the dependence of spatially and temporally averaged vortical part of rms wall-normal vorticity fluctuations, Ω_{R, rms}, on Wi similar to f/f_lam and $I_{\mathit{el}}/w_{\mathit{rms}}^2$. Thus, the conclusion followed from the data shown in Fig. 3 is a correlation in the Wi dependence of the three main flow properties, f/f_lam, $I_{\mathit{el}}/w_{\mathit{rms}}^2$, and Ω_{R, rms}, in three flow regimes, transition, ET, and DR, above the elastic instability value Wi_c=150. Finally, in Fig. 4, we present the same data as in Fig. 3 for f/f_lam and Ω_{R, rms}, as a function of $I_{\mathit{el}}/w_{\mathit{rms}}^2$ separately in three flow regimes. Both flow characteristics reveal a linear dependence on $I_{\mathit{el}}/w_{\mathit{rms}}^2$ with about the same slope in three flow regimes. Here, we notice that contrary to a linear growth of both f/f_lam and Ω_{R, rms} with $I_{\mathit{el}}/w_{\mathit{rms}}^2$ increase with Wi in the transition and ET regimes, they linearly decrease with $I_{\mathit{el}}/w_{\mathit{rms}}^2$ decay in DR at still increasing Wi. Thus, these plots clearly illustrate a strong correlation of f/f_lam and Ω_{R, rms} with $I_{\mathit{el}}/w_{\mathit{rms}}^2$ in the transition, ET, and DR regimes. The larger (smaller) $I_{\mathit{el}}/w_{\mathit{rms}}^2$, the larger (smaller) both f/f_lam and Ω_{R, rms}.

Fig. 4.

Linear dependence of the friction factor f /f_lam, (A–C) and rms normalized vortical part of wall-normal vorticity (Liutex) fluctuations Ω_R,rms, (D–F) on the spatially and temporally averaged, normalized elastic wave energy discovers their unambiguous correlation in transition (A and D), ET (B and E), and DR (C and F) flow regimes. The dashed lines are linear fits. The direction of the green arrows indicates the increase of Wi. (The Pearson correlation coefficients are 0.982, 0.991, 0.971, 0.980, 0.822, and 0.945, respectively).

Mechanism of Vorticity Amplification by Elastic Waves.

A strong correlation of the friction factor and normalized elastic wave energy variations are presented first in ref. 38, where the linear dependence of f/f_lam on $I_{\mathit{el}}/w_{\mathit{rms}}^2$ is shown in ET and DR. Here, we obtain the experimental results on the linear dependence of two main properties of chaotic flow, f/f_lam and Ω_{R, rms}, on $I_{\mathit{el}}/w_{\mathit{rms}}^2$, which extends the correlation to three flow properties and hint on the key role of the elastic waves in pumping energy, withdrawn from the main shear flow, into the vorticity fluctuations presented in Fig. 4. Indeed, the growing energy of the elastic waves amplifies the spatially and temporally averaged vorticity fluctuations, Ω_{R, rms}, resulting in the friction factor growth in the transition and ET regimes, whereas the decay of the elastic wave energy leads to suppression of Ω_{R, rms} followed by the friction factor decay in DR.

As discussed in ref. 38, the elastic wave frequency grows with increasing Wi that leads to strong attenuation of the elastic wave energy due to sharply increasing viscous dissipation. Thus, at a sufficiently high frequency, it leads to a sharp change from the growth to decay of the elastic wave energy, resulting in transition from ET to DR (38). Then, the central question regarding the energy transfer from the elastic waves to the vorticity fluctuations is the following: What is the physical mechanism of the vorticity amplification by the elastic waves? Here, we suggest that interaction of the transverse elastic wave with the wall-normal vortex fluctuations leads to their amplification in the transition and ET regimes, where elastic wave energy grows with Wi, and their subsequent decay in DR, where the elastic wave energy decreases with Wi due to elastic wave frequency growth causing a sharp increase of viscous dissipation (see schematics of such interaction in Fig. 3D). The mechanism of the interaction of the elastic waves with the wall-normal vortex fluctuations results in the resonant pumping of the elastic wave energy, withdrawn from the main shear flow, into the wall-normal vortex. Moreover, the energy pumping by the elastic waves into the fluctuating vortices at the same time causes significant damping of the elastic waves. It recalls the Landau damping mechanism, which occurs due to the resonant interaction of electromagnetic waves with fast electrons in relativistic plasma (41). When velocity of electrons approaches the speed of light, the resonant energy pumping takes place resulting in damping of electromagnetic waves. Similarly, resonant interaction of acoustic waves with gas bubbles located in a bubbly fluid leads to strong energy damping of propagating sound waves (42). Furthermore, it is also similar to the mechanism of the amplification of wall-normal fluctuating vortices by the elastic wave in elastically driven Kelvin–Helmholtz-like instability, suggested and discussed in ref. 39.

To summarize, the proposed physical mechanism of the wall-normal vortex amplification by the resonant energy pumping of the transverse elastic waves explains the friction factor growth at increasing elastic waves energy in transition and ET regimes and its decrease at decaying elastic waves energy in DR, in accord with the variation of wall-normal vorticity fluctuations. Moreover, similar to the Landau damping mechanism, the proposed mechanism could be generally relevant to flows exhibiting both transverse waves and vortices, such as Alfven waves interacting with vortices in turbulent magnetized plasma (43), and Tollmien–Schlichting waves amplifying vorticity in both channel Newtonian and elasto-inertial fluid flows (22). We anticipate that our results inspire the development of a theoretical description of the suggested mechanism.

Materials and Methods

Experimental Setup and Flow Discharge Measurements.

The experiments are conducted in a straight channel of 500(L) × 3.5(W) × 0.5(h) mm³ dimensions, shown in SI Appendix, Fig. S1. The fluid is driven by N₂ gas at pressure up to 100 psi. The fluid discharge is weighed instantaneously, m(t), by a PC-interfaced balance (BPS-1000-C2-V2, MRC) to measure the time-averaged fluid discharge rate Q = ⟨Δm/Δt⟩ to get the mean velocity U = Q/ρWh. Then Wi = λU/h and Re = ρUh/η vary in the ranges (30, 6,000) and (0.005, 0.9), respectively. High-resolution (0.1% of full scale) differential pressure sensors (HSC series, Honeywell) of different ranges, 5, 15, 60, and 150 psi, are used to measure the pressure drop ΔP and its fluctuations.

Preparation and Characterization of Polymer Solution.

As a working fluid, a dilute polymer solution of high-molecule-weight Polyacrylamide (Polysciences, M_w= 18 MDa) at concentration c = 80 ppm [c/c * ≈0.4 with the overlap polymer concentration c * ≈200 ppm (44)] is prepared using a water–sucrose solvent with 64% sugar weight fraction. The solution properties are the solution density ρ=1,320 kg/m³, the solvent and solution viscosity η_s = 0.13 Pa s and η = η_s + η_p = 0.17 Pa s, respectively, η_s/(η_s + η_p)=0.765, where η_p is the polymer contribution into the solution viscosity, and the longest polymer relaxation time λ = 13 s obtained by the stress relaxation method (44).

Imaging System and PIV Measurements.

We conduct velocity field measurements at various distances l/h downstream from the inlet, using the particle image velocimetry (PIV) method. 3.2 μm latex fluorescent particles (and red-fluorescent 0.2 μm) tracers of ∼1% (and ∼0.15%) w/w concentration (Thermo Scientific) illuminated by a laser sheet of ≈50 μm thickness over the middle channel plane. A high-speed camera (Mini WX100 FASTCAM, Photron) with a high spatial resolution (up to 2,048 × 2,048 pxl²) using from 500 up to 15,000 fps is used to capture images of tracer pairs. The OpenPIV software (45) is employed to analyze u(x, z, t) and w(x, z, t) in 2D x–z plane to record data for ∼𝒪(15) min, or ∼𝒪(50λ), for each Wi to get sufficient statistics. For velocity fluctuations in Fig. 1 and elastic wave energy in Fig. 3, we use spatial averaging over small windows with resolution 256 × 96 pxl² with 4x Objective lens, serving as a single-point velocity measurement at the channel center. The window size for PIV is 32 × 32 pxl² with 50% overlap. However, for vorticity visualization in Figs. 2 and 3, we take 20x lens with 0.2 μm particles at different resolutions (as large as we can for each increasing frame rate). At Wi <  500, resolution is 512 × 512 pxl² at 4,000 to 8,000 fps; at 500 < Wi< 1,000, resolution 256 × 256 at 10,000 to 18,000 fps; at 1,000 < Wi< 3,000, 256 × 96 at 20,000 to 40,000 fps; and at Wi> 3,000, 256 × 64 at 54,000 fps. To increase the spatial resolution, the PIV window size is set to 8 pxl with 50% overlap.

Supplementary Material

Appendix 01 (PDF)

Movie S1.

The dynamics of fluctuating streamlines (left) and corresponding vortical part of wall-normal vorticity fluctuations, Ω_R, (right) fields at Wi = 1035 at the channel center.

Movie S2.

central quarter of SI Movie S1 for better visualization.

Data, Materials, and Software Availability

All study data are included in the article and/or supporting information.